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Vibration of Planetary Gears Having an Elastic Continuum Ring Gear Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Xionghua Wu, B.S., M.S. Graduate Program in Mechanical Engineering The Ohio State University 2010 Dissertation Committee: Professor Robert G. Parker, Advisor Professor Daniel Mendelsohn, Co-Advisor Professor Chia-Hsiang Menq Professor Ahmet Kahraman Copyright by Xionghua Wu 2010 Abstract The primary goal of this work is to develop mathematical models for planetary gears having an elastic ring gear to conduct numerical and analytical studies to understand the modal properties, parametric instability, and nonlinear gear dynamic behaviors. First, natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description through perturbation and Galerkin analyses. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated. Second, the distinctive modal properties of equally spaced planetary gears with elastic ring gears are analytically studied through perturbation and a candidate mode method based on an elastic-discrete model. Two perturbations are used to obtain closed-form expressions of all the eigenfunctions. In the Discrete Planetary Perturbation (DPP), the unperturbed system is a discrete planetary gear with a rigid ring. In the Elastic Ring Perturbation (ERP), the unperturbed system is an elastic ring supported by the ring-planet ii mesh springs; the sun, planet and carrier motions are treated as small perturbations. All vibration modes are classified into rotational, translational, planet and purely ring modes. The well defined properties of each type of mode are analytically determined. All modal properties are verified numerically. Also the modal properties of planetary gears having diametrically opposed planets and an elastic ring gear are studied through the candidate mode method. Two types of modes are found: rotational and translational modes. Fourth, the parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on the elastic-discrete model. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple expressions in terms of mesh parameters. Instability existence rules for in-phase and sequentially phased planet meshes are also discovered. For in-phase planet meshes, instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. The instability boundaries are validated numerically. Finally, the nonlinear dynamics of planetary gears having an elastic ring gear is studied. The frequency-response functions are presented for primary, subharmonic, and superharmonic resonances. The tooth separation phase between meshes depends on the mode type excited and the position of the planets. The resonances associated with a translational or planet mode are investigated. Parametric instability rule and mesh deflection phase expression are confirmed in the numerical simulation for both in-phase and sequentially phased planets. Response of planetary gears having commensurate natural frequencies is also studied. iii Dedication Dedicated to my wife, Fanyi Zeng to my kids, Susan Wu and Alex Wu and to my parents iv Acknowledgments I wish to express my sincere thanks to my advisor, Robert G. Parker, for his guidance and assistance throughout the research project and the dissertation. I also want to thank Prof. Parker for his patience and time in educating the knowledge, correcting my scientific errors, and editing the manuscripts, which made this dissertation possible. I wish to thank Prof. Mendelsohn for being my co-advisor, and Prof. Menq, Prof. Kaharaman for being committee members of my dissertation. Without their supports, my defense will not happen in Spring quarter 2010. I am grateful for thank Gang Liu for stimulating discussions about the nonlinear dynamics of gears. I also want to say thanks to my laboratory mates: Lingyuan Kong, Farong Zhu, Tugan Eternal, Yi Guo, and Yang Wang for their kindness help. My thanks also go to the Prof. Subramaniam, Prof. Lempert, Janeen Sands and staff in OIA and Graduate School for helping me all the paperwork. Finally, I want to thank my wife and my parents for their constant help and supports, without those the Ph.D degree would not have been possible. v Vita August 29, 1975…………………………… Born, Yushan Jiangxi. China Sept. 1993 – July 1997……………………. B.S. Mechanical Engineering Harbin Institute of Technology, China Sept. 1997 – July. 2000……………………..M.S. Engineering Mechanics Tsinghua University, China Sept. 2000 – present…………………….. Mechanical Engineering The Ohio State University PUBLICATIONS X. Wu and R. G. Parker, Vibration of rings on a general elastic foundation, Journal of Sound and Vibration (2006), V295, p194-213. X. Wu and R. G. Parker, Modal properties of planetary gears with an elastic continuum ring gear, Journal of Applied Mechanics, (2008), V75, 031014. R. G. Parker and X. Wu Structured eigensolution properties of planetary gears with elastically deformable ring gears (2009), Proceeding of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, DETC2009-87340. R. G. Parker and X. Wu Vibration modes of planetary gears with unequally spaced planets and an elastic ring gear, (2010), V329, p2265-2275. FIELDS OF STUDY Major Field: Mechanical Engineering vi Table of Contents Abstract ............................................................................................................................... ii Dedication .......................................................................................................................... iv Acknowledgments............................................................................................................... v Vita..................................................................................................................................... vi Table of Contents .............................................................................................................. vii List of Figures ................................................................................................................... xii List of Tables ................................................................................................................. xviii Nomenclature .................................................................................................................... xx Chapter 1: Introduction ....................................................................................................... 1 1.1. Motivation and Objectives.................................................................................................... 1 1.2. Literature Review ................................................................................................................. 5 1.2.1. Literature on Vibration of the Elastic Ring ................................................................... 5 1.2.2. Literature on Modal Analysis of Planetary Gears ......................................................... 6 1.2.3. Literature on Parametric Instabilities and Nonlinear Dynamics ................................... 7 1.3. Scope of Investigation .......................................................................................................... 9 Chapter 2: Vibration of Rings on a General Elastic Foundation ...................................... 12 vii 2.1. Introduction ........................................................................................................................ 13 2.2. Mathematical Formulation ................................................................................................. 15 2.2.1. Perturbation Method.................................................................................................... 18 2.2.2. Galerkin Method ......................................................................................................... 24 2.3. Method Validation .............................................................................................................. 25 2.4. Vibration of Rings with Identical, Equally Spaced Springs ............................................... 30 2.4.1. Perturbation Solution................................................................................................... 30 2.4.2. Mode Classification for Rings Having Identical, Equally Spaced Spring Supports ... 35 2.4.3. Effect of Support Number ........................................................................................... 37 2.4.4. Effect of Stiffness ........................................................................................................ 39 2.5. Vibration of Rings on Unequally Spaced Spring Supports ................................................ 44 2.6. Vibration of Rings on Distributed Elastic Foundation ....................................................... 48 Chapter 3: Modal Properties of Planetary Gears with an Elastic Continuum Ring Gear . 50 3.1. Introduction ........................................................................................................................ 51 3.2. Modeling and Equations of Motion .................................................................................... 52 3.3. Perturbation Analysis ......................................................................................................... 63 3.3.1. Discrete Planetary Perturbation ................................................................................... 63 3.3.2. Elastic Ring Perturbation ............................................................................................ 75 3.4. Candidate Mode Method .................................................................................................... 79 Chapter 4: Natural Modes of Planetary Gears with Unequally Spaced Planets and an Elastic Ring Gear .............................................................................................................. 87 viii 4.1. Introduction ........................................................................................................................ 87 4.2. Diametrically Opposed Planet Pair Modal Properties ........................................................ 89 4.2.1. Rotational Modes ........................................................................................................ 90 4.2.2. Translational Modes .................................................................................................... 92 4.3. Relationships between Modes for Equally Spaced and Diametrically Opposed Planets ... 94 4.4. Example ............................................................................................................................ 105 Chapter 5: Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears ............................................................................................................................... 108 5.1. Introduction ...................................................................................................................... 109 5.2. Mathematical Formulation ............................................................................................... 110 5.3. Modal Properties and Gear Mesh Deformations .............................................................. 118 5.3.1. Gear Mesh Deflection of Rotational Modes.............................................................. 118 5.3.2. Gear Mesh Deflection of Translational Modes ......................................................... 119 5.3.3. Gear Mesh Deflection of Planet Modes .................................................................... 121 5.3.4. Gear Mesh Deflection of Purely Ring Modes ........................................................... 122 5.3.5. Parametric Instabilities for In-Phase Meshes ............................................................ 123 5.3.6. Parametric Instabilities for Sequentially Phased Meshes .......................................... 131 Chapter 6: Nonlinear Dynamics of Planetary Gears Having an Elastic Ring................. 137 6.1. Introduction ...................................................................................................................... 138 6.2. Mathematical Formulation and Assumptions ................................................................... 141 6.2.1. Mesh Stiffness Representation .................................................................................. 142 ix 6.2.2. Mesh Deflection Representation and Mesh Deflection Phase .................................. 144 6.2.3. Tooth Separation Representation .............................................................................. 145 6.2.4. Equation of Motion ................................................................................................... 147 6.2.5. One Mode Dominant Response................................................................................. 148 6.2.6. Tooth Separation of Degenerate Modes .................................................................... 151 6.3. Method of Multiple Scales ............................................................................................... 153 6.3.1. Primary Resonance Ω = ω p + εσ ............................................................................ 155 6.3.2. Sub-harmonic Resonance = Ω 2ω p + 2εσ .............................................................. 159 6.3.3. Second-harmonic Excitation Resonance = Ω ω p / 2 + εσ / 2 ................................. 159 6.3.4. Super-harmonic Resonance = Ω ω p / 2 + εσ / 2 ..................................................... 160 6.4. Results and Discussions.................................................................................................... 161 6.4.1. Comparison of Numerical and Analytical Results .................................................... 162 6.4.2. Effect of Torque ........................................................................................................ 164 6.4.3. Onset of Tooth Separation......................................................................................... 165 ( L) 6.4.4. Properties of D pq f q ................................................................................................. 165 6.4.5. Effect of the Mesh Stiffness Truncation.................................................................... 167 6.4.6. Response of Translational and Planet modes ............................................................ 167 6.4.7. Nonlinear Response for Planetary Gears with Commensurate Natural Frequencies 182 6.4.8. Nonlinear Response for Planetary Gears with Sequentially Phased Planets ............. 186 Chapter 7: Summary, Conclusions, and Future Work .................................................... 191 x 7.1. Summary........................................................................................................................... 191 7.2. Conclusions ...................................................................................................................... 192 7.2.1. Vibration of Rings on a General Elastic Foundation ................................................ 192 7.2.2. Modal Properties of Planetary Gears with an Elastic Continuum Ring Gear ........... 193 7.2.3. Natural Modes of Planetary Gears with Unequally Spaced Planets and an Elastic Ring Gear ..................................................................................................................................... 194 7.2.4. Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears ... 195 7.2.5. Nonlinear Dynamics of Planetary Gears Having an Elastic Ring ............................. 196 7.3. Future Work...................................................................................................................... 198 7.3.1. Future Work for the Current Model .......................................................................... 198 7.3.2. Planetary Gear Modeled with Multiple Elastic Bodies ............................................. 201 7.3.3. Profile Modification .................................................................................................. 204 References ....................................................................................................................... 205 Appendix A. Eigensolution Properties of Rings with Identical, Equally Spaced Spring Supports .......................................................................................................................... 212 Appendix B. Nondimensional Operators ....................................................................... 215 xi List of Figures Figure 1.1. Deformation of a planetary gear with eight equally spaced planets. ................ 2 Figure 2.1. Ring vibration with elastic foundation. .......................................................... 16 Figure 2.2. Ring segment. ................................................................................................. 17 Figure 2.3. Natural frequencies of rings having four equally spaced radial springs. The symbols denote values calculated by the Galerkin method; solid lines denote results from perturbation. ..................................................................................................... 26 Figure 2.4. Fourier coefficients of vibration modes for a ring with five identical, equally spaced springs with ε = 500 , α = π / 4 , β = 0 . Black ■ ( As ) denotes the mode having cosine components, white □ ( Bs ) denotes the mode having sine components. (a) Three nodal diameter modes. (b) Five nodal diameter modes. ........................... 34 Figure 2.5. Natural frequencies versus number of supports (equally spaced) with ε = 10 , α = π / 2 , β = 0 . ...................................................................................................... 38 Figure 2.6. Natural frequencies for varying spring stiffness ε with l = 4 , β = 0 by the Galerkin method. Solid lines denote natural frequencies for tangential stiffness xii springs ( α = 0 ), dashed lines denote natural frequencies for radial springs ( α = π / 2 )................................................................................................................. 40 Figure 2.7. Mode evolution of a ring with four equally spaced springs radial indicated by circles with α = π / 2 , β = 0 for ε = 10 , 100 and 1000. (a-b) 1 nodal diameter mode, (c-d) 2 nodal diameter mode, (e-f) 3 nodal diameter mode, (g-h) 4 nodal diameter mode. The values are natural frequencies. ................................................................ 42 Figure 2.8. Modes of a ring on four unequally spaced springs with α = π / 2 , β = 65.4ο and ε = 53.9279 . Unequally spaced springs are located at θ1 = 0 , θ 2 = 32π / 63 , θ3 = π and θ 4 = 95π / 63 . ......................................................................................... 46 Figure 2.9. Natural frequencies for varying modulation γ with l = 4 , α = π / 2 , β = 65.4ο and ε = 53.9279 . ..................................................................................... 48 Figure 3.1. Elastic-discrete model of a planetary gear and corresponding system coordinates. The distributed springs around the ring circumference are not shown. 54 Figure 3.2. Typical modes of a planetary gear. The system parameters are given in Table 3.1. Distinct planet modes as in (d) only exist for an even number of planets. ........ 59 Figure 4.1. Mode comparison of a planetary gear with: (a) a planet mode for equally spaced planets ( ω4,5 = 0.5539 ), and (b) the corresponding rotational mode for diametrically opposed planets ( ω5 = 0.5561 ). Parameters are given in Table 4.1. For the diametrically opposed case, the positions of the planets are ψ 1 = 0 , ψ 2 = 2π / 5 , xiii ψ 3 = 2π / 3 , ψ 4 = π , ψ= ψ 2 + π , ψ= ψ 3 + π . ...................................................... 98 5 6 Figure 4.2. Mode comparison of a planetary gear with: (a) a purely ring mode for equally spaced planets ( ω15 = 1.518 ), and (b) the corresponding translational mode for diametrically opposed planets ( ω15 = 1.519 ). Parameters are given in Table 4.1. For the diametrically opposed case, the positions of the planets are ψ 1 = 0 , ψ 2 = 2π / 5 , ψ 3 = 2π / 3 , ψ 4 = π , ψ= ψ 2 + π , ψ= ψ 3 + π . .................................................... 100 5 6 Figure 4.3. Dimensionless natural frequencies of planetary gears when the position of one pair of diametrically opposed planets deviates an angle θ from the equally spaced position. The six-planet system is defined in Table 4.1. For equally spaced planets ( θ = 0 ), the designations R, T denote rotational and translational modes, and the designations P2, P3 denote planet modes having jN ± 2 , jN ± 3 nodal diameter components for equally spaced planets. The designations r, t denote rotational and translational modes for diametrically opposed planets. .......................................... 107 Figure 5.1. Mesh stiffness variations for the n th (a) sun-planet and (b) ring-planet meshes. cs , cr are contact ratios, γ sn , γ rn are mesh phases, and ρ s , ρ r are trapezoid wave slope coefficients. ................................................................................................... 112 Figure 5.2. Instability regions for a planetary gear with in-phase meshes as γ sn = 0 , γ sr = 12 , cs = 1.4 , cr = 1.6 , ε = µ , and other parameters in Table 5.1. ─ ─ , analytical solution; ***, numerical solution. .......................................................... 130 xiv Figure 5.3. Instability regions for a planetary gear with sequentially phased meshes as γ sn = 0, 14 , 12 , 43 γ sr = 12 cs = 1.4 cr = 1.6 ε = µ , , , , , and other parameters in Table 5.1. ──, analytical solution; ***, numerical solution. ................................................ 136 Figure 6.1. Mesh stiffness variations for the n th (a) sun-planet and (b) ring-planet meshes. cs , cr are contact ratios, and γ sn , γ rn , γ sr are mesh phases. ................................. 144 Figure 6.2. Tooth separation function and mesh deflection δ [jnp ] . ................................... 150 Figure 6.3. RMS comparison for ring-planet mesh deflection of a planetary gear defined in Table 6.1. Lines without marker denote analytical solution. Lines with marker * denote numerical solution for full representation of the mesh stiffnesses, and lines with marker ○ denote numerical solution for the first harmonic of the mesh stiffnesses. ............................................................................................................... 163 Figure 6.4. Response of the translational modes ( Ω ≈ ω2,3 ) for a planetary gear defined in Table 6.1 with modal damping ratios as ζ 2,3,5,6 = 0.01 and the damping ratio for the remaining degrees of freedom is 0.05. (a) translations of the ring, (b) ring-planet mesh deflections, (c) sun-planet mesh deflections, (d) spectra of ring-planet mesh deflection................................................................................................................. 170 Figure 6.5. Subharmonic resonance of the translational modes for a planetary gear defined in Table 6.1. Lines without marker denote analytical solution. Lines with marker * denote speed up from subharmonic resonance zone of the translational xv modes. Lines with marker ○ denote speed up away from subharmonic resonance zone of the translational modes. Lines without marker denote analytical solution for distinct modes. ........................................................................................................ 173 Figure 6.6. Subharmonic resonance spectra (speed up) of a translational mode ω6,7 for the first ring-planet mesh deflection of system in Table 6.1. The corresponding RMS plot in Figure 6.5 is the lines with marker *. .......................................................... 174 Figure 6.7. Time response for subharmonic resonance of a translational mode ω6,7 when Ω =3768.8 Hz. The corresponding RMS plot in Figure 6.5 is the lines with marker *. (a) ring-planet mesh deflections. (b) ring-planet mesh forces. ............................... 176 Figure 6.8. Time response of the planet modes ( Ω ≈ ω10 ) for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.005 , ζ remain = 0.05 , and the mesh stiffnesses only have the second harmonic. (a) ring-planet mesh deflections, (b) sun-planet mesh deflections...................................................................................................... 178 Figure 6.9. Time response of the planet modes ( Ω ≈ ω10 ) for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.005 , ζ remain = 0.05 , and trapezoid mesh stiffness variations. (a) ring-planet mesh deflections, (b) sun-planet mesh deflections. .............................................................................................................. 180 Figure 6.10. Response of planet mode ω10 for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.02 , ζ remain = 0.05 . Lines with marker * denote results from xvi numerical integration. Lines without the marker denote result from perturbation solution.................................................................................................................... 181 Figure 6.11. Effect of modal damping on the subharmonic resonance for a planetary gear defined in Table 6.1................................................................................................. 182 Figure 6.12. Effect of modal damping on the subharmonic resonance for a planetary gear defined in Table 6.2................................................................................................. 184 Figure 6.13. Waterfall of ring-planet mesh for a planetary gear system defined in Table 6.2 and modal damping ζ = 0.05 . .......................................................................... 185 Figure 6.14. Response for ring-planet mesh deflection of a planetary gear defined in Table 6.2 and modal damping, ζ remain = 0.05 . (a) RMS of ring-planet mesh deflection, (b) Waterfall of ring-planet mesh deflection ......................................... 187 Figure 6.15. Response of ring-planet mesh deflection for a sequentially phased planetary gear defined in Table 6.1 with zs = 22 , ζ = 0.04 , and γ sr = 0.5 . Lines with marker * and ○ denote the numerical solution for speed up and down. ............................... 189 Figure 6.16. Time response for ring-planet mesh deflections at the condition of point A in Figure 6.15 when the mesh frequency ( Ω =1400.6 Hz) is away from the peak of the resonance................................................................................................................. 190 Figure 7.1. Planetary gear elastic sun and ring model. ................................................... 203 xvii List of Tables Table 2.1. Nondimensional natural frequencies of a ring with two identical, equally spaced, radial springs compared to values from [9]. For ε → ∞ , results with * come from [12]. Note: -- means no result is provided. ν = 0 is assumed, so that ε is the same as in [9]. ........................................................................................................... 28 Table 2.2. Nondimensional natural frequencies of a ring with three identical, equally spaced radial springs compared to values from [12]. ............................................... 29 Table 2.3. Ring gear parameters and material properties of OH-58. ................................ 45 Table 2.4. Comparison of dimensionless natural frequencies for an elastic ring with four equally and unequally spaced springs with ε = 53.9279 . Unequally spaced springs are located at θ1 = 0 , θ 2 = 32π / 63 , θ 3 = π and θ 4 = 95π / 63 . .............................. 45 Table 2.5. Natural frequency splitting for rings having the elastic foundations in (2.69). S denotes split natural frequencies, and R denotes repeated natural frequencies. ....... 49 Table 3.1. Dimensional parameters and dimensionless natural frequencies of a planetary gear with six equally spaced planets. The designations R, T, P and PR denote rotational, translational, planet and purely ring modes. ............................................ 60 Table 3.2. Number of planet modes in different sub-types for different number of planets N , where × denotes not applicable. ......................................................................... 62 Table 4.1. Dimensional parameters of a planetary gear with six equally spaced planets. 97 Table 4.2. Modal property comparison of planetary gears for four different cases. × denotes not applicable, and j = 0,1, 2, . ............................................................... 103 Table 4.3. Number of planet and purely ring modes that evolve into rotational or translational modes when the planets deviate from equally spaced to diametrically opposed. Designations t, r denote translational and rotational modes for diametrically opposed planets. ................................................................................ 105 Table 5.1. Dimensional parameters and dimensionless natural frequencies of an example planetary gear with four equally spaced planets. The designations T, R, P and PR denote translational, rotational, planet and purely ring modes. .............................. 131 Table 6.1. Dimensional parameters for a planetary gear with four equally spaced planets. k rbs and k rus denote radial and tangential distributed stiffnesses. γ sr = 0 . ............ 164 Table 6.2. Dimensional parameters for a planetary gear with four equally spaced planets. k rbs and k rus denote radial and tangential distributed stiffnesses. γ sr = 0.5 . .......... 184 ( L) Table 6.3. Property of D pq f q for different type of modes and phase conditions. .......... 188 xix Nomenclature Ω Mesh frequency R Neutral radius of the ring gear αr Ring-planet pressure angle cs , cr Contact ratios of sun-planet and αs Sun-planet pressure angle ψn Location of the n-th planet ψ rn ring-planet meshes k j , k ju Translational and rotational stiffness of supports/bearing for = ψ n + αr the carrier and sun, j = c, s ψ sn = ψ n − αs kbend γ sr Ring bending stiffness Mesh phase between sun-planet krp Ring-planet mesh stiffness mean and ring-planet mesh ν ρ ζ krn Poisson’s ratio Mesh stiffness for the nth ringplanet mesh Mass density per unit length ksp Sun-planet mesh stiffness mean ksn Mesh stiffness for the nth sun- Modal damping ratio E Young’s modulus J Area moment of inertia N Number of planets planet mesh krbs , krus Radial, tangential distributed ring elastic foundation stiffnesses xx Ij Mass moment of inertia for the v ring, carrier, and sun, j = r , c, s xj , yj ,uj and carrier, j = r , c, s j = r , c, s ξ n ,η n , u n Base radius for the ring and sun, rj Inner, outer radii of the ring gear u,w Ring tangential, radial Radial, tangential, and rotational displacements of the j = r , c, s r1 , r2 Translational and rotational displace-ments of the ring, sun, Mass of the ring, carrier, and sun, mj Ring elastic tangential deflection nth z s , zr planet Tooth numbers of the sun and ring deflections Superscript 0 and 1 of a, q, v denote the unperturbed and the first order perturbation eigenfunctions, respectively. Subscripts c, r , s, n denote the carrier, ring, sun and the nth planet. xxi Chapter 1: Introduction 1.1. Motivation and Objectives Planetary gears are widely used in automotive and aerospace transmissions due to advantages such as compactness, high torque/weight ratio, low bearing load, and high transmission ratio. The compactness, high stiffness, and low bearing load allow planetary gears be quieter than parallel shaft drives. While, in practical planetary gear systems reducing the noise and vibration remains a key concern. Large deformation of the elastic ring gear has been observed for planetary gears in wind turbines, and the structure-born noise of the planetary gear in helicopters can exceed 100dB [1]. Figure 1.1. Deformation of a planetary gear with eight equally spaced planets. shows the deformation of a typical planetary gear. As can be seen, ring gear’s elastic deformation is significant. This is especially true for planetary gears with thin rims or planetary gears with a floating ring gear, including those used in aerospace applications. From a power density point of view, the gear rims must be thin to reduce weight, and a thin rim introduces compliance that improves load sharing among the planets. Thus, planetary gears with an elastic ring are used in many applications, and reducing the noise and vibration of those applications 1 requires a sequence of investigations, such as modal analysis, parametric instabilities, nonlinear dynamics, etc. Figure 1.1. Deformation of a planetary gear with eight equally spaced planets. Gear tooth meshes usually are modeled as elastic springs due to tooth compliance. These tooth meshes are not necessarily equally spaced and are inclined relative to the radial direction as tooth meshing occurs along the line of action. An elastic ring on two perpendicular distributed foundations is adopted to simulate the vibration of the planetary ring gear. Modal properties, natural frequency splitting and mode contamination rules of 2 an elastic ring on a general foundation [2] are presented in simple expressions. It builds a solid base to analyze the modal properties of planetary gears with an elastic continuum ring gear. An elastic ring on equally spaced planets can be used as an unperturbed system for planetary gears having equally spaced planets and an elastic ring [3, 4]. A systematical investigation on modal properties is essential because almost all the vibration concerns require understanding modal properties. Modal properties are prominent in the following studies: identifying which modes are the most concern for noise and dynamics loads; calculating the forced responses; performing modal tests; suppressing resonances, investigating parametric instabilities; extending the study to other models, such as compounded planetary gears or planetary gears with diametrically opposed planets. While planetary gears usually have equally spaced planets, some planetary gears have unequally spaced planets due to the limitations of assembly conditions or special applications. This is because diametrically opposed planets have the benefits of reducing the bearing force, maintaining static and dynamic balance, and improving the load sharing. This research project investigates the modal properties of planetary gears having diametrically opposed planets. A planetary gear system is a time-varying system because the sun-planet and ring– planet meshes stiffnesses change as the number of teeth in contact changes. Mesh stiffness variations combines with geometric errors and tooth micro-geometry modifications are the main source of parametric instability. The objective of this study is to analyze parametric instability. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple 3 expressions in terms of mesh parameters. Instability existence rules for in-phase and sequentially phased planet meshes are also discovered. For in-phase planet meshes, the instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. When planetary gears are under the condition of parametric instabilities, tooth separations occur due to large deflections. Accordingly, phenomena such as subharmonic, super-harmonic resonances, jumps, period-doubling, bifurcation, or even chaos can be observed. The objective of this study is to analyze nonlinear dynamics of planetary gears having an elastic ring gear. In summary, the purpose of this research is building an accurate analytical model to reveal the intrinsic reasons of vibration phenomena for planetary gears having an elastic continuum ring gear through a sequence of analytical investigations, such as modal analysis, parametric instability, and nonlinear dynamics. The research also studies the modal properties of a planetary gear ring on a general foundation and planetary gears having diametrically opposed planets. This research addresses problems of practical applications, predicts of the dynamic behavior, and provides guidance of adjusting the mesh phases and system parameters to suppress the parametric instabilities and reduce the nonlinear response. 4 1.2. Literature Review This section is divided into three parts. Part 1 focuses on the literature of the elastic ring. Part 2 reviews the literature on the modal analysis of planetary gears. Part 3 concentrates on literature of the time-varying planetary gear system, which includes studies on parametric instabilities and nonlinear dynamics of planetary gears. 1.2.1. Literature on Vibration of the Elastic Ring Structures having ring geometry are widely used in mechanical systems, such as gears, tires, bearings and rotors. For some purposes, these structures are modeled as rings with elastic spring connections to the mating components. Ring-planet gear tooth meshes of planetary gears are modeled as elastic springs due to tooth compliance [5]. Rao and Sundararajan [6] investigated the natural frequencies and vibration modes of rings on rigid radial supports by separating the ring into several parts with supports located at the ends of each segment. This method is cumbersome for cases with many supports. Sahay and Sundararajan [7] developed a method that can be applied to rings having many springs but with the limitation that cyclic symmetry is required. Many other methods such as the complementary transfer matrix method [8], transfer matrix method [9, 10] and wave approach [11] have been applied to investigate vibration characteristics of rings on multiple radial springs supports. Detinko [12] investigated the free vibration of a thick ring on multiple radial springs by the Galerkin method, where the extension of the neutral axis, shear deformation and rotary inertia are considered. The effect of these factors on natural frequencies is negligible for thin rings. In Detinko’s paper, the springs do not 5 have to be equally spaced but need at least one axis of symmetry. Most of the above works restrict their studies to the problem of rings with discrete radial springs. In this study, the distributed elastic foundation is more general. 1.2.2. Literature on Modal Analysis of Planetary Gears A detail review on mathematical models for parallel axis gear dynamics was presented by Ozguven and Houser [5]. More researchers interested in the vibration of planetary gears in the last two decades. The free vibration of planetary gears with equally spaced planets received intensive attentions [13-20], where all the planetary gear components were treated as rigid bodies. Lin and Parker [21] established a lumped parameter model that includes both transverse and torsional motion. The modal properties were obtained analytically, and the vibration modes are classified into rotational, translational and planet modes. In this study, these modes are called discrete rotational, translational and planet modes. Lin and Parker used this discrete model to study natural frequency and vibration mode sensitivity [22], natural frequency veering [23], and parametric instability caused by changing contact conditions at the multiple tooth meshes [24]. The influence of gear rim flexibility on static and dynamic behavior of planetary gears has been studied by a few researchers. Kahraman and Vijayakar [25] investigated the impact of ring gear rim flexibility on gear stresses and planet load sharing under static conditions using a deformable-body planetary gear model. The study indicates that reducing rim thickness minimizes the adverse effects of gear and carrier manufacturing errors and improves the planet load sharing. Kahraman et al. [26] further studied the 6 effects of gear rim flexibility on dynamic behavior of planetary gears. These studies exposed the importance of the ring deformation for planetary gears, especially when the ring is floated or the rim of ring gear is thin. Some planetary gears have unequally spaced planets due to the limitations of assembly conditions or special applications. Lin and Parker [18] studied the modal properties of planetary gears with diametrically opposed planets using a discrete model. Kiracofe and Parker [27] extended that research to compound planetary gears with equally spaced or diametrically opposed planets. 1.2.3. Literature on Parametric Instabilities and Nonlinear Dynamics Parametric instabilities in single-pair gears have been investigated in [28-32]. Only a few studies exist on parametric instabilities of multiple mesh gear systems. Tordion and Gauvin [33] and Benton and Seireg [34] analyzed the instabilities of two-stage gear systems but with contradictory conclusions. This was clarified by Lin and Parker [35], who derived formulae that allow designers to suppress particular instabilities by choice of contact ratios and mesh phasing. Lin and Parker [24] analytically investigated the parametric instability of planetary gears using a purely rotational model. The impact of mesh stiffness variation on tooth loads and load sharing in planetary gears was studied by August and Kasuba [17] and Velex and Flamand [36]. They numerically computed the dynamic response of planetary gears with three sequentially phased meshes and found the impact of mesh stiffness variations on dynamic response is significant.. The nonlinear gear dynamics, especially single-mesh gears, received intensive attentions [20, 37-42]. As a result of softening nonlinearity, phenomena such as jumps, 7 period-doubling, bifurcation, and chaos can be observed. Kahraman and Singh [37] studied the interactions between time-varying mesh stiffness and contact loss nonlinearity of single-mesh gears. Theodossiades and Natisiavas [39] investigated periodic responses of a gear pair by using a piecewise linear technique and perturbation analysis. Vijayakar and Parker [38] analyzed the dynamic response of a spur gear pair by using a finite element/contact mechanics model. Al-shyyab and Kahraman [40] adopted harmonic balance and continuation methods to investigate subharmonic and chaotic motions of a multi-mesh gear train of two gear pairs. Liu and Parker [41] studied the nonlinear dynamics of idler gear systems by perturbation, harmonic balance/arclength continuation and numerical integration methods. Changing the mesh phasing is frequently adopted to suppress the resonance. The mesh phasing relations of planetary gears [43] and rules of suppressing a given type of mode’s response [44] were presented by Parker. Ambarisha and Parker [20] calculated the nonlinear response of planetary gears by using both lumped-parameter and finite element models. Mesh phasing rule to suppress rotational and translational vibrations in planetary gears was provided, but the rule can not apply to chaotic and period-doubling regions. Bahk and Parker [42] studied the nonlinear dynamics of planetary gears with a purely rotational model. Perturbation, harmonic balance/arclength continuation and numerical integration methods were used. Frequencyresponse functions were presented in closed-form expressions. Nonlinear phenomena, such as gear tooth contact loss, period-doubling, and chaos, have been observed in the experiments conducted by Kahraman and Blankenship [45-47] on a specially designed gear pair subjecting the parametric excitation. The noise sources were identified from the gear tooth deflections and geometric deviations that vary 8 periodically at the mesh frequency. Due to the structured complexity and coupling effects, experiments on the planetary gears nonlinear dynamics have not been found. 1.3. Scope of Investigation The current project addresses the systematic analytical studies of planetary gears having an elastic ring gear through a sequence of investigations. A crucial characteristic of this research is the incorporation of elastic deformation of the ring gear into the planetary gears. Chapter 2 investigates the vibration of a single elastic ring on an elastic foundation [2]. Chapter 3 systematically identifies modal properties of planetary gears having an elastic ring gear [3]. Chapter 4 extends the modal study in Chapter 3 to planetary gears having diametrically opposed planets and an elastic ring [48]. Chapter 5 discusses parametric instability of planetary gears having an elastic ring [49]. Chapter 6 investigates the nonlinear dynamics of planetary gears having an elastic ring. The contributions of this study are summarized as follows. A model for elastic rings on a general foundation is developed. The natural frequency splitting and mode contamination rules are obtained in general, compact forms involving the Fourier coefficients of the elastic foundation stiffness distribution functions. The mode contamination rules and modal properties are crucial to predict the nodal diameter components of the elastic ring for the whole planetary gear system [5052]. The general natural frequency splitting and mode contamination rules are valid for the axisymmetric systems, such as tires, disks and planetary gears. 9 An elastic-discrete planetary gear model is developed, where the ring gear is modeled as an elastic body. The systematic analysis of the elastic-discrete coupled system in extended operator form provides the knowledge and guidance for solving similar problems. Closed-form expressions of the vibration modes for planetary gears with an elastic ring gear are derived in detail by perturbation and candidate mode methods. It not only identifies the effects of the elastic ring deformation on the modal properties of planetary gears, but also provides a solid foundation of solving the subsequent studies (parametric instability and nonlinear dynamics) analytically. All vibration modes are classified into rotational, translational, planet, and purely ring modes according to their unique characteristics. The reasons and paths of modal evolving are presented to explain how four types of modes for equally spaced planetary gears having an elastic ring evolve into two types as the planets deviate to diametrically opposed. All the planet modes and purely ring modes of equally spaced planets evolve into either rotational or translational modes when the planets change to diametrically opposed. The rule of modal evolving is: Any mode of equally spaced planets having odd (even) nodal diameter components of the elastic ring evolves into a translational (rotational) mode as the planets deviates to diametrically opposed The parametric instability of planetary gears having an elastic ring gear is studied. Parametric instability boundaries are presented in closed-form expressions in terms of gear parameters. Furthermore, the parametric instability existence rules for any two modes are simplified to only depending on the teeth number of the sun (or ring) and the 10 mode type of each mode which allows one to predict instabilities without solving the problem. This rule provides significant convenience and freedom for the designer choosing the parameters to suppress the response. For in-phase meshes, instability existence rule is further simplified as it only depends on the type of gear mesh deformation. The nonlinear dynamics of planetary gears having an elastic ring gear is studied by perturbation method and verified by numerical simulation. The phases caused by mesh stiffness, mesh deflection, and number of teeth are considered. The closed-form frequency-response equations for primary, subharmonic, superharmonic, and the second harmonic excited resonances are obtained. They provide a guidance of how the system parameters affect the nonlinear response. The effects of mesh stiffnesses truncation and modal damping are discussed. Resonances associated with translational and planet modes are found. The nonlinear response of a sequentially phased planetary gear is also studied. Most conclusions can also be applied to planetary gears with a rigid ring. 11 Chapter 2: Vibration of Rings on a General Elastic Foundation Equation Chapter 2 Section 1 The free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and Galerkin analyses. Natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description. The elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle. The foundation stiffnesses vary circumferentially. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. Discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters: non-dimensional spring stiffness, stiffness angle, support angle, number of springs, and location of the springs. The predicted effects of these parameters on the natural frequencies are verified numerically. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated. 12 2.1. Introduction Structures having ring geometry are widely used in mechanical systems, such as gears, tires, bearings and rotors. For some purposes, these structures are modeled as rings with elastic spring connections to the mating components. This work is motivated by planetary gear systems, where the internal (or ring) gear is an elastic ring. Ring-planet gear tooth meshes are modeled as elastic springs due to tooth compliance [5]. These tooth meshes are not necessarily equally spaced and are inclined relative to the radial direction as tooth meshing occurs along the line of action. Planetary gears used in helicopters and aircraft engines usually have thin ring gears, justifying use thin ring theory. From a power density point of view, the ring gear must be thin to reduce weight, and a thin ring introduces compliance that improves load sharing among the planets [21, 53-55]. Kahraman et al. [26] verified the importance of ring gear compliance when they computationally studied the vibration of a planetary gear used in an automotive transmission. The ring vibration literature contains a variety of approaches for analyzing the influence of supports. Rao and Sundararajan [6] investigated the natural frequencies and vibration modes of rings on rigid radial supports by separating the ring into several parts with supports located at the ends of each segment. This method is cumbersome for cases with many supports. Sahay and Sundararajan [7] developed a method that can be applied to rings having many springs but with the limitation that cyclic symmetry is required. Many other methods such as the complementary transfer matrix method [8], transfer matrix method [9, 10] and wave approach [11] have been applied to investigate vibration characteristics of rings on multiple radial springs supports. Detinko [12] investigated the 13 free vibration of a thick ring on multiple radial springs by the Galerkin method, where the extension of the neutral axis, shear deformation and rotary inertia are considered. The effect of these factors on natural frequencies is negligible for thin rings. In Detinko’s paper, the springs do not have to be equally spaced but need at least one axis of symmetry. Most of the above works restrict their studies to the problem of rings with discrete radial springs. In this paper, the distributed elastic foundation is more general. When a structure deviates from axisymmetry, its natural frequencies and vibration modes can change significantly. This attracted many researchers to investigate problems of axisymmetric structures with asymmetric features. The asymmetries may come from manufacturing errors, dimensional variations, material non-uniformity, or attached masses and springs. Allaei et al. [56] analyzed the natural frequencies and vibration modes of a ring with radial spring attachments. They formulated the characteristic equation by the receptance method, obtained natural frequencies from the roots of the characteristic equation, and achieved mode shapes by the mode expansion method. Yu and Mote [57] studied the effects of radial slots of circular plates with rotating load and provided a rule for natural frequency splitting in circular plates with equally spaced, identical radial slots. Tseng and Wickert [58] studied the vibration of an eccentrically clamped annular plate and pointed out splitting of the degenerate natural frequencies. Parker and Mote [59-61] used perturbation analysis to investigate the eigensolutions for plate vibration and the wave equation on annular domains with boundary shape or stiffness variations. Natural frequency splitting rules were generalized to arbitrary distributed asymmetric deviations. Kim et al. [62] presented a natural frequency splitting rule for general rotationally periodic structures and investigated the effects of 14 imperfection on both repeated and split natural frequency modes. A natural frequency splitting rule and a mode contamination rule for axisymmetric structures with identical, evenly spaced asymmetries were obtained by Chang and Wickert [63, 64] when they studied the vibration of rotationally periodic structures. This work investigates the dynamic characteristics of rings with asymmetry from an attached elastic foundation through perturbation and Galerkin methods. General rules of natural frequency splitting and mode contamination are obtained, and the vibration properties are examined for changes in the major parameters. 2.2. Mathematical Formulation Figure 2.1 shows a thin ring with two perpendicular distributed foundations: one has stiffness kd (θ ) and the other has stiffness ke(θ ) , where k is a dimensional stiffness, and d (θ ) and e(θ ) are dimensionless O (1) stiffness distribution functions. To retain full generality, the foundation at each circumferential location is oriented at an arbitrary inclination angle β with the radial direction. This permits natural application to planetary gears where tooth meshing occurs along the line of action defined by the pressure angle, and the ring gear may be supported on its outside by angled spline teeth. From in-plane force and moment balances of this ring segment (Figure 2.2) and the constitutive relation, one obtains the equations (see Nomenclature) ρ ru d θ − ∂N11 d θ − Q13 d θ − Ft r d θ = 0, ∂θ (2.1) d θ − ρ rw ∂Q13 d θ + N11 d θ − Fr r d θ = 0, ∂θ (2.2) 15 ∂M 11 ∂2w EJ ∂u == Q13 , M − , 11 r∂θ 1 −ν 2 r 2 ∂θ r 2 ∂θ 2 (2.3) where u (θ , t ) is the tangential displacement. The elastic foundation appears through the Fr and Ft terms. For inextensible in-plane vibration u and w are related by the constraint [65] Figure 2.1. Ring vibration with elastic foundation. 16 Figure 2.2. Ring segment. ∂u + w = 0. ∂θ (2.4) Equations (2.1)-(2.4) yield the equation of motion ρ ∂ 6u ∂2 ∂ 2u ∂ 4u ∂ 2u ∂Fr EJ − − + + 2 =+ u Ft 2 . 2 2 4 2 6 4 ∂t ∂θ r (1 −ν ) ∂θ ∂θ ∂θ ∂θ (2.5) The tangential and radial distributed forces Ft and Fr are F = −T−1KTx , cos β F F = t ,T = sin β Fr − sin β 0 u kd (θ ) , K= , x= . cos β ke(θ ) w 0 (2.6) (2.7) Introducing dimensionless quantities by the definitions u* = u t ρ r 4 (1 −ν 2 ) r 4 (1 −ν 2 ) Ft r 4 (1 −ν 2 ) Fr kr 3 (1 − ν 2 ) , τ = ,T = , ft = , fr = ,ε= ,(2.8) T r EJ EJ EJ EJ where ε is the dimensionless foundation stiffness, the equation of motion is 17 Muττ* + Lu* + ε L1u* = 0, M =1− ∂2 , ∂θ 2 (2.9) 2 ∂6 ∂4 ∂ 2 L =g + dh − q ∂ − dq ∂ , , − 6 + 2 4 + 2 L= 1 dθ ∂θ 2 dθ ∂θ ∂θ ∂θ ∂θ = g (θ ) d (θ ) cos2 β + e(θ )sin 2 β , (2.10) (2.11) h= (θ ) [e(θ ) − d (θ )]sin β cos= β , q(θ ) d (θ )sin 2 β + e(θ ) cos2 β , (2.12) 2π The operators L and M are self-adjoint with the inner product < u, v >= ∫ uv dθ , with ~ 0 denoting complex conjugate. In what follows the superscript * on u* is omitted. 2.2.1. Perturbation Method With the solution u(θ , t ) → u(θ )eiωt , the eigenvalue problem from (2.9) is Lu − ω 2 Mu = −ε L1u , (2.13) When the foundation stiffness is small compared to the ring bending stiffness, ε << 1 , the perturbed eigensolutions un and ωn2 are represented as asymptotic expansions in ε un= un + ε vn + ε 2ηn + Ο(ε 3 ) , (2.14) 2 ω= ωn2 + ελn + ε 2γ n + Ο(ε 3 ) . n (2.15) Substitution of (2.14) and (2.15) into (2.13) generates the sequence of perturbation problems Lun − ωn2 Mun = 0, Lvn − ωn2 Mvn = − L1un + λn Mun , Lη n − ωn2 Mη n = − L1vn + λn Mvn + γ n Mun . 18 (2.16) (2.17) (2.18) The bending natural frequencies ωn of a free ring are degenerate with multiplicity two. These and the associated n nodal diameter unperturbed eigenfunctions that satisfy (2.16) are ωn = n 2 (n 2 − 1) 2 , n2 + 1 un ,1 = 1 2π (1 + n ) 2 1 einθ , un ,2 = 2π (1 + n ) 2 e − inθ , (2.19) where < un ,i , Mun , j >= δ ij . The general unperturbed eigenfunction un of (2.16) is a linear combination of un ,1 and un ,2 un an ,1un ,1 + an ,2un ,2 , = (2.20) where an ,1 and an ,2 are determined subsequently to ensure continuous change in un of (2.14) as ε → 0 . For the self-adjoint problems (2.17) and (2.18) the solvability conditions are < − L1un + λn Mun , un= 0,= i 1, 2 , ,i > (2.21) < − L1vn + λn Mvn + γ n Mun , un= i 1, 2 . 0,= ,i > (2.22) With the normalization condition < Mun , un >= 1 , (2.20) and the solvability conditions (2.21) form the algebraic eigenvalue problem Dan = λn an , D= 1 1 + n2 g0 + n 2 q0 2 g −2 n − i 2n h−2 n − n q−2 n (2.23) an ,1 g 2 n + i 2n h2 n − n 2 q2 n , a = a , n g0 + n 2 q0 n ,2 (2.24) where g m , hm and qm in (2.24) are from the Fourier expansions g (θ ) = ∞ ∑ g m e imθ , h(θ ) = m = −∞ ∞ ∑ hm eimθ , q(θ ) = m = −∞ 19 ∞ ∑q m = −∞ m eimθ , (2.25) ( em − d m )sin β cos β , qm d m sin 2 β + em cos2 β , g m d m cos2 β + em sin 2 β , h= = = m (2.26) and d m , em are the complex Fourier coefficients of d (θ ) and e(θ ) . D is Hermitian. Solution of the eigenvalue problem for D gives the first order eigenvalue perturbations in (2.14) as λn ,1 λn ,2 = 1 d (cos 2 β + n 2 sin 2 β ) + e0 (sin 2 β + n 2 cos 2 β ) 2 0 1+ n 1 ± d 2 n (cos β − in sin β ) 2 + e2 n (sin β + in cos β ) 2 . 2 1+ n (2.27) If D has distinct eigenvalues λn ,1 ≠ λn ,2 , the degenerate unperturbed eigenvalues split. The two eigenvectors an in (2.24) are then determined following the normalization < Mun , un >= 1 . This establishes an ,1 and an ,2 for each mode (2.20) of the split natural frequencies, and the unperturbed eigenfunctions un are determinate at the first order. If D has repeated eigenvalues λn ,1 = λn ,2 , however, the perturbed eigenvalues do not split at this order of perturbation, any an satisfies (2.24), and the unperturbed eigenfunctions un in (2.20) remain indeterminate. The n nodal diameter natural frequencies split at first order if the second term of (2.27) is non-zero; otherwise they remain repeated. Obviously, the n nodal diameter natural e= 0 . If one or both of d 2n and e2n are non-zero, frequencies remain repeated for d= 2n 2n the natural frequencies split at first order except when d 2n and e2n satisfy d 2 n (cos β − in sin β ) 2 + e2 n (sin β + in cos β ) 2 = 0. (2.28) In these cases (an example is given in the later), the natural frequencies split if either of the individual distribution are present separately but remain repeated with both 20 distributed springs acting simultaneously. Repeated and split natural frequencies experience different influence from the asymmetry. The impact on some of the split natural frequencies is larger than on the repeated natural frequencies. The sum of the first order eigenvalue perturbations for an arbitrary eigenvalue pair is λn ,1= + λn ,2 2 d 0 (cos2 β + n 2 sin 2 β ) + e0 (sin 2 β + n 2 cos2 β ) /(1 + n 2 ) . (2.29) For a split natural frequency pair, one natural frequency changes less than ε (λn ,1 + λn ,2 ) /(4ωn ) , and the other one changes more. A limit case is a ring having equally spaced supports with one spring in each support. In that case, the natural frequency change is maximal for one split natural frequency while it is zero for the other. Natural frequencies that remain repeated always change by ελn ,1 /(2ωn ) = ελn ,2 /(2ωn ) . The first order eigenfunction perturbation vn governed by (2.17) is expanded as a series of the complete unperturbed eigenfunctions v= n ∞ ∑ rn,seis=θ s = −∞ ∞ ∑ s = −∞ , s ≠± n rn ,s eisθ + rn ,n einθ + rn ,− n e − inθ . (2.30) Substituting (2.30) into (2.17) and forming the inner product of (2.17) with eimθ yields rn , s = 1 + n2 1 Cn , s ( Pn , s an ,1 + P− n , s an ,2 ) , Cn ,s = , s ≠ ±n , 2 2 2π (1 + n )(ωn − ωs2 )(1 + s 2 ) (2.31) Pn ,s= d s −n (cos β + in sin β )(cos β − is sin β ) + es −n (sin β − in cos β )(sin β + is cos β ) . (2.32) The coefficients rn ,n and rn ,− n remain indeterminate from this process because the inner product of the left and right sides of (2.17) with e ± inθ vanish (one must use λn an = Dan 21 from (2.23) to show the right sides vanishes). Indeed, rn ,n and rn ,− n are governed by the normalization condition < Mun , vn >= 0 as an ,1rn ,n + an ,2 rn ,− n = 0. (2.33) For eigenvalues that split at first order, rn ,n and rn ,− n are determined by (2.33) and the solvability conditions (2.22) of the second order perturbation equation (2.18). These equations form the Hermitian algebraic problem 2 − Pn ,n + (1 + n )λn − Pn ,− n 1 + n2 an ,1 2 π 1 + n2 an ,1 2π ∑ Ps ,n rn , s rn ,n s ≠± n 2 1+ n . an ,2 rn ,− n =∑ Ps ,− n rn , s s ≠± n 2π γ n 0 0 − P− n ,n − Pn ,n + (1 + n 2 )λn 1 + n2 an ,2 2π (2.34) Because an ,1 and an ,2 are known when the eigenvalues split at first order perturbation, two solutions for rn ,n , rn ,− n and γ n are solved from (2.34) for the two eigensolutions of (2.23). For eigenvalues that remain degenerate at first order, an ,1 and an ,2 are arbitrary. This and (2.33) yield = rn ,n rn= 0 . The solvability conditions of (2.22) generate an algebraic ,− n eigenvalue problem for γ n Ea n = γ n a n , ∞ ∑ Cn,s Pn,s Ps ,n s = −∞ , s ≠± n E= ∞ ∑ Cn,s Pn,s Ps ,− n s = −∞,s ≠± n (2.35) Cn,s P− n,s Ps ,n s = −∞ , s ≠± n . ∞ Cn,s P− n,s Ps ,− n ∑ s = −∞ , s ≠± n ∞ ∑ 22 (2.36) Ps ,n is the complex conjugate of Pn ,s . The eigenvalues of (36) yield the second order eigenvalue perturbations γ n γn = ∞ ∑ s =−∞ , s ≠± n 2 Cn ,s Pn ,s ± ∞ ∑ s =−∞ , s ≠± n Cn ,s P− n ,s Ps ,n . (2.37) The properties of P− n ,s Ps ,n , which is governed by the foundation’s Fourier coefficients d m and em , dictate whether the natural frequencies split at second order perturbation. The properties of P− n ,s Ps ,n for identical, equally spaced discrete supports are obtained in Appendix A; these guarantee that all eigenvalues that are degenerate at first order remain degenerate for higher order perturbations. The closed-form eigenfunction approximation is un = an ,1un ,1 + an ,2un ,2 + ε ∞ ∑r s = −∞ n ,s eisθ . (2.38) Because of asymmetry from the circumferentially varying elastic foundation, the vibration modes no longer consist of pure n nodal diameter sinusoidal variations as they do for a free ring. The base n nodal diameter mode (or unperturbed eigenfunction (2.20)) is contaminated by additional nodal diameter components. The last term in (2.38) governs how the elastic foundation introduces s nodal diameter contaminants into an unperturbed n nodal diameter base mode. The coefficients rn ,s in (2.31) determine the presence and magnitude of each contaminant for given base mode and asymmetric foundation e(θ ) and d (θ ) . These contaminants can change the dynamic characteristics, and especially the forced response, dramatically. The mode contamination rule states: an n nodal diameter base mode will be contaminated with the s nodal diameter component if s satisfies 23 s±n = p, (2.39) where p is any index (positive or negative) of nonzero complex Fourier coefficients for either of the foundation stiffness distribution functions d (θ ) , e(θ ) . The s nodal diameter component will disappear, however, for the unusual case where one or more of d s ± n and es ±n are non-zero but occur such that rn ,s = 0 . The ascending sequence of all nodal diameter components for a given n nodal diameter base mode is called the contamination sequence for that mode. 2.2.2. Galerkin Method An alternative solution spatially discretizes (2.9) by expanding u as a series of basis functions u (θ ,τ ) = N ∑U n= − N n (τ )einθ . Galerkin discretization yields (1 + n 2 )Un + n 2 ( n 2 − 1) 2U n + ε N ∑ [g m= − N m + n ( n − m ) qn + imhm ]U m = 0. (2.40) The matrix form of (2.40) and the associated eigenvalue problem are + Ψ MU ( + λ)U = 0 , −ω 2 Mv + Ψ ( + λ)v = 0 , (2.41) = M diag[1 + = n 2 ] , Ψ diag[n 2 (n 2 − 1) 2 ] , (2.42) λn + N +1,m + N +1 = ε [ g m + n( n − m) qn + imhm ] , m, n = − N , , N . (2.43) For identical, equally spaced springs with no inclination ( β = 0 ), one has l l i =1 i =1 = d (θ ) sin α ∑ δ (θ − ψ= e(θ ) cos α ∑ δ (θ − ψ i ) , i), = ψ i 2π (i − 1) / l , i = 1, , l , 24 (2.44) (2.45) where α is a stiffness angle that can be varied to change the relative stiffness between d (θ ) and e(θ ) . In this case (2.40) simplifies to ε (1 + n 2 )Un + n 2 (n 2 − 1)2U n + 2π N ∑U l i ( m − n )θ = 0 , m, n = − N , , N . (2.46) m (sin α + mn cos α ) ∑ e −N 1 m= i= For a planetary ring gear, β is no longer zero. Each ring-planet gear tooth mesh with β π / 2 −ϕ . pressure angle ϕ is modeled as a single spring with α = π / 2 and= 2.3. Method Validation Figure 2.3 shows the natural frequencies of a ring with four equally spaced radial springs obtained by perturbation and Galerkin discretization. For a wide range of nondimensional spring stiffness ε , the natural frequencies match well with differences less than 1%. 25 Figure 2.3. Natural frequencies of rings having four equally spaced radial springs. The symbols denote values calculated by the Galerkin method; solid lines denote results from perturbation. 26 The vibration of free rings having two identical, equally spaced springs has been investigated by prior researchers [9, 11, 12]. The problem is briefly considered to confirm the accuracy and convergence of the numerical method. Table 2.1 gives the natural frequencies of a ring having two radial springs located at 0 and π. The results with N = 50 are compared with those of [9], where N is the number of Galerkin expansion terms. Some natural frequencies, such as ω2 , ω4 , ω6 , match very well for a wide range of ε , but others do not. When the radial springs are identical and equally spaced there always exist some natural frequencies that are independent of spring stiffness due to all the springs being located at nodes of these modes. They are also natural frequencies of the free ring. As revealed in Table 2.1, large discrepancies occur exactly for these natural frequencies. For instance, 2.6833 and 7.5895 are the two and three nodal diameter natural frequencies of a free ring without springs, as expected. In [9], however, the corresponding values are incorrectly given as 4.3843 and 9.6542, which are natural frequencies for two rigid radial springs (see ω 2 and ω 4 for ε → ∞ ). The present results agree well with known results and previous research [12] for two special cases: one case is ε → 0 , which corresponds to rings with no springs; the other is ε → ∞ , which corresponds to rigid springs. 27 ε 1 nodal diameter 2 nodal diameter 3 nodal diameter 28 ω1 ω1 [9] ω2 ω2 [9] ω3 ω 3 [9] ω4 ω 4 [9] ω5 ω 5 [9] ω6 ω 6 [9] 0 0 -- 0 -- 2.6833 -- 2.6833 -- 7.5895 -- 7.5895 -- 0.1 0 -- 0.1783 0.1779 2.6833 4.3843 2.6928 2.6928 7.5895 9.6542 7.5932 7.5933 1 0 -- 0.5609 0.5610 2.6833 4.3843 2.7762 2.7762 7.5895 9.6542 7.6273 7.6283 10 0 -- 1.6854 1.6854 2.6833 4.3845 3.4792 3.4792 7.5895 9.6615 7.9702 7.9706 100 0 -- 3.6071 3.6071 2.6833 4.3844 6.5267 6.5279 7.5895 9.6449 10.935 10.915 600 0 -- 4.2364 4.2359 2.6833 4.3845 8.9087 8.9090 7.5895 9.6511 15.868 -- ∞ 0 -- 4.3844 4.385* 2.6833 -- 9.6519 9.657* 7.5895 -- 17.922 -- Table 2.1. Nondimensional natural frequencies of a ring with two identical, equally spaced, radial springs compared to values from [9]. For ε → ∞ , results with * come from [12]. Note: -- means no result is provided. ν = 0 is assumed, so that ε is the same as in [9]. 1 nodal diameter 2 nodal diameter ω1 [12 ω2 [12 ω1 , ω2 ] ] ω3 , ω4 0 0 0 0 1 0.4750 0.4750 10 1.1871 100 1.5855 ε 3 nodal diameter ω3 [12 ω4 [12 ] ] ω5 2.6833 2.683 2.683 0.6012 2.7555 2.755 1.187 1.330 3.4329 1.586 1.606 6.9712 ω5 [12 ω6 [12 ] ω6 7.589 7.589 7.589 7.589 2.804 5 7.589 7.589 5 7.645 14.57 3.432 3.872 5 7.589 7.589 8 8.130 14.71 6.973 7.795 5 7.589 7.589 4 11.61 16.08 7.589 4 19.60 19.61 7.589 5 20.29 20.30 100 1.6378 1.638 1.640 9.9941 10.00 10.15 5 7.589 0 ∞ 1.6437 1.644 1.644 10.465 10.47 10.47 5 7.589 5 ] 6 Table 2.2. Nondimensional natural frequencies of a ring with three identical, equally spaced radial springs compared to values from [12]. Natural frequencies of a ring with three identical, equally spaced radial springs are shown in Table 2.2 with comparisons to results in [12]. Many values agree well, although differences arise due to splitting of degenerate (repeated) natural frequencies. Published research about axisymmetric structures with identical, equally spaced asymmetries [56-64, 66, 67] ensure the following natural frequency splitting rule holds: When the nodal diameter n and the number of supports l satisfy n = ml / 2 for even l or n = ml for odd l ( m = 1, 2,3, ), then the degenerate natural frequencies split. The natural frequency splitting behavior for the present solution in Table 2.2 obeys this rule. When l = 3 the first and second pairs of natural frequencies are repeated, and the third pair of natural frequencies splits. Results in [12] violate this rule. 29 2.4. Vibration of Rings with Identical, Equally Spaced Springs The distributed elastic foundation modeled in this study encompasses a broad range of possibilities. Discrete spring supports are of primary interest because this work is motivated by planetary gear systems, where the ring gear is an elastic ring acted on by discrete ring-planet mesh stiffnesses. 2.4.1. Perturbation Solution For identical, equally spaced springs with no inclination ( β = 0 ), the stiffness distribution functions are governed by (2.44). The Fourier coefficients of d (θ ) , e(θ ) are 0 dm = , l sin α /(2π ) 0 em = l cos α /(2π ) m / l ≠ int m / l = int (2.47) Two cases must be considered depending on the relationship between the base nodal diameter and the number of equally spaced springs. Case 1, 2n / l ≠ integer: Substituting (2.47) into (2.27), the first order eigenvalue perturbations are repeated λ= λ= n ,1 n ,2 l (sin α + n 2 cos α ) . 2π (1 + n 2 ) (2.48) The eigenvalues do not split at the first order perturbation. The eigenvectors a n of D are arbitrary unit vectors. Substitution of (2.48) into (2.30) and (2.31) yields the first order eigenfunction perturbations = vn l 1 + n 2 ±∞ ∑ {an,1Cn,ml +n [sin α + (n + ml )n cos α ] ei(ml +n)θ + an,2Cn,ml −n [sin α + (n − ml )n cos α ] ei(ml −n)θ } (2.49) 2π 2π m = ±1 30 where an ,1 and an ,2 are arbitrary. The second order eigenvalue perturbations are obtained from the same substitution into (2.37) = γn ±∞ l2 2 C sin α + (ml + n)n cos α ] . ∑ 2 n , ml + n [ m = ±1 4π (2.50) Note that the second order eigenvalue perturbation is calculated even though vn in (2.49) is not fully determined as a n is an arbitrary unit vector. The eigenvalues do not split at the second order either. In fact, the eigenvalues associated with the n nodal diameter base mode will not split at any order of perturbation for rings on equally spaced springs when 2n / l ≠ integer, as shown in Appendix A. Case 2, 2n / l = integer: The eigenvalues split at the first order. The eigensolutions at first order are λn ,1 = l sin α , π (1 + n 2 ) λn ,2 = n 2l cos α 2 1 2 −1 , a = . ,2 n 2 , a n ,1 = π (1 + n ) 2 1 2 1 (2.51) Substitution of (2.51) and (2.31) into (2.34) yields = rn ,n rn= 0 for each split mode. In ,− n integer if and using (2.47) to reduce (2.31) and (2.34) for this case, note that ( s − n ) / l = integer . The eigensolutions are only if ( s + n ) / l = vn ,1 = sin α 1 + n 2 ∑ 2n 2π π Cn,ml +nlei ( ml +n)θ , m = −∞ , m ≠ 0, − ∞ (2.52) l vn ,2 = l cos α 1 + n 2 Cn ,ml + n (ml + n)nei ( ml + n )θ , ∑ 2n 2π π m = −∞ , m ≠ 0, − ∞ (2.53) l γ n ,1 = l 2 sin α ∑ 2n 2π 2 Cn,n+ml [sin α + (ml + n)n cos α ] , m = −∞ , m ≠ 0, − ∞ l 31 (2.54) γ n ,2 = ∞ ∑ m = −∞ , m ≠ 0, − 2n l l 2 cos α Cn ,ml + n (ml + n)n[sin α + (ml + n)n co α ] . s 2π 2 (2.55) These expressions show the relationships between parameters and eigensolutions, which is convenient for modal analysis, system identification, response analyses and design. The nonzero Fourier coefficient indices for equally spaced springs are integer multiples of the number of springs. Substitution of this into the general contamination rule (2.39) yields the mode contamination rule for rings having equally spaced springs s±n = ml , m = 1, 2,3, . (2.56) This rule is consistent with the mode contamination rule in previous works where discrete, equally spaced attachments are added to an axisymmetric structure [62-64]. The magnitude of each contaminating component is evident from (2.49), (2.52) and (2.53). Substitution of (2.19) and (2.30) into (2.38) gives the expression for un ∞ ∞ un =( an ,1 + an ,2 ) ∑ As cos sθ + cos nθ + i ( an ,1 − an ,2 ) ∑ Bs sin sθ + sin nθ , (2.57) 1, s ≠ n 1, s ≠ n s= s= 1 + n2 1 + n2 As = Cn , s ( Pn , s + Pn ,− s ) , Bs = Cn , s ( Pn , s − Pn ,− s ) . −ε −ε 2π 2π (2.58) For repeated natural frequencies ( 2n / l ≠ int ), an are arbitrary unit vectors, and (2.57) is an arbitrary linear combination of the expressions in square brackets. Thus, one can regard the terms in the first square brackets as one mode of the degenerate pair and terms in the second square brackets as the other mode. For split natural frequencies, the two an are determinate, but substitution (separately) of an = 2 / 2 (1, 1) = and an Τ 2 / 2 (1, − 1) Τ from (2.51) into (2.57) also yields the bracketed expressions in (2.57) as the modes. Thus, 32 the bracketed expressions in (2.57) are the two vibration modes associated with the base n nodal diameter mode, regardless of whether the springs split the degenerate natural frequency. One mode is called the cos nθ mode and has only cosine components, while the other is called the sin nθ mode and has only sine components. The distinction between the cases of splitting and no splitting lies in the properties of As and Bs , the coefficients of the contaminating components for the cos nθ and sin nθ modes. For no splitting of the degenerate unperturbed eigenvalues, the condition 2n / l ≠ int guarantees n − s and −n − s cannot simultaneously be integer multiples of l because their difference is 2n . This and (2.32) ensure either Pn ,s or Pn ,− s is zero. ml ( Pn ,− s = 0 ), then As = Bs . In contrast, if s + n = ml ( Pn ,s = 0 ), then Therefore, if s − n = As = − Bs . One of these two conditions holds. This structure of the base and contaminating Fourier coefficients for repeated natural frequencies does not hold when the natural frequencies split because both of Pn ,s and Pn ,− s are nonzero in (2.58). As an example, the mode contamination properties are considered for a ring with five identical, equally spaced springs. Figure 2.4a depicts the Galerkin method Fourier coefficients for the three nodal diameter mode, where the natural frequencies do not split. The mode contamination rule (2.57) is confirmed with the contaminants s = 2, 7, 8,12,13 . Furthermore, the coefficients As and Bs satisfy As = Bs for s − n = ml ml , as predicted by perturbation. In contrast, Figure 2.4b shows and As = − Bs for s + n = the contamination behavior of the five nodal diameter split natural frequency modes. Both the sine and cosine components of the fifth mode are contaminated by s = 0,10,15, 20, , but their magnitudes are different, as predicted. There is a special case 33 for modes with split natural frequencies: When the discrete supports are uniform, equally spaced radial springs, only the cos nθ split modes are contaminated; the sin nθ split modes are associated with natural frequencies that are independent of the supports, so they have no contamination. On the contrary, cos nθ modes have no contamination for uniform, equally spaced tangential springs. Figure 2.4. Fourier coefficients of vibration modes for a ring with five identical, equally spaced springs with ε = 500 , α = π / 4 , β = 0 . Black ■ ( As ) denotes the mode having cosine components, white□ ( Bs ) denotes the mode having sine components. (a) Three nodal diameter modes. (b) Five nodal diameter modes. 34 From (ωs2 − ωn2 ) in the denominator of (2.31), the amplitude of a particular mode contamination component depends on the proximity s − n of the contaminating nodal diameter s to the base nodal diameter n. For repeated modes, the contaminations are prominent because the nodal diameters s of the contaminating components can be close to n . For split natural frequencies ωn , however, the difference between s and n satisfies s − n ≥ l , so rn , s is relatively small. Therefore, in what may seem a counter-intuitive result, split modes are less contaminated compared to repeated modes even though they have greater natural frequency change. 2.4.2. Mode Classification for Rings Having Identical, Equally Spaced Spring Supports As shown in Appendix B for the case of identical, equally spaced spring supports, the natural frequency splitting rule does not change for the second or higher order perturbations. In other words, if a degenerate natural frequency does not split at first order, it does not split at any order. A similar conclusion exists for the mode contamination rule; the sets of nodal diameter components present in (2.49), (2.52) and (2.53) do not change for the entire range of ε > 0 . The modes of a ring having equally spaced springs are classified into distinct and degenerate modes as reduced from (2.49), (2.52) and (2.53). The distinct modes have distinct natural frequencies and reduce from (2.52) and (2.53). They evolve from the n nodal diameter free ring modes where 2n / l = integer . = m 0,1,, ∞ They are linear combinations of the ml nodal diameter components, where 35 ∞ (u n )1 = ∑ U mln ,1 cos mlθ , m=0 ∞ (u n ) 2 = ∑ U mln ,2 sin mlθ , m=0 n = 0, l , 2l , . (2.59) One such pair of modes exists for every n = 0, l , 2l , , where n indicates the dominant nodal diameter component (as it does for the mode types to follow). These modes exist for even or odd l. For even l, a second pair of modes, each with distinct natural frequency, l 3l 5l exists for every n = , , , 2 2 2 = (u n )1 ∞ l + )θ , (u n ) 2 l cos( ml = ml + ,1 2 2 ∑U n m=0 ∞ ∑U m=0 n l ml + ,2 2 l l 3l 5l sin(ml + )θ , n = , , , . (2.60) 2 2 2 2 All split natural frequency modes are of the form (2.59) or (2.60). For example, with l = 4 , representative modes of the form (2.59) and (2.60) are 4 4 4 (u 4 )1 = U 0,1 cos 4θ + U 8,1 cos 8θ + , + U 4,1 2 2 (u 2 )1 = U 2,1 cos 2θ + U 6,1 cos 6θ + U102 ,1 cos10θ + . (2.61) (2.62) The degenerate modes have degenerate frequencies and reduce from (2.49). They evolve from the n nodal diameter free ring modes where 2n / l ≠ integer , which implies that n can be written as = n jl + s , where j is an arbitrary integer and s is one of the integers belonging to [1,int((l − 1) / 2)] . For a given s in this range, these degenerate modes are arbitrary linear combinations of the following two independent modes = (u n )1 ∞ + s )θ , (u n )2 ∑ U mln +s cos(ml= m = −∞ ∞ ∑U m = −∞ n ml + s sin( ml + s )θ , n =s, l ± s, 2l ± s, . (2.63) With l = 4 , representative modes of the form (2.63) are u1= c1 U11 cos θ + U −13 cos 3θ + U 51 cos 5θ + U −17 cos 7θ + +c2 U11 sin θ − U −13 sin 3θ + U 51 sin 5θ − U −17 sin 7θ + , 36 (2.64) u 3= c1 U13 cos θ + U 33 cos 3θ + U 53 cos 5θ + U 73 cos 7θ + +c2 −U −31 sin θ + U 33 sin 3θ − U −35 sin 5θ + U 73 sin 7θ + , (2.65) where c1 and c2 are arbitrary. When each support has only one spring ( α = 0 or α = π / 2 ), some of the distinct modes have only a single nodal diameter component. These are called ring modes. Their natural frequencies are the same as for the corresponding nodal diameter mode for a free ring. When each support has only one tangential spring ( α = 0 ), the modes governed by the first of (2.59) and (2.60) devolve into the ring modes (u n )1 = 1 π (1 + n 2 ) cos nθ , n = l , 2l , for odd l n = l / 2, l ,3l / 2 , for even l (2.66) Similarly, when each support has only one radial spring ( α = π / 2 ), the modes governed by the second of (2.59) and (2.60) devolve into the ring modes (u n ) 2 = 1 π (1 + n 2 ) sin nθ , n = l , 2l , for odd l n = l / 2, l ,3l / 2 , for even l (2.67) Of all split modes of a ring having identical, equally spaced supports with one spring at each support, half are ring modes and the others are modes of the form (u n )1 (for α = π / 2 ) or (u n ) 2 (for α = 0 ) governed by (2.59) and (2.60). 2.4.3. Effect of Support Number Figure 2.5 shows the relationship between natural frequencies and number of supports obtained by the Galerkin method. 37 Figure 2.5. Natural frequencies versus number of supports (equally spaced) with ε = 10 , α = π /2 , β = 0. All natural frequencies are degenerate for the free ring, and they all split for one or two springs. Figure 2.5 confirms the natural frequency splitting rule for rings with l equally spaced springs: when the nodal diameter n and the number of supports l satisfies n = ml / 2 for even l or n = ml for odd l ( m = 1, 2,3, ), then the natural frequencies split; otherwise they do not. For split modes, the split modes with smallest nodal diameter (m=1) have the largest difference between split natural frequencies. For a given l , the maximum discrepancy between split natural frequencies occurs for n = l / 2 for even l 38 and n = l for odd l . For a given n nodal diameter mode, the maximum difference between split natural frequencies always occurs for l = 2n . 2.4.4. Effect of Stiffness The natural frequencies and vibration modes of rings on radial springs were investigated by Allaei et al. [56], which is a special case of the present model with α = π / 2 , β = 0 . In this paper, radial springs, tangential springs and their combination are considered. As an example, the dynamic characteristics of a ring with four elastic springs are presented. Figure 2.6 shows the natural frequency dependence on stiffness for equally spaced radial or tangential springs ( β = 0 ). There is a range for each mode where the associated natural frequency is sensitive to spring stiffness, whether radial or tangential. The number of supports l determines the sensitivity of each mode (as discussed in Figure 2.5). Comparison of the dashed and solid lines for the same mode shows that tangential stiffness and radial stiffness dominate in different ranges. For small stiffness, the effect of radial springs is stronger than that of tangential springs, while tangential springs are more dominant for high stiffness. For purely radial springs, a rigid body rotational mode ( n = 0 ) always exists. When the supports have tangential components, this natural frequency is no longer zero, and its mode shape is contaminated with other nodal diameter modes according to (2.56). As seen in Figure 2.6, some natural frequency loci cross each other while others veer away. For example, the loci (solid lines) of n = 3 and n = 5 veer away as they approach each other. This is because the three and five nodal diameter modes have the same mode 39 components as they both contain 1,3,5,7, nodal diameter components, which creates strong coupling of the two modes in the veering region. Figure 2.6. Natural frequencies for varying spring stiffness ε with l = 4 , β = 0 by the Galerkin method. Solid lines denote natural frequencies for tangential stiffness springs ( α = 0 ), dashed lines denote natural frequencies for radial springs ( α = π / 2 ). Figure 2.7 reveals the effects of stiffness on vibration modes from the Galerkin solution. The first eight modes are shown for four identical, equally spaced radial springs 40 with ε = 10, 100, 1000 . The vibration modes for ε = 1000 are close to the modes of a ring with rigid springs. Vibration modes with repeated natural frequencies are always heavily contaminated by other numbers of nodal diameters, much moreso than for split natural frequency modes, as predicted by perturbation. The contamination is the same for each mode in a repeated natural frequency pair, while a split natural frequency pair has different contamination in each split mode. When the supports are radial springs, one split mode is exactly the free ring vibration mode with no contamination (see Figure 2.7c, g), while the other mode suffers significant mode contamination (see Figure 2.7d, h), as predicted by perturbation. 41 (a) (b) (c) (d) Continued Figure 2.7. Mode evolution of a ring with four equally spaced springs radial indicated by circles with α = π / 2 , β = 0 for ε = 10 , 100 and 1000. (a-b) 1 nodal diameter mode, (c-d) 2 nodal diameter mode, (e-f) 3 nodal diameter mode, (g-h) 4 nodal diameter mode. The values are natural frequencies. 42 Figure 2.7 continued (e) (f) (g) (h) For rings with equally spaced tangential or radial springs, each natural frequency is bounded in a specific range as the stiffness increases. The upper bound of natural frequency is determined by the mode contamination sequence for the chosen mode. For a 43 given n nodal diameter base mode, if the contamination sequence is , n, p, , then the natural frequency is bounded according to ωn ≤ ωn < ω p (2.68) where ωn and ω p are the n and p nodal diameter natural frequencies for a free ring and ωn is the corresponding natural frequency of a ring with equally spaced springs. In the above example with l = 4 , the natural frequencies are bounded as: ω0 ≤ ω0 < ω4 , ω1 ≤ ω1 < ω3 , ω2 ≤ ω2 < ω6 , ω3 ≤ ω3 < ω5 , and so on. 2.5. Vibration of Rings on Unequally Spaced Spring Supports The ring gear of a planetary gearset with unequally spaced planets can be modeled as a ring with several unequally spaced spring supports. Ring-planet gear tooth meshes are modeled as elastic springs due to tooth compliance. As a concrete example of a ring with unequally spaced spring supports, the dynamic characteristics of the ring gear in a US Army OH-58 Kiowa helicopter planetary gear are studied. Ring gear parameters and material properties are listed in Table 2.3. The mesh stiffnesses between the planets and the ring gear are modeled as four unequally spaced springs located at θ1 = 0 , θ 2 = 32π / 63 , θ 3 = π and θ 4 = 95π / 63 with α = π / 2 and = β 90ο − 24.6ο , where 24.6ο is the pressure angle. The mesh stiffness = is k 3.237 × 107 N/m and the ring kbend bending stiffness is= EJ = 6.0025 × 105 N/m, so the non-dimensional mesh 2 r (1 − ν ) 3 44 stiffness is ε = 53.9279 . The dimensional natural frequencies are found from ωdim = 1266.8ω . = k 3.237 × 107 N/m Mass 2.35 kg Bending stiffness Ring radius r = 0.014415 m Young’s Modulus E = 2.0717 × 1011 N/m Ring thickness h = 0.01551 m Poisson’s ratio ν = 0.3 Face width b = 0.0254 m Pressure Angle 24.6 deg Table 2.3. Ring gear parameters and material properties of OH-58. Natural frequency ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 Unequally spaced 2.1469 2.6952 3.1769 3.3201 7.2362 9.2249 9.4470 14.5577 Equally spaced 2.1560 2.6833 3.2477 3.2477 7.2401 9.3367 9.3367 14.5521 Table 2.4. Comparison of dimensionless natural frequencies for an elastic ring with four equally and unequally spaced springs with ε = 53.9279 . Unequally spaced springs are located at θ1 = 0 , θ 2 = 32π / 63 , θ 3 = π and θ 4 = 95π / 63 . When the springs are unequally spaced, the natural frequencies and vibration modes change significantly. Table 2.4 compares natural frequencies for the unequally and 45 equally spaced cases. All natural frequencies split for rings with asymmetric springs. The modes are shown in Figure 2.8. The first two modes shown in Figure 2.8 correspond to the two nodal diameter base mode with contamination from the 0, 4, 6, nodal diameter components. ω1 = 2.1467 ω2 = 2.6952 ω3 = 3.1766 ω4 = 3.3199 ω5 = 7.2354 ω6 = 9.2246 ω7 = 9.4465 ω8 = 14.5577 Figure 2.8. Modes of a ring on four unequally spaced springs with α = π / 2 , β = 65.4ο and ε = 53.9279 . Unequally spaced springs are located at θ1 = 0 , θ 2 = 32π / 63 , θ 3 = π and θ 4 = 95π / 63 . 46 In planetary gears with unequally spaced planets, planet pairs typically lie on diameters, although these diameters are not equally spaced. This is done for load sharing and bearing force considerations. For four planets, the locations of the planets can be θ2 π / 2 + γ , θ3 = π = represented as θ1 = 0 ,= , θ 4 3π / 2 + γ with γ being a modulation of the equally spaced springs. The influence of modulation γ on natural frequencies is shown in Figure 2.9. When γ = ±π / 2 , the system is equivalent to rings having two equally spaced springs (the stiffness is doubled at each support). When γ = 0 , the equally spaced case is recovered. According to the natural frequency splitting rule of rings on equally spaced springs, two loci for each of n = 1,3, meet at γ = 0 , which corresponds to repeated natural frequencies. As the modulation γ changes from −π / 2 to π / 2 the loci of the n nodal diameter base natural frequency pair cross n times. 47 Figure 2.9. Natural frequencies for varying modulation γ with l = 4 , α = π / 2 , β = 65.4ο and ε = 53.9279 . 2.6. Vibration of Rings on Distributed Elastic Foundation A distributed elastic foundation is an appropriate model in certain rotor/stator systems or bearings. Consider the foundation stiffness distributions 1 1 1 1 d (θ ) = 1 + cos θ + sin θ + sin 4θ − cos 6θ , e(θ ) = d (θ ) , 4 4 3 4 (2.69) with β = 0 . Fourier expansion of d (θ ) ensures that only d 0 , d1 , d 4 , d 6 and their complex conjugates are non-zero. The Fourier coefficients of e(θ ) have the same 48 properties. The set of nontrivial Fourier coefficient indices is ϒ= {0, ±1, ±4, ±6} . Natural frequency splitting properties at the first and second orders are shown in Table 2.5. n =1 n=2 n=3 n=4 n=5 n≥6 βn R S R R R R γn S S S S R Table 2.5. Natural frequency splitting for rings having the elastic foundations in (2.69). S denotes split natural frequencies, and R denotes repeated natural frequencies. Natural frequencies split at the first order for n = 3 because 2n ∈ ϒ . For n = 2 , the natural frequency pair splits at first order if either of the individual distribution is considered because 2n ∈ ϒ , but it remains repeated with both d (θ ) and e(θ ) acting 0 . Thus, the combined effects of d (θ ) and e(θ ) simultaneously due to d 4 − 4e4 = neutralize the individual effects. Natural frequencies split at second order for n = 1,2,4,5,6 due to P− n , s Ps ,n ≠ 0 for certain number of s, such as when n = 1 and s = −5 , or when n = 4 and s = 0 , both n − s and −n − s belong to ϒ . Natural frequencies do not split at either order for n > 6 . According to the mode contamination rule (2.39), the s nodal diameter components contaminate the n nodal diameter vibration mode for all s satisfying s ± n∈ϒ. 49 Chapter 3: Modal Properties of Planetary Gears with an Elastic Continuum Ring Gear EQUATION CHAPTER 3 SECTION 1 The distinctive modal properties of equally spaced planetary gears with elastic ring gears are studied through perturbation and a candidate mode method. All eigenfunctions fall into one of four mode types whose structured properties are derived analytically. Two perturbations are used to obtain closed-form expressions of all the eigenfunctions. In the Discrete Planetary Perturbation (DPP), the unperturbed system is a discrete planetary gear with a rigid ring. The stiffness of the ring is perturbed from infinite to a finite number. In the Elastic Ring Perturbation (ERP), the unperturbed system is an elastic ring supported by the ring-planet mesh springs; the sun, planet and carrier motions are treated as small perturbations. A subsequent candidate mode method analysis proves the perturbation results and removes any reliance on perturbation parameters being small. All vibration modes are classified into rotational, translational, planet and purely ring modes. The well defined properties of each type of mode are analytically determined. All modal properties are verified numerically. 50 3.1. Introduction Planetary gears are widely used in automotive and aerospace transmissions due to advantages such as compactness, high torque/weight ratio, low bearing load and high transmission ratio. In practical systems where planetary gear vibration is a key concern, ring gear elastic deformation is significant. This is especially true for planetary gears with thin rims, including those used in aerospace applications. The free vibration of planetary gears with equally spaced planets has typically been studied by treating all the planetary gear components as rigid bodies [13-20]. Lin and Parker [21] established a lumped parameter model that includes both transverse and torsional motion. The modal properties were obtained analytically, and the vibration modes are classified into rotational, translational and planet modes. In the present paper, these modes are called discrete rotational, translational and planet modes. Lin and Parker used this discrete model to study natural frequency and vibration mode sensitivity [22], natural frequency veering [23], and parametric instability caused by changing contact conditions at the multiple tooth meshes [24]. The influence of gear rim flexibility on static and dynamic behavior of planetary gears has been studied by a few researchers. Kahraman and Vijayakar [25] investigated the impact of ring gear rim flexibility on gear stresses and planet load sharing under static conditions using a deformable-body planetary gear model. The study indicates that reducing rim thickness minimizes the adverse effects of gear and carrier manufacturing errors and improves the planet load sharing. Kahraman et al. [26] studied the effect of 51 gear rim flexibility on dynamic behavior of planetary gears using the finite element method, exposing the importance of ring deformation for practical systems. This study analytically addresses the dynamics of planetary gears having elastic ring gears. An elastic-discrete model is developed, where the ring gear is modeled as an elastic body while all other gears are represented as rigid bodies. Modal properties are derived in detail using eigenvalue perturbation and a candidate mode method. Two unperturbed systems are considered to form a complete representation for all modes. This yields closed-form expressions for all the eigenfunctions and a systematic characterization of planetary gears’ highly structured modal properties. All vibration modes are classified in detail into four different types according to their unique characteristics. These perturbation results are proved by a mathematically rigorous approach where vibration modes having the form revealed by perturbation are assumed and then shown to satisfy all equations of the elastic-discrete eigenvalue problem. This builds a base for subsequent studies such as dynamic response, parametric instability, and contact loss nonlinearity, all of which commonly use modal expansion methods. 3.2. Modeling and Equations of Motion An elastic-discrete model of a planetary gear is shown in Figure 3.1. All gear meshes are represented by linear springs. The sun, carrier and planets are considered as rigid bodies, while the ring gear is modeled as a thin elastic body. The bearings and supports of the sun, carrier and planets are modeled as two perpendicular springs of equal stiffness. The bearings and supports of the ring gear are represented as an elastic foundation with 52 uniform radial and tangential distributed stiffnesses per unit length krbs and krus , respectively. The planets are identical and equally spaced. All ring-planet mesh stiffnesses are equal ( krp ), and all sun-planet mesh stiffnesses are equal ( k sp ), where krp and k sp are averages over a mesh cycle. The angular speeds are assumed to be small, so gyroscopic effects are neglected. The coordinates are shown in Figure 3.1. The deformation of the sun and carrier T = p j = x j , y j , u j , j s, c are described relative to the fixed basis {i, j, k} ; the tangential displacement of the ring is u(θ , t ) ; the ring radial deflection is determined from the inextensibility condition w = −∂u / ∂θ [2] ; and, the deflections of the planets are p n = [ξ n ,ηn , un ] , n = 1, , N . The symbol u j denotes rotational (or tangential) deflection T (rotation in radians times the gear base radii rs , rr , rp or radius of the carrier rc ). The equations of motion for the sun and carrier are the same as those in the discrete model [18], while the equations of motion for the ring and planets change. The equation of motion for the elastic ring gear is [2] N M eu + kbend L1u + k rp L2u + k rp ∑ Ln3 (ξ n sin α r − ηn cos α r − un ) = 0, (3.1) n =1 = M e ρ R(1 − EJ ∂6 ∂4 ∂2 ∂2 = k , , = − + + L ( 2 ), ) bend 3 2 R (1 − ν ) 1 ∂θ 2 ∂θ 6 ∂θ 4 ∂θ 2 N ∂δ (θ −ψ n ) ∂2 ∂ −∑ (sin 2 α r − cos 2 α r )δ (θ −ψ n ) + (sin α r + cos α r ) sin α r L2 = 2 ∂θ ∂θ ∂θ n =1 (3.2) 2 ∂ +(krus R − krbs R 2 ) / krp , ∂θ 53 Ln3 cos α rδ (θ −ψ n ) − sin α r = ∂δ (θ −ψ n ) , ∂θ (3.3) where kbend is the ring bending stiffness (see Nomenclature). L1 , L2 and Ln3 are dimensionless operators. The first two terms of (3.1) represent the in-plane vibration of a free ring; the last two terms incorporate the effects of gear meshes and elastic supports. Figure 3.1. Elastic-discrete model of a planetary gear and corresponding system coordinates. The distributed springs around the ring circumference are not shown. 54 Separation of the ring rigid body motions from the elastic deformation v (θ , t ) is achieved with the expansion ±∞ u (θ , t ) =v(θ , t ) + U1 (t )eiθ + U −1 (t )e − iθ + U 0 (t ) =∑ Vm (t )eimθ + U1 (t )eiθ + U −1 (t )e −iθ + U 0 (t ). (3.4) m = ±2 Thus, v is orthogonal to the rigid body motions ∫ 2π 0 v dθ = 0 , ∫ 2π 0 veiθ d θ = 0 , ∫ 2π 0 ve − iθ d θ = 0 . (3.5) Substituting (3.4) into (3.1) and forming the inner product of the result with eimθ yields the discretized equations of motion. Comparison of the equations for the rigid ring motions U1 ,U −1 ,U 0 to the equations of motion for a rigid ring planetary gear model with variables p r = ( xr , yr , ur )T [18] yields the relations xr = −i (U1 − U −1 ) , y= U1 + U −1 , ur = U 0 cos α r , I r = mr R 2 . r (3.6) The true moment of inertia expression for a ring is = I r ,true mr ( r22 + r12 ) / 2 , where r1 and r2 are the inner and outer radii of the ring gear, respectively. The difference between I r and I r ,true is small when the ring is thin. We introduce the following dimensionless quantities v = k R ki mr t v = , i r, c, s, p, rp, sp, bend , krbs = rbs , (3.7) ,τ= ,T= , ki k= krp rp k rp R T Ij mj k R krus = rus , m j == , j r , c, s , n . , Ij = krp mr rj2 mr 55 (3.8) In what follows the ~ on all variables is omitted, and the equations of motion remain the same except that k rp is replaced by 1, M e is replaced by 1 2π (1 − ∂∂θ 2 ) , and k rbs R , k rus R 2 are replaced by k rbs , k rus . The displacement of the whole system is separated into v (θ ,τ ) and q(τ ) . v is the elastic deformation of the ring gear, and q is a vector of the deflections for the discrete elements including the ring rigid body motions q = [ xr , yr , ur , xc , yc , uc , xs , ys , us , ξ1 ,η1 , u1 , , ξ N ,η N , u N ]T . pr pc p1 ps (3.9) pN The dimensionless equations of motion and the associated eigenvalue problem in extended operator forms are M a + Ka = 0, (3.10) −ωi2 Mai + Kai = 0, (3.11) T where a = v, qT is referred to as an extended variable, ωi is a natural frequency, and M, K are extended stiffness and inertia operators defined by their action on elements of the space of extended variables according to 1 (1 − ∂ 2 )v (kbend L1 + L2 )v + L3q ∂θ Ma = 2π , Ka = , L4 v + Kq Mq 2 = L3q N ∑ cos α δ (θ −ψ n =1 r n ) − sin α r (3.12) ∂δ (θ − ψ n ) δ n , ∂θ (3.13) N T T T T , χ ∂v sin α r + v cos α r , (3.14) ), 0, 0, b p χ θ ψ= , , b p χ θ ψ= L4 v = ∑ (b r χ θ =ψ n = 1 N ∂θ n =1 br = [ − sinψ rn , cosψ rn , 1] T 56 , b p = [sin α r , − cos α r , − 1] , T (3.15) − xr sinψ rn + yr cosψ rn + ur + ξ n sin α r − ηn cos α r − un . δn = ∫ M and K are self-adjoint with the inner product = < a1 , a2 > 2π 0 (3.16) v1v2dθ + q1T q 2 , where an overbar denotes complex conjugate. M and K (see Appendix B for details) are the dimensionless mass and stiffness matrices for planetary gears based on a discrete model. Their dimensional forms are identical to the mass and stiffness matrices in [18] with the only difference in M r and K rb as M r = diag(1,1,1/ cos 2 α r ) , K rb = diag(4krbs + 4krus , 4krbs + 4krus , 2π krus / cos 2 α r ) . (3.17) Expansion of (3.11) into N + 4 groups of equations associated with the individual components yields − ωi2 ∂2 (1 − 2 )v + kbend L1v + L2v + L3q = 0, 2π ∂θ (3.18) −ωi2 M r p r + (K rb + ∑ K nr1 )p r + ∑ K nr 2p n + ∑ (b r χ θ =ψ ) = 0, (3.19) n n n n −ωi2 M c p c + (K cb + ∑ K cn1 )p c + ∑ K cn2p n = 0, (3.20) −ωi2 M s p s + (K sb + ∑ K ns1 )p s + ∑ K ns 2p n = 0, (3.21) n n n n −ωi2 M p p n + (K cn2 )T p c + (K rn2 )T p r + (K ns 2 )T p s + K p ppn + b p χ θ =ψ = 0 , n = 1, , N . (3.22) n Equation (3.11) is cast entirely in discrete form with modal expansion of v as v (θ ,τ ) = ± JN ∑V m = ±2 m (τ )eimθ , (3.23) where J ≥ 1 is an integer. The basis functions eimθ are complete, (3.23) converges, and J is arbitrarily large. Thus, the error in (3.23) can be made as small as desired. No 57 restriction is put on J in what follows, so the findings apply to the continuum ring model without any limitation introduced by the expansion (3.23). A discretized model results from substitution of (3.23) into (3.18)-(3.22) and then forming the inner product of (3.18), (3.19) and (3.22) with eipθ . Numerical experiments on the discretized equations confirm that ring elastic deformation alters the natural frequencies and vibration modes compared to the lumped parameter model and introduces additional natural frequencies associated with modes dominated by ring elastic deformation. The numerical solutions indicate that all vibration modes of this elasticdiscrete model are classified into four types: rotational, translational, planet and purely ring modes. For example, a planetary gear with six equally spaced planets is analyzed with J = 3 in (3.23). The system parameters and the dimensionless natural frequencies are listed in Table 3.1. The natural frequencies in Table 3.1 include all four mode types: ω1 , ω9 and ω14 are for rotational modes; ω2,3 and ω7,8 are for translational modes; ω4,5 and ω10,11 are for degenerate planet modes (type 2) and ω6 , ω13 are for distinct planet modes (type 3); ω12 is for a purely ring mode. Figure 3.2a shows the vibration mode of a rotational mode ( ω1 ). From the numerical simulations, a rotational mode has the following characteristics: (a) The discrete elements q have the same properties as a discrete rotational mode, where the translations of the sun, carrier, and ring rigid motion are zero, and all planets have identical deflections, (b) The associated natural frequency is distinct, (c) The elastic deformation of the ring contains only jN, j = 1, 2, , J nodal diameter components. 58 (a) Rotational mode ( ω1 ) (b) Translational mode ( ω2,3 ) (c) Planet mode: degenerate ( ω4,5 ) (d) Planet mode: distinct ( ω6 ) (e) Purely ring mode ( ω12 ) Figure 3.2. Typical modes of a planetary gear. The system parameters are given in Table 3.1. Distinct planet modes as in (d) only exist for an even number of planets. 59 Inertias (kg) I r / rr2 = 8.891 , I c / rc2 = 6.000 , I s / rs2 = 2.500 , I p / rp2 = 2.000 Masses (kg) mr = 7.350 , mc = 5.430 , ms = 0.400 , m p = 1.000 Stiffnesses (N/m) k= k= 108 rp sp Pressure angle (deg) α= α= 24.60 r s Dimensionless natural ω1 = 0.1520 (R), ω2,3 = 0.1871 (T), ω4,5 = 0.6472 (P), ω6 = 1.0227 (P), frequencies ω7,8 = 1.0231 (T), ω9 = 1.1009 (R), ω10,11 = 1.1695 (P), ω12 = 1.6971 (PR), , k= k= 0 , k = 5 × 106 , k = k = 5 × 107 , k = k = 5 × 1011 , rbs rus bend s su c cu ω13 = 1.8549 (P), ω14 = 1.9161 (R) Table 3.1. Dimensional parameters and dimensionless natural frequencies of a planetary gear with six equally spaced planets. The designations R, T, P and PR denote rotational, translational, planet and purely ring modes. Figure 3.2b shows the vibration mode of a translational mode ( ω2,3 ). A translational mode has the following characteristics: (a) The discrete elements q have the same properties as a discrete translational mode, where the rotations of the sun, carrier, and ring rigid motion are zero, and the deflections of the planets are related by a rotation matrix, (b) The associated natural frequency is repeated with multiplicity two, (c) The elastic deformation of the ring contains only jN ± 1 nodal diameter components, where j is any nonzero integer satisfying jN ± 1 ∈ {− JN , − JN + 1, , JN } (a condition imposed by the ± JN limits in (3.23)). Planet modes are classified into two sub-types according to the degeneracy of the natural frequencies. For odd N all planet modes are degenerate, as are the majority of planet modes for even N. Degenerate planet modes have the following characteristics: (a) 60 The discrete elements q have the same properties as a discrete planet mode, where the deflections of the sun, carrier, and ring rigid motion are zero, and the deflections of the planets are scalar multiples of the first planet’s deflection, (b) The associated natural frequency is repeated with multiplicity two, (c) Each mode is associated with a particular s ∈ {2,3, ,int( N2−1 )} . For that particular s, the elastic deformation of the ring contains only jN ± s nodal diameter components, where j is any integer satisfying jN ± s ∈ {− JN , − JN + 1, , JN } . Figure 3.2c shows a degenerate planet mode ( ω4,5 ) where the two nodal diameter component is the dominant ring deformation. For even N, the remaining planet modes have distinct natural frequencies. Their discrete elements behave as in (a) above, but their elastic ring deflection contains only jN + N / 2 nodal diameter components, where j is any integer satisfying jN ± N / 2 ∈ {− JN , , JN } (see Figure 3.2d for a distinct planet mode). Thus, planet modes are classified into int ( N2 ) − 1 subtypes according to the ring nodal diameter components they contain. Planet modes having jN ± s nodal diameter components are named type s planet modes. Each planet mode belongs to a unique type. For the example where N = 6 , two types exist: the degenerate planet modes are type 2 ( s = 2, , int ( N2−1 ) ), and the distinct planet modes are type 3 ( s = N 2 ) which only exist for even N. There are no planet modes outside of these two types for N = 6 . Table 3.2 summarizes the number of degenerate/distinct planet modes and their types for varying numbers of planets. 61 Planet mode N 4 5 6 7 8 9 10 J+3 0 J+3 0 J+3 0 J+3 0 2(2J+3) 2(2J+3) 4(2J+3) 4(2J+3) 6(2J+3) 6(2J+3) Type 2 planet modes J+3 2(2J+3) 2(2J+3) 2(2J+3) 2(2J+3) 2(2J+3) 2(2J+3) Type 3 planet modes × × J+3 2(2J+3) 2(2J+3) 2(2J+3) 2(2J+3) Type 4 planet modes × × × × J+3 2(2J+3) 2(2J+3) Type 5 planet modes × × × × × × J+3 category Distinct planet modes Degenerate planet modes Table 3.2. Number of planet modes in different sub-types for different number of planets N , where × denotes not applicable. Figure 3.2e shows a purely ring mode ( ω12 ). A purely ring mode has the following characteristics: (a) The discrete elements q are all zero, (b) The natural frequency is distinct, (c) The elastic deformation of the ring contains only a single nodal diameter component. 61 eigensolutions are obtained In this example ( N = 6 , J = 3 ), 3 N + 2 JN + 7 = numerically: J +6= 9 rotational modes, 4 J + 10 = 22 translational modes, (2 JN − 7 J ) + (3N − 9) = 24 planet modes divided as 2 J + 3 = 9 degenerate pairs and J +3= 6 distinct modes, and 2 J = 6 purely ring modes. The remainder of this paper analytically proves that these properties (natural frequency multiplicity, modal properties and the number of each type of mode) hold for general planetary gears. 62 3.3. Perturbation Analysis To find all natural frequencies and vibration modes of the elastic-discrete model of a planetary gear, two perturbations are used for different ranges of parameters. For the chosen nondimensional variables, the stiffness of the ring-planet mesh is always unity ( k rp ≡ 1 ). The first perturbation is termed Discrete Planetary Perturbation (DPP), with the unperturbed system being a discrete planetary gear having a nearly rigid ring gear where the bending stiffness is O (1/ ε ) while the stiffnesses of all remaining meshes/supports are O (1) . The small quantity ε is the ring bending compliance. The opposite case of DPP is Elastic Ring Perturbation (ERP). In this case the bending stiffness is O (1) and the stiffnesses of the remaining meshes/supports (except k rp ≡ 1 ) are O (1/ ε ) . The unperturbed system for the ERP is an elastic ring having multiple springs with the elimination of the rigid body motions. The attached springs represent the ringplanet gear meshes. The combined eigensolutions from the DPP and ERP form a complete set of eigensolutions for planetary gears having elastic rings without any redundancy (as proved in a subsequent Candidate Mode Method solution). This process leads to analytical results that mathematically expose the system’s highly structured modal properties. 3.3.1. Discrete Planetary Perturbation In DPP, the ring bending stiffness is much larger than the mesh and bearing stiffnesses. The mesh and bearing stiffnesses are O (1) , while the ring bending stiffness 63 kbend = 1/ ε , where ε is a small parameter. The eigenvalue problem in extended operator form is −ω 2 (1 − ∂ 2 )v 1 L v L v + L q 2 2 3 Ma + Kˆ a 2π 2 ∂θ + ε 1 + = −ω = 0, −ω Mq 0 L4 v + Kq 2 (3.24) where M and K̂ are self-adjoint operators. The eigensolutions of (3.24) are represented as v 0 1 v1 a =a + ε a + O (ε ) , ω = ω + εµ + O (ε ) , a = 0 , a = 1 . q q 0 1 2 2 2 0 2 0 (3.25) Substitution of (3.25) into (3.24) gives the perturbation equations. The perturbation equation of order ε −1 is L1v 0 = 0 . L1 is positive definite, giving v0 = 0 . (3.26) Substitution of (3.26) into the remaining perturbation equations yields −ω02 Mq 0 + Kq 0 = 0, −ω02 Mq1 + Kq1= µ Mq 0 − L4 v1 . L1v1 = − L3q 0 , (3.27) (3.28) Equation (3.27) is the eigenvalue problem for a discrete (rigid ring) planetary gear model [18]. From (3.26) and (3.27), the unperturbed eigenfunction is 0 a0 = 0 . q (3.29) The structured properties of the discrete model unperturbed eigensolutions are proven analytically in [18], where the discrete system vibration modes q 0 are classified into rotational, translational and planet modes. In this study, they are called discrete rotational, translational and planet modes. In the elastic-discrete model, similar mode types are 64 found; they are called rotational, translational, and planet modes. The different mode types are considered separately. Common to each mode type, v1 is solved from the first equation of (3.28) by expanding v1 as v1 = ± JN ∑V e m = ±2 1 imθ m , multiplying (3.28) by e − imθ , and integrating from 0 to 2π . This yields Vm1 = − cos α r − im sin α r 2π m 2 ( m 2 − 1) 2 N ∑δ n =1 0 − imψ n n e , (3.30) where δ n0 is the nth ring-planet mesh deflection without considering the elastic deformation of the ring gear, as given by (3.16). According to (3.30), V−1m = Vm1 . 3.3.1.1. Rotational Modes When the unperturbed eigenfunction q 0 from (3.27) is a discrete rotational mode, the translational motions of the sun, carrier and ring are zero and all the planets have the same deflections [18] T q 0 = 0, 0, ur0 , 0, 0, uc0 0, 0, us0 , ξ10,η10 , u10 , , ξ10 ,η10 , u10 . (3.31) In the absence of any rigid constraints on any degrees of freedom (e.g., fixed carrier rotation), six such modes exist, each having a distinct natural frequency. Application of these properties to (3.30) yields Vm1 = − cos α r − im sin α r 0 N − imψ n δ r1 ∑ e ur0 + ξ10 sin α r − η10 cos α r − u10 . (3.32) , δ r01 = 2 2 2 2π m ( m − 1) n =1 65 ψ n 2π (n − 1) / N , the identity Because the planets are equally spaced with= N ∑e − imψ n =0 n =1 holds for m ≠ jN , where j is an arbitrary nonzero integer. Thus, for q 0 being a discrete rotational mode, the elastic deformation of the ring in the perturbed system contains only the jN nodal diameter components Vm1 = − cos α r − im sin α r N δ r01 , m = ± N , , ± JN . 2π m 2 (m 2 − 1) 2 (3.33) The eigenvalue perturbation µ is determined by the solvability condition of the second of (3.28) as (with < Mq 0 , q 0 >= 1 ) µ =< L4 v1 , q 0 >= − N 2 (δ r01 ) 2 π j =1,, J ∑ m = jN γm , γm = cos2 α r + m 2 sin 2 α r . m 2 ( m 2 − 1) 2 (3.34) A candidate solution of the second of (3.28) is proposed as T q1 = 0, 0, u1r , 0, 0, uc1 0, 0, u1s , ξ11,η11 , u11 , , ξ11,η11 , u11 . (3.35) Note that q1 has the same form as q 0 . Use of (3.35) and the known discrete rotational mode properties reduce (3.28) to 1 (2π krus / cos 2 α r + N − ω02 I r )u1r + N δ= µ I r ur0 − πN δ r01 r1 j =1,, J ∑ m = jN γm , (kcu + Nk p − ω 02 I c )uc1 − Nk pη11 = µ I c uc0 , (3.37) (k su + Nk sp − ω 02 I s )u1s − Nk sp (u11 − ξ11 sin α s + η11 cos α s ) = µ I s us0 , 1 0 0 N N (K1c 2 )Τ p1c + (K1r 2 )Τ p1r + (K1s 2 )Τ p1s + (K pp − ω02 M p )p= 1 µ NM p p1 − b p π δ r1 (3.36) j =1,, J ∑ m = jN γm . (3.38) (3.39) Expressing (3.36)-(3.39) in matrix form yields the 6×6 linear system A rot p1rot = b rot , 66 (3.40) T p1rot = u1r , uc1 , u1s , ξ11 ,η11 , u11 , b rot µ M rot p 0rot − πN δ r01 = j =1,, J ∑ m = jN γ m 1, 0, 0, bTp , M rot = diag( I r , I c , I s , NM p ) . (3.41) T (3.42) One can show that the solvability condition of (3.40) is identical to (3.34), so it is satisfied. This guarantees that the solution of the second of (3.28) has the assumed form (3.35). The normalization condition < q1 , Mq 0 >= 0 becomes < p1rot , M rot p 0rot >= 0 in this problem. This and (3.40) yield A rot 1 b rot ˆ p1 = bˆ . (M p 0 )T p rot = 0 ⇒ A rot rot rot rot rot (3.43) ˆˆˆT A ) −1 A T bˆ . This completes the solution for q1 in The solution of (3.43) is p1rot = ( A rot rot rot rot (3.28). Collecting results, we have six eigenfunctions a in (3.25) with the form ±1,, ± J j= ε ∑ V 1eimθ m . a = m = jN 0 1 q + ε q (3.44) The discrete elements of the planetary gear (including the ring rigid body motion) deflect as in the discrete rotational modes described in [18]. The elastic ring deflection contains only the jN nodal diameter components. The natural frequencies of these modes are distinct. 67 3.3.1.2. Translational Modes When the unperturbed eigenfunction from (3.27) is a discrete translational mode, the eigenvalues are repeated with multiplicity two and the rotational motions of the carrier, sun and ring are zero [18]. The pair of degenerate vibration modes q 0 and q̂ 0 satisfy T 0 0 0 0 0 T 垐 q 0 = p 0r , p c0 , p 0s , p10 , , p 0N , q垐 = p 0r , p垐 c , p s , p1 , , p N , (3.45) T T p 0j = x 0j , y 0j , 0 , = pˆ 0j y 0j , − x 0j , 0 , j = r , c, s . (3.46) ψ n 2π (n − 1) / N , the nth planet displacements p0n , pˆ 0n When the planets are located at= are related as p 0n cosψ n I sinψ n I p10 0 = 0 , n = 1, 2, , N , pˆ n − sinψ n I cosψ n I pˆ 1 (3.47) where I is a 3 × 3 identity matrix. Six such eigensolution pairs exist. The degenerate unperturbed eigenvalue ω02 of multiplicity two in (3.25) has two orthonormal, unperturbed eigenfunctions a 0 and â 0 of the extended operator form (3.11). As a consequence, the unperturbed eigenfunction a 0 is a linear combination of a 0 and â 0 0 0 = a 0 c1a 0 + c2aˆ 0 , a 0 = 0 , aˆ 0 = 0 , q qˆ (3.48) where c1 and c2 are constants. Analogous to the procedure for the rotational mode, use of the discrete translational mode properties reduces (3.30) to N cos α r − im sin α r N − i ( m −1)ψ n (c1 ic2 )( A1 − iA2 ) , − + Vm1 = υ e υ e − i ( m +1)ψ n , υ =+ ∑ ∑ 2 2 2 m − 1) n 1 =n 1 4π m (= A1 = yr0 cos α r − xr0 sin α r + ξ10 sin α r − η10 cos α r − u10 , 68 A2 = − yr0 sin α r − xr0 cos α r + ξˆ10 sin α r − ηˆ10 cos α r − uˆ10 , where υ is the complex conjugate of υ . N ∑e − i ( m −1)ψ n (3.49) being zero requires m ≠ jN + 1 , n =1 N where j is an arbitrary integer; ∑e n =1 − i ( m +1)ψ n being zero requires m ≠ jN − 1 . Thus, Vm1 vanishes if and only if m ≠ jN ± 1 . This yields the following rule: The elastic deformation of the ring for elastic translational modes contains only jN ± 1 nodal diameter components. The solvability conditions of the second equation of (3.28) form a 2 × 2 algebraic eigenvalue problem Dr c = µ c , where c = (c1 , c2 )T . Dr is diagonal with the repeated eigenvalues µ1 = µ2 = − N 2 ( A12 + A22 ) ∑ γm , 4π = m jN +1 (3.50) where here (and in all subsequent summations) j is an integer such that m takes only values within the range specified in (3.23), i.e., − JN ≤ m ≤ JN and m ≠ −1, 0,1 . Thus, the eigenvalues for the elastic ring model remain degenerate and c1 , c2 are indeterminate. The eigenfunction perturbation is proposed as = q1 c1q1 + c2qˆ 1 , where q1 and q̂1 are a pair of vectors having the same properties (3.45)-(3.47) as the discrete translational modes. Substitution of q1 into (3.28) yields a set of simplified equations that, if satisfied, ensures (3.28) is satisfied for any c1 and c2 . The perturbation equations from (3.28) for the sun, carrier and ring rigid motion reduce to the six equations (3.51)-(3.56), and the perturbation equations for all the planets reduce to (3.57) and (3.58) 69 µ mr xr0 + β1 , (π krbs + π krus + N2 − ω02 mr ) x1r − N2 (σ r1 sin α r + σˆ r1 cos α= r) (3.51) µ mr yr0 + β 2 , = (π krbs + π krus + N2 − ω02 mr ) y1r + N2 (σ r1 cos α r − σˆ r1 sin α r) (3.52) ( kc + Nk pn − ω02mc ) xc1 + N2 k p ( −ξ11 + ηˆ11 ) = µ mc xc0 , (3.53) ( kc + Nk pn − ω02mc ) yc1 + N2 k p ( −ξˆ11 − η11 ) = µ mc yc0 , (3.54) (ks + N 2 k sp − ω02ms ) x1s + N 2 k sp ( −σ s1 sin α s + σˆ s1 cos α s ) = µ ms xs0 , (3.55) (ks + N 2 k sp − ω02ms ) y1s + N 2 k sp ( −σ s1 cos α s − σˆ s1 sin α s ) = µ ms ys0 , (3.56) k pnp1c + K1r 4p1r + K1s 4p1s + (K pp − ω02 M p )p11= µ M p p10 + β3b p , (3.57) 1 1 1 1 1 2 1 0 垐 k pnp?垐 c + K r 4 p r + K s 4 p s + (K pp − ω0 M p )p1= µ M p p1 + β 4 b p , (3.58) where σ r1 = ξ11 sin α r − η11 cos α r − u11 , σˆˆr1 = ξˆ11 sin α r − ηˆ11 cos α r − u11 , 1 σ s1 = −ξ11 sin α s − η11 cos α s + u11 , σ垐 −ξˆ11 sin α s − ηˆ11 cos α s + u11 , s = β1 = K (3.60) − N 2 ( A1 sin α r + A2 cos α r ) N 2 ( A1 cos α r − A2 sin α r ) γ β = , ∑ m 2 ∑ γ m , (3.61) 4π 4π = = m jN +1 m jN +1 β3 = 1 r4 (3.59) − sin 2 α r = sin α r cos α r sin α r sin α r cos α r cos2 α r cos α r NA1 NA2 γ m , β4 = ∑ 2π =m jN +1 2π − sin 2 α s 0 k sp − sin α s cos α s 0 , K 1s 4 = sin α s 0 ∑ γm , (3.62) − sin α s cos α s − cos2 α s cos α s 0 0 , 0 = m jN +1 σ r1 and σ s1 are the deflections of the first planet in the direction of the lines of action for the ring-planet and sun-planet meshes, respectively. The superscript 1 denotes the first order perturbation. 70 Expressing (3.51)-(3.58) in matrix form after multiplying (3.57) and (3.58) by N / 2 , the second equation of (3.28) reduces to the 12×12 linear system A trnp1trn = btrn , (3.63) T p1trn = x1r , y1r , xc1 , yc1 , x1s , y1s , ξ11 ,η11 , u11 , ξˆ11 ,ηˆ11 , uˆ11 , (3.64) T 0 = btrn µ M trnptrn + βt1 , βt 2 , 0, 0, 0, 0, βt 3bTp , β t 4bTp , M trn = diag(1,1, mc , mc , ms , ms , N2 M p , N2 M p ) . (3.65) (3.66) One can show that the solvability condition of (3.63) is identical to (3.50), so it is satisfied. Thus, (3.28) is satisfied for the given q1 independent of c1 and c2 (which remain indeterminate), and the perturbation q1 has the same form as q 0 . In summary, there are six degenerate pairs of eigenfunctions a in (3.25) with the form ε ∑ Vm1eimθ + c.c. = a = m= jN +1 q 0 + ε q1 1 imθ ε ∑ Vm e jN ±1 m= . 0 1 q + ε q (3.67) m jN − 1 in (3.67) are the complex conjugate terms for Note that terms associated with = = m jN + 1 . The discrete elements of the planetary gear (including the ring rigid body motion) deflect as in the translational modes described in [18]. The elastic ring deflection contains only the jN ± 1 nodal diameter components. The natural frequencies of these modes are degenerate. 71 3.3.1.3. Planet Modes For N ≥ 4 , the unperturbed system has three unperturbed eigenvalues associated with the discrete planet modes, and each of them is degenerate with multiplicity N − 3 . For these modes, the sun, carrier and rigid ring motions are zero. The deflections of the planets are proportional with p 0n = wnl p10 , where the N − 3 sets of coefficients satisfy [18] N ∑ wnl = 0 , n =1 N ∑ wnl cosψ n = 0 , n =1 N ∑w n =1 l n sinψ n == 0 , l 1, , N − 3 . (3.68) When N is odd, the N − 3 solutions of (3.68) are wn2 s −3 = cos sψ n , wn2 s − 2 = sin sψ n , s = 2, , N2−1 . (3.69) When N is even, the N − 3 solutions of (3.68) consist of (3.69) for s = 2, ,int( N2−1 ) and the additional solution wnN −3 = cos N2 ψ n . (3.70) A general discrete planet mode of the unperturbed system is the linear combination N −3 q 0 = ∑ dl ql0 , with l =1 T ql0 = 0, 0, 0, w1l p10 , , wNl p10 . (3.71) With this mode, reduction of (3.30) yields the elastic deformation of the ring as Vm1 = −σ r0 N cos α r − im sin α r N −3 l − imψ n ( d ) , σ r0 = ξ10 sin α r − η10 cos α r − u10 . (3.72) l ∑ wn e 2 2 2 ∑ 2π m (m − 1)=l 1 =n 1 The N − 3 solvability conditions for the second of (3.28) give D p d = µ d , d = [ d1 , d 2 , , d N -3 ]T , 72 (3.73) D p = Dtj ( N −3)×( N −3) N N . (3.74) (σ r0 ) 2 ± JN t − imψ n j imψ n = − w e w e γ ( )( ) ∑ ∑ ∑ m n n 2π m = ±2 1 1 n= n= ( N −3)×( N −3) Although the elements Dtj of D p appear complicated, use of the solutions (3.69) and (3.70) simplifies them. D p is diagonal, yielding closed-form expressions for µ . When N is odd, the first-order eigenvalue perturbations are µ 2 s −3 = µ 2 s − 2 = − N 2 (σ r0 ) 2 4π ∑ = m jN + s γ m , s = 2, , N2−1 , (3.75) When N is even, (3.75) holds = for s 2, , N2 − 1 , and the remaining eigenvalue perturbation is µ N −3 = − N 2 (σ r0 ) 2 4π ∑ = m jN + N2 γm . (3.76) For each of the three unperturbed discrete planet modes with multiplicity N − 3 , the corresponding perturbed eigenfunctions evolve into int( N2−3 ) pairs of degenerate planet modes for arbitrary N and one additional distinct planet mode for even N. For degenerate planet modes with natural frequency perturbation from (3.75), the unperturbed eigenfunction is a linear combination of two instead of N − 3 modes in (3.71). According to this and (3.69), (3.72) reduces to cos α r − im sin α r N Vm1 = −σ r0 ∑ ( d 2 s−3 cos sψ ne−imψ n + d 2 s−2 sin sψ ne−imψ n ) . (3.77) 2π m 2 ( m 2 − 1) 2 n =1 Vm1 is zero when m ≠ jN ± s . This yields a rule governing the nodal diameter components of the ring modal deflections for a mode with given s ∈ {2, ,int( N2−1 )} : the elastic deformation of the ring for degenerate planet modes contains only jN ± s nodal diameter components. The nonzero nodal diameter components are 73 cos α r − im sin α r Vm1 = − Nσ r0 (d 2 s −3 − id 2 s − 2 ) , = m jN + s , 4π m 2 (m 2 − 1) 2 (3.78) cos α r − im sin α r Vm1 = − Nσ r0 (d 2 s −3 + id 2 s − 2 ) , = m jN − s . 4π m 2 (m 2 − 1) 2 (3.79) 2 For distinct planet modes whose natural frequency is ω= ω02 + εµ N (exist only for 2 even N), (3.72) reduces to cos α r − im sin α r N cos( N2 ψ n )e − imψ n 2 2 2 ∑ 2π m (m − 1) n =1 (3.80) cos α r − im sin α r N = − Nσ r0 m = jN ± for all . 2 4π m 2 (m 2 − 1) 2 Vm1 = −σ r0 Terms in the first expression for Vm1 in (3.80) vanish for m ≠ jN ± N2 . Accordingly, the perturbed eigenfunction contains only jN ± N2 nodal diameter components ( s = N 2 ). For the degenerate eigensolution µ2 s −3 = µ2 s − 2 with specified s ∈ {2,3, ,int( N2−1 )} , the eigenfunction perturbation q1 is proposed as the linear combination = q1 d 2 s −3q1 + d 2 s − 2qˆ 1 , (3.81) T T q1 = 0, 0, 0, z11 (p11 )T , , z1N (p11 )T , qˆ 1 = 0, 0, 0, z12 (p11 )T , , z N2 (p11 )T , (3.82) where q1 and q̂1 have the same form as the discrete planet modes in (3.71). Substituting this form of q1 into (3.28), the equations associated with the sun, carrier and ring rigid motion lead to three equations identical to (3.68) except wnl → znl , but here l = 1, 2 . The solutions for z1n , zn2 are z1n = cos sψ n , zn2 = sin sψ n , n = 1, 2, , N . 74 (3.83) The remaining equations of (3.28) (the ones associated with deflections of the planets) yield Nσ r0 (K pp − ω M p )p = µ2 s −3M p + bp ∑ γm . 2π = m jN + s 2 0 1 1 0 p 1 (3.84) One can show that the solvability condition of (3.84) is identical to (3.75) and (3.76). Thus, p11 is solved from (3.84), which, with (3.83), completes the solution for q1 . This ensures q1 has the structure of a discrete planet mode. For the distinct eigenvalue µ N −3 in (3.76), one can similarly show that the eigenfunction perturbation q1 has the form of a discrete planet mode. In summary, for each s ∈ {2,3, ,int( N2−1 )} there are three degenerate pairs of eigenfunctions a s in (3.25) with the form ε ∑ Vm1eimθ + c.c. a s = =m jN + s . 0 1 q + εq (3.85) For even N, an additional three distinct eigenfunctions are present with the form of (3.85) and s = N 2 . The discrete elements of the planetary gear (including the ring rigid body motion) deflect as in the planet modes described in [18]. The elastic ring deflection contains only the jN ± s nodal diameter components. 3.3.2. Elastic Ring Perturbation ERP is the complementary case of DPP. The stiffness of the ring-planet mesh is unity in both cases (from (3.7)). In ERP, the ring bending stiffness is O (1) while in DPP it is 75 O (1/ ε ) ; the stiffnesses of all the remaining meshes/bearings are O (1/ ε ) while in DPP they are O (1) . The perturbation parameter is defined by ε = 1/ k sp . A perturbation process similar to (3.25)-(3.28) yields the perturbation equations for a 0 −ω02 (1 + ∂ 2 / ∂θ 2 )v 0 /(2π ) + kbend L1v 0 + L2v 0 + L3q 0 = 0 , 0 n 0 0, K rbp0r = 0 , K cbp c + ∑ K c 2p n = n (3.86) (3.87) ( K sb + ∑ K ns1 )p0s + ∑ K ns 2p0n = 0 , (K n )T p 0 + (K n )T p 0 + K p 0 = 0. (3.88) c2 c s2 s pp n n n Equations (3.87)-(3.88) form a problem as Aq 0 = 0 . One can prove that A is positive definite so q 0 = 0 . Accordingly, the last item in (3.86) vanishes, so the unperturbed system is an elastic ring having equally spaced spring supports with elimination of the three rigid body motions as indicated in (3.5). The unperturbed eigenfunction is T v 0 a0 = . 0 (3.89) Equations (3.29) and (3.89) are the unperturbed eigenfunctions from DPP and ERP, respectively. Together they form a non-overlapping, complete (in the mathematical sense) T basis for the linear space of extended variables a = v, qT . This suggests the set of perturbed eigenfunctions from DPP and ERP forms a complete set of vibration modes for planetary gears having elastic ring gears. This conclusion is made rigorous subsequently. The perturbation equations for a1 are ∂v 0 −∑ b r ( sin α r + v 0 cos α r ) K rbp1r = , ∂ θ n θ =ψ n 76 (3.90) 0 , (3.91) K cbp1c + ∑ K nc 2p1n = 0 , ( K sb + ∑ K ns1 )p1s + ∑ K ns 2p1n = n n n ∂v 0 −b p ( (K cn2 )T p1c + (K ns 2 )T p1s + K pp p1n = sin α r + v 0 cos α r ) , (3.92) ∂θ θ =ψ n − ω02 µ 0 ∂ 2v0 ∂2 (1 + 2 )v1 + kbend L1v1 + L2 v1= (v + ) − L3q1 . 2 ∂θ ∂θ 2π 2π (3.93) We draw on the modal properties of a ring on a general elastic foundation as determined analytically in [2], where the modal expressions for rings having equally spaced springs are given. In the unperturbed problem (3.86), each spring is oriented with an angle of π / 2 − α r to the radial direction. With elimination of the ring rigid body motions, the ring deflection is represented as (3.23). Thus, 2 JN − 2 unperturbed modes exist. For a free ring with no supports, all the natural frequencies are degenerate with multiplicity two. When the ring has equally spaced springs, some natural frequencies split and the others remain degenerate. The unperturbed modes of the ERP are classified into four types based on the nodal diameter components they contain: type 0, type 1, type s and single nodal diameter component modes [2]. For brevity, only type 0 modes are considered. They are linear combinations of the jN J nodal diameter components, v 0 = ∑ V jNd cos jNθ . Such a mode exists for each of the J j =1 values of d = N , 2 N , , JN , where d indicates the dominant nodal diameter component. Substitution of this expression for v 0 into (3.90) yields j =1,, J T d K p = 0, 0, − N ∑ (m sin α r + cos α r )Vm . m = jN 1 rb r 77 (3.94) Because K rb is diagonal, the first two elements of p1r corresponding to ring rigid translations are zero, which is the same as for a discrete rotational mode. Similar analysis of (3.91) and (3.92) for the sun, carrier and planets reveals that q1 has the form (3.31) of a discrete rotational mode. The eigenvalue perturbation µ is obtained from the solvability condition of (3.93). Following lengthy algebra, the solution v1 of (3.93) has the same form as v 0 . These results show that the perturbed eigenfunction has the properties of a rotational mode as defined earlier. Similar processes show that when the unperturbed mode v 0 is of type 1 from [2], the perturbed mode of the elastic-discrete model is a translational mode. When the unperturbed mode v 0 is of type s from [2], the perturbed mode is a planet mode. When the unperturbed mode is a single nodal diameter component mode, the perturbed mode is a purely ring mode. Thus, every unperturbed ERP mode evolves into one of the four modal categories of the elastic-discrete model. The same is true for DPP. The numbers of modes obtained from each of DPP and ERP are 3N+9 and 2 JN − 2 , respectively. The total number of eigenfunctions obtained from the perturbation analyses is 3 N + 2 JN + 7 , which equals the number of degrees of freedom for arbitrary J in (3.23). The modal property classification from perturbation analysis exactly matches the properties of the numerical results in Figure 3.2, Table 3.1, and Table 3.2. Evidently, all modes have been included and categorized from the two perturbations. 78 3.4. Candidate Mode Method The foregoing perturbation analysis derives the modal properties by combining two perturbation problems, each having a different perturbation parameter and unperturbed problem. The method appeals to physical reasoning where the elastic-discrete system modes are seen to evolve from known simpler systems. A plausible argument given above heuristically concludes that this approach captures all modes of the general system. Nevertheless, perturbation is inherently linked to small values of the perturbation parameter, and the use of two separate perturbation problems to conclude that all modes are accounted for is not rigorous mathematically. Guided by the foregoing perturbation results, this section derives the general elastic-discrete system modal properties in a rigorous way that is free from any reliance on a small parameter. This alternate derivation assumes eigensolutions having the properties of the four mode types from perturbation and then confirms such eigensolutions satisfy the eignevalue problem. An accounting at the end ensures this approach captures all possible vibration modes. A candidate rotational mode has the ring deflection J vrot = ∑V jN cos jNθ , (3.95) j =1 and discrete element deflection q rot having the form (3.31). Substituting (3.95) into (3.18), multiplying by cos lθ , and integrating from 0 to 2π yields − J c 1+ l2 2 0 , l = N , 2 N , , JN , (3.96) ω Vl + l Vl + N cos 2 α r ∑ V jN + Nσ r cos α r = 2 2 j =1 = cl 2π kbend l 2 (l 2 − 1) 2 + 2π k rus + 2π l 2k rbs , σ r = ξ1 sin α r − η1 cos α r − u1 . 79 (3.97) Use of the assumed modal properties to reduce (3.19) yields only one equation for the ring rigid motion J . (2π krus / cos 2 α r + N − ω 2 / cos 2 α r )ur + Nσ r + cos α r ∑V jN = 0 (3.98) j =1 The remaining equations in (3.19) vanish. Similarly, (3.20) and (3.21) reduce to (kcu + Nk p − ω 2 I c )uc − Nk pη1 = 0, (3.99) (k su + Nk sp − ω 2 I s )us + Nksp (−ξ1 sin α s − η1 cos α s + u1 ) = 0 . (3.100) With the assumed modal form and algebraic manipulation, (3.22) becomes J 0 . (3.101) (K ) p c + (K ) p r + (K ) p s + (K pp − ω M p )p1 + b p co α r ∑ s V jN = 1 T c2 1 T r2 1 T s2 2 j =1 Equations (3.96)-(3.101) form a reduced eigenvalue problem of order J + 6 with the eigenvector (VN , , VJN ,ur , us , uc , ξ1 ,η1 , u1 )T . In general, the eigenvalues are all distinct (except for specially chosen parameters). From the eigenvectors of the reduced eigenvalue problem, J + 6 rotational modes of the full system are constructed from (3.95) and q rot . A pair of candidate translational modes is T T , T , â imθ imθ T = ., q trn a ∑ Vm e + c.c= ∑ iVm e + c.c., qˆ trn =m jN +1 =m jN +1 (3.102) where q trn , qˆ trn are a pair of discrete translational modes having the same form as described in (3.45)-(3.47). (Recall the note below (3.50) regarding allowable values of m.) Guided by the perturbation solution in = Vm (cos α r − im sin α r )U m , where U m is complex. 80 (3.49), Vm is expressed as Substituting a and â into (3.18), multiplying by e − ilθ , and integrating from 0 l jN + 1 , there are 2 J − 1 equations to 2π yields the equations governing U m . When= −(1 + l 2 )ω 2U l + clU l + N2 ( A1 − iA2 ) + N ∑ = m jN +1 , (3.103) (cos 2 α r + m 2 sin 2 α r )U m = 0 l jN − 1 , where A1 and A2 have the form in (3.49), and cl is defined in (3.97). When= the following 2 J − 1 equations result −(1 + l 2 )ω 2U l + clU l + N2 ( A1 + iA2 ) + N ∑ = m jN −1 . (cos 2 α r + m 2 sin 2 α r )U m = 0 (3.104) For other values of l the resulting equations from (3.18) vanish. For each l in (3.103), there is a corresponding −l in (3.104) whose equation is the complex conjugate of (3.103). Thus, equations (3.104) and (3.103) are equivalent. One obtains 4 J − 2 real equations because U l in (3.103) is complex. Substitution of a and â into (3.19)-(3.22) generates an additional 12 real equations similar to (3.51)-(3.58) with the elimination of superscripts 0 or 1, substitution of µ = 0 , and replacement of β1 , β 2 , β3 , β 4 by β5 , β 6 , β 7 , β8 , respectively, β5 Ne − iα r = β 2 6 β7 = β 8 = m = m , i (cos 2 α r + m 2 sin 2 α r )U m ⋅ + c.c. jN +1 −1 ∑ . 1 (cos 2 α r + m 2 sin 2 α r )U m ⋅ + c.c. jN +1 i ∑ (3.105) (3.106) The resulting 4 J + 10 real equations form a reduced order eigenvalue problem. Because a and â are interchangeable, all eigensolutions of the reduced order problem must occur as degenerate eigenvalues with multiplicity two. With these eigensolutions, 2 J + 5 pairs of degenerate translational modes are constructed from (3.102). 81 A pair of candidate planet modes for a selected s ∈ {2,3, ,int( N2−1 )} is T , = a s1 ∑ Vm eimθ + c.c., qTplt , s =m jN + s (3.107) T , = a s 2 ∑ iVm eimθ + c.c., qˆ Tplt , s =m jN + s (3.108) qTplt , s = 0, 0, 0, cos sψ 1p1T , , cos sψ N p1T , qˆ Tplt , s = 0, 0, 0,sin sψ 1p1T , ,sin sψ N p1T , (3.109) where q plt ,s , qˆ plt ,s are a pair of discrete planet modes having the same form as described in (3.71). The linear combination d 2 s −3a s1 + d 2 s − 2a s 2 gives the elastic deformation of the ring in the form = v ∑ = m jN + s Vm (d 2 s −3 + id 2 s − 2 )eimθ + c.c. . (3.110) Comparing (3.110) to the perturbation solution (3.78) and (3.79) suggests the Vm in Vm (cos α r − im sin α r )U m with real U m and = (3.107) and (3.108) can be written as U − m = U m . This is adopted in the candidate modes (3.107) and (3.108). Substituting a s1 into (3.18), multiplying by e − ilθ , and integrating from 0 to 2π yields l jN + s , there are 2J equations the equations for U m . When= −(1 + l 2 )ω 2U l + clU l + N2 σ r + N ∑ = m jN + s (cos 2 α r + m 2 sin 2 α r )U m = 0 , (3.111) l jN − s , 2J equations result that are equivalent to where σ r is defined in (3.72). When= (3.111). For other values of l the resulting equations vanish. With the properties of a s1 , 82 equations (3.19)-(3.21) are satisfied. Substitution of a s1 into (3.22) yields the same equation for each n (K pp − ω 2 M p )p1 = −2 ∑ (cos 2 α r + m 2 sin 2 α r )U m . (3.112) = m jN + s The resulting 2 J + 3 equations in (3.111) and (3.112) form a reduced order eigenvalue problem with 2 J + 3 eigensolutions. Substitution of a s 2 into (3.11) yields the same 2 J + 3 order eigenvalue problem. Therefore, each of the 2 J + 3 eigensolutions corresponds to a pair of planet modes. Thus, for each s, 2 J + 3 pairs of degenerate modes are constructed from (3.107)-(3.109). When N is odd there are N −3 2 different values of s ∈ {2,3, , N2−1} , so ( N − 3)(2 J + 3) / 2 degenerate pairs of planet modes are constructed from (3.107)-(3.109). When N is even, there are N / 2 − 2 different values of s ∈ {2,3, , N2 − 1} , so ( N / 2 − 2)(2 J + 3) degenerate pairs of planet modes are similarly constructed. When N is even, besides the degenerate planet modes, there are additional distinct planet modes. They have the same form as the degenerate planet mode in (3.107) with s = N / 2 . With some algebraic manipulation of (3.107), the distinct planet modes have the form T a = ∑ Vm cos mθ , qTplt , N , qTplt , N = 0, 0, 0, p1T , −p1T , , p1T , −p1T , (3.113) 2 2 =m jN + N2 =j 0,, J −1 where Vm is real. A similar reduction as above yields a J + 3 order eigenvalue problem for the Vm and p1 with eigenvalue ω 2 from equations (3.18) and (3.22). This gives J + 3 planet modes with distinct eigenvalues from (3.113). Totally, for even N, there are 83 (2 JN − 7 J ) + (3N − 9) planet modes constructed from (3.107)-(3.109) and (3.113). Table 3.2 summarizes the different numbers and types of planet modes. The final mode type is that of purely ring modes having the (not normalized) form = a [(cos α r sin mθ − m sin α r cos mθ )Vm , 0] T jN , j = 1, J for odd or even N . ,m= (3.114) jN + N / 2, j =0, J − 1 for even N . In such modes, only the ring gear deforms, and the ring has nodes at all ring-planet mesh locations. All purely ring mode natural frequencies are distinct. Note that a purely ring mode with m = jN in (3.114) has the same structure as a rotational mode except many elements of the rotational mode are zero. Similarly, a purely ring mode with = m jN + N / 2 in (3.114) has the same structure as a distinct planet mode. Rotational modes and purely ring modes with m = jN emerge as split modes of the degenerate eigensolution pairs of a free ring; distinct planet modes and purely ring modes with = m jN + N / 2 similarly emerge as split modes for even N [2]. Substitution of (3.114) into (3.18), multiplication by e − ilθ , and integration from 0 to 2π yields the following J order diagonal eigenvalue problem for odd N (2J order for even N ) with eigenvalue ω 2 jN , j = 1, J for odd or even N , −(1 + l 2 )ω 2 + cl Vl = 0, l = jN + N / 2, j =0, J − 1 for even N , (3.115) where cl is from (3.97). The remaining equations (3.19)-(3.22) vanish for the a in (3.114). According to (3.115), the closed-form natural frequencies expressions are = ω 2 cl /(1 + l 2 ) , where cl depends on the ring bending stiffness ( kbend ) and the distributed stiffnesses around the ring circumference ( krus and krbs ). Thus, the natural 84 frequencies of purely ring modes are independent of mesh stiffnesses ( ksp and krp ). This can also be explained through the gear mesh deflections. The general expressions of sunplanet and ring planet mesh deflections ( δ sn and δ rn ) are = δ sn ys cosψ sn − xs sinψ sn − ξn sin α s − ηn cos α s + us + un = δ rn (v cos α r + ∂v − xr sinψ rn + yr cosψ rn + ur + ξ n sin α r − η n cos α r − un sin α r ) ∂θ θ =ψ n (3.116) (3.117) Substitution of (3.114) into (3.116) and (3.117) ensures both the sun-planet and ringplanet mesh deformations are zero. Overall, four types of modes are identified. For odd N, the numbers of modes for rotational, translational, planet and purely ring modes are J + 6, 4 J + 10 , (2 JN − 6 J ) + (3N − 9) , and J, respectively. For even N, they are J + 6, 4 J + 10 , (2 JN − 7 J ) + (3N − 9) , and 2J, respectively. While, the numbers of planet and purely ring modes are different for odd and even numbers of planets, the total number of modes is (2 J + 3) N + 7 for either odd or even N. This total equals the total degrees of freedom with v (θ , t ) from (3.23) and J arbitrarily large. Thus, all modes have been categorized. Furthermore, the numbers of rotational and translational modes are independent of the number of planets N. Changing the number of planets N, while retaining the same J in the ring deformation expansion (3.23), only changes the numbers of planet modes and purely ring modes. Table 3.2 lists how the number of planets N affects the number of degenerate and distinct planet modes, and it specifies the planet mode type breakdown for each N. If N increases by one, the total number degrees of freedom increases by 2 J + 3 as the total degrees of freedom is (2 J + 3) N + 7 . If N increases by one from odd to even, 85 J + 3 of the additional modes are distinct planet modes, and the remaining J additional modes are purely ring modes (the total number of purely ring modes becomes 2J ). If N increases by one from even to odd, all 2 J + 3 additional modes are planet modes; furthermore, J purely ring modes change into planet modes. Therefore, the number of planet modes increases by 3 J + 3 , and the number of purely ring modes decreases by J. The natural frequency multiplicities and all modal properties from the candidate mode method match the numerical solution in Table 3.1, Table 3.2 and Figure 3.2 (as well as the perturbation results) for arbitrary N and J. 86 Chapter 4: Natural Modes of Planetary Gears with Unequally Spaced Planets and an Elastic Ring Gear Equation Chapter 4 Section 1 The highly structured modal properties of planetary gears having diametrically opposed planets and an elastic ring gear are illustrated and mathematically proved in this work. Two types of modes are found: rotational and translational modes. The properties of each mode type are given mathematically. A rule for how the modes of planetary gears having equally spaced planets evolve as the planets deviate to diametrically opposed is presented and discussed. 4.1. Introduction While planetary gearboxes usually have equally spaced planets, some planetary gears have unequally spaced planets due to the limitations of assembly conditions or special applications. If the first planet is located at angle zero, the possible locations of the = remaining planets are at ψ 2π j / ( zs + zr ) , where j is an arbitrary integer and zs , zr are the tooth numbers of the sun and ring, respectively. In the case that equal spacing is not achievable, such as when ( zs + zr ) / N is not an integer where N is the number of planets, 87 the planets are diametrically opposed in almost all applications. This is because diametrically opposed planets have the benefits of reducing the bearing force, maintaining static and dynamic balance, and improving the load sharing. Several helicopter planetary gears have diametrically opposed planets, as do some automotive transmission planetary gears. Ring gears have the largest radii in planetary gears. Often for purposes of weight reduction and increased power density, ring gears are designed to be thin in the radial direction. Some planetary gears are designed with thin rings to introduce compliance that improves load sharing among the planets. Thin rings with large radius deform elastically in applications like helicopters, wind turbines, and cars. This is evident by measured data on an OH-58 helicopter planetary gear [68], cracks observed in ring gears, and finite element analyses that shows elastic deformation of the ring under operating loads. In the literature, many studies have been found in the vibration of elastic rings [2, 6, 7, 9, 12] and the free vibration of planetary gears [13-15, 18-20, 69] with lumped-parameter models and equally spaced planets. Lin and Parker [18, 21] analytically studied the modal properties of planetary gears using a lumped-parameter model with consideration of both translational and rotational motion. Wu and Parker [3] analytically investigated the vibration of planetary gears with equally spaced planets based on an elastic-discrete model, in which the ring gear was modeled as an elastic body and the remaining components were modeled as rigid bodies. The well-defined modal properties were characterized for all possible modes. Guo and Parker [70] studied the modal properties of planetary gears using a model with only rotational degrees of freedom. Parker and Ambarisha [43, 71] showed how these modal properties are important for dynamic 88 response suppression. Wu and Parker [49] used the modal properties in [3] to examine parametric instabilities for planetary gears having an elastic ring under periodic mesh stiffness variations. Kiracofe and Parker [27] extended that research to compound planetary gears with equally spaced or diametrically opposed planets. Recently, Bartelmus and Zimroz examined planetary gear condition monitoring [72], which can take advantage of the modal properties derived in this work. The natural frequencies and vibration modes are fundamental when dealing with an existing vibration problem or designing new systems to avoid resonant vibration, as gear engineers routinely need to do. This paper provides detailed and rigorously derived properties of the natural frequencies and modes for general planetary gears with unequally spaced planets. The work provides knowledge engineers can use in practice as well as modal properties critical to further research on resonant vibration response, nonlinearity, diagnostics, and the like. 4.2. Diametrically Opposed Planet Pair Modal Properties Details of the model in Figure 3.1 including nomenclature, dimensionless variables, extended operators, and equations of motion are given in chapter 3 and adopted here. Equations associated with the individual components in the sequence of the elastic ring, ring rigid motion, carrier, sun, and planets are given in (3.18)-(3.22). The natural frequencies of planetary gears with equally spaced planets and an elastic ring gear are either distinct or degenerate with multiplicity two. When the planets are diametrically opposed, it destroys the cyclic symmetry of the equal spacing. The 89 asymmetry of diametrically opposed planets causes all the repeated natural frequencies to split into distinct ones. Certain modal properties remain, however, and all modes are classified into two types: rotational and translational modes. Fourier expansion of the elastic deformation of the ring gives = v(θ ) JN ∑V e m=2 m imθ + cc , (4.1) where J ≥ 1 is an arbitrarily large integer, and m= 0, ±1 are contained in the rigid body motion of the ring p r . cc denotes the complex conjugate of all proceeding terms, and V− m = Vm . 4.2.1. Rotational Modes Rotational modes for diametrically opposed planets contain only even numbered nodal diameter components of the elastic ring, and the translations (but not rotations) of the ring rigid motion, sun, and carrier are zero. A candidate rotational mode has the form T JN = a ∑ Vm eimθ + cc, p r , p c , p s , p1 , , p N /2 , p1 , , p N /2 , m = 2,4, = p h [0, = 0, uh ]T , h r , c, s , p z + N /2 = p z , z = 1, , N / 2 . (4.2) (4.3) Compared to rotational modes of equally spaced planets, deflections of the planets are no longer identical for all the planets. Instead, they are identical for the two planets of every diametrically opposed pair. Furthermore, the elastic deformation of the ring contains all even numbered nodal diameter components, where the equally spaced case contains only the JN nodal diameter components. 90 Substituting (4.2) and (4.3) into (3.18), multiplying by e − ijθ , and integrating from 0 to 2π yields − N /2 ± JN c 1+ j2 2 ω V j + j V j + 2 ∑ ∑ (cos α r + im sin α r )(cos α r − ij sin α r )Vm ei ( m − j )ψ n 2 2 1 ±2, ±4, n = m= N /2 0 +2(cos α r − ij sin α r )∑ (ξ n sin α r − η n cos α r − un )e − ijψ n = n =1 j =±2, ±4, , ± JN , (4.4) c j 2π kbend j 2 ( j 2 − 1) 2 + 2π krus + 2π j 2 krbs , = (4.5) where krus and krbs are the tangential and radial distributed stiffnesses per unit length around the circumference of the ring. Use of the specified modal properties (4.2) and (4.3) to reduce (3.19) yields only one equation for the ring rigid motion N /2 (2π krus / cos 2 α r + N − ω 2 / cos 2 α r )ur + 2∑ (ξ n sin α r − η n cos α r − un ) n =1 +2 JN N /2 ∑ ∑ (cos α = = n 1 m 2,4, r + im sin α r )Vm eimψ n + cc = 0. (4.6) The remaining equations in (3.19) vanish. Similarly, (3.20) and (3.21) reduce to the following equations, respectively, N (kcu + Nk p − ω 2 I c )uc − k p ∑ηn = 0, (4.7) n =1 N (k su + Nksp − ω 2 I s )us + ksp ∑ (−ξ n sin α s − ηn cos α s + un ) = 0. (4.8) n =1 With the modal expressions (4.2)-(4.3) and straight forward manipulation, (3.22) becomes (K1c 2 )T p c + (K1r 2 )T p r + (K1s 2 )T p s + (K pp − ω 2 M p )p n +b p JN ∑ m = 2,4, (cos α r + im sin α r )Vm eimψ n + cc = 0, 91 n = 1, , N / 2 . (4.9) Equations (4.4)-(4.9) form a reduced eigenvalue problem of order JN + 3 N / 2 + 3 as: JN equations from (4.4) for the ring elastic deformation, three equations from (4.6)-(4.8) for the ring, carrier, and sun rotations, and 3 N / 2 equations from (3.101) for the planet motions. Thus, one can construct JN + 3 N / 2 + 3 homogeneous equations with undetermined eigenvalue ω 2 . This algebraic eigenvalue problem yields JN + 3 N / 2 + 3 rotational modes when each eigenvector is substituted into (4.2) and (4.3). Compared to the rotational modes of planetary gears with equally spaced planets [3], the number of rotational modes increases from 6 + J to JN + 3 N / 2 + 3 . Where these additional rotational modes come from is discussed subsequently. 4.2.2. Translational Modes Translational modes for diametrically opposed planet pairs contain only odd numbered nodal diameter components of the elastic ring, and the rotations (but not translations) of the ring rigid motion, sun, and carrier are zero. A candidate translational mode has the form T JN −1 = a ∑ Vm eimθ + cc, p r , p c , p s , p1 , , p N /2 , −p1 , , −p N /2 , m =3,5, −p z , z = 1, , N / 2 . = ph [ = xh , yh , 0]Τ , h r , c, s , p z + N /2 = (4.10) (4.11) For translational modes with equally spaced planets, the nth planet’s motion is calculable from the arbitrarily chosen first planet’s motion. For diametrically opposed planets, only the deflections of a diametrically opposed pair of planets are related as p z + N /2 = −p z . Furthermore, the elastic deformation of the ring contains all odd numbered nodal 92 diameter components instead of only the JN ± 1 nodal diameter components for equally spaced planets. Substituting (4.10) and (4.11) into (3.18), multiplying by e − ijθ , and integrating from 0 to 2π yields the equations governing V j as − ± ( JN −1) N /2 c 1+ j2 2 ω V j + j V j + 2 ∑ ∑ (cos α r + im sin α r )(cos α r − ij sin α r )Vm ei ( m − j )ψ n 2 2 ±3, ±5, n = 1 m= N /2 0, +2(cos α r − ij sin α r )∑ (ξ n sin α r − η n cos α r − un )e − ijψ n = n =1 j =±3, ±5, , ±( JN − 1) . (4.12) Substitution of (4.10) and (4.11) into (3.19)-(3.21) generates the following six equations N /2 N /2 (4.13) (krx + 2krp ∑ sin 2 ψ rn − ω 2 mr ) xr + 2∑ sinψ rnδ n = 0, = n 1= n 1 N /2 N /2 (4.14) (krx + 2krp ∑ cos 2 ψ rn − ω 2 mr ) yr − 2∑ cosψ rnδ n = 0, = n 1= n 1 N /2 (kcx + Nk pn − ω 2 mc ) xc + 2k pn ∑ (−ζ n cosψ n + η n sinψ n ) = 0, (4.15) n =1 N /2 (kcx + Nk pn − ω 2 mc ) yc − 2k pn ∑ (ζ n sinψ n + η n cosψ n ) = 0, (4.16) n =1 N /2 N /2 (k sx + 2ksp ∑ sin 2 ψ sn − ω 2 ms ) xs + 2ksn ∑ sinψ sn (ζ n sin α s + ηn cos α s − un ) = 0 , (4.17) = n 1= n 1 N /2 N /2 (k sx + 2ksp ∑ cos 2 ψ sn − ω 2 ms ) ys − 2k sn ∑ cosψ sn (ζ n sin α s + η n cos α s − un ) = 0 , (4.18) = n 1= n 1 where −ζ n sin α r + η n cos α r + un − δn = JN −1 ∑ m =3,5, 93 (cos α r + im sin α r )Vm eimψ n + cc . (4.19) The specified modal expressions (4.10)-(4.11) and straight forward manipulation reduce (3.22) to (K1c 2 )T p c + (K1r 2 )T p r + (K1s 2 )T p s + (K pp − ω 2 M p )p n +b p JN −1 ∑ m =3,5, (cos α r + im sin α r )Vm eimψ n + cc = 0 , n = 1, , N / 2 . (4.20) Eqs. (3.103)-(4.20) form a reduced eigenvalue problem of order JN + 3 N / 2 + 4 as: JN − 2 equations from (3.103) for the ring elastic deformation, six equations (4.13)-(4.18) for the ring, carrier, and sun translations, and 3 N / 2 equations (4.20) for the planet motions. For each eigensolution of this reduced eigenvalue problem, the full system mode is constructed from (4.10) and (4.11). Generally, all the eigenvalues are distinct. Compared to the translational modes of planetary gears with equally spaced planets [3], the number of translational modes increases from 4 J + 10 to JN + 3 N / 2 + 4 . These additional translational modes are discussed below. 4.3. Relationships between Modes for Equally Spaced and Diametrically Opposed Planets Four types of modes exist for planetary gears with equally spaced planets and an elastic ring: rotational, translational, planet, and purely ring modes [3]. Rotational and purely ring modes have distinct eigenvalues. Translational modes have degenerate eigenvalues with multiplicity two. If the number of planets is odd, all the planet modes are degenerate with multiplicity two, otherwise, the planet modes are either degenerate with multiplicity two or distinct. Planet modes exist only when the number of planets N ≥ 4 , and distinct planet modes always contain the jN + N / 2 ( j = 0,1, ) nodal 94 diameter components. For rotational modes, the translations of the ring rigid motion, sun, and carrier are zero. For translational modes, the rotations for the ring rigid motion, sun, and carrier are zero. The deflections of the ring rigid motion, sun, and carrier are zero for all planet modes. Rotational, translational, planet, and purely ring modes contain the jN , jN ± 1 , jN ± s , and jN or jN + N / 2 nodal diameter components of the ring, respectively, where s is selected from 2,3, ,int( N / 2) and j is an integer. An interesting question is: When the planet spacing changes from equally spaced (with an even number of planets) to diametrically opposed, how do the planet and purely ring modes, which exist only for equally spaced planets, evolve into rotational or translational modes? The rule is: If a mode for equally spaced planets has odd nodal diameter ring components (and so p z + N /2 = −p z , z = 1, , N / 2 ), it evolves into a translational mode when the planets are diametrically opposed; if a mode has even nodal diameter ring components (and so p z + N /2 = p z ), it evolves into a rotational mode. To apply this rule, note that for equally spaced planets with even N, the nodal diameter components of any mode are all even or all odd; for diametrically opposed planets, translational modes have all odd nodal diameter components while rotational modes have all even nodal diameter components. Some clues guide the justification of the above rule. Every mode for equally spaced planets must evolve into either a rotational mode or a translational mode as the planets deviate to the diametrically opposed case. All diametrically opposed modes satisfy either p z + N /2 = −p z (translational mode) or p z + N /2 = p z (rotational mode). Because equal spacing is a special case of diametrically opposed planets, all equally spaced modes also 95 satisfy one of these conditions. Because of the continuity of the modes for changes in planet spacing, equally spaced modes where p z + N /2 = −p z holds retain this property when the planet spacing changes to diametrically opposed (rather than discontinuously jumping to the alternate diametrically opposed possibility that p z + N /2 = p z ). Similar arguments apply to equally spaced modes where p z + N /2 = p z holds. Because purely ring modes where p z = 0 satisfy both p z + N /2 = −p z and p z + N /2 = p z , one cannot use the simpler criteria p z + N /2 = −p z or p z + N /2 = p z as the conditions for the mode evolution rule given above (as discussed subsequently). A pair of degenerate planet modes for equally spaced planets [3] has the form T = a s1 ∑ Vm eimθ + cc, qTp , s , =m jN + s (4.21) T = a s 2 ∑ iVm eimθ + cc, qˆ Tp , s , =m jN + s (4.22) qTp , s = 0, 0, 0, cos sψ 1p1T , , cos sψ N p1T , qˆ Tp , s = 0, 0, 0,sin sψ 1p1T , ,sin sψ N p1T , (4.23) where s is an integer selected from 2,3, ,int[( N − 1) / 2] and j is any integer satisfying jN ± s ∈{− JN , − JN + 1, , JN } . Notice that cos sψ z π) cos sψ z= cos s (ψ z += + N /2 − cos sψ z for even s for odd s (4.24) sin sψ z sin sψ z += sin s (ψ z += π) N /2 − sin sψ z for even s for odd s (4.25) Thus, according to (3.109)-(4.25), p z + N /2 = −p z holds for odd s, and p z + N /2 = p z holds for even s. For distinct planet modes, one can substitute s = N / 2 into (3.107) and the first of 96 (3.109) to find p z + N /2 = −p z holds for odd N / 2 , and p z + N /2 = p z holds for even N / 2 . This indicates planet modes, whether distinct or degenerate, having odd (even) nodal diameter components will evolve into translational (rotational) modes as the planets change from equally spaced to diametrically opposed. As an example, Figure 4.1a shows a planet mode having jN ± 2 (even numbered) nodal diameter components with degenerate natural frequency ω4,5 = 0.5539 for equally spaced planets. The system parameters are listed in Table 4.1. When the planets are diametrically opposed the natural frequency pair splits into two rotational modes. One of the split rotational modes is shown in Figure 4.1b with natural frequency ω5 = 0.5561 . Inertias (kg) I r / rr2 = 8.891 , I c / rc2 = 6.000 , I s / rs2 = 2.500 , I p / rp2 = 2.000 Masses (kg) mr = 7.350 , mc = 5.430 , ms = 0.400 , m p = 1.000 k= k= 100 × 106 rp sp , k= k= 0 rbs rus , kbend = 4 × 106 , k= 10 × 106 s , Stiffnesses (N/m) k= 500 × 109 , = k su= 50 × 106 , k= k p 200 × 106 c cu Pressure angle (deg) α= α= 24.60 r s Table 4.1. Dimensional parameters of a planetary gear with six equally spaced planets. 97 (a) (b) Figure 4.1. Mode comparison of a planetary gear with: (a) a planet mode for equally spaced planets ( ω4,5 = 0.5539 ), and (b) the corresponding rotational mode for diametrically opposed planets ( ω5 = 0.5561 ). Parameters are given in Table 4.1. For the diametrically opposed case, the positions of the planets are ψ 1 = 0 , ψ 2 = 2π / 5 , ψ 3 = 2π / 3 , ψ 4 = π , ψ= ψ 2 + π , ψ= ψ3 +π . 6 5 A purely ring mode for equally spaced planets has the form = a [(cos α r sin mθ − m sin α r cos mθ )Vm , 0] T jN , for odd or even N , m= (4.26) jN + N / 2, for even N The deflections of the planets (and all other rigid components) are zero for a purely ring mode. Thus, using the deflections of the planets as the condition to determine to which 98 type of mode it will evolve into does not work. A purely ring mode has one and only one nodal diameter component. For continuity of the modes, the mode that a purely ring mode evolves into should contain at least that specific nodal diameter component. Thus, if the purely ring mode has an odd nodal diameter component, the corresponding diametrically opposed mode will contain that (and other) odd nodal diameter components; this means the purely ring mode evolves into a translational mode. Similarly, if the purely ring mode has an even nodal diameter component, it evolves into a rotational mode. The nodal diameter component in a purely ring mode is jN or jN + N / 2 . Whether jN or jN + N / 2 is odd or even for given j governs the mode type to which it will evolve. Figure 4.2a shows a three nodal diameter purely ring mode with ω6 = 1.518 for equally spaced planets. The system parameters are list in Table 4.1. The corresponding mode for diametrically opposed planets is a translational mode with natural frequency ω 15 = 1.519 (Table 4.3b). The deflections of the planets and sun are significant, and the dominant (but not only) elastic ring deformation is the three nodal diameter component. 99 (a) (b) Figure 4.2. Mode comparison of a planetary gear with: (a) a purely ring mode for equally spaced planets ( ω15 = 1.518 ), and (b) the corresponding translational mode for diametrically opposed planets ( ω15 = 1.519 ). Parameters are given in Table 4.1. For the diametrically opposed case, the positions of the planets are ψ 1 = 0 , ψ 2 = 2π / 5 , ψ 3 = 2π / 3 , ψ 4 = π , ψ= ψ 2 + π , ψ= ψ3 +π . 5 6 For rotational modes of equally spaced planets, the deflections of all the planets are identical [3]. This guarantees p z + N /2 = p z for all z, so they are rotational modes when the planets deviate to the diametrically opposed case. Considering the ring nodal diameter components, a rotational mode of equally spaced planets contains only the jN nodal 100 diameter components. Because N is even, all the numbers of nodal diameter components are even. Thus, one can identify the corresponding modes for diametrically opposed planets are rotational modes using the rule based on even/odd nodal diameter components, and this agrees with the conclusion immediately above from p z + N /2 = p z . For translational modes of equally spaced planets, the planet deflections for a pair of = n N / 2 + z into (3.47) yields translational modes satisfy (3.47) Substitution of p z + N /2 = −p z , pöz + N / 2 = −pöz . (4.27) According to (4.27), translational modes of equally spaced planets remain translational modes when the planets become diametrically opposed. Because translational modes of equally spaced planets contain the jN ± 1 nodal diameter components with even N, all the ring nodal diameter components are odd. Thus, one can identify it as a translational mode for diametrically opposed planets based on the even/odd condition as well. In summary, for any mode of equally spaced planets, whether the elastic ring nodal diameter components are even or odd determines the mode type to which it evolves as the planets deviate to diametrically opposed. For equally spaced or diametrically opposed planets, the total number of degrees of freedom (i.e., modes) is the same: (2 J + 3) N + 7 , where J is the user-selected upper limit in (4.1). For equally spaced planets with even N, the numbers of modes for rotational, translational, planet and purely ring modes are J + 6, 4 J + 10 , (2 JN − 7 J ) + (3 N − 9) , and 2J, respectively. For diametrically opposed planets (obviously with even N), all the modes fall into two types: JN + 3 N / 2 + 3 rotational modes plus JN + 3 N / 2 + 4 101 translational modes. The total number of rotational and translational modes equals the total degrees of freedom, so no other mode types are possible. Table 4.2 summarizes the modal properties of four cases of planetary gears with either equally spaced or diametrically opposed planets based on the elastic-discrete [3] and lumped-parameter models [18, 21]. For the lumped-parameter model, when the planets change from equally spaced to diametrically opposed, most of the modal properties in Table 4.2 are retained. For the elastic-discrete model, the changes are more dramatic: the number of mode types reduce from four to two and the properties of each mode type change. Compared to the other three cases in Table 4.2, planetary gears with an elastic-discrete model and diametrically opposed planets is the only case without any planet modes. In the lumped-parameter model, however, no matter if the planets are equally spaced or diametrically opposed 3 N − 9 planet modes are always present. The number of planet and purely ring modes that evolve into rotational or translational modes can be determined. According to [3], a pair of degenerate planet modes is given in (3.107)-(3.109) with s selected from 2,3, , N / 2 − 1 when the number of planets N is even. For each s in [2, N / 2 − 1] , there are 2 J + 3 pairs of degenerate planet modes. For s = N / 2 , there are J+3 distinct planet modes [3]. To identify how many of these planet modes evolve into either rotational or translational modes, we must consider two cases. 102 Planet mode Translational mode Rotational mode Elastic-discrete model Lumped-parameter model Equally spaced Diametrically Equally spaced Diametrically (even N) opposed (even or odd N) opposed Multiplicity 1 1 1 1 Number 6+J JN+3N/2+3 6 6 0 nodal 0 nodal diameter diameter (rotation) (rotation) Ring nodal diameter components jN 2j Planet deflections p n = p1 pn+ N / 2 = pn p n = p1 p n = p1 Multiplicity 2 1 2 1 Number 10+4J JN+3N/2+4 12 12 1 nodal 1 nodal diameter diameter (translation) (translation) Ring nodal diameter components jN ± 1 2j+1 p n + N / 2 = −p n p n sinψ 2 = p1 sin(ψ 2 −ψ n ) Eq. (3.47) +p1 sinψ n Planet deflections Eq. (3.47) Multiplicity 2 or 1 N-3 N-3 Number 2JN-7J+3N-9 3N-9 3N-9 Ring nodal diameter jN ± s × × p n = wn p1 p n = wn p1 × × components s = 2, , N / 2 Planet deflections p n = wn p1 Number of purely ring modes × × 2J Table 4.2. Modal property comparison of planetary gears for four different cases. × denotes not applicable, and j = 0,1, 2, . 103 Case 1: When N / 2 is even, N / 2 − 1 is odd. Accordingly, half of the s in [2, N / 2 − 1] are odd and the other half are even. Thus, ( N / 2 − 2)(2 J + 3) planet modes contain odd nodal diameter components, and they evolve into translational modes; ( N / 2 − 2)(2 J + 3) + J + 3 planet modes evolve into rotational modes, where the additional J+3 modes come from the distinct planet modes. Case 2: When N / 2 is odd, there are ( N − 2) / 4 even s and ( N − 6) / 4 odd s for s in [2, N / 2 − 1] . Thus, [( N − 2) / 4](2 J + 3)(2) = ( N / 2 − 1)(2 J + 3) degenerate planet modes contain even nodal diameter components, and they evolve into rotational modes; ( N / 2 − 3)(2 J + 3) + J + 3 planet modes evolve into translational modes, where J+3 modes come from the distinct planet modes. According to (4.26), for even N there are 2J purely ring modes. When N / 2 is even all the purely ring modes have an even nodal diameter component, thus they evolve into rotational modes. When N / 2 is odd, half of the purely ring mode evolve into rotational modes, and the other half evolve into translational modes. Table 4.3 lists the number of planet and purely ring modes evolving into translational and rotational modes as the planets deviate from equally spaced to diametrically opposed. As indicated in Table 4.3，whether N / 2 is even or odd, there are the same number, ( N / 2 − 1)(2 J + 3) + J , of total planet and purely ring modes evolving into rotational modes. Similarly, whether N / 2 is even or odd, there are the same number, ( N / 2 − 2)(2 J + 3) , of total planet and purely ring modes evolving into translational modes. 104 N/2 Planet modes Purely ring modes ( N / 2 − 2)(2 J + 3) → t Even Odd ( N / 2 − 1)(2 J + 3) − J → r 2J → r ( N / 2 − 2)(2 J + 3) − J → t J→ t ( N / 2 − 1)(2 J + 3) → r J→ r Table 4.3. Number of planet and purely ring modes that evolve into rotational or translational modes when the planets deviate from equally spaced to diametrically opposed. Designations t, r denote translational and rotational modes for diametrically opposed planets. 4.4. Example As an example, a planetary gear with six equally spaced planets is analyzed The system parameters are given in given in Table 4.1. One of the three pairs of diametrically opposed planets deviates from the equally spaced position by an angle θ . Figure 4.3 shows the effects of θ on the natural frequencies. The designations R, T denote rotational and translational modes, and P2, P3 denote planet modes having jN ± 2 , jN ± 3 nodal diameter components for equally spaced planets, respectively. From Figure 4.3 one can numerically verify how each type of mode evolves when the planets deviate from equally spaced to diametrically opposed. Natural frequency splitting is observed, such as the translational modes ω6,7 and planet modes ω4,5 and ω10,11 . Note that the limiting case of 105 θ = ±π / 3 is not practically meaningful because two pairs of diametrically opposed planets lie along the same diameter. In the elastic-discrete model, the elastic deformation of the ring is highly coupled with the positions of the planets. Thus, deviation of planet positions from equal spacing yields significant changes to some natural frequencies (Figure 4.3). For certain rotational modes with small elastic ring deformation, the natural frequencies are insensitive to θ (see ω3 in Figure 4.3). For rotational modes in the lumped-parameter, rigid ring model, all rotational modes are independent of the positions of the planets [18]. This is because changing planet positions does not change the projection of mesh stiffnesses in the tangential direction. For the elastic-discrete model, the ring-planet mesh stiffnesses have a radial component that couples to elastic ring deformation. Thus, rotational modes for the elastic-discrete model are affected by planet position, with greater effect for modes with large relative amplitude of ring deformation. Translational modes are, in general, more sensitive to planet position. They experience the same ring-planet mesh stiffness interaction with ring deformation as above for rotational modes. In addition, changing the positions of the planets alters the support and mesh stiffness forces between the sun, planets, carrier, and ring in the horizontal and vertical directions, and this directly affects the translational modes even if the elastic deformation of the ring is negligible. 106 Figure 4.3. Dimensionless natural frequencies of planetary gears when the position of one pair of diametrically opposed planets deviates an angle θ from the equally spaced position. The six-planet system is defined in Table 4.1. For equally spaced planets ( θ = 0 ), the designations R, T denote rotational and translational modes, and the designations P2, P3 denote planet modes having jN ± 2 , jN ± 3 nodal diameter components for equally spaced planets. The designations r, t denote rotational and translational modes for diametrically opposed planets. 107 Chapter 5: Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears Equation Chapter 5 Section 1 The parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on a hybrid continuous-discrete model. Mesh stiffness variations of the sun-planet and ring-planet meshes caused by the changing number of teeth in contact are the source of parametric instability. The natural frequencies of the time invariant system are either distinct or degenerate with multiplicity two, which indicates three types of combination instabilities: distinct-distinct, distinct-degenerate and degenerate-degenerate instabilities. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple expressions in terms of mesh parameters. Instability existence rules for in-phase and sequentially phased planet meshes are also discovered. For in-phase planet meshes, instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. The instability boundaries are validated numerically. 108 5.1. Introduction In order to maximize the power density and improve load sharing among the planets, planetary gears in numerous industries are designed to have thin rims, and this leads to elastic deflection of the gear bodies, especially the ring gear [3, 15, 26]. The vibration of planetary gears has been studied in [13-20]. These works model the ring gear as a rigid body. Wu and Parker [3] established an elastic-discrete model that includes planetary gear discrete degrees of freedom (rotational and translational) and ring gear elastic deflection. The modal properties are systematically identified, and the vibration modes are classified into rotational, translational, planet and purely ring modes. The well-defined properties of each mode type provide a crucial foundation for the current problem of parametric instability with an elastically deformable ring gear. Parametric instability in single-pair gears has been investigated in [28-30, 32]. The parametric instabilities were experimentally observed by Kahraman and Blankenship [47] through a spur gear pair. Nonlinear phenomena such as gear tooth contact loss, perioddoubling and chaos were also observed. Only a few studies exist on parametric instabilities of multiple mesh gear systems. Tordion and Gauvin [33] and Benton and Seireg [34] analyzed the instabilities of two-stage gear systems but with contradictory conclusions. This was clarified by Lin and Parker [35], who derived formulae that allow designers to suppress particular instabilities by choice of contact ratios and mesh phasing. Liu and Parker [41] analytically investigated the nonlinear resonant vibration of idler gears parametrically excited by mesh stiffness variation. The impact of mesh stiffness variation on tooth loads and load sharing in planetary gears was studied by August and 109 Kasuba [17] and Velex and Flamand [36]. They numerically computed the dynamic response of planetary gears with three sequentially phased meshes and found the impact of mesh stiffness variations on dynamic response is significant. Lin and Parker [24] analytically investigated the parametric instability of planetary gears using a purely rotational model, and Bahk and Parker [42] extend this to examine the nonlinear dynamics. All of these works adopt a rigid ring model. This work examines planetary gear parametric instability using a model that includes the translational vibration of all components and the elastic deformation of the ring gear. With the modal expressions of the elastic-discrete model from [3], the instability boundaries are obtained as simple expressions. We show that many modes can not interact to create combination instabilities, and general instability existence rules are obtained for equally spaced planets. By adjusting the tooth numbers, contact ratios, and mesh phases one can minimize or completely suppress many potential instabilities. 5.2. Mathematical Formulation Details of the elastic-discrete model are presented in Chapter 3. The motion of the ring u (θ , t ) is separated into two parts: the rigid body motion ( xr , yr , ur ) and the elastic tangential deformation v(θ , t ) , which is related to the elastic radial deflection by the inextensibility condition w(θ , t ) = −∂v(θ , t ) / ∂θ . With the assumption of the mesh stiffness being independent of the load, the timevarying sun-planet and ring-planet mesh stiffnesses for the nth planet are represented as k sn (= t ) k sp + k1n (t ) , krn (t ) = krp + k2 n (t ) = 1 + k2 n (t ) , 110 (5.1) where k sp = O(1) and krp = 1 (a result of non-dimensionalization) are mean values. k1n (t ) , k2 n (t ) are the zero-mean mesh stiffness variations as the number of tooth pairs in contact changes for the sun-planet and ring planet meshes. Trapezoidal waves in Figure 5.1 are adopted to approximate the mesh stiffness variations. Compared to rectangular waves, trapezoidal waves are better approximations if there is corner contact. The non-parallel sides of the trapezoid have the same slope coefficients ρ j , j = s, r . The slope coefficients ρ j are in the range of [0 1/ 4] , where the two special cases ρ j = 0 and ρ j = 1/ 4 correspond to rectangular and triangle waves, respectively. Fourier expansion of the mesh stiffness variations k1n (t ) and k2 n (t ) gives ∞ k1n (t ) 2 µ ∑ [asn( L ) sin LΩt + bsn( L ) cos= LΩt ] , k2 n (t ) 2ε = L =1 ∞ ∑ [a L =1 ( L) rn sin LΩt + brn( L ) cos LΩt ] , (5.2) = asn( L ) sin(2 Lπρ s ) sin(2 Lπρ s ) γ sn )] sin( Lπ cs ) , bsn( L ) sin [ Lπ (cs + 2= cos [ Lπ (cs + 2γ sn ) ] sin( Lπ cs ) , (5.3) 2 2 ρs L π ρ s L2π 2 arn( L ) = sin(2 Lπρ r ) sin(2 Lπρ r ) ( L) cos [ Lπ (cr + 2γ sn + 2γ sr ) ] sin( Lπ cr ) sin [ Lπ (cr + 2γ sn + 2γ sr ) ]= sin( Lπ cr ) , brn 2 2 ρ r L2π 2 ρr L π , (5.4) where 2 µ and 2ε are peak-to-peak values of k1n (t ) and k2 n (t ) , respectively, cs and cr are contact ratios of the sun-planet and ring-planet meshes, and Ω is the mesh frequency. γ sn + γ sr , where γ sn is the phase of the The phases of k1n (t ) and k2 n (t ) are γ sn and γ= rn nth sun-planet mesh and γ sr is the mesh phase between the sun-planet and ring-planet meshes for a given planet, which is the same for each planet [35, 44]. Note that γ s1 = 0 .. 111 (a) (b) Figure 5.1. Mesh stiffness variations for the nth (a) sun-planet and (b) ring-planet meshes. cs , cr are contact ratios, γ sn , γ rn are mesh phases, and ρ s , ρ r are trapezoid wave slope coefficients. The displacement of the whole system a is the combination of the elastic deformation of the ring v(θ , t ) and the discrete body deflections q as aT = [ v , q T ] . (5.5) Based on the equations of motion for the time-invariant system in Chapter 3, the dimensionless equation of motion for the time-varying system is −ω 2 Ma + K (t )a = 0 (5.6) where M and K (t ) are extended inertia and stiffness operators. M and K (t ) are given in Appendix B, where the operator L2 is different from the one given in (3.2). 112 The ratios of the amplitudes of sun-planet and ring-planet mesh stiffness variations to their mean values are µ / k sp and ε , respectively. We assume µ / k sp is of the same order as ε , i.e., µ / k sp = gε , where g = O(1) . The extended stiffness operator K (t ) is separated into time-varying and time-invariant parts as ∞ K (t )= K 0 + 2ε ∑ [ K v(1L ) sin LΩt + K v( 2L ) cos LΩt ] , (5.7) L =1 where K 0 is the stiffness operator for the time-invariant system, which has the same form as K (t ) with k sn (t ) , krn (t ) substituted by k sp and 1. The Fourier coefficient operators K v(1L ) and K v( 2L ) also have the same form as K (t ) with the substitutions of all non mesh stiffness being zero and the mesh stiffnesses k sn (t ) , krn (t ) in K (t ) are substituted by ksp gasn( L ) , arn( L ) for K v(1L ) and by ksp gbsn( L ) , brn( L ) for K v( 2L ) . The method of multiple scales is used with the introduction of the slow time τ = ε t , and the dynamic response is represented as v0 v1 a =a 0 (t ,τ ) + ε a1 (t ,τ ) + O(ε 2 ) , a0 = , a1 = . q1 q 0 Substitution of (3.25) into (5.6), use of (5.8) d ∂ ∂ → +ε , and separation of powers in ε dt ∂t ∂τ lead to ∂ 2a0 M 2 + K 0a 0 = 0, ∂t M ∞ ∂ 2a 0 ∂ 2a1 K M 2 2 ( K v(1L ) sin LΩt +K v( 2L ) cos LΩt )a 0 . + a =− − ∑ 0 1 2 ∂t ∂t ∂τ L =1 The general solution for the time-invariant system (5.9) is 113 (5.9) (5.10) a0 = ∞ ∑ c (τ )Y (θ )e ω i s =1 s st s + cc , (5.11) where Ys (θ ) are the eigenfunctions in extended variable form and cc refers to the complex conjugate of preceding terms. The natural modes Ys (θ ) in (5.11) are analytically solved by Wu and Parker [3]. The multiplicity of the natural frequencies ωs are summarized as: (i) Rotational and purely ring modes have distinct natural frequencies; (ii) Translational modes are degenerate with multiplicity two; and (iii) If the number of planets is odd, all the planet modes are degenerate with multiplicity two, otherwise, the planet modes may be either degenerate with multiplicity two or distinct; note that planet modes exist only when the number of planets N ≥ 4 . Therefore, all natural frequencies of planetary gears with an elastic ring gear are either distinct or degenerate with multiplicity two. When a harmonic of the mesh frequency is close to the sum of two natural frequencies secular terms exist in (5.10), which leads to sum type parametric instability. This condition is LΩ= ω p + ωq + εσ . (5.12) When a harmonic of mesh frequency is close to the difference of two natural frequencies, such as LΩ= ω p − ωq + εσ , one can show that difference type parametric instabilities are not possible. According to the degeneracy of ω p and ωq , the solvability conditions of (5.10) are classified into three cases: Case 1, both eigenvalues are distinct; Case 2, one eigenvalue is distinct and the other is degenerate; Case 3, both eigenvalues are degenerate. Single mode instabilities are treated as a special case of combination instabilities with 114 ω p = ωq . Case 1: Instability boundaries for two distinct eigenvalues Consider the case where ω p and ωq are distinct natural frequencies. When (5.11) is used in (5.10) and condition (5.12) is invoked, the two solvability conditions of (5.10) for s = p and s = q are i 2ω p ∂c p ∂τ ( L) ( L) + ( E pq − iD pq )cq eiστ = 0 , i 2ωq ( L) D pq = Dqp( L ) = < Yp , K v(1L ) Yq > , ∂cq ∂τ ( L) ( L) (5.13) + ( Eqp − iDqp )c p eiστ = 0, ( L) ( L) E pq = Eqp = < Yp , K v( 2L ) Yq > , (5.14) ( L) ( L) where D pq = Eqp( L ) because K v(1L ) , K v( 2L ) are self-adjoint and Yp , Yq are = Dqp( L ) and E pq real. For all s ≠ p, q in (5.11), the solvability conditions of (5.10) lead to cs (τ ) = constant (and cs (τ ) = 0 if any damping is present) . The solutions of (5.13) have the form τ ) ( R p + iI p )eλτ +iστ / 2 , cq (= τ ) ( Rq + iI q )eλτ +iστ / 2 , c p (= (5.15) where R p , I p , Rq , I q are real coefficients independent of τ . Substitution of (5.15) into (5.13) yields an eigenvalue problem for R p , I p , Rq , I q . The eigenvalues λ of the coefficient matrix have non-positive real part only when σ ≥ Λ (pqL ) /(ω pωq ) , (5.16) (D ) + (E ) (L) 2 pq ( L) ( L) ( L) ( L) ( L) Λ= D pq Dqp + E pq E= pq qp (L) 2 pq . (5.17) The instability boundaries for two distinct natural frequency modes are thus = Ω Combination instabilities: ω p + ωq L 115 ± 1 2 ( L) ε Λ pq /(ω pωq ) , L (5.18) 2ω p Single mode instabilities:= Ω L ± 1 Lω p ε 2 Λ (ppL ) . (5.19) Case 2: Instability boundaries for distinct-degenerate eigenvalues If one of the natural frequencies in (5.12) is degenerate with multiplicity two (such as ωq = ωm ) and the other remains distinct, the solvability conditions of (5.10) generate three equations i 2ω p ∂c p ∂τ ( L) ( L) ( L) ( L) + ( E pq − iD pq − iD pm )cq eiστ + ( E pm )cm eiστ = 0, i 2ωq i 2ωm ∂cq (5.20) ( L) ( L) + ( Eqp − iDqp )c p eiστ = 0, (5.21) ∂cm ( L) ( L) + ( Emp − iDmp )c p eiστ = 0. ∂τ (5.22) ∂τ The solutions of equations (5.20)-(5.22) have the same form as (5.15), and an analogous 6 × 6 eigenvalue problem results. With some algebraic manipulation, the combination instability boundaries for modes with distinct-degenerate eigenvalues are obtained in closed-form as in (5.18) with ( L) Λ= pq (D ) + (D ) + (E ) + (E ) ( L) 2 pq ( L) 2 pm ( L) 2 pq ( L) 2 pm . (5.23) Case 3: Instability boundaries for two degenerate eigenvalues When both natural frequencies in (5.12) are degenerate such that ω p = ωq and ωr = ωm , the solvability conditions of (5.10) generate four equations i 2ω p i 2ωq ∂c p ∂τ ∂cq ∂τ ( L) ( L) ( L) ( L) )cr eiστ + ( E pm )cm eiστ = 0, + ( E pr − iD pr − iD pm (5.24) ( L) ( L) )cm eiστ = 0, + ( Eqr( L ) − iDqr( L ) )cr eiστ + ( Eqm − iDqm (5.25) 116 i 2ωr ∂cr 0, + ( Erp( L ) − iDrp( L ) )c p eiστ + ( Erq( L ) − iDrq( L ) )cq eiστ = ∂τ (5.26) i 2ωm ∂cm ( L) ( L) ( L) ( L) )c p eiστ + ( Emq )cq eiστ = 0. + ( Emp − iDmp − iDmq ∂τ (5.27) Similar to cases 1 and 2, the solutions of (5.24)-(5.27) have the form (5.15), yielding an 8 × 8 eigenvalue problem. Due to the size of the coefficient matrix, the instability boundaries can not be expressed in simple expressions unless one finds simplifying ( L) ( L) ( L) ( L) properties of D pq , E pq , D pm , E pm , . ( L) ( L) All three cases above require further analysis of D pq and E pq in (5.14), which expand to give N N ( L) = D pq k sp g ∑ asn( L )δ sn[ p ]δ sn[ q ] + ∑ arn( L )δ rn[ p ]δ rn[ q ] , (5.28) = n 1= n 1 N N ( L) = E pq k sp g ∑ bsn( L )δ sn[ p ]δ sn[ q ] + ∑ brn( L )δ rn[ p ]δ rn[ q ] , (5.29) = n 1= n 1 = δ sn ys cosψ sn − xs sinψ sn − ξn sin α s − ηn cos α s + us + un , = δ rn (v cos α r + ∂v − xr sinψ rn + yr cosψ rn + ur + ξ n sin α r − η n cos α r − un , sin α r ) ∂θ θ =ψ n (5.30) (5.31) where δ sn[ p ] and δ rn[ p ] are deformations of the nth sun-planet and ring-planet meshes for the p th mode, α s and α r are the pressure angles of the sun-planet and ring-planet meshes, ψ n −αs , ψ= ψ n + α r , where= ψ n 2π (n − 1) / N is the circumferential planet and ψ = sn rn ( L) ( L) location. From (5.28) and (5.29), D pq and E pq depend on modal gear mesh deformations and the Fourier coefficients of the mesh stiffness variations. Substitution of (5.3)-(5.4) and (5.28)-(5.29) into (5.17)-(5.19) for case 1 and (5.23) 117 for case 2 gives expressions for the instability boundaries in terms of mesh parameters, such as contact ratios and phasing. With these expressions, one can predict the effects of mesh parameters on the instability regions. For each of the three cases, however, the ( L) and instability boundaries clarify further after discovering the intrinsic properties of D pq ( L) ( L) ( L) (and D pm , E pm E pq , ) by studying gear mesh deformations for each type of mode. 5.3. Modal Properties and Gear Mesh Deformations Fourier expansion of the elastic deflection of the ring gives v(θ ) = ± JN ∑V e m = ±2 imθ m , (5.32) where J ≥ 1 is an arbitrarily large integer. For all vibration modes, the deformations of the ring-planet and sun-planet meshes are expressed in a general compact form as = δ jn c1 cos Tψ n + c2 sin Tψ n , j = s, r , (5.33) where c1 and c2 are coefficients independent of ψ n , and T is an integer to be determined for each type of mode (i.e., rotational , translational, planet and purely ring modes). A mesh deformation having the form (5.33) is called type T deformation. For each type of mode, T is determined below. 5.3.1. Gear Mesh Deflection of Rotational Modes A rotational mode (mode p) in extended variable form has the structure [3] T [ p] a J imNθ = + c.c. Vm[ p ] , qTrot , ∑ (cos α r − imN sin α r )e m =1 118 (5.34) T q rot = 0, 0, ur[ p ] , 0, 0, uc[ p ] 0, 0, us[ p ] , ξ1[ p ] ,η1[ p ] , u1[ p ] , , ξ1[ p ] ,η1[ p ] , u1[ p ] , (5.35) where the Vm[ p ] are real. According to (5.34) and (5.35), the translations of the sun, carrier and ring rigid motion are zero, the displacements of all planets are identical, and the elastic deflection of the ring is a linear combination of the mN nodal diameter components. With these properties, substitution of (5.35) into (3.116) ensures δ sn is independent of ψ n as δ sn[ p ] = δ s[1p ] = −ξ1[ p ] sin α s − η1[ p ] cos α s + u1[ p ] + us[ p ] . (5.36) Similarly, δ rn[ p ] is independent of ψ n due to the modal properties and the identities cos lNψ n = 1 and sin lNψ n = 0 , δ rn[ p ] = δ r[1p ] = ∞ ∑ 2U l =1 m (cos 2 α r + m 2 sin 2 α r ) + ur[ p ] + ξ1[ p ] sin α r − η1[ p ] cos α r − u1[ p ] . (5.37) δ sn[ p ] and δ rn[ p ] in (5.36) and (5.37) have the form of (5.33) with T = 0 . Accordingly, they are called type 0 deformations. 5.3.2. Gear Mesh Deflection of Translational Modes Natural frequencies of translational modes are degenerate with multiplicity two. A degenerate pair of translational modes has the form T [ p] a = ∑ Vm[ p ]eimθ , qTtrn , m= lN ±1 (5.38) T [q] a = ∑ iVm[ p ]eimθ , qˆ Ttrn . m= lN ±1 119 (5.39) where the Vm[ p ] are complex and qtrn , qˆ trn are a pair of translational modes of a planetary gear modeled with only discrete degree of freedom [21]. Here and in all that follows, l in the summations is such that m always falls within the range defined in (4.1). The deflections of the sun, carrier and ring rigid motions are related as p j = ( x[j p ] , y[j p ] , 0)T , = pˆ j ( y[j p ] , − x[j p ] , 0)T , j = r , c, s . The deflections of the nth planet p n , pˆ n are known in terms of p1 and p̂1 as p n cosψ n I sinψ n I p1 pˆ = − sinψ I cosψ I pˆ , n n 1 n (5.40) where I is a 3 × 3 identity matrix. According to (5.38)-(5.40), the deflections of the nth sun-planet and ring-planet meshes have simple relations defined by the deflections of the first meshes = δ [jnp ] δ [j1p ] cosψ n + δ [j1q ] sinψ= δ [jnq ] δ [j1q ] cosψ n − δ [j1p ] sinψ n , j = s, r , n, (5.41) where the deflections δ s[1p ] , δ s[1q ] , δ r[1p ] , δ r[1q ] of the first gear meshes for the p th and q th translational modes are (with ψ 1 = 0 ) δ s[1p ] = ys[ p ] cos α s + xs[ p ] sin α s − ξ1[ p ] sin α s − η1[ p ] cos α s + u1[ p ] , (5.42) δ s[1q ] = ys[ p ] sin α s − xs[ p ] cos α s − ξˆ1[ p ] sin α s − ηˆ1[ p ] cos α s + uˆ1[ p ] , (5.43) p] δ r[= 1 ∑ = lN ±1 m +ξ [ p] 1 [q] δ r= 1 ∑ (cos α r + im sin α r )Vm[ p ] − xr[ p ] sin α r + yr[ p ] cos α r sin α r − η = lN ±1 m [ p] 1 cos α r − u , (5.44) [ p] 1 i (cos α r + im sin α r )Vm[ p ] − xr[ p ] cos α r − yr[ p ] sin α r +ξˆ1[ p ] sin α r − ηˆ1[ p ] cos α r − uˆ1[ p ] , 120 (5.45) Because the sun-planet and ring-planet gear deformations in (5.41) have the form of (5.33) with T = 1 , they are called type 1 deformations. 5.3.3. Gear Mesh Deflection of Planet Modes For planet modes, the sun, carrier and rigid ring motion are zero, and only the elastic ring and planet motions are non-zero. The planet modes are classified into int ( N2 ) − 1 subtypes according to the ring nodal diameter components they contain [3]. Planet modes having lN ± d , l = 0,1, 2, nodal diameter components are named type d planet modes, where d is an integer selected from {2, , int ( N2 )} . Each planet mode belongs to a unique type. For instance, when N = 8 , three types exist: degenerate planet modes are either type 2 or 3 ( d = 2, , int ( N2−1 ) ), and distinct planet modes are type 4 ( d = N 2 ) which only exist for even N. There are no planet modes outside of these three types. A degenerate pair of type d planet modes has the form T [ p] a = ∑ Vm[ p ]eimθ , qTplt ,d , m= lN ± d (5.46) T [q] a = ∑ iVm[ p ]eimθ , qˆ Tplt ,d , m= lN ± d (5.47) qTplt ,d = 0, 0, 0, p1T cos dψ 1 , , p1T cos dψ N , qˆ Tplt ,d = 0, 0, 0, p1T sin dψ 1 , , p1T sin dψ N , (5.48) where q plt ,d , qˆ plt ,d are a pair of discrete (i.e., rigid ring model) planet modes [21]. With some algebraic manipulation, the deflections of the nth sun-planet and ring-planet meshes have simple relations with the first mesh deflections as δ sn[ p ] = δ s[1p ] cos dψ n , 121 δ sn[ q ] = −δ s[1p ] sin dψ n , (5.49) = δ rn[ p ] δ r[1p ] cos dψ n + δ r[1q ] sin d= ψ n , δ rn[ q ] δ r[1q ] cos dψ n − δ r[1p ] sin dψ n , (5.50) where the deflections of the first gear meshes for the p th and q th planet modes are δ s[1p ] = −ξ1[ p ] sin α s − η1[ p ] cos α s + u1[ p ] , p] δ r[= 1 ∑ = lN ± d m (cos α r + im sin α r )Vm[ p ] + ξ1[ p ] sin α r − η1[ p ] cos α r − u1[ p ] , = δ r[1q ] ∑ = lN ± d m i (cos α r + im sin α r )Vm[ p ] . (5.51) (5.52) (5.53) Therefore, for a pair of type d planet modes in (3.107)-(3.109), the sun-planet and ringplanet mesh deflections are type d deformations as defined in (5.33). These results are for any N. When N is even, extra distinct planet modes exist. They have the form of (3.107) with d = N / 2 . The corresponding gear mesh deformations are of type N / 2 . 5.3.4. Gear Mesh Deflection of Purely Ring Modes A purely ring mode, which always has distinct natural frequency, has the structure for odd or even N T m = lN (5.54) = a[ p ] (cos α r sin mθ − m sin α r cos mθ )Vm[ p ] , 0 , = lN + N / 2 for even N m where Vm[ p ] is real. The sun-planet mesh deformations of purely ring modes are zero. The ring-planet mesh deformations of a purely ring mode are also zero according to (3.117) ( L) ( L) , E pq in (5.28) and (5.29) always vanish and (5.54). For a purely ring mode p, D pq because the gear mesh deflections are zero. Thus, combination and single mode instabilities associated with a purely ring mode in (5.17)-(5.19) and (5.23) always vanish for either in-phase or sequentially phased meshes. 122 5.3.4.1. Parametric Instabilities for In-Phase and Sequentially Phased Meshes Equal planet spacing requires ( zs + zr ) / N to be an integer, where zs and zr are the tooth numbers of the sun and ring. The gear mesh phases [44] are defined as γ sn= (n − 1) zs / N for clockwise planet rotation. Equal planet spacing has only two possible phase conditions: in-phase and sequentially phased. When z s and z r are not each integer multiples of N, the sun-planet meshes are sequentially phased, as are the ring-planet meshes; otherwise, all the sun-planet meshes are in-phase ( γ sn = 0 ) and all the ring-planet meshes are in-phase. 5.3.5. Parametric Instabilities for In-Phase Meshes For in-phase planet meshes, the Fourier coefficients of the mesh stiffness variations in (5.3) and (5.4) are independent of the planet index n, so asn( L ) , arn( L ) , bsn( L ) , brn( L ) in (5.28) and (5.29) are moved outside of the summations to give N N ( L) = D pq k sp gas(1L ) ∑ δ sn[ p ]δ sn[ q ] + ar(1L ) ∑ δ rn[ p ]δ rn[ q ] , (5.55) = n 1= n 1 N N ( L) = E pq k sp gbs(1L ) ∑ δ sn[ p ]δ sn[ q ] + br(1L ) ∑ δ rn[ p ]δ rn[ q ] . (5.56) = n 1= n 1 With the general expression of the mesh deflection in (5.33), the summation terms in (5.55) and (5.56) are written as N N ∑ δ [jnp ]δ [jnq ] = ∑ (c1 cos Tpψ n + c2 sin Tpψ n )(c3 cos Tqψ n + c4 sin Tqψ n ) , j = s, r , (5.57) = n 1= n 1 where ci , i = 1, 2,3, 4 are known coefficients for the given modes p and q. Tp and Tq are indices of mesh deformation types for modes p and q as defined in (5.33). 123 For equally spaced planets, the following identities hold for integer l N sin Tpψ n sinTqψ n = ∑ N sin Tpψ n cosTqψ n ∑= N cos T ψ cos T ψ ∑= = n 1= n 1= n 1 p n q n 0 , for any Tp ± Tq ≠ lN (5.58) For all types of modes, Tp and Tq are integers within [0, int ( N / 2)] , so the condition Tp ± Tq ≠ lN in (5.58) is equivalent to Tp ≠ Tq . Three cases must be considered: 1. ( L) ( L) When Tp ≠ Tq , D= E= 0 according to (5.55), (5.56), (5.57) and (5.58). Thus, pq pq for any potential distinct-distinct instability with Tp ≠ Tq , Λ (pqL ) in (5.17) vanishes, which ensures that no such instability is possible. For example, no combination instability can occur between a rotational mode and a distinct planet mode. 2. For distinct-degenerate combination instabilities, it is always the case that Tp ≠ Tq ( L) ( L) ( L) ( L) and Tm = Tq , from which D pq , E pq , D pm , E pm in (5.23) vanish and so does Λ (pqL ) ; no distinct-degenerate instabilities occur. As one example, combination instabilities between a rotational mode and any pair of translational modes can not occur. 3. For degenerate-degenerate instabilities where Tp ≠ Tr (and, of course, Tq = Tp , ( L) ( L) ( L) ( L) Tm = Tr ), all terms associated with D pr , E pr , D pm , E pm , in (5.24)-(5.27) vanish, and so do these combination instabilities. Thus, degenerate-degenerate instabilities can occur only between two degenerate pairs of the same type (i.e., two pairs of translational modes, or two pairs of planet modes). Therefore, one can state a key rule governing the existence or elimination of parametric instabilities for in-phase meshes: If two modes have the same type of gear 124 mesh deformation their combination instability exists, otherwise their combination instability vanishes. This result is independent of which mesh frequency harmonic L is the source of potential instability (see (5.12)). Consequently, all distinct-degenerate instabilities vanish for in-phase planet meshes. The following sections examine the remaining cases of distinct-distinct and degeneratedegenerate instabilities that do not vanish. 5.3.5.1. Distinct-Distinct Combination Instabilities for In-Phase Meshes The possible distinct-distinct combination instabilities involve rotational, purely ring and distinct planet modes. As shown above, however, all distinct-distinct instability boundaries vanish for in-phase meshes except for two cases: (a) two rotational modes; (b) two distinct planet modes. For the purpose of the simplification, we assume the sun-planet and ring-planet mesh has the same trapezoid wave slope coefficients ρ s = ρ r . For both cases of distinct-distinct combination instability, reduction of (5.17), (5.28) and (5.29) using the properties of each mode type yield a common form defining the instability boundaries as 2 ε Λ 2 ( L) p N sin(2 Lπρ s ) 2 [ p] [q] 2 2 2 [ p] [q] 2 2 µ (δ s1 δ s1 ) sin ( Lπ cs ) + ε (δ r1 δ r1 ) sin ( Lπ cr ) q 2 2 ρs L π [ p] [ p] [q] [q] +2 µεδ s1 δ r1 δ s1 δ r1 sin( Lπ cs ) sin( Lπ cr ) cos Lπ (cr + 2γ sr − cs ) . (5.59) For two rotational modes, δ s[1p ] , δ s[1q ] , δ r[1p ] , δ r[1q ] in (5.59) are given by (5.36) and (5.37). For two distinct planet modes, they are given by (5.51)-(5.53). Distinct planet modes exist only for even N, and substitution of d = N / 2 into (5.49) and (5.50) yields expressions for the nth mesh deflections in terms of the first mesh deflections 125 δ sn[ p ] = ( −1) n δ s[1p ] , δ rn[ p ] = ( −1) n δ r[1p ] , δ sn[ q ] = ( −1) n δ s[1q ] , δ rn[ q ] = ( −1) n δ r[1q ] . (5.60) Equation (5.60) indicates that the product of two mesh deflections is independent of n, which is needed to establish (5.59) as the instability boundary for this case. The contact ratios cs , cr and the phase difference between the sun-planet and ringplanet meshes γ sr are governed by the gear geometries. The expression for γ sr is given in [43]. When both Lcs and Lcr are integers, all potential instabilities driven by the Lth harmonic of mesh frequency vanish. When neither of Lcs , Lcr are integers, a minimum instability region can be achieved by adjusting cs , cr and γ sr in the third term of (5.59) such that this term is negative with absolute value comparable to the sum of the first two terms. For any two given modes, the sign of δ s[1p ]δ r[1p ]δ s[1q ]δ r[1q ] in (5.59) is unambiguous because multiplication of a mode by -1 does not change the products δ s[1p ]δ r[1p ] and δ s[1q ]δ r[1q ] . According to (5.18), the width of the instability region of two distinct modes is = ∆Ω 2 2 ( L) ε Λ pq /(ω pωq ) , which is proportional to the number of planets N and inversely L proportional to L2 . 5.3.5.2. Degenerate-Degenerate Combination Instabilities for In-Phase Meshes According to the instability existence rule for in-phase meshes, combination instabilities between a pair of translational modes and a pair of degenerate planet modes always vanish because they have different mesh deformations as defined in (5.33). Combination instabilities for degenerate modes exist only between two pairs of translational modes or between two pairs of type d planet modes. Closed-form expressions of instability boundaries for two pairs of degenerate modes are obtained as 126 follows. For two degenerate translational mode pairs such as ω p = ωq and ωr = ωm , δ sn[ p ] and δ sn[ q ] are determined by (5.41) and δ sn[ r ] , δ sn[ m ] are given by analogous expressions. According to (5.41) and (5.58), the following equations hold N N ∑ δ [jnp ]δ [jnr ] = ∑ δ [jnq ]δ [jnm] , N N ∑ δ [jnp ]δ [jnm] = −∑ δ [jnq ]δ [jnr ] , j = s, r . (5.61) 1 = n 1= n = n 1= n 1 Substitution of (5.61) into (5.28) and (5.29) gives D pr = Dqm , E pr = Eqm , D pm = − Dqr , E pm = − Eqr . (5.62) Substitution of (5.62) into the coefficient matrix of (5.24)-(5.27) yields a simple expression for the 8 × 8 eigenvalue problem. Requiring the eigenvalues of the coefficient matrix to have non-positive real parts yields the combination instability boundaries ) = Ω (ω p + ωr ) / L ± ε 2 Γ (prL ) /(ω pωr ) / L , Γ(prL= max( ∆1( L ) , ∆ (2L ) ) , ) ( L) ( L) 2 ( L) (L) 2 ) ( L) ( L) 2 ( L) ( L) 2 ( D pr ) + ( D pm ) . + E pm − E pr ( D pr ) + ( D pm ) , ∆ (2L= ∆1( L= − E pm + E pr (5.63) (5.64) With the properties of gear mesh deformations in (5.41), ∆1( L ) and ∆ (2L ) for two translational modes are written in terms of the mesh parameters as 2 ε 2 ∆1( L ) N sin(2 Lπρ s ) 2 2 2 2 2 2 = {( β1 + β3 ) sin ( Lπ cs ) + ( β 2 + β 4 ) sin ( Lπ cr ) + 2sin( Lπ cs ) sin( Lπ cr ) 2 2 2 ( L) ε ∆2 2ρs L π ⋅ [ ( β1β 2 + β3 β 4 ) co Lπ s(cr + 2γ sr − cs ) ± ( β1β 4 − β 2 β3 ) sin Lπ (cr + 2γ sr − cs ) ]} , (5.65) m] = β1 µ (δ s[1p ]δ s[1r ] + δ s[1q ]δ s[= ε (δ r[1p ]δ r[1r ] + δ r[1q ]δ r[1m ] ) , 1 ) , β2 [q] = β3 µ (δ s[1p ]δ s[1m ] − δ s[1r ]δ= ε (δ r[1p ]δ r[1m ] − δ r[1r ]δ r[1q ] ) , s1 ) , β 4 127 (5.66) (5.67) where the gear mesh deflections δ s[1p ] , δ s[1q ] , δ r[1p ] , δ r[1q ] are governed by (5.42)-(5.45) (with similar equations for modes r and m). The instability boundaries for a single pair of translational modes are obtained by the replacements r → p and m → q in (5.66)-(5.67) to give 2 ε 2 ∆1( L ) N sin(2 Lπρ s ) 2 2 2 2 = β1 sin ( Lπ cs ) + β 2 sin ( Lπ cr ) 2 2 2 ( L) ε ∆2 2ρs L π (5.68) + 2 β1β 2 sin( Lπ cs ) sin( Lπ cr ) cos Lπ (cr + 2γ sr − cs ) ] , = ) 2 , β 2 ε (δ r[1p ] ) 2 + (δ r[1q ] ) 2 . with β1 µ (δ s[1p ] ) 2 + (δ s[1q ] = Similarly, for two pairs of degenerate planet modes of the same type, the same properties as (5.61)-(5.62) are found. Therefore, the instability boundaries of two pairs of type d planet modes are also governed by (5.63)-(5.64). With the properties of gear mesh deformations, ∆1( L ) and ∆ (2L ) for two pairs of planet modes of the same type governed by (5.49) and (5.50) simplify to 2 ε 2 ∆1( L ) N sin(2 Lπρ s ) 2 2 2 2 2 = {β5 sin ( Lπ cs ) + ( β 6 + β 7 ) sin ( Lπ cr ) + 2 β5 sin( Lπ cs ) sin( Lπ cr ) ⋅ 2 2 2 ( L) ε ∆2 2ρs L π [ β6 co Lπ s(cr + 2γ sr − cs ) ± β 7 sin Lπ (cr + 2γ sr − cs ) ]} , (5.69) ] [r ] [m] β5 = µδ s[1p= δ s1 , β 6 ε (δ r[1p ]δ r[1r ] + δ r[1q ]δ= ε (δ r[1p ]δ r[1m ] − δ r[1q ]δ r[1r ] ) , r1 ) , β 7 (5.70) where the gear mesh deflections δ s[1p ] , δ r[1p ] , δ r[1q ] are governed by (5.51)-(5.53) (with similar equations for modes r and m). The single degenerate planet mode pair instability boundaries are obtained by the replacements r → p and m → q in (5.69)-(5.70) to give 128 2 ε 2 ∆1( L ) N sin(2 Lπρ s ) 2 2 2 2 = {β 5 sin ( Lπ cs ) + β 6 sin ( Lπ cr ) 2 2 2 ( L) ε ∆2 2ρs L π (5.71) +2 β 5 β 6 sin( Lπ cs ) sin( Lπ cr ) [ cos Lπ (cr + 2γ sr − cs ) ]} , p] 2 ε (δ r[1p ] ) 2 + (δ r[1q ] ) 2 . with β5 = µ (δ s[= 1 ) , β6 Similar to (5.59), one can minimize the degenerate-degenerate instability regions by adjusting cs , cr and γ sr based on (5.65), (5.68), (5.69) and (5.71). 5.3.5.3. Numerical Verification for In-Phase Meshes The foregoing closed-form expressions for the instability boundaries are compared to numerical solutions from Floquet theory. Figure 5.2 shows the instability boundaries for a planetary gear with four in-phase planet meshes. The parameters and the first 11 natural frequencies are given in Table 5.1. ω3 and ω7 are for rotational modes, ω4 and ω9 are for planet modes, ω8 is for a purely ring mode, and the remaining natural frequencies in Table 5.1 are for translational modes. The numerical and analytical instability boundaries in Figure 5.2 match well. The seemingly larger numerical instability region near ω5 results from a higher order instability for which the analytical solution is not shown. As can be seen from Figure 5.2, single mode instabilities near ω5 , ω7 , ω9 , 2ω4 , 2ω5 , 2ω7 always exist. Combination instabilities exist only for modes with the same type of gear mesh deformation such as ω1 + ω5 (two translational modes), ω3 + ω7 (two rotational modes) and (ω4 + ω9 ) / 2 (two planet modes). The predicted absence of instabilities for two modes with different types of gear mesh deformations, such as ω1 + ω3 and ω3 + ω4 , is numerically verified. 129 Figure 5.2. Instability regions for a planetary gear with in-phase meshes as γ sn = 0 , γ sr = 12 , cs = 1.4 , cr = 1.6 , ε = µ , and other parameters in Table 5.1. ──, analytical solution; ***, numerical solution. 130 Inertias (kg) I r / rr2 = 7.6810 , I c / rc2 = 6 , I s / rs2 = 2.5 , I p / rp2 = 2 Masses (kg) mr = 6.35 , mc = 5.43 , ms = 0.4 , m p = 4 k= k= 108 , kbend = 5 × 107 , k p = kc = k s = k su = 5 × 1011 , kcu = 5 × 1014 , sp rp Stiffnesses (N/m) krbs = 0 , krus = 0 Pressure angle (deg) α= α= 24.6 s r Dimensionless natural ω= 2.733 (T), ω= ω= 0.9184 (T), ω3 = 1.116 (R), ω4 = 2.438 (P), ω= 5 6 1 2 frequencies ω7 = 2.896 (R), ω8 = 4.756 (PR), ω9 = 5.045 (P) Table 5.1. Dimensional parameters and dimensionless natural frequencies of an example planetary gear with four equally spaced planets. The designations T, R, P and PR denote translational, rotational, planet and purely ring modes. 5.3.6. Parametric Instabilities for Sequentially Phased Meshes When the planets are equally spaced and z s , z r are not integer multiples of N, all the sun-planet and ring-planet meshes are sequentially phased. With γ sn= ( n − 1) zs / N and = ψ n 2π (n − 1) / N , 2 Lπγ sn in (5.3) and (5.4) can be written as Lzsψ n . Expanding with trigonometric identities, the mesh stiffness Fourier coefficients in (5.3) and (5.4) are linear combinations of sin( Lzsψ n ) and cos( Lzsψ n ) . Because N ∑ cos( Lz ψ n =1 s n N ∑ sin( Lz ψ n =1 s n ) and ) vanish for equally spaced planets if Lzs ≠ jN , one can show that the 131 following identities hold N asn( L ) ∑= N N arn( L ) ∑= = bsn( L ) ∑ N = brn( L ) 0 , for any Lzs ≠ jN . ∑ (5.72) = n 1= n 1= n 1= n 1 ( L) For any two modes whose gear mesh deformations are of type Tp and Tq , D pq in (5.28) has the form = D p( Lq) N 4 ∑∑ c = n 1 =i 1 s ,i sin( Riψ n + ϕ s ,i ) + cr ,i sin( Riψ n + ϕ r ,i ) N + ( cos cos ) c c ϕ ϕ ∑ s ,i s ,i r ,i r ,i ∑ sin Riψ n =i 1 = n 1 = 4 (5.73) 4 N + ∑ (cs ,i sin ϕ s ,i + cr ,i sin ϕ r ,i ) ∑ cos Riψ n =i 1 = n 1 R1 = Lzs + Tp + Tq , R2 = Lzs + Tp − Tq , R3 = Lzs − Tp + Tq , R4 = Lzs − Tp − Tq , (5.74) where cs ,i , cr ,i , ϕ s ,i , ϕr ,i are coefficients independent of n that are known for given modes ( L) has the form (5.73). From (5.17) and (5.18), the instability p and q. Similarly, E pq ( L) ( L) boundaries for two distinct modes vanish as long as D= E= 0 . For distinctpq pq degenerate or degenerate-degenerate combination instabilities, the stability boundaries vanish as long as those terms associated with D and E in (5.23) or (5.24)-(5.27) vanish. N sin Riψ n = According to (5.73) and ∑ N cos R ψ ∑= n 1= n 1 = i n ( L) ( L) 0 for Ri ≠ jN , D pq and E pq vanish if none of the Ri is an integer multiple of N. Note that the four conditions Ri ≠ jN ( i = 1, 2,3, 4 ) can be written as Lzs ± (Tp ± Tq ) ≠ jN . Therefore, the instability existence rule for sequentially phased meshes is identified: if [ Lzs ± (Tp ± Tq )]/ N ≠ integer, 132 (5.75) the combination instability vanishes. Considering the converse, is it true that ( L) ( L) [ Lzs ± (Tp ± Tq )]/ N = integer (which gives D pq ≠ 0 and E pq ≠ 0 ) implies the combination instabilities for mode p and q exist? This can mostly be answered. For p and q denoting ( L) ( L) two distinct modes or a distinct and a degenerate mode, then D pq ≠ 0 and E pq ≠ 0 ensures that the corresponding instability exists according to (5.17) and (5.23). When p and r each denote a degenerate mode, lack of an analytical reduction analogous to (5.63)-(5.64) for in-phase meshes prevents rigorously concluding that the instability exists for ( L) [ Lzs ± (Tp ± Tr )]/ N = integer ( D pr( L ) ≠ 0 , D pm ≠ 0, ). In contrast to in-phase meshes, the rule in (5.75) depends on which mesh frequency harmonic L (see (5.12)) drives the potential instability. In applying (5.75), recall that for rotational, translational and planet modes, the deformation type indices are 0, 1, and d, respectively. The deformation type indices for purely ring modes are 0 for odd N and 0 or N/2 for even N, respectively. The instability existence rule for in-phase meshes as presented earlier can be viewed as a special case of the sequentially phased mesh condition because when zs is an integer multiple of N, (5.75) is equivalent to Tp ≠ Tq after consideration of Tp and Tq being integers within [0, int ( N / 2)] . Therefore, the in-phase and sequentially phased instability existence rules can be generalized as the single equation (5.75). The general instability existence rule (5.75) reveals some interesting properties. Consider two different sun tooth numbers zs1 and zs 2 , for example. If mod( Lzs1 / N ) = mod( Lzs 2 / N ) or mod( Lzs1 / N= ) N − mod( Lzs 2 / N ) , where mod( Lzs1 / N ) is the integer remainder of Lzs1 divided by N, then according to (5.75) changing the sun 133 tooth number from zs1 to zs 2 does not change the instability existence properties. To see the range of potential instabilities that can be eliminated, consider a planetary gear with 6 j + 2 , where j is an arbitrary integer. One can find many instability six planets and z= s existence properties, such as: a) The fundamental ( L = 1 ), secondary ( L = 2 ) and tertiary ( L = 3 ) instabilities between any translational mode and any rotational mode always vanish ( Tp ± Tq = ±1 , so (5.75) yields (2 L ± 1) / 6 ≠ integer ); b) The fundamental and secondary (but not tertiary) instabilities between any two rotational modes always vanish ( Tp ± Tq = 0 , so (5.75) yields (2 L ± 0) / 6 ≠ integer ); c) The fundamental instabilities between any rotational mode and any type 2 planet mode exist ( Tp ± Tq = ±2 , so (5.75) 2 / 3, 0 ), etc. Analogous properties are easily derived for z= 6j+k yields (2 ± 2) / 6 = s and k ∈ {0,1,3, 4,5} . Thus, one can quickly identify all possible gear tooth numbers with desirable instability existence properties that avoid potential instabilities. This provides designers with multiple options. This information is useful for the common situation where the natural frequencies and operating speeds (and so the harmful instabilities) are calculable in the design stage, yet the design admits small changes in tooth numbers that only slightly change the natural frequencies and mesh frequency. Figure 5.3 shows the instability boundaries for a similar planetary gear as in Table 5.1 with the difference that the planet meshes are sequentially phased such that γ sn = 0, 14 , 12 , 43 . This phasing does not alter the natural frequencies or mode types in Table 5.1. The analytical and numerical instability boundaries match well. The spurious numerical instabilities just above Ω =4 are from higher order ( L > 2 ) instabilities for 134 which the analytical solutions are not shown in Figure 5.3. According to the gear phases, the tooth numbers of the sun and ring satisfy = zs jN + 1 , = zr kN + 3 , (5.76) where j and k are arbitrary integers. Invoking the instability existence rule for sequentially phased meshes, all fundamental ( L = 1 ) instabilities vanish except for a rotational mode interacting with a pair of translational modes (such as ω5 + ω7 in Figure 5.3) or a type 2 planet mode interacting with a pair of translational modes (such as ω4 + ω5 in Figure 5.3). All secondary instabilities ( L = 2 ) vanish except for a rotational mode interacting with a type 2 planet mode (such as (ω3 + ω9 ) / 2 in Figure 5.3) or two translational modes (such as ω5 in Figure 5.3). 135 Figure 5.3. Instability regions for a planetary gear with sequentially phased meshes as γ sn = 0, 14 , 12 , 43 , γ sr = 12 , cs = 1.4 , cr = 1.6 , ε = µ , and other parameters in Table 5.1. ──, analytical solution; ***, numerical solution. 136 Chapter 6: Nonlinear Dynamics of Planetary Gears Having an Elastic Ring Equation Chapter 6 Section 1 Nonlinear dynamics of planetary gears having an elastic ring gear excited by the mesh stiffness variations is studied. The phases caused by mesh stiffness, mesh deflection, and gear teeth number (in-phase or sequentially phased) are considered. For a planetary gear having N planets, the tooth separations of the sun-plant and ring-planet meshes may or may not occur simultaneously. With the assumptions of small mesh stiffness variations, small tooth separations, and one mode dominant response, closed-form expressions for the frequency response functions of primary, subharmonic, second harmonic excited, and superharmonic resonances are obtained for equally spaced planets with either in-phase or sequentially phased planets. The tooth separation phase between meshes depends on the mode type excited and the position of the planets. The tooth separation angle is independent of the torque, and the amplitude of response is proportional to the torque. The effect of mesh stiffness truncation is discussed. For in-phase planets the truncation to the first harmonic of the mesh stiffness variation is inaccurate for the response associated with the translational and planet modes. Due to the assumption of one mode domination response, the analytical solutions only work for distinct modes. The resonances 137 associated with a translational or planet mode are investigated. Parametric instability rule and mesh deflection phase are confirmed in the numerical simulation for both in-phase and sequentially phased planets. Response of planetary gears having commensurate natural frequencies is also studied. 6.1. Introduction Due to the compactness, high torque to weight ratio and high transmission ratio, planetary gears are widely used. In order to reduce the cost, maximize the power density and improve load sharing among the planets, gears are designed with thin rims. The thin rims also lead to elastic deflections of the gear bodies, especially the ring gear. Thin rings with large radius deform elastically under the forces experienced in applications like helicopters, wind turbines, and cars. Wu and Parker [3] presented the closed-form expressions for the structure of all four types of modes by establishing an elastic-discrete model that includes the translational and rotational vibration of all components and the elastic deformation of the ring gear. Gear mesh stiffness variations, as a result of changes for the number of teeth in contact, are the primary source of structure borne noise. August and Kasuba [73] and Velex and Flamand [36] studied the impact of mesh stiffness variation on dynamic response, tooth loads and load sharing among planets. They found the impact of mesh stiffness variations on dynamic response is significant for a planetary gear with three sequentially phased meshes. Lin and Parker [24] analyzed the parametric instability of planetary gears by using a purely rotational model. Wu and Parker [49] studied the 138 parametric instability for planetary gears having an elastic ring under periodic mesh stiffness variation. With the modal expressions of the elastic-discrete model from [3], general instability existence rules were presented for both in-phase and sequentially phased meshes, and the instability boundaries are obtained as simple expressions. The parametric instabilities can be neutralized (or suppressed) by adjusting the tooth numbers, contact ratios, and mesh phases. For planetary gears having N planets under the condition of parametric instability, 2N tooth separation may occur due to large gear deflections. Linear models can only identify the instability of the system but fail to predict the response. The nonlinear gear dynamics, especially single-mesh gears, received intensive attentions [20, 38, 39, 41, 74]. As a result of softening nonlinearity, phenomena such as jumps, period-doubling, bifurcation, and chaos can be observed. Kahraman and Singh [74] studied the interactions between time-varying mesh stiffness and contact loss nonlinearity of single-mesh gears. Vijayakar and Parker [38] analyzed the dynamic response of a spur gear pair by using a finite element/contact mechanics model. Theodossiades and Natisiavas [39] investigated periodic response of a gear pair by using a piecewise linear technique and perturbation analysis. Al-shyyab and Kahraman [40] adopted harmonic balance and continuation methods to investigate subharmonic and chaotic motions of a multi-mesh gear train of two gear pairs. Liu and Parker [41] studied the nonlinear dynamics of idler gear systems by perturbation, harmonic balance/arclength continuation and numerical integration methods. Ambarisha and Parker [20] calculated the nonlinear response of planetary gears by using both lumped-parameter and finite element models. Mesh phasing rule to suppress rotational and translational vibrations in planetary gears were provided, but it is 139 invalid for chaotic and period-doubling regions. Bahk and Parker [42] studied the nonlinear dynamics of planetary gears with a purely rotational model. Perturbation, harmonic balance/arclength continuation and numerical integration methods were used. Frequency-response functions were presented in closed-form expressions. Mesh phase, the phase of the mesh force, previously was used to describe the phase caused by mesh stiffnesses [43, 44, 71, 75]. In this work, the phase difference caused by mesh stiffnesses is called mesh stiffness phase, and the phase difference caused by mesh deflection is called mesh deflection phase. Mesh phase is the sum of mesh stiffness phase and mesh deflection phase. Contradictory mode suppression rules have been found for planetary gears having time-varying mesh stiffnesses. Parker [43] provided the physical explanation for planet phasing to suppress planetary gear. Ambarisha and Parker [71] found similar mode suppression conclusions in the study of the nonlinear vibration of planetary gears. Wu and Parker found [49] the combination existence rule for the parametric instabilities of planetary gears having an elastic ring gear, where the phases caused by the mesh stiffnesses and mesh deflections are both considered. Due to without consideration of the mesh deflection phase (or without having the assumption that the magnitudes of the mesh forces are independent of the planet positioning ψ n ), method of mode suppression in [43] and [71] actually are ways of response reductions. This work confirms the combination existence rule in [49] .The disappearance of some nonrotational modes in the response is because the damping effect is more significant than the effect of mesh stiffness variations. This work examines nonlinear dynamics of planetary gears with an elastic ring gear. Both planetary gear discrete degrees of freedom (rotational and translational) and ring 140 gear elastic deflection are considered. Closed-form expressions of frequency-response functions are obtained through the method of multiple scales. This work studies effects of mesh stiffness truncation, damping, the phases caused by mesh stiffness, mesh deflection, and number of gear teeth on the nonlinear response of planetary gears. 6.2. Mathematical Formulation and Assumptions The elastic-discrete model of planetary gears was built in [3]. The sun, carrier and planets are treated as rigid bodies, and the ring gear is modeled as a thin elastic body. The same model is adopted here with dimensional variables (see Appendix B for definitions of dimensionless operators). The planetary gear is axisymmetric because: All planets are identical and equally spaced, all planet bearing stiffnesses are equal, and the bearings and supports of the ring gear are axisymmetric and represented by an elastic foundation with distributed tangential and radial stiffnesses. The extended variable a is aT = [v, qT ] , q = [ xr , yr , ur , xc , yc , uc , xs , ys , us , ξ1 ,η1 , u1 , , ξ N ,η N , u N ]T , pr pc ps p1 (6.1) pN where q is the discrete body deflections, and v is the elastic tangential deformation of the ring. The combination of v and p r (the rigid motion of the ring) is the total deflection of the ring gear [3]. N is the number of planets; x j , y j , j = r , c, s are the translations of the ring rigid motion, carrier and sun; ξ n ,ηn , n = 1, N denote the radial and tangential translations of the planets; and u j = rjθ j , where j = r , c, s , 1, , N , are rotational deflections (rotation in radians times the gear base radii rr , rs , rp or radius of the carrier 141 rc ). 6.2.1. Mesh Stiffness Representation In [3], all the mesh stiffnesses are time independent, while in this work, the mesh stiffnesses not only are time-varying but also nonlinear. Tooth separation may occur due to the large gear deflections if the system is in the condition of parametric instability. The sun-planet and ring-planet mesh stiffnesses with consideration of tooth separations are represented as K sn (δ sn , t ) = k sn (t ) H (δ sn ) , K rn (δ rn , t ) = krn (t ) H (δ rn ) , n = 1, , N (6.2) (t ) k sp (1 + k2 n (t )) , n = 1, , N ksn= (t ) k sp [1 + k1n (t ) ] , k rn= (6.3) δ sn = − xs sinψ sn + ys cosψ sn + us − ξ n sin α s − ηn cos α r + un , ∂v = − xr sinψ rn + yr cosψ rn + ur + ξ n sin α r − η n cos α r − un , δ rn v cos α r + sin α r ∂ θ θ =ψ n (6.4) (6.5) where the Heaviside step functions H (δ sn ), H (δ rn ) represent the contact losses of sunplanet and ring-planet meshes; δ sn and δ rn are compressive relative gear mesh deflections; ksn (t ) , krn (t ) are the time-varying sun-planet and ring-planet mesh (n − 1)2π / N ; k sp stiffnesses for the nth planet; ψ n is the position of the nth planet ψ = n and k rp are mean values; k1n (t ) , k2 n (t ) are the zero-mean mesh stiffness variations between s and s + 1 pairs of teeth in contact for the sun-planet and ring-planet meshes. Equations (6.4)-(6.5) omit n = 1, , N . In what follows, any equation having index n and N without the summation ∑ indicates N equations, where n = 1, , N is omitted in the n=1 142 equation. The mesh stiffnesses calculated from the finite element method is close to a trapezoidal wave [41]. Thus, trapezoidal waves [49] are used to approximate the mesh stiffness variations k1n (t ) , k2 n (t ) (see Figure 6.1). The phases of the mesh stiffness are γ sn and γ sn + γ sr , where γ sn is phase of the nth sun-planet mesh stiffness, and γ sr is the mesh stiffness phase between the sun-planet and ring-planet mesh stiffnesses for a given planet. γ sn and γ sr are mesh stiffness phases instead of mesh phases. The non-parallel sides of the trapezoid have the same slope coefficients ρ j , j = s, r . The slope coefficients ρ j are in the range of [0 14 ] , where special cases ρ j = 0, 14 correspond to rectangular and triangle waves. Complex Fourier expansion of the trapezoidal mesh stiffness variations leads to = k1n (t ) k2 n (t ) = ∞ ∑C L =1 ∞ ( L ) iLΩt sn ∑C L =1 e + cc , ( L ) iLΩt rn e + cc , Csn( L ) = µ sin(2 Lπρ s ) sin Lπ cs e − iLπ ( c + 2γ ) , 2 2 ρsL π Crn( L ) = ε sin(2 Lπρ r ) sin Lπ cr e − iLπ ( c + 2γ ρ r L2π 2 s r (6.6) sn sn + 2 γ sr ) , (6.7) where Ω is the mesh frequency, 2 µ , 2ε are peak-to-peak values, cs , cr are contact ratios of the sun-planet and ring-planet meshes, and cc refers to the complex conjugate of all the preceding terms. The peak-to-peak values 2 µ , 2ε are assumed small when compared to the mean values of mesh stiffnesses. We assume µ / k sp is of the same order as ε , i.e., µ / k sp = gε , where g is a coefficient of order 1. 143 (a) (b) Figure 6.1. Mesh stiffness variations for the nth (a) sun-planet and (b) ring-planet meshes. cs , cr are contact ratios, and γ sn , γ rn , γ sr are mesh phases. 6.2.2. Mesh Deflection Representation and Mesh Deflection Phase In [49], the mesh deflections of the ring-planet and sun-planet meshes are expressed in a general compact form as δ [jnp ] =+ (a[j p ] ib[j p ] )e iTpψ n + cc , j = s, r , (6.8) where a[j p ] , b[j p ] are real coefficients independent of ψ n , and Tp is called mesh deflection type [49]. For rotational, translational, and planet modes, Tp is 0, 1, and d, respectively, where d is an integer selected from {2, , int ( N2 )} . The values of a[j p ] , b[j p ] can be obtained from [49]. Representing (5.33) in the form of cosine function gives 144 2 (a[j p ] ) 2 + (b[j p ] ) 2 cos(Tpψ n + φ [j p ] ) , j = s, r , δ [jnp ] = φ [ p] j arctan(b[j p ] / a[j p ] ) = [ p] [ p] arctan(b j / a j ) + π when a[j p ] ≥ 0 when a[j p ] < 0 (6.9) (6.10) Tpψ n + φ [j p ] is the phase caused by the mesh deflections. It is called mesh deflection phase, and we call the phase between the first and nth sun-planet (ring-planet) meshes caused by the mesh deflection as mesh deflection phase difference α n[ p ] = Tpψ n , (ψ 1 =0 ). Similar to the phase γ sr representing between the sun-planet and ring-planet mesh stiffnesses, [ p] α= φs[ p ] − φr[ p ] denotes the phase between the sun-planet and ring-planet mesh sr deflections for mode p. As the mesh force is the product of the mesh stiffness and mesh deflection, mesh phase is the sum of mesh stiffness phase and mesh deflection phase, where mesh stiffness phase is the phase caused by the mesh stiffness and planetary gear configruation, and mesh deflection is the phase caused by the mesh deflection only. Previous studies [44, 71] on mesh phase of planetary gear only focused on the mesh stiffness phase, results only focus on the mesh stiffness phase is not accurate if the mesh deflection phase of the response is nonzero. Details of mesh deflection phase of the response is discussed later. 6.2.3. Tooth Separation Representation When the response is periodic but not necessary be one mode dominant, the tooth separation period is assumed small (order of ε ) as compared to the response period 145 2π ω , where ω is the response frequency. The tooth separation functions H (δ jn ) are expanded in Fourier series as ∞ (m) 1 + ε Hˆ (δ jn ) = 1 + ε h (0) H (δ jn ) = jn + ε ∑ [ h jn (θ jn ) e im ( ωt − χ +ϕ jn ) + cc ] , j = s, r , (6.11) m =1 where χ is the response phase angle, ϕ jn is called the phase of tooth separation, which is phase between the response and the tooth separation of the nth sun-planet or ring planet mesh, and θ jn is the tooth separation angle to be solved later. The term e im ( ωt − χ +ϕ jn ) in the expansion (6.11) is different for different meshes. This causes difficulty in simplifying the expressions of the subsequent study. Use of the representation imϕ h (jnm ) (θ jn ) = h (jnm ) (θ jn )e jn in (6.11) yields the uniform expression for the tooth separation functions, where the effect of phase difference is considered in the coefficient h (jnm ) (θ jn ) ∞ (m) im ( ωt − χ ) H (δ jn ) = 1 + ε h (0) + cc ], j = s, r , jn + ε ∑ [ h jn (θ jn ) e (6.12) m =1 The Fourier coefficient h (jnm ) (θ jn ) is a function of the tooth separation angle θ jn . One can find the expression of the Fourier coefficient once the tooth separation angle is determinant. Substitution of (6.2)-(6.3), (6.6)-(6.7), and (6.12) into (6.2) yields Fourier expansion forms of mesh stiffnesses with consideration of the tooth separations ∞ ∞ K sn (δ sn , t ) = k sp + ε k sp ∑ Csn( L ) eiLΩt + ∑ hsn( m ) eim (ωt − χ ) + cc + hsn(0) + O (ε 2 ) , = L 1 =m 1 ∞ ∞ K rn (δ rn , t ) = krp + ε krp ∑ Crn( L ) eiLΩt + ∑ hrn( m ) eim (ωt − χ ) + cc + hrn(0) + O(ε 2 ) . = L 1 =m 1 146 (6.13) (6.14) 6.2.4. Equation of Motion The equation of motion for time varying system with steady state external force T is M a + K (δ , t )a = T, ∞ (6.15) ∞ K (δ , t ) = K 0 + ε ∑ K v( L ) eiLΩt +ε ∑ K h( m ) eim (ωt − χ ) + cc + ε K h(0) , (6.16) = L 1= m 1 where M, K (δ , t ) are self-adjoint extended inertia and stiffness operators with the inner product <= Y1 , Y2 > ∫ 2π 0 v1v2 dθ + (q1 )T q 2 , where the overbar denotes complex conjugation. M and K (δ , t ) are defined in Appendix B. The external force is the torques added on the rotations of the sun, carrier or ring, which means T only has nonzero elements in the rotations of the sun, carrier or ring. In the examples of this work, the input is added on the sun, the output is added on the ring, and the rotation of the carrier is fixed. The general solution for the time invariant system of (6.15) is = a0 ∞ ∑ b (τ )Y e q =1 q fq = i ( ωq t + β q (τ )) q < Yq , T > ωq2 , + cc + f q Yq , (6.17) (6.18) where bq (τ ) , β q (τ ) are the leading order response amplitude and phase, bq (τ ) , β q (τ ) , and f q are real, Yq is the eigenfunction in the extended variable form, and ωq is the natural frequency of the time invariant system. For equally spaced planets, four types of modes are found for planetary gears having an elastic ring [3]. According to the modal expression of each type, T is orthogonal to all the modes except rotational modes, i.e., 147 < Yq , T >= 0 for all the translational, planet, and purely ring modes. This indicates the modal force element f q ≠ 0 only for rotational modes. 6.2.5. One Mode Dominant Response Large response is excited due to the parametric excitation from the mesh stiffness variations. The large deflections of gear meshes may trigger gear tooth contact losses, and the nonlinearities of the tooth separations and the damping in return bound the amplitudes of the parametric instabilities. As one mode dominates the response, the response is nearly harmonic [20, 37, 39, 76]. The assumption of one mode dominant response works only for distinct natural frequency. One mode dominant response indicates that for each gear mesh only one contact loss at most in a period. The infinite eigenfunctions in (6.17) are usually truncated to a finite integer M with a small truncation error (M should include all degrees of freedom for the rigid bodies and large enough Fourier expansion terms of the elastic ring). For one mode dominant response, the general solution for the linear system in (6.17) becomes = a0 bp (τ ) Yp ( e i [ω p t + β p (τ )] M + cc ) + ∑ f m Ym . (6.19) m =1 The starting point of the contact loss is the moment of the mesh deflection being zero. Substituting the leading order solution (6.19) into the expressions of tooth deflections (6.4)-(6.5) and letting them being zero yield the expression for the critical point of tooth separation bp (τ )δ [jnp ] e i [ω p t + β p (τ )] M + cc + ∑ f mδ [jnm ] = 0 , j = r, s . m =1 148 (6.20) M ∑f m =1 δ [jnm ] in (6.20) is identical for different n because f m is nonzero only for rotational m modes and for rotational modes δ [jnm ] = δ [j1m ] . Substitution of (5.33) into (6.20) yields M 2 ( a [j p ] ) 2 + (b[j p ] ) 2 cos ω p t + β p (τ ) + φ [j p ] + Tpψ n + ∑ f mδ [j1m ] / bp (τ ) = 0 , j = r, s , (6.21) m =1 The relationship of tooth separation and response is shown in Figure 6.2. According to (6.21), When 1 2 M ∑f m =1 δ [j1m ] / bp (τ ) ≥ (a[j p ] ) 2 + (b[j p ] ) 2 there is no tooth m separation, which means the mesh deflections caused by the steady torque are more significant than the mesh deflections caused by mesh stiffness variations; Otherwise, tooth separations occur. The separation angle is found as θ [jnp ] M π − arccos − ∑ f mδ [j1m ] / (2bp ) / ( a [j p ] ) 2 + (b[j p ] ) 2 , = 2 m =1 j = r, s . (6.22) Because the right hand side of (6.22) is independent of n, the tooth separation angles for all the sun-planet (ring-planet) meshes are the same θ [jnp ] = θ [j1p ] . The response frequency and phase given in (6.11) and the leading order response in (6.17) indicates ωt − χ = ω p t + β p . Comparison of (6.20) with (6.11) yields the phase of tooth separation is ϕ= φ [j p ] + Tpψ n . jn 149 (6.23) [ p] Figure 6.2. Tooth separation function and mesh deflection δ jn . The phase between tooth separations for different sun-planet (ring-planet) meshes is called tooth separation phase difference. According to (6.23), tooth separation phase difference is the mesh deflection phase difference Tpψ n when the response is one mode dominant. This indicates that for response of rotational modes, the tooth separations occur simultaneously for all the sun-planet (or ring-planet) meshes; For response of translational modes, the tooth separations occur in the sequence of ψ n ; and for response of type s planet modes, the tooth separations occurs in the sequences of sψ n . Besides the phase between different sun-planet (or ring planet) meshes, there is still a phase 150 φs[ p ] − φr[ p ] between the contact losses of the nth sun-planet and ring planet meshes. For rotational modes [49], φs[ p ] − φr[ p ] is either 0 or π because b[j p ] in (6.8) is zero. As can be seen in (6.21), the tooth separation functions are even functions. Thus, Fourier expansion coefficients for the tooth separation governed by (6.11) are real θ j1 sin m(π − θ j1 / 2) , h (j1m ) = , h (0) j1 = − mπε 2πε (6.24) Accordingly, the Fourier coefficients in (6.12) are (0) h (0) jn = h j1 = − θ j1 sin m(π − θ j1 / 2) im[T ψ , h (jnm ) = e mπε 2πε [ p] p n +φ j ] . (6.25) 6.2.6. Tooth Separation of Degenerate Modes The natural frequencies of time-invariant system are either distinct or degenerate with multiplicity two [3]. For degenerate natural frequency with multiplicity two, the response is no longer one mode dominant, both the degenerate modes will be excited. The general form of the leading order response for degenerate modes p, l, is a0 bp (τ ) Yp e = Substituting M (6.26) into the i (ω pt + β p ) + bl (τ ) Yl e expressions i (ω pt + βl ) M + cc + ∑ f m Ym . (6.26) m =1 of tooth deflections, considering M ∑ f mδ [jnm ] = ∑ f mδ [j1m ] , and letting them being zero yield = m 1= m 1 2bpδ M [ p] jn cos(ω p t + β p ) + 2blδ cos(ω p t + β l ) + ∑ f mδ [j1m ] = 0 , j = s, r . = a0 bp (τ ) Yp e [l ] jn i (ω pt + β p ) (6.27) m =1 + bl (τ ) Yl e 151 i (ω pt + βl ) M + cc + ∑ f m Ym . m =1 (6.28) Substitution of (6.28) into (6.4) and (6.5) yields 2N equations M 2bpδ [jnp ] cos(ω p t + β p ) + 2blδ [jnl ] cos(ω p t + β p + φ p ) + ∑ f mδ [jnm ] = 0 , , j = s, r (6.29) m =1 where φ= βl − β p is the phase between the degenerate modes. p M 2 B jn cos(ω p t + β p + ϕ jn ) + ∑ f mδ [jnm ] = 0 , n = 1, , N , (6.30) m =1 B jn = (bpδ [jnp ] + blδ [jnl ] cos φ p ) 2 + (blδ [jnl ] sin φ p ) 2 , where B jn is functions of bp , bl , φ p but independent of time t and ψ n . Equation (6.30) indicates the phase between the contact losses of the first and nth sun-planet (ring-planet) meshes is still Tpψ n . When bpδ [jnp ] + blδ [jnl ] cos φ p > 0 ϕ jn = arctg −blδ [jnl ] sin φ p b pδ [ p] jn + blδ cos φ p [l ] jn , ϕ jn ∈ (− π2 , π2 ) . (6.31) , ϕ jn ∈ ( π2 , 32π ) . (6.32) When bpδ [jnp ] + blδ [jnl ] cos φ p < 0 ϕ jn= π + arctg If −blδ [jnl ] sin φ p bpδ [jnp ] + blδ [jnl ] cos φ p 1 M f mδ [jnm ] > B jn , there is no tooth separation, otherwise ∑ 2 m =1 M ∑f ,l ] θ [jnp= 2π − 2 arccos m =1 δ [jnm ] m 2 B jn . (6.33) One can also show that the right hand side of (6.33) is independent of n, which indicates the tooth separation angles for all the sun-planet (ring-planet) meshes are the same θ [jnp ,l ] = θ [j1p ,l ] . 152 6.3. Method of Multiple Scales The method of multiple scales is used with the introduction of the slow time τ = ε t , and the eigenfunctions are represented in asymptotic expansion v v v a =a0 + ε a1 + ε a2 + O (ε 3 ) , a0 = 0 , a1 = 1 , a2 = 2 . q1 q 2 q 0 (6.34) Substitution of (3.25) into (6.15), consideration of the small modal damping (order of ε ), use of d ∂ ∂ ∂ → +ε + ε 2 , and separation of powers in ε lead to dt ∂t ∂τ ∂ι ∂ 2a 0 M 2 + K 0a 0 = T, ∂t M (6.35) ∂ 2a 0 C ∂a 0 ∞ ( L ) iLΩt ∞ ( m ) im (ωt − χ ) ∂ 2a1 K M 2 a + = − − − ∑ Kv e +∑ K h e + cc + K h(0) a 0 , 0 1 2 ε ∂t L 1 =m 1 ∂t ∂t ∂τ = (6.36) M ∂ 2a 2 ∂ 2a1 C ∂a1 C ∂a0 ∞ ( L ) iLΩt ∞ ( m ) im (ωt − χ ) a + = − − − − ∑ Kv e +∑ Kh e + cc + K h(0) a1 K M 2 0 2 2 ∂t ∂t∂τ ε ∂t = ε ∂τ L 1 =m 1 −M ∂ a0 ∂ a0 − 2M 2 ∂τ ∂t∂ι 2 . (6.37) 2 where C is the modal damping operator. The operator C has the following property < Yp , 0 Yq >= ε 2ζ pω p C q≠ p q= p (6.38) , q 1, ∞} where ζ p is the damping ratio. The completeness of the eigenfunctions {Yq= allows the general solution of (5.10) be written as ∞ a1 = ∑ cl (t ,τ ) Y j , l =1 153 (6.39) where cl (t ,τ ) is real. a1 usually is also truncated to M terms as a 0 . For linear time-invariant system without the damping and torque, (5.10) reduces to ∂ 2a 0 ∞ ( L ) iLΩt ∂ 2a1 M 2 + K 0a1 = −2 M − ∑ K v e a0 . ∂t ∂t ∂τ L =1 (6.40) The parametric instability problem described by (6.40) was solved by Wu and Parker [49]. When the mesh frequency satisfies LΩ ≈ ω p + ωq combination instabilities may exist. Parametric combination instability rules for both in-phase and sequentially phased planets were given. For in-phase planets, parametric instabilities only exit for the two modes of the same type [49]. For in-phase planets, the fundamental ( L = 1, Ω ≈ 2ω p ) and secondary ( L= 2, Ω ≈ ω p ) instabilities exist for rotational, translational, and planet modes. Substitution of (6.17) and (6.39) into (5.10), and the inner product of Y j with (5.10) lead to cl (t ,τ ) + ωl2cl (t ,τ ) = −i 2ωl ei (ωl t + βl ) (bl′ + ibl β ′) − i 2ζ lωl2bl ei (ωl t + βl ) ∞ M i ( LΩt +ωq t + β q ) i ( LΩt −ωq t − β q ) − ∑∑ Dlq( L )bq e +e L 1= q 1 = ∞ M i ( mωt − m χ +ωq t + β q ) i ( mωt − m χ −ωq t − β q ) i ( ωq t + β q ) − E (0) − ∑∑ Elq( m )bq e +e jq bq e (6.41) m 1= q 1 = ∞ ∞ M M M − ∑∑ Dlq( L ) f q eiLΩt − ∑∑ Elq( m ) f q eim (ωt − χ ) + cc − ∑ Elq(0) f q . L 1= q 1 = m 1= q 1 = N q 1 = N Dlq( L ) =< Yl , K v( L ) Yq >= k sp ∑ Csn( L )δ sn[ l ]δ sn[ q ] + k rp ∑ Crn( L )δ rn[ l ]δ rn[ q ] , (6.42) = n 1= n 1 N N Elq( L ) =< Yl , K h( L ) Yq >= k sp ∑ hsn( L ) (θ sn )δ sn[ l ]δ sn[ q ] + k rp ∑ hrn( L ) (θ sn )δ rn[ l ]δ rn[ q ] . = n 1= n 1 154 (6.43) Terms associated with Elq( L ) in (6.41) incorporate effects of tooth separations and terms associated with Dlq( L ) incorporate effects of mesh stiffness variations. Generally, Dlq and Elq are complex. hsn( L ) (θ sn ) and hrn( L ) (θ rn ) in (6.43) are Fourier coefficients of tooth separation given in (6.25). Mesh stiffness variations are the source of the parametric instabilities and contact losses. As the mesh frequency Ω is close to the harmonics of a natural frequency, several types of resonances are found. 6.3.1. Primary Resonance Ω= ω p + εσ When the mesh frequency Ω is close to a natural frequency ω p as Ω= ω p + εσ , it causes primary resonance at the mesh frequency, where σ is the detuning. Substitution of these into (6.41) yields cl (t ,τ ) + ωl2cl (t ,τ ) = −i 2ωl ei (ωl t + βl ) (bl′ + ibl β l′) − i 2ζ lωl2bl ei (ωl t + βl ) ∞ M i ( Lω t + Lστ +ωq t + β q ) i ( Lω t + Lστ −ωq t − β q ) − ∑∑ Dlq( L )bq e p +e p L 1= q 1 = ∞ M M i ( mωt − m χ +ωq t + β q ) i ( mωt − m χ −ωq t − β q ) − ∑ Elq(0)bq ei (ωqt + βq ) − ∑∑ Elq( m )bq e +e q =1 m 1= q 1 = ∞ M − ∑∑ Dlq( L ) f q e = L 1= q 1 (6.44) i ( Lω p t + Lστ ) ∞ M M − ∑∑ Elq( m ) f q ei ( mωt −mχ ) + cc − ∑ Elq(0) f q = m 1= q 1 = q 1 For any natural frequencies ω p , ωq and ωl , generally, they are incommensurate as mω p ≠ ωl or not related as mω p ± ωq ≠ ωl . where m is a integer. When l ≠ p , a stable solution of (6.44) requires the secular terms of (6.44) being zero 0, −i 2ωl (bl′ + ibl β l′)eiβl − i 2ζ lωl2bl eiβl = l≠ p. 155 (6.45) The real and imaginary parts of (6.45) being zero gives 0 , 2bl β ′ωl eiβl = 0 . −iωl eiβl 2(bl′ + ζ lωl bl ) = (6.46) For steady state response, bl′ = 0 and βl′ = 0 . The first of (6.46) indicates bl = 0 for l ≠ p . Thus, when the response natural frequency is close to a natural frequency ω p , and the natural frequencies do not satisfy mω p ± ωq ≈ ωl , mω p ≈ ωl , the response of all the other natural frequencies (not the response near ω p ) vanishes. Response around the natural frequency ω p is found by substitution of l = p into (6.44). For one mode dominant response, the response frequency and phase given in (6.11) and the leading order response in (6.17) imply ωt − χ = ω p t + β p . With substitution = ω p + εσ into (6.44), the secular terms in (6.44) being zero gives of ω iβ i 2ω p e p (b′p + ibp β p′ ) + i 2ζ pω p2bp e (2) bp e + E pp i (2στ − 2 χ − β p ) iβ p (2) bp e + D pp M i (2στ − β p ) (0) bp e + E pp iβ p (6.47) M (1) iστ (1) i (στ − χ ) 0. e f q + ∑ E pq e fq = + ∑ D pq q 1= q 1 = The response phase χ and the leading order response phase − β p are related as = χ στ − β p . Multiplying (6.47) by e − iβ p (6.48) , substituting (6.48) into (6.47) yield M M (2) (0) (1) (2) (1) i 2ω p b′p + ibp (σ − γ ′) + i 2ζ pω p2bp + D pp bp ei 2 χ + E pp bp + ∑ E pq f q + E pp bp + ei χ ∑ D pq fq = 0. = q 1= q 1 (6.49) (2) The term associated with the second harmonic of the mesh stiffness variations D pp b p ei 2 χ is usually ignored [41, 42] in the following mathematic manipulations for the purpose of the simplification. Sometimes this term, however, is significant for the nonlinear response 156 because it is the parametric excitation of the secondary instability ( L= 2, Ω ≈ ω p ) for the linear system [49]. For steady state response ( b′p = 0 , χ ′ = 0 ) and without consideration the second harmonic of the mesh stiffness variations , (6.49) is reduced to M M (0) (1) (2) (1) −2ω p bpσ + i 2ζ pω p2bp + E pp bp + ∑ E pq f q + E pp bp + ei χ ∑ D pq fq = 0, (6.50) = q 1= q 1 (0) (1) (2) , E pq , and E pp are Before solving the problem in (6.50), one needs to know whether E pp real or complex. Substitution of (6.25) into (6.43) yields ( L) E pq ) iLφs k sp g s(1L ) (θ s1he [ p] N ∑e iLT pψ n δ sn[ p ]δ sn[ q ] + k rp hrn( L ) (θ r1 )eiLφ [ p] r N ∑e iLT pψ n = n 1= n 1 Every term in (6.51) is real except eiLφs [ p] and e iLT pψ n δ rn[ p ]δ rn[ q ] (6.51) . For rotational modes, b[j p ] = 0 according to the mesh deflection of given in [49]. This indicates φ [j p ] in (6.51) is either 0 ( L) or π . Thus, E pq is real because e iLφ [j p ] is either 1 or −1 and e iLT pψ n = 1. (0) (1) (2) For purely ring modes, E pp , E pq , and E pp vanish because the mesh deflections are zero. This prevents any effects from the mesh stiffness variations. Thus, no parametric instability or response will be found for purely ring modes. For distinct planet modes, b[j p ] = 0 according to the mesh deflection of given in [49]. This indicates e iLφ [j p ] (0) (2) is either 1 or −1 . When L = 0, 2 , eiLN /2ψ n = 1 , so E pq and E pq are (1) real. When L = 1 , eiLN /2ψ n = (−1)( n −1) , so E pq is real. 157 (0) (1) (2) (1) Therefore, E pp , E pq , and E pp are real for all distinct modes. D pq is the only M (1) complex variable in (6.50). With movement of eiγ ∑ D pq f q to the right hand side of the q =1 equation, the norm of (6.50) indicates M M (0) (1) (2) 2 2 4 2 p p pp p pq q pp p p p p = q 1= q 1 2 ( −2ω b σ + E b + ∑ E f + E b ) + 4ζ ω b = ∑D f (1) pq q . (6.52) Mathematic manipulation of (6.52) yields 2 M 1 (0) M (1) (2) (1) 2 4 E pp + ∑ E pq f q / bp + E pp ± ∑ D pq f q / bp − 4ζ pω p . σ = 2ω p = q 1= q 1 (6.53) Equation (6.53) is the frequency response function with considering only the first harmonic of the time-varying mesh stiffnesses. By this truncation the second harmonics of the mesh stiffness variations are lost. The effect of the truncation is unacceptable for certain modes and it is discussed later. The frequency response function of the primary resonances in (6.53) explicitly shows the response amplitude bp and natural frequency (1) detuning σ is controlled by several parameters: D pq , f q , ν p , and tooth separation terms (1) (0) (2) ( E pp , E pq , E pp ). The peak of the response is bp ,max = M ∑D q =1 f / (2ζ pω p2 ) , (1) pq q (6.54) bp ,max in (6.54) is in terms of known system parameters, which is ready for solving. Once (0) (1) (2) bp is known, the sequence of finding the detuning is: bp → ϕ jn → E pp →σ . , E pq , E pp 158 6.3.2. Sub-harmonic Resonance = Ω 2ω p + 2εσ When the mesh frequency is close to period two of a natural frequency = Ω 2ω p + 2εσ , it causes subharmonic resonance ω = Ω / 2 . The primary source of the subharmonic resonances is the fundamental single mode instability [49] ( L = 1 , Ω ≈ 2ω p ). = ω p + εσ into (6.41) yield the solvability Ω 2ω p + 2εσ and ω Substitutions of = condition M (1) (0) (2) (1) i 2ω p (b′p + ibp β p′ ) + i 2ζ pω p2bp + D pp bp ei 2 χ + E pp bp + E pp bp + ∑ E pq fq = 0 (6.55) q =1 With similar mathematic manipulations in primary resonance, for steady state response, equation (6.55) is reduced as M (1) (0) (1) (2) 0. −2ω p bpσ + i 2ζ pω p2bp + D pp bp ei 2 χ + E pp bp + ∑ E pq f q + E pp bp = (6.56) q =1 (0) (1) (2) As mentioned in the above section, E pp , E pq , and E pp are real for all distinct modes. Separation of the real and imaginary parts of (6.56) yields the frequency response function for the subharmonic resonance σ = 1 (0) M (1) (2) E pp + ∑ E pq f q / bp + E pp ± 2ω p q =1 (1) 2 D pp − 4ζ p2ω p4 . (6.57) 6.3.3. Second-harmonic Excitation Resonance= Ω ω p / 2 + εσ / 2 When the mesh frequency is near the half of certain natural frequency = Ω ω p / 2 + εσ / 2 , it causes second-harmonic excitation resonance and super-harmonic 159 resonance at ω = 2Ω . The frequency response function of the second harmonic resonance is studied as followings. For steady state response, b′p = 0 , χ ′ = 0 yields M M (4) (0) (1) (2) (2) (6.58) 0. bp ei 2 χ + E pp bp + ∑ E pq f q + E pp bp + ∑ D pq f q ei χ = −2ω p bpσ + i 2ζ pω p2bp + D pp = q 1= q 1 (4) Ignoring the effect of the fourth harmonic ( D pp bp ei 2γ ) of the mesh stiffness variation, M moving ∑D q =1 (2) iγ pq q f e to the right hand side, and taking the norm of (6.58) yield 2 M (2) (0) pq q p p pp p q 1= q 1 M ∑D f (1) (2) = f q + E pp bp ) 2 + 4ζ p2ω p4bp2 . ( −2ω b σ + E b + ∑ E pq (6.59) Similar procedures of solving (6.59) yield the frequency response function for the second harmonic excited resonance 2 M 1 (0) M (1) (2) (2) 2 4 σ E pp + ∑ E pq f q / bp + E pp ± ∑ D pq f q / bp − 4ζ pω p . = 2ω p = q 1= q 1 (6.60) (2) (1) Expression (6.60) is similar to (6.53) with the difference of D pq in (6.60) and D pq in (6.53). One can get similar properties and conclusions as the primary resonances. Note that the effect of tooth separations on the detuning is the same for (6.53), (6.57) and (6.60). 6.3.4. Super-harmonic Resonance= Ω ω p / 2 + εσ / 2 The super-harmonic resonance at ω = 2Ω is obtained through the second order Ω ω p / 2 + εσ / 2 with consideration of only one harmonic of the perturbation (6.37) for= 160 excitation. As the mesh frequency is away from any natural frequency (no harmonics of mesh frequency), the leading order solution only contains the static deflection M a0 = ∑ f m Ym . (6.61) m =1 The tooth separation does not occur for the leading order solution. Thus, K h( m ) = 0 in (5.10), and the particular solution of (5.10) a1* is found with substitution of (6.61) into (5.10) as = a1* M ∑ N m Ym (θ )eiΩt + cc , m =1 Nm = (1) Dmm fm . 2 ωm − Ω 2 (6.62) The solution of (5.10) is the combination of the general and particular solutions = a1 M ∑ b m =1 m (τ ) Ym ei (ωmt + βm ) + N m Ym eiΩt + cc . (6.63) Consideration of only the first harmonic of the mesh stiffness, substitution of (6.61) and (6.63) into (6.37), and the secular terms being zero give the solvability condition M (0) (2) (1) (6.64) −2ω p bpσ + i 2ζ pω p2bp + E pp bp + E pp bp + ∑ D pq N q ei χ = 0. q =1 The mathematic manipulation of (6.64) gives the frequency response function for the super-harmonic resonance 1 (0) (2) σ E pp + E pp = ± 2ω p 2 M ∑D q =1 (1) pq N q / bp − 4ζ ω . 2 p 4 p (6.65) 6.4. Results and Discussions This section discusses the results from perturbation analysis and numerical simulation. 161 6.4.1. Comparison of Numerical and Analytical Results As an example, a planetary gear with four equally spaced planets is studied. The system parameters are given in Table 6.1. The first 10 natural frequencies in Hz of the ω= 947.1 (T), ω4 = 1471.9 (P), time invariant system are ω1 = 875.6 (R), ω= 2 3 2007.0 (T), ω7 = 2290.8 (R), ω8 = 2875.6 (PR), ω9 = 3097.5 (R), and ω= ω= 5 6 ω10 = 3305.5 (P), where R, T, P, PR denote rotational, translational, planet and purely ring modes. All the examples in this paper use the trapezoid wave to represent the sun- ρ= 0.04 . Figure 6.3 planet and ring-planet mesh stiffnesses with coefficients ρ= s r shows both numerical and analytical Root Mean Square (RMS) of the ring-planet mesh deflection for the response. For numerical solution, two cases are considered: the full harmonics and only the first harmonic of the mesh stiffnesses. As shown in Figure 6.3, numerical and analytical solution with consideration of only the first harmonic of the mesh stiffnesses have a good comparison for both primary and subharmonic resonances of ω 7 . When one considers only the first harmonic of the mesh stiffnesses, only the responses of rotational modes are found. Response from the numerical simulation confirms the tooth separation occurs simultaneously for rotational modes. The responses of non-rotational modes are discussed later. 162 Figure 6.3. RMS comparison for ring-planet mesh deflection of a planetary gear defined in Table 6.1. Lines without marker denote analytical solution. Lines with marker * denote numerical solution for full representation of the mesh stiffnesses, and lines with marker ○ denote numerical solution for the first harmonic of the mesh stiffnesses. Four types of frequency response functions are given in (6.53), (6.57), (6.60), and (6.65). Comparison of (6.53) with (6.60) indicates the frequency response functions of primary and second harmonic excited resonances are similar. In what follows, we focus 163 on the discussion of primary resonance. One can easily obtain the similar behavior for the second harmonic excited resonance. Inertias (kg) 2 I r / rr2 = 3.8408 , I c / rc2 = 6.000 , I s / rs2 = 2.500 , I p / rp = 2.000 Masses (kg) mr = 2.85 , mc = 5.430 , ms = 0.400 , m p = 4.000 k= k= 100 × 106 rp sp , k= 100 × 103 rbs , k rus= 20 × 106 kbend= 20 × 106 , Stiffnesses (N/m) k p 500 × 109 = k s 500 × 109 , k= kc 100 × 109 , k= 100 × 1012 , = 500 × 106 , = cu su Pressure angle (deg) α= α= 24.60 r s Torque (N·m) Tr = 5 , Ts = 5 Table 6.1. Dimensional parameters for a planetary gear with four equally spaced planets. krbs and krus denote radial and tangential distributed stiffnesses. γ sr = 0 . 6.4.2. Effect of Torque (0) (1) (2) The effects of the contact loss are included in E pp , E pq , E pp . From (6.22), it is M obvious that the tooth separation angle depends on ∑f m =1 m / bpδ [jnm ] , where δ [jnm ] is constant. f m and bp always in the form of f m / bp in (6.22), (6.53), (6.57), (6.60) and (6.65). This indicates the response amplitude is proportional to the torque T as f m is the linear combination of T . It also implies the tooth separation angle depending on the ratio of f m / bp is independent of the torque T . This conclusion is based on the assumption of 164 changing of the torque does not affect the mesh stiffnesses or contact ratios. In fact, when the torque changes in a wide range, the variations of the contact ratios caused by large tooth deflections under high torque are too big to ignore (larger contact ratio indicates larger mesh stiffness in measurement). 6.4.3. Onset of Tooth Separation The tooth separation angle describes the length of tooth separation, and the onset of tooth separation describes the starting moment of tooth separation. When there is no tooth ( L) separation, E pq governed in (6.43) is always zero. Thus, in primary resonance, the critical point of natural frequency detuning for the starting of the tooth separation is 2 M 1 σc = ± ∑ D pq(1) f q − 4ζ p2ω p4bp2 2ω p bp q =1 (6.66) Obviously, one can find similar expression as (6.66) for the second harmonic excited resonance. ( L) 6.4.4. Properties of D pq fq For equally spaced in-phase planets, the coefficients of the mesh stiffness variations M can be removed out of the summation. Thus, ∑D q =1 ( L) pq q f is written as N N ( L) [ p] [q] ( L) [ p] [q] k g Cf δ δ k C f + ∑ rp rn q ∑ δ rn δ rn . sp sn q ∑ sn sn = = q 1= q 1 = n 1 n 1 M = ∑ Dpq( L) f q M 165 (6.67) ( L) If D pq f q need to be nonzero, mode q should be a rotational mode so that f q is nonzero. N According to the properties of mesh deflection in [49], ∑δ n =1 δ [ p] [q] sn sn ≡ 0 when the planets are in phase, and modes p, q are of different types (different Tp ). Thus, for translational, planet, and purely ring modes M ∑D q =1 M According to (6.52) and (6.59), ∑D q =1 f = 0. ( L) pq q (6.68) f = 0 indicates no primary and second harmonic ( L) pq q excited response at the first order perturbation. The mode suppression rules found in [43] M and [71] might based on ∑D q =1 f = 0 and claimed the translational and planet modes ( L) pq q M are suppressed for in-phased planets. Although ∑D q =1 M degenerate planet modes, one cannot use ∑D q =1 f = 0 holds for translational and ( L) pq q f = 0 to identify the response of ( L) pq q degenerate modes being zero due to the dissatisfaction of one mode dominant assumption. Actually, responses of degenerate modes are found in the numerical simulation even when the mesh stiffnesses only have the first harmonic of the Fourier expansion. Also, the truncation of the terms associated with the second harmonic of the mesh stiffness is M unacceptable when ∑D q =1 f = 0 . The mesh stiffness truncation is discussed in the next ( L) pq q section. 166 6.4.5. Effect of the Mesh Stiffness Truncation The truncations of the higher harmonics of the mesh stiffness variations are performed in the analysis of the primary and the second harmonic excited resonances. (2) bp e The elimination of the term D pp i (2στ − β p ) M in (6.47) is unacceptable when ∑D q =1 f = 0. (1) pq q According to (6.47) and the algebraic manipulation, frequency response function for the primary response caused by the second harmonic of the mesh stiffness variation is σ = 1 (0) M (1) (2) E pp + ∑ E pq f q / bp + E pp ± 2ω p q =1 (2) 2 D pp − 4ζ p2ω p4 . (6.69) Equation (6.69) fits for distinct planet modes. One can find similar expression for the second harmonic excited response. Response of distinct planet mode is excited when (2) D pp > 2ζ pω p2 . The secondary instabilities are observed for linear planetary gear systems without the (2) damping [24, 49]. D pp in (6.69) contains the effect of the second harmonics of the mesh stiffness variations. It is this term that causes the secondary instabilities. 6.4.6. Response of Translational and Planet modes Damping not only governs the amplitude of the response in the primary and second harmonic excited resonances, but also controls the existence of the response. The interaction between the mesh stiffness variations and damping is governed by the terms in the square root of equations (6.53), (6.57), (6.60), (6.65), and (6.69). If the damping ratio is so large that the terms in the square root of equations (6.53), (6.57), (6.60), (6.65), and (6.69) is less than zero, the response disappears (see Figure 6.3). 167 The responses of rotational modes usually are excited when the planets are in-phase. The frequency response functions for different kinds of resonance of rotational modes are given in (6.53), (6.57), (6.60), and (6.65). The conclusions given on mode suppression in [42, 43, 71] is inconsistent with results of this work and the combination existence rule obtained in the parametric instabilities of planetary gears [49]. Actually, the disappearance of mode response happens for distinct planet modes when the effect of mesh stiffness variation (terms associated with Dll(2) ) is smaller than the effect of damping. As all the analytical studies in this work requires the assumption of one mode dominant response, conclusions for the response of degenerate modes are based on the results of numerical simulation. 6.4.6.1. Response of Translational Modes As an example, the damping of the system in Table 6.1 is: ζ 2,3,5,6 = 1% and the damping ratio for the remaining degrees of freedom is 5%. Only the second harmonic of mesh stiffnesses is considered. In this example, all the planets are in-phase, and the first two harmonics of the mesh stiffness are considered. When the mesh frequency is close to the natural frequency of a translational mode, such = as Ω 947 Hz ≈ ω2,3 , the primary resonance of the translational modes is excited. The translation of the rigid ring motion is shown in Figure 6.4a, and the ring-planet mesh deflections are shown in Figure 6.4b with consideration of only the second harmonic of the mesh stiffnesses. From the modal expression of planetary gears having an elastic ring gear in [3], the translations of the ring only associate with the translational modes. Thus, once the response has ring translations, a translational mode must be excited. The mesh 168 deflections of the diametrically opposed pairs in Figure 6.4 satisfy δ r1 = −δ r 3 , δ r 2 = −δ r 4 , which confirms the mesh phase of translational modes is Tpψ n . Furthermore, the response magnitude for all the sun-planet or ring-planet mesh deflections are the same, and particularly they overlap as δ r1 = δ r 4 , δ r 2 = δ r 3 . The general expression of the mesh deflection is given in (5.33). When the degenerate modes are equally excited ( a[j p ] = b[j p ] ), the mesh deflections of the translational modes are δ [jnp ] π [ p] [ p] 2 2a j cos(ψ n + 4 ) for a j > 0 , j = s, r , = π p p [ ] [ ] −2 2a j cos(ψ n + ) for a j < 0 4 (6.70) From (6.70), it’s straightforward to find the mesh deflections as δ j1 = −δ j 3 , δ j 2 = −δ j 4 , and δ r1 = δ r 4 . This confirms the degenerate modes not only are excited but also dominate in the response. When the degenerate modes is equally excited, degenerate modes become a distinct mode in response calculation. The equally excited modes phenomena is a special case, in other cases for the response of translational modes, this may be different. 169 (a) (b) Continued Figure 6.4. Response of the translational modes ( Ω ≈ ω2,3 ) for a planetary gear defined in Table 6.1 with modal damping ratios as ζ 2,3,5,6 = 0.01 and the damping ratio for the remaining degrees of freedom is 0.05. (a) translations of the ring, (b) ring-planet mesh deflections, (c) sun-planet mesh deflections, (d) spectra of ring-planet mesh deflection. 170 Figure 6.4 continued (c) (d) 171 If the modal damping is increased to ζ 2,3,5,6 = 5% , the translational mode ω2,3 that was excited for ζ 2,3,5,6 = 1% disappear when the mesh frequency is close to ω2,3 . This is because the effect of damping is larger than the effect of the mesh stiffness variations. When the mesh frequency is closed to a natural frequency of the translational modes, but the translational mode is not excited, the response is the combination of rotational modes because the adjacent resonances are rotational modes. The subharmonic resonances of degenerate modes ω5,6 usually are excited even for a large damping. One may notice that the RMS response show in Figure 6.5 does not have the subharmonic resonance of ω5,6 . This is because the subharmonic resonance of ω5,6 depends on the initial condition, and the response in Figure 6.3 does not have the condition to trig the response of translational mode when the frequency sweep (speed down) starts far way from ω5,6 and uses the last time step solution of the previous frequency as the initial condition of the next frequency. For speed up frequency sweep, there is no jump up around 3692.4Hz. However, when the speed sweep starts from 3692.4Hz with certain initial conditions, the subharmonic resonance of the translational mode ω5,6 is excited (see lines with marker * in Figure 6.5 for the RMS response for the ring-planet mesh deflection). Similar situation was found for speed down case (not shown in the figure). 172 Figure 6.5. Subharmonic resonance of the translational modes for a planetary gear defined in Table 6.1. Lines without marker denote analytical solution. Lines with marker * denote speed up from subharmonic resonance zone of the translational modes. Lines with marker ○ denote speed up away from subharmonic resonance zone of the translational modes. Lines without marker denote analytical solution for distinct modes. The waterfall plot of the first (or third) ring-planet mesh deflection for speed up frequency sweep (from 3692.4 Hz to 4202 Hz) is given in Figure 6.6, it’s clear that a 173 subharmonic resonance is excited when the mesh frequency Ω is close to the natural frequency 2ω5,6 . (a) Figure 6.6. Subharmonic resonance spectra (speed up) of a translational mode ω6,7 for the first ring-planet mesh deflection of system in Table 6.1. The corresponding RMS plot in Figure 6.5 is the lines with marker *. While the waterfall plot of the second (or fourth) ring-planet mesh deflection indicates the subharmonic resonance is not excited (not shown). The mesh deflections and mesh forces of all the ring-planet meshes are shown in Figure 6.7. As can be seen in Figure 6.7a, tooth separations occur for the first and third ring-planet meshes, but not for the second and fourth ring-planet meshes. The phase between the first and third ring174 planet meshes is close to π because the translational mode is dominant. The phase between the second and fourth meshes is 0 because they only contain rotational mode. This can be explained as only one mode of the translational mode pair is excited in the subharmonic resonance. The mesh deflection for translational modes in (6.8) is reduced to δ [jnp ] = 2a [j p ] cosψ n (6.71) For in-phase planets, when the resonance associated with a translational mode is excited, the response is the combination of the translational modes and rotational modes. The mesh deflection for the response in the subharmonic zone of the translational modes can be represented as = δ [jnresponse ] 2a [j p ] cosψ n + δ [j1rot ] (6.72) where 2a [j p ] cosψ n is the mesh deflection from the translational mode and δ [j1rot ] is the mesh deflection from the rotational mode. According to (6.72), when 2a [j p ] δ [j1rot ] , the phase between the first and third meshes is close to π . The second and fourth mesh deflections are the same because cos = ψ 2 cos = ψ4 0. 175 (a) (b) Figure 6.7. Time response for subharmonic resonance of a translational mode ω6,7 when Ω =3768.8 Hz. The corresponding RMS plot in Figure 6.5 is the lines with marker *. (a) ring-planet mesh deflections. (b) ring-planet mesh forces. 176 When the response only contains one type of mode (for in-phase planets, it can only be the rotational mode; for sequentially phases planets, it can either be translational or planet mode), the load is equally distributed among the planets. When the response contains more than one type of modes, the load is unequally distributed among the planets according to the expression of mesh deflections (6.8). The mean values of the mesh forces for different meshes are shown in Figure 6.7b. The load is unequally distributed among the planets. A special case of response having more than one type of modes is the response containing the equally excited degenerate mode and the rotational mode (see (6.70) ), where the load is equally distributed among the planets. 6.4.6.2. Response of Planet Modes Figure 6.8 shows the ring-planet and sun-planet mesh deflections of the primary resonance for the response of a planet mode ω10 for a planetary gear defined in Table 6.1, where the damping is: ζ 10 = 0.005 and the damping ratio for the remaining degrees of freedom is 0.05 and only the second harmonic of mesh stiffnesses is considered. As shown in Figure 6.8, δ j1 = δ j 3 , and δ j 2 = δ j 4 . This confirms the mesh deflection phase is Tpψ n , and Tp = 2 for the planet mode. 177 a) b) Figure 6.8. Time response of the planet modes ( Ω ≈ ω10 ) for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.005 , ζ remain = 0.05 , and the mesh stiffnesses only have the second harmonic. (a) ring-planet mesh deflections, (b) sun-planet mesh deflections. 178 For comparison, the first two harmonics of the trapezoid wave is considered, and the RMS of ring-planet and sun-ring mesh deflections is Figure 6.9. The response in time domain indicates the both rotational and planet modes are excited, where the planet mode response is mainly excited by the second harmonic of the mesh stiffness and the rotational mode response is mainly excited by the first harmonic of the mesh stiffness. Due to the combination of the two types of modes, the phase between mesh deflections is no longer in simple expression as Tpψ n . The mesh deflections of a diametrically opposed pair are the same for both rotational and planet modes. Thus, δ j1 = δ j 3 , and δ j 2 = δ j 4 still hold for the response. For in-phase planets, when the resonance associated with a planet mode is excited, the response is the combination of the planet modes and rotational modes. As can be seen in Figure 6.9, phases between the first and second mesh deflections are different for sunplanet and ring-planet meshes due to combination of rotational and planet modes in the response. This also indicates the load is unequally distributed among the planets. 179 a) b) Figure 6.9. Time response of the planet modes ( Ω ≈ ω10 ) for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.005 , ζ remain = 0.05 , and trapezoid mesh stiffness variations. (a) ring-planet mesh deflections, (b) sun-planet mesh deflections. 180 The subharmonic resonance of the planet mode can also be excited. The subharmonic resonance of the planet mode ω10 is suppressed when the damping ratio ζ 10 = 0.05 . Changing the damping ratio to ζ 10 = 0.02 , the primary resonance of the planet mode ω10 is suppressed but the subharmonic resonance is excited when ζ 10 = 0.02 . One can find the subharmonic resonance of the planet modes as shown in .. Numerical and analytical solutions show a good agreement. Figure 6.10. Response of planet mode ω10 for a planetary gear defined in Table 6.1 with modal damping ζ 10 = 0.02 , ζ remain = 0.05 . Lines with marker * denote results from numerical integration. Lines without the marker denote result from perturbation solution. 181 6.4.7. Nonlinear Response for Planetary Gears with Commensurate Natural Frequencies The effect of damping on subharmonic resonance given in the frequency response function (6.57), the numerical simulation result is shown in Figure 6.11. Figure 6.11 indicates that increasing the damping of also narrows the instability region furthermore, when the damping ratio ζ 7 is bigger enough the subharmonic response disappears. Figure 6.11. Effect of modal damping on the subharmonic resonance for a planetary gear defined in Table 6.1. For planetary gears with commensurate natural frequencies, damping ratio of other natural frequencies ζ remain can affect the subharmonic. As an example, a planetary gear 182 with commensurate natural frequencies is studied. The system parameters are listed in 808.7 (T), ω= Table 6.2. The first 10 natural frequencies in Hz are: ω1 = 581.4 (R), ω= 2 3 ω4 = 1196.9 (R), ω5 = 1438.2 (P) ω= 2022.8 (T), ω8 = 2359.4 (R), ω9 = 2705.8 ω= 6 7 (PR), ω10 = 3117.4 (R). There are several commensurate natural frequencies: ω8 ≈ 2ω2 ≈ 4ω1 , ω6,7 ≈ 2.5ω2,3 . As seen in Figure 6.12, if the damping ratios for non subharmonic natural frequencies increase the range of subharmonic increases. The waterfall plot in Figure 6.13 indicates that 1/6 and 1/3 subharmonic resonances are excited in the subharmonic resonance of ω8 due to the commensuration ω8 ≈ 2ω2 ≈ 4ω1 . All the non subharmonic components can be treated as perturbations of the subharmonic resonance. Thus, when the damping of the non subharmonic natural frequencies increases it reduces the perturbations. This indicates the subharmonic resonance is more stable with larger ζ remain and the jumping down point is farer away from the jumping up point. Chaos is also observed in the subharmonic resonance of the rotational mode. In the region of chaos, all types of modes are excited. 183 Figure 6.12. Effect of modal damping on the subharmonic resonance for a planetary gear defined in Table 6.2. Inertias (kg) 2 I r / rr2 = 3.1669 , I c / rc2 = 6.000 , I s / rs2 = 2.500 , I p / rp = 2.000 Masses (kg) mr = 2.35 , mc = 5.430 , ms = 0.400 , m p = 4.000 100 × 106 k= k= rp sp , k= 100 × 103 rbs , k= 100 × 103 rus kbend= 15 × 106 , Stiffnesses (N/m) k p 500 × 109 500 × 106 ,= kc 500 × 109 , kcu = 109 , = = k s 500 × 109 , k= su Pressure angle (deg) α= α= 24.60 r s Torque (N·m) Tr = 100 , Ts = 100 Table 6.2. Dimensional parameters for a planetary gear with four equally spaced planets. krbs and krus denote radial and tangential distributed stiffnesses. γ sr = 0.5 . 184 The spectra in Figure 6.13 is of great interesting, while it is difficult to explain why as the subharmonic resonance of the translational mode ω6,7 may also contribute to the complexity Changing the modal damping ratio of ω8 to ζ 8 = 0.08 can suppress the subharmonic resonance of ω8 . The RMS and waterfall plots for the ring-planet mesh deflection are presented in Figure 6.14. Without the effect the subharmonic resonance of ω8 , one can clearly see the subharmonic resonance of the translational mode ω6,7 is excited. Although ω2,3 is the commensurate natural frequency of ω6,7 , the translational mode ω2,3 is not excited. Figure 6.13. Waterfall of ring-planet mesh for a planetary gear system defined in Table 6.2 and modal damping ζ = 0.05 . 185 Another interesting phenomena is observed for the subharmonic resonance of the translational modes The peak of the subharmonic resonance is on the right corner of the subharmonic resonance zone instead of on the left corner as usual (see Figure 6.14a). Figure 6.14b shows the spectra of the subharmonic resonance of the translational mode ω6,7 and the spectra of the combination resonance when the mesh frequency is close to the sum of ω1 , ω4 for speed down This combination resonance is caused by the commensuration of the natural frequencies ω1 , ω4 . For speed-up the combination resonance is still excited but with smaller amplitude. Explanation the observed interesting behaviors requires further analytical studies. 6.4.8. Nonlinear Response for Planetary Gears with Sequentially Phased Planets For equally spaced planets, if the tooth number of the sun or ring is not the integer multiple of the number of planets N, all the planets are sequentially phased. The phases of the planets are defined as γ sn= ( n − 1) zs / N for clockwise planet rotation. The derivation ( L) in section 3 is also valid for sequentially phased planets. The properties of D pq fq depends on mod( Lzs / N ) , where mod( Lzs / N ) is the integer remainder of Lzs divided by ( L) N. Table 6.3 lists the cases that D pq f q being zero for different mode types and phase ( L) conditions. D pq f q being zero indicates a high possibility of the disappearance for the response. 186 (a) (b) Figure 6.14. Response for ring-planet mesh deflection of a planetary gear defined in Table 6.2 and modal damping, ζ remain = 0.05 . (a) RMS of ring-planet mesh deflection, (b) Waterfall of ring-planet mesh deflection 187 Mode p is a type of ( L) D pq fq rotational translational purely ring planet mode (type s) mode mode mode mod( Lzs / N ) = 0 ≠0 0 0 0 mod( Lzs / = N ) 1, N − 1 0 ≠0 0 0 mod( Lzs /= N ) d, N − d 0 0 d, N − d s = s ≠ d , N − d ( L) D pq fq ≠ 0 ( L) D pq f q = 0 0 ( L) Table 6.3. Property of D pq f q for different type of modes and phase conditions. ( L) According to the properties of D pq f q , for in-phase planets, all the responses have the component of rotational modes, and the rotational modes are dominant in the response when the mesh frequency is away from the resonance peak. Similarly, for sequentially phased planets with mod( zs / N ) = 1 (or mod( zs / N ) >= 2 ) all the responses have the component of translational (or planet) modes, and when the mesh frequency is away from the resonance peak, translational (or planet) modes are dominant in the response. As an example, a four sequentially phased planetary gear in Table 6.1 with zs = 22 and ζ = 0.04 is studied. Figure 6.15 shows the RMS of the ring-planet mesh deflection for the sequentially phased planetary gear. The primary resonance of the planet mode ω4 is ( L) excited because D pq f q is nonzero for two nodal diameter planet modes. Figure 6.15 also indicates subharmonic resonance of the translational mode ω2,3 is excited. 188 Figure 6.15. Response of ring-planet mesh deflection for a sequentially phased planetary gear defined in Table 6.1 with zs = 22 , ζ = 0.04 , and γ sr = 0.5 . Lines with marker * and○ denote the numerical solution for speed up and down. Figure 6.16 shows the time response when the mesh frequency is away from the resonance peak. δ r1 = δ r 3 and δ r 2 = δ r 4 in Figure 6.16 confirm the response is a planet mode dominant mode. 189 Figure 6.16. Time response for ring-planet mesh deflections at the condition of point A in Figure 6.15 when the mesh frequency ( Ω =1400.6 Hz) is away from the peak of the resonance. 190 Chapter 7: Summary, Conclusions, and Future Work Equation Chapter 7 Section 1 7.1. Summary This work analytically investigates several key issues in the dynamics of planetary gears having an elastic ring gear. Before investigating the dynamics of the planetary gears having an elastic ring gear, the vibration of rings on a general elastic foundation is studied. The mode contamination and natural frequency splitting rules provides a solid base for understanding the modal properties of planetary gears. An elastic-discrete planetary gear model is built with the ring gear being modeled as an elastic body and the remaining parts as the rigid bodies. Through the perturbation and candidate mode methods, closed-form modal expressions are obtained for each types of mode. The structure modal deflections are similar to the modes of discrete planetary gears, and the elastic deformation of the ring shares similar contamination properties as the modes of rings on a general elastic foundation. In the case that equal spacing is not achievable, the planets are diametrically opposed in almost all applications. Modal properties and mode involving rule are obtained through the subsequent study. With the consideration of the mesh stiffnesses as time-varying variables, the parametric instability problem is addressed. Furthermore, with the consideration of the contact losses of gear meshes, the 191 nonlinear dynamics of planetary gears having an elastic ring gear is investigated. Properties of mesh deflections, which are obtained from the modal expressions, are important for the investigation of the parametric instabilities and the nonlinear dynamics. 7.2. Conclusions The main conclusions are summarized for each of the following topic. 7.2.1. Vibration of Rings on a General Elastic Foundation The eigensolutions of rings on a general elastic foundation are derived through perturbation and Galerkin analyses. The main conclusions for this topic are: 1) Closedform expressions for the natural frequencies and vibration modes of rings on a general elastic foundation are formulated through perturbation. This includes discrete and distributed foundations that vary circumferentially in radial, tangential, or inclined orientations. 2) The natural frequency splitting and mode contamination rules are obtained in general, compact forms involving the Fourier coefficients of the elastic foundation stiffness distribution functions. Splitting of the n nodal diameter natural frequency at first order rule is determined by whether or not the 2n th Fourier coefficients of the foundation vanish. The n nodal diameter mode of the free ring will be p for any non-zero contaminated with an s nodal diameter component if s ± n = p th coefficient in the foundation’s Fourier expansion. 3) For rings with identical, equally spaced springs, all modes are described in closed-form. The effects of the number of springs, spring stiffness, and location of the springs are studied. The influence of the 192 springs on vibration modes is more significant for repeated modes than split modes, while the effect on natural frequencies is more significant for split modes than repeated modes. 4) For rings with two sets of orthogonally oriented distributed springs, the combined effects may neutralize the effects of the individual foundations on certain modes. 7.2.2. Modal Properties of Planetary Gears with an Elastic Continuum Ring Gear The distinctive modal properties of planetary gears having equally spaced planets and an elastic continuum ring gear are derived using perturbation analysis and proved using a candidate mode method. The main conclusions for this topic are: 1) All vibration modes of equally spaced planetary gears having an elastic ring gear are classified into rotational, translational, planet and purely ring modes. For each mode type, the deflections of each planetary gear component, including the elastic ring, are derived in closed-form. In addition, the number of each mode type and multiplicity of the natural frequencies are determined. 2) The modal deflection properties of the sun, carrier and planets for rotational, translational and planet modes are the same as for the discrete model, while the deformation of the ring gear is governed by simple analytical rules dictating which nodal diameter components are present in each mode type. 3) Rotational modes contain jN nodal diameter ring deformation components, while the sun, carrier and ring rigid motion have only rotational motion. All planets have the same displacement. The natural frequencies are distinct. 4) Translational modes contain jN ± 1 nodal diameter ring deformation components, while the sun, carrier and ring rigid motion have only translational motion. The deflections of individual planets are related by a rotation matrix. The natural frequencies have multiplicity two. 5) Planet modes contain jN ± s nodal 193 diameter ring deformation components, where s is one of 2,3, ,int( N / 2) . The translation and rotation of the sun, carrier and rigid ring are zero, and the deflections of the planets are proportional to each other. Most of these natural frequencies have multiplicity two, but some natural frequencies are distinct for an even number of planets. 6) A purely ring mode has only a single nodal diameter ring deformation component. The deflections of all the discrete elements, including the ring rigid motion, are zero. The natural frequencies are distinct. 7) Changing the number of planets N does not affect the number of rotational and translational modes. How the vibration modes are distributed between purely ring modes and planet modes with the addition of a planet depends on whether N changes from odd to even or vice versa. 7.2.3. Natural Modes of Planetary Gears with Unequally Spaced Planets and an Elastic Ring Gear This work analytically identifies the modal properties of planetary gears with diametrically opposed planets and an elastic ring gear. The elastic-discrete model represents the ring gear as an elastic body free to deform radially while the remaining components are rigid bodies. The elastic continuum ring model leads to an infinitedimensional system. Relationships between the modal properties of planetary gears with equally spaced and diametrically opposed planets are examined in detail. The following conclusions are obtained: 1) All the modes are classified into rotational or translational modes with distinct natural frequencies. Closed-form expressions are provided for the structure of each mode type. A rotational mode contains only even numbered nodal diameter components of the elastic ring, and a translational mode contains only odd 194 numbered nodal diameter components. The planet and purely ring modes present when the planets are equally spaced no longer exist. 2) For rotational modes, the translations for the ring rigid motion, sun, and carrier are zero. For translational modes, the rotations for the ring rigid motion, sun, and carrier are zero. The motions (displacements and rotation) of the two planets of every diametrically opposed pair are identical for rotational modes and opposite for translational modes. 3) All the planet and purely ring modes of equally spaced planets evolve into either rotational or translational modes when the planets change to diametrically opposed. The rule governing this modal evolution is: Any mode for equally spaced planets having odd (even) nodal diameter components evolves into a translational (rotational) mode as the planets deviate to diametrically opposed. The exact numbers of planet and purely ring modes evolving to each of rotational and translational modes are given. 7.2.4. Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears This work analytically derives the parametric instability regions for planetary gears with equally spaced planets and an elastic continuum ring gear. For equal planet spacing, in-phase and sequentially phased mesh conditions are possible, and both phase conditions are considered. The main conclusions for this topic are: 1) The possible parametric instabilities of planetary gears having an elastic ring are classified as: distinct-distinct, distinct-degenerate, and degenerate-degenerate depending on the natural frequency multiplicity of the unstable modes. Using the well defined modal properties of the elasticdiscrete model, closed-formed expressions for distinct-distinct, distinct-degenerate and 195 degenerate-degenerate instability boundaries are obtained for both in-phase and sequentially phased meshes. 2) Every mode can be classified by the nature of its mesh deformation using an integer index Tp . For any two modes, if their gear mesh deformation type indices satisfy Lzs ± (Tp ± Tq ) ≠ jN for given mesh frequency harmonic L, their combination instabilities vanish. When the gear meshes are in-phase, this simplifies such that combination instabilities for modes p and q vanish for any L if Tp ≠ Tq and exist for any L if Tp = Tq . This result generates useful design information evident by inspection of a simple formula. 3) In-phase and sequentially phased planet meshes have significantly different instability boundaries. Switching between in-phase and sequentially phased meshes can eliminate certain instabilities with minimal change in the natural frequencies and vibration modes. 4) For in-phase meshes, the instability boundaries are in terms of the contact ratios ( cs and cr ), the mesh phase between the sunplanet and ring-planet meshes γ sr and the modal mesh deformations. For any possible single-mode or combination instability, a minimum instability region can be achieved by adjusting the contact ratios and γ sr . 7.2.5. Nonlinear Dynamics of Planetary Gears Having an Elastic Ring In this work, the nonlinear dynamics of planetary gears having equally spaced planets and an elastic ring gear is investigated through the method of multiple scales and numerical simulation. Nonlinear behaviors such as jump up, jump down, chaos are observed. The main conclusions are: 1) The mesh stiffness variations are modeled as trapezoid waves. With the assumption of one mode dominant response, closed-form 196 expressions for the frequency response functions for primary, subharmonic, second harmonic excited, and superharmonic resonances are obtained for distinct modes through the method of multiple scales. The tooth separation angle is independent of the torque, and the amplitude of response is proportional to the torque. 2) For a gear mesh, the mesh phase is the combination of the mesh stiffness phase and mesh deflection phase. For one mode dominant response, the mesh deflection phase or tooth separation phase between two meshes are governed by the mesh deflection index Tp and the position of the planets ψ n as Tpψ n , where the mesh deflection index Tp is 0,1, s for rotational, translational and type s planet modes, where s is an integer selected from 2 to int(N/2). 3) Solutions in this study verify the parametric instability existence and find the nonlinear response for rotational, translational and planet modes for both in-phase and sequentially phased planets. 4) For in-phase planets, all the responses contain rotational mode components although for different resonances, the dominant mode may not be a rotational mode. For N ) 1, N − 1 , all the responses contain sequentially phased planets with mod( zs / = N ) s, N − s , all the translational modes. For sequentially phased planets with mod( zs / = responses contain type s planet modes. 5) When response only contains one type of mode, the load is equally distributed among the planets. When response contains more than one type of modes, the loads usually are unequally distributed among the planets. 6) When the natural frequencies are commensurate, combination resonance and chaos in the subharmonic resonance are observed in numerical simulation. 197 7.3. Future Work This research focus on a series studies on the vibration of planetary gears with an elastic ring gear. It builds solid foundations for further analysis. The future work in the following challenging areas is recommended for the advanced dynamics of planetary gears. 7.3.1. Future Work for the Current Model Method of multiple scales and numerical simulation are used to find the nonlinear dynamics of the planetary gears having an elastic ring gear. Other analytical studies such as harmonic balance and Arc-length continuity methods can be performed. For degenerate modes, due to the complexity of the work, the analytical solution is not performed in this work. One may find the analytical solution for degenerate modes with a similar procedure of finding the frequency response functions for distinct natural frequencies, which can be used to support and explain the numerical solution obtained in this research. For degenerate modes the separation function depends on the combination of two degenerated modes. The inner product of right hand side of (6.36) with Yp (θ ) gives 198 < Yp , RHS >= −i 2ω p e ∞ −∑ iω p t (b′p e iβ p iβ + ibp β ′e p ) − iζ pω p2bp e i (ω pt + β p ) M ∑ < Yp , K v( L ) Yq > bq ei ( LΩt +ωqt + βq ) + < Yp , K v( L ) Yq > bq ei ( LΩt −ωqt − βq ) = L 1= q 1,q ≠ p ,l ∞ M −∑ = m ∑ < Yp , K h( m ) Yq > bq ei ( mωt −mγ +ωqt + βq ) + < Yp , K h( m ) Yq > bq ei ( mωt −mγ −ω pt − β p ) q 1,q ≠ p ,l 0= ∞ i ( LΩt +ω p t + β p ) i ( LΩt −ω p t − β p ) − ∑ < Yp , K v( L ) Yp > bp e + < Yp , K v( L ) Yp > bp e L =1 ∞ i ( mωt − mγ +ω p t + β p ) i ( mωt − mγ −ω p t − β p ) − ∑ < Yp , K h( m ) Yp > bp e + < Yp , K h( m ) Yp > bp e m =0 ∞ i ( LΩt −ω p t − β l ) i ( LΩt +ω p t + β l ) − ∑ < Yp , K v( L ) Yl > bl e + < Yp , K v( L ) Yl > bl e L =1 ∞ i ( mωt − mγ −ω p t − β l ) − ∑ < Yp , K h( m ) Yl > bl ei ( mωt −mγ +ωl t + βl ) + < Yp , K h( m ) Yl > bl e m =0 ∞ ∞ M − ∑∑ < Yp , K v( L ) Yq > f q eiLΩt − ∑ L 1= q 1 = M ∑ m q 1,q ≠l = 0= < Yp , K h( m ) Yq > f q eim (ωt −γ ) (7.1) ∞ + ∑ < Yp , K h( m ) Yl > f q eim (ωt −γ ) +c.c. m =0 For the primary resonance ω = ω p + εσ , the following equation is obtained from one of the solvability conditions when mesh frequency Ω is closed to a degenerate natural frequency ω p = ωl . i 2ω p (b′p + ibp β p′ )e (2) bp e +( E pp iβ p i (2σ − 2γ p − β p ) + iζ pω p2bp e iβ p (2) bp e + ( D pp i (2σ − β p ) (0) bp e + D pl(2)bl ei (2σ − βl ) ) + ( E pp M (1) f q eiσ + + E pl(2)bl ei (2σ −2γ l − βl ) ) + ∑ D pq = q 1 M ∑ = q 1,q ≠l (1) E pq f qe i (σ −γ p ) iβ p + E pl(0)bl eiβl ) + E pl(1) f q ei (σ −γ l ) = 0 (7.2) The simplification of (7.2) yields e iβ p M M i 2γ p iγ p 2 (2) (0) (2) (1) (1) ′ ′ i ω b ib β i ζ ω b D b e E b E b D f e E pq fq 2 ( ) + + + + + + + ∑ ∑ p p p p p p p pp p pp p pp p pq q = = q 1 q 1,q ≠l 0 + eiβl D pl(2)bl ei (2σ −2 βl ) + E pl(2)bl + E pl(0)bl + E pl(1) f l = (7.3) Substitution of φ= βl − β p into (7.3) gives p 199 (2) i 2ω p (b′p + ibp β p′ ) + iζ pω p2bp + D pp bp e + M ∑ E f +e (1) pq q = q 1, q ≠l iφ p i 2γ l (2) pl l D b e i 2γ p M (0) (2) (1) bp + E pp bp + ∑ D pq f qe + E pp iγ p q =1 (7.4) 0 + E b + E b + E f = (2) pl l (0) pl l (1) pl l Similarly, the other solvability condition gives rise to M i 2ω p (bl′ + ibl β l′) + iζ pω p2bl + Dll(2)bl ei 2γ l + Ell(0)bp + Ell(2)bp + ∑ Dlq(1) f q eiγ l + = q 1 +e − iφ p i 2γ p (2) lp p D b e M ∑ = q 1,q ≠ p Elq(1) f q (7.5) 0 + E b + E b + E f p = (2) lp p (0) lp p (1) lp In single mode response, hsn( L ) , hrn( L ) in (6.43) are the functions of separation angles that caused by mode p. For degenerate mode, (6.43) becomes N N ( L) E pq =< Yp , K h( L ) Yq >= k sp g ∑ hsn( L ) (θ sn[ p ,l ] )δ sn[ p ]δ sn[ q ] + krp ∑ hrn( L ) (θ rn[ p ,l ] )δ rn[ p ]δ rn[ q ] (7.6) = n 1= n 1 where θ sn[ p ,l ] and θ rn[ p ,l ] are governed by (6.33). According to (7.6), p, q in (7.6) are interchangeable. This indicates E pl( L ) = Elp( L ) . Ignoring terms with e i 2γ p , e i 2γ l in (7.4) and (7.5), considering the steady state response ( b′p = 0 , bl′ = 0 ), (7.4) and (7.5) yield M −2ω p bpσ + iζ pω p2bp + ∑ D p(1) qf q e q =1 iγ p M −2ω p blσ + iζ pω p2bl + ∑ Dlq(1) f q eiγ l + χ l1 + e q =1 (0) (2) χ p1 = E pp bp + E pp bp + χ l1 = Ell(0)bl + Ell(2)bl + M ∑ = q 1, q ≠ l M ∑ = q 1, q ≠ p iφ + χ p1 + e p χ p 2 = 0 − iφ p 0 χl 2 = (1) E pq f q , χ p 2 = E pl(2)bl + E pl(0)bl + E pl(1) fl Elq(1) f q , χ l 2 = E pl(2)bp + E pl(0)bp + E pl(1) f p (7.7) (7.8) (7.9) (7.10) The real and imaginary parts of (7.7) and (7.8) yields four equations M M (1) (1) Re(∑ D pq f q ) cos γ p − Im(∑ D pq f q ) sin = γ p 2ω p bpσ − χ p1 − χ p 2 cos φ p = q 1= q 1 200 (7.11) M M (1) pq q p = q 1= q 1 (1) Re( ∑ D f )sin γ + Im( ∑ D pq f q ) cos γ p = −ζ pω p2bp − χ p 2 sin φ p M (7.12) M Re(∑ Dlq(1) f q ) cos γ l − Im(∑ Dlq(1) f q ) sin= γ l 2ω p blσ − χ l1 − χ l 2 cos φ p (7.13) = q 1= q 1 M M Re( ∑ Dlq(1) f q )sin γ l + Im( ∑ Dlq(1) f q ) cos γ l = −ζ pω p2bl − χ l 2 sin φ p (7.14) = q 1= q 1 According to (7.11)-(7.14), five independent variables are bp , bl , γ p , φ p , σ , one can find an equation relating the amplitudes bp , bl . 2 M ∑D q =1 M = f (2ω p bpσ − χ p1 − χ p 2 cos φ p ) 2 + (ν pω p2bp + χ p 2 sin φ p ) 2 (1) p qq 2 ∑D q =1 (7.15) = f (2ω p blσ − χ l1 − χ l 2 cos φ p ) 2 + (ν pω p2bl + χ l 2 sin φ p ) 2 (1) lq q (7.16) Equation (7.15) and (7.16) are coupled. The author has not found a method to solve the problem analytically, while the numerical solution can be found through the iteration method. Interesting numerical results have been found for planetary gears with commensurate natural frequencies. Numerical and analytical studies on this topic are also recommended for the further study. 7.3.2. Planetary Gear Modeled with Multiple Elastic Bodies The Multiple Elastic Bodies model is a better approximation for planetary gears having thin rims in multiple bodies. The interaction between two or more elastic bodies is of great interest. 201 Besides the ring gear being treated as an elastic ring, the sun and planets can also be modeled as elastic rings. Due to the structural complexity of the carrier, the elastic representation of the carrier should not be modeled as an elastic ring. Finite Element Model is recommended for planetary gears having significant carrier elastic deformation, otherwise, the carrier is modeled as a rigid body. The simplest multiple elastic bodies model for planetary gears is the elastic sun and ring model, where both the sun and ring are treated as elastic rings while all the other components are modeled as rigid bodies (see Figure 7.1). Due to the similarity between the planetary gear elastic sun and ring model and the planetary gear elastic ring model, it is straight forward for one to follows the same procedure of finding the modal properties in Chapter 3 to obtain the following modal properties for the elastic sun and ring model: • The vibration modes of the planetary gears elastic sun and ring model can be classified as five types: rotational, translational, planet, purely ring, and purely sung modes. • The natural frequency multiplicities for rotational, translational, planet, and purely ring modes are the same as those in the elastic ring model. Purely sun modes have distinct natural frequencies. They have the same modal properties as the purely ring modes. • Most modal properties for rotational, translational, and plant modes remain unchanged. For rotational modes, the translations of the carrier are zero, and the planets have the same deflections. For translational modes, the rotation of the carrier is zero, and displacements of all the planets can be represented by 202 two planets. For planet modes, the deflection of the carrier is zero, and the displacements of the planets are proportional to each other. • The elastic deformations for the sun and ring always exist for rotational, translational, and planet modes. For any of these three types of modes, the sun and ring have the same nodal diameter component rule, furthermore the nodal diameter component rule is the same as the rule for the elastic ring model. Figure 7.1. Planetary gear elastic sun and ring model. 203 Other Multiple Elastic Bodies models require the planets being modeled as elastic bodies. The vibration modes may still be classified as several types. The number of mode types depends on the model. But for any model with elastic ring deformation, it should include rotational, translational, planet, and purely ring modes. It is interesting to discover how planets deform for the “planet modes”. 7.3.3. Profile Modification Gear tooth profile modification, the microgeometry modification of the involute gear teeth, usually is introduced to improve the static and dynamic performance of the gear systems by reducing the mesh stiffness variation and transmission error. Previous studies on tooth profile modifications mainly focus on the single pair gear mesh [77, 78]. Kahraman and Blankenship [79] experimentally analyzed the impact of tip relief on rotational vibration of a gear pair. Liu and Parker [80] investigated the tooth profile modification on the nonlinear dynamics for multi-mesh gear systems. 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Kasuba, Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System, Journal of Vibration, Acoustics, Stress, and Reliability in Design, 108 (1986), pp. 348353. [74]. A. Kahraman and R. Singh, Interactions between Time-Varying Mesh Stiffness and Clearance Non-Linear in a Geared System, Journal of Sound and Vibration, (1991), pp. 135-156. [75]. A. Kahraman and G. W. Blankenship, Planet Mesh Phasing in Epicyclic Gear Sets, Proc. of International Gearing Conference, Newcastle, UK, (1994), pp. 99-104. [76]. G. W. Blankenship and A. Kahraman, Gear Dynamics Experiments, Part-I: Characterization of Forced Response, ASME Power Transmission and Gearing Conference, San Diego (1996). [77]. P. Velex and M. Maatar, A Mathematical-Model for Analyzing the Influence of Shape Deviations and Mounting Errors on Gear Dynamic Behavior, Journal of Sound and Vibration, 191 (1996), pp. 629-660. [78]. F. B. Oswald and D. P. Townsend, Influence of Tooth Profile Modification on Spur Gear Dynamic Tooth Strain, The 31st Joint Propulsion Conference and Exhibit., (1995). [79]. A. Kahraman and G. W. Blankenship, Gear Dynamics Experiments, Part-III: Effect of Involute Tip Relief, ASME Power Transmission and Gearing Conference, (1996). [80]. G. Liu and R. G. Parker, Dynamic Modeling and Analysis of Tooth Profile Modification of MultiMesh Gears, ASME Journal of Mechanical Design, 130 (2008), pp. 121402-1. [81]. C. Yuksel and A. Kahraman, Dynamic Tooth Loads of Planetary Gear Sets Having Tooth Profile Wear, Mechanism and Machine Theory, 39 (2004), pp. 695–715. 211 Appendix A. Eigensolution Properties of Rings with Identical, Equally Spaced Spring Supports Equation Section 1 We are interested in the natural frequency splitting and mode contamination rules at higher order perturbations for a ring having identical, equally spaced spring supports. The eigensolutions are in terms of Q order perturbations Q Q j =1 j =1 2 u n + ∑ ε jϕ n , j . ω= ωn2 + ∑ ε jσ n , j , u= n n (A.1) Substitution of (A.1) into the eigenvalue problem (2.13) and collection of like powers of ε give j Lϕn , j − ω M ϕn , j = − L1ϕ n , j + ∑ σ n ,m Mϕ n , j − m , 2 n j = 1, 2, Q (A.2) m =1 The natural frequency splitting rule and the mode contamination rule has been obtained at first order perturbation. When 2n / l = int, the natural frequencies split at the first order; when 2n / l ≠ int , the natural frequencies remain repeated at first order. The eigensolutions of the second order perturbation are governed by (2.34) and (2.37). For 2n / l ≠ int, n − s and −n − s cannot simultaneously be integer multiples of l because their difference is 2n . This and (2.47) indicate d s − n d − s − n = 0 , d s − n e− s − n = 0 , es − n e− s − n = 0 212 (A.3) Letting f 1 = d (θ ) and f 2 = e(θ ) , (A.3) is written in a simple form as f si− n f −js − n = 0 , i, j = 1, 2 (A.4) According to (A.4), P− n ,s Ps ,n = 0 for all s, so the natural frequencies do not split at second order. The second order eigenfunction perturbation ϕn ,2 ( ηn in the main text) is rn , s Pst eitθ ∑ ∑ (1 + t ϕ n= η= ,2 n t ≠± n s ≠± n 2 (A.5) )(ωn2 − ωt2 ) where rn ,s and Pst are defined in (2.31) and (2.32). Equation (A.5) yields the same mode contamination rule as (2.56). The solvability conditions of the third order perturbation yield ∑ ∑ c3 Pn , s Ps ,t Pt ,n σ n ,3a n = t ≠± n s ≠± n ∑ c3 Pn,s Ps,t Pt ,− n t∑ ≠± n s ≠± n c3 = ∑ ∑cP ∑ ∑cP t ≠± n s ≠± n P P a n − n , s Ps ,t Pt , − n 3 − n , s s ,t t , n t ≠± n s ≠± n 3 Cn , s (A.6) (A.7) (1 + t )(ωt2 − ωn2 ) 2 where Cn ,s is defined in the second of (2.36). The two diagonal terms in (A.6) are identical and real, and the two off diagonal terms are complex conjugate. Consequently, = σ n ,3 ∑ ∑cP t ≠± n s ≠± n P P ± 3 n , s s ,t t , n ∑ ∑cP t ≠± n s ≠± n P P 3 − n , s s ,t t , − n (A.8) One can prove that s − n , t − s and −t − n cannot simultaneously be integer multiples of l for 2n / l ≠ int . A similar equation as (A.3) is obtained f si− n f t −j s f −kt − n = 0 , i, j , k = 1, 2 (A.9) Thus, the second term of (A.8) vanishes, and the natural frequencies remain repeated at third order. Comparison of (2.37) and (A.8) indicates that the eigenvalue expressions 213 have the same pattern. For each higher order of perturbation, the number of the summation increases by one. Thus, one can anticipate the eigenvalue expression of for the Q th order perturbation σ n ,Q ∑ ∑ ∑ ∑ c t ≠± n s ≠± n w ≠± n y ≠± n M Pn ,s Ps ,t Pw, y Py ,n ± Q ∑ ∑ ∑ c t ≠± n s ≠± n y ≠± n M Pn ,s Ps ,t Pw, y Py ,− n (A.10) Q Similarly, one can show that f si− n ft −j s f yo− w f −zy − n = 0 , i, j , , o, z = 1, 2 (A.11) Equation (A.11) guarantees that the second term of (A.10) vanishes, so the natural frequencies are still repeated at the Q th order of perturbation for 2n / l ≠ int . The natural frequency splitting rule obtained from the first order holds for any order of perturbation. Similar conclusion can be drawn for the mode contamination rule. 214 Appendix B. Nondimensional Operators The operators M and K (t ) in (5.6) are defined as 1 ∂2 ∂ 2v (kbend L1 + krus − krbs 2 )v + krn (t ) L2 v + krn (t ) L3q (v − 2 ) Ma = 2π ∂θ ∂θ , K (t )a = krn (t ) L4 v + K (t )q Mq ∂6 ∂4 ∂2 −( 6 + 2 4 + 2 ) , L1 = ∂θ ∂θ ∂θ N ∂δ (θ −ψ n ) ∂2 ∂ −∑ (sin 2 α r 2 − cos 2 α r )δ (θ −ψ n ) + (sin α r + cos α r ) sin α r L2 = ∂θ ∂θ ∂θ n =1 = L3q N ∑ cos α δ (θ −ψ n =1 r n ) − sin α r ∂δ (θ − ψ n ) δ n ∂θ N ∂v b r ∑ ( ∂θ sin α r + v cos α r ) θ =ψ n n =1 0 0 L4v = ∂v b p ( sin α r + v cos α r ) ∂θ θ =ψ 1 ∂v b p ( ∂θ sin α r + v cos α r ) θ =ψ N b p = (sin α r , − cos α r , − 1)T , b r = (− sinψ rn , cos ψ rn , 1)T δn = − xr sinψ rn + yr cosψ rn + ur + ξ n sin α r − ηn cos α r − un 215 M = diag( M r , M c , M s , M1 , , M N ) M j = diag( m j , m j , I j / rj2 ) , j = c, s,1, , N , M r = diag[1,1,1/ cos2 α r ] ∑ K nr1 + K rb K (t ) = ∑K n c1 + K cb ∑K n s1 + K sb symmetric K 1r 2 K rN2 K 1c 2 K cN2 K 1s 2 K sN2 K 1pp K Npp K jb = diag( k jx , k jy , k ju ) , j = c, s , K rb = R diag(4krbs + 4krus , 4krbs + 4krus , 2π krus / cos 2 α r ) k rbs and k rus are uniform radial and tangential distributed stiffnesses, respectively, R is the pitch radius of the ring, and k *ju is torsional stiffness with units F − L / rad and k ju = k *ju / rj2 with units F / L . K npp = K nr 3 + K nc 3 + K ns 3 sin 2 ψ rn − cosψ rn sinψ rn n K r1 = krn (t ) cos 2 ψ rn symmetric − sinψ rn sin α r K = k rn (t ) cosψ rn sin α r sin α r n r2 sinψ rn cos α r − cosψ rn cos α r − cos α r sin 2 α r − cos α r sin α r n K r 3 = krn (t ) cos 2 α r symmetric − sinψ rn cosψ rn 1 sinψ rn − cosψ rn −1 − sin α r cos α r 1 1 0 − sinψ n − cosψ n n k pn − sinψ n K = k pn 1 cosψ n , K c 2 = symmetric 1 0 n c1 216 sinψ n − cosψ n −1 0 0 0 K c 3 = diag( k pn , k pn ,0) sin 2 ψ sn − cosψ sn sinψ sn n K s1 = k sn (t ) cos 2 ψ sn symmetric K n s2 sinψ sn sin α s = k sn (t ) − cosψ sn sin α s − sin α s sinψ sn cos α s − cosψ sn cos α s − cos α s sin 2 α s cos α s sin α s K ns3 k sn (t ) = cos 2 α s symmetric ψ= ψ n − αs , ψ= ψ n + αr sn rn 217 − sinψ sn cosψ sn 1 − sinψ sn cosψ sn 1 − sin α s − cos α s 1