Development of 2D to 3D FEA Conversion Factors for Natural Frequencies of Beams Under Transverse Vibrations by James Austin Cosby An Engineering Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING Major Subject: Mechanical Engineering Approved: _________________________________________ Dr. Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, CT 11, 2013 i CONTENTS LIST OF TABLES ............................................................................................................ iii LIST OF FIGURES .......................................................................................................... iv LIST OF SYMBOLS ......................................................................................................... v KEYWORDS .................................................................................................................... vi ACKNOWLEDGMENT ................................................................................................. vii ABSTRACT ................................................................................................................... viii 1. INTRODUCTION ....................................................................................................... 1 2. THEORY/METHODOLOGY ..................................................................................... 4 2.1 FEA Analysis and Modeling .............................................................................. 8 3. RESULTS and DISCUSSIONS ................................................................................ 16 3.1 Results Verification and Validation ................................................................. 16 3.2 Overview of Results ......................................................................................... 17 4. Conclusions................................................................................................................ 22 5. References.................................................................................................................. 23 6. Appendix A................................................................................................................ 24 6.1 Fixed – Fixed Solid element ANSYS INPUT file .......................................... 24 6.2 Fixed Fixed Beam Elemen ANSYS input file ................................................. 27 6.3 Fixed – Free Solid Element ANSYS Input File ............................................... 31 6.4 Fixed – Free Beam Element ANSYS Input File .............................................. 34 6.5 Simply Supported Solid Element ANSYS Input File ...................................... 38 6.6 Simply Supported Beam Element ANSYS Input File ..................................... 42 7. Appendix B ................................................................................................................ 46 8. Appendix C ................................................................................................................ 49 9. Appendix D................................................................................................................ 51 10. Appendix E ................................................................................................................ 52 ii LIST OF TABLES Table 1 Beam Boundary Conditions and Their Constraints .............................................. 7 Table 2 Titanium Material Properties Used for Beam..................................................... 12 Table 3 Geometric Input for Each FEA Case .................................................................. 13 Table 4 Number of Modes and Frequency Range for Each Case.................................... 14 Table 5 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Fixed Beam ......................................................................................................................................... 16 Table 6 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Free Beam ......................................................................................................................................... 16 Table 7 Frequency Comparison of Hand Calcs vs. FEA for a Simply Supported Beam 16 Table 8 Conversion Equations for Modes 1 and 2 .......................................................... 19 Table 9 Conversion Equations for Modes 3 and 4 .......................................................... 19 Table 10 Frequency Comparison of Mode 1 for a Fixed-Fixed Beam 12 Inches Long .. 20 Table 11 Test Case 2 Frequency Comparison for Mode 4 .............................................. 21 iii LIST OF FIGURES Figure 1 Typical Spring-Mass System .............................................................................. 1 Figure 2 Spring-Mass System with a Harmonic Force Input ............................................ 2 Figure 3 An Aircraft wing structure [1] and a Truss Bridge [2] Both Utilize Beams ....... 3 Figure 4 Beam under Transverse Loading......................................................................... 4 Figure 5 Beam Boundary Conditions and Closed-Form Natural Frequency Solutions .... 7 Figure 6 Beam Element ..................................................................................................... 8 Figure 7 Example of 3D FEA Element............................................................................ 11 Figure 8 Example of Beam Element Model .................................................................... 13 Figure 9 Example of Solid Element Model ..................................................................... 14 Figure 10 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Solid Elements ......................................................................................................................................... 15 Figure 11 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Beam Elements ......................................................................................................................................... 15 Figure 12 Frequency Comparisons W/ Fixed-Fixed Boundary Conditions .................... 17 Figure 13 Frequency Comparisons W/ Fixed-Free Boundary Conditions ...................... 18 Figure 14 Frequency Comparisons W/ Simply Supported Boundary Conditions ........... 18 Figure 15 Test Case 2 I-Beam Cross-section .................................................................. 20 iv LIST OF SYMBOLS h Height of the Beam πΌππβππ L Length of the Beam πΌππβππ M Bending Moment ππ β ππ F Force Applied to beam π Density A Cross-Sectional Area ππ2 u Distance ππ3 E Elastic Modulus I Area Moment of Inertia V Shear Force ππ/ππ2 θ Rotation radians m Mass πππ’ππ k Spring Constant πΏπ/ππ ππ Natural Frequency π»πππ‘π§ π Forcing Frequency π»πππ‘π§ π Frequency Ratio π Displacement ππ ππ/ππ3 ππ/ππ2 ππ4 ππππ‘πππ π πΌππβππ v KEYWORDS Modal Analysis, Euler-Bernulli Beam Theory, ANSYS, Beam Elements, Solid Elements, Conversion Factor, Finite Element Method vi ACKNOWLEDGMENT Thank you to my wonderful wife and darling daughter for supporting me through this process. vii ABSTRACT In this project conversion factors for natural frequency predictions from 2D beam FEA beam elements to 3D solid FEA element analysis was obtained. The conversion factors allow an engineer to run a model using beam elements and convert the frequency solution to a more accurate answer based on geometry while keeping solution time to a minimum. To do so, FEA cases were run for beams with slenderness ratios between 20 and 4. The beams studied had a square cross-section and had fixed-fixed, fixed-free, and simplysupported boundary conditions. To check the accuracy of the conversion factors 2 test cases were ran. The first test had a square cross-section and simply supported boundary conditions: the second test case had an I-Beam cross-section and fixed free boundary conditions. In both cases the element error for beams with slenderness ratios below 10 was decreased to below 10% from as high as 40%. Classical beam vibration theory assumes that shear deformation and stress through the thickness of the beam is very small. This assumption is good for very smaller slenderness ratios, but classical beam theory hand calculations for natural frequencies diverge from the true solution for thicker beams. viii 1. INTRODUCTION When designing engineering structures, be it an airfoil in a gas turbine or a ventilation stack of a factory, one must be mindful of vibratory failure modes. Vibration is the oscillation of a mechanical structure. These vibratory oscillations can occur either freely or forced. One of the simplest systems that can be used to describe free and forced vibration is a spring-mass system shown in Figure 1. Figure 1 Typical Spring-Mass System The equation of motion for the system in Figure 1 is Equation 1, where k is the spring constant and m is the mass of the system. . ππ₯Μ + ππ₯ = 0 [1] The frequency at which the system will oscillate, or natural frequency, is described in Equation 2. ππ = √ π π [2] The natural frequency of a system is a key parameter in the design process of a structure. It becomes more important when an external force is applied to the system. Figure 2 shows a spring-mass system with an external, harmonic input also applied where πΉ0 is the amplitude, ω is the forcing frequency and t is time. 1 Figure 2 Spring-Mass System with a Harmonic Force Input The equation of motion for this system is Equation 2. ππ₯Μ + ππ₯ = πΉ0 sin ππ‘ [3] The solution to this differential equation is Equation 3 assuming the initial boundary conditions of π₯(0) = π₯Μ (0) = 0. π₯(π‘) = πΉ0 1 (sin π€π‘ − π sin π€π π‘) π 1 − π2 [4] Where r, also known as the frequency ratio, is: π= π ππ It is noticed that if r=1, the forcing frequency and the natural frequency are equal, then the displacement solution goes to infinity. This occurrence is known as a resonance and is a major concern when designing engineering structures. Knowing the natural frequencies of your structure is crucial to robust design that will not fail. The simple spring-mass system that has been discussed up until now is also known as a lumped-parameter system. In reality most structures, such as beams, are continuous systems that have more complex equations of motion and have multiple natural frequencies. Beams are a fundamental engineering structures used in multiple disciplines of engineering such as civil and aerospace. The primary use of beam is to withstand loading that induces bending or buckling. Figure 3 shows two such examples of how beams are used in engineering structures. 2 Figure 3 An Aircraft wing structure [1] and a Truss Bridge [2] Both Utilize Beams To solve these more complex problems finite element analysis software is used. Finite element analysis software packages, such as ANSYS, have beam element classes. These classes of elements are governed by Euler-Bernoulli beam theory. Beam elements take much less time to solve than solid elements. However, as the beam becomes thicker the accuracy of the solution decreases due to the assumptions of beam theory. To avoid inaccuracy, it is recommended to use solid elements to model thick beams. This is very computationally expensive compared to using beam elements. It is the purpose of this project to find conversion factors between beam elements and solid elements using ANSYS. The conversion factors will be for transverse vibration of beams with simply supported, fixed-fixed and fixed-free boundary conditions. This would allow an engineer to convert his beam-element solution based on the geometry of the beam, thus reducing solution time without sacrificing solution accuracy. 3 2. THEORY/METHODOLOGY Euler-Bernoulli beam theory for transverse vibration was developed as a simplification of the theory of elasticity. The assumptions for classical beam theory are as follows [Ref 3]: 1. 2. 3. 4. 5. 6. 7. The beam has uniform properties The beam is slender (L/h is small) The beam obeys Hooke’s Law There is no axial load Plane sections remain plane during motion The plane of motion is the same as the beam symmetry plane Shear Deformation is Negligible Figure 4 shows a beam under transverse loading. Figure 4 Beam under Transverse Loading 4 The bending moment is described by Equation 5. π2 π’ π(π₯, π‘) = πΈπΌ(π₯) 2 ππ₯ [5] Where u is a distance from its equilibrium position at point x on the beam, M is the bending moment, E is the elastic modulus and I is the area moment of inertia. Summing the forces in the Y direction yields. ππ΄ π 2π’ ππ = (π + ππ₯) − π + πππ₯ 2 ππ‘ ππ₯ [6] Where V is the shear force, f is the applied force, π is the density and A is the crosssectional area. Carrying out the subtraction, Equation 6 becomes Equation 7 ππ΄ π 2π’ ππ = (π + ) ππ₯ 2 ππ‘ ππ₯ [7] Summing the moments about point O yields 8 (π + ππ ππ ππ₯ ) ππ₯ − π + (π + ππ₯) ππ₯ + (πππ₯) =0 ππ₯ ππ₯ 2 [8] If (ππ₯)2is much smaller than ππ₯, Equation 8 becomes Equation 9 assuming A and I are both constant. π 2π’ π 4π’ 2 + π =0 ππ‘ 2 ππ₯ 4 Where π2 = 5 πΈπΌ ππ΄ [9] Equation 8 were simplified by assuming that shear deformation is negligible compared to π’(π₯, π‘). This assumption holds well with slender beams, however, when β/πΏ or π€/πΏ is less than 10, this assumption becomes less valid. Using the method of separation of variables to solve Equation 9 yields Equation 10 where u(x, t) = F(x)G(t). π2 πΉ′′′′(π₯) πΊΜ (π‘) =− = π2 πΉ(π₯) πΊ(π‘) [10] πΉ ′′′′ (π₯) − π½ 4 πΉ(π₯) = 0 [11] This yields Equation 11 Where, π½4 = π2 π2 [12] Assuming the solution of the form πΉ(π₯) = πΆπ ππ₯ gives Equation 13. πΉ(π₯) = π·1 πππ βπ½π₯ + π·2 π ππβπ½π₯ + π·3 πππ π½π₯ + π·4 π πππ½π₯ [13] To obtain the answer to Equation 9, the boundary conditions of the end of the beams must be taken into account. The 4 different types of boundary condition and their constraints are shown in Table 1. 6 Table 1 Beam Boundary Conditions and Their Constraints End Condition Free Fixed Simply Supported Sliding Deflection Slope Bending Moment Unconstrained Unconstrained 0 0 0 Unconstrained Shear Force 0 Unconstrained 0 Unconstrained 0 Unconstrained Unconstrained 0 Unconstrained 0 Where Deflection is π’(π₯, π‘), slope is ππ’/ππ₯, bending moment is πΈπΌπ 2 π’/ππ₯ 2 and shear force is π(πΈπΌπ 2 π’/ππ₯ 2 )/ππ₯. For this project, 3 different beams will be considered using fixed-fixed, fixed-free, and simply-supported boundary conditions. Using Equation 13 and the boundary conditions in table 1, solutions for the 3 different beams can be found in Figure 5. Figure 5 Beam Boundary Conditions and Closed-Form Natural Frequency Solutions 7 2.1 FEA Analysis and Modeling For this project ANSYS FEA models were created using both beam elements and solid elements. Beam element equations are derived very similarly to that hand calculation method seen in the in Methodology portion of this paper. A beam element is shown in Figure 6. Figure 6 Beam Element The displacement vector for a beam element is defined in Equation 10. π£1 π1 π½ = [π£ ] 2 π2 8 [14] Boundary Conditions for the beam element are as follows: π£(0, π‘) = π£1 (π‘) ππ£(0, π‘) = π1 (π‘) ππ₯ π£(πΏ, π‘) = π£2 (π‘) ππ£(0, π‘) = π2 (π‘) ππ₯ The solution form for the deflection of the beam element is assumed as follows: π£(π₯, π‘) = π(π‘) + π(π‘)π₯ + π(π‘)π₯ 2 + π(π‘)π₯ 3 [15] Applying the boundary conditions and solving, the following is obtained: π(π‘) = π£1(π‘) π(π‘) = π1 π‘ 1 [−3π£1 (π‘) − 2π1 (π‘) + 3π£2 (π‘) − πΏπ2 (π‘)] πΏ2 1 π(π‘) = 3 [2π£1 (π‘) + πΏπ1 (π‘) − 2π£2 (π‘) + πΏπ2 (π‘)] πΏ π(π‘) = Equation 15 now becomes as follows: π£(π₯, π‘) = π1 (π₯)π£1 (π‘) + π2 (π₯)π2 (π‘) + π3 (π₯)π£2 (π‘) + π4 (π₯)π2 (π‘) Where, π₯ 2 π₯ 3 π1 (π₯) = 1 − 3 ( ) + 2 ( ) πΏ πΏ π₯ 2 π₯ 3 π2 (π₯) = π₯ − 2πΏ ( ) + πΏ ( ) πΏ πΏ 2 π₯ π₯ 3 π3 (π₯) = 1 − 3 ( ) + 2 ( ) πΏ πΏ 9 [16] π₯ 2 π₯ 3 π4 (π₯) = −πΏ ( ) + πΏ ( ) πΏ πΏ The kinetic energy of the element can expressed using the following equation: 1 ππ£ 2 πΎπΈ = ∫ ππ΄ ( ) ππ₯ ππ‘ 0 2 πΏ [17] Substituting for ππ£/ππ‘ yields: πΏ 1 2 πΎπΈ = ππ΄ ∫ [π1 (π₯)π£Μ 1 (π‘)+π2 (π₯)πΜ1 (π‘) + +π3 (π₯)π£Μ 2 (π‘) + +π4 (π₯)πΜ2 (π‘)] ππ₯ 2 0 [18] Integrating over the length of the beam yields: πΎπΈ = ππ΄πΏ π» πΜ π΄πΜ 420 [19] Where, 156 22πΏ π΄=[ 54 −13πΏ 22πΏ 4πΏ2 13πΏ −3πΏ2 54 13πΏ 156 −22πΏ −13πΏ −3πΏ2 ] −22πΏ 4πΏ2 [20] And π£Μ1 πΜ π£Μ = 1 π£Μ 2 [πΜ2 ] The potential energy of the beam can be expressed as the following: πΏ 2 π 2π£ ππΈ = 1/2πΈπΌ ∫ ( 2 ) ππ₯ ππ₯ 0 10 [21] Substituting Equation 16 into Equation 21 in and integrating over the length of the beam yields the stiffness matrix which is shown in Equation 22. 12 6πΏ πΈπΌ 6πΏ 4πΏ2 π²= 3[ πΏ −12 −6πΏ 6πΏ 2πΏ2 −12 6πΏ −6πΏ 2πΏ2 ] 12 −6πΏ −6πΏ 4πΏ2 [22] Finally, the equation of motion can be expressed as the following: π΄πΜ + π²π = 0 [23] 3D finite element takes a structure, such as a beam, and discretizes it into a finite number of elements. An example 3D element is shown in Figure 7. Figure 7 Example of 3D FEA Element 11 Each node is allowed to move in all three directions (x,y, and z). For free vibration analysis of a 3D mesh, the equation of motion can be described as Equation 24. π΄π πΏΜ + π²π πΏ = 0 [24] Where π΄π and π²π are the global mass and stiffness matrix of the entire system. These are developed from the individual elements and nodes. The models considered for this project utilize 4 types of boundary conditions commonly used in engineering. The types of boundary conditions and their effects can be seen in Figure 5. For each set of boundary conditions both a beam and solid element model will be run for 13 cases. Cases were selected such that geometries had slenderness ratios between 5 and 20. The geometric inputs for each case are shown in Table 3. For this analysis Ti 6-4 material, a very common aerospace alloy, will be used. Ti 6-4 material properties are summarized in Table 2 Titanium Material Properties Used for Beam. An ANSYS input file was created to for each boundary condition and element type to automate the model generation process. All the input files can be found in Appendix A. Table 2 Titanium Material Properties Used for Beam 12 Table 3 Geometric Input for Each FEA Case BEAM3 elements will be used for the beam cases and SOLID45 elements for the solid cases. Examples of each type model can be seen in Figure 8 and Figure 9. Figure 8 Example of Beam Element Model 13 Figure 9 Example of Solid Element Model In addition to the boundary conditions above, the solid element cases will have roller boundary conditions on the vertical faces of the beams. As transverse vibrations are the only mode shapes being considered for this project, the roller boundary conditions eliminates extraneous, non-transverse modes shapes. This assists with modal identification later in the process. The number of mode shapes and frequency range for each type of boundary conditions are shown in Table 4. Table 4 Number of Modes and Frequency Range for Each Case After the solutions are obtained, further modal identification was required to find the first 4 transverse vibration mode shapes. This was done by visual observation of the mode shapes themselves. The first four mode shapes of the fixed-fixed boundary condition using solid and beam elements can be seen in Figure 10 and Figure 11. 14 Figure 10 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Solid Elements Figure 11 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Beam Elements 15 3. RESULTS and DISCUSSIONS 3.1 Results Verification and Validation To ensure that our FEA modeling techniques are correct the closed form solutions for each case was obtained using the results listed in Figure 5. A frequency comparison of mode 1 for each type of boundary conditions is shown in Table 5 and Table 6. Mode 2 through 4 frequency comparisons can be seen in Appendix B. Table 5 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Fixed Beam Table 6 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Free Beam Table 7 Frequency Comparison of Hand Calcs vs. FEA for a Simply Supported Beam The hand calculations follow the beam element and solid element solutions closely between slenderness ratios of 20 and 10. For example, the percent difference for the solid 16 element and beam element solution compared to the hand calculations for a fixed-fixed beam with a slenderness ratio equal to 20 are 2.5% and 0.5% respectively. For slenderness ratios below 10, the hand calculations and beam elements are still close, but both diverge from the solid element solution. The percent difference for the solid element and beam element compared to the hand calculations for a fixed-fixed beam with a slenderness ratio, L/W, equal to 5.7 are -12.5% and -1.9% respectively for mode 1. 3.2 Overview of Results All cases were run for each set of boundary conditions and the first 4 modes for each case were identified. Comparisons of beam element and solid element frequencies for each mode and boundary condition can be seen in Figure 12 through Figure 14 . Figure 12 Frequency Comparisons W/ Fixed-Fixed Boundary Conditions It is observed from the frequency comparisons for the fixed-fixed beam that as the slenderness ratio decreases below 10, the 2 types of elements solutions diverge. The higher order modes have a larger difference between solutions at slenderness ratios greater than 10 than the lower order modes. It is also noticed that the difference between mode 1 17 frequencies is less than the higher order modes throughout the slenderness ratio regime considered. This is due to the mode shapes being more complicated in the higher order modes. The more complicated the mode shapes the less the assumption that plane sections remain plane holds. The same observations hold for the fixed-free and simply supported beams frequency comparisons which can be seen in Figure 13 and Figure 14 Figure 13 Frequency Comparisons W/ Fixed-Free Boundary Conditions Figure 14 Frequency Comparisons W/ Simply Supported Boundary Conditions 18 To obtain a conversion factor between the solid and beam element models, the solid element frequencies are divided by the beam element frequencies for each mode. This non-dimensional parameter is then plotted against the slenderness ratio. Using excel, a curve fit was obtained and a convsersion factor equation based on slenderness ratio is generated. The polynomial orders of the equations correspond to the lowest order that had a π 2 value of greater than .99 and the best visual fit. The conversion factor equations can be seen in Table 8 and Table 9 where y is the conversion factor and x is the slenderness ratio. Table 8 Conversion Equations for Modes 1 and 2 Table 9 Conversion Equations for Modes 3 and 4 To ensure that the conversion factors are accurate two test cases were ran. The first test case was a fixed-fixed beam with the same geometric inputs seen in Table 3 except the length was reduced by half to 12 inches. This simple test case was used as a first pass attempt at proving the assumption that the conversion factors obtained would be valid for different geometries. Frequency comparisons between solid element frequencies, beam element frequencies, and converted beam element frequencies for mode 1 of test case 1 can be seen in Table 10. A frequency comparison for modes 2 through 4 are documented in Appendix D. 19 Table 10 Frequency Comparison of Mode 1 for a Fixed-Fixed Beam 12 Inches Long Comparing the unconverted beam element error with the converted error shows an improvement throughout the entire slenderness ratio band considered. For the smallest slenderness ratio considered, which has the large amount of error associated with it, the percent error decreased from 37.82% to 6.65% using the conversion factors obtained. The second test case was selected to test the bounds with which the conversion factors would hold. This case utilizes fixed-free boundary conditions, an I-beam crosssection and a length increase from 24 inches to 32 inches. The cross-sectional geometry is shown in Figure 15. Figure 15 Test Case 2 I-Beam Cross-section A frequency comparison of mode 4 for test case 4 is shown in Table 11. A frequency comparison for modes 1 through 3 are documented in Appendix E. 20 Table 11 Test Case 2 Frequency Comparison for Mode 4 Once again a significant improvement in frequency predictions is noticed. Converting the frequencies using the conversion factors decreases the error from a max of 47% to 7%. It is also interesting that the Beam element solution is, once again, higher than the solid element solution throughout the range of slenderness ratios considered. It is also of note that for a slenderness ratio above 10 the beam element solution for test case 2 has an error associated with it above 10%. This increase in error is most likely due to the more complex cross-sectional geometry. This just further expands the regime of slenderness ratios where the conversion factors obtained in this project are useful. 21 4. Conclusions The ability to accurately predict natural frequencies of engineering structures is vital in any design. Having a resonance within the operating parameters of a structure during its life will lead to failure. Beams are a fundamental engineering structure with multiple applications, be it in civil, aerospace, or mechanical engineering. Euler-Bernoulli beam theory was developed to aid in analysis of beams, but due to its basic assumptions it becomes less valid as the beam thickness increases. FEA software packages have elements that are based on beam theory and have very fast solution time. However, these elements have the same limitations for thicker beams so solid elements are advised. Using solid elements greatly increase the solution times for solving a model. Large models can be extremely computationally expensive. Three different sets of boundary conditions were used in this project which was discussed previously which all had a square cross-section. The beam element solutions were very close to hand calculation predictions throughout the slenderness ratio regime considered. In each case, a large increase in solution accuracy using solid elements was observed for beams with slenderness ratios less than 10. It was also noticed that for all boundary conditions the higher order mode shapes had a greater amount of error than the lower order modes shapes. The conversion factors obtained with this project can improve the beam element natural frequency predictions to within 10% error from as high as 38%. The validity of the conversion factors were checked using 2 test cases. The first test case considered was very similar to the original fixed-fixed case but with a decrease in length. The second test had an I-Beam cross-section and an increase in length compared to the original fixed-free boundary condition case. Both of the test cases beam element solution error decreased to below 10% for the mode when converted using the conversion factors. While this project only covered free transverse vibration of beams using a select group of boundary conditions, further work could be completed for the remaining conditions. Also, conversion factors for more complex structures based on beams such as trusses could be done. 22 5. References 1. http://thexodirectory.com/wp-content/uploads/2011/05/Air-to-air-overhead-frontview-of-an-SR-71A-460×361.jpg; 11/05/2013 2. http://www.texasescapes.com/TexasTowns/Burkett-Texas.htm; 11/05/2013 3. Palm, W. [2007], Mechanical Vibrations, Hoboken: Wiley & Sons 23 6. Appendix A 6.1 Fixed – Fixed Solid element ANSYS INPUT file /prep7 et,1,45 ! sets element type 1 to beam3, the 2d beam element in ANSYS mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) ESIZE,0.1,0, mode_num=10 !mode numbers to extract /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 *DIM, frequencies, ARRAY, mode_num + 1, rw, *do,j,1,rw fini run_name='modal_analysis_%j%' 24 /FILNAME,run_name,0 /syp,mkdir, run_name /CWD,W:\m310186\RPI_project\fx-fx-solid\%run_name% /prep7 BLOCK,0,24,0,inputs(j,1),.25,-.25, vmesh,1 ! creates a mesh of elements on volume 1 asel,s,area,,5 asel,a,area,,6 nsla,s,1 cm,nn,node d,all,all,0 ! constrains displacements on forward face to zero d,all,all,0 !constrains displacements on rear face to zero asel,s,area,,1 asel,a,area,,2 nsla,s,1 cmsel,u,nn d,all,uz,0 !constrains the side faces in teh z direction to zero allsel,all /solu antype,2 LUMPM,0 PSTRES,0 MODOPT,LANB,mode_num,0,20000,,OFF 25 ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' 26 r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,W:\m310186\RPI_project\fx-fx-solid /prep7 vclear,1 vdele,all,all,,1 *enddo *cfopen,freq_tot,out *vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies( 1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10 ),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1, 15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) *cfclose 6.2 Fixed Fixed Beam Elemen ANSYS input file /prep7 k,2,24,0 ! sets a keypoint at (24,0) l,1,2,36 ! creates a line with 5 divisions from keypoints 1 to 2 et,1,3 ! sets element type 1 to beam3, the 2d beam element in ANSYS 27 mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 r,1,.75,9, ! defines property set 1 for the beam element (A,I,height) mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) lmesh,1 d,1,all,0 ! creates a mesh of elements on line1 (undo with lclear) ! constrains displacements at node 1 to zero d,2,all,0 /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 mode_num=10 *DIM, frequencies, ARRAY, mode_num + 1, rw, *do,j,1,rw fini run_name='modal_analysis_%j%' /FILNAME,run_name,0 /syp,mkdir, run_name 28 /CWD,S:\csmc_struct_64\RPI_project\fx-fx-beam\%run_name% /solu r,1,inputs(j,3),inputs(j,2), ! defines property set 1 for the beam element (A,I,height) antype,2 LUMPM,0 PSTRES,0 MODOPT,LANB,mode_num,0,25000,,OFF ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, 29 frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,S:\csmc_struct_64\RPI_project\fx-fx-beam *enddo *cfopen,freq_tot,out *vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies( 1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10 ),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1, 15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) *cfclose 30 !*get,name,active,,jobnam 6.3 Fixed – Free Solid Element ANSYS Input File /prep7 et,1,45 ! sets element type 1 to beam3, the 2d beam element in ANSYS mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) ESIZE,0.1,0, mode_num=10 !mode numbers to extract /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 *DIM, frequencies, ARRAY, mode_num + 1, rw, 31 *do,j,1,rw fini run_name='modal_analysis_%j%' /FILNAME,run_name,0 /syp,mkdir, run_name /CWD,W:\m310186\RPI_project\fx-free-solid\%run_name% /prep7 BLOCK,0,24,0,inputs(j,1),.25,-.25, vmesh,1 ! creates a mesh of elements on volume 1 asel,s,area,,5 nsla,s,1 cm,nn,node d,all,all,0 ! constrains displacements on forward face to zero asel,s,area,,1 asel,s,area,,2 nsla,s,1 cmsel,u,nn d,all,uz,0 !constrains the side faces in teh z direction to zero allsel,all /solu antype,2 32 LUMPM,0 PSTRES,0 MODOPT,LANB,mode_num,0,20000,,OFF ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum 33 fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,W:\m310186\RPI_project\fx-free-solid /prep7 vclear,1 vdele,all,all,,1 *enddo *cfopen,freq_tot,out *vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies( 1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10 ),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1, 15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) *cfclose 6.4 Fixed – Free Beam Element ANSYS Input File /prep7 k,1,0,0 ! sets a keypoint at (0,0) k,2,24,0 ! sets a keypoint at (24,0) l,1,2,36 ! creates a line with 5 divisions from keypoints 1 to 2 34 et,1,3 ! sets element type 1 to beam3, the 2d beam element in ANSYS mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 r,1,.75,9, ! defines property set 1 for the beam element (A,I,height) mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) lmesh,1 d,1,all,0 ! creates a mesh of elements on line1 (undo with lclear) ! constrains displacements at node 1 to zero /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 mode_num=10 *DIM, frequencies, ARRAY, mode_num + 1, rw, *do,j,1,rw fini run_name='modal_analysis_%j%' 35 /FILNAME,run_name,0 /syp,mkdir, run_name /CWD,S:\csmc_struct_64\RPI_project\fx-free-beam\%run_name% /solu r,1,inputs(j,3),inputs(j,2), ! defines property set 1 for the beam element (A,I,height) antype,2 LUMPM,0 PSTRES,0 MODOPT,LANB,mode_num,0,20000,,OFF ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' 36 r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,S:\csmc_struct_64\RPI_project\fx-free-beam *enddo *cfopen,freq_tot,out *vwrite,frequencies(1,1),frequencies(1,2), frequencies(1,3),frequencies(1,4),frequencies(1,5),frequencies(1,6),frequencies(1,7),freq uencies(1,8),frequencies(1,9),frequencies(1,10),frequencies(1,11),frequencies(1,12),freq uencies(1,13),frequencies(1,14),frequencies(1,15),frequencies(1,16),frequencies(1,17),fr equencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) 37 *cfclose 6.5 Simply Supported Solid Element ANSYS Input File /prep7 et,1,45 ! sets element type 1 to beam3, the 2d beam element in ANSYS mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) ESIZE,0.1,0, mode_num=10 !mode numbers to extract /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 *DIM, frequencies, ARRAY, mode_num + 1, rw, *do,j,1,rw fini run_name='modal_analysis_%j%' 38 /FILNAME,run_name,0 /syp,mkdir, run_name /CWD,/pw/data/csmc_struct_64/RPI_project/simply-supportedsolid/%run_name% /prep7 CSYS,0 WPAVE,0,0,0 wpcsys,,, BLOCK,0,24,0,inputs(j,1),.25,-.25, CSYS,0 WPAVE,0,0,0 CSYS,0 ofst=inputs(j,1)/2 wpoff,,ofst, wpro,,90.000000, VSBW, 1 vmesh,2 vmesh,3 ! creates a mesh of elements on volume 1 lsel,s,line,,19 lsel,a,line,,20 nsll,s,0 cm,nn,node d,all,ux,0,,,,uy, ! constrains displacements at node 1 to zero d,all,ux,0,,,,uy, 39 asel,s,area,,7 asel,a,area,,8 asel,a,area,,13 asel,a,area,,12 nsla,s,1 cmsel,u,nn d,all,uz,0 !constrains the side faces in teh z direction to zero allsel,all /solu antype,2 LUMPM,0 PSTRES,0 MODOPT,LANB,mode_num,0,20000,,OFF ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit 40 /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,/pw/data/csmc_struct_64/RPI_project/simply-supported-solid /prep7 vclear,all vdele,all,all,,1 *enddo *cfopen,freq_tot,out 41 *vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies( 1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10 ),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1, 15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) *cfclose 6.6 Simply Supported Beam Element ANSYS Input File /prep7 k,1,0,0 ! sets a keypoint at (0,0) k,2,24,0 ! sets a keypoint at (24,0) l,1,2,36 ! creates a line with 5 divisions from keypoints 1 to 2 et,1,3 ! sets element type 1 to beam3, the 2d beam element in ANSYS mp,ex,1,16e6 ! sets modulus of mtl 1 mp,prxy,1,0.3 ! sets poissons ratio of mtl 1 r,1,.75,9, ! defines property set 1 for the beam element (A,I,height) mp,dens,1,0.41443E-03 ! defines density mat,1 ! sets material to 1, not needed as it defaults to 1 real,1 ! sets the property set to 1 (this is the default) lmesh,1 d,1,ux,0,,,,uy ! creates a mesh of elements on line1 (undo with lclear) ! constrains displacements at node 1 to zero d,2,ux,0,,,,uy /INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized *DIM, input_parms, TABLE, rw-1, 3 !dimension table *DIM, inputs, ARRAY, rw-1, 4 !dimension array 42 *TREAD, input_parms,input_parms, csv,,1 !reads input file into table *MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array fini rw=rw-2 mode_num=10 *DIM, frequencies, ARRAY, mode_num + 1, rw, *do,j,1,rw fini run_name='modal_analysis_%j%' /FILNAME,run_name,0 /syp,mkdir, run_name /CWD,S:\csmc_struct_64\RPI_project\simply_supported_beam\%run_name% /solu r,1,inputs(j,3),inputs(j,2), ! defines property set 1 for the beam element (A,I,height) antype,2 LUMPM,0 PSTRES,0 mode_num=10 43 MODOPT,LANB,mode_num,0,20000,,OFF ematwrite,yes solve psolve,eiglanb ematwrite,yes MXPAND,mode_num, , ,0 PSOLVE,EIGEXP fit /post1 /output,run_name,freq set,list /output set,first plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_1.jpg' r_r='%run_name%000.jpg' *GET,reqf, ACTIVE, 0, set, frequency, frequencies(1,j)=j frequencies(2,j)=reqf /syp,ren ,r_r,i_i looper=mode_num-1 *do,k,1,looper l=k+1 z=k+2 set,next plnsol,u,sum fit /ui,copy,save,jpeg,grap,colo,vid,port,,100 i_i='mode_%l%.jpg' 44 r_r='%run_name%000.jpg' /syp,ren ,r_r,i_i *GET,reqf, ACTIVE, 0, set, frequency, frequencies(z,j)=reqf *enddo /CWD,S:\csmc_struct_64\RPI_project\simply_supported_beam *enddo *cfopen,freq_tot,out *vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies( 1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10 ),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1, 15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19) (f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f 10.1,f10.1,f10.1,f10.1) *cfclose 45 7. Appendix B 46 47 48 8. Appendix C 49 50 9. Appendix D 51 10.Appendix E 52