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PURDUE UNIVERSITY G rad u ate School Form 9 (Revised8'89) GRADUATE SCHOOL T hesis A cceptance This is to certify that the thesis prepared P« .Adams______________________________________ B y ______ Glynn_ Entitled Modelling and Computer Simulation of Rotor Chatter and Oscillating Bearing Forces in Twin Screw Compessors Complies with University regulations and meets the standards of the Graduate School for originality and quality For the degree o f Doctor of Philosophy________________________ Signed by the final examining committee: , chair Prof. Werner Soedel Prof. C.M. Krousgrily / s i . ' [ / Prof. Oleg Wasynczufc Approved by: /T/? < -/— vt /Departcrfent Head (J Date (School of Mechanical Engineering) IS is This thesis □ is not to be regarded as confidential Major Professor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MODELLING AND CO M PU TER SIMULATION O F R O T O R CH A TTER AND OSCILLATING BEARING FORCES IN TW IN SCREW COM PRESSORS A Thesis Subm itted to the Faculty of Purdue University by Glynn P. Adams In Partial Fulfillment of the Requirem ents for the Degree of Doctor of Philosophy December 1993 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI N u m b e r : 9638268 UMI Microform 9638268 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOW LEDGM ENTS The author would like to acknowledge the efforts of several people. F irst and fore­ m ost is Prof. W erner Soedel, not only for his technical input, b u t for his guidance. Special acknowledgments are also due to my rem aining com m ittee m em bers, Prof. J.D . Jones, Prof. C.M. Krousgrill and Prof. Oleg W asynczuk. This research effort was m ade possible through financial sponsorship by U nited Technologies Carrier Cor­ poration. Erric H eitm ann of Carrier deserves special recognition for his tim ely and extrem ely valuable help. T he work was conducted w ith the resources m ade available at the Ray W. Herrick Laboratories. Personal “thanks” are extended to all the fac­ ulty and student m em bers of Herrick Labs who were around during m y tim e there. Finally, I would be remiss if I did not single out John Huff. He has been my technical advisor, editor, handym an, babysitter, listener and above all friend. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page LIST O F T A B L E S ........................................................................................................... vii LIST O F F I G U R E S ....................................................................................................... viii N O M E N C L A T U R E ....................................................................................................... xiii A B S T R A C T .................................................................................................................... xv 1. IN T R O D U C T IO N .................................................................................................... 1 1.1 1.2 1.3 M o tiv a tio n ......................................................................................................... Research O b jectives........................................................................................ Screw Compressor Design and O p e r a tio n ................................................. 1.3.1 Compression P r o c e s s .......................................................................... 1.3.2 Basic Compressor D e sig n ................................................................... 2. LITERATU RE SURVEY 2.1 ...................................................................................... 2 2 3 3 5 18 Background on Screw C o m p re s s o rs ........................................................... 2.1.1 History of the Screw C o m p r e s s o r ................................................... 2.1.2 Advantages of Screw C om pressors................................................... Compressor Geom etry and Performance Analysis ................................ 2.2.1 Rotor Profiles and G eo m etry ............................................................. 2.2.2 Compressor P e rfo rm a n c e .................................................................... Gear M o d e ls...................................................................................................... 2.3.1 Spur Gear M o d els................................................................................. 2.3.2 Helical Gear M o d e l s ........................................................................... 2.3.3 Rigid Gear M o d e l................................................................................. Im pact Models ................................................................................... S u m m a r y .......................................................................................................... 18 18 21 22 22 24 28 28 33 35 38 43 3. R O TO R K IN E M A T IC S .......................................................................................... 45 2.2 2.3 2.4 2.5 3.1 Fundam entals of Gear K in e m a tic s.............................................................. 3.1.1 Angular Velocity R a t i o ....................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 47 V Page 3.1.2 Generating and Conjugate S h a p e s .............................................. . 3.1.3 Pressure Angle and Direction of Force T ra n sm issio n .................. 3.1.4 Definition of Driving P o i n t s .............................................................. 3.1.5 P a th of C o n ta c t..................................................................................... Fundam entals Applied to Original Rotor P ro files..................................... 3.2.1 K inem atic Inversion of the Rotor P ro file s ..................................... 3.2.2 Com putation of Male Profile N o rm a ls ........................................... 3.2.3 Generating and Conjugate Rotor P ro file s .................................... 3.2.4 Iterative M ethod for Profile G e n e ra tio n ......................................... Im plem entation of Profile Generation M e t h o d ........................................ 3.3.1 Final Profile Shapes and Contact L i n e ........................................ 3.3.2 Direction of Force Transmission for Final P r o f i l e s .................... Rotor C ontact C oefficien ts............................................................................ 3.4.1 Derivation of R otor C ontact C o e ffic ie n ts ..................................... 3.4.2 Contact Coefficients for New P r o f i l e s ........................................... Conclusions Based on K inem atic A n a ly s is ................................................ 49 53 55 57 59 60 61 63 63 64 66 66 67 67 72 73 R O TO R LOADING DUE T O GAS C O M P R E S S IO N ..................................... 94 3.2 3.3 3.4 3.5 4. 4.1 4.2 4.3 Overview of M e t h o d ....................................................................................... Integration of a Scalar over a S u r f a c e ......................................................... Integration M ethod Applied to the Screw Compressor R o to r s ............. 4.3.1 M apping the 3-D Rotor Surface to a 2-D R e g io n ........................ 4.3.2 Resulting I n t e g r a l s ............................................................................. 4.3.3 Evaluation of Integral L im its............................................................ 4.3.4 Revisions Required for Application to the Fem ale R otor . . . . Volume and Pressure C o m p u ta tio n s............................................................ Com putations Applied to Research D a t a ................................................... 94 96 98 99 100 101 104 105 107 5. MODEL AND C O M PU TE R S IM U L A T IO N ...................................................... 120 4.4 4.5 5.1 5.2 5.3 Overview of M o d e l ........................................................................................... 5.1.1 Basic Assum ptions U t i l i z e d ............................................................. 5.1.2 R otor C o n ta c t....................................................................................... 5.1.3 Im pact M o d e l ....................................................................................... 5.1.4 Load P a r a m e te r s ................................................................................. Equations of M o t i o n ....................................................................................... 5.2.1 Equations of M otion for the C ontact M o d e .................................. 5.2.2 Equations of M otion for the Independent M o d e ........................ D eterm ination of T ra n sitio n s.......................................................................... 5.3.1 Transition from C ontact to Independent M o d e s ........................ 120 120 121 121 122 123 123 125 126 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi Page 5.3.2 Im pact, Transition from Independent to C ontact Modes . . . . Com puter S im u la tio n ...................................................................................... Numerical Integration ................................................................................. Evaluation of C ontact Mode and T ra n s itio n s ............................................. Test C a s e s .......................................................................................................... 126 129 130 131 132 6. SIMULATION R E S U L T S ....................................................................................... 138 5.4 5.5 5.6 5.7 6.1 6.2 O perating Conditions E v a lu a te d .................................................................... Bearing Forces Com puted from Simulations ............................................ 138 139 7. CONCLUSIONS AND R E C O M M E N D A T IO N S ............................................... 170 7.1 7.2 Kinematics ....................................................................................................... Bearing Forces ................................................................................................ 170 171 LIST O F R E F E R E N C E S .............................................................................................. 175 VITA 183 .................................................................................................................................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table Page 6.1 Port tim ing, Ospc and &d p o , for various compressor capacities....................... 141 6.2 Sim ulated operating conditions, capacity, V,, compression condition, Pduc- 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST O F FIGURES Figure Page 1.1 Typical screw compressor rotors with one cham ber em phasized......... 9 1.2 Suction process in a screw compressor; (a) 6m = 60.0 degrees; (b) 6m = 120.0 degrees; (c) Bm = 240.0 degrees; (d) 6m = 300.0 degrees....................................... 10 1.2 (c o n tin u e d ) ....................................................................................................... 11 1.3 Compression process in a screw compressor; (a) 0m = 360.0 degrees; (b) 6m = 420.0 degrees; (c) 8m = 480.0 degrees; (d) 0m = 540.0 degrees....................................... 12 1.3 (c o n tin u e d ) ....................................................................................................... 13 1.4 Compression process shown w ith interlobe seal curve on m ale rotor. . . . 1.5 Typical screw compressor rotor profiles...................................................... 15 1.6 End view of compressor housing with rotor profiles shown................... 16 1.7 (a) Suction porting; (b) discharge porting................................................. 17 3.1 Fundam ental condition for constant angular velocity ratio ................... 74 3.2 Reference position of pitch circles for generating a conjugate shape. . . . 3.3 R otated position of pitch circles where point Q is generating th e conjugate point S ............................................................................................................... 76 3.4 Condition required for a point to transm it force from gear a to gear b .. 3.5 Original profiles from d a ta supplied for research; (a) m ale profile; (b) female profile.............................................................. 3.6 14 78 Kinem atic inversion m ethod applied using th e m ale profile to generate the fem ale................................................................................................................. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 77 ^ Figure 3.7 — Page Comparison of one flute of the specified female profile w ith one flute of the profile generated by kinem atic inversion..................................................... 80 3.8 Normal vectors to the original m ale profile, x and y com ponents............ 81 3.9 Correspondence between discrete points on th e original m ale profile and the generated conjugate female profile............................................................... 82 3.10 Conjugate female profile generated w ith original m ale profile.................... 83 3.11 Schematic of iterative procedure used for profile generation....................... 84 3.12 Revised m ale profile obtained through iterative generation m ethod. . . . 85 3.13 Revised female profile obtained through iterative generation m ethod. . . 85 3.14 Contact curve associated w ith final profiles..................................................... 86 3.15 Driving points on the final m ale profile............................................................ 87 3.16 Direction cosines as a function of 8m for the final profiles.............................. 88 3.17 Typical bearing configurations for a single compressor ro to r...................... 89 3.18 Contact coefficients vs 0m, m ale suction bearing; (- ) C B l Xm; ( - - - ) C B l ym-, ( - . - ) C B l Zm...................................................... 90 C ontact coefficients vs 6m, m ale discharge bearing; (- ) C B 2 Xm; ( - - - ) C B 2 ym................................................................................... 91 Contact coefficients vs 6m, female suction bearing; ( - ) C B 1 X}] ( - - - ) C B l yj] ( - . - ) C B l Zf........................................................ 92 Contact coefficients vs 6m, female discharge bearing; (— ) C B 2 Xj\ ( - - - ) C B 2 yj .................................................................................... 93 4.1 Male rotor with one section emphasized........................................................... 109 4.2 Interlobe Seal Curve on Male Rotor .............................................................. 110 4.3 Progression of Interlobe Seal Curve on Male R o t o r .................................... Ill 4.4 M apping from 3 dimensional surface to 2 dimensional region..........................112 3.19 3.20 3.21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.5 Page Coordinate system used to m ap 3 dimensional rotor surface to a 2 dim en­ sional region. D em onstrated on a cylindrical surface...................................... 113 Cham ber pressure vs 6m for 100% capacity, Pauct = 45psia; (— ) Pdiac = 152psia (ideal); ( )P<n*c — 117psia (over-pressure); (— • —)Pdi,c = 253psia (under-pressure)............................................................ 114 Suction bearing forces due to compression loads, m ale rotor, 50% capacity, Pauct = 45psia, Pdisc = 152psia (under-pressure); ( - ) B l Xm , (--------) B l ym, ( - • —) B l Zm......................................................... 115 Discharge bearing forces due to compression loads, m ale rotor, 50% capacity, Pauct = 45psia, Pdisc = 152psia (under-pressure); ( - ) B 2 Xm , (--------)B 2 ym.................................................................................... 116 Suction bearing forces due to compression loads, female rotor, 50% capacity, Pauct = 45psia, Pdisc —152psia (under-pressure); (-)£ !« / , ( ) B1 UJ, ( - - - ) B l Zf........................................................... 117 4.10 Discharge bearing forces due to compression loads, female rotor, 50% capacity, Psuct = 45psia, Pdisc = 152psia (under-pressure); ( - ) B 2 XJ , ( )B 2 y i..................................................................................... 118 4.11 M oments about the 2-axis due to compression loads, 50% capacity, Pauct = 45psia, Pdi„c = 152psia (under-pressure); ( - ) M Zm , ( ) MZf....................................................................................... 119 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.4 (a) Clearance between rotors, (b) schem atic diagram of rotor contact m odel............................................................................................................ 134 Exam ple of alternating contact chatter vibrations, (— ) envelope representing th e female contact points; (--------- ) displacem ent of the m ale contact point.............................................. 135 Exam ple of one-sided contact chatter vibrations, (— ) envelope representing the female contact points; ( --------- ) displacem ent of th e m ale contact point.............................................. 136 Exam ple of m ultiple contact chatter vibrations, (—) envelope representing the female contact points; ( ) displacement of the m ale contact point.............................................. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.1 Male rotor suction bearing force, B l Xm vs tim e, 50% capacity, Pauct = 45psza, Pdisc — 152psia(under —p ressu re); (a) approxim ately 60 revolutions; (b) approxim ately 9 revolutions. Page . . . 143 Male rotor suction bearing force, B l Xm, 50% capacity, Pauct = 45psia, Pdi,c — 152psia(under —p ressu re); ( -------- ) force due to rotor contact, (— ) force due to compression loads. 144 Frequency spectra, B l Xm and B 2Xm, 50% capacity, P aU c t = 45 psia, P d i „ c = 91 psia (ideal); (a) B l Xm\ (b) B 2 Xm........................................................ 145 Frequency spectra, B l ym and B 2 ym, 50% capacity, P , u c t = 45 psia, P d i s c = 91 psia (ideal); (a) B l ym\ (b) B 2 ym........................................................ 146 Frequency spectra, B l Xf and B 2 XJ, 50% capacity, P 3U c t = 45 psia, P d i s c = 91 psia (ideal); (a) B l Xj\ (b) B 2 X}......................................................... 147 Frequency spectra, B l yj and B 2 yj , 50% capacity, Pauct = 45 psia, P d i s c = 91psza (ideal); (a) B l yf\ (b) B 2 yj.......................................................... 148 Frequency spectra, B l Zm and B 12/, 50% capacity, P ,uct = 45 psia, P d i s c = 91psia (ideal); (a) B \ Zm\ (b) B l Zj......................................................... 149 Frequency spectra, B \ Xm and B 2Xm, 50% capacity, P , u c t = 45 psia, P d i s c = 70psia (over-pressure); (a) B l Xm; (b) B 2 Xm........................................ 150 Frequency spectra, B l ym and B 2 ym, 50% capacity, P,uct = 45 psia, = 70psia (over-pressure); (a) B l ym] (b) B 2 Vm......................................... 151 6.10 Frequency spectra, B \ Xj and B 2 Xf, 50% capacity, P SUct = 45 psia, P d i s c = 70psza (over-pressure); (a) B 1X/; (b) B 2 Xf.......................................... 152 6.11 Frequency spectra, B l yj and B 2 y}, 50% capacity, P S u c t = 45 psia, P d i s c — 70psia (over-pressure); (a) B l yj\ (b) B 2 yj.......................................... 153 6.12 Frequency spectra, J91*m and B \ zn 50% capacity, Pauct = 45 psia, P d i s c = 70psza (over-pressure); (a) B l 2m; (b) B l Zf.......................................... 154 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 P d is c 6.13 Frequency spectra, B l*m and B 2 Xm, 50% capacity, Pauct = 45 psia, P d i s c = I52psia (under-pressure); (a) B l Xjn\ (b) B 2 Xm........................................155 6.14 Frequency spectra, B l ym and B 2 Vm, 50% capacity, P 3Uc t = 45 psia, P d i s c = 152psza (under-pressure); (a) B l ym\ (b) B 2 ym.................................... 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 6.15 Frequency spectra, B 1 XJ and B 2 XJ, 50% capacity, P ,uct = 45 psia, P d i s c = 152psia (under-pressure); (a) B \ Xj; (b) B 2 X/....................................... 157 6.16 Frequency spectra, B 1 VJ and B 2 yj, 50% capacity, P„uct = 45 psia, Pdisc = 152psia (under-pressure); (a) B l y/; (b) B 2 yj ....................................... 158 6.17 Frequency spectra, B l Zm and B 1Z/, 50% capacity, P ,uct = 45 psia, P d i s c = 152psia (under-pressure); (a) B \ Zm\ (b) B 1 Z/....................................... 159 6.18 Frequency spectra, B l Xm and B 2 Xm, 75% capacity, P i U c t — 45 psia, P d i s c = 215psia (under-pressure); (a) B l Xm; (b) B 2 Xm................................. 160 6.19 Frequency spectra, B l ym and B 2 ym, 75% capacity, P ,uct = 45 psia, P d i s c = 215psia (under-pressure); (a) B l ym\ (b) B 2 Vm.................................. 161 6.20 Frequency spectra, B l x/ and B 2 Xf, 75% capacity, P,uct = 45 psia, P d i s c = 215psia (under-pressure); (a) B l Xf] (b) B 2 X}................................... 162 6.21 Frequency spectra, B l yj and B 2 yj, 75% capacity, P3Uct — 45 psia, Pdisc = 215psia (under-pressure); (a) B \ yj \ (b) B 2 yf ................................... 163 6.22 Frequency spectra, B l 2m and B l Z/, 75% capacity, P a u c t = 45 psia, P d i s c = 215psia (under-pressure); (a) B l Zm; (b) B 1 ZJ................................... 164 6.23 Frequency spectra, 2?lIm and B 2 Xm, 100% capacity, P , u c t = 45 psia, P d i s c = 253psia (under-pressure); (a) B l Im; (b) B 2 Xm.................................. 165 6.24 Frequency spectra, B l ym and B 2 ym, 100% capacity, P ,uct = 45 psia, Pdisc = 253psia (under-pressure); (a) jBlWm; (b) JB2„m....................................... 166 6.25 Frequency spectra, B l XJ and B 2 Xl, 100% capacity, P aU c t = 45 psia, P d i s c = 253psia (under-pressure); (a) B l Xj\ (b) B 2 Xf................................... 167 6.26 Frequency spectra, B 1 VJ and B 2 yf, 100% capacity, Psuct = 45 psia, P d i s c = 253psia (under-pressure); (a) B l yj\ (b) B 2 VJ.................................... 168 6.27 Frequency spectra, B l Zm and B 1 Z{, 100% capacity, Pauct = 45 psia, P d i s c — 253psia (under-pressure); (a) B l 2m; (b) B \ Zf.................................... 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NOM ENCLATURE B l Xmj ^-com ponent of suction bearing force, m ale and fem ale rotors B l y mj y-component of suction bearing force, m ale and fem ale rotors B l Zm l 2 -com ponent B 2 Xm J x-com ponent of discharge bearing force, m ale and fem ale rotors B 2 Vm I y-component of discharge bearing force, m ale and fem ale rotors of suction bearing force, m ale and fem ale rotors Cfii- m,/ , contact coefficient associated with B l xmjw Cbi„»m,/ contact coefficient associated with B l vm * / Cr i, m , f contact coefficient associated w ith B l zm r Cj9 2 r , contact coefficient associated w ith B 2 xmj Cs 2 ym contact coefficient associated w ith B 2 xmj C capacity rating of compressor, percentage cc center-to-center distance between m ale and female rotors FXin t x-com ponent of the compression forces, m ale and fem ale rotors Fymj y-component of the compression forces, m ale and fem ale rotors FZmJ z-com ponent of th e compression forces, m ale and fem ale rotors L length of m ale and fem ale rotors M XmJ m om ent about the m ale x-axis due to compression, m ale and fem ale rotors MymJ m om ent about the m ale y-axis due to compression, m ale and female rotors M ZmJ m om ent about the m ale 2-axis due to compression, m ale and fem ale rotors mag m agnitude of normalized surface norm al Nm num ber of lobes on m ale rotor Nj num ber of flutes on fem ale rotor Npy y-component of contact force between rotors Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiv n ratio of specific heats for compressed gas Ttcp num ber of contact points P chamber pressure p em chamber pressure as a function of P suct suction pressure P disc discharge pressure Pax axial pitch of rotors Rm discrete radial coordinate of m ale profile Rj discrete radial coordinate of female profile RmP male rotor pitch circle R fP female rotor pitch circle Vem cham ber volume as a function of 6m 6m Greek Symbols a direction cosine associated with x direction P direction cosine associated w ith y direction Pm change in rotor radius as a function of 0 m 7 direction cosine associated with z direction average angular velocity of m ale rotor Uf average angular velocity of female rotor <t> integration variable associated w ith helical twist pressure angle for gear pair 7*m wrap angle of m ale rotor Ts wrap angle of fem ale rotor 6 integration variable associated w ith polar coordinates 0m angular position of m ale rotor Of angular position of female rotor Qm discrete polar coordinate of m ale profile ©/ discrete polar coordinate of female profile Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. XV ABSTRACT Adams, Glynn P., Ph.D ., P urdue University, December 1993. Modelling and Com­ puter Simulation of Rotor C h atter and Oscillating Bearing Forces in Twin Screw Compressors. M ajor Professor: Prof. W erner Soedel, School of Mechanical Engineer­ ing. A model of the rotor interaction in twin screw compressors is developed, and im plem ented via a com puter sim ulation. Geometric param eters of the rotors are used to develop a kinem atic analysis of the rotor m otion, and to com pute the loads associated w ith the compression process. The m ain objectives are to provide the ability to predict backlash type rotor vibrations (chatter) and com pute the bearing forces. The rotor surfaces are defined by th e 2-dimensional rotor profiles and the helix angle associated w ith each rotor. This geometry is used to identify the kinem atic constraints which apply to the rotor m otion based on gear theory. The nature of the contact between the rotors and the resulting force transm ission is investigated. The rotor profiles are designed to obey the laws of conjugacy. This fact is used in the development of an iterative procedure for generating m ating rotor profiles. The forces and m om ents due to gas compression are com puted using vector cal­ culus principles to integrate th e cham ber pressure over th e rotor surfaces. T he 3dimensional surface of each rotor is m apped to a 2-dimensional region and the pres­ sure integrated over the region. The general m ethod is presented, w ith the details for a specific compressor configuration given. Of prim ary interest is the accurate com putation of the m om ent load, due to compression, about th e axis of rotation for each rotor. Through these com putations, it is dem onstrated th a t the m om ent load induced on the female is approxim ately 10% of th a t induced on th e male. Therefore, the female rotor m otion approaches th a t of an idler gear. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xvi The bearing forces are com puted for a specific compressor configuration for various operating conditions. The total bearing force is separated into com ponents caused by the contact force between the rotors and components caused by compression loads. The component due to compression loads dom inate the bearing forces. A frequency analysis dem onstrates th a t the frequency content of the bearing forces does not de­ crease substantially until beyond approxim ately the 8th harm onic of th e fundam ental screw frequency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1. INTRODUCTION The origins of the twin screw compressor development can be traced to 1934. To­ day the screw compressor is utilized throughout the refrigeration and gas compression industries. Recent applications include the use of a screw compressor in implem enting a Miller cycle on an autom otive engine [1]. For m any applications, the current screw compressor designs have proven to be m ore reliable, more efficient and quieter than the competing designs. The twin screw compressor developm ent, from conception to m arket, has been unique. One of the most im portant param eters required for de­ signing an effective screw compressor is the shape of the rotor profiles. However, the patent for the rotor shapes which are most often utilized is held by a company which does not m anufacture the compressors. Instead, several companies have been licensed to m anufacture and sell compressors which use the rotor designs. This circum stance has created an atm osphere of com petition among the m anufacturers to refine their basic products and m ake them available to industry as quickly as possible. There­ fore, the early engineering efforts concentrated m ore on production techniques and evaluation of overall compressor designs, rather than studies and refinement of the basic profiles. In recent years, screw compressor m anufacturers and researchers have initiated analytical studies to determ ine improvements to the basic profile and overall com­ pressor designs. Many of these studies are presented in th e literature survey. A rather extensive am ount of work has been published which addresses th e role of the geometric param eters of the screw compressor rotors and rem aining components. The effects of variations in these param eters on leakage paths, volum etric and adiabatic efficiencies, and gas pulsations have been evaluated through analytical modelling and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 i |^ com puter sim ulations, and m easurem ents. In addition, a considerable am ount of work has been performed to address the development of therm odynam ic models to predict efficiencies, leakage flow rates, and gas pulsations once the geom etric param eters of the compressor are defined. 1.1 M otivation As w ith m ost machinery, reduction of radiated noise is an im portant concern. Noise levels which are potentially hazardous m ust be reduced for obvious reasons. However, generation of noise in mechanisms usually stem s from sources which also cause wear and inevitably failure. Recent studies have been conducted to investigate the characteristics of the noise radiated from an operating screw compressor [2]. These studies suggested th a t the rotor interaction, and subsequent vibration mechanisms, are a significant source of noise a t low load operating conditions. An understanding of the rotor interaction m ust be achieved so th a t noise generated through rotor interaction can be reduced. Once the fundam entals of the rotor inter^ action, and effects of the param eters which influence this interaction, are understood, the compressor designs can be improved to prevent vibrations and corresponding radiated noise. This provides th e m otivation for th e current research efforts. 1.2 Research Objectives There are two m ajor objectives for this research. The first is to provide a basic model of the rotor interaction. This model will include th e m ain physical param eters of the rotors, as well as the loading param eters which affect th e rotor m otion. The m ain purpose is to provide a m odel which can be utilized to obtain insight into how each of th e rotor param eters and loading param eters affect th e rotor m otion. One aspect of this objective is to provide the ability to predict ch atter vibrations. As defined here, chatter refers to the loss of contact and subsequent im pact between the male and female rotors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 The second m ajor objective for this research is to develop a m odel of th e bearing loads which result from the compression loadings and rotor interaction. D eterm ina­ tion of the bearing loads will provide further insight into noise sources which exist at the interface of the bearings and compressor housing. 1.3 Screw Compressor Design and Operation Various types of screw compressors have been designed and developed. T he work contained in this docum ent is directly applied to tw in screw compressor technology. Unless otherwise stated, the term “screw compressor” as used here, refers to th e twin screw compressor design. In order to appreciate the work presented, one m ust first have a basic understanding of th e design and operation of a tw in screw compressor. In this section an overview of the compression process is first presented. T he compressor components and their general arrangem ent are then described. The objective is to provide an understanding of the basic compression process which occurs in th e twin screw compressor. 1.3.1 Compression Process T he twin screw compressor is a positive displacem ent, axial flow, rotary compres­ sor w ith helical compression chambers. T he m ale and fem ale rotors are th e m ain components of a twin screw compressor. These are helical rotors which function as both a compression m echanism and a gear pair. Shaw [3] presents a concise description of the compression process for screw com­ pressors using schematics to represent unwrapped versions of the helical rotors. A brief description of the compression process is given here using com plete representa­ tions of th e 3D rotors. A typical set of screw compressor rotors is shown in Figure 1.1. Each rotor resembles a helical gear w ith a low tooth num ber. T he m ale rotor is iden­ tified as having 6 teeth, referred to as “lobes” . T he female rotor for this configuration has 7 teeth, referred to as “flutes” . In this figure, th e m ale rotor rotates clockwise, as viewed from th e suction end, and drives the female in th e counter-clockwise direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Helical cham bers provide the compression m echanism. Each cham ber consists of two helical sections, one on each rotor. These helical sections are bounded by th e tips of the lobes on the male rotor and the tips of the flutes on the female rotor. A single compression chamber is em phasized in Figure 1.1. The operation of a twin screw compressor can be understood with the aid of Figures 1.2 and 1.3. In these figures, the angular position of the m ale rotor is designated as 0m. T he suction process is presented in Figure 1.2. For this rotor configuration, the suction process occurs along the top of the rotors. Several cham bers exist at various stages of compression simultaneously. For clarity, a single cham ber is emphasized through the suction process. The contact curve formed by th e meshing of the rotors is commonly referred to as the interlobe seal curve. The curve seals adjacent compres­ sion chambers as well as sealing the top section of the rotors from the bottom section. The interlobe seal curve is shown in Figure 1.4. A compression cham ber begins a t the suction plane and forms along the rotor axis on the top of th e rotors. T he chamber is bounded by the suction plane and the upper portion of th e interlobe seal curve. As the rotors rotate, the interlobe seal curve progresses towards the discharge plane. Therefore, the volume of the cham ber increases. The cham ber is exposed to suction porting throughout this process and rem ains at the suction pressure. T he suction process is completed when th e interlobe seal curve reaches th e discharge plane and the cham ber is formed throughout the length of the rotor. T he compression process is presented in Figure 1.3. During the suction process, the cham ber forms on the top of the rotors. The chamber is bounded by the suction plane and the upper portion of the interlobe seal curve. T he compression process occurs in a sim ilar m anner on the bottom section of the rotors. The boundaries of the compression chamber are defined by the lower portion of th e interlobe seal curve and the discharge plane. As th e rotors continue to rotate, the helical chambers wrap around the circumference of th e rotors. T he interlobe seal curve progresses along the rotor axis, towards the discharge plane. Therefore, the volume of the chamber Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 decreases, providing compression. Discharge occurs when the compression chamber is exposed to the discharge porting. Figure 1.4 is an additional representation of the compression process. In this fig­ ure, the female rotor has been removed. One compression cham ber on the m ale rotor is emphasized with the interlobe seal curve imposed. The interlobe seal progresses from the suction plane to th e discharge plane as the rotor rotates. T he top portion of th e cham ber, from th e suction plane to th e seal curve, is exposed to suction pressure and fills w ith gas at th e suction pressure as the volume of the cham ber increases. The bottom portion of the cham ber, from the seal curve to the discharge plane, is exposed to neither suction or discharge porting. As the seal progresses towards the discharge plane, the volume of this cham ber decreases and th e gas is compressed. The capacity of the compressor is defined in term s of the volume of the helical chambers. In addition, the volume ratio, V;, is specified as th e ratio of the volume at the beginning of compression to the volume a t th e onset of discharge. These param eters are dependent upon the geometry of the rotors and th e design of the rem aining compressor components. 1.3.2 Basic Compressor Design Knowledge of th e basic compression process which occurs in a screw compressor is key to understanding the design of the basic compressor components. These basic components of a twin screw compressor are: 1. m ale and female rotors, 2. rotor housing, 3. suction and discharge planes and porting, 4. volume ratio and capacity control mechanism, 5. drive unit. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 The m ale and female rotors are the key components of a screw compressor. The geometry of the rotors is one of the m ost im portant factors in determ ining th e com­ pressor performance. This geometry affects the shape of the interlobe seal curve, leakage paths, the torque transmission, etc. In the oil-injected/flooded screw com­ pressor, the rotors not only serve to form the compression cham bers, b u t also function as a gear pair. Typically the m ale rotor drives the female rotor. The male and female rotors of a screw compressor are conjugate helical shapes. The rotors have a low num ber of teeth in comparison with helical gears commonly used for m otion and force transmission. The geometry of each rotor is completely defined by the 2D profile, taken as a cross section perpendicular to th e rotor axis of rotation, the rotor length and the wrap angle. The 2D profile of a typical set of screw compressor rotors is presented in Figure 1.5. In this example, th e m ale rotor consists of 6 lobes and the female rotor consists of 7 flutes. It should be noted th a t th e lobes and flutes are asymm etrical. T he profiles are not defined by any of th e classical gear profiles, such as involute, circular or elliptical. The sym m etry of th e entire rotor is obtained by repeating the definition of a single lobe or flute around th e circumference of the rotor. A typical pair of screw compressor rotors are shown in Figure 1.1. T he geometry of each rotor is completely described by the two dimensional profiles, th e num ber of rotor lobes, the rotor length and the wrap angle. T he m ale and female rotors are conjugate, helical shapes. Therefore, if these geometric param eters are given for one of the rotors, along w ith the center-to-center distance, th e other rotor is completely defined. During a compression cycle, helical chambers are form ed between th e tips of adjacent lobes on the m ale rotor and between the tips of adjacent flutes on the female rotor, as presented in Figures 1.2 to 1.3. T he compressor housing is shown in Figure 1.6. T he housing is formed by two partial cylinders which encase each of the rotors. Each cylinder extends th e length of the rotors, from the suction plane to the discharge plane. Vertices are formed at the intersection of the two cylinders. The meshing zone betw een th e m ale and female Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 rotors is the region between the vertices. The interlobe seal curve is defined by the contact between the rotors which occurs in the meshing zone. A leakage path exists due to clearances between the housing vertices and th e rotor lobe and flute tips. This leakage path is commonly referred to as the blow hole. The interface between the housing and the lobe and flute tips around the rem aining circumference of each rotor also serves as a seal between adjacent compression chambers. T he clearance at this interface is an additional leakage path. T he suction and discharge end planes are located a t the corresponding ends of the rotors. These planes serve two purposes. F irst, they provide th e sealing a t the rotor faces required to develop the compression chambers. In addition, suction and discharge ports are provided w ithin these end planes. These axe referred to as axial ports, Figures 1.7. W hen a compression chamber becomes exposed to th e suction or discharge port, the cham ber pressure is affected by the respective pressure. The capacity and volume ratio of th e screw compressor depend upon th e design of the suction and discharge ports. In order to design a compressor to operate within a range of conditions, both th e volume ratio and capacity m ust be variable. One means of providing a variable effective volume ratio and capacity control is w ith valves installed in the suction and discharge porting. Unlike valves designed for rotary vane and reciprocating type compressors, these valves do not open and close during each compression cycle. Instead, the effective axial port area is varied by setting th e valves in either the open or closed position. The effective volume ratio is varied by changing the axial discharge port area. By opening the valve, the compression cham ber is exposed to the discharge pressure at an earlier stage of compression. T he cham ber volume a t the onset of dischaxge is therefore increased, causing a related decrease in th e effective volume ratio. Capacity control is obtained w ith valves in the axial suction port area. In the early stage of compression, a cham ber is opened to the suction porting by setting the valve in th e open position. T he gas contained within the cham ber is bypassed to the suction pressure. This reduces the effective capacity of the compressor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Due to the helical geometry of the cham ber, the effective porting is not only in the plane of the axial port area, but also extends inward along th e rotor axis. This effect provides radial porting. The locations of th e housing vertices along th e rotor axis define the am ount of radial porting. The m inim um am ount of radial porting occurs when the vertices of th e housing extend the full length of th e rotors, from the suction to the discharge planes. As these vertices are moved inward, away from the respective planes, the radial port area is increased. An additional means of providing capacity control and variable volume ratio is to adjust the axial locations of the housing vertices with the use of a slide valve. The slide valve is located a t the bottom of the meshing zone of the rotors and has a shape sim ilar to th e compressor housing. The difference between the slide valve and the housing is th a t th e slide valve is movable. The effective volume ratio of the compressor is varied by moving the slide valve, and therefore the housing vertex, inward from th e discharge plane. This affects the tim ing of exposure to discharge pressure and therefore changes the chamber volume a t the onset of compression. This directly affects the volume ratio. Capacity control is provided by moving the slide valve inward from th e suction plane. This allows partially compressed gas to bypass to the suction pressure early in the compression cycle, therefore reducing the effective capacity of th e compressor. As was stated earlier, the rotors act as a gear pair, with th e m ale rotor usually driving the female rotor. This can be accomplished by m ounting th e electrom otor rotor directly to the m ale rotor shaft, or by driving the rotor through an external drive. In the case of direct m ounting, the entire electrom otor is norm ally enclosed within the compressor shell. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Female Figure 1.1 Typical screw compressor rotors with one chamber emphasized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Discharge Plane Suction Plane (b) Discharge Plane Suction Plane Figure 1 2 Suction process in a screw compressor; (a) 6m 60.0 degrees; (b) 0m — 120.0 degrees; 240.0 degrees; (d) 6m = 300.0 degrees. (c) Om Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 (C) Discharge Plane Suction Plane (d) Discharge Plane Suction Plane Figure 1.2 (continued) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 (a) Suction Plane Discharge Plane Suction Plane Discharge Plane Figure 1.3 Compression process in a screw compressor; (a) 6m = 360.0 degrees; (b) 0m = 420.0 degrees; (c) 8m = 480.0 degrees; (d) 9m = 540.0 degrees. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 (C) Suction Plane Discharge Plane (d) Suction Plane Discharge Plane Figure 1.3 (continued) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Figure 1.4 Compression process shown with interlobe seal curve on m ale rotor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Lobe Flute Male Rotor Female Rotor Figure 1.5 Typical screw compressor rotor profiles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Figure 1.6 End view of compressor housing with rotor profiles shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 (a) <b) Figure 1.7 (a) Suction porting; (b) discharge porting. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 2. LITERATURE SURVEY The literature survey is divided into four m ain sections. In the first section a brief history of the development of the twin screw compressor is presented, along with recent publications concerning advantages and advancem ent of screw compressor applications. The second section is a review of studies of the geometric param eters of compressor rotors. These consist m ainly of various m ethods used to represent the rotor profiles m athem atically, and m ethods to com pute the overall effect of various profiles on compressor performance. Also included in this section, are publications in which therm odynam ic models of the compression process are presented for analyzing compressor performance and gas flow characteristics. T he th ird section is a review of studies of gear vibration models. This section includes models of complete gear systems along with models of single gear pair interactions. In addition studies of the effects of dynamic loads on gear systems are included, as are studies in which helical gears in particular are addressed. T he last section is a review of models used to study mechanisms which involve im pact and experim ents performed to m easure im pact phenomena. 2.1 Background on Screw Compressors This section includes a brief history of the twin screw compressor. In addition, some advantages of the screw compressor are also presented. 2.1.1 History of the Screw Compressor The use of the screw for transferring fluids can be traced back to Archimedes and his use of the screw for pum ping irrigation water. However, the m odern screw Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 I compressor has a more recent history. In 1878 a Germ an engineer obtained the first patent for a twin screw compressor. It was a dry running type and was capable of pressure ratios of 2:1. O ’Brien [4] credits this to an engineer nam ed Grigar, while Laing [5] credits this to Kruger, the former seemingly being a translation problem. The invention of the m odern screw compressor is generally credited to Alf Lysholm [4, 6]. In 1934 he was chief engineer at Svenska Rotor M askiner AB (SRM), when he conceived the idea as part of gas turbine development. Lysholm’s original design was a dry running type compressor. The m ale and female rotors did not contact each other, rather their motion was constrained using a pair of tim ing gears. SRM did not m anufacture the compressor, but controlled the patent and licensing arrangem ents for the technology. They began licensing the twin screw compressor technology in 1946. In 1952, Hans Nilson was chief engineer at SRM. At this tim e he introduced the circular profiles. These provided higher pressure ratios and improved the compressor efficiency. Also at this tim e a Holyrod cutter was used by Howden Company, one of the licensees of the screw compressor technology. This cu tter increased the speed at which the rotors could be m anufactured, thereby reducing th e m anufacturing cost. Two developments in the early 1950’s increased the range of compression applica­ tion for which screw compressors were suitable, th e introduction of the slide valve and the development of oil injection. The introduction of the slide valve m ade it possible to vary the pressure ratio. Until this tim e each compressor had a fixed pressure ratio which was inherent to the compressor geometry. W ith th e slide valve, the pressure ratio is varied by physically changing the geometry of the porting areas and through bypassing some of the compressed gas back to the suction side of the compressor. The second development which occurred at this tim e was the development of oil injected compressors. Initially, oil free compressors had been used in th e refrigeration industry by enclosing them within pressure vessels to contain the seal leakage, [5], This was no longer necessary with oil injected compressors. Unlike the dry running compres­ sor, the rotors of an oil injected compressor form a gear pair. Therefore, the tim ing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 gears are no longer necessary, decreasing the num ber of moving parts required. In addition, the oil plays an im portant role in the compression process. The oil provides additional sealing between the compression chambers. This increases the volumetric efficiency by reducing the am ount of leakage. This sealing property of the oil reduces the am ount of energy utilized for compressing gas which eventually leaks back to a lower pressure chamber. The oil also provides a source for cooling the compressed gas. This cooling effect helps decrease the discharge tem perature, m aking the compressor suitable for a wider range of applications. The first oil injected compressor tests were carried out by Howden in 1955 [7]. These were tests of an oil-free type compressor with oil sprayed into the suction port. The oil injected compressor was first used in industry in 1957, by Atlas Copco for air compression [6]. In the early 1960’s the developments of the slide valve and oil injection were combined for use in the refrigeration industry. One of the most recent m ajor developments was the use of asym m etric profiles in 1969. The leakage path area and sealing line length are both reduced as compared to the sym m etric profile designs. These reductions translate into improved compressor efficiency. Nearly all m odern screw compressors m anufactured today utilize an asym ­ m etric design, most of which are some derivative of the SRM “D” profile which was released in 1982. In addition recent developments in rotor m anufacturing, credited to Rune Nilson of SRM, have reduced the machining tim e required to m anufacture the rotors, from hours to m inutes, by the use of precision carbide form cutting tools [6]. The development of the screw compressor has been unique. Although SRM holds the patent on the technology, they do not m anufacture screw compressors or the rotors. The technology has been licensed to m anufacturers who have achieved much of the development in cooperation w ith SRM. Most of the early work concentrated on developing the technology for application to suitable m arkets. Com petition has rem ained a driving force between the licensees for being first to produce suitable compressors and first to update designs as new concepts were introduced. In recent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 years, however, more attention has been focused on developing analytical models of the compressors and defining improvements to the basic design. 2.1.2 Advantages of Screw Compressors T he use of screw compressors in the refrigeration industry and gas processing industries has continued to increase since it was first introduced. In m any cases the screw compressor has replaced reciprocating and rotary type compressors. This is due to the advantages which screw compressors have over other types of compressors under the same operating conditions [6, 7, 4, 8, 9, 10]. T he advantages are: • T he high operating speeds which can be obtained w ith screw compressors allow for com pact designs for given volumetric flow rates. • Due to the rotary, m ulti-pocket, compression process, th e compressor is rela­ tively quiet and produces small vibrations. • T he screw compressor design incorporates few moving parts, m ainly the rotors and slide valve, making it a reliable, low m aintenance machine. • T here are no valves to control gas flow, elim inating the losses, noise and m ain­ tenance associated with valving. • For m ost applications, the screw compressor is more efficient th an either the reciprocating or rotary type compressors. This is especially tru e of applications involving partial load conditions, which can be satisfied by a screw compressor equipped with a slide valve. The increased efficiency results directly in power reduction due to the elim ination of over pressure and under pressure conditions. These advantages have lead to the use of screw compressors in various applica­ tions. Klein [11] gives a description of how the screw compressor can be applied to heat pum p applications. Fukazawa and Ozawa [12] discuss the potential for using screw compressors in the autom obile industry. Price [13] enum erates the advantages Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 associated with the use of screw compressors in the gas processing, refining and chem­ ical production industries. It should be noted th a t “screw compressor” , as used throughout the rem ainder of this docum ent, refers to twin screw type compressors. A nother type of screw compressor, the star or globoidal type, consists of one screw type rotor w ith either one or two meshing star shaped rotors. T he axes of rotation of th e screw and th e stars are arrange perpendicular to each other. Lundberg [14, 15] performed a comparison of the twin screw compressor and the star type compressor a t full load and part load operating conditions. Both theoretical analysis and experim ental tests were performed. Lundberg concluded th a t the twin screw compressor was more efficient throughout the full operating range. 2.2 Compressor Geometry and Perform ance Analysis This section is a review of studies of the geometric param eters of compressor rotors. These consist mainly of m ethods used to represent the rotor profiles m athem atically and m ethods to compute the overall effect of variations in the profiles on compressor performance. Also included in this section are reviews of therm odynam ic models of the compression process utilized for analyzing compressor perform ance and gas flow characteristics. 2.2.1 Rotor Profiles and Geom etry The geometry of the screw compressor rotors is the key factor in determ ining the performance and reliability of the machine. The rotor geom etry can be completely described by the two dimensional shape of the end profiles, th e wrap angle and the length to diam eter ratio. T he shape of the end profiles is the m ost im portant, and most difficult to design, of the geometric param eters. Profile shapes affect the leakage paths, seal line lengths and blow hole area. Each of these geometric param eters has a direct effect on the overall efficiency of the compressor. The development of methods for generating the profiles in an efficient, simple m anner and evaluating the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 « corresponding performance is an essential step required to adequately study various rotor profiles and their application to screw compressors. Today nearly all screw compressors are oil injected or oil flooded types. In this design the rotors acts as a gear pair. Therefore, the profile shapes m ust not only generate the chambers required for efficient compression, but m ust also be conjugate shapes w ith desirable torque transmission characteristics. For these reasons, the m ajority of research associated with screw profile geometry has been in two areas. The first is the development of m athem atical models and interactive com puter aided systems for generating profile shapes. T he other area is the development of models and com puter aided systems for analyzing the performance of these shapes when used for gas compression. Vinogradov et al. [16] divide the geometric param eters associated with the screw compressor into three separate classes; the outer radii and num ber of lobes on each rotor, the points describing an individual lobe profile for one rotor, and the helix angle and rotor length. The authors then present realistic ranges for values of each param eter which are applicable to screw compressor design. Singh and Onuschak [17] present a comprehensive com puter aided approach to profile development and analysis. The m ethod consists of an interactive design pro­ cess in which the designer starts with a rough draft of th e profile. W ith the aid of com puter programs, the profile is refined and the conjugate profile is generated. In addition, com puter program s are utilized to com pute the leakage path, seal line lengths, blow-hole area and discharge-port area. In a separate publication, Singh and Patel [18] present the com puter code required to analyze the performance of the resulting profiles. Singh and Schwartz [19] present a technique to define the rotor surfaces exactly in three dimensions. The exact m athem atical definition of the rotors can then be used to compute the chamber volume as a function of the male rotor rotation angle, along with the blow hole area. This m ethod eliminates the need for dealing with the geometry problem in the typical piecewise fashion where cross sections of the rotors, along the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 rotor axis, are used to extend the geometry to three dimensions. This m ethod is b etter suited to describing such geometric param eters as the blow hole area, which is highly three-dimensional. C om puter com putation of the geometric param eters can be accomplished through num erical integration of volume and surface integrals, once the profile is defined. Singh and Bowman [20] describe two approxim ate m ethods for com puting the blow hole area, in addition to the exact m ethod following the representation outlined in reference [19]. Zhang and Ham ilton [21] present a m athem atical derivation of the compression volume curve, sealing line length, flute area factor, wrap angle factor, and blow hole area, given the profile end shapes. Here the flute area and wrap angle factors are used to com pute the discharge volume of the compressor. In addition, Zhang and Ham ilton [22] provide an example com putation of the sealing line length. M easurement of the actual gap between a rotor pair in a screw compressor is also im portant. The theoretical com putations may incorporate m achining tolerances and therm al expansion, but only through m easurem ent can the actual interlobe gap be determ ined. Vinogradov et al. [23] present the use of pneum atic measuring m ethods. These m ethods have the advantages of high resistance to disturbances, reliability, wide m easuring range, high accuracy and low required m easuring tim e. A specific measuring procedure is presented where air flow and a standing tube are utilized to m easure the effective interlobe gap in an air compressor application. The m ethod allows the m easurem ent to be performed efficiently on individual rotor pairs, providing the possibility of selective assembly at production tim e. Zenan et al. [24] give an overview of a system used to m easure “space curve surface errors” of the screw rotors. The paper provides only an overview of the m ethod and a comparison to previously established “magneto-railing techniques.” 2.2.2 Compressor Performance The previous section discussed m ethods for modeling and generating screw pro­ files. In order to determ ine the quality of a specific pair of rotors w ithout actually Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 m anufacturing a compressor, models have been developed to predict th e performance of a compressor, given the rotor geometry. Screw compressors are classified as pos­ itive displacement compressors. Soedel [25] and Ham ilton [26] give basic techniques for modeling and com puter sim ulation of positive displacem ent compressors. Fujiwara et at. [27] present the development of a com puter program for predict­ ing the performance of an oil free type screw compressor. It was assumed th a t the internal leakage power loss would be th e dom inating factor. T he effects of clearances, blow hole, built-in pressure ratio, operating pressure ratio, and rotational speed on performance are predicted. Results are presented for an air compressor with circular sym m etric profiles. The volum etric efficiency was found to decrease w ith a decrease in speed. This can be attrib u ted to the fact th at the internal leakage flow is a func­ tion of the pressure differential across adjacent chambers. This pressure differential does not vary with speed. However, the volumetric flow rate increases with speed. Therefore, the volumetric efficiency also increases w ith speed. Firnhaber and Szarkowicz [28] developed a m athem atical m odel and com puter simulation for the overall compression process involved w ith screw compressors. The model included the expansion process through the suction porting, compression in the chamber volumes and expansion through the discharge porting. Flows are m od­ eled by compressible orifice flow equations, using experim entally determ ined orifice coefficients. The compression process is assumed to be polytropic. T he model did not include the leakage paths internal to the individual compressor chambers or th e effects of coolant injection. Brablik [29] developed an analytical model of an oil free compressor which in­ cluded the effects of rotational speed, compression losses and variations in operating conditions as compared to built-in pressure ratios. His model also included the dy­ namic effects of the discharge process, although the dynamics of the suction process were assumed to be negligible. T he model was applied to analyze a SRM type screw compressor. Predictions of the pressure, flow rates, efficiencies and discharge tem per­ atures are presented. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Sangfors [30] presents a model for both oil free and oil flooded screw compressors. His model includes the dynamics of both the suction and discharge processes, along with internal leakage and the therm odynam ics within the cham ber volumes. Coeffi­ cients required to determ ine leakage, heat transfer and dynam ic losses were obtained by comparison with SRM laboratory tests. This model was used to analyze several compressors. The results of the analyses are presented in the form of pressure ver­ sus male rotor rotation rate, volumetric and adiabatic efficiencies versus oil injection flow rate and power consumption and adiabatic efficiency versus volume flow rate. In all cases the model was found to be in good agreement w ith results obtained in laboratory tests. T he volum etric efficiency was found to increase w ith an increase in rotational speed, as was discussed earlier. However, the adiabatic efficiency decreased once speed was increased beyond a certain lim it, due to dynam ic losses. Therefore, an optim um speed exists which is dependent upon both volum etric and adiabatic efficiencies. More details of this model are presented in a later publication [31]. Fujiwara et al. [32] extended the model which had been previously developed [27] to incorporate oil injected compressors. The model was sim ilar to th a t of Sangfors [30], with the exception th at the volume curve is obtained from the sealing line shape using the principle of virtual work. This procedure simplified th e com putation of the volume curve as a function of the male rotor rotation angle. T he results of applying the model are presented in term s of various rotor geometries and the subsequent effects on the efficiencies and P-V diagrams. Singh and Patel [18] present a performance model which was derived as an exten­ sion of earlier work by Singh and Onuschak [17]. The model and com puter program presented here attem pts to account for all reasonable capacity and power losses. Em ­ pirical coefficients are used when necessary. T he program was checked against test d ata for several compressors. Comparisons of the predicted and experim ental d ata are presented. The model predicts input power, capacity, and efficiencies accurately as compared to the test data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Another im portant area associated with the performance of screw compressors is the proper design and use of the slide valve to vary the volume ratio, V i, and capacity for a given compressor. The Vi is defined as the ratio of the cham ber volume a t the sta rt of the compression process to the cham ber volume a t th e onset of discharge. The Vi is normally varied by decreasing the volume a t th e onset of th e discharge process. This is usually accomplished by varying the radial discharge porting. The capacity control is achieved by allowing gas to bypass back to the suction side of the compressor. The point in the compression cycle at which this bypass occurs deter­ mines the reduction in capacity. Several authors have addressed th e various designs available, and how these designs can be most effectively utilized [33, 3, 34]. T he m ain concern with designing slide valves and selecting appropriate settings is to avoid over­ pressure and under-pressure conditions. Over-pressure conditions reduce efficiency by compressing the gas beyond the required pressure. U nder-pressure conditions reduce efficiency due to back flow at the discharge and the corresponding recompression of gas. Both conditions increase gas pulsations associated with th e discharge process. Shaw [3] presented a “conflict” involved with proper selection of the Vi when a slide valve and movable stop, as opposed to a fixed stop, is utilized. W ith this type of design, th e bypass to the suction side is regulated by the movable stop and occurs at a higher pressure, further into the compression cycle, th an a sim ilar design w ith a fixed stop. Therefore, some of the efficiency savings which are obtained by a b etter matching of the Vi may be lost by allowing higher pressure gas to flow back to suction pressure. The subject of gas pulsations in screw compressors was addressed by Koai and Soedel,[35, 36, 37]. The pulsations due to the compression process were initially investigated [35], In addition, the effects of the compression process were combined with a typical discharge system [36], The discharge system was modeled using a Helmholtz model, a one-dimensional wave m otion model, and a three-dimensional finite elem ent model. Comparisons of the results, obtained using each of the discharge models, are made. The accuracy of any particular model depends not only on the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 physical characteristics of the discharge system itself, but also on the frequency range of interest. A theoretical analysis of the pulsations, as they relate to the use of capacity control and variable V;, in a screw compressor designed w ith a movable stop and slide valve, are presented in reference [37]. 2.3 Gear Models The rotor pair in an oil injected/flooded screw compressor is a gear pair, the male usually being the driver and the female the driven gear. Therefore, it is only natural to investigate the work which has been published on models of gear dynamic interaction. The author was unsuccessful in finding published literature on dynamic models applied directly to the screw compressor rotors. Several texts present and discuss the basics of gear kinem atics and dynam ic inter­ action [38, 39, 40, 41, 42]. However, these references generally give basic analytical models and derivations. More specific models and details can be found in journals and conference proceedings. This section is a review of studies of gear vibration models. T he section includes models of single gear pair interactions along with models of com plete gear systems. Most of these studies include effects of dynamic loads on gear systems due to tooth stiffness variations and tooth profile errors. In addition, studies are included in which helical gears in particular are addressed. A model developed by C.C. Wang [43, 44] is closely related to the work proposed by the author. This model is given special attention. 2.3.1 Spur Gear Models Ozguven and Houser [45] give a review of m athem atical models used for gear dynamics. This review includes literature in which a range of topics are covered. Several models were developed to empirically determ ine a dynam ic load factor based on the static load and gear speed. A large portion of the work present in the literature concerns the development of models which incorporate tooth compliance, with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 rem ainder of the system rem aining rigid. These models are used to account for the variations in tooth stiffness due to the path of contact and the meshing frequency. The tooth compliance models were later extended by accounting for the mass and flexibility of the shafts and th e compliance in the bearings. More recent advances include the modeling of the gear-shaft systems to study shaft whirl and rotor dynamic effects. Ozguven and Houser also review gear models in which the gear teeth were assumed to be rigid. Wang [43, 44] used this type of model to study the dynamic interaction of the gear pair and estim ate the loads associated w ith tooth separation and subsequent im pact. His models were found to be useful and are reviewed later in this chapter. Ozguven and Houser referenced 188 separate publications. Harris [46] addressed the dynam ic loads on the teeth of spur gears. He cited three internal sources of vibration: 1. Periodic variations in the velocity ratio due either to running the gears above or below the designed load, or small m anufacturing errors. 2. Periodic variation in the tooth stiffness due to progression of contact along the contact path and meshing frequency. 3. Non-linearity in apparent tooth stiffness due to loss of contact. Harris noted th a t the vibration caused by the variations in velocity ratio can only be elim inated if the gears are operated a t exactly the design load, a condition which is not practically realizable. The variation in velocity ratio occurs due to the tooth deflection. The rem aining two sources cause vibration which occurs only when the dam ping is below an unspecified lower lim it. Dam ping above this lim it will suppress the motion. Harris presents experim ental results which display the vibration due to velocity ratio variations. These variations were induced m ainly by profile errors. However, the system dam ping exceeded the lower lim it needed to allow vibrations due to periodic tooth stiffness and loss of contact. Harris concludes th at although theoretical models give insight to possible modes of vibration, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 actual relationship between profile errors and vibration can only be obtained with m easurem ents performed under actual load conditions. M ahalingam and Bishop [47] present a m athem atical model in which the dynamic effects are shown to be com parable to th a t of a cam-follower system . A good definition of the static transmission error is given here as follows: T he static transm ission error is defined, for any instantaneous position of one gear, as the departure of the m ating gear from the position it would occupy if the system were perfect, with an unvarying velocity ratio, and the teeth were rigid. The static transm ission error thus encompasses all the effects such as eccentric m ounting, errors of m anufacture, elastic deform ation, etc. The static transmission error causes an oscillatory m otion to be superimposed on the constant velocity of the m ating gear pair. M ahalingam and Bishop model this motion as analogous to the placement of a wedge between two rigid components of a mechanical system. The shape of the wedge is such th a t its m otion produces the oscillatory component defined by the static transmission error. M ahalingam and Bishop improve previously developed sim ilar models by using m odal analysis and allowing for static transm ission errors which are not periodic. Nakam ura [48] presents some theoretical analyses and experim ental investigations of the loss of contact of gear teeth. The model includes variations in tooth stiffness and loss of contact due to profile errors, and operating conditions. The theoretical model of the gear interaction determ ines whether one or two teeth pair are meshing when loss of contact occurs. These two conditions are treated differently. However, the to tal gear interaction is treated in a piecewise m ethod, with regions of contact and separation solved separately. Nakam ura performed experim ents to test his theoretical model and to m easure noise generation from the resulting gear vibrations. He found th at the theoretical model and the experim ental results agreed qualitatively. Nakam ura concluded th a t a specific speed exists at which tooth separation occurs. The value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 of this speed is dependent upon the amount of transm ission error and the am ount of load applied to the gear pair. He also observed a subharm onic noise component in his experim ents, which he attrib u ted to abnorm al meshing of every second or third tooth. Tobe and Takatsu [49] exam ined the dynamic loads due to im pact. They present two different models for the tooth stiffness. In the first model, th e tooth is reduced to a single degree of freedom, mass-spring system. This model is justified since previous studies, which modeled the tooth as a cantilever beam using the Euler-Bernoulli equations, showed th at the response was dom inated by th e first mode. Therefore, proper specification of the mass and spring stiffness can produce accurate results. In addition, Tobe and Takatsu examined the effects of shear deform ation and rotary inertia using a Timoshenko type cantilever beam model of the tooth. These effects were shown to decrease the calculated im pact load, by a m axim um of 20%, while increasing the duration of im pact. Finally, it was shown th a t th e m axim um value of the bending m oment at th e root of the tooth during im pact is proportional to the impact load, a t any im pact condition. Fukuma [50] presents a m ethod for modeling the generating mechanisms of vibra­ tion in spur gears. In this model the stiffness of the shafts and bearings is accounted for, as is the mass of the shafts. Fukum a dem onstrates the coupling of vibration components in the circumferential, radial and axial directions. Coupling coefficients, which relate the rms acceleration values of the axial and radial vibrations to the cir­ cumferential vibrations, are determ ined. The coefficients are found to be functions of the gear teeth stiffness and th e param eters which define th e shaft-bearing systems. He concludes th at the effects of the gyroscopic action are negligible. The theoretical model agrees qualitatively w ith the experimentally determ ined radial vibration coef­ ficients. However, the theoretically computed axial vibration coefficients are not in agreement with experim ental results. Fukum a contributes this to the fact th a t he did not account for effects of unbalances and non-linearities in the bearings. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Bahgat [51] studied the effects of bearing clearances on dynam ic tooth loading. In this study, he developed a model for com puting the dynam ic tooth loading, torques, bearing forces and separation occurrences as affected by th e bearing clearances. He concludes th a t the bearing clearances can have significant effects and therefore should also be considered when studying the dynam ic tooth loading and gear interaction. Umezawa et al. [52] investigated the effects of transm ission errors on rotational vibration by including actual m easured errors. T he m easurem ent procedure was de­ veloped by Ishikawa et al., references are given by Umezawa [52]. Various tooth stiffness functions were studied and compared to experim ental results. A stiffness function was derived to define the onset of tooth meshing. A sim ulation of th e gear interaction was developed based on the m easured profile and errors and the stiffness function. T he model was found to accurately predict experim entally obtained behav­ iors. In a later publication Umezawa and Sato [53] used the m odel to sim ulate profile errors with wave forms and the subsequent effect on vibration of power transm ission spur gears. The influence of the error was determ ined in term s of the rotational speed and the contact ratio. They concluded th a t th e vibration of th e gear was more influ­ enced by the speed than by th e contact ratio. The sim ulator was able to detect the speed at which loss of tooth contact occurred, but was not designed to handle the resulting motion. Singh et al. [54] present a comprehensive analysis of gear vibration as applied to autom otive gear rattle. Both linear and non-linear models for predicting loss of contact and subsequent m otion are investigated. A criterion is developed for pre­ dicting rattle based on the acceleration of the driver as opposed to th e driven gear. Numerical techniques for integration of the equations of m otion, such as R unge-K utta and G ear’s m ethod, are studied. Numerical sim ulation techniques for th e gear ra ttle problem are found to be error prone and tim e consuming for param etric studies. It is suggested th a t either linear analytical or simple non-linear models be employed. The model is developed for application to autom otive transm issions and considerable attention is given to the design of the individual components. Com parin and Singh Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 [55] later developed a reduced order, non-linear model for the autom otive transm is­ sion which provided an analytical base for the non-linear behavior reported in various publications of experim ental results. 2.3.2 Helical Gear Models The analytical models used for spur gears have been extended to helical gears. Kiyono et al. [56] developed a theoretical model of a helical gear-shaft system . He considered the vibrations in four directions: torsional, lateral, longitudinal and rota­ tional. The vibrations in these four directions are shown to be coupled through the stiffness m atrix, due to the helix angle. The source of excitation for the model is the periodic fluctuations in th e stiffness m atrix. Kiyono concludes th a t th e torsional component of vibration is the most significant, when considering the dynam ic tooth loads, and th a t the rotational component of vibration is more im portant th an the longitudinal one for helical gears. In a later publication [57], Kiyono et al. present experim ental investigations of the m easurem ents of vibrations. Dynam ic strain and accelerations of three-directional vibrations of helical gears are m easured and their natural frequencies, modes and exciting forces are discussed in comparison to spur gears. These experim ents resulted in the following conclusions. Both the axial vibra­ tion level and dynam ic load in helical gears increase as compared to spur gears. This is due to natural frequencies and modes found in helical gears which are not found for spur gears. The exciting force associated with periodic variations of tooth stiffness decreases with an increase of the helix angle. Therefore, a reduction of vibration can be obtained by increasing the helix angle. However, the ratio of the level of axial vibration to th at of torsional vibration also increases with an increase in the helix angle. The experim ental results confirm the conclusion of the analytical analysis, th at the natural mode which has the greatest effect on the dynamic load is the one in which the torsional vibrations dom inate. Kubo [58] presents a general m ethod for predicting the effects which m anufactur­ ing and alignm ent errors have on load sharing among meshing teeth and subsequent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 stiffness functions, fillet stress, and contact stress. He investigates various forms of profile errors and allows for lack of contact due to profile errors. T he predicted results were compared with experim entally m easured values for several gear pairs and were found to be qualitatively accurate. Kubo and Kiyono [59] extend the relation between profile errors, tooth stiffness variations and vibration excitation for helical gears. It was found th a t an increase of total contact ratio or helix angle can be utilized to pro­ duce a reduction in the total vibrational exciting force due to both periodic variations in tooth stiffness and transverse tooth form error. Relationships sim ilar to those for spur gears were also observed. The concave type of tooth error was shown to produce increases in vibrational excitation. The convex type of tooth error could be used to serve as a correction, if properly chosen. Errors associated w ith m isalignm ent tended to increase vibrational excitations 'aused by profile errors. Umezawa et al. [60] developed an approxim ation for th e tooth stiffness equation and the associated meshing resonance frequency for helical gears based on a one degree of freedom system analysis. Only a rotational degree of freedom is included. The associated theoretical deflection of the teeth is used to com pute the stiffness variations. From this stiffness model, a meshing resonance frequency is obtained. Rotational velocity variations due to static transm ission errors are not considered. The m ethod is applied to evaluate four types of gear pairs by com puting the meshing resonance frequency. The results of the sim ulation agree in form but not in detail. Neriya et al. [61] present a m ethod for exam ining th e response of the helical gear system subjected to static transmission errors. In this work th e content of the transmission error is allowed to include both determ inistic and filtered white noise components. The corresponding m athem atical model and valid num erical solution techniques are presented. T he model uses a piecewise approach to represent the contact and loss of contact situations, and determ ine the subsequent m otion. Tsay [62] presents an analytical treatm ent of helical gears in which he addresses the development of a m athem atical model for generation of the surfaces of involute helical gears, and a com puter sim ulation for the meshing and bearing contact. In addition, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 he investigates the sensitivity of the gears to m anufacturing and assembly errors, and performs a stress analysis of the gear pair utilizing a finite elem ent model. The results obtained using his m odel and sim ulation are not com pared w ith experimentally obtained values. However, some conclusions are drawn from th e computations. Tsay points out th a t center distance variation from design does not cause kinem atic errors. This is due to the involute shape. However, axes m isalignm ent does produce kinem atic errors. According to Tsay, the level of kinem atic errors is one of the m ain sources of gear noise and vibration. R autert and Kollmann [63] describe a com puter sim ulation for the dynamic forces in spur or helical gears with one or two stages. Dynam ic load on gear teeth and all bearing forces are com puted from a lumped mass and spring model. The periodical change of tooth stiffness due to single and double contact, the inertias and masses of shafts and gears, and the bearing stiffnesses are considered. T he derivation results in a set of ordinary differential equations which are solved in the tim e domain. The associated force spectra are obtained using Fourier transform s. T he scope of the paper is the development of a tool for the com putation of dynam ic bearing forces of different types of gears. These forces can then be used to evaluate design modifications of the inner gearbox elements as they relate to the radiated noise. 2.3.3 Rigid Gear Model Many of the models reviewed in the previous sections were developed to investigate gear vibrations which occur when the gears are operated at load conditions above, or near, the design loads. Tooth deflections at these higher loads induce oscillations in the velocity ratio between the gears and variations in the stiffness function related to the tooth contact. However, excessive noise, wear and failure are known to exist under load conditions below the design loads. Wang [43] presents m athem atical models, com puter simulations and experim ental testing of a spur gear pair in which the driven gear is subjected to a load which is substantially less than the design load. The main features of these models are described here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 ^ In W ang’s models, the teeth of both the driver and the driven gears are assumed to be rigid, neglecting deflections due to loads. In addition th e inertia of th e driver is assumed to be infinite. Therefore the m otion of the driver is not affected by changes in the contact force between th e gears. The transmission error is incorporated into the model as a variation in th e profile of the driver teeth. Backlash is modeled as a tim e varying angle used to define the distance between adjacent contacts on the teeth of the driver gear. The overall result of these assum ptions is the form ation of an envelope, attached to the driver gear, defined by the transmission error and backlash. The motion of the driven gear is then restricted to fall within this envelope. V ibration excitation is provided by the static transmission errors assigned to the driver gear and the tim e varying backlash. Wang uses only the transm ission errors related to the eccentricity of the gear m otion and not those associated with profile errors. The only loads on each gear are assumed to be constant torque loads. Two separate models, along with the associated sim ulations and experim ental tests, are presented. The first model, a two-mass model, assumes th a t th e driven gear is rigidly connected to the output load. Therefore the inertia of the gear and shaft can be lumped into one quantity. The total angular m otion of the driver is defined by the m ean rotational speed and th e transm ission error, since the driver is assumed to have infinite inertia. The system m otion is then completely defined if the motion of the driven gear is com puted. The tim e varying backlash and driver m otion define an envelope which constrains the m otion of the driven gear. Three possible contact modes can occur. These modes are: • The driven gear can be in contact with the driving side of the envelope, its m otion being determ ined by the transmission error. • The driven gear can be in contact with the trailing side of the envelope, its motion being determ ined by the transmission error and the backlash. • The driven gear can be between the envelope contacts, its m otion not affected by the driver. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 These three conditions are represented by three separate sets of differential equa­ tions. T he m otion of th e driven gear is then solved by treatin g the to tal m otion in a piecewise linear fashion, w ith each contact m ode solved using the corresponding dif­ ferential equations. W hen the contact mode changes, the final velocity and position com puted for the ending mode are used as initial conditions for th e m ode which is beginning. Since this approach does not violate any natural laws of m otion, whatever non-linear characteristics the system may have are preserved and dem onstrated in the response. In the second model, a three-m ass model, the driven gear is connected by means of a low stiffness shaft to a pulley which is loaded by the ou tp u t torque. The shaft and pulley are lum ped together as one mass. Therefore, the m otion of three masses, the driver and driven gears and pulley-shaft system , m ust be determ ined to completely define the motion of the system. The equations of m otion become somewhat more involved due to the flexible connection between the driven gear and pulley. However, the basic approach to the solution still applies. Wang modeled the im pact between gear teeth using conservation of m om entum laws and a coefficient of restitution. The im pact tim e was assum ed to be infinitesimal. T he coefficient of restitution was assumed to be constant and th e num ber of rebounds after initial im pact was lim ited by requiring a m inim um tim e increm ent between subsequent im pacts on one side of the envelope. Predicted results using the two-mass model and a sinusoidal transm ission error were com pared with experim ental test results. The experim ental results included the basic response pattern m easured using m agnetic transducers. In addition a m ethod of detecting and recording loss of contact between the gears was established. Predicted and experim ental results were found to be sim ilar in the basic form of the response pattern of the driven gear and the prediction of loss of contact. T he discrepancies can be attrib u ted to the approxim ation of the im pact tim e as infinitesimal and the assum ptions m ade concerning th e rigidity of the system . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 fi Wang was able to draw some general conclusions from the two-mass model. The model could be used to determ ine the basic response p attern of the driven gear for specified transmission errors and backlash. Due to the im pacts which exist under light load conditions, the dynam ic load can not be considered proportional to the static load. The dynamic load is a function of shaft speed, transm ission error, back­ lash and m om ents of inertia. A resisting torque applied to th e output shaft of the driven gear can be used to reduce loss of tooth contact. T he value of this resisting torque is a function of the transmission error and angular velocity of th e driver gear. The assum ption of an infinitesimal duration of im pact is valid a t lower shaft speeds. However, as the shaft speed increases, the duration of im pact rem ains nearly constant while the meshing period of the gear teeth decreases. Therefore, the duration of im ­ pact becomes a larger percentage of the meshing period as the shaft speed increases. For this reason, the im pact condition should be handled more accurately for higher shaft speeds. Wang also presents the experim ental results for the tests of th e three-m ass model [44]. Addition of the low stiffness shaft, which connects the pinion to the load, m eant th at the operating speed of the test rig was in the range of the critical speed associated with the natural frequency of the pinion-shaft assembly. The com puter predictions agree well with actual responses in various speed ranges, despite a series of bold assumptions th at were m ade to simplify both the physical system and the numerical process. Wang concluded th a t the modelling concept is valid. 2.4 Im pact Models Select studies of phenom ena regarding im pact models and experim ental studies are presented in this section. The main objective is to review the various models which have been studied and to determ ine the effectiveness of each model. Due to the complex nature of im pact and the associated dynamic characteristics, the models which are presented are based on simple physical systems. The premise is th at these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 simple systems can be utilized to obtain some insight into real mechanical systems involving im pact. Goldsmith [64] devotes an entire text to the analysis of im pact. In this tex t, he reviews the classical theory of im pact, which involves the com putation of the velocities after im pact and the applied impulses. This theory requires application of the law of impulse-mom entum for rigid bodies and conservation of m echanical energy, with the use of a coefficient of restitution. The advantage of the classical theory is th a t it is m athem atically simple to apply. However, using the classical theory, the transient stresses, forces or deform ations produced by th e im pact can not be obtained. Lifshitz and Kolsky [65] present the results of experim ents on rebound phenom ­ ena. M easurements were obtained using a steel ball impinging on a steel block. In this work they studied the effects of surface finish and approach velocities on the coefficient of restitution and duration of contact tim e. T he m easurem ents were also used to determ ine the effect which plastic deformation had on th e coefficient of resti­ tution and duration of contact. Lifshitz and Kolsky concluded th a t the surface finish plays a m ajor role in determ ining the coefficient of restitution for this specific im pact pair. They attrib u ted this to the fact th at the contact radius for a sphere on a flat surface, as predicted by H ertzian contact law, is relatively sm all com pared to sur­ face imperfections. Therefore, these imperfections have severe effects on th e nature of the im pact. In addition they also concluded th a t th e coefficient of restitution is independent of the approach velocity, up to a specific lim it. A t this lim it, increasing the approach velocity results in a rapid decrease in the coefficient of restitution. The approach velocity lim it a t which the decrease begins can then be utilized to deter­ mine the yield stress of the m aterial. In contrast, the duration of contact did not vary with the approach velocity, even after the lim it a t which the coefficient of restitution decreased. Laird, Ashley and Kelly [66, 67, 68] present sim ulations of m echanical im pact using analog com puters. Laird gives a general m ethod for sim ulating mechanical im pact in which his model is analogous to a “bouncing ball on a pogo stick.” He uses Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 a spring-dam per model and allows for clearance between the fixed foundation and the bouncing mechanism. In this sim ulation, the impulses and deform ations due to the collision are not accurately com puted. However, the net effect which the im pact has on the overall motion of the m echanism is accurately sim ulated. Ashley, improves upon Laird’s sim ulation by adding lim it stop damping, in which the bouncing m echanism is allowed to im pact a lim it stop a t some point during the deflection of the spring. Kelly further enhances the sim ulation to include pressure differentials across the mechanism. This was done to allow for modelling of valves and pistons. Dubowsky and Freudenstein [69, 70] present a dynam ic analysis of m echanical systems w ith clearances, concentrating on pin joint connections. In this work, they utilize a one dimensional model of two masses w ith joint clearances, which they refer to as an “im pact pair” . The joint clearances are modelled by allowing one of the masses, the internal mass, to move within fixed stops which are attached to th e other mass, the external mass. T he surface compliance between the masses is modelled using a linearized version of H ertzian contact. In P art 1 of the paper [69], th e equa­ tions of m otion for the contact and non-contact positions are derived in term s of both absolute and relative coordinates. The analytical solutions of th e subsequent m otions are presented for three separate cases; free vibrations, constant load conditions, and prescribed displacement of the external mass. In P art 2 of the paper [70], Dubowsky and Freudenstein present results of simulations of the dynam ic response of the im ­ pact pair using numerical techniques on digital com puters as well as approxim ate m ethods. It was found th a t a num erical approach using a predictor-corrector m ethod yielded accurate results when compared to the analytical solutions. T he results of the approxim ate solutions obtained using the describing-function technique were not as accurate. The main conclusion expressed by the authors was th a t the simple im pact pair model could be used to study mechanical systems w ith clearances and backlash. Hunt and Crossley [71] present a derivation whereby the coefficient of restitution can be interpreted as dam ping in vibro-impact systems. The authors use an energy approach, and develop a form of hysteresis loop for the forcing during im pact, to argue Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 th at the spring-dam per model has inherent inaccuracies. Using the spring-dam per model, where damping is proportional to velocity, the dam ping effect is greatest at the instant of im pact, producing an impulse. In addition, the forcing during the resti­ tution of the im pact seems to represent a tensile force between th e two bodies. These two effects have been shown to be inaccurate. Therefore, a m odel for the coefficient of restitution is developed using a dam ping term which is related to both the relative velocity, x, and the displacement of the im pacting objects, x. The dam ping is of the form \ x px q. W here A is the dam ping coefficient. Digital com puter simulations utilizing this type of dam ping were performed and the system responses were found to vary significantly with th at of equivalent linearized models. Veluswami and Crossley [72, 73] present results of experim ental investigations of a ball bouncing between two plates. The objective of the experim entation was to provide a datum for developing a m athem atical model for a single im pact. In the experim ental set up, a steel ball was arranged between two plates, w ith clearances between the ball and each plate. T he m otion of the plates was prescribed, with various am plitudes and frequency ranges available. T he effects of various m aterials were studied by changing the plate m aterial. In addition, th e mass of the steel ball was also varied. The results are presented as contact patterns of m ultiple and single strikes of the ball on the plates, along with the duration of im pact. The following conclusions were obtained from the experim ental results. It was found th at as the contact duration increases (lower velocity), the loss of energy due to damping decreases. This is contrary to predictions obtained using linear dam ping coefficients. It was also found th at the change in the am plitude of the prescribed m otion is more significant than the change in the frequency, as associated w ith the various contact patterns. In part 2 of this publication [73], m athem atical models are presented to predict the behavior obtained in the experim ental work. The m athem atical model which best predicts the behavior found in the experim ental work is one in which the damping is of the form \ x px q, with p = 1.5 and q = 1.0, and th e H ertzian compliance is of the form k x 1,s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Wood and Byrne [74] studied a simple model of a random repeated im pact process. They examined the response of a ball to a vertically vibrating table, whose excitation is random. It was assumed th a t the mass of the table was m uch greater th an th a t of the ball. Therefore, the velocity of the table was unaffected by the im pacts. The displacement of the table is assumed to be much less th an th a t of the ball. This assum ption allows the velocity of the ball at the current im pact to be estim ated as the final velocity a t the end of the previous im pact. A constant coefficient of restitution for the repeated im pacts is assumed and the tim e of im pact is assum ed to be negligible when compared to the period of m otion of the ball. The results of the study were in the form of probability density functions of the tim e between im pacts, table velocity and peak im pact force. In a subsequent paper, [75], Wood and Byrne compared experim ental results to those predicted by the analytical model. They found the probability density functions for the tim e between im pacts and table velocity at im pact to be in good agreement with the predicted functions. T he probability density function for th e peak im pact force was not as accurately predicted. In their analytical study [74], Wood and Byrne m entioned experim ental evidence which suggests th at random , non-periodic motions of the ball are possible, w ith sinu­ soidal excitation, if the am plitude is sufficient. Holmes [76] extended their studies to results obtained using a simple model of a ball bouncing on a sinusoidally vibrating table. In this study, Holmes is concerned with the effects of energy dissipation on the dynamics of the m otion. He presents the non-linear aspects of the response of the ball, to periodic m otion of the table, as represented by the patterns of successive strikes on the table. Holmes concludes th at for sufficiently large excitation velocities, and a coefficient of restitution close to 1.0 , several complex motions can be obtained, even though the foundation of the computations is a rather sim ple difference equation and basic assumptions. In several publications, Shaw and Holmes [77, 78] and Shaw [79], present studies of the motion of piecewise linear oscillators. The details of the non-linearities, and discontinuities which separate linear portions of the m otion, are different in each case Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 and therefore various specific issues are addressed in each case. However, each of the models utilizes a simple im pact rule with a constant coefficient of restitution. Although different specific issues are addressed in each case, the overall complex motion of each system can be obtained with digital sim ulations using these simplified models. 2.5 Summary The literature review, as presented above, supports the au th o r’s research efforts. The general review of the screw compressor geometry and perform ance evaluations provides a good working knowledge of the twin screw compressor. In addition, this search reveals the areas of concentration of recent research efforts directed towards screw compressors. The lack of published literature directly related to the dynamic interaction of the rotors provides considerable m otivation for th e au th o r’s research. As part of this research effort, a m easurem ent technique was developed to measure rotor chatter in twin screw compressors. This work was conducted by John Huff and is presented in his m aster’s thesis, [80]. The review of gear dynamics models provides direction for the development of a model for the compressor rotor interaction. This research effort is directed a t obtain­ ing a model which dem onstrates the overall m otion of th e rotors. The com putation of the elastic dynamic loads and stresses is not w ithin the scope of this research. However, a model in which the rotor lobes are assumed to be rigid will be useful. In addition, a piecewise linear solution can be utilized w ithout sacrificing the influence of the non-linear characteristics involved. The review of im pact models supports the use of a constant coefficient of resti­ tution. The dynamics of the rotors in the screw compressor are affected by many complicated factors. The introduction of an estim ated dam ping relationship for im ­ pact would serve only to complicate the model and lessen the am ount of insight which can be gained through sim ulations. In addition, several authors dem onstrated th at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 the complicated motions associated with im pact can be predicted with the use of a coefficient of restitution to model th e im pact. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 3. RO TO R KINEMATICS T he earliest twin screw compressor designs included tim ing gears to control the relative rotation of the male and female rotors, (see C hapter 2). Therefore, the shape of the rotors did not affect the kinem atics, but served only to produce th e chambers required for compression. As th e technology progressed, th e tim ing gears were elimi­ nated and the rotors were required to perform as both a gear pair and a compression m echanism. Therefore, th e shapes of th e m ale and fem ale rotor profiles m ust be de­ signed such th a t one of the rotors can effectively drive th e other. Historically, the m ale rotor has been designated as the driver gear and th e female as the driven gear. In this chapter, a kinem atic analysis of the screw compressor rotors is presented. The m ain objectives are to develop m ethods for determ ining how the screw geometry affects both the dynamics of th e rotational m otion and the bearing loads which result from the rotor contact forces. Planar theory is applied to the 2-dimensional profiles of th e rotors. The effects associated w ith the 3-dimensional helical geom etry of the rotors are also included. This analysis is essential to understanding th e angular velocity ratio associated w ith th e profile geometries, the nature of force transm ission between th e rotors, and th e definition of th e contact curve a t th e interface between the rotors. In addition, the kinem atic analysis has lead to th e development of an iterative m ethod for defining th e profile shapes required for a twin screw compressor. Some fundam entals of the kinem atics of gears, and th e relationships between gen­ erating and conjugate shapes are first reviewed. T he fundam entals are applied in the investigation of profile d a ta which was provided for this research. This original profile d a ta was intentionally modified by the sponsor, before it was provided, such th a t it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 did not represent a reliable profile geometry. In applying these fundam entals, an iter­ ative m ethod for generating rotor profiles is developed. This m ethod is im plem ented to revise the original d a ta and obtain m ore reliable profile data. The direction of force transmission between the m ale and female rotors is com puted as a function of the rotation angle of the m ale rotor. This inform ation, along w ith definition of the rotor contact curve, is used in deriving coefficients necessary to com pute the bearing loads which result from the contact force between th e rotors. 3.1 Fundam entals of Gear Kinem atics The rotors in a twin screw compressor are helically shaped screws w ith non- classical 2-dimensional profiles. In order to understand th e kinem atics of the com­ plex screw geometry, one m ust first understand th e kinem atics of planar mechanisms based on th e 2-dimensional rotor profiles. Therefore, fundam entals of planar gear kinem atics are presented here. In addition, the fundam entals of th e kinem atics of the 2-dimensional profiles are key to understanding the conjugacy relationship between the male and female rotors. Throughout this chapter, references to a gear pair imply two planar mechanisms which rotate relative to one another about fixed axes, located a t their respective centers. Each gear is defined by individual tooth profiles, th e profile of one gear m ating with th a t of the other. Each gear is defined for a com plete circumference of 2ir radians. In the case of a planar, external contact, spur gear pair, th e 2-dimensional profile of each gear is uniform along the length of th e axis of rotation. Therefore, the kinem atics which govern th e gear pair m otion can be determ ined by th e 2 -dimensional profiles of th e gears. The surface, or face, of th e gear is projected to a 2-dimensional curve. The line of contact along the tooth face is projected to a point. T he contact point traces a path, in the plane of the 2-dimensional gear profile, as th e gears are rotated. These profile and contact curves can be used to analyze th e gear m otion. T he concepts developed for th e planar gear can then be extended to helical gears. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 The kinem atics which govern the relative m otion of a rotating gear pair are best described using several geometric constructions. The gear profiles are the physical shapes which define each m ating tooth pair. The pitch curve associated with each gear, and the path of contact between the gears, are curves which m ay not physically exist, but are useful in defining the kinem atics of the gear pair. The relative motion of the two gears, as each rotates about a fixed axis, is described by attaching each of the gears to its associated pitch curve and rolling th e curves upon each other. The path of contact is described in an inertial reference fram e, and defines th e curve along which point contact between the profiles occurs during rotation of the gears. 3.1.1 Angular Velocity R atio The angular velocity ratio is defined as the relative rate of rotation for a gear pair. This param eter is an im portant consideration in gear design since the angular acceleration is the tim e derivative of the angular velocity. The angular acceleration is a factor in determ ining the nature of the contact between the gears. It is necessary to distinguish between the average angular velocity ratio and the instantaneous angular velocity ratio. Consider a gear pair, a and 6, with N a and Nb num ber of teeth, respectively. The average angular velocity ratio can be expressed as — w = jNra [3 -i] u a = the average angular velocity of gear a [3.2] u>b = the average angular velocity of gear b. [3.3] 6 where T he average angular velocity ratio is an integer ratio defined over an integral num ­ ber of complete rotations of th e driver. T he average angular velocity ratio rem ains constant. However, the instantaneous angular velocity ratio defines th e velocity rela­ tionship during the m ating of an individual pair of teeth, instead of over an integral Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 num ber of revolutions. The instantaneous velocity ratio therefore depends on the tooth profiles and m ay not be constant. In gear designs for transm ission of m otion and force, it is often desired to pro­ duce a constant instantaneous angular velocity ratio between th e gears. A constant instantaneous angular velocity ratio produces constant output angular velocity given a constant input angular velocity. This insures th a t if the driver gear is rotated at a constant rate, it will not induce angular accelerations on th e driven gear. The angular velocity ratio between the gears can be defined by the radii of the pitch curves associated with each gear. In order for the instantaneous angular velocity ratio to rem ain constant, the pitch radii of the two gears m ust rem ain constant through­ out the rotational motion. Therefore, the pitch curves become circles. Steeds [81] gives a derivation of th e fundam ental condition required for a gear pair to transm it m otion with a constant angular velocity ratio. This derivation is an im plem entation of Kennedy’s Theorem , which states th a t if three independent bodies have relative planar m otion, their instantaneous centers lie on a straight line, [82]. The basics of this fundam ental condition are presented here, (refer to Figure 3.1). Consider two arbitrary gear profiles, a and b. The position of each profile is defined by the angles 0a and 0b, respectively. The angles are referenced to the axis x c, which is coincident w ith th e line of centers, 0 a —Ob. For a given angular position, 0a\, of gear a, gear b is positioned a t the angle 0&i. At this instant the contact between the two profiles is defined by the points Q on gear a and S on gear b. T he angular positions of the gears are such th a t a common norm al, N — N , exists between th e two profiles a t the contact point. In order for the gears to transm it rotational m otion, without loss of contact or deflection of th e profiles, the component of th e velocity of points Q and S along the common norm al, N — N , m ust be equal. T he pitch point, P , is defined for this angular position by th e intersection of the common norm al, N — N , and the line of centers for the gears, Oa — Ob. It can be shown, using constructions of sim ilar triangles, th a t the distance from th e respective centers of rotation to the pitch point defines the instantaneous radii of th e pitch Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 curves for each of the gears. Therefore, at this instant, the radius of the pitch curve for gear a, R a, is the distance 0 a — P and th a t for gear b, Rb, is the distance Ob — P. These pitch radii are dependent upon the profile geometries and m ay not rem ain constant for all rotational positions. The pitch curves can be com pleted by repeating this construction for the entire range of rotation required for the gear pair. The instantaneous angular velocity ratio, 6b/0a, is dependent upon the radii of the pitch curves. This relationship is given by In order for the instantaneous angular velocity ratio to rem ain constant throughout the range of rotation, the radius of each pitch curve m ust rem ain constant. T he pitch curves m ust therefore be circles. The location of the pitch point, P , along th e line of centers, m ust rem ain constant for all values of 9a within the desired range of rotation. An infinite num ber of profile shapes can fulfill this requirem ent. T he m ost commonly used is the involute to a base circle. The involute gear is an especially robust design, due to the fact th a t the constant instantaneous angular velocity ratio is independent of the gear pair center-to-center distance. The instantaneous angular velocity ratio for a gear pair can be defined by the geom etry of the pitch curves associated with the gear pair. A gear pair which has pitch curves defined as circles will rotate with a constant instantaneous angular velocity. Therefore, the kinem atics alone will not induce angular accelerations on th e driven gear if the driver is rotated a t a constant rate. 3.1.2 Generating and Conjugate Shapes Gears are m ost commonly designed to transm it motion and force. Therefore, one of the gears, the driver gear, serves to drive the other, the driven gear. A segment of the tooth profile on the driver gear contacts a corresponding segment on the tooth profile of the driven gear. It is along these contacting segments th a t force transm ission Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 occurs. These segments of the tooth profiles also function as generating and conjugate shapes. One can assume th a t the segment of the driver gear profile which does not transm it force m ust rem ain in point contact with the driven gear profile. In this case, one of the profiles m ay be used to generate the complete m ating profile. In order to accomplish this, either of the profile shapes is designated as the generating shape, or generator. The m otion of the generator is then determ ined by th e kinem atics which govern the desired relative m otion of the gears, as defined by the pitch curves. The shape of the conjugate profile is defined by the envelope of the generator as it completes this m otion. This conjugate shape is the profile required to produce a m ating profile to the generator, and therefore obtain a working gear pair. T he pitch curves used to define the generating and conjugate shapes also define the kinem atics of the resulting gear pair. In Section 3.1.1, the kinem atic constraints governing the m otion of a gear pair were described in term s of the pitch curves associated with the gears. T he generating and conjugate shapes can be produced using these pitch curves. Two m ethods are described here. The m ethods are equivalent and each will be applied to the screw compressor rotors. The first m ethod is the kinem atic inversion of the gear pair system . Normally, each gear rotates about a fixed axis located a t its center. In order to invert this system , one of the gears, gear b in this case, is held fixed and gear a orbits about it. The orbiting m otion of gear a is constrained to pure rolling contact of the pitch curve associated with gear a on the pitch curve associated with gear b. For a gear pair with a constant angular velocity ratio, this consists of one circle rolling on another fixed circle. T he profile of gear a is attached to its orbiting pitch circle and rotated completely around one circumference of the pitch circle for gear b. The profile of gear a is plotted a t increm ental values of its orbiting position. The result of the superposition of each of these plots is an envelope which describes th e conjugate gear profile for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 gear b. If this conjugate gear profile is used, gear pair a and b will have the constant angular velocity ratio defined by the radii of the pitch circles. The advantage of this kinem atic inversion m ethod, when applied to the screw compressor rotors, is the visual representation of the conjugate shape which results. This m ethod is applied to the screw compressor rotors and the results presented in Section 3.2.1. Throughout the rem ainder of this docum ent, this is referred to as th e kinem atic inversion m ethod. The second m ethod is equivalent to th e first. T he m ethod is based on generating procedures presented individually by B eard,[83], and Colbourne,[84] and implem ented by the author, [85]. It shall be referred to as th e generation m ethod. T he im plem enta­ tion varies from the kinem atic inversion m ethod due to the definition of the coordinate systems. Two Cartesian coordinate systems (A'a, Ya) and (X/,, Y/,) are defined with their origins a t the fixed centers of the pitch circles, O a and O b, respectively. The coordinate systems are attached to the pitch circles and ro tate about their origins from the reference positions, 0aO and 6bo. These angles axe referenced to th e positive x c axis. The center of each coordinate systems is fixed. Therefore th e rotation of the coordinate systems is constrained by the rolling contact of the pitch circles, Ob ~0bO = ~ ( ^ ) (0a ~ 0aO) • [3.5] The profile for gear a is attached to its pitch circle and used as the generator. Let be located a t the reference position, 0ao = 0. T he corre­ coordinate system sponding reference position of system (Xb, Y&) is Obo = 0, see Figure 3.2. The norm al to th e profile is known, from th e description of th e profile geometry, for each point defined on profile a. The norm al to profile a, a t an arbitrary point, Q, in the reference position is given by N q . Profile a is then rotated away from th e reference position to a value of 6q , see Figure 3.3. A t this position, th e norm al to th e profile a t point Q is directed through the pitch point, P. Therefore, point Q on profile a is serving as a generating point. The corresponding point, S, on the conjugate profile, b, is obtained by transform ing the coordinates of point Q from system ( X a,Y a) to (Xb, Yb). The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 rotation of profile 6, corresponding to 0q is 0s = - (R a /R b ) 6 Q. [3.6] The transform ation required to obtain the coordinates of point S is given by Sx = Qx cos (0q — 0s) — Qy sin(0q —Os) + cccos( 7r — Os) [3.7] Sy = Qx sin(0g - 0s) + Qy cos(0q — 0s) + ccsin( 7r — 0s) [3.8] where cc = the center to center distance, Oa — Ob- [3.9] Using the angular velocity ratio relationship, the relative rotations can be expressed as 0q - Os = ( j ^ Oq - [3.10] Therefore, the coordinate transform ation becomes Sx = Qxcos ~ QyStn + cccos( 7r - 0s) [3.11] Sy = Qxs in + Qycos + ccsin( 7r - 0s). [3.12] This procedure is repeated for each point on the generating profile. T he value of Oq for each point is defined as the rotation a t which the profile norm al at point Q passes through the pitch point, P. This angle is used to transform th e generating point, Q, to a conjugate point, S. Not all conjugate points com puted in this m anner may be valid for creating a m ating gear, depending upon the geom etry of the generating shape. The invalid points are those which would interfere w ith the envelope defined using the kinem atic inversion m ethod described above. T he kinem atic inversion and generation m ethods can be useful in designing a gear pair with a desired kinem atic relationship. This relationship is defined by the pitch curves associated with each gear. In this work, the kinem atic inversion m ethod is used Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 to determ ine the accuracy of assuming a constant instantaneous angular velocity ratio for the screw compressor rotors. In addition, an iterative procedure for defining rotor profiles is based on the generation m ethod and is im plem ented to revise the original profile d a ta which was provided. 3.1.3 Pressure Angle and Direction of Force Transmission Another im portant consideration in the analysis of a gear pair is th e direction of the force transmission. T he contact force between the gears can be resolved into components along the common normal and the common tangent to the profiles. In this study, the effects of friction are considered negligible. Therefore, the force com­ ponent along the common tangent can be neglected, leaving only the component of the contact force in the direction of th e common normal. This is th e direction of force transm ission, in the case of the 2 dimensional gear profiles, and can be specified using the pressure angle,ip. The pressure angle is the angle between the norm al to the profile at the point of contact and a line tangent to the pitch curves a t the pitch point, as shown in Figure 3.3. The value of ip can be com puted, using th e notation of Section 3.1.2 as follows. N q = the profile norm al a t point Q [3.13] “ flxi “f" n yj [3.14] where unit vectors i and j are defined in the inertial coordinate system , ( X c, Yc). The normal vector, N q , is rotated by the am ount Oq to obtain n'x — tix cos(0q ) — n y sin( 0g) Tiy = nx sin(0Q) + n y cos(0Q). [3.16] ip = arctan f—f ). [3.17] [3.15] The value of ip is then defined as nx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 The pressure angle is therefore dependent upon th e profile norm al a t the point of contact and m ay vary as a function of the rotation of the driver gear. The pressure angle for helical gears m ust include th e z com ponent of th e normal to the profile surface. This is achieved by com puting th e direction cosines associated with the normal. Force transm ission between the gears is in the direction of th e profile normal and can be com puted using the direction cosines. These direction cosines are defined as cos(a) = cos(/3) = cos (7 ) = Tlx m nv \\N\\ nz [3.18] [3.19] [3.20] 1m In addition to the common norm al at the point of contact, th e direction of rotation is also im portant in determ ining the direction of force transm ission. For gear pairs in which the driver is designed to ro tate both clockwise and counter-clockwise, the entire profile shape m ust be considered when determ ining the angular velocity ratio and the direction of force transmission. W hen only one direction of rotation is required, a portion of the profile servers only to m aintain contact and does not transm it force. The twin screw compressor rotors are a gear pair designed to ro tate in one direction only with the m ale rotor serving as the driver gear and the fem ale as th e driven gear. Therefore, only a portion of each profile is involved in force transm ission. The rem ainder of each profile shape is designed to m aintain the seal a t the contact between the rotors and to optim ize th e cham ber geometry required to provide compression. Therefore, the direction of force transm ission is required for th e portion of the profile which transm its force. However, the integrity of th e cham ber seals is determ ined in p art by the accuracy with which the rem aining portions of the profiles are designed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 3.1.4 Definition of Driving Points In order for a point on the driver gear to transm it a force to the driven gear, the contact force m ust produce a compressive stress on both gears. A tensile contact stress is not physically possible and indicates a separation of the gears. T he condition required for gear a to transm it force to gear b, while rotating counter-clockwise can be defined as follows. Let point Q be a point on the profile of gear a w ith the vector from Oa to Q defined as Q in the ( X a,Y a) coordinate frame. The norm al to the profile a t Q is defined by the vector N q . An inertial reference fram e ( X c, Yc) is located with its origin a t Oa. Assume th a t Q is the contact point when gear a is rotated to vector Q and the normal, N q , N q , Nq = The are expressed in the inertial reference fram e as Q' and respectively, corresponding to the rotation of Q' - Oq . Oq , by [Qx cos(0q) — Qj, sin( 0<g)] i*+ [Q*sin( 0Q) 4 - cos( 0q ) ] J [Aqx cos( 0q) - N qy sin(0o )] i + [Nqy sin(0g) + N Qx cos( 0Q)] J. Assume th a t the m ale rotor is positioned at Oq [3.21] [3.22] as defined above. At this instant, for the m ale rotor to transm it a force to the female rotor a t Q, th e absolute velocity of Q m ust have a component in the positive direction of N q . The com putations required to accomplish this determ ination follow. The velocity of Q at this instant is defined as Vq , refer to Figure 3.4, where VQ = Oq x Q'. [3.23] The component of the velocity along the norm al direction, Vqn, is com puted as the dot product of the velocity vector and the normal, N q , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 V Qn = Vq -N' q = [Nqx c o s (O q ) + [3.24] — N qy [ N q x s in (0g ) + sin(0g)] [—0 a Q x sin(flg) - N q y c o s ( 0 q ) ] [0 a Q x c o s ( 0 a ) - 0aQ y cos(flg)] 0 a Q y s in (0 Q ) ] . [3 . 2 5 ] Simplifying the right hand side of equation 3.25 gives Vq„ = 0a (Q xN qv — Q y N q ^ . Only the sign of Vqn [3.26] is im portant in determ ining w hether or not this point transm its force from gear a to gear b. Therefore, the m agnitude of 6a can be elim inated. The dot product is then com puted as th e variable, D r iv e r , 6a D river -> ± 1.0 [3.27] = ± ( QxN Qy - QyN q S ) • [3.28] If the sign of D riv er is the sam e as the sign of 6a, then point Q transm its force from the driver to the driven gear at the instant when it is th e contact point. From th e above equations, it is noted th a t the value of D r iv e r is independent of the value of Oq, the instantaneous position of gear a. Therefore th e triple product rule can be applied to obtain th e value of D riv er by performing th e cross product w «■* between the vector Q, and the norm al N q. VQn = = (fQ x Q j ■ Oq • (Q1 x N (k ) [3.29] This fact is used in defining th e driving points for th e screw compressors profiles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 3.1.5 P ath of Contact In this study, gear pairs and the m ating gear teeth are considered rigid bodies. It is assumed th a t line contact occurs on the faces of m ating gear teeth. In the case of spur gears, this line of contact along the tooth face projects to a point on the 2-dimensional gear profiles. The path of contact is defined as the sequence of contact points which occur as the gear pair rotates through its m otion. This path is commonly defined in an inertial coordinate system ( X C,Y C), w ith its origin a t the center of the driver gear. For a planar gear pair, th e path of contact can be defined by transform ing th e contact points, Q, to the inertial coordinate system . In this case the transform ation is simply a rotation by —0q, the negative of the angle associated with the position a t which Q serves as a contact point. Cx = Qx cos(—Oq ) — Q ysin(—Oq ) [3.30] Cy — Qx sin(—6q ) + <3yCos(—Oq ). [3.31] where C = the contact point with origin a t the center of the inertial coordinate system. [3.32] For a planar gear pair, th e contact between the 2-dimensional profiles occurs a t one point for a specific position. Therefore, the p a th of contact is the curve traced as the contact point changes location along the profile during th e rotational m otion. Let Q\ and Q 2 be the contact point on the profile of gear a for th e positions de­ fined by 6a = 0qi and 0a — 0q3, respectively. The coordinates of contact points C\ and C 2 are defined in th e inertial reference fram e by the transform ations of equa­ tions 3.30 and 3.31. As gear a rotates from 6q 1 to 0q2, th e p a th of contact will be the trace from Ci to C2. In the case of helical gears, as for th e screw compressor rotors, th e path of contact is a 3-dimensional curve. A rotation of the helical gear about its center is analogous to translation along the gear axis. Therefore, following the exam ple above, contact Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 occurs at points C\ and C 2 simultaneously, The helical gear differs in th a t the contact occurs in different planes along the gear axis. Let contact point Ci be located in the plane with a z coordinate of z c x• Then contact point C 2 is located in the plane with a z coordinate of zc3, where zc 2 = (0 q 2 - 0Ql) ) [3.33] and L r = the length of the gears along the rotational axis = the wrap angle of the gear. [3.34] [3.35] Therefore, contact occurs along a 3-dimensional curve for a helical gear as opposed to a line of contact on the tooth face of a planar gear. The contact line on the tooth of a planar gear projects to a point on the 2 dimensional gear profile.This is no longer true for the 3-dimensional contact curve associated w ith a helical gear. As a consequence, the contact between helical gears is defined by determ ining the location of the contact curve along the axis of rotation. T he location of this curve is a function of the angular position of the helical gear. This location can be defined as a translation from a reference location, corresponding to a reference angular position. Assume th a t for helical gear A at angular position 0ao the contact curve betw een two m ating teeth is a 3 dimensional curve defined by endpoints C\ and C2. Let this be the reference contact curve. If gear A is rotated to angular position 6a x, th e reference contact curve translates along the rotor axis such th a t the endpoints are located a t C[ and C'2 given by C[ = C'2 = Cx + (0M - 0 A O) (£ ) C2 + (0Ai - 0 A o) ( j )- [3.36] [3.37] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 This m ethod of com puting th e contact curve is applied to the rotor profiles which are defined in the following sections. In the context of screw compressor applications, the entire contact curve is referred to as the interlobe seal curve. A portion of the curve corresponds to the profile segments in which force is transm itted from the male to the female rotor. However, the rem ainder of the curve is equally im portant since seals m ust be m aintained along the entire contact curve. This interlobe seal prevents leakage between the compression chambers of the m echanism and therefore directly affects th e compressor performance. 3.2 Fundam entals Applied to Original Rotor Profiles In this section, the fundam entals described in Section 3.1 are applied to the screw profile d a ta originally supplied by the sponsor for this research. T he d a ta describing the profile shapes consists of a series of discrete polar coordinates. One lobe of the m ale rotor and one flute of the female rotor are defined. This data, as originally supplied, is presented in Figure 3.5. The kinem atic inversion m ethod is first applied to determ ine the accuracy of assuming a constant angular velocity ratio for the rotor pair. T he normals to the male rotor are then com puted for each of th e discrete points. It is assumed th a t the radius of the profile varies linearly in th e regions between adjacent points. Using these normals and the m ale profile as a generating shape, the conjugate female profile is obtained by im plem enting th e generation m ethod presented in Section 3.1.2. The m ethod for determ ining which points on th e m ale profile serve to transm it force to the female is then applied. An iterative m ethod for generating profile shapes, based on the kinem atics presented here, is discussed. An im plem entation of this m ethod to improve the original profile shapes is presented in Section 3.3. The p ath of contact and direction of force transm ission are com puted for th e improved profiles. Male and female profiles were provided by an industrial m anufacturer of screw compressors. These were representative of screw compressor profiles currently in use, however they were intentionally not 100% accurate. The d a ta describing th e female Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 profile contained a cusp near the tip of the flute. This is not physically realizable. Both male and female profiles are available as discrete polar coordinates, with the center-to-center distance specified. In the following analyses, only the m ale profile is considered to be a known shape. A female profile is generated using th e male. T he female profile provided by th e m anufacturer is used only for comparison with the generated female shape. In the discussions and figures included in this section, the compressor profiles are viewed from the suction end of the compressor. For th e configuration used, the male rotates in a counter-clockwise direction and drives the female in a clockwise direction. The original param eters are used to define the rotors. These param eters are Nm = num ber of m ale lobes [3.38] 7 num ber of female flutes [3.39] cc = 56.77(mm) center-to-center distance [3.40] L 99/77(m m ) length of the rotors [3.41] 214.20(degrees) wrap angle of the m ale rotor. [3.42] Nf = = Tm = 3.2.1 6 K inem atic Inversion of th e Rotor Profiles The angular velocity ratio of the m ating rotors is an im portant consideration in performing an analysis of the rotor dynamics. This param eter is initially investigated by conducting the kinem atic inversion m ethod discussed in Section 3.1.2. T he kine­ m atic inversion m ethod is applied to the screw compressor profiles by allowing the m ale profile to be the generating shape and determ ining th e associated conjugate shape of the female. pitch The pitch curves are assumed to have constant radii, therefore circles. T he radius of the pitch circles for the m ale rotor, Rmp, and the female rotor, R f p , are defined as follows. cc = \Rmp0m\ = RmP + R j p [3.43] \R/p0f\ [3-44] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 |JV m 0 m | R JP = \N fO f \ = N j_ R mp [3 .4 5 ] r . « , N m R jp [ 3 -4 6 ] = R m p ( ;^ r ) CC = Rm p + “ " M *5^ ) R n I3 -4 7 ] (J ^ j [3 . 4 8 ] [3'491 * - = “ hcTNj) M R jp [3.51] = cc — Rmp Comparison of the generated female conjugate shape with the specified female profile, reveals the accuracy of the assumed kinem atic relationship. Figure 3.6 shows the result of performing the kinem atic inversion. The male rotor is plotted a t several positions as it orbits around the circumference of the female pitch circle. The envelope formed by this plot represents the conjugate female rotor. Figure 3.7 is a view of one flute of the conjugate female shape w ith one flute of the specified female profile shown. This figure shows th a t the conjugate female shape obtained by assuming constant pitch radii, RmP and R / p , is sim ilar to the specified female profile. This supports the assum ption of constant pitch radii w ith the resulting constant angular velocity ratio. In accordance w ith this assum ption, if the male rotates with a constant angular velocity, it does not induce angular accelerations in the female rotation. 3.2.2 C om putation of Male Profile Normals In order to complete the kinem atic analysis of the compressor rotors, th e normals to the male profile m ust be computed. The profile consists of a series of discrete polar coordinates. In C hapter 4, compression loads are com puted by m apping the 3-dimensional screw geom etry into a 2-dimensional region and integrating over the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 rotor surface. One of the steps required in performing the integration is th e compu­ tation of the normal to the surface, (see Sections 4.2 and 4.3). The results of these com putations are used here. The components of th e profile normal can be expressed as _ ( £ ) ( y i°(ii)+ji” cosW) f L \ I dRm [3.52] cos( 0 ) — Rm sin( 0 ) [3.53] * - - m - | 3 '5 4 ! The normals to the rotor surface are com puted at each of the discrete profile coordinates. For the i th point, the required variables are defined as Rm = 0 = dRm dO Rm(i) [3.55] e m(i) Rm{i + 1) - Rm(i - 1) 0 m(* + 1 ) - 0 m(i ~ 1) [3.56] [3.57] These definitions are consistent with the assum ption th a t the radius varies linearly between adjacent points. T he x and y components of the profile normals, com puted for a single lobe of the m ale rotor, are presented in Figure 3.8. The vectors are normalized to unit vectors, and the m agnitudes are scaled to provide a m ore easily viewed plot. The z components of the profile norm als do not contribute to th e planar kinem atics analysis. However, the z component is required to com pute the direction of force transmission and resulting bearing loads. This is accomplished through th e definition and use of the direction cosines associated with the profile norm al, as presented in Section 3.1.3. The application to the screw compressor geometry is presented in Section 3.3.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 3.2.3 Generating and Conjugate Rotor Profiles Two m ethods for producing a conjugate shape from a generating shape were pre­ sented in Section 3.1.2. The kinem atic inversion m ethod was employed in Section 3.2.1 to dem onstrate th a t the assum ption of constant radii pitch curves is accurate. In this section, the generation m ethod which was presented is applied to the compressor rotors. Coordinate system s, (AmjFm) and (X j ,Y j ), are attached to the centers of the male and female rotors, respectively, and allowed to ro tate about their fixed origins according to the kinem atic constraint of rolling contact betw een th e pitch circles. The m ale profile is designated as the generating shape and is attached to the origin of (X m, Vjn). For each of the discrete points defining the m ale profile, a corresponding point on the female profile is defined according to the m ethods presented in Sec­ tion 3.1.2. Figure 3.9 shows th e correspondence between the generating points on the male and the conjugate female points. The conjugate female profile is presented in Figure 3.10. In Figure 3.10 a cusp exists in the female profile, near th e portion of th e profile where the radius is a m axim um , the flute tip. T he specified fem ale profile, which was provided as representative data, contained a cusp in th e sam e region. This cusp does not appear in the conjugate shapes derived by th e kinem atic inversion m ethod, Figures 3.6 and 3.7. The cusp in th e female profile is not physically realizable w ithout producing an interference between th e rotors. In order to conduct load com putations and analysis of the rotors, the d a ta was modified to remove th e cusp in th e female profile. The modifications were obtained using the iterative approach described in Section 3.2.4, below. 3.2.4 Iterative M ethod for Profile Generation In order to remove th e cusp in th e original profile data, an iterative approach to the generation m ethod presented in Section 3.1.2 was im plem ented. This im plem entation also resulted in m ore reliable contact line d a ta and values of th e pressure angle. In Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 this section, a brief description of the iterative approach is given. The final profiles which were obtained and used throughout the rem ainder of this research are presented in Section 3.3. A schem atic of the iterative procedure is presented in Figure 3.11. This procedure is best described via an example. The procedure is initiated by m anually defining one of the profiles, in this case the male, as a set of discrete coordinates. Designate this initial generating profile as M l. T he profile definition dictates th e num ber of lobes on the male rotor by defining the circumferential pitch of th e rotor. In addition, the num ber of female flutes is m anually defined, as is the center-to-center distance between the rotors. This enables com putation of the pitch circles associated with each rotor using equations 3.43 - 3.51. T he normals to profile M l are com puted. The kinem atic m ethod for defining th e conjugate shape is then employed using M l as the generating shape to define the m ating conjugate female profile, F I. This conjugate profile is revised by m anually redefining the coordinates of the discrete points for the female profile. The revised profile shape is designated as F2. T he normals to profile F2 are com puted. Profile F2, and the assumed pitch curves are then used to generate the m ating m ale conjugate, M2. This procedure is iterated until satisfactory profiles are defined. 3.3 Im plem entation of Profile Generation M ethod As was stated in Section 3.2, the original profiles and associated contact line which were provided for this research were not accurate. In order to revise th e data, the iterative m ethod described above was implem ented. T he objectives of th e implemen­ tation were to remove the cusp in the female profile, to increase th e confidence in the accuracy of the profile shapes and the associated contact line, and to validate the m ethod of profile generation. T he results of the im plem entation are presented in this section. T he profile generation m ethod is initiated by defining the geom etry of th e gen­ erating profile shape. During the iteration, m anual revisions to the generating and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 conjugate shapes are required. In this im plem entation, these revisions are restricted to the use of circular arcs and parabolic param etric curves. Some details for imple­ m enting each of these are provided below. Circular arcs are used by identifying three existing profile points which define the desired arc. These three points are used to com pute the radius and center of the prescribed circle. The two exterior points are used to com pute th e arc length subtended by the segment. In addition, the indices associated w ith th e exterior points are used to com pute the num ber of points on the profile segment which lie on the arc. U pdated profile points are then distributed evenly along th e length of the circular arc. T he normals associated with these revised profile points are com puted based on the radius and center of the arc. The parabolic param etric curves are im plem ented by identifying the end points of the profile segment which is to be revised. All profile points which lie between these endpoints are then updated using a param etric curve. In implem enting the param etric curve definition, th e two points which are adjacent to the endpoints, on the exterior of th e segment, are used to define the curve. For example, assum e th a t the 20th and 29th profile points are identified as the endpoints of the segment to be revised. T he coefficients of th e param etric curve are defined using these points and the 19th and 30th points. The param etric curve is defined as a function of param eter, s, such th a t s = 0 at point 20 and s = 1 a t point 29. T he coordinates of points 21 through 28 are then redefined using uniform increm ents of s. The norm als to the updated profile segment are com puted using the definition of th e param etric curve. T he im plem entation of the profile generation m ethod is lim ited by th e num ber of options which are available for redefining profile segments. However, this was not the m ain focus of the research, and satisfactory results were obtained. These results are presented in the next section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 3.3.1 Final Profile Shapes and Contact Line The profile generation m ethod was used to revise the profile d a ta which was orig­ inally provided for this research. The original geometric param eters were used with the m ale profile as the generating shape. The female conjugate shape was generated. As was stated in Section 3.2.3, this female conjugate shape contains a cusp near the flute tip. The cusp in the tip of the resulting female shape was removed using the circular segments and param etric curves as described above. This revised female was then used to generate a male profile. The procedure was iterated until satisfactory profile shapes were obtained. These profile shapes are shown in Figures 3.12 and 3.13. In Section 3.1.5 a m ethod for defining the contact curve associated w ith two profiles is presented. This m ethod was used to produce the contact curve associated with the final male and female profiles. The contact curve is shown in Figure 3.14. 3.3.2 Direction of Force Transmission for Final Profiles The direction of force transmission associated with the final profiles can be deter­ m ined once the normals to the male profile and the contact curve have been defined. In Section 3.2.2, the normals to th e m ale profile are defined for each of the discrete polar coordinates. The contact curve, is defined in Section 3.1.5. T he m ethod for defining the driving points on th e male is presented in Section 3.1.4. These param eters are used to define the direction of force transmission. These derivations were applied to each of the discrete points of the final male profile. Figure 3.15 shows the portion of the m ale profile which serves to transm it force to the female rotor. The points are shown at th e reference angle, 0m = 0. The value of the direction cosines associated w ith the i th profile point depend upon the position of the male rotor when the i th point is the contact point. The values of the direction cosines are shown in Figure 3.16. From these figures it is evident th a t the value of the direction cosines, and therefore the direction of force transm ission, vary throughout the motion of the rotors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 3.4 Rotor Contact Coefficients It has been shown th a t a kinem atic analysis of the screw compressor rotor profiles is im portant in determ ining the angular velocity ratio, contact curve and direction of force transmission associated with specific profile designs. This analysis has also served as the basis for a profile generation m ethod. Another application of the kine­ m atic analysis results is the resolution of th e contact force between the rotors to the bearing locations. This development is addressed in this section. A m athem atical model and associated com puter sim ulation of the rotor dynamics is presented in C hapter 5. One of the results com puted during the sim ulation is the contact force between the m ale and female rotors. The sim ulation is based on a 2 -degree-of-freedom model in which the motion of each rotor is defined as rotation about its central axis. The pitch circle associated with each rotor is used to define the rotational motion. In addition, contact between the rotors is assumed to take place a t the pitch point, along the line of centers. Therefore, the contact force which is com puted during th e sim ulation is in th e direction of the common tangent to each of the pitch circles. This force represents the y component of the to ta l contact force. This y component, along with the contact curve and the direction of force transmission associated w ith the profiles, is used to com pute the total contact force. The total force is expressed in term s of the forces and moments a t the bearing locations. The approach is to derive coefficients for each of the bearing forces. T he individual bearing forces can be com puted by m ultiplying the respective coefficient by the y component of the contact force which results from the dynam ic sim ulation. The coefficients are defined in term s of the m ale rotor rotation angle. 3.4.1 Derivation of Rotor Contact Coefficients In order to compute the contact coefficients, the num ber and location of individual contact points m ust be obtained as a function of the rotation of th e m ale rotor. Through the kinem atic analysis, the contact curve is defined for one lobe of the m ale Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 rotor, for a reference rotation value of 0m = 0mo. The contact curve for each of the rem aining lobes is obtained by translating the reference contact curve along the rotor axis. Therefore, for a specified rotation value, the contact curve for the entire length of the rotor can be obtained. T he derivation is given below. Let (CXo, Cyo, Cz,,) represent the coordinates of the contact curve associated with lobe 1 when the male is rotated to 0m = 6mo. The origin of this curve is the center of the m ale rotor a t the suction plane. It is defined in an inertial coordinate system. This is the reference contact curve. The contact curve associated with the rem aining lobes have the same x and y coordinates as th e reference contact curve. However, the contact curve z coordinates for the remaining lobes are shifted along the rotor axis. For the k th lobe the 2 coordinates are given by, c- - « • - * ( £ ) ( £ ) • M In a sim ilar m anner, the reference contact curve can be translated to account for rotation of the male rotor. Let C '0 represent the z coordinates of the reference contact curve translated to account for a male rotation of 0m "• = 0'm . The values of C'ZQ are given by, \ ‘m / The values [3.59] are then replaced by C '0 in equation 3.58 to com pute th e z coor­ dinates for the rem aining lobes, for a rotation of 0m = 0'm. In order for a contact point to be valid, th e z coordinate of th e point m ust be within the lim its defined by th e rotor length. Therefore, the condition — L < CZk < 0.0 [3.60] m ust be m et for all valid contact points. T he num ber of contact points, location of each contact point, and direction of force transm ission associated with each contact point, can therefore be derived as a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 function of the rotation of the male rotor. Once this is accomplished, the contact coefficients can be computed. Let Npv be the force com puted during the sim ulation based on the 2-degree-offreedom model. This force is th e y component of the to tal contact force, \Np\. In this developm ent, it isassumed th a t the to tal contact force is evenly distributed among each of the contact points. Therefore, the contact force at the i th contact point is, \Ni\ = ^ [3.61] flop where n cp = the total num ber of contact points. [3.62] At each contact point, the direction of force transm ission can be defined using the direction cosines com puted from the inward normals to the rotor profile at the i th contact point, cos(a,) = cos (/?,) = cos(7 ,) = m ag = nXi mag n vi mag n Zi mag [3.63] [3.64] [3.65] \A*x;2 + Tlyi2 + n 2).2. [3.66] T he components of the contact force at each contact point are then defined using equations 3.61 - 3.66 as, Nix = ( i r ^ ) cos(a,) [3,67] = ( ^ ) cos(A) [3.68] = (1——^ cos(7 .) [3.69] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 The m agnitude of the total force, \Np\, is com puted using th e sum m ation of the y components a t the individual contact points. Tlcp [3.70] N Py = [3.71] *> ■ f e ) s [3.72] Since the contact force m ust always produce a compressive stress, the force can only occur in one direction w ith respect to the surface normal. Therefore, the absolute value of Npv is used in the com putations. [3.73] \ n cp ) t=l ( cp IJVH = K | Tlcp \ [3.74] E cos (/?,) li=l This definition of \Np\ is then used to com pute the com ponents of th e contact force a t each contact location. = [3.75] K | E cos(/3i) Ni, [3.76] = E c o st# ) N i. ITlcp E cos(/?,) [3.77] 1=1 T he components of th e contact force at each location are therefore known as a function of |^ p y| and the direction cosines. T he locations of each of th e individual Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 contact forces is determ ined by the contact curve. The contact coefficients related to each bearing force are com puted by resolving the individual contact forces to the bearing locations and factoring out |-Npy|. The typical configuration of th e bearings for a single rotor is shown in Figure 3.17. The equivalent bearing forces, com puted from the contact force, are also designated. Radial bearings located near th e suction and discharge planes support transverse forces in the x and y directions. T hrust bearings located near th e suction planes, on each rotor, support axial forces. The bearings are not designed to support a moment about the z axis. Therefore, th e contribution of the contact force to a z m om ent is com puted and shown near th e suction end bearing. The distances L I and L2 are specified as positive quantities. Resolution of the individual contact forces to the bearing locations results in th e following equations. Tlcp Li N B 2 Xm t=i — Kl L \ B 2 ym = TlCp cos(a<) + Tlcp - ^]cos(a,)C 'Zi *=1 cos t'=i Tlcp (L\ + L + L 2 ) Y , cos(A ) t=l T lcp TlCp E c o s ( / ? i) + X ^ c o :’( l i ) C y , «= i •= ! T lcp ~ + L + L 2 ) Y 1= 1 T lcp ( L \ [3.78] Y COS{ P i) C v [3.79] c ° s ( / 5i) 1=1 Tlcp £ c o s (a .) B l x„ = K KTlcp--------------- b i „ [3.80] £ c o s ( /? i) B 1Vm = \Np„\ sign ^ Y COS(/?;) j - B 2 ym [3.81] T lcp Y cos( 7 .) K M Tlcp [3.82] Y cos(/?,•) [3.83] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Coefficients for each of the bearing forces and the m om ent about the z axis are defined by factoring out the y component of the contact force, \Npy |. This results in a coefficient which can be m ultiplied by |-/Vp„| to obtain the bearing force due to the contact. Tic p Cb 2x T icp T lcp L x ^ 2 cos(a,) + cos(7 ,)C Xi - ^ c o s ta ,) ^ , ■=1_________ i=l i=l T lcp [3.84] (Lx + L + L2) ]T c o s (/?,) 1=1 Lx Y2 C0S(Pi) + 1=1 CBlym - i=l cos(li)Cyi - X I cos(/?i)C2i 1=1 T lcp [3.85] {Lx + L + L 2 ) 2 cos (/?,) t '= l T lcp X > s ( a ,- ) CBlIm = —Cb 2x T lcp [3.86] Ecos(A) Csiym = Sign ^ E COS(/5,) j ~ C B2y [3.87] T lcp E cos(7i) CBUm = [3.88] Tlcp E cos(A ) [3.89] The bearing forces and m om ent about the z axis which result from the rotor contact force are obtained by m ultiplying the corresponding coefficients by the y component of the contact force, |-Npv|. As example, JBlXm = \N pv \ * Cfii*m- These coefficients depend upon the coordinates of the contact points and therefore on the position of the male rotor, specified by the value of 9m. 3.4.2 Contact Coefficients for New Profiles T he contact coefficients for the final m ale and female profiles are presented in Figures 3.18 to 3.21. In com puting these coefficients, it was assum ed th a t contact Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 occurs a t each point which can potentially serve to transm it force between the male and female rotors. As was discussed above, these coefficients depend on th e num ber of contact points which exist at any given rotation angle. By lim iting the num ber of potential contact points per lobe, the contact coefficients can be varied to accommo­ date different models of th e contact between the m ale and female rotors. As example, it was assumed th a t only one point on each lobe can transm it force from th e m ale to the female rotor. 3.5 Conclusions Based on Kinem atic Analysis The following conclusions can be based on th e kinem atic analysis of the screw compressor rotors as a gear pair. The models used to analyze the rotor m otion and bearing loads are based on these conclusions. The assum ption of a constant angular velocity ratio is accurate for the profiles under investigation. The envelope produced by th e kinem atic inversion m ethod, Fig­ ure 3.6, supports this assum ption. The conjugate female profile shape produced using the m ale profile as a generating shape also supports this assum ption. Although a cusp was evident in the female profile, it was in a region not associated with the transm ission of force. Therefore, this cusp does not affect the kinem atic analysis. The direction of force transmission between the rotors varies significantly for the 2-dimensional profiles analyzed. In addition, m anufacturing tolerances will influence the direction of force transmission. These effects are not included in this work. The kinem atic analysis provided here is based solely on th e 2-dimensional profiles of the screw compressor rotors. These rotors are actually helical surfaces wrapped around the center of the profiles. In theory, contact between the rotors occurs along a continuous p ath which extends the length of the rotors. This has an overall effect of averaging the param eters analyzed here. This aspect of th e compressor design lends more support to th e accuracy of assuming a constant velocity ratio. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 Profile a Profile b Figure 3.1 Fundam ental condition for constant angular velocity ratio. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 C i \xb xc ■Profile b Profile a Figure 3.2 Reference position of pitch circles for generating a conjugate shape. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 £ Ya Yb Profile a Profile b Figure 3.3 R otated position of pitch circles where point Q is generating the conjugate point S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 ■77T Profile a Figure 3.4 Condition required for a point to transm it force from gear a to gear b. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 (a) 30 20 10 0 (b) 15 10 5 -20 Figure 3.5 Original profiles from d a ta supplied for research; (a) m ale profile; (b) female profile. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.6 Kinem atic inversion m ethod applied using the m ale profile to generate the female. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Figure 3.7 Comparison of one flute of the specified female profile with one flute of the profile generated by kinem atic inversion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 Figure 3.8 Norm al vectors to th e original m ale profile, x and y components. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 35 10 15 20 25 30 35 40 45 50 Figure 3.9 Correspondence between discrete points on the original m ale profile and the generated conjugate female profile. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 o o o o -30 -25 -15 -10 Figure 3.10 Conjugate female profile generated w ith original m ale profile. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 Define Generating Shape (Ml) Revise / Improve Conjugate / Generating Shapes Compute Normals to Generating Shape ( i ( Compute Angles at which Conjugate Shape is Generated ( Are Shapes Acceptable ?) Define Conjugate Shape (FI) Figure 3.11 Schematic of iterative procedure used for profile generation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 25 25 30 40 45 Figure 3.12 Revised male profile obtained through iterative generation m ethod. 18 16 o 14 o o 12 10 8 6 o 4 2 -30 -2 5 -20 -1 5 -10 Figure 3.13 Revised female profile obtained through iterative generation m ethod. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 (b) (a) -10 ? £-20 N -20 -3 0 x (m m ) 20 z (m m ) 40 x (m m ) (c) (d) E E £ £ >. >. -10 -10 -20 z (m m ) 0 20 30 40 x (m m ) Figure 3.14 Contact curve associated with final profiles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 20 35 40 45 Figure 3.15 Driving points on the final m ale profile. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 1 o 1 0 10 20 30 40 50 60 70 50 60 70 Profile Index 1 .□ Q) 0 1 0 10 20 30 40 Profile Index Profile Index Figure 3.16 Direction cosines as a function of 0m for the final profiles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B2y B2x Discharge Plane Bly Suction Plane Biz Blx Figure 3.17 Typical bearing configurations for a single compressor rotor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 0.8 0.6 0.4 c o -0.2 O -0.4 - 0.6 - 0.8 50 100 150I 200 :250 300 350 400 thetam (degrees) Figure 3.18 Contact coefficients vs 6m, m ale suction bearing; ( - ) C 5 1 Xm; (-----) C B l Vm; ( - • - ) C B \ Zm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 0.8 0.6 0.4 0.2 ® • 0.2 -0.4 - 0.6 - 0.8 100 150» 200 :250 300 350 400 thetam (degrees) Figure 3.19 Contact coefficients vs 6m, m ale discharge bearing; ( ~ ) C B 2 Xm] ( - - - ) C B 2 ym. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 0.8 0.6 0.4 .2 0.2 - 0.2 -0.4 - 0.6 - 0.8 100 150I 200 ; 250 300 350 400 thetam (degrees) Figure 3.20 Contact coefficients vs 0m, female suction bearing; { - ) C B 1*,; (-----) CBlyt \ ( - . - ) C B l Z{. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 0.8 0.6 0.4 0.2 - 0.2 -0.4 - 0.6 - 0.8 50 100 150I 200 ! 250 300 350 400 thetam (degrees) Figure 3.21 Contact coefficients vs 6m, female discharge bearing; ( - ) C B 2 b /; ( - - - ) C B 2 yj. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. R O T O R LOADING DUE TO GAS COM PRESSION In this chapter, the m ethodology developed for com puting th e m om ents and forces on each rotor due to the compression process is presented. Throughout this docu­ m ent, these are referred to as the compression loads. T he author and Soedel [86] have previously published related work in which some of th e compression loads are com puted. In th a t work, th e interlobe seal line is approxim ated as a straight line parallel to the rotor axis of rotation. This approxim ation results in m om ent loads, about the rotor axes, of nearly equal m agnitude for both the m ale and female rotors. In this work, th e actual interlobe seal geom etry is used. Therefore, the inaccuracy associated w ith the straight line approxim ation is avoided. An overview of the m ethod is first presented, followed by a review of the vector calculus which is used to com pute the compression loads. T he details of the im ple­ m entation are provided for the geom etry associated w ith tw in screw compressors. The compression loads are com puted for capacity ratings of 100%, 75% and 50%. These are the capacity ratings which can be obtained on th e screw compressor a t the Herrick Laboratories. Results of experim ental m easurem ents of gas pulsations and rotor chatter are available for this compressor. O perating conditions which represent over-pressure, under-pressure and ideal conditions are sim ulated for each capacity rating. 4.1 Overview of M ethod The compression process is described in C hapter 1. T he m ale rotor is used to dem onstrate th e m ethod used to com pute the compression loads. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 Three geometric constructions, sections, chambers and strips, are used in com­ puting the compression loads for the screw compressor rotors. T he m ale rotor is divided into N m equivalent sections, where N m is the num ber of lobes on the male rotor. These sections are bounded by adjacent lobe tips, and extend from the suction plane to th e discharge plane. Figure 4.1 is a view of th e m ale rotor w ith a single section emphasized. The contact points between the m ale and female rotors form an interlobe seal curve along the male rotor. This interlobe seal curve, shown for one section in Figure 4.2, further divides the N m sections of th e m ale rotor into separate compression chambers. A low pressure cham ber extends from th e suction plane to the seal curve along the top of the rotor. A high pressure cham ber extends from the seal curve to the discharge plane along the bottom of the rotor. T he compression loads are com puted by integrating the pressure over the surface defined by each chamber. The required integrations are performed by further dividing each cham ber into helical strips and sum m ing the integrals com puted over each strip. As the m ale rotor rotates, th e interlobe seal for a specific section translates along the rotor axis, from the suction plane to the discharge plane, Figure 1.4, repeated in Figure 4.3. T he resulting decrease in the chamber volume causes compression of the enclosed gas. The compression loads are com puted by integrating the pressure over each individual chamber and summ ing the results. Due to th e sym m etry of the rotors, the compression cycle is a periodic function of 6m, th e rotation of the m ale rotor. The period is 2 n / N m radians. Therefore, the compression loads are com puted for values of 0m from 0.0 to / N m. This completely defines th e compression loads as a function of 0m. The compression in each chamber is assumed to be a polytropic process. In addition, for a specified value of 6m, the pressure is assumed to be constant throughout each individual chamber. Therefore, the integrands used to com pute the compression loads are functions only of th e chamber geometry, as defined by th e rotor geometry and the cham ber boundaries. The rotor geometry consists of: 1. the 2-dimensional rotor profile, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 2. the rotor length, and 3. the rotor wrap angle. The chamber boundaries are: 1. the interfaces between th e rotors and the suction and discharge planes, 2. the rotor lobe tip and housing interfaces, and 3. th e interlobe seal curve between the male and female rotors. These boundaries define the lim its of integration for each chamber. The integrations are accomplished using vector calculus principles. T he pressure due to compression is a scalar which can be integrated over the 3-dimensional surface of the rotor. Using the definition of the norm al to th e rotor surface allows the com­ putation of the principal components, (a:, y, and z) of the compression loads. The integration is performed by m apping each 3-dimensional compression cham ber into a 2-dimensional integrating region. 4.2 Integration of a Scalar over a Surface The vector calculus principles used are reviewed in this section. A graphical representation of the required m apping is shown in Figure 4.4. Let a surface, S, be defined in term s of two variables, u and v. The integral of a scalar function over the surface can be defined over a 2-dimensional region as / / / f d s = j J f \ \ T u x f v\ \d u d v s R [4.1] where S ds = th e surface definition; S x i + S yj + S z k = elem ental surface area [4.2] [4.3] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 R = the integrating region [4.4] f = scalar value [4.5] 2-D m apping param eters [4.6] surface tangent vectors. [4.7] u and v = Tu and Tv = The tangent vectors to S are defined using Cartesian coordinates as - asx* dSyl , as, t r . C1 dv 3 1 J du dv The unit norm al to the surface is defined as n = ^ X^ l |r .x r .||. [4.10] 1 J The components of the normal vector are then defined as U j IX * z ny = n ■j nz = n ■k. In order to implem ent this m ethod to com pute the compression loads, th e scalar quan­ tity, / , becomes the pressure, P , in a given chamber, m ultiplied by th e appropriate component of the surface norm al vector n. The component forces due to compression are com puted as Fx = P J J (Tu x Tv)x du dv [4.11] Fy = P j J (Tu x f v)y dudv [4.12] Fz = [4.13] P J J (Tu x Tv)z du dv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 where ( f u x Tv)x = (Tu x T v) - i (Tu X f„ ) y = ( f U X f„ ) • 7 (f„ x f„)* ( f u x f„ ) • k. = The component m oments due to compression are com puted from the forces as Mx = j j J {FzS y - F yS z)d s [4.14] My = 1 1 J (F*S * ~ F zS x)ds [4.15] Mz = JJJ [4.16] (FyS* ~ FxS y) ds where the component m om ents are about the origin of th e C artesian coordinate sys­ tem . T he values of Sx, S y and Sz are defined in term s of th e param eters, u and v. These equations lead to the following integrations for th e com ponent m om ents due to compression. Mx = P J J (Tu x f v)z S y — ( f x Tv)y S z du dv [4.17] My = P J J (Tu x Tv)x S z - (Tu x Tv)z S x du dv [4.18] Mz = P J J (fu x f v)y Sx — (Tu x Tv)x S y d u d v [4.19] These concepts are im plem ented by developing an appropriate m apping from the 3-dimensional rotor surfaces to 2-dimensional integration planes. 4.3 Integration M ethod Applied to the Screw Compressor Rotors The integration m ethod described above is applied to th e screw compressor rotors. A suitable m apping is defined. Assum ptions are m ade about th e n atu re of the discrete polar coordinates which define the rotor profiles. T he integrals axe defined. The details are presented here for the male rotor only, w ith the revisions required for application to the female described. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 4.3.1 M apping the 3-D Rotor Surface to a 2-D Region The m apping from the 3-D rotor surface to the 2-D integration region is defined using the param eters which define the rotor geometry. T he C artesian coordinate system used for the m apping, and the required variable definitions are shown in Figure 4.5. The origin of the Cartesian coordinate system is th e point defined by the rotational axis of the male rotor, the center of the 2-D profile, a t the suction plane. The 3-dimensional rotor geom etry is completely defined by th e 2-dimensional pro­ file and th e wrap angle of the rotor. The 2-dimensional profile is defined by th e discrete polar coordinates, Rm and 0 m, of one lobe. This is referred to as th e profile data. The sym m etric profile is com pleted by repeating these coordinates, increm enting the 0 m values appropriately, to com plete one full revolution and define N m lobes. The wrap angle is the am ount of helical twist which the m ale exhibits through the length of the rotor, from the suction plane to the discharge plane. Using these geometric param eters, the 2-dimensional integration variables can be defined. The param eters 6 and (f>are used as the integrating variables. The variable 6 is th e unique polar coordinate associated w ith each point on the 2-dimensional rotor profile. This coordinate is defined in term s of the profile d ata coordinate 0 m, the specific section involved, and the current rotation of the rotor. = 0m (O + m + e c i -- 11)) 1 00 = + 00m + ^(sect 2?r [4.20] where 0 m(i) sect = the polar coordinate of the ith point in the profile d a ta = the current section num ber; from 1 to N m 9m = the rotation of the m ale rotor about its axis. The 2-dimensional profile is not known in closed form polar coordinates. but only as a set of discrete T he radius of th e rotor is defined in term s of th e discrete polar coordinates by assuming th a t the radius varies linearly as a function of 0 m between adjacent points. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 R m(6) = R m(i) + /3m( i ) ( 6 - G m(i)) [4.21] where 0m (l) n PmK) < S < _ 0 m( i + l ) Rm(i + 1) ~ Rm{i) 0m (< + l)-0m (») This relation can be used to define R(6) for th e entire 2-D profile. 4.3.2 Resulting Integrals The integration variables, 0 and <f>, completely define th e 3-dimensional rotor surface. The Cartesian coordinates of the rotor surface are defined in term s of 6 and <t> as Sx = Rm(0) cos(0 + <l>) [4.22] Sy = Rm{0) sin(0 + <j>) [4.23] Sz = ( L /r m) x <j>. [4.24] In addition, th e cross product of th e tangent vectors, (Tu x Tv), becomes T^xTe = sin(0 + <j>) + i?m(0) cos(0 + <f>)}i [4.25] + ( £ / r m){ An cos(0 + (j>) - Rm(0) sin(0 + (f>)}J [4.26] + pm R m{6)k. [4.27] ( ~ L / T m ){ Using this m apping scheme, the compression loads can be com puted. The resulting integrations are J J{P FXm = - P ( £ / r m) FVm = P ( L / r m) J J { / 3 cos(6 + <!>)- Rm(0) sin(0 + <j>) }d<f>d0 F*m = -P MXm = sin(0 + <j>) + 11^(6) cos(6 + <f>) }d<f>d6 J J PRm(0)d4>d0 [4.28] [4.29] [4.30] - P j f p Rm(0) {Rm {0)am (0 + 4)} d<f>d0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 - P ( L / r mf / / { f t , cos(0 4- <j>) — Rm{0) sin(0 + <^)} <f>d(f>d0 M ym = p [4.31] J J P R m{ 9 ){ R m{e) cos{e + 4>)}d<t>de - P ( L / r mr J j { 0 m sin(0 + (j>) + Rm(9) cos(0 + </>)} (f>dtpd0 [4.32] M im = P(L/rm) J J P R m(9)d<f>d9 [4.33] P = the pressure of the cham ber being integrated. [4.34] where 4.3.3 Evaluation of Integral Limits The integration is performed by dividing each cham ber into helical strips along the rotor surface. The boundaries which define these strips are 1. The arc subtended by adjacent discrete profile points a t th e suction plane, 2. Helical lines extending from the adjacent discrete profile points a t th e suction plane to the location of th e corresponding points a t th e discharge plane, 3. The arc subtended by adjacent discrete profile points a t th e discharge plane. The integration lim its are defined by the boundaries of th e cham ber being inte­ grated. Each pair of adjacent discrete points in the 2-dimensional profile d a ta are used to define a separate helical strip for integration. For a single helical strip, the lim its on 9 are simply the polar coordinates of adjacent points on the 2-dimensional profile. The lim its on <j> are defined by the suction and discharge planes and the interlobe seal curve. The variable </>defines the am ount of twist associated w ith a given helical inte­ gration strip. T he lim its on <f>are defined by the endpoints of th e helical lines which bound an integration strip. These lim its are directly related to the z coordinates of the boundaries by -(£ )* |4 '351 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 where L = rotor length rm = wrap angle. If a specific helical line does not intersect the interlobe seal curve, the endpoints of the line are the suction and discharge planes. These endpoints occur a t z = 0 and 2 = —L, giving integration lim its of <j>= 0 and <j>= r m, respectively. The helical lines which form the boundary of the integration strip m ay intersect the interlobe seal curve, a t some distance along the rotor axis. If this occurs, the section is divided into two chambers. The seal curve serves as an additional boundary. Separate integrations are performed for the low pressure cham ber from the suction plane to th e seal curve, and the high pressure cham ber from th e seal curve to the discharge plane. Therefore, the value of (j>associated with the seal curve intersection serves as one of the lim its in each of these integrations and m ust be determ ined. The value of (f>at the intersection between a helical line extending along the rotor surface and the interlobe seal curve is com puted by establishing the relationship between the profile data and the seal curve data. A one-to-one correspondence exists between the discrete 2-dimensional profile points and the discrete 3-dimensional seal data. T he ith point of th e seal d a ta represents th e contact point between th e ith points of the m ale and female profiles. Therefore, each point on the 2-dimensional rotor profile corresponds to a specific reference seal point. The 2-dimensional profile d a ta is defined for a single lobe of the rotor. The corresponding reference seal data for this single lobe extends for one axial pitch, pax, along the rotor axis, where p ., = -L • [4.36] The correspondence between reference seal points and profile points is valid for a specific rotation angle, 0som> of the m ale rotor. At this rotation angle of the male rotor, each discrete point in the 2-dimensional profile d a ta is associated w ith a specific reference seal point. The z coordinate, and related <f>value, of the i th reference seal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 points are designated by the as z Som{i) and 0som(O- The correspondence between reference seal coordinates and profile coordinates is shown for one lobe of the male rotor and one flute of the female rotor in Figures 4.4 and 4.5. As the m ale rotor is rotated to a value of 0m, the reference seal points can be viewed as translating along the rotor axis, from the suction plane to the discharge plane. The distance, dz, which the reference seal points move in the z direction, can be obtained from the relation dz = (em - 6 s o m) ( ^ ) . [4.37] For the ith point, this distance is added to the value zsom(i) to determ ine the current z ( i ) value of the intersection w ith the interlobe seal curve. The associated value of <f>{i) can then be determ ined as = (*so„(0 + dz) ■ This process is repeated, defining values of z(i) and [4-38] for each profile point. A value of z(i), for a given value of 0m which is greater th an 0.0 or less th an —L, signifies th a t the com puted intersection w ith the interlobe seal curve is beyond the boundaries of the suction or discharge planes, respectively. In this case, the value of ij>(i) is set to either 0.0, for the suction plane, or —r m, for th e discharge plane. The integrands and lim its of integration required to com pute the compression loads are dependent on the specific value of 0m, the rotation of th e m ale rotor. The integrations are performed a t increm ental values of 6m, from 0.0 to 2 r / N m , for the series of helical strips which define each individual cham ber, w ithin th e N m sections of the rotor. T he sum m ation of th e integrations for each cham ber results in the compression loads for the entire rotor at each increm ental value of 6m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 4.3.4 Revisions Required for Application to the Female Rotor The details for com puting th e compression loads have been presented for th e m ale rotor. The m ethod can be applied directly to the female rotor, with some revisions to the details of conducting th e com putations. These revisions are presented here. As with the m ale rotor, a right-hand coordinate system is used in performing the integrations required to obtain the compression loads on th e female rotor. The origin of the system is at the center of the rotor profile, located axially a t th e suction plane. The rotor is defined such th a t the z values lie between z = 0.0 a t the suction plane and z = —L at the discharge plane. This is sim ilar to th a t of the m ale rotor. However, the wrap angle of the female rotor, r / , is opposite to th a t of the m ale rotor. Therefore, equation 4.35 becomes for the female rotor. W ith the coordinate system as defined above, the interlobe seal curve for the female rotor is in th e region containing 6 — ■ k (the 2nd and 3rd quadrants), as opposed to 0 = 0.0 (the 4th and 15< quadrants) for the male. This m ust be accounted for in the details of im plem enting the integration schemes. Finally, each compression cham ber consists of helical sections on both the male and female rotors. The relationship between a single helical section on the m ale and one on the female is dependent upon the num ber of lobes on th e male, N m, and the num ber of flutes on the female, N j. For example, assume th a t section 1 on th e male and section 1 on the female form a compression chamber for a value of 0m = 0mi. Then a t 0m = 0mi + 27t, m ale section 1 will form a compression cham ber w ith the female section, (N m + 1 ). Since the chamber pressures are com puted for values of 6m, some logic m ust be included to incorporate this aspect of the compressor operation in determ ining th e pressure values for each cham ber of the fem ale rotor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 4.4 Volume and Pressure Com putations The twin screw compressor is a positive displacem ent compressor. Compression in an individual cham ber is achieved by reducing the cham ber volume from a m axim um value a t the suction port closing to a m inimum volume a t the discharge port opening. This compression is assumed to be a polytropic process, in each cham ber. Therefore, the pressure and volume are related by P(6m) x V(0m)n = C [4.40] where P(Qm) = chamber pressure as a function of 0m V(flm) = chamber volume as a function of 0m n = the polytropic constant for the gas being compressed C = constant. It is assumed th a t the polytropic constant, n, is known for the gas being compressed. Therefore, P(0m) can be com puted if the cham ber volume is known as a function of 0m, and the value for C is known for a reference value of 6m. [ 4 '4 1 ) The compression process takes place between the tim e th e cham ber volume is closed to the suction porting and the tim e the cham ber volume is opened to the discharge porting. This tim ing is specified by th e m ale rotor port closing, &spcm, and a t discharge port opening, 0 DPOm • rotation angles a t suction For 6m values less than 6spcm the pressure is equal to the suction pressure. For 6m values greater th an 0 DPOm the pressure is equal to the discharge pressure. Therefore, it is necessary only to know the cham ber volume as a function of 0m between 0spcm and 0 DPOm - A compression chamber consists of two helical sections, one on the m ale rotor and one on the female rotor. These helical sections are th e volume contained between Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 adjacent lobe tips on the m ale and adjacent flute tips on the female. The boundaries of the sections are defined by th e suction and discharge planes and th e interlobe seal curve, as described in Section 4.1. If a section of the m ale rotor does not intersect the interlobe seal curve, its boundaries are defined by the suction and discharge planes. At this position, the m axim um possible volume is contained in this section. This condition can occur simultaneously for th e m ale and female sections which form a compression chamber. Therefore, the m axim um possible cham ber volume, Vmax, is the sum m ation of the m axim um section volume which can be obtained for each rotor. The value of the m ale rotor rotation angle associated w ith Vmax is designated as 9pom. This angle occurs a t the instant in which the interlobe seal curve begins to form at the suction plane. It is assumed th a t the cham ber volume decreases linearly as a function of 0m, from Vm ax to 0.0. The 0m value, at which the cham ber volume is 0.0 is com puted 6vendm , as a function of the compressor geometry. 27r Ovendm = 9pom + T 7 - + d'm [4.42] Tm T he cham ber volume is therefore com puted as a linear function of 0m, V {em) = Vmax + dV ^ - ( 0 m d v m 0POm) [4.43] where Qspcm < 9m < dV _ dBm 9opom ( 0.0 — Vmax [4.44] \ Uy.ndm - 0 p o j ‘ [ 1 T he volume a t the instant th a t the cham ber closes to the suction porting can be com puted using equation 4.43 by setting 6m = 9spc m• Vs p c - V max + ( 9spcm - 9p0m) [4.46] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 The chamber is a t suction pressure, PiUCt , at the instant th a t it is no longer exposed to the suction porting. Therefore, the value of C can be com puted using th e param eters defined a t suction port closing. c = (P3uct) X {V s p c T [4.47] The values of P{0m) are then computed using the reference value for C and the function for V{0m). P^ = v { o my t4,48l The cham ber pressure is set to the discharge pressure when the compression chamber becomes exposed to the discharge porting and remains a t th e discharge pressure throughout the discharge process. Pressure pulsations associated with the discharge process are not included in this work. However, the addition of the pulsations can be im plem ented, since the cham ber pressure is specified as a function of 6m. 4.5 Com putations Applied to Research D ata Experim ental m easurem ents of gas pulsations, sound pressure levels and rotor chatter type vibrations were conducted on a test compressor a t the Herrick Labo­ ratories, [2, 35, 36, 80]. The test compressor is configured to operate a t discrete capacity ratings of 100%, 75% and 50%. The volume ratio, Vi, is set to 2.75 a t the 100% capacity and 1.8 at the 50% capacity. T he V a t the 75% capacity is known to be approxim ately 2.4. T he experim ental m easurem ents have been conducted at several operating conditions which represent ideal, over-pressure and under-pressure conditions for each capacity rating. T he capacity ratings and operating conditions for th e com putations and sim ulations conducted in this work are designed to enable a qualitative comparison of the results w ith the previously obtained experim ental m easurem ents. T he capacity and V{ for each sim ulation are determ ined by the tim ing of the suction port closing and discharge port opening. For a capacity rating of C%, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 tim ing of suction port closing is set by specifying &spcm such th at Vspc = (C%) VmaxT he desired volume ratio, V{is then determ ined by correctly specifying [4.49] th e tim ing of the discharge port opening. This is accomplished by specifying 0DPOmsuch th a t VDPO = ^ [4.50] The compression loads on both the male and the female rotors are com puted for capacity ratings of 100%, 75% and 50%. At each of these capacity ratings, 3 separate operating conditions are specified. These operating conditions represent ideal, over­ pressure, and under-pressure conditions. The ideal condition occurs when the m axi­ m um cham ber pressure equals the discharge pressure. An over-compression condition occurs when the m axim um chamber pressure is greater th an the discharge pressure. An under-compression condition, occurs when the m axim um cham ber pressure is less th an the discharge pressure. An exam ple of the cham ber pressures obtained for a single compression cham ber for each of the loading conditions at 100% capacity is presented in Figure 4.6. Typical bearing loads which result from th e compression forces are presented in Figures 4.9 to 4.11. T he m om ents about the rotor axes, M Zm and M Zf, exhibit an im portant charac­ teristic of the compression loadings present in the screw compressor. The m agnitude of M Zj is approxim ately 12% of the m agnitude of M Zm. This effect is caused by the shape of the interlobe seal line and the resulting projected area of each rotor which is exposed to the various chamber pressures. The effect is also evident in th e axial forces, FZm and FZ}. Due to the significant difference in the axial m om ent values, the female rotor tends to behave more like an idler, effectively un-loaded, than a m em ber of a gear train. This effect is the m ain mechanism by which the compression loads contribute to chatter vibrations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 Figure 4.1 Male rotor with one section emphasized. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Discharge Plane Suction Plane 110 Figure 4.2 Interlobe Seal Curve on Male Rotor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Figure 4.3 Progression of Interlobe Seal Curve on Male Rotor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 S(u,v) x v R(u,v) ►u Figure 4.4 M apping from 3 dimensional surface to 2 dimensional region. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 (a) Interlobe seal intersections Helical lines defining a strip Suction Plane (<Pl = 0) (b) Suction Plane «Pl=0) Helical lines defining a strip Discharge Plane «P4“ *m> * Interlobe seal intersections <pu Figure 4.5 Coordinate system used to m ap 3 dimensional rotor surface to a 2 dim en­ sional region. D em onstrated on a cylindrical surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 300 250 200 150 Q. 100 100 200 300 400 500 600 thetam (degrees) Figure 4.6 Cham ber pressure vs 0m for 100% capacity, Psuct = 45psia; (— ) Pdinc = 152psia (ideal); ( )Pnae = 111psia (over-pressure); (— • —)Pdisc = 253psia (under-pressure). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 400 300 200 z 100 a> 2 ,o Li. -100 -200 -300, 100 150) 200 250 : 300 350 400 thetam (degrees) Figure 4.7 Suction bearing forces due to compression loads, male rotor, 50% capacity, Psuct = A5psia, P ^ = 152psia (under-pressure); ( ~ ) B i Bm, ( ) B i ym, ( - - - ) m Zrn. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 100 -100 8 -200 -3 0 0 -4 0 0 -5 0 0 . 50 100 i 150 200 : 250 300 350 400 thetam (degrees) Figure 4.8 Discharge bearing forces due to compression loads, m ale rotor, 50% capacity, Pauct = 45psia, Pduc — 152psia (under-pressure); ( - ) B 2 Xm , ( )B 2 ym. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 200 180 160 140 40 20. 50 100 150I 200 ; 250 300 350 400 thetam (degrees) Figure 4.9 Suction bearing forces due to compression loads, fem ale rotor, 50% capacity, Psuct = 45psia, Pdisc = 152psia (under-pressure); (-)£ !* , , ( ) B ly „ ( - - - ) B l z r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 600 550 500 450 400 350 300 250 200 150 100. 100 150I 200 250 : 300 350 400 thetam (degrees) Figure 4.10 Discharge bearing forces due to compression loads, fem ale rotor, 50% capacity, Psuct = 45psia, Pdi,c = 152psia (under-pressure); ( ) B 2 Xj , ( )B 2 yj. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I\ i. _ E z E <s E o 5 • 10, 100 150i 200 250 : 300 350 400 thetam (degrees) Figure 4.11 M oments about the z-axis due to compression loads, 50% capacity, P ,uct = 45psia, Pdi„c = 152psia (under-pressure); ( - ) M Zm , (-------- )M Z}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 5. MODEL AND CO M PU TER SIMULATION This chapter is a discussion of the analytical model used to sim ulate the interaction between the screw compressor rotors. The chapter includes the assum ptions used in developing the model, along w ith the details used for the contact between the rotors and im pacts which result from loss of contact. Also included is th e description of the loadings which are applied to the compressor rotors. 5.1 Overview of Model An overview of the approach to modelling is given in this section. Details of the development are provided in subsequent sections. 5.1.1 Basic Assumptions Utilized T he following basic assum ptions are applied in developing th e model of the rotor interaction. 1. The rotors are assumed to be rigid bodies. The flexibility of the m ale lobes and female flutes is neglected. 2. T he rotor shafts are assumed to be rigid, w ith no torsional or lateral flexibility. 3. The bearing m ounts are assumed to be rigid, with no lateral displacements. These assum ptions result in a 2-degree-of-freedom model, each rotor having only a single rotational degree of freedom about its central axis. Based on the literature reviewed earlier, with these simplifying assum ptions, the model can be utilized to accurately predict the interaction of the compressor rotors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 5.1.2 Rotor Contact The contact between the rotors is modelled using points on the rotor pitch circles, as seen in Figure 5.1. T he pitch circles determ ine the kinem atic relationship between the rotors as described in C hapter 3. The location of the physical contact points on the rotor profiles vary continually along a path of contact. However, th e radii of the pitch circles can be assumed to rem ain constant. Therefore, the contact point on each pitch circle rem ains a t a fixed radius. A backlash type of clearance is modelled by allowing only one contact point, C , to exist on the m ale rotor pitch circle, while two contact points, A and B , exist on the female pitch circle. The backlash clearance is then specified as the angle between the two contact points on the female. This can most easily be realized by allowing point C on the m ale to be in contact with point A on the female. W ith the male held fixed, the clearance is the angle through which the female can rotate before the m ale comes into contact w ith point B . 5.1.3 Im pact Model The interaction of the compressor rotors m ay involve loss of contact between the m ale lobes and the female flutes. The continuation of the m otion after the loss of contact will result in an im pact between the rotors. Therefore, th e model m ust also include the ability to deal with rotational im pact. In accordance w ith th e contact m odel established above, the points on the pitch circles are used to represent the loss of contact and subsequent im pact. The dissipation of energy during im pact is modelled by a constant coefficient of restitution, w ith a value between 0 and 1, which relates the relative velocity after im pact to the approach velocity. T he equations resulting from the im plem entation of the coefficient of restitution and th e conservation of m om entum are utilized to determ ine the subsequent m otion after im pact. More details of the equations and their application are give in Section 5.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 5.1.4 Load Param eters Completion of the model is obtained by including the loads which are imposed on the rotors. These include the electrom otor input torque, the compression loads, and damping loads. The particular compressor is being modelled as a direct drive type. T he rotor of the electrom otor is m ounted directly to the shaft of the m ale rotor, and the stator is m ounted to the compressor housing. During normal operation, instantaneous vari­ ations in the m otor torque occur, due to changes in the compression loads. These torque variations are related to variations in the rotational speed of the compressor rotor. The electrom otor is an induction m otor, designed to operate a t steady state in the range where the average torque input is linearly related to the rotational speed. This relationship between average torque and rotational speed is assumed to apply to the instantaneous torque also. The line describing this relationship between the m otor torque and rotational speed is approxim ated, and its param eters are included in the differential equations of motion for the compressor rotors. The compression loads on the rotors m ust also be included in th e model. A m ethod for com puting the compression loads is presented in C hapter 4. These loads include axial, and transverse forces and moments. Each of these components is required to com pute the resulting bearing loads. However, for the 2-degree-of-freedom model described above, only the m om ent load about the rotor axis is required. The specific geometric param eters of th e compressor, and especially th e rotor profiles, determines the m agnitude of the m om ent compression loads. These loads are represented in the model as a function of the male rotation angle, and are incorporated into the num erical solution of the equations of motion. Dam ping loads are included in the model for two reasons. First, some am ount of rotational dam ping from the bearings is inherent in the physical system . In addition, the dam ping can be used to aid the numerical evaluation in reaching steady state conditions by dam ping the transients associated with the initial conditions. However, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 the dam ping m ust be within reasonable lim its. High dam ping values can possibly suppress m otion which m ight otherwise occur. 5.2 Equations of Motion In this section, the equations of m otion for the model described above are pre­ sented. T he overall m otion of the system can be obtained by deriving th e angular acceleration of each of the rotors about its central axis of rotation. T he m otion is defined by separate, linear modes and the constraints which apply to the transitions between the modes. The two modes are defined by the interaction between the rotors. T he “contact mode”refers to the state when point C on the m ale rotor is in contact w ith either point A or B on th e female. The “independent m ode” refers to the state when point C on the male lies between points A and B on the fem ale and the rotors are not interacting. One set of coupled differential equations defines the rotor m otion during the con­ tac t mode. A separate set of independent differential equations defines the m otion during the independent mode. The transition between modes is defined by th e im pact model and the loss of contact conditions. The details of the equations of m otion and the transition equations follow. 5.2.1 Equations of Motion for the Contact Mode Figure 5.1 is a schem atic diagram of the rotors representing contact between the m ale lobes and the female flutes. In this diagram , contact is assumed a t point A on the female rotor. Euler’s equations are applied to each of the rotors. This results in coupled equations for the angular acceleration of each rotor. Imfirn = T l - M mz - CmOm - N rm I j Oj = —M f z — cjOj + N r j [5.1] [5.2] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 A kinem atic constraint exists between the rotors when they are in contact. The constraint defines the angular velocity ratio for the rotors. In addition, if the pitch radii of the rotors is assumed constant, a sim ilar constraint applies to the angular acceleration of the rotors. These kinem atic constraints, given by equations 5.3 and 5.4, provide the coupling between the equations of motion. Of = — 6m r/ [5.3] Of = ^r 6 m [5.4] Tf The contact force, N , which acts between th e rotors m ust have the sam e value in each of equations 5.1and 5.2. Therefore, by solving each of the equations for the contact force and equating the results, one can obtain a relationship for th e angular acceleration of the m ale rotor in term s of th e angular acceleration and velocity of the female rotor. T he kinem atic constraints, equations 5.3 and 5.4, can then be applied to obtain an equation for the angular acceleration of the m ale rotor in term s of the applied forces, th e angular velocity of the m ale rotor and th e physical param eters of the two rotors. This derivation is shown below. N = (Tl ~ M mz - CmOm ~ ImOm) [5-5] N = ( ^ j ( M f2 + cf 6f + I f 0f ) ( J ~ ) (Tl - M mz - CmOm - ImOm) = ^ i k +3 )- ( ^ [5.6] ( M fz + CjB, + Ij'6j) ) - ( ^ ) - *■ (£ *■ + rt < ) [5.7] M Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 «» = ^ ’ Y L J r_ V — /„ + -y /; rm »/ ^ [5.9] W ith the assum ption th a t the instantaneous torque is a linear function of the rotor speed, the input torque can be modelled as: TI = T0 - A T e m [5.10] Substituting equation 5.10 into equation 5.9 gives the equation for the angular accel­ eration of the m ale rotor. ( Xo-MmA _ {M jA _ + r em =------------ — V _ V V J — ^ --------tj . — Im + - y / / rm rj + azA W [5.H] Equation 5.11 is solved numerically. The m ethod is described in Section 5.4. Once the angular acceleration and velocity of th e m ale rotor are com puted, equations 5.3 and 5.4 are used to com pute th e angular velocity and acceleration of the female rotor. 5.2.2 Equations of Motion for the Independent Mode W hen the rotors 5.4 no longer apply. are not in contact the kinem atic constraints of equations 5.3 and Therefore, the equations of m otion are not coupled.The angular acceleration of each of the rotors is independent of the other. Equations 5.13 define the angular accelerations for the m ale 5.12 and and female rotors respectively. 0m = To - M m. - ( & T + cm)6m [5.12] -*771 9, = ~ M'- ~ > [5.13] 1S These equations can be integrated separately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 5.3 D eterm ination of Transitions The m ethods used to handle th e transitions between th e two modes are presented in this section. This includes the detection of the transition and th e com putation of the m otion which follows. A transition defines the change in contact from the “current” m ode to the “next” mode. The basic approach is to determ ine the final conditions which result from th e end of th e current m ode. These are then used as the initial conditions for the transition to the next mode. 5.3.1 Transition from Contact to Independent Modes The first transition to consider is th a t from the contact m ode to the independent mode. W hile in contact, the rotors m ust comply with the kinem atic constraints defined by equations 5.3 and 5.4. In addition, th e contact force between th e rotors m ust be compressive, as a tensile force is not physically possible. This is th e key to defining the loss of contact between the rotors. In the derivations presented above, equations 5.5 and 5.6 define th e contact force on the male and female rotors, respectively. A compressive contact force is represented by a positive value. The loss of contact can be determ ined by solving either of these equations for the contact force, using the current values of angular acceleration and angular velocity. If the solution results in a contact force, N , on the m ale rotor such th at, N < 0, then a tensile force would be required to m aintain the m otion. Therefore, th e true solution requires th a t the rotors are no longer in contact. W hen a transition from the contact m ode to th e independent m ode occurs, the system of differential equations used to solve the m otion changes. Therefore, the angular acceleration and angular velocity a t the transition tim e are used as th e initial conditions for the new system of differential equations. 5.3.2 Im pact, Transition from Independent to C ontact Modes The transition from the independent m ode to th e contact m ode is somewhat more difficult to handle, due to the fact th a t it involves an im pact situation. This transition Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 is detected by m onitoring the position and velocity of the contact points on each rotor during the independent mode. As the m otion progresses, an interference check is m ade between the contact point on the male rotor and each of th e contact points on the female rotor. The condition for interference is th a t both the displacem ent and the relative velocity of the contact points being considered m ust be w ithin an established tolerance. The displacem ent of the contact points is used to determ ine their proximity. However, the state of the system is not completely defined by the proxim ity only. The relative velociy is required to determ ine the difference between an im pending im pact and m otion im m ediately following loss of contact. An im pending im pact is defined by a relative velocity which displays th at the points are approaching one another. Motion im m ediately following lossof contact is defined by a relative velocity which displays th a t the points are separating. These conditions are represented by th e equations below. For interference a t contact point A on th e female rotor, th e conditions are: rjO} — rm6m < 6 [5.14] rjOj - rm6m < e [5.15] For interference at contact point B on the female rotor, the conditions are: rmem - rs {6} - AOf) < 6 [5.16] rm9m - rjOj < e [5.17] T he interference conditions are utilized to detect th e transition from the independent to the contact mode. An appropriate model m ust then be applied to com pute the final conditions which result from the im pact. These are then used as the initial conditions in solving the differential equations of m otion for the subsequent contact mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 As was described in Section 5.1.3, the im pact model used here consists of applying the impulse m om entum equations along with a coefficient of restitution. T he required derivation is shown below. In this derivation the subscripts “1” and “2” apply to the conditions im m ediately before and after im pact, respectively. T he im pact is assumed to occur on contact point A of the female rotor, a t tim e t, w ith infinitesim al duration, dt. The conservation of angular m om entum applied to th e m ale and female rotors is shown in equations 5.18 and 5.19, respectively. t+dt Im8m3 = Im 6m 1 + J VmN dt [5.18] t IjOj .2 The ~ t+dt Ij9 h —J rf N dt t coefficientof restitution is used to accom m odate for [5.19] th e energy absorbed during th e im pact. T he coefficient relates the final velocity of the contact points to their approach velocity. C ontact is assumed to occur at the pitch point, th e theoretical contact point along the line which joins the centers of th e rotors. The deviation of the contact from the pitch point due to th e backlash clearance is neglected. T he result is given as equation 5.20 below, where e designates the coefficient of restitution. «= [5.20] (rm0m - 1'}0})l Equation 5.20 can be solved for 0j2. • e {rm0mi + rm0m2 ° h = -----------------: -----------------r} 5.21 Equations 5.18, 5.19 and 5.21 can be m anipulated to give an equation for the velocity of the male rotor after im pact. The first step is to m ultiply equation 5.18 by r j and equation 5.19 by r m, and sum the results. This elim inates th e impulsive force of the im pact. The result is given in equation 5.22 below: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 rmI f (Oh - e h ) + rjlm (0m2 - O mi) = 0 [5.22] Substituting equation 5.21 into equation 5.22 and solving for 0m2 gives equation 5.23. \lm- e L , = ----------— I f ) emi + (1 + e) ( ^ ) I f 9h ----- 1 , Equation 5.23 can be used to com pute tion 5.21 to solve for 0 j 2. 0 m2. . 2------------------------- [5.23] This value can then be used in equa­ C om putation of the angular velocities of each rotor after im pact allows for the solution of the differential equations of m otion for the subse­ quent contact mode. Once these com putations are completed, the impulse, FaveA t , can be obtained from the change in angular m om entum of either the m ale or fem ale rotors. FaveA t = (0m 2 - 0 mi) FaveA t = ( ^ j ( 0 h - 0 h ) [5.24] [5.25] The im pact is assumed to occur instantaneously. Therefore, the change in veloc­ ities occurs w ithout a change in position. To accom m odate for this in the com puta­ tions, the displacem ents of the contact points at th e instant of im pact are averaged, to establish a common position. This average position is th en used as th e initial condition for the subsequent contact mode. 5.4 Com puter Simulation This chapter is a description of the com puter sim ulation of th e m athem atical model, presented above. The discussion centers on some of the details of th e com puter sim ulation required to realize the model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 The com puter sim ulation is designed to be flexible. The required d a ta is input via d a ta files, which the user can easily modify. These input files include: 1. physical param eters of the compressor rotors, 2. dam ping coefficients, 3. m otor torque param eters, 4. compression process variables, 5. variables to define step size and term ination tim e. The sim ulation is based on the use of a R unge-K utta routine to perform the numerical integration. The m ain program initializes the sim ulation and is used to com pute required loading param eters at each tim e step. In addition, th e m ain routine is used to evaluate the conditions obtained a t the end of the m ost recent tim e step and adjust the next iteration accordingly. 5.5 Numerical Integration The equations which describe the m otion of the rotors are second order, ordinary differential equations. In the event th a t the loading param eters, m otor torque and compression m om ents, can be represented in closed form, these equations can be solved directly. However, th e m om ent loadings due to compression are generally more complicated. These loadings can be expressed as discrete functions of th e male rotor rotation angle. Therefore, in order to develop a m ore generalized sim ulation, the solutions are obtained using num erical integration. The current sim ulation utilizes IVPRK from the IMSL library. This is a RungeK u tta integration routine, w ith adaptable step size. T he second order equations are reduced to first order, using th e following scheme Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 3/(1) = Om [5.26] 3/(2) = e} [5.27] 3/(3) = L [5.28] 3/(4) = 6j. [5.29] T he derivative functions required to evaluate th e interm ediate steps in th e integration are then defined as 3/'(l) = 3/(3) [5.30] y'(2) = 2/(4) [5.31] ?/(3) = function defined by the contact m ode [5.32] j/'(4) = function defined by the contact mode. [5.33] Initially, th e rotors are in th e independent mode with the external loads are ap­ plied. The dam ping coefficients allow the transients to dim inish so th a t the system reaches a steady state condition. 5.6 Evaluation of Contact Mode and Transitions The logic utilized to determ ine the contact m ode and th e transitions is imple­ m ented in the following m anner. Assume th a t the integration routine is currently evaluating a tim e step from t\ to <2, where in the integration, between <1 and £2 = + dt. At several interm ediate steps the R unge-K utta routine calls the derivative subroutine. A t this tim e, the variables used in evaluating th e derivatives are also used to determ ine if a transition to a different m ode has taken place. If a change in the m ode occurs during the evaluation of the interm ediate steps, flags are set within the derivative subroutine, and the R unge-K utta routine is allowed to continue. These flags are evaluated when the R unge-K utta routine completes th e integration for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 entire tim e step, and returns to the main program. If no change in the mode has occurred, the end tim e for the next step is set to t 2 + dt and th e integration continues. However, if a change in the mode has occurred, the beginning tim e for the next step is reset to t\, and the end tim e is set to ti + In addition, the value of dt is reset to 4r • This algorithm continues until it converges to dt < tolerance, where the tolerance is user defined. A t this point, ti « t 2 and the transition tim e is taken as the average value. Once the transition is determ ined, the current contact m ode is updated and th e sim ulation continues. 5.7 Test Cases In order to test and validate the com puter sim ulation, physical param eters and loading param eters were specified which would produce known results. These values of the param eters are not associated with the screw compressor designs and are intended only to dem onstrate th a t the sim ulation is able to predict chatter if the conditions allow it. In each case the input torque associated with the electrom otor is set to 0. Therefore, the displacement of the contact points due to the m ean angular velocity is not present. This allows th e rotor m otion to be defined only by the external m om ent which represents the compression loads. The resulting m otion is m ore easily predicted. The displacem ent of the contact points for these test cases is presented in Figures 5.2 to 5.4. This displacement is the linear m otion of th e contact points. An envelope is created by the two points, A and B on the female rotor. The contact point, C on the m ale rem ains within the boundaries of the envelope. T he cases dem onstrate three general types of chatter m otion, alternating contact chatter, one-sided chatter and m ultiple contact chatter. For th e initial test case, th e inertia of both rotors is equal. The compression m om ent on each rotor is specified as a pure sinusoidal function of tim e. T he frequency of these two forcing functions is the same. The m om ent on th e fem ale rotor is specified to have a phase opposing the m om ent on the male. Since th e forcing and inertias are equivalent, th e im pacts cause each rotor to come to rest. The resulting displacement Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 of the contact points is shown in Figure 5.2. The sim ulation accurately predicts the rotor motion. The rotor contact alternates between th e envelope created by contact points A and B on the female rotor. This type of m otion is referred to as alternating contact chatter. The one-sided chatter vibration is dem onstrated in Figure 5.3. T he forcing on each rotor and the clearance between the female contact points is set so th a t the m ale only contacts the side of the envelope created by female point A . Loss of contact between the rotors occurs. However, before the m ale contacts female point B , th e external forcing causes the direction to be reversed. Therefore, the m ale only im pacts contact point A on the female. The final test sim ulation is performed by specifying different frequencies for the forcing functions on each of the rotors. Contact occurs on both sides of the envelope created by th e female contact points. However, the m ale contact point im pacts the A side m ultiple tim es before im pacting the B side. This type of chatter is referred to as m ultiple contact chatter. The bearing forces corresponding to these cases are not presented, since the pa­ ram eters are not related to realistic compressor configurations and conditions. The sim ulations discussed above were conducted to validate th a t the model and associated com puter sim ulation can predict chatter type vibrations, if it exists. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 (a) (b) zm A 0 Pitch Circles Figure 5.1 (a) Clearance between rotors, (b) schem atic diagram of rotor contact model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 0.25 0.2 0.15 0.1 0.05 ■5-0.05 - 0.1 -0.15 - 0.2 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 time Figure 5.2 Exam ple of alternating contact chatter vibrations, (— ) envelope representing the female contact points; (---------) displacem ent of th e m ale contact point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 0.2 0 .1 5 0.1 0 .0 5 g -0 .0 5 - 0.1 -0 .1 5 5 .7 5 .8 6.1 5 .9 6.2 6 .3 t im e Figure 5.3 Exam ple of one-sided contact chatter vibrations, (— ) envelope representing the female contact points; (-------- ) displacem ent of the m ale contact point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 (0 Q > C C a> E a) o J5 a. W D 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7 tim e Figure 5.4 Exam ple of m ultiple contact chatter vibrations, (— ) envelope representing th e female contact points; (-------- ) displacement of the m ale contact point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 6. SIMULATION RESULTS A m ethod for com puting th e compression loads on each rotor was presented in C hapter 4. In addition, a m athem atical model and corresponding com puter simu­ lation of the rotor m otion was presented in C hapter 5. In this chapter, the results of combining the compression load com putations and the com puter sim ulation are given. One of the m ain objectives of the simulations was to be able to predict rotor chatter in an operating screw compressor. However, the m odel did not predict the occurrence of rotor chatter, for the specified compressor profiles and param eters, at any of the operating conditions which were sim ulated. This is discussed further in C hapter 7. The other m ain objective was to com pute the bearing forces which exist in an operating screw compressor. Some typical examples of th e bearing forces are pre­ sented as a function of tim e. The m ajority of the results are presented in term s of the frequency content of the bearing forces. 6.1 O perating Conditions Evaluated The com puter simulations were conducted for the capacities and operating con­ ditions discussed in Section 4.5. These represent typical capacities and operating conditions for the test compressor a t the Herrick Laboratories. Table 6.2 contains the values of the suction and discharge p ort tim ing required to obtain the desired capacities for the specific compressor being modelled. The capacity, C, is a volum etric param eter defined as a percentage of the m axim um volume capacity available. T he capacity is determ ined by setting the value of &s p c such th a t the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 cham ber volume when the suction port closes is the specified percentage. A specific Vj ratio, Vs p c / V d p o , is associated with each capacity rating. This Vi ratio can be obtained once the value of is defined by properly setting the value of Os p c Od p o - Therefore, the angular positions associated with the port tim ing can be set to obtain the desired capacity and corresponding Vi. The required equations are shown below. Vspc = CVmax Ss p c = (VSPC, [ 6 .1 ] ~ [6 .2] 5 ( ) Vd p o = [6.3] Sy ^ ° , BPO = [6.4] 5 ( ) In addition to setting the specific capacity and Vi ratio, the compression condition, ideal, under-pressure, or over-pressure, m ust also be defined. This is accomplished by allowing the suction pressure to rem ain constant, P auct = 45.0p.sia, and varying the discharge pressure. For a given suction pressure, the m axim um pressure attained in the compression cham ber is determ ined by the Vi ratio, P m ax = P ,u c t( V i) n [6.5] where n is the polytropic constant for the compressed gas. Therefore, th e discharge pressure can be set to obtain the desired compression condition. Pdisc > Pmax results in an under-pressure condition. Pduc = Pmax results in an ideal condition. And Pdisc < Pmax results in an over-pressure condition. T he values of Pdi,c used for the sim ulations are presented in Table 6.2. 6.2 Bearing Forces Com puted from Simulations The forces at each bearing are com puted as a function of tim e during the num er­ ical integration of the equations of motion for the two rotors. In Figure 6.1, the x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 component of the bearing forces on the male suction bearing as a function of tim e are presented for 50% capacity at the under-pressure condition. T he shape of the curve suggests th a t the forces contain high frequency components. During the simulation of th e rotor m otion, the bearing forces are com puted in term s of the components due to rotor contact and the components due to compression loads. In Figure 6.2 these components are shown separately. T he m agnitude of the bearing force due to contact is not discernable when plotted w ith th e bearing force associated w ith the compression loads. A lim ited am ount of information can be obtained from the bearing forces repre­ sented in the tim e domain. In order to gain more insight, a frequency analysis of the bearing forces as a function of tim e (the tim e d ata) was conducted. T he analysis was performed on the final portion of the tim e data. This ensures th a t the the system has reached steady state. The transients due to initial conditions are no longer influencing the system. The bearing forces are com puted by numerically solving the differential equations of motion. The algorithm used in this solution is a R unge-K utta m ethod w ith variable step size. Therefore, the bearing forces may not be com puted a t uniform tim e inter­ vals. However, in m ost cases, the tim e steps do not deviate severely from a uniform am ount. Uniform tim e steps are required to perform a Fourier transform of th e tim e data. Therefore, a cubic spline function is used to fit th e bearing forces to uniform tim e data. The m agnitude of the resulting uniform tim e step, A t, can be used to determ ine the associated sampling frequency, f a, The Nyquist frequency, J n , is half the sampling frequency, / n — The num ber of points used for the cubic spline is the same as the num ber used to perform the frequency analysis. Therefore, this num ber is defined to be an integer power of 2 to insure the use of the more efficient Fast Fourier Transform (F F T ) algorithm . For the d a ta presented, 2048 points were used in performing the FFT . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 The tim e d a ta is filtered using a low-pass filter with a cut-off frequency of 0.9 */wA Hanning window is applied to the tim e data. The power spectra is com puted and the am plitude adjusted to represent the am plitude of the bearing force. The results of the frequency analysis are presented in Figures 6.3 to 6.27. Each of the compression conditions is presented for the 50% capacity case. The under-pressure condition is presented for the 75% and 100% capacities. In Figures 6.3 to 6.7, the bearing forces for the 50% capacity, ideal condition are presented. This condition dem onstrates the lowest m ean force m agnitude and the lowest oscillatory force m agnitudes. The 50% capacity, over-pressure condition, Figures 6.8 to 6.12, dem onstrates an increase in the m ean and oscillatory force m ag­ nitudes, as com pared with the ideal case. In addition, significant contributions to the oscillations exist at higher harmonics. The 50% capacity, under-pressure condition, Figures 6.13 to 6.17, dem onstrates the largest m ean and oscillatory forces for the 50% capacity rating. This trend was also realized for the 75% capacity and 100% capacity ratings. The 75% capacity, under-pressure condition, Figures 6.18 to 6.22, dem onstrates an increase in the m ean and oscillatory force m agnitudes as com pared to the 50% capac­ ity, under-pressure condition. As capacity increases the m ean and oscillatory bearing forces increase. T he 100% capacity, under-pressure condition, Figures 6.23 to 6.27, dem onstrates this trend further. Extended discussion of these results is presented in C hapter 7. Capacity (%) Bspc (degrees) Bdpo (degrees) 100 304 469 75 358 478 50 427 488 Table 6.1 Port tim ing, Bspc and Bd p o >for various compressor capacities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Capacity (%) / Vi 100 / 2.75 75 / 2.4 50 / 1.8 Compression / ■Pd.ac (psia) * diac Over-pressure / 1.3 117 Ideal / 1.0 152 Under-pressure / 0.6 253 Over-pressure / 1.3 99 Ideal / (1.0) 129 Under-pressure / 0.6 215 Over-pressure / 1.3 70 Ideal / (1.0) 91 Under-pressure / 0.6 152 Table 6.2 Sim ulated operating conditions, capacity, V i, compression condition, P < a ,c - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 530223532348482353482353235348235323534823534890235348235348235323 168999999999 t -500 lime (secs) •300 •350 -400 •450 E-S00 •550 •600 •650 0.1 0.15 02 Figure 6.1 Male rotor suction bearing force, B l Xm vs tim e, 50% capacity, Pauct = 45psia, Pdiac = 152psia(under — p ressu re); (a) approxim ately 60 revolutions; (b) approxim ately 9 revolutions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 -100- - 200- 5--300 - -7001-----------------------------------'------------------------------------ 1-----------------------------------0.05 0.1 0.15 0.2 time (secs) Figure 6.2 Male rotor suction bearing force, B l Xm, 50% capacity, Psuct = 45psia, Pdisc = 152psia(under — p ressu re ); (-------- ) force due to rotor contact, (— ) force due to compression loads. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 <10 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 Li. 720 1440 2160 2880 Freq (Hz) Figure 6.3 Frequency spectra, B l Xm and B 2 Xm, 50% capacity, Pauci = 45 psia, = 91psta (ideal); (a) B l Xm; (b) B 2 Xm. P d isc Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 < 10 ' u. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) a < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.4 Frequency spectra, B l Vm and B 2 Vm, 50% capacity, Pauct = 45 psia, Pdisc = 91psia (ideal); (a) B l ym; (b) B 2 ym. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 CL < 10' 11. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) < 10 720 1440 2160 2880 Freq (Hz) Figure 6.5 Frequency spectra, B \ Xj and B 2 Xf, 50% capacity, PaUct = 45 psia, Pdisc — Olpszct (ideal), (a) B \ x^ (b) B 2 Xj * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 Q. 1 10' u. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) Q. < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.6 Frequency spectra, B l yj and B 2 yf, 50% capacity, P auct = 45 psia, Piiac = 91 psia (ideal); (a) B l yj ; (b) B 2 yj. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 Q. < 10' 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) Q. < 10 ' LL 720 1440 2160 2880 Freq (Hz) Figure 6.7 Frequency spectra, B l Zm and B l Zj, 50% capacity, Psuct = 45 psia, Pdisc = 91psia (ideal); (a) J51Zm; (b) B 1 Z}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 <10 u. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 720 1440 2160 2880 Freq (Hz) Figure 6.8 Frequency spectra, J91Xm and B 2 Xm, 50% capacity, Pauct = 45 psia, Pdiso = 70psia (over-pressure); (a) B \ Xm\ (b) B 2 Xm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 < 10' U. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) a < 10 ' Li. 720 1440 2160 2880 Freq (Hz) Figure 6.9 Frequency spectra, B l ym and B 2 ym, 50% capacity, P d i s c — 70psia (over-pressure); (a) B l Vm\ (b) B 2 ym. P , uct = 45 psia, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 <10 Li- 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) a. < 10 ' Li- 720 1440 2160 2880 Freq (Hz) Figure 6.10 Frequency spectra, B l X) and B 2 XJ, 50% capacity, Pauct = 45 psia, Pdisc = 70psia (over-pressure); (a) B 1X/; (b) B 2 X/. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 Q. 1 10' u. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) Q. < 10' LL 720 1440 2160 2880 Freq (Hz) Figure 6.11 Frequency spectra, B l y} and B 2 y}, 50% capacity, P3UCt = 45 psia, Pdiic = 70psia (over-pressure); (a) B l yj\ (b) B 2 yj. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 CL I IQ- 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) Q. < 10 ' u. 720 1440 2160 2880 Freq (Hz) Figure 6.12 Frequency spectra, J312m and B \ Zj, 50% capacity, Pauct = 45 psia, Pdiac = 70psia (over-pressure); (a) (b) B 12/. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 a. < 10 ' LL 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.13 Frequency spectra, B \ Xm and B 2 Xm, 50% capacity, Pauct = 45 psia, Pdiic = 152psia (under-pressure); (a) B l Xm; (b) B 2 Xm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 <10 U- < 720 1440 2160 2880 3600 4320 5040 720 1440 2160 2880 3600 4320 5040 Freq (Hz) 10‘ U. Freq (Hz) Figure 6.14 Frequency spectra, B l„m and B 2 Vm, 50% capacity, Psuct = 45 psia, Pdisc = 152psia (under-pressure); (a) B l ym; (b) B 2 Vm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 Q. • I 10' 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) Q. 1 10' 10 720 1440 2160 2880 Freq (Hz) Figure 6.15 Frequency spectra, B1 X} and B 2 Xf, 50% capacity, P ,uct — 45 psia, Pdisc = 152psia (under-pressure); (a) B l Xj\ (b) B 2 X/. Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 158 < 10 Li. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 LL 720 1440 2160 2880 Freq (Hz) Figure 6.16 Frequency spectra, B l yj and B 2 y}, 50% capacity, Psuci = 45 psia, Pdisc = 152psia (under-pressure); (a) B l yf\ (b) B 2 yj . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 Q. < 10 ' Li. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 LL 720 1440 2160 2880 Freq (Hz) Figure 6.17 Frequency spectra, B l Zm and B l ZJ, 50% capacity, P„ucl = 45 psia, Pdisc = 152psia (under-pressure); (a) B l Zm; (b) B 1 Z). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 < 10 ' 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) CL < 10' LL 720 1440 2160 2880 Freq (Hz) Figure 6.18 Frequency spectra, B l Xm and B 2Xm, 75% capacity, P ,Uct = 45 psia, Pdiac = 215psia (under-pressure); (a) B l Xm; (b) B 2 Xm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 <10 LL 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 LL 720 1440 2160 2880 Freq (Hz) Figure 6.19 Frequency spectra, B \ ym and B 2 Vm, 75% capacity, Psuct = 45 psia, Pdisc = 215psia (under-pressure); (a) B l Vm; (b) B 2 Vm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 <10 LL. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.20 Frequency spectra, B l x/ and JB2X/, 75% capacity, Pauct = 45 psia, Pdisc = 215psia (under-pressure); (a) B 1X/; (b) B 2 X}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 <10 LL 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) a < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.21 Frequency spectra, B l y} and B 2y/, 75% capacity, P , uct = 45 psia, Pdisc = 215psm (under-pressure); (a) B l y/; (b) B 2 y / . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 nmti <10 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) CL < 10 ' 720 1440 2160 2880 Freq (Hz) Figure 6.22 Frequency spectra, J51Zm and B 1 Z}, 75% capacity, Psuct = 45 psia, Pdisc = 215psia (under-pressure); (a) B l Zm] (b) B 1 ZJ. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 < 10 ‘ LL 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 LL 720 1440 2160 2880 Freq (Hz) Figure 6.23 Frequency spectra, B l Xm and B 2 Xm, 100% capacity, P3UCt = 45 psia, Pdisc — 25‘i p sia (under-pressure); (a) B l Xm; (b) B 2 Xm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 <10 u. 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) a. < 10 ' LL 720 1440 2160 2880 Freq (Hz) Figure 6.24 Frequency spectra, B l ym and B 2 ym, 100% capacity, P ,uct = 45 psia, $ \ y my (b) Bdisc — 253p«s2G (under-pressure), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 <10 LL 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 LL 720 1440 2160 2880 Freq (Hz) Figure 6.25 Frequency spectra, B l Xf and B 2 X], 100% capacity, Pauct = 45 psia, Pdisc = 253psia (under-pressure); (a) B l r / ; (b) B 2 Xf. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 a. 1 10 ' 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) <10 u. 720 1440 2160 2880 Freq (Hz) Figure 6.26 Frequency spectra, B l y} and B 2y/, 100% capacity, Psuct = 45 psia, Pdiac = 253psm (under-pressure); (a) B l yj ; (b) B 2 yj . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 (a) 10' 10 10“ CL E ...2 < 10 ® a £ 10' 10 10 720 1440 2160 2880 3600 4320 5040 3600 4320 5040 Freq (Hz) CL < 10 ' Ll 720 1440 2160 2880 Freq (Hz) Figure 6.27 Frequency spectra, B l Zm and B l z /, 100% capacity, P3UCt = 45 psia, Pdisc = 253psia (under-pressure); (a) B l 2m; (b) B 1 Z}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 7. CONCLUSIONS AND RECOM M ENDATIONS This chapter is divided into two sections. In the first section, the kinem atic aspects of this work are addressed. This section includes com m ents about the profile generation m ethod described in this work as well as th e overall effects of the kinem atics defined by the rotor profile geometry. In the second section, th e bearing forces are addressed. 7.1 Kinematics The kinem atic analysis conducted in this work dem onstrates how the male and female rotors can be designed as a conjugate pair which ro tate w ith constant angular velocity. However, these shapes are extrem ely sensitive to tolerances in th e specified profiles and the center-to-center distance associated w ith the profiles. Small variations in either of these dimensions cause the profiles to no longer ro tate w ith a constant instantaneous angular velocity. For example, while the m ale rotor m ay ro tate at a constant angular velocity, the female rotor m ay experience angular accelerations. M anufacturing tolerances and rotor wear are sources of profile variations. In addition, the rotors in an operating compressor are displaced from th e design center-to-center distance by compression loads. It is believed th a t the fluctuating fem ale rotor speeds m easured by Huff [80] are due to these types of profile variations and center-to-center distance variations. These variations are not included in th e m athem atical m odel of th e rotor m otion, which explains why no chatter vibrations were predicted at operating conditions at which chatter was expected to occur. A profile generation m ethod was presented in this work. This was not initially intended as a m ain direction of the research efforts. However, reliable profile data Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 ^ was not available. The com putation of the compression loads depends upon reliable geometric representations of the rotor profiles. Therefore, the generation m ethod was developed. The m ethod is based on fundam ental gear theory and can be applied to develop or revise rotor profiles. However, the im plem entation developed in this work is not intended as a robust, commercial product. Several enhancem ents would provide a more usable technique. Some of these are reviewed here. The m ethod is based on using the rotors as generating and conjugate shapes. Therefore, it is dependent upon the definition of the profile surface, in particular the normals to the surface. A m ethod should be provided for insuring th e accuracy of the com putation of the normals, and for com puting th e deviation of th e normals based on acceptable m anufacturing tolerances. In implementing the m ethod, the author applied the constraints of generation and conjugacy to the entire profile. This provided some difficulty in producing acceptable profiles. Since the rotors rotate in one direction only, these constraints need apply only to the portion of the rotors involved in force transmission. T he rem aining portion of the rotors serve only to seal the adjacent compression chambers. Therefore, a more relaxed constraint should be used for the portion of the profiles which serves as the seal only. Due to the history of the screw compressor and the licensing policies of the patent holder, the compressor m anufacturers have only recently become involved in the de­ sign of rotor profiles. The developed basic kinem atic analysis and generation m ethod should contribute significantly to th a t process. 7.2 Bearing Forces The bearing forces in the compressor have been com puted based on th e com­ pression loads and the rotor contact force. The presentation of the bearing forces versus tim e shows th a t the compression loads are the dom inant influence. T he bear­ ing forces due to the rotor contact are negligible in comparison (Figure 6.2). This is attrib u ted to the fact th a t the m om ent, due to the compression process, about the Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 172 rotor axis of the female rotor is approxim ately 10% of the m om ent about the male rotor. Therefore, the female rotor functions nearly as an idler. This confirms, through com putation, what was intuitively expected by Shaw, [87]. One m ethod of reducing the overall vibration effects in th e screw compressor may be to design rotor profiles such th a t th e compression loads are more evenly distributed between the rotors, at the cost of increased contact forces between the rotors. The nature of the compression loads are determ ined by the compression condition. At severe over-pressure and under-pressure conditions, each cham ber experiences a significant change in pressure when it is exposed to the discharge port. This causes an associated change in the compression loads developed w ithin th a t chamber. The effect is evident in th e plots of the bearing forces caused by compression loads only. The forces are shown as a function of the male rotor angle (Figures 4.7 to 4.11). Significant changes occur each tim e a chamber is exposed to th e discharge port. The effects of the compression condition on the bearing forces are evident in the frequency plots of the bearing forces. Some general notes concerning the frequency spectra of the bearing forces are presented here. 1. The fundam ental screw frequency (FSF) of the modelled compressor is approx­ im ately 360 Hz. 2. The peaks in the frequency spectra occur a t harm onics of the FSF, w ith the m agnitude decreasing for the harmonics. 3. The force on the suction bearing is larger than th a t on the discharge bearing, for a specific force component, capacity and compression condition. 4. T he a:-component of the oscillating bearing forces has th e largest m agnitude for each of the conditions. 5. T he ideal pressure condition produces bearing force oscillations of the lowest m agnitude. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 6. The under-pressure condition produces oscillating forces of higher m agnitude th an the over-pressure condition, for each capacity. 7. For the under-pressure condition, the m agnitude of th e bearing force oscillations increases with increasing capacity. At the 50% capacity, ideal condition, only the x-com ponent of the oscillating bearing forces was above 1 newton. This was true for both th e m ale and female rotor. The 100% capacity, under-pressure condition dem onstrated the largest bearing force oscillations of approxim ately 1000 newtons for the x-com ponent. The x components of the bearing forces are typically larger th an the y components. The reason is th a t the projected area of th e compression cham bers is larger in the x direction th an in the y direction. Therefore, the forces due to th e cham ber pressure are larger in the x direction. The under-pressure conditions consistently dem onstrate larger bearing force oscil­ lations than the over-pressure conditions. This can be explained as follows. Assume th at 6 separate chambers exist along the axis of the male rotor, each a t a different stage of compression. The m ale rotor is positioned a t th e instant th a t cham ber 6 becomes exposed to the discharge port. As the male rotor rotates, the pressure in chambers 1 through 5 increases according to the assum ption of a polytropic compres­ sion process. T he pressure in cham ber 6, however rem ains constant a t th e discharge pressure. This phenomenon alone contributes to oscillations in the bearing forces, as can be seen when the ideal compression condition exists. T he m ale rotor has a nearly constant angular m om entum during the compression process. For th e pur­ poses required in this discussion, it can be assumed constant. In the over-pressure condition, the exposure of chamber 6 to the discharge pressure serves to reduce the m om ent about th e z axis due to compression loads. This reduction serves to decrease the angular m om entum of the rotor and is “cancelled” by the m om ents produced in the rem aining chambers. In the under-pressure condition, the exposure of chamber Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 6 to the discharge pressure increases the m oment about the z axis due to compres­ sion loads. This increase is in the same direction as the m om ents produced in the rem aining chambers. Therefore, the effects of the discharge process which occur in the under-pressure condition are not “cancelled” by the rem aining compression ef­ fects. For this reason the under-pressure condition dem onstrates larger bearing force oscillations. As w ith any large machinery, the bearing forces in the tw in screw compressor are affected by m any phenomena. This research effort has established th at, for per­ fect rotor profiles, the compression loads are the dom inant factor in determ ining the bearing forces. In addition, th e sound m easured for an operating screw compressor shows a rich harm onic content in m ultiples of the fundam ental screw frequency. The work presented here suggests th a t the compression loads contribute significantly to the higher harm onics in the tim e varying bearing forces. These bearing forces are a mechanical source of noise and therefore the presence of rotor chatter is not necessary to explain the rich harm onic content of the noise spectra. A significant contribution of this work has been the developm ent of a m ethod for com puting the compression loads for arbitrary profiles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST O F REFERENCES Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF REFERENCES [1] Leo O ’Connor, inputoutput: M anipulating the autom obile engine. Mechanical Engineering, 115(10):114, October 1993. [2] R obert W. Andrews. Noise source identification in twin-screw compressors. Mas­ te r’s thesis, Purdue University, W. Lafayette, IN 47907, August 1990. [3] David N. Shaw. Screw compressors control of vi and capacity: The conflict. In Proceedings of the 1988 International Compressor Engineering Conference at Purdue, pages 236-243, W .Lafayette, IN 47907, July 1988. Purdue University. [4] John A. O ’Brien. Course outline for screw compressors. Internal short course given at United Technologies Carrier in Syracuse, NY, M arch 1987. [5] P.D. Laing. The place of the screw compressor in refrigeration. In Symposium on an Engineering Review of Refrigeration Technology and Equipment, pages 91-105, London, March 1968. 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[76] P.J. Holmes. The dynamics of repeated im pacts with a sinusoidally vibrating table. Journal o f Sound and Vibration, 84(2): 173-189, 1982. [77] S.W. Shaw and P.J. Holmes. A periodically forced piecewise linear oscillator. Journal o f Sound and Vibration, 90(1):129—155, 1983. [78] S.W. Shaw and P.J. Holmes. A periodically forced im pact oscillator with large dissipation. Journal o f Applied Mechanics, Transactions o f the ASM E, 50(l):849-857, December 1983. [79] S.W. Shaw. The dynamics of a harmonically excited system having rigid ampli­ tude constraints p a rti: Subharmonic motions and local bifurcations. Journal o f Applied Mechanics, Transcations of the ASM E, 52:453-458, June 1985. [80] John E. Jr. Huff. Development of a m easurem ent technique to evaluate ro­ tor chatter in twin screw compressors. M aster’s thesis, P urdue University, W. Lafayette, IN 47907, December 1992. with permission of the copyright owner. Further reproduction prohibited without permission. 182 [81] W. Steeds. Involute Gears. Longmans, Green and Co., 215 V ictoria Street, Toronto, first edition, 1948. [82] H.H. Mabie and F. W. Ocvirk. M ehcanisms and Dynam ics o f Machinery. John W iley and Sons, New York , NY, third edition, 1978. [83] J. E. Beard. Kinem atic Analysis o f Gerotor Type Pumps, Engines and Compres­ sors. PhD thesis, Purdue University, Purdue University W . Lafayette, IN 47907, December 1985. [84] J. R. Colbourne. Gear shape and theoritical flow rate in internal gear pumps. Transactions of the Canadian Society o f Mechanical Engineering, 3(4):215—223, 1975. [85] G. P. Adams. Perform ance characteristics of gerotors with non-uniform cross sec­ tional areas perpendicular to the rotor axis of rotation. M aster’s thesis, Louisiana S tate University, Louisiana S tate University Baton Rouge, LA 70803, December 1989. [86] G.P. Adams and W erner Soedel. Rem arks on oscillating bearing loads in twin screw compressors. In Jam es F. Ham ilton, editor, 1992 International Compressor Engineering Conference at Purdue, pages 439-447, Herrick Laboratories, West Lafayette, IN, 47907, July 1992. Ray W . Herrick Laboratories, School of Me­ chanical Engineering, Purdue University. [87] David N. Shaw, personal conversation w ith Dave Shaw, 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 VITA Glynn Adams was born on November 13th, 1960 in W hite C astle, Louisiana. He was the youngest of seven children of M arie Therese Richard Adam s and L.G. Adams II. Glynn was raised on a sugar cane farm in south Louisiana by parents w ith a strong devotion to the Catholic church and a firm belief in the lim itless benefits of a good education. Following 12 years of “straight A’s ” in parochial school, G lynn enrolled at Louisiana S tate University. There he tested both his academ ic abilities and his parents’ will. After obtaining a B.S.M.E. from LSU in 1982, Glynn worked in the petro-chemical industry in Louisiana. On January 3rd, 1987 he m arried Julie Michelle Blondeau. Six months later he re-entered LSU as a p art-tim e graduate student. Glynn was awarded an M.S.M.E. degree from LSU in December of 1989. Four years and two children later, Glynn is being awarded a Ph.D . from Purdue University. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.