r». .iv.!.l ‘t‘ — '- X 1. ~ I .3... 1:: $5.». .1..fi.hfi . :2 e \nxu.r ”2.3.x- . ‘4anqu .l . 15v 3.»? $.qu $5.1..st... ed. .;.u.»flbu.1i. . .. .. . If. 14% u @Mmmmmr‘ .. 9mg... flaw . flannnhsuuumh 1 in. 9 .‘tjhhwv . 8...... ahlfi‘mfi. h. . . 3%.fimrtafiq - t .. fjowhwwW? ; o l‘tlulln I 739' ' 1:! v _ m '13. NVIJII “cu-{In v. ‘ .. {34. s A. o! .2 2‘ : 3.4%-. ‘32.. #3th :2... :v it. Jim I I 1 A53 v Vb... 1 fl I vantlc I...'.1' 2 l’ V 3 7|.» 1: 1.. 1I.\. .. I - m. vq-adru ‘ brruh: 1.3.9.4. l a t \Uflmwuf. . Lu...‘l,.. ‘ . , 2. lid-3 . , . ‘ 3;. 15...... man... . , .. lit-:1 slug fir; ‘va-hr . 55.4. . . . v t X. a..- . .. .nLr - Withfiii ., . ‘ £11.; ,. 5A..» 1.. a... 333.qu u- A 1.1.. e - .2... .. 4 .1: L... 9-... .. 4 db. .uul Hds‘hAfloIo qr‘z-HQ. .- ., t It 51v. 1!"! .ns 5...: o 4..“ bit; 4 I x. . .\;.. .8 .I 5) 138. [\t 1 {thkmnmw .wwhwn... .3...‘ .. . yak-had». .nhdhfli. v.._§hy1%.a. 10.5.5. c .23.». . ' .33.» 1...; .u..§.vb.$ to; ‘3. ix. 1... , avzxudafi rind“... It lint atrhrld$ .toaahiafiill'lfitfluunfitfi v! . .rdbll.‘ .3491. It... 1.3!. .. 5... i1: l....P..un>.\.onMat ihhtdhvflnl . Inhaviiuh 3i .mr-vUMow . (bury 0 r!!! u .00. QIV i 013“; !I.. « 1\I I: .lel..xtl... ‘ ‘33:, .3... I r I v x. .‘o 2on9: h . . Vii?“ g V’C"? “$3.411... I 1.. if, ........i.. . v . 3-. .v Ill... 3 I: .. 91.9w. . I. . u... .I'vutw will v.1. 4 "45k. lump... 1.5...." h r. Hun“?! .u b I... .13. .11.; - .x..v tvv i~x ‘0‘. v'. '-|¢ n 00. I: lit X . 1.-» {I rl‘. . v . . (3133.1. hm... .. 5.5. 1cm ‘ . .05.... ‘st,dfahh.1 .. . ‘ ‘ .33... {4.1. .v‘..!.voirvdu. at. . ‘ . . . . v .Fflvhycur I xdlpmb. .51.. 5'... . - . x d. . . . . .wh‘v... . . y 1:15.-.. :.!... . It‘lr- ...l..l. 3.. . -Wb4hnuam‘nhotu t v... . b». .ILN'UJNTIW n. ,2 . _ . . » luau.“ «MHz... . . . y . .. . . .. 5 . . v9. . 1r..- . . 1.:Ivylv‘l! . . ‘ ! «larvae: . . ‘ . ‘. VI ‘91. 4.... . . ‘ . otqultflCfl. ¢_.‘\.||.1 . ‘ YD- t . nut-I ‘ .. . . . : l . u. up. - . r3551.-.) . .nmmwmh it. viiitm. huh? I . ‘ It! - . .1- a... Mir... .. it. 2.0;... xvi... .65»... .nhu. , , . 9 rwmuunfl.m.....xa.nm{.: nu“. l3... v.‘¢: it \t’ I . ‘ ‘I . n ‘32.!“ :1 642"? ”4660!.- . . . 1 .HI I 1.11% MI...- (11... Q... In! Xv ‘tvlntlu. ‘ noch.o"I1|JA-,tv0!.osflvn¢¢b\-OOI . H1.“ul~n—..As‘..‘t. ‘ 09‘ Q'Ci‘l'zu ‘ til-r311. it»! 1. VI 398:1..nvbttlobzlbb3s \ .u1\3‘c o!t‘!.‘u i. . wahkc‘lgbn fl E Jul! ‘ “9‘"; 11 . vilzcu OI! . t I. ~t”..v0ls:v ; 13‘ {N1}. “viva-lid. \~....!..~mm«.mul Irismmhfinrx: Qt. nllcl.‘u\. 5' 1.. .hMIJaII: u] ; a]. if. a. .1 t ) .NVvPsIID. nu. . ninth .nq n... 2 33.3.. x 'b'It‘ng-r . LU.“ Huflnumvfiu .2 1s - .. :3» .o it ‘0 ‘ 5‘. ‘u‘vr II- 1 ‘c. tl. . Al. 8! ‘ It; I h: v.1. -ul. ob! In .3... . (v!!- I000}!- :04 ulb .. ”PERU .9453 . .r, - .3 a . Lfifiqqkfirfiu: Wm" 185?... Lot-"3t .}IV .1 I: 19 ‘. u}! . y t .. ha: Uihnhbflvhv.‘ 1.. II in _ § .3 y ‘nuhrnwltr‘fiil .I! - . yntkb‘uflu‘l '\.| ‘\I.\ “i g Illiliiiiiili“ LIBRARY Michigan State Unhuusfly This is to certify that the dissertation entitled W (7 PK’YQN ‘0‘“ (.9 K; M \A‘tl? ] o PQAA cl ml \AY“ V“ kmL (AA m3 saw \‘b A VPV‘ “5 l“ R ”at “3 é) Jam 5 presented by (3/ x e to Cm 0 has been accepted towards fulfillment of the requirements for M— degree in WV 6‘7 ‘vyé ET‘VJKV Y we, Major professor DateAMz I3, 2712 2 urn;- n- Arr 0-1277! PLACE N RETURN BOX to remove thb checkout 1mm your record. TO AVOID FINES return on or before dete due. DATE DUE DATE DUE DATE DUE Ms} M 2005 MSU le An Afflmetlve Action/Ewe! Opportunity lnetitulon mama-M THE PERFORMANCE OF MULTIPLE PENDULUM VIBRATION ABSORBERS APPLIED TO ROTATING SYSTEMS By Chang-Po Chao A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1997 ABSTRACT THE PERFORMANCE OF MULTIPLE PENDULUM VIBRATION ABSORBERS APPLIED TO ROTATING SYSTEMS By Chang-Po Chao A centrifugal pendulum vibration absorber (CPVA) is a device used for reducing torsional vibrations in rotating machinery. It consists of a movable mass, the cen- ter of gravity of which is restricted to follow a prescribed path. When this path is properly designed, the motion of the CPVA is tuned so as to generate a torque that reduces torsional vibrations. CPVA’s are currently widely employed to suppress tor- sional vibrations in light aircraft engines, and are receiving attention in experimental automotive studies. Existing CPVA designs are based on several assumptions, including the following two: First, for a multiple-absorber system, the set of identical absorbers moves in unison, and, second, the absorber paths are manufactured exactly as designed. The present study aims to re-assess absorber performance in terms of relaxing these two assumptions. This necessitates consideration of the nonlinear dynamics of mulitple- CPVA systems. This study starts with an overview of the operation of CPVA’s and a description of some existing CPVA designs. A mathematical model is then derived that captures the nonlinear dynamics of a multi-absorber/ rotor system response. Using a generic methodology which combines asymptotic techniques (averaging) and bifurcation the- ory, the mathematical model is analyzed for two representative absorber systems: tautochronic and subharmom'c. Analysis is first conducted for the tautochrom'c sys- tem, which enables one to: (1) predict the instability/ bifurcation point of the unison motion, (2) investigate the dependence of the post-critical dynamics on various sys- tem parameters, and (3) assess the absorber performance in terms of two quantitative measures: the rotor acceleration and the feasible range of the applied torque. A sim- ilar analysis is carried out for system comprised of multiple pairs of subharmom'c absorbers. In addition, uncertainties and intentional mistuning are incorporated into the absorber path configurations, which permits one to consider design robustness issues. It is found that the system dynamics and absorber performance measures are accurately predicted by the analyses, as verified by extensive numerical simulations for both absorber systems. Based on these predictions, design guidelines are distilled for various system parameters, including absorber damping, the number of absorbers, and intentional mistuning of the path. Chang-Po Chao 1997 All Rights Reserved To my wife and family ACKNOWLEDGMENTS It has been six years since I first arrived in the States. During this important period of my life, I have obtained a great deal of valuable academic and non-academic assistance from a lot of generous people, especially, my advisor, Professor Steven W. Shaw. Thanks to the opportunity to work under his guidance in the past two and half years, I have experienced a wonderful research life in East Lansing. Not only have I freedom on the direction of my research, but also ground-breaking insights into problems and solid financial support throughout my Ph.D. studies. I truly believe this experience will have great impact at some points of the rest of my life. I also owe great debt to my former advisor, Dr. Philip M. FitzSimons, who introduced me to the fascinating world of control. Working under his guidance on a real mechanical model for my master’s thesis and developing a novel theoretical methodology for the ensuing project give me a complete knowledge of control theory, which, I believe, will have tremendous contribution on my potential for an extended academic career. In addition, I would like to express my grateful thanks to the members of my committee, Professors Clark J. Radcliffe, Sheldon Newhouse Alan G. Haddow and Brian B. Feeny, for providing helpful comments on this study. Michigan State University has been a joyful place for me, especially, this is the place where I met my wife, a intelligent young woman, Hsing-Yuh Chen, who has been supporting me as a family member, friend and moral booster for a long period of time. My family, of course, provided me crucial financial support in the early stages at MSU and continues to encourage me to pursue my dream. I would like to vi ‘ 1. dedicate this work to my wife and family. Special thanks go to the members of Dynamics and Vibration Group for providing persistent encouragement and fostering an intellectual, stimulating atmosphere in the laboratory. They are Professors Shyh-Leh Chen and J in-Wei Liang, Dr. Cheng-Tang Lee, Dr. Ramana Kappagantu, Mr. Chris Hause, Mr. Wei Li, Ms. Yihong Zhang, Mr. Choong-Ming Jung, ...... Working in such a supportive environment will always be a pleasant memory for me. vii TABLE OF CONTENTS LIST OF FIGURES ............................... xii LIST OF TABLES ................................ xv LIST OF APPENDICES ............................ xvi CHAPTER 1. INTRODUCTION ............................ 1 1.1 Operation of CPVA’s 1.2 History and Literature Review 1.3 Motivation 1.4 Organization of this Dissertation 2. PRELIMIN ARIES ............................ 11 2.1 Assumptions 2.2 Equations of Motion 2.3 The Absorber System 2.4 Limitations on Absorber Motions 2.5 Absorber Dampings 2.6 Symmetry Identification 2.7 Measures of Absorber Performance viii 3. 5. STABILITY OF THE UNISON RESPONSE FOR A ROTATING SYSTEM WITH MULTIPLE TAUTOCHRONIC ABSORBERS . . . 3.0 Equations of Motion 3.1 Scaling and Reduction of the Equations of Motion 3.2 The Averaged Equations 3.3 Stability Criterion 3.4 Concluding Remarks NON—UNISON DYNAMICS OF A ROTATING SYSTEM WITH MULTIPLE TAUTOCHRONIC ABSORBERS ............. 4.1 Preparation of Equations of Motion 4.2 The Averaged Equations 4.3 Steady-State Responses 4.4 The Post-Bifurcation Dynamics 4.5 Absorber Performance 4.6 Numerical and Simulation Results 4.7 Concluding Remarks THE EFFECTS OF IMPERFECTIONS AND MISTUNING ON THE PERFORMANCE OF THE PAIRED, SUBHARMONIC CPVA SYS- TEM .................................... 5.1 The Subharmonic Absorber System 5.2 Measures of Performance 5.3 Scaling and Reduction of the Equations of Motion 5.4 The Averaged Equations ix 21 38 66 5.5 Approximate Steady-State Solutions 5.6 Absorber Performance and Design Guidelines 5.7 Concluding Remarks 6. NONLINEAR DYNAMICS OF A MULTIPLE SUBHARMONIC CPVA SYSTEM ................................. 110 6.1 The Multiple Subharmonic Absorber System 6.2 Reduction Of the Equations Of Motion 6.3 The Averaged Equations 6.4 Case Studies 6.5 Remarks and Design Guidelines 7. CONCLUSIONS AND FUTURE WORK ................ 132 APPENDIX B. ON THE EIGENVALUES OF C ..................... 138 C. ON THE EIGENVALUES OF [A + (N — 1)B] .............. 140 D. PROOF OF EQUATION (3.17) ...................... 142 E. JUSTIFICATION OF it, 2 17),, v 2 g i, j _<_ N .............. 143 F PROOF OF Trace[A + (N — 2)B] < 0 AND 0qu + (N — 2)B] > 0 AS i3—> 0+ ...................................... ' . 145 G. THE LOW-ORDER APPROXIMATION OF yy'(0) ........... 146 H. AVERAGED EQUATIONS IN CARTESIAN COORDINATES ..... 148 I. STABILITY OF SOLUTIONS ON SMl+ ................ 150 J. STABILITY OF SOLUTIONS ON 8M2+ ................ 151 K. THE NUMERICAL ALGORITHM FOR COUPLED-MODE SOLUTIONS 152 BIBLIOGRAPHY ................................ 153 xi LIST OF FIGURES Figure 1.1 The CPVA carrier assembly from the cross-section view ......... 1.2 The Schematic diagram for the CPVA’s and the rotor from the cross-section view .......................................... 2.1 Cross-sectional schematic diagram Of the rotor and absorbers ....... 3.1 The critical torque level, F3, versus the number of absorbers, N with 3°35 2: 0.0013. “e” represents the bifurcation point derived from “Det[A — B] = 0”. “*” represents the bifurcation point derived from the simplified criterion (3.23). “x” represents an non-unison motion from numerical simulations. “0” represents a unison motion from numerical simulations ..................... 3.2 The critical torque level, F3, versus various absorber damping [10. The solid line represents bifurcation points derived from “Det[A — B] = 0”. The dashed line represents bifurcation points derived from the simplified criterion (3.23). “ x” represents an non-unison motion from numerical simulations. “0” represents a unison motion from numerical simulations ..................... 3.3 Absorber motions before and after the bifurcation point for N = 7. . . . 4.1 Post—bifurcation steady:state responses Of the absorbers for N = 4 (four absorbers), [1,, = 0.0026 and F9 = 0.048. Solid lines: Simulation; Dotted lines: Truncated; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response. ...................................... 4.2 Post-bifurcation steady-state responses of the rotor acceleration for N = 4 (four absorbers), [to = 0.0026 and P9 = 0.048. Solid lines: Simulation; Dotted lines: The 2nd-order approximation; Triangles: The 3rd-order approximation; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response. 4.3 The ||s||ss’s derived by different approximations versus the applied torque level. The system parameters used are N = 4 (four absorbers) and [1,, = 0.0026. Solid lines: Simulation; Dotted lines: Truncated; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response .................. xii 12 32 34 36 56 58 59 4.4 The percent reduction in torque range, relative to the unison motion, versus the number Of absorbers for [la = 0.0026. “+”: Simulation; “e”: Truncated; “D”: Non—truncated. ................................... 61 4.5 The ||yy'||85’s derived by different approximations versus the applied torque level, for system parameters N = 4 (four absorbers) and [1,. = 0.0026. Solid lines: Simulation; Dotted lines: The 2nd—order approximation; Triangles: The 3rd-order approximation; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response. ............................. 62 4.6 The ratio of Ilyy'llss to that for the unison response versus the number of absorbers for I‘o- — 0.0555 and A, = 0. 0026. “”2+ Simulation;e “ ” .The 2nd-order approximation; “A”: The 3rd- order approximation; “[3”: N on-truncated Imposed unison response. ............................. 63 5.1 The bifurcation diagram for n = 2, [1,, = 0.05, 652 = 0.06, 654 = 0, 6,,2 = 0, 6,,.; = 0 . The solid lines represent stable solutions and the dashed lines represent unstable solutions. ................................. 81 5.2 The bifurcation set of the zero solution for 6,,2 = 6,,., = 0 and [2,, = 0.05. 83 5.3 The curves represent the relationship between in and F5 in coupled-mode solutions for n = 2, 652 = 0.05 and 654 = 0. The feasible absorber motions lie inside of triangle OAB ............................... 90 5.4 The response bifurcation diagram for SM2 with n = 2, 6,54 = 0, 6,,2 = 0, 6,,4 = 0, [1,, = 0.0083 and various 652: (A) 0.03 (B) 0.02 (C) 0.01 (D) 0.00 (E) -0.01 (F) -0.02 (C) -0.03 ............................... 96 5. 5 The response bifurcation diagram for SM2 with n- — 2, 652— — 0.02, 6,”: 0, 6 n4 = 0, [10 = 0. 0083 and various 654: (A) -0. 03 (B) -.0 02 (C) 0. 01 (D) 0.00 (E) 0. 01 (F) 0.02 (G) 0. 03. ............................... 98 5. 6 The response bifurcation diagram for n - -2, [la =0. 05, 662: 0. 01, 6,,2 = 0. 01, 654 = 0 and 6,,.;- — 0. The solid lines represent stable solutions and the dashed lines represent unstable solutions. .................. 99 5. 7 The response bifurcation diagram of SM2 for n- — 2, [ta— — 0. 0083, 6(2— — 0.02 and 654- — 0. 01, 6,,4- — 0 and various 6,72: (A) 0.0 (B) 0. 01 (C) 0.02 ........ 101 5. 8 The simulated and analytical responses on SM2 for n- — 2,11,, = 0. 0083, 6g" — (102,653—— 011196772: 0. 005 and 6,,4" — 0. 005. ............... 103 5.9 The absorber responses and rotor accelerations for F9 = 0.038, 0.055, 0.085, 0.11, corresponding to points (A), (B), (C) and (D) in Figure 5.8, respectively. . . . 108 xiii 6.1 The graphical interpretation Of the distorted ellipsoids ........... 121 6.2 The mode shapes Of the steady—state solutions with various isotropy groups 122 6.3 The stability and feasibility boundaries Of the solutions with isotropy subgroup S; x S; for P9 = 0.035, [1,, = 0.005, n = 2 and V = 0.1662 ....... 124 6.4 The stability and feasibility boundaries of the solutions with isotropy subgroup S; x S, for P9 = 0.035, [1,, = 0.005, n = 2 and V = 0.1662 ....... 125 6.5 The stability and feasibility boundaries of the solutions with isotropy subgroup S; x S; x S; for Pa = 0.035, it; = 0.005, n = 2 and V = 0.1662. . . . 126 6.6 The contours of the rotor accelerations for F9 = 0.035, [1,, = 0.005, n = 2 and V = 0.1662 .................................... 127 6.7 The feasible disturbing torque range Of the S; x S; solution branch for [La = 0.005, n = 2 and V = 0.1662 .......................... 128 6.8 The feasible disturbing torque range of the S; x 8;, solution branch for [1,, = 0.005, n = 2 and V = 0.1662 .......................... 129 xiv LIST OF TABLES fhble 6.1 The solutions branches classified by their isotropy subgroups and their mode shapes. .................................... 121 XV LIST OF APPENDICES AM A. SSS EXPRESSIONS SSS R: .222 S—r ... S3 ----- 137 B. ON THE EIGENVALUES OF C ....................... 138 C. ON THE EIGENVALUES OF [A + (N — 1)B] ............... 140 D. PROOF OF EQUATION (3.17) ....................... 142 E. JUSTIFICATION OF .23, 2 (2,, v 2 _<_ 2', j g N ............... 143 F. PROOF OF Trace[A + (N — 2)B] < 0 AND 0qu + (N — 2)B] > 0 AS ,‘I—> 0+ ....................................... 145 G. THE LOW-ORDER APPROXIMATION OF yy'w) ............ 146 H. AVERAGED EQUATIONS IN CARTESIAN COORDINATES ...... 148 I. STABILITY OF SOLUTIONS ON SM1+ .................. 150 J. STABILITY OF SOLUTIONS ON SM2+ .................. 151 K. THE NUMERICAL ALGORITHM FOR COUPLED-MODE SOLUTIONS 152 xvi CHAPTER 1 INTRODUCTION Torsional vibrations in rotating systems are induced primarily by torques trans- mitted to a rotor from forces applied to attached components. For example, in internal combustion (IC) engines, cylinder gas pressure, friction and slider-crank in- ertia cause these torques, while in helicopter rotors aerodynamic loads on blades are the primary source. These torsional vibrations can propagate through the system and often cause fatigue and NVH (Noise, Vibrations and Harshness) difficulties. A centrifugal pendulum vibration absorber (CPVA) is a device used for reducing these torsional vibrations. It consists essentially of a mass that is restricted tO move along a prescribed path relative to the base rotating system. The absorber is driven by the centrifugal field generated by rotation, and its motion provides a restoring torque which is designed to reduce torsional vibrations of the rotating system. CPVA’s were invented for the use in internal combustion engines as early as 1929 [5] and have been successfully employed to suppress torsional vibrations in light aircraft engines [27]. A number of previous works have concentrated on sizing the absorber inertia and designing the absorber path by analyzing the linear or nonlinear dynamics of the absorber system under a given order Of excitation. All these designs are based on the following two assumptions: FIRST, each absorber system consists of only a single dynamic mass; SECOND, the absorber paths are exactly tuned and manufactured exactly as desired. The present study aims to re-assess the absorber performance along the lines of relaxing these two assumptions. This chapter starts with an elaboration Of the operation Of CPVA’s in section 1.1. In section 1.2, previous designs of the CPVA’s are described in order to motivate the 1 2 Objectives of the present study, which are described in section 1.3. Finally, the organization of the rest of this dissertation is outlined in section 1.4. 1.1 Operation of CPVA’S In a reciprocating internal combustion engine, combustion in the cylinders and in- ertial loads of the connecting rods and pistons generate oscillatory torques and forces that act on the crankshaft. These result in torsional oscillations of the crankshaft, which lead to several undesirable consequences, including vibration excitation of aux- iliary components and fatigue failure. Several Options are available to remedy this problem, including the addition Of flywheels [60], torsional friction dampers [42, 27], or tuned vibration absorbers [44, 27]. These devices Offer effective means of vibration reduction for rotating machines, and have the benefit Of Operating in an Open loop manner, thus achieving a cost-favorable solution when compared to systems which employ sensors and actuators. However, each also has some shortcomings. The ad- dition of a flywheel increases the total mass and rotational inertia of the system, thereby reducing system responsiveness. Torsional friction dampers consume energy and generate heat. Conventional tuned vibration absorbers that use elastic elements can be tuned only to a single frequency, and therefore are not useful except at one rotation rate, and may lead to detrimental effects at other rotation rates. Centrifu- gal pendulum vibration absorbers have many desirable features when compared to these solutions. Their main drawback is system complexity in terms of the number of moving components required. In the following the basic Operation of CPVA’s is described through a particular implementation. The favorable features of the CPVA are then compared to the aforementioned devices. Figure 1.1 shows one type of physical realization of CPVA’S using a carrier assem- bly which, in application, is bolted onto a crankshaft at some location. This general Figure 1.1: The CPVA carrier assembly from the cross-section view configuration was employed by Borowski et a1 [3] for an experimental study on an automotive engine. This carrier contains three bifilar pendulums [27] which move relative to the carrier along a prescribed path as the crankshaft rotates. By using identical contact curves cut on the carrier and the absorber masses and using the circular rollers between them, the CPVA masses undergo pure translation relative to the carrier. Their centers of gravity (C.G.) will follow the path shown in Figure 1.1, which can be specified by the shape of the contact curves on the CPVA’s and the carrier. Note that due to the pure translation of the CPVA’s, the dynamic effect of the CPVA’s on the crankshaft is equivalent to that of point masses moving along the the C.G. paths as shown in Figure 1.2, while their moments of inertia about their own C.G.’s simply add to the overall moment of inertia of the rotating system. As the CPVA’s are driven by the rotation of the carrier, their motions provide restoring torques on the carrier which, when the absorber C.G. paths are properly designed, reduce the level of torsional oscillations of the crankshaft. In the absorber configuration in Figure 1.1, the absorbers are used to replace The rotor Applied Torque Absorber mass . Vertex of the Path Absorber path Figure 1.2: The Schematic diagram for the CPVA’s and the rotor from the cross-section view. the usual counterweights, and can thus be implemented without increasing the net mass or moment of inertia of the crankshaft. Hence, the absorber is considered to be favorable over a heavy flywheel for reducing torsional oscillations. In addition, since an insignificant amount Of energy dissipated from the dynamic contact between the absorbers, rollers and the carriers, the absorber system generates much less heat than a friction damper during Operation. Most importantly, the oscillating frequency of the CPVA can be tuned, by proper design of the paths, to be the same as that Of the applied torque over a continuous range Of rotation speeds, such that it renders much more efficient reduction on torsional vibrations than elastic, tuned vibration absorbers. The mechanism behind this favorable property of the CPVA is further elaborated in the following paragraph. In most applications, the input torque for a rotating system can be considered as a nominal constant torque, which keeps the system running at a nominally constant speed, (1, plus a periodic fluctuating part whose base frequency is n times that of rotating system rotation; i.e., n0. Such a torque is referred to as a torque of order n, and this torque is typically approximated by its low-order harmonic components. As 5 shown in the derivation of the equations Of motion for the CPVA system, one has the freedom to tune the linear oscillating frequency of the CPVA (the frequency of small amplitude motion) to n!) by designing the radius Of curvature in the neighborhood of the path vertices. This unique feature of the CPVA is critical since, as the speed Of the system varies, the torque generated by the motion of the CPVA always contains the frequency n0, which is used to counteract that Of the fluctuating component of the applied torque at all speeds. 1.2 History and Literature Review For a thorough history Of the CPVA up to 1960, one can refer to [27]. A brief history including designs of interests for the present study is provided herein for a better understanding Of the content of this dissertation. In the earliest stage of CPVA development, implementation of the CPVA into the assembly Of rotating systems drew the most attention. The first conceptual design of these absorbers can be dated back tO 1911 when Kutzbach (as referred to Wilson in [27]) proposed a special mechanical arrangement which consists of moving fluid in U-shaped channels mounted in some component of the rotating system. Carter [5] in 1929 introduced a assembly consisting of absorber masses in roller- forms in a. British Patent. In 1930, Meissner first demonstrated the effectiveness of the CPVA via experiments in a conference paper [33]. This study started an intensive development of CPVA arrangements over the next ten years in Europe. This included various inventions of the bifilar-and-roller suspensions of CPVA’S by Sarazin [50, 51] and Salomon [49]. The Chilton and Reed Propeller Company [9] was also granted a patent in 1935 for a particular bifilar suspension system applied to radial aero- engines, which is believed to be the first implementation of the bifilar CPVA in industry. In the United States Taylor [61] introduced the CPVA in order to eliminate torsional vibrations of geared radial aircraft-engine-propeller systems. Moore [38] 6 in 1942 realized a design of the CPVA by incorporating it into the assembly Of a crankshaft for various applications. Only the use Of CPVA’s allowed for the practical implementation of the light, but powerful aircraft engines used in many aircraft during World War II [66]. Without these CPVA’S many Of the most popular engines of that era could never have been put into service. In the second stage, with the existing maturity of the hardware designs, tun- ing of the pendulum absorbers (riding on circular paths) and various applications became intensive research issues. Stieglitz [58] in 1938 identified the basis for pendu- lum tuning. Zdanowich and Wilson [69] refined the tuning basis of Stieglitz [58] in 1940. Meyer and Saldin [34] in 1942 presented an experimental study of absorbers applied to turbine blades. Harker [23] in 1944 lists charts and design guidelines for the absorber tuning. In 1949, Reed [48] pointed out the use of CPVA’s for reduction of translational vibrations. Pluntkett [46] gives a short review on the usage of the CPVA up to 1953. Most of the above studies were aimed at developing the tuning methods for absorbers riding on circular paths and were based on small amplitude, linear theory. Circular paths were widely used simply because they could be easily manufactured and the tuning methods are valid in the range Of small oscillations. However, such designs often fail for moderate amplitude motions [41, 54] due to the fact that the pendulum frequency generally changes as a function Of its amplitude (for example, it decreases as amplitude increases for the common circular path). In the third stage, the research was extended to account for effects of nonlinear dynamics of the absorbers. Lame-amplitude motions and the detrimental effects for circular paths were first discussed by Den Hartog [14] in 1938 and Porter [47] in 1945. Crossley in 1952 and 1953 gives more complete investigations of the associated undamped systems for free [12] and forced [13] responses, respectively. Newland [41] in 1964 identified possible catastrophic failure Of circular paths due to the mistun- 7 ing of the CPVA. TO solve this problem, both Den Hartog [14] and Newland [41] suggested that one can intentionally over-tune the linear oscillating frequency of the absorber so that it comes into a more favorable tuning for larger amplitudes. Most recently, the effects of damping, moderate amplitude motion and motion-limiting stops (snubbers) were studied by Sharif-Bakhtiar and Shaw [53, 52, 54]. Also, Shaw and Wiggins [57] found in 1988 that without motion-limiting stops, chaotic motions can exist for certain ranges of parameter values in a pendulum configuration which allows the absorber to undergo complete rotations. Along another line of research in the third stage, efforts were dedicated to the reduction of torsional vibrations and shake forces in helicopter rotors. Kelley [26] in 1962 described the potential use of absorbers for reducing torsional vibrations Of helicopter rotors. Paul [43] in 1969 recorded experimental data indicating significant success of the rotor absorbers for reducing vibrations of the helicopter mainframe. Wachs [64] in 1973 investigated the effects Of the absorbers on helicopter reliability and maintainability by tracking repair costs over a period of time in helicopters with and without CPVA’s. He found that CPVA’s retrofitted to rotors saved an average of $367,311 per aircraft per 10 years. Miao and Mouzakis [36] (1980) presented an experimental study of the nonlinear dynamics Of the absorbers mounted on a rotor. Murthy and Hammond [40] (1981), Hamouda and Pierce [21] (1984) explored the use of absorbers for reducing vibrations of helicopter rotor blades. Recently, Wang et al. [65] investigated transverse vibration of rotating beams fitted with CPVA’s. Also, shake reduction using multiple CPVA’s has been studied by Miao and Mouzakis [35], Lim [31] and Cronin [11]. The bifilar suspension makes non-circular paths realizable. In the most recent stage, a number of works were devoted to the design and analyses of CPVA’S riding on non-circular paths in order to improve performance of the absorbers at large amplitudes. As early as 1938, Bulter [4] recognized the potential value of noncircular 8 paths for absorber C.G.’s via bifilar suspensions. Madden [32] in 1980 first proposed cycloidal paths, which have favorable tuning behaviors at large amplitudes. Pre- sumably Madden used the cycloidal path since it is known to be the solution of the tautochrone problem, that is, it Offers the path for which a point mass, moving under a gravitational field, exhibits oscillations that are independent of the amplitude of motion. (Note that the solution of the tautochrone problem is identical to that of the more well-known brachristichrone problem [15].) This motivated an experimental study by Borowski and co-workers [3] that used absorbers riding on cycloidal paths, which was carried out by the Ford Motor Company. In this study, second-order ab- sorbers were found to be effective in reducing the second-order torsional oscillations in an in-line, four-cylinder, four-stroke, 2.5L engine. However, the fourth-order oscil- lations were significantly magnified. The remedy Offered for this problem was to use a combination of second and fourth order absorbers [3]. The cycloid does improve performance, but is not optimal in avoiding the mistuning problem [16, 56]. This follows since, in the centrifugal field, the force on a mass is not constant, as in a gravitational field, but is rather proportional to the radial distance from the center of rotation. The solution for the tautochronic path for this case is known to be a certain epicycloid [15, 68]. This tautochronic path enables the absorber to possess a constant oscillatory frequency, regardless of its amplitude of oscillation. Tuned to have the same frequency as the disturbing torque of order n, the motion Of the absorber, said to be of order 12, provides a periodic torque which counteracts most of the order n harmonic Of the disturbing torque, thereby reducing torsional oscillations at that order. This promising property of the tautochronic paths launched theoreti— cal studies [16, 11] which explored the effectiveness of epicycloid path, tautochronic, absorbers. It should be noted at this point that all of the above absorber designs are capable Of only partially counteracting the torsional vibrations that arise from a harmonic 9 torque, even in an ideal setting [28, 29, 7, 6, 55]. There always exist residual vibra— tions, especially at higher order harmonics, that arise from nonlinear effects. How- ever, Lee and Shaw [30] recently proposed a novel absorber design which consists Of a pair Of identical absorbers riding on special paths tuned to one — half the order of the disturbing torque. Such a configuration is referred to as the subharmonic absorber system. It was shown in [30] that the restoring torque generated by an ideal, perfectly tuned, undamped pair Of subharmonic absorbers is exactly a pure harmonic over a wide range of amplitudes. This has significant potential advantages over conventional designs, since it generates no higher-harmonic torques, even when accounting for nonlinear effects. However, this type of absorber system has yet tO be experimentally tested. 1.3 Motivation All designs stated in the last section for CPVA’S are based on two assumptions: first, the absorber system used for addressing a given harmonic consists Of only a single dynamic mass; second, the absorber paths are perfectly manufactured as desired. In practice, it is necessary to choose the total absorber inertia to be sufficiently large such that the absorbers do not hit motion-limiting stops, or snubbers, under severe Operating conditions. This is typically accomplished by stationing several absorber masses along and around the axis of rotation, which also is beneficial for balancing considerations. In addition, ideal path shapes can never be realized in practice. There always exist manufacturing tolerances, thermal and direct stress deformations, and distortions due to wear. The goal of the present study is to introduce a generic methodology that can be used to re-evaluate the performance of an absorber system by exploring the dynamics of a rotating system with multiple absorbers which follow paths that include imperfections and mistuning. The use 10 of this method is demonstrated herein through two cases of particular interest: (1) a system composed Of a rigid rotor and N tautochronic absorbers, as proposed by Denman [16], and (2) multiple pairs Of subharmonic absorbers as proposed by Lee et al [30]. 1.4 Organization of this Dissertation Tha remainder of this dissertation is organized as follows. Chapter 2 covers the following topics that are common throughout the thesis: the mathematical forms of the path configurations are given; the assumptions on the system are listed; the equations of motion for general absorber paths are derived for a multiple absorber system; the symmetry characteristics Of the system, which are critical in the analy- sis, are identified; two performance measures for an absorber system are defined; and characteristics of the absorber damping are described. In chapter 3, a stability crite- rion of the unison motion for a system Of N multiple tautochronic absorbers Of order n is derived. This is accomplished by using some scaling assumptions and trans- formations that massage the equations of motion into a form amenable to asymp- totic analysis. In chapter 4, the performance of the tautochronic absorber system is re—assessed in the post-stable parameter range by carrying out a post-bifurcation analyses Of the response. In chapter 5, the effects Of imperfection and mistuning, and the nonlinear dynamics of a single pair of subharmonic absorbers proposed by Lee et al [30] are investigated in order to re-evaluate the absorber system performance. In chapter 6, the analysis in chapter 5 is extended to the case of multiple pairs of absorbers. In section 7 some conclusions and directions for future work are given. CHAPTER 2 PRELIMINARIES This chapter aims to establish a mathematical model based on some physical assumptions, which is followed by a description Of limitations imposed on absorber motions, the characteristics of absorber dampings, and an identification of symmetry properties of the system. These results will be used in subsequent chapters. 2.1 Assumptions The equations of motion are derived for an idealized model that consists of a rigid rotor spinning about a fixed axis, subjected to an applied torque, and fitted with N general-path point-mass absorbers. The system is shown schematically by the cross sectional view of the rotor in Figure 2.1. This dynamical system consists Of a rotor of moment of inertia L; with respect to the center Of rotation, O, and N absorbers moving freely on prescribed paths relative to the rotor. Each individual absorber, denoted by subscript i for the ith absorber, is considered to be a point mass with mass m;. (In the common bifilar configuration, one can account for the moments of inertia Of the absorbers about their respective C.G.’s by simply including them in 14, since they rotate identically with the rotor.) The path for each absorber mass is specified by a function R.- = R,-(S'.'), where R,- is the distance from the C.G. of the absorber to point 0 and S.- is an arc-length variable measured along the path defined relative to a frame of reference that rotates with the rotor. The origin of each 5',- is taken to be at the path vertex, that is, the point where R,- reaches its maximum. value, Rio = R;(0). The nominal moment Of inertia for each absorber with respect 11 12 Id: The moment of inertia of the rotor \ m;: Absorber’s mass \ \S;: Arc variable Absorber’s path Figure 2.1: Cross-sectional schematic diagram Of the rotor and ab- sorbers. to 0 is defined by I,- = ng30. The absorber path is designed to be symmetric with respect to S.- = 0; i.e., R,(S;) = R,(—S.~). The damping between the ith absorber and the rotor is assumed to be an equivalent viscous damping with coefficient cag. Resistance between the rotor and ground is also modeled as equivalent linear viscous damping, with coefficient co. Let 6 denote the angular displacement Of the rotor. The net applied torque (in- cluding load torques) is assumed tO be a nominal constant, To, plus a disturbing torque T9(6) which is periodic in 6. These torques arise from a variety Of sources, including attached linkages, etc., and are generally periodic with several harmonics. They may also depend on 6 and 6. Here a simple, single harmonic model for the applied torque is considered, as there is typically one dominant harmonic and the absorber system will be designed to address it. Thus, the disturbing torque is as- sumed to be of order n, as follows, T9(6) = Tosin(n6), where T9 > 0. (This leaves Open the potentionally large issue of nonlinear resonances that may arise from other 13 harmonics in the excitation. This is left for future work, but see [29] for some results along these lines.) 2.2 Equations of Motion With these assumptions, the overall system kinetic energy can be formulated, which is given by K.E. = {1.62 + i mpg-(sat2 + 5'3 + 26,-(S.)éS’.-]} (2.1) i=1 NIH where () denotes 1129, t is time and X;(S.') = 123(5.) and G.(S.') = \/X;(S.') - i(%(5i))2. (2.2) In the expression for the kinetic energy (2.1), %Id62 is the rotational energy of the rotor, -;-m,-[X,-(S.-)62 + SE] is sum of the rotational energy relative the fixed frame and the translational energy relative to the rotating frame for the ith absorber, and the term 2G;(S,-)6S,- arises from Coriolis effects. By considering the ratio g/(fiflz), it can be shown that gravitational effects are small compared to rotational effects for any reasonable rotating mechanical system. Then, by assuming that the corresponding potential energy is negligible, the gov- erning equations Of motion are determined by applying Lagrange’s method to the kinetic energy and to the generalized forces associated with the dampings and the applied torque. The results are: 1% m.[S + GAS-)6 - 2 d5. (5..)9'2] = -c..s'.-, 1 s 2' s N (Na) -1 N (1X; - ° '° " ' ' 1016 + Zm,[F(S.-)S;9 + Xi(5i)6 + Gi(Si)Si + EGOSH i=1 ‘ d5: N = Z: cagG,-(S.°)Sg — C00 + To + Tgsin(n6), (2.31)) i=1 where Co and Col are damping coefficients for the rotor and the ith absorber respec- tively. Note that equation (2.3a) describes the dynamics of the ith absorber, which 14 are coupled to the dynamics of the rotor through the terms 6 and 62. Equation (2.3b) results from the dynamic balance among the applied torque To + Tgsin(n6), the rotor damping CO6, the damping and resistive torques caused by the motions of all absorbers — as described in the two summation terms, respectively — and the inertial resistance of the rotor [.16. A nondimensionalization and a change of independent variable are performed on the equations of motion for simplification. To facilitate this process, the nominal steady-state rotational speed Of the rotor, fl, is taken to be the speed at which the constant torque To balances the mean component of the torque which arises from rotational damping friction; thus, To Q = —. 2.4 CO ( ) Also, a new dimensionless dependent variable y, representing the rotor speed, is defined as (2.5) 61 Then, assuming that 6 is a smooth and invertible function Of t, applying the chain rule and using (2.5), one can obtain the following relationships between derivatives with respect to t and 6, 5: myy', (3) = nym’ and ('5) = nan/(I + 92.21%)". (2.6) Where (-)' denotes 5%}. With the relationships (2.5) and (2.6), the resulting equations of motion (2.3) can be transformed into a set Of periodically forced, non-autonomous equations with the independent variable 6 replacing t. This step transforms the nonlinearity, Tosin(n6), into a periodic forcing term. The following steps are then performed: the equations of motion are transformed to the form that has 6 as the independent variable; they are divided through by the inertia terms, m,- and Id, respectively, and by 02; and the absorber displacement S.- 15 rescaled in terms of R“. The resulting dynamical system that describes the dynamics of the N absorbers and the rotor are then given as follows, II I I 1dr.- , I . 313; +13.+ 9£(3:‘)l - 5195690?! = _flaisiv 1 S 2 S N. (2721) I II di i ' Zb,{j“ iii-3y 1'.+x(8.)yy +g.(s .i')3yy +g.-(S .)S. y2+%8.2y’l i=1 N +yy = Z Maia-(808w - [Soy + 1"o + F0 sin(n9) (2-7b) i=1 3. I d . where () denotes 71%, s, = FL _1..__ ,o’b‘"7:’#°'_m.f2’f”°=14mm):Ian’I‘9=7_SQT’" d and $43,.) = 51%}23—0 and g;(s,-) = \]x,-(s;) — Z 1(::(s,-)) , (2.8) are functions set by the path of the absorber C.G. Note that in terms of these dimensionless quantities, the steady rotation condition (2.4) becomes F0 = filo. (2.9) 2.3 The Absorber System It is assumed that the system is composed Of N absorbers with identical individual masses m; = 3%,“ and identical damping coefficients [Im- = [1,, for each 2'. Two different absorber systems will be considered in the subsequent investigations. The first system consists of N tautochronic absorbers riding on standard epicyloids tuned to order n (the same as the order of the applied torque), which can be specified in the ideal case by 22,-(s,-) = 1 — nzsf, with Rm = R0 for each i. (2.10) The second is composed of N tautochronic absorbers riding on subharmonic epicy- loids tuned to order 1;- (One-half the order of the applied torque), which can be specified in the ideal case by 2 :r;(s,-) =1 — (g) s? with Rm = R0 for each i. (2.11) 16 Note that the path functions (2.10) and (2.11) presented herein assume that the paths are perfectly manufactured as desired. The formulation of imperfections will be introduced in section 5.1.2, particularly for the subharmonic absorber system. The equations of motion for a system with N identical absorbers is then given by II I I 1d ;' A . 315.- + [3.- + 9480] — --i.(8.)y = -/Sa8.-. 1 g 2 S N, (2.12a) dz,- I d t S, I _Zl—w +$.(8 )yy +gs(8 )8.yy +9S(8 ”)8 y + yd: )8.2y2] Ni=13‘ + ill/I: — NZ #a9i(8 )~‘5.-y- [toy + F0 + F9 sin(n6) (2.12b) where Io = moRg, and V = fl. It will be shown in section 3.1 that the standard epicycloidal paths tune the oscillating frequency of each absorber to be equal to that of the disturbing torque, even when the absorbers undergo large motions. This favorable property motivates the investigation on this type of path. Also, the subharmonic epicycloidal paths tune the oscillating frequency of each absorber to be one-half that of the disturbing torque, again over a large amplitude range. The merits Of this path design will be further elaborated in section 5.1. Also, it should be noted that in practice, the inertia of the entire absorber system is much smaller than that of the overall rotary system, typically on the order of 1%- 10%. This implies that V is generally a small parameter in these systems. 2.4 Limitations on Absorber Motions The value of the function g;(s,-) must be kept real during absorber motions, and this leads to a restriction on the amplitudes Of the absorber motions, given by 125(35) - l (il-ECSJ) > 0, V 9 and t. (2.13) 17 This restriction keeps the absorber from passing any cusp point that may exist on the path. For example, the epicycloid and cycloid paths have cusps at fairly large amplitudes. The explicit form for the right-hand-side of equation (2.13) depends on the type of the absorber path used, which will be derived for each particular absorber path considered in the ensuing analyses. Note that since the absorber amplitude grows as the torque level is increased, the restriction in inequality (2.13) imposes a finite operating range on the disturbing torque level, which is an important measure of the absorber system performance. 2.5 Absorber Dampings A discussion Of the damping models employed in the equations of motion is per- tinent at this point. The damping on the main rotor system and the mean torque simply set the nominal speed and play no other vital role in the the system dynamics. However, the damping in the absorbers plays a central role in the performance of the system, and this is a notoriously difficult effect to qualitatively determine or quanti- tatively measure. First, the source of the damping is complicated and depends on the specifics of how the absorbers are implemented. Some sources of damping include: rolling resistance, resistance due to movement through oil-saturated air, slippage, and pumping Of fluid. Second, even if one knows the basic damping mechanisms, the physical constants are difficult to measure and they will vary with operating condi- tions (such as temperature). We have assumed a form of equivalent viscous damping in our equations, and will consider two different types of damping, viscous and hys- teretic. Note that if the damping is viscous, the associated coefficient ca (herein we assume that ca,- = c,- for each i) in equation (2.3a) is assumed to be independent of the mass of the absorber, as in the linear case, then . N Ca. 11.. m0“ (2.14) 18 which shows that the nondimensional “effective” damping coefficient [2,; is pro- portional to the number Of absorbers. However, an experimental result given by (Cronin [11], 1992) indicates that there exists a hysteretic damping factor that is independent of the absorber mass, implying that ca is proportional to mass Of the absorber. Therefore, if is a constant in this case. According to this result, intro- ducing the quantity coo as the coefficient for a single absorber, we can express [Ia 8.5 it _ COD a moat which renders the damping ratio to be independent Of the number of absorbers. (2.15) 2.6 Symmetry Identification Intuitively, due to the identical nature of each absorber, it is expected that the system described by equations (2.12a) and (2.12b) will enjoy some special properties. These properties can be mathematically characterized by transformations among the state variables that yield new sets of system equations which are both structurally and mathematically invariant from the original system equations. Such transforma- tions are symmetries of the system. Identifying the symmetry of the system allows one to search for and characterize many solutions in an efficient way. To mathemat- ically characterize the symmetries of the system, conventional notation from group theory is employed. (See [20] for details.) Let 2': = h(:r,/\) (2.16) be a system of first-order differential equations, where :r is a generalized state vector, A is a system parameter, and h : R" x R —+ R", is a smooth transformation. Let 7 be an invertible k x 1: matrix representing a transformation among the state variables. It is said that 7 is a symmetry of the system (2.16) if h(7:r, A) = 7h(:c, A) Va: 6 R”. (2.17) 19 It can be shown that system (2.16) is invariant subject to 7 if equation (2.17) is satisfied. If there exists a group G such that the equation (2.17) is satisfied for each 7 E G, then G is called a symmetry group of the system, or, equivalently, that the function h is called G-equivariant. To identify the symmetry group of the present model, first consider equation (2.12b), which describes the dynamics of the rotor. It is seen that the speed of the rotor, y(6), is invariant subject to any permutation among the absorbers. Furthermore, from equation (2.12a), it can be confirmed that each absorber is coupled with all other absorbers only through y. Therefore, any permutation of absorbers should result in a system that is indistinguishable from the original. One can easily transform equations (2.12a) into 2N first-order differential equations and use condition (2.17) to show that the symmetry group of the system is SN, known as the “symmetric group”, which is a group containing all permutations on N symbols [18]. Based on group theory [20], there exist invariant subspaces in the absorber system due to the embedding symmetry SN. A partition Of particular utility in the present work is offered by splitting the phase space into components that capture the unison, or synchronous, response of the system, and its complement. In mathematical terms, we define V={e€RN I s =[v,v,...,v]T} and W=RN—V (2.18) where V is the subspace spanned by the unison mode and W is its complement. For any given initial conditions 8(0) 6 V or 8(0) 6 W, the system dynamics will stay in V or W, respectively, for all time. It should be pointed out that bifurcations in systems with this level of symmetry can be extremely rich. In fact, due to the fact that many eigenvalues associated with W are identical, and thus may become simultaneously unstable (which is always true for perfect abosorber system), the corresponding bifurcation problem is highly 20 degenerate and there may exist numerous branches Of solutions emanating from a single bifurcation point. It is not always possible to determine all these branches, let alone their stability types. In the present study, measures Of absorber performance are used in conjunction with symmetric bifurcation theory in order to get a handle on the most important branches, and in particular, the dynamically stable ones that define and limit the post-bifurcation steady-state system behavior. 2.7 Measures of Absorber Performance Two measures will be used to quantify the effectiveness of an absorber system. The first is the amplitude of torsional oscillations of the rotor, here represented by its peak angular acceleration. The nondimensionalized angular acceleration of the rotor is given by 6(t)/92, and is represented in terms of the variable y(6) by yy'(6). The corresponding measure of absorber performance is given by the peak value (that is, the infinity norm) of yy'(6) during a steady-state response. This quantity is denoted by llyy'llss- The second performance measure used is the range of the applied torque ampli- tude over which the absorber can operate, denoted by F 9. This is imposed by the limiting cusps on the absorber paths, as stated in condition (2.13) above. (It should be noted that in practice, the geometry of the bifilar configuration commonly used when implementing these absorbers will impose even stricter limits than those given by the cusp.) The general aim of an absorber system is to minimize ||yy'||ss over the largest possible range, 0 < F9 < F9. It will be seen that these goals oppose one another, and the information Obtained from the present study can be used to make informed judgments for the selection of the number Of absorbers and the path parameters. CHAPTER 3 STABILITY OF THE UNISON RESPONSE FOR A ROTATING SYSTEM WITH MULTIPLE TAUTOCHRONIC ABSORBERS Due to spatial and balancing considerations, the implementation Of CPVA’s in- variably requires that the total absorber mass be divided into several absorber masses that are stationed about the center of rotation and along the rotating shaft. In order to achieve the designed-for performance, a system of like-tuned, identical CPVA’s is assumed to move in an exact unison response. However, due to nonlinear dynamic effects, the absorbers may undergo non-unison steady-state motions, even under a moderate level of applied torque. The study in this chapter and the next is an investigation of the dynamic stability and bifurcation of the unison response of a system of identical CPVA’s operating on a rotating system. This chapter is organized as follows. Section 3.1 gives a preparation for asymp- totic analysis by re-arranging the equations of motion into a form of N weakly- coupled and weakly-nonlinear oscillators. This special formulation will also be uti- lized in chapter 4 for further investigation of post-bifurcation responses. Section 3.2 presents a derivation of a stability criterion for the unison motion. It indicates that for small levels of absorber damping (a condition required for satisfactory perfor- mance), the critical torque level is proportional to the square root of the absorber damping level. 21 22 3.0 Equations of Motion Results in this and the next chapter are obtained under the assumptions that the absorber paths for the N identical absorbers are tuned epicycloids Of order n (Denman [16], 1992). These paths are specified by :r;(s,-)=1- nzsf, 1S i S N. (3.1) Applying the path configuration (3.1) into equations (2.12) yields the equations of motion which describe the dynamics of the rotor and N absorbers riding on epicy- cloids of order n, as follows, ys: + [8:- + g(s.°)]yl + nzsgy = —fias:-, 1 S i S N, (3.2a) V N 2 I 2 2 2 I I I II 2 dg(3;') I2 2 321-2" 8.8.31 + (1 - n 8.)yy +g(8.)8.-yy +g(8.)8.y + 778.- y ] i=1 I I V N A I a + 311/ = 7)? Z #a9(~9i)3;y - [toy + I‘o + I‘198in(n9) (32b) i=1 where g(.,) = \/1 — (n2 + n4).2 “9“") - “"2 + ””3" (3.3) I I (13; _ \/1 _ ("2 + n4)3?. Note that the value of the function g(s,-) must be kept real during absorber motions, and this leads to a restriction on the amplitudes of the absorber motions, given by l i0< mu:-———, \7’6 d‘v’ '. . s()_s n n2+1 an I (34) 3.1 Scaling and Reduction of the Equations of Motion Approximate steady-state solutions of the system are sought by making some scaling assumptions and employing asymptotic analysis techniques. A series approx- imation for the equations of motion is derived below, and this leads to a form that is amenable to asymptotic analysis. 23 3.1.1 Scaling Assumptions In applications the total nominal moment of inertia of all absorbers about point 0 is much smaller than that of the entire rotating system. This motivates the definition of the small parameter, c E V, (3.5) the ratio of absorber inertia to rotor inertia, which is used for the asymptotic analysis. With this definition, many of the system parameters can be scaled such that the desired system behavior can be captured by asymptotic analysis. It is assumed that the nondimensional damping and excitation parameters, {16, fig, f0 and f9, are also small such that they can be scaled as follows: [2,, = cfia, [to = silo, P0 = trio, and f9 = fife. (3.6) The unperturbed system dynamics for this scaling are determined by considering equation (3.2b) with e = 0, that is, u = 0, which yields y = 1. Using this in equation (3.2a) with [1,, = 0 yields a linear oscillator with frequency n for the absorber motion. Thus, the steady-state solution of the unperturbed system is simply a constant rotor speed, y = 1, and the absorber motion is harmonic with frequency n and arbitrary amplitude. This limiting case can be imagined as that with a very large flywheel attached to the rotor, in which the absorbers move in a harmonic manner but have no effect on the rotor. Since the rotor speed will change smoothly as the absorber mass, the applied torque and the absorber damping are increased from zero, 3; will be smooth in e and can be expanded as follows, y(0) = 1+ «11(0) + 0(62). (3.7) where yl captures the speed fluctuations induced by the net interaction of the applied torque, damping effects, and the torques induced by the motions of the absorbers. 24 Note that condition (2.9) is assumed to maintain as c is increased from zero, thereby keeping the mean rotational rate near y = 1. 3.1.2 The Rotor Angular Acceleration It is convenient to have an explicit expression for the rotor acceleration, since it is a measure of the torsional vibration amplitude of the rotor. This can be derived by first noting that since 6 << 1 and 311 is bounded, y(0) oscillates about unity and is never zero. Therefore, equation (3.2a) can be divided through by y in order to obtain an expression for 3;, in terms of 3,, s;- and y. Substitution of this expression into equation (3.2b) and utilization of equation (2.8) gives an exact expression for yy'(0), as follows, N -l ill/Te) = [1+ {72714.53} [-fioy+F0+ngin(n0) i=1 N I d 1’ I I + ‘V— : (27128.3:‘3/2 - $63312 + n28.9(8e)y2 + 2flas.g(s.-)y) ](3.8) N i=1 d3; Utilizing the definition 5 E V, the scalings in equation (3.6), the expansion in (3.7), and condition (2.9), a series approximation for 313/, in terms of 6 can be obtained as follows, j=l d3] +0052). (3.9) I 1 N I d ' I ~ yy (0) = —e {N z(-2n23,-3j — n2g(s,-)sj + £9,232) — I‘gsin(n0)} The above equation shows that the nondimensionalized angular acceleration is of order 6, a result consistent with the known limiting case as c —> 0. 3.1.3 The Absorber Dynamics The method of averaging is used in the next section to determine the dynamic response for 0 < 6 << 1. To obtain equations in the correct form for the application 25 of averaging, some modifications of the equations of motion are carried out. First, based on the expansions in equations (3.7) and (3.9), one can show that it, is the same as 313/ to leading order in 6. Then by dividing equation (3.2a) through by y, a modified equation describing the absorber dynamics is obtained, into which the 6- series approximation of ”y: is substituted. Expanding the result in terms of 6 yields a set of weakly coupled, weakly nonlinear oscillators for the absorber dynamics. These oscillators, in which the dynamics of the rotor has been eliminated to first order, are as follows, 3;, +1123,- = ef,(31,....,sN,s'l,....,s}v,0) + 0(62), 1 S i S N (3.10) where f;(sl,....,sN,s'1,....,s}v,9) = —[Ias:- I 1 N I d 8‘ I +[3i + 9(3i)l[N:(-2n23j3j " "29(Sjlsj + if: {)33'2) 1:1 J —f‘93in(n0)]. Remarks: 0 These equations are weakly coupled. The weak coupling arises due to the fact that the absorbers are not directly coupled in a physical sense, but only indirectly so through the rotor, and each absorber has only a small effect on the rotor due to its small relative inertia. o The equations of motion are weakly nonlinear, even though the amplitude of motion of the absorbers is not assumed to be small. The weak nonlinearity is due to the epicycloidal path used for the absorbers, which renders a linear equation of motion valid for all feasible absorber amplitudes when the rotor speed is constant. Again, due to the relative smallness of the absorbers’ inertias, the rotor speed is nearly constant (cf. equations (3.7)), rendering nearly linear equations of motion. 26 o The symmetry SN is evident in equations (3.10), as the absorbers appear in a completely interchangeable manner. 3.2 The Averaged Equations The method of averaging is employed to analyze these equations. To this end, the following transformation to amplitude and phase variables is introduced, 3.- = a,cos(d>,- — n0), .9:- = na,sin(d>,- — n0), 1 S i S N, where a,- and 45.- are slowly-varying due to the form of equation (3.10). Substituting the above transformations into equation (3.10), a set of ODE’s are derived which govern the dynamics of a,- and (15,-, of the form, d i 2‘;— = if,(a1cos(¢1 — n0), ....,aNcos(¢N — n0) , nalsin(¢1 — n0), ...., naNsin(¢N -— 120)) sin(¢,- — n9) + 0(62) d i 2% = if, (alcos(¢1 — n0), ....,aNCOS(¢N - n0) , nalsin(¢1 — n9), ....,naNsin(¢N - 729)) cos(¢,- — n0) + C(62). (3.11) We now average the first order terms in the R.H.S. above over one period of the excitation, 27". The resulting averaged equations are expressed in terms of the new variables r.- and 90,-, (that is, the first order averaged quantities of a,- and (15,-, respec- tively), as follows, dr; “'1 - f E = £{7pa1‘g+';9COSQOgF1(T{) l 1 . +75 glznanrfsmflaj.) - "71010;, 13, 01,-.) — ”(n2 + n4)er1(r.-, "J" 01")” J I +002”) dcp.’ _ “f0 . 1 1 5 2 1 d0 - 4 WWW + w " a”) 1 —1 3 2 nrj 2 4 "i +N§[-4—n rjcos(2a,-.-) - TG2(Ti,rj,aji) ’ n(n + n ):H2(ri’ri’aii)]} J t ' ' +0052) (3.12) 27 where ajI' = I sin 2:r[l — (722 +12 4,2)7‘ cos 2.2]2dx, ‘pl F1(T,') = 2—0 271' 1 1 rgcoszx] 2 -:9. F2(r,-) = 1]: cos 2:z:[1— (n 2-+-n“)r;"cos"’a:]idx, Gl(r,-, rj,aj,-) = 21rcos(;r)sin(:1: — aj,)[1 — (n2 + 11") 21ro [1 — (n2 + n4)r?cosz(z — a,,~)]%d:c 1 21r G2(7'.°,7'j,aj.') = 2;./o cos(a:)cos(x — 01,-.)[1 — (n2 + n4)r§coszx]i’ [1 - (n2 + n4)r?cosz($ — aj,)]%dx 1 21r . 2 . 1 — (n 2 + n‘)r?cosz(:c —xa,-,)1 . . .. = _ _ .i d H.(r.,r.,a..) 2, f0 cossm (some: a. )1 ,_ (n. + was 1 x 1 2" , 1—(n2 +n 4,?)1" cosz(z— x-ag) H2(r,,rj,aj,) = -2_7r-./o cos(x)31n2(:r)c08(:r—Oji)[ 1_ (n2+n4)r2c032 a: J 1% 1WI .13. These equations govern the first-order, slow-time dynamics of the absorber motions, from which the first-order rotor dynamics can be obtained from equation (3.9). While these equations cover quite general motion, the present interest is in the existence and stability of the important unison motion of the absorbers, to which we now turn. 3.3 Stability Criterion In order to express the unison steady-state solution for the averaged equations (3.12), we introduce new variables 1' and cp as the steady-state amplitude and phase of unison oscillation, respectively. When the system undergoes unison motions, 1": r, (p,- = (p for 1 S 3,] S N (313) The steady versions of equations (3.12) can be solved for sin( 1. The results regarding 1‘ _<_ 1"incur = (am) the instability and the limiting absorber amplitude, equations (3.19) and (3.20), respectively, can be used to obtain a condition on the system parameters which will insure that the unison motion will be stable over the entire feasible operating range. This holds if r" > rm”, and provides the following condition on the level of damping that will ensure the stability of the unison motion for all amplitudes up to rm”: nu 2‘, > —— . . ” — 2(1+ n2) (3 21) In terms of the dimensional damping coefficients, this yields the following conditions for the cases of viscous and hysteretic damping, respectively: ca 2 filgfigz—z), if (2.14) is assumed (3.22) can 2 fi%%, if (2.15) is assumed It is observed that the only difference in these results is that the minimal damping level required for the viscous model to remain in unison motion decreases as the number of absorbers is increased (while holding the total absorber mass fixed), while 31 for the hysteretic model the damping level required is independent of the number of absorbers. If condition (3.21) does not hold, then there will exist a critical torque level at which the absorbers, moving in unison, will reach an amplitude of r“, and the instability will occur. In order to determine this critical torque we use the relationship between To and r given in equation (3.14), expanded for 1' << 1 and [1,, << n, which yields f9 2 n21" (this is simply the undamped, linear response relationship, which is a sufficient approximation for motions with amplitudes nearly up to rma$)- With this result, equation (3.19) can be re—expressed in term of the critical level of the disturbing torque as1 I‘; '2 (Ml/nil“. (3.23) The corresponding dimensional forms of the critical torque for the cases of viscous and hysteretic damping are, respectively, . 2nNcaIdQ3Rg, if (2.14 is assumed T; 2 \/ ) (3.24) \/2ncaoIdQ3R3, if (2.15) is assumed. This result indicates that the critical torque level for the viscous damping case de- pends on the number of absorbers N, and is raised by splitting a given total absorber mass into more absorbers. In contrast, for hysteretic damping, the critical torque level is independent of the number of absorbers used. (Recall that this result is valid only if the instability occurs before the absorbers reach the maximum amplitudes of the unison motion.) For the case of linear viscous damping, the effective damping flu in (2.14) is 1 See the work by Lee and Shaw (1996) [29] for an alternate derivation of the critical torque level in which a pair of absorbers is considered and one assumes from the outset that the amplitude, r, is small, but allows for arbitrary values of the mass ratio V. In this case, the critical torque level contains a correction term involving 1/2 — a correction that can be captured in the present analysis only by carrying out second order averaging, a nontrivial task for this system. However, note that their result cannot account for the finite amplitude effects captured here. 32 Bifurcation Set No.1 0.1 1 1 I I I 1 1r 0.09 - .. * x 0.08 "" * fl X o 1: x o 0.07 - x o O ‘ I? ' ° 0.06 ~ 1 . ° - X 0 o O 0.05 - ’5 - O 0.04 - 0.03 l l l l l l 2 3 4 5 6 7 8 10 N Figure 3.1: The critical torque level, 1‘3, versus the number of ab sorbers, N with 777:5" ~ 0. 0013. “”0 represents the bifur- cation point derived from “Det[A— B] = 0”. “”1: rep- resents the bifurcation point derived from the simplified criterion (3.23). “x” represents an non-unison motion from numerical simulations. “0” represents a unison mo- tion from numerical simulations. 33 proportional to the the number of absorbers and the quantity 3°35 = % is a fixed physical quantity. Based on this assumption, Figure 3.1 shows the critical disturbing torques levels Fg’s for different numbers of absorbers, N. The specific numerical values used for parameters are: 1/ = 0.1662, n = 2 (values taken from the 2.5 liter, in-line, four-stroke, four-cylinder engine considered by Denman [16] (1992)), and 51$ = (1‘7" x V = 0.008 x V 2 0.0013. The symbol ’0’ in Figure 3.1 denotes the critical torque levels determined by numerically solving Det[A — B] = 0 for the critical r value and using equation (3.14), while ’19 denotes the results derived from the simplified, small—amplitude criterion (3.23). In this figure, we also show simulation results obtained at selected “check points”, using a bearing damping level given by {to = 0.005. Points marked by ’ x ’ denote the upper check points, which lie at torque levels 5% above the critical values; here the absorbers undergo non-unison motions after bifurcation — more on this below. The symbols ’0’ denote the lower check points, which lie at torque levels 5% below the critical values; here the absorbers undergo unison motion. The responses near the check points converge very slowly to their respective steady states since they are close to the instability. One can see that the predicted critical torque levels derived from “Det[A—B] = 0” are within the range bounded by the corresponding upper and lower check points, while the simplified criterion diverges away from the more accurate result as N increases. Figure 3.2 shows the approximations for the critical disturbing torque level 1‘; versus absorber damping [2“, as given by the simplified criterion (3.23) and by “Det[A — B] = 0”. Note that by the results obtained from using equation (3.18), the curves in this figure are independent of the number of absorbers N. The remaining system parameters used are the same as those in Figure 3.1. The upper bound of fia=0.01 is chosen to prevent any absorber from hitting a cusp after bifurcation for N S 10, as based on observations from simulations. This figure also shows simulation 34 Bifurcation Set No.2 1 1 1 1 0.08 0.07 0.06 0.05 f3 0.04 0.03 0.02 0.01 0 0.002 0.004 0.006 0.008 0.01 Figure 3.2: The critical torque level, F3, versus various absorber damping [10. The solid line represents bifurcation points derived from “Det[A — B] = 0”. The dashed line repre- sents bifurcation points derived from the simplified crite- rion (3.23). “x” represents an non-unison motion from numerical simulations. “0” represents a unison motion from numerical simulations. 35 results at check points ’x’ and ’0’, which indicate that utilizing “Det[A — B] = 0” gives a very accurate prediction of the bifurcation. Figure 3.3 shows the absorber responses for N = 7 at the two check points. The unison motion is obvious. It is interesting to note that the post-critical, non-unison motion involves six absorbers moving together while the seventh has a different mo- tion with a larger amplitude. This super-critical bifurcation was the only type ob- served in many simulations involving different values of system parameters, different N, and different initial conditions. Note that there will be N such responses possible that are virtually identical, since the symmetry of the problem allows any one of the absorbers to be the one that is “out of step”. The specific absorber that steps out is dictated soley by initial conditions. The determination of this response and its effect on the performance of the absorber system is the topic of the next chapter. 3.4 Concluding Remarks This study investigated the stability of unison motions for multiple identical tau- tochronic vibration absorbers (CPVA’s). This problem has practical importance, as systems of CPVA’s are typically divided into N identical masses due to spatial restrictions and in order to balance the rotating system. However, the selection of the total absorber mass is generally done based on the assumption of unison mo- tion. Using the method of averaging, a stability criterion for unison motions was derived in terms of the disturbance torque level and the system parameters. The results indicate that the number of absorbers has an effect on the system stability if the absorber damping is viscous. If the absorber damping is hysteretic, however, the number of absorbers does not affect the instability. In practice, this damping, while critical for good operation, is extremely difficult to determine, either in type or magnitude. Designers with the aim of improving performance by lowering absorber damping 36 Absorber motions before bifurcation with f9 = 0.0655 (at the lower check point) 0.25 1 1 1 I 1 1 1 1 1 0.2 0.15 0.1 0.05 85(9) 0 —0.05 —0.1 —0.15 - -0.2 '- l _025 1 1 1 1 1 1 J 1 1 Absorber motions after bifurcation with F9 = 0.0724 (at the upper check point) 0.25 1 1 1 l 1 I 1 l 1 0.15 - 0.1 - 0.05 - 3,-(0) 0 — — — — — -0.05 - -0.1 - —0.15 -0.2 - _0.25 l l l l l i l l I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0/7r Figure 3.3: Absorber motions before and after the bifurcation point for N = 7. 37 should keep this instability in mind, as it is more likely to occur with smaller damp- ing. The net result of lowering the damping is that the system will have improved performance, but over a smaller operating range, since the post-critical motion of the absorbers will reach amplitude limits at a smaller torque level than the unison motion. This is considered in detail in the next chapter. inc $1111 01:] CHAPTER 4 NON-UNISON DYNAMICS OF A ROTATING SYSTEM WITH MULTIPLE TAUTOCHRONIC ABSORBERS It was shown in the previous chapter that for epicycloidal absorber paths, the unison motion of N identical absorbers may become unstable at a moderate level of the disturbing torque, in which case the performance of the absorbers in the post- bifurcation stage becomes of interest. In addition, it was observed in simulations that the post-critical response involved N - 1 absorbers moving in relative unison with the remaining absorber undergoing a larger amplitude motion. This chapter aims to uncover the source of this response and to determine the effects it has on system performance. To this end, the dynamic response of the model considered in the previous chapter, which consists of a rigid rotor fitted with N identical absorbers using epicycloidal paths and subjected to a harmonic torque, is investigated. The primary goal of this effort is to determine the nature and stabilities of the post-bifurcation, non-unison solutions in order to estimate the effects that the bifurcation has on the two performance measures: the rotor acceleration and the feasible operating torque range. It is determined that the torsional oscillation level is reduced in the post-bifurcation regime, which improves the absorber performance. However, it is also found that the feasible operating torque range is reduced due to the bifurcation, since the absolute peak of all steady state absorbers’ motions is increased. It should be noted that the present results are only the first step in such a study, as some important issues must be considered in subsequent work in order for the results to be of any practical use. These matters are taken up in the conclusions of this chapter. 38 39 4.1 Preparation of Equations of Motion ' In the previous chapter, the method of averaging was employed to find a criterion that determines the point at which the unison motion becomes unstable. However, this approach fails to characterize the post-bifurcation dynamics in a convenient form, since the averaged equations are highly nonlinear and coupled in terms of the amplitudes and phases. Essentially, while it is possible to find the post—bifurcation solutions using numerical methods, it is not possible to predict the behavior of the post—bifurcation dynamics in terms of system parameters. To solve this problem, a linear coordinate transformation among absorber displacements is used herein that splits the dynamics into two invariant subspaces, representing the unison motion and its complement, respectively. 1 This transformation is given by 1 N 1 . {1 = Egg, 6,- = N(81 — s.) for 2 S 2 S N. (4.1) Remarks: 0 This transformation enables one to separate the dynamics in the subspace of the unison mode V, with attendant coordinate {1, from the dynamics in the complement space W, with coordinates 5,, 2 g i S N. From the structure of the Jacobian C, it is known that when the unison response bifurcates, (N — 1) eigenvalues of this system response, which correspond to the system dynamics in W, cross the imaginary axis through zero. Therefore, in order to determine the post-bifurcation behavior, the dynamics in W must be analyzed. 0 Note that for a response in which a group of p absorbers move in unison, with 31 included in that group, there will be (p — 1) {i’s with zero amplitudes and (N — p) 65,8 with nonzero amplitude (for 2 S i S N). Furthermore, if the 1 A similar transformation was used in [11] in order to put linearized equations in a useful form. 40 remaining (N — p) absorbers move together, the nonzero {g’s (2 S 2' S N) will be equal to one another. 0 Each 5,- (2 S i S N), is orthogonal to {1 but they are not orthogonal to one another. A standard block diagonalization technique (see [24]) suggests that one choose a set of orthogonal coordinates to characterize the dynamics in W in order to find the linearized solutions near the bifurcation point. In contrast, herein the special transformation (4.1) is chosen for convenience in estimating the feasible operating range of the applied torque. o The inverse of the transformation exists and is given by N' .N 81:26” 35=Z€j—N€;, for 2_<_ZSN. (4.2) '=l j=l o For efficiency of presentation, the matrix T is defined such that a = T5 where a = [31,32, ..., 3N]T and £ = [€1,52, ...,{N]T. The final form for averaging is obtained by applying transformation (4.2) to the equations of motion (3.10) and implementing a transformation to polar coordinates. First, substituting transformation (4.2) into equations (3.10) yields the following transformed equations of motion {1’ + "251 = €f1(£,€'19) + C(52)» 61", + nzéi = efi(£a£'10) + 0(62)1 2 S 2 S N: (43) where fl(£a£'1 6) = —fia£1 +61 Y(T£, 0) 1 N N N +-,\—, [:(m) +Zg(Z£.-Ne.) j=l i=2 '=l —pa€i + £iY(T£a 0) +% [a (£261) -9 (£611 N 61) i=1 i=1 Y(T€,9). fi(£a 5'1 0) y(T£,9),2 S i S N, 41 1 N d ‘ I " . Y(s = N 2; —2n2 sis; —n2g(s,-)sj + figs-1%,?) — F951n(n0). j]: J The polar transformation 13 then given by {g = p;cos(¢,~ — n0), and 5:- = np;sin(1,b,~ - n9), 1 S i S N. (4.4) Note that this transformation is singular when 6; is zero, and it is therefore not ap- propriate for determining the stability of the unison mode. However, of interest here is the system dynamics in the post-bifurcation stage. Substituting transformation (4.4) into equations (4.3) results in a set of first-order differential equations which describe the dynamics of p,- and 112,-, 1 S i S N, as follows, 11:- = 51:01. ,p~,¢.,....¢~, 61.11101 .--n61+0( 1 (Ma) 11.11; = gs1p.,...,p~,:1.....,¢~,o1cos(¢».-no1+0181, 199114.511) where the function F,- is simply f; expressed in terms of coordinates p,- and 1,11,, as obtained by incorporating transformation (4.4) into fg. Equations (4.5a) and (4.5b) are in the desired form for averaging. It should be pointed out that, when expressed in terms of the coordinates pg’s and ng’s, the subspace of the unison mode V is spanned by [p1,1b1,0, 0, ..... ,0,0]T and the complement W is spanned by [0,0, p2, 1122, ..... , pN, ¢N]T. 4.2 The Averaged Equations Considering only the first order terms in e in equations (4.5 a) and (4.5b), aver- aging is performed in 0 over one period of the excitation, 2,11. The resulting aver- aged equations are expressed in terms of the first—order averaged variables ,6,- and 113,-, 1 S i S N. Due to the complicated nature of the system, this process results in many terms in the forms of integrals, which render closed-form solutions unachievable. In order to obtain simplified, approximate estimates of the rotor acceleration and the operating torque range, it is assumed that the oscillation amplitudes of the 42 absorbers, that is, the fig’S, are small and of the same order, denoted 0(5). Then the averaged equations can be expanded in terms of the p",-’s. This yields a set of truncated, averaged equations in terms of p; and 15,-, 1 S 2' S N, as follows, where each is expanded to the desired order (more on this below): ~ dfil _ "flafil F9 ‘ —3 dé - 2 + 2nc08¢11+ 0(10 ), _ dIBI _ f0 . - ”/31 -3 P1 dé "" —2n81n¢ll — 2 +O(P )9 dfii __ _flafii ”3 -2 _ . - - d6 '— 2 + 4 P1P:Sln(21/)1 2w!) 3.. n pg (4.6a) (4.6b) +— Z {2fi1fijsin0/7a- 2131-1—(N — 11fi§sin[2(13. — 1311]} 4 11513 n '1 _ _ . - — — +710 )3 pjpksm(2¢.- - t/Jj - 1&1.) 1.1.11..- & #1: 16n j=l 2 4 -. ~ N ~ - - +M {NFgfigcoswg — Z2rgfijcos(2¢g — 7.0,) N anfilfiiSin(J’l — 1171) - 2 27103115163051 - 21/71 + 1131)} + 0035), (Me) j=1 _ (1—1 3 _ - _ _ N — 1 Aid—16 = —nTp,p§cos(2tl)1 - 21A.) — ( 4 )n3p"? n35.- - - ‘ ‘ ‘2 _ - +-4— Z {2p1PjCOSWi - 7%) ‘ (N " 1)pjcos[2(1b; — “5)” 19513 +7136.- 2 —.— ‘. '. ‘ T . . _ pkaCOS(22,[!. — Kb, — We) J,k¢1,a a: #1: n2 + n4 '1' ~ — - N ” ' _ — ~ ' _ +——————( )p {—3NI‘9pgs1n1fl.’ + Z [2I‘9fijsln(21/Ji " $1) + 4F96jsmwjl 16n ‘3Nn2/31P3'COSW—H — 1131) N + Z [2n’filfijcos(2¢3. — «13. — «131) + 4nzmfijcoswl — 13.)] '=l 0035) i=1 where2SiSNand6260. }+ (4.6d) 43 4.3 Steady-State Responses Note that equations (4.6c) and (4.6d) are expanded out to third order, while terms out to fifth order are retained in the remaining equations. This is consistent for obtaining steady-state solutions, as the C(53) terms in the dynamics of 51 and 1/31 contribute at 0(1’15) in the dynamics of p,- and 173,-, 2 S 2' S N. Since only the first-order nonlinear terms in the dynamics of p, and 113,-, 2 S i S N, are needed to find the desired approximate solutions, the 0(1’13) terms in [1'1 and 1131 are not needed. This fact implies that only the linear dynamics of the unison response are needed in order to determine the first-order nonlinear steady-state solution of the non-unison component (this is most easily seen by making use of the coordinates employed here). In the above suggested method for finding the approximate solutions, it is assumed that the C(53) and 0(6) terms in the averaged equations will dominate the 0(62) terms that would result from second order averaging (which is not considered here). The validity of this assumption depends on the actual values of e and p, which depend in turn on the level of the disturbing torque. It is shown in the simulations to follow that the present expansion method provides satisfactory results for the system dynamics well beyond the bifurcation. To find a simple approximation for the steady-state solution for [11 and 1,51, it is assumed that [2,, is small compared to n (this is true in most applications), and that the C(53) terms in equations (4.6a) and (4.6b) are neglected. Setting equations (4.6a) and (4.6b) equal to zero yields the following approximate steady-state solutions for [11 and 11—21, denoted by £31 and 17),, P1 = — and 1A1 = "‘- (4-7) This is nothing more than the linear undamped response, but a reasonable approx- imation of the unison mode at steady state, even up to amplitudes for which the bifurcation occurs; this is verified in simulations. Substituting the above solutions 44 into equations (4.6c) and (4.6d), setting their derivatives equal to zero, and ignoring the 0(55) terms, a set of stationary equations obtains which can be solved for the approximate steady-state solutions of 5,- and 15,-, 2 S 2' S N, denoted here as 5,- and ~ 111,-, respectively. These equations are _ ”flaBi fgfii - ~‘_ 0 — —, + —4n slum.) 79291015131162. 41,-) - (N - 1)5fsin[2(171. - 13,11} #13 n3 P; z. z . 2: ‘ - +—2—— Z: . p,p1.sm( 10.- -¢’,- -1/)1.)1 (4:861) j,k¢l,1 8!. J¢k _ f2 93$ (N - 1)n3 :3 0 — 4113 c——os(211),-) 4 ,0,- +_3_p_, z {21.1.cos(15- — 11)) — (N— 1)5 Eco—s[2(11—) 15 )]} 4 j¢1£ pup) 1 J "3131' “- 2‘ ”- ‘ 2<'< SN_p solution branches generically exist for all p, 1 S p S N. Therefore, the existence of the solution branches on the surface of the ellipsoid with identical am— plitudes p.- and phases $.- for each i 6 N is generically ensured for the non-truncated averaged equations. Note that traveling wave types of solutions are also possible in the generic case for this bifurcation, but these have not been observed for the system under consideration. They may not exist, or may be dynamically unstable for this system. Attention is now turned to the most important of these solution branches, 81 X SN_1 . 2 Working to first nonlinear order predicts the existence of this invariant ellipsoid, but it does not provide the dynamics on it. This could presumably be obtained by using higher order averaging. However, for present purposes this is not necessary. 3 These two conditions are (also see [20]): (1) The symmetric group SN acts on W irreducibly. (2) The critical eigenvalues cross the imaginary axis with non-zero speed as the parameter of interest is varied. These conditions can be verified in the present case. However, one still needs to prove that the present bifurcation problem is generic. It is not the author’s intention to complete such a rigorous proof here. The “Equivariant Branching Lemma” is simply used as a “road map” to search for possible solution branches, and their existence can be confirmed by numerically solving the non-truncated averaged equations given in equations (3.12). 48 4.4.2 Search for the Solution Branch Leading to the Maximum IISIIss Instead of finding all possible solution branches, a search for the branch leading to the maximum ||s|]ss is conducted in order to estimate the feasible torque range. This is accomplished by substituting the polar form of the absorber responses given in equation (4.4) into the absorber displacements in terms of the 6 coordinates given in equations (4.2), and assuming identical phases for each absorber during steady- state Operation (this is the assumption justified in Appendix E). From this, one can express the steady-state peak value of the first absorber motion by ||31||ss E max{ 31(0) I 60 S 0 S 00 +27r, 00 -> 00} N 2 N ~'1 ”2 ~ 21‘ , ~— ._ I‘ 2 [(g...) -_;.....)(z )+.—:] , {‘2 i=2 which is a square root of a positive quadratic function of 2H), 5,. (Note that it is implied from equations (4.8) and Appendix E that sing/3,) is independent of a, 2 S i S N.) Subject to the ellipsoid in equation (4.10), ||31||ss will reach its maximum value when 25.2 ,5,- reaches its extremum. Since 2Z2 Z3,- = 0 is a principal axis for the ellipsoid, 2&2 )3, reaches its extrema at the direction of the associated eigenvector where :3, = ’31-, 2 S i, j S N. Hence, among all the possible post- bifurcation solutions, the one with identical f5, and 1%,, 2 S 2' S N, leads to the maximum ”Slugs. It can be easily shown that the maximum ||s,-||ss for all 2 S i S N is equal to the maximum H31 ”95, since the results are preserved under different choices of the first absorber ( since all absorbers are identical). As a result, among all the possible post—bifurcation solutions, the one with identical ,3,- and J),- for 2 S i S N leads to the maximum IISHss of all possible absorber motions on the steady-state el- lipsoid. This solution corresponds to the isotropy subgroup S; X SN_1, wherein one absorber moves out-of-step relative to all other absorbers, which remain in relative unison. This is also the post-bifurcation solution observed in simulations. Based on the Equivariant Branching Lemma, at least one such solution branch is 49 expected to exist (and it contains N identical steady state responses). The Newton- Raphson method was employed to numerically determine from the non-truncated averaged equations (given in equations (3.12)) that such branches indeed exist in the post-bifurcation stage over a wide range of parameter values. 4.4.3 Stability of the 51 x SN_1 Solution Branch With the existence of the 51 x SN-1 solution in hand, a stability analysis is carried out based on the truncated equations (4.6). Consider equations (4.6a) and (4.6b), in which [61 and 1,51 capture the dynamics of the unison mode. The steady-state solutions of [31 and 151 can be approximated by pl = ,3, + 0(i53) and tan 1/31 = tan 121 + 0(fi3) (4.13) where 1 f9 A“ -71 P1 _ "(fig + n2)1/2 and tan 1/21 — [ta , (4.14) which, when truncated, is simply the linear, damped steady-state unison solution. Note that when compared to the approximate solutions in equation (4.7), here the effect of damping is required since it is crucial to the stability analysis of the 81 x SN_1 branch. This approximate solution is independent of )5.- and 113,-, 2 S i S N, up to 0(fi3) (that is, the unison dynamics are independent of the non-unison dynamics to second nonlinear order). By treating the 0033) terms as non-vanishing perturbations in equations (4.6a) and (4.6b), it can be shown (using Lyapunov techniques) that there exists a positive number 9, independent of 5,-(0), 1,5;(0), 2 S 2' S N, such that [[51 (0),:Z11(0)]T is ultimately bounded in an 0(fi3) neighborhood of [§1(0),1Z1(9)]T for 0 _>_ 9. Hence, the stability of the 51 x SN_1 branch can be examined by 50 incorporating the approximate solution from equation (4.13) in equations (4.6c) and (4.6d), which govern the dynamics of fig, 213,-, 2 S i S N, and in which the 0(fi3) terms in (4.13) only contribute to the terms of 0(fi5). The subsystem consisting of equations (4.6c) and (4.6d), governing the dynamics of f).- and 1%, 2 S i S N, is considered for the stability analysis. The Jacobian of this subsystem is first derived and evaluated on the 81 x SN_1 branch. Due to the symmetry of the subsystem and this solution, this Jacobian, denoted by J, has the form 1 A2x2 B2x2 B2x2 B2x2 A2x2 B2x2 J(2N—2)x(2N—2) = . (4.15) . o . 82x2 B2x2 B2x2 B2x2 A2x2 J It can be shown that all eigenvalues of J are eigenvalues of one of the 2 x 2 matrices, [A + (N — 2)B] or [A — B]. This result is a consequence of the symmetry and does not depend on the actual values of A and B. The nature of the eigenvalues of [A — B] are first determined by the well-known fact that both eigenvalues of a 2 x 2 matrix possess negative real parts if and only if the trace is negative and the determinant is positive. By incorporating approximation (4.13) into the Jacobian J, the determinant and trace of [A — B] are determined to be, Trace[A - B] = —fla, (4.16a) N~a 722 + 124 11 1‘— WM — B] = " (256 )pp. [(12 - 5mm? + mm), +(4N - 12W + 1241235. com: — 2123.) +16 c0502 — 72.21)] + C(56) (4.16b) Where )7 and 1; are used to denote the steady-state amplitudes and phases of [3.- and ii, 2 S i S N, respectively, on the S; x SN-1 branch. Since the trace is always negative, only the sign of Det[A - B] needs to be determined. Letting f3 —> 0*, The bran new as} 51 it is found that the sign of Det[A — B] is dominated by the sign of c030,; — 17),) near the bifurcation point. Based on equations (4.8), it can be shown that near the bifurcation point the phase must satisfy —37r or *— 71' The above two solutions for the phases provide two different S; x SN_; solution branches. With the approximate value of gb; given in equation (4.7), one has cosh/.2 — 'Z’l) ‘3 2, near the bifurcation point. The first branch, with {Z 2 32—”, leads to Det[A — B] > 0 as )3 -> 0+ while the other branch similarly leads to Det[A — B] < 0. As for the [A + (N — 2)B] matrix, in Appendix F it is proved that the branch with {IS 2 "—21 leads to Trace[A + (N — 2)B] < 0 and Det[A + (N — 2)B] > 0 as 2') -> 0"“ near the bifurcation point. Hence, the S; x SN-; branch with I; close to ~31r / 4 is stable. Henceforth, this branch will be designated as “the stable S; x SN_; branch.” 4.5 Absorber Performance In this section, two important measures of absorber system performance, the feasible operating range of the applied torque and the angular acceleration of the rotor, are estimated. 4.5.1 Estimate of the Feasible Torque Range As the amplitude of the applied torque is increased, the absorbers’ amplitudes likewise increase, until a cusp limit (3.4) is reached for one or more absorbers. T here- fore, the feasible torque range can be determined if one combines the relationship 52 between the torque amplitude and ||s||ss with the absorber amplitude limit. This process is described here for both the truncated and non-truncated versions of the equations. From the analytical results obtained in the previous section, is known that there exists a stable S; x SN-; solution which yields the maximum ||s||ss (based on the truncated equations). This implies that for any initial conditions, the system will converge to a solution branch that renders an ||s||ss which is less than or equal to that resulting from the stable S; x SN-; branch. Therefore, this solution branch can be used to predict the maximum ||s||ss, which is used in turn to determine the feasible torque range. By using the fact that the steady—state amplitudes, pg, 2 S 2' S N, are all equal on the stable S; x SN_; solution branch, the ellipsoid prescribed in equation (4.10) can be used to determine the steady-state amplitudes, yielding ... l - - 1 P4 4~2 ‘ . pap,=—(—9— p“), 2ngN. (4.18) n8 n6 Similarly, by using equations (4.8), the equal steady-state phases on this solution branch are found to be “— *— 1 . 2 ~a . 1/2 E (bi = —[sm-1(_%g_)] — 7r, 2 S 2 S N. (4.19) o It has been shown that ||s||ss can be derived from ||s;||ss. To determine [[s;||ss, the expression for s; in terms of {,,1 S i S N, given in equation (4.2) is utilized. Substituting the angular transformation (4.4) into this expression, using the stable S; x SN_; branch and the approximate steady-state unison solution for p; and d); given in equations (4.7), one obtains Hsllss, as follows, ”Sllss E lrsngx‘fl 35(0) | 00 S 0 S 00 + 27r, 00 —> oo} - 1 1‘3 2 2:2 2 —— —(N—1)1”‘9i;sinzZ+(N— 1) p (4.20) n4 n2 53 where Z3 and 17) are given by equations (4.18) and (4.19), respectively. It is now possible to estimate the feasible operating range of the applied torque level I}; by recalling inequality (3.4) and using the approximate expression for Ilsllss in equation (4.20). This can be carried out to an analytical equation, which is not presented here since it is not easily solved for an explicit expression for the maximum torque. Note that since this estimate is based on the truncated equations in equation (4.8), it will deteriorate near the singularity of the absorber path. In order to determine a more accurate estimate for the torque range, one can numerically solve the non-truncated equations (3.12) (as described in section 3.2) for a more accurate estimate of the S; x SN-; solution. Numerical results for the torque range are given in Section 4.6. 4.5.2 Estimate of the Rotor Acceleration An approximate expression for the angular acceleration is first formulated to leading nonlinear order, after which more accurate estimates are computed. Taking the nondimensionalized acceleration yy'(0) stated in equation (3.9), considering only the 0(6) terms in yy'(6), expanding yy'(0) in terms of 3,,1 S 2' S N, and then using the definition 6 E V and the transformation (4.1), yields I r2712 N I ~ yy (6) = u T zgsjsj + n25; + P9 sin(n0) + 0(p3) (4.21) J: where only the first and second order amplitude terms are considered. Utilizing the truncated stationary equations (4.8), a nontrivial calculation (outlined in Ap- pendix G) yields the following lower-order approximation for yy'(0), , 15[n3;3:sin(21721 — 2n9)] = g; sin(2n0), before bifurcation. 1111(9) 2’ 22) 4241,, cos(2172 —- 2710)] = 242,, cos(21Z - 2110), after bifurcation. where the approximate solution for Ii; and 121 in equations (4.7) have been used and 54 =_1 ,_1 271141,, 44 u. )1 « An interesting feature of this result is that the peak value of yy'(0); i.e., Hyy'llss, where is quadratic in terms of the applied torque level in the pre—bifurcation stage — this is due to the fact that the absorber is tuned to eliminate the acceleration at linear order. An even more interesting result is that in the post-bifurcation stage, ||yy’||ss is independent of the torque level; i.e., it saturates after bifurcation. Furthermore, the acceleration yy'(0) vanishes as [4,, goes to zero. (Recall that the bifurcation torque level also goes to zero as [4,, goes to zero.) Since the acceleration predicted by equations (4.22) saturates after bifurcation, higher order terms in p will become dominant when the applied torque level begins to go beyond the bifurcation level. In order to obtain a more accurate estimate, one can use the acceleration approximated to the next order, which is given by 2 N I 2 ' ~ . N , 2 2 4 N yy(6) 2 V %:sjsj+n2§;+I‘gs1n(n0)+(n2+n4)ZsJ-sj2—Big—qui j=l j=1 j=l (4.23) where s,-,1 S 2' S N are approximated by equations (4.7), (4.18) and (4.19). An even more accurate estimate can be obtained by numerically solving the non- truncated equations (3.12) given in section 3.2 for the stable S; x SN_; branch and substituting the resulting 3,, l S 2' S N, into equation (3.9). These results are found to match simulations very closely over the entire feasible torque range. 4.6 Numerical and Simulation Results In this section, existence and stability results for steady-state solutions are pre- sented, along with simulation results, which are used to confirm the analytical results and to examine the accuracy of the various levels of approximations used in this study. 55 In addition to the approximate results obtained in the previous sections, included here are numerical solutions of the non-truncated averaged equations (3.12) given in section 3.2. The system parameters used throughout this section are: u = 0.1662 and n = 2; these were taken from the 2.5 liter, in-line, four-cylinder, four-stroke engine considered by Denman [16]. Recall that our approximations are based on a small V assumption, and the value considered here is a relatively large ratio for absorber systems; typical values are in the range 0.01— 0.1. The absorber damping [1,, is taken to be independent of the number of the absorbers, N. The Newton-Raphson method was employed to solve the non-truncated averaged equations (3.12) for the post-bifurcation branches. This process was repeated for the following parameter ranges: N = 2 to 10 with increments of one, [1,, = 0.0013 to 0.013 with increments of 0.0001, F; = 0.03 to 0.08 with increments of 0.0001. In order to determine as many solutions as possible, several starting points were randomly chosen in the range 1‘,- = 0 to 0.22 (the cusp level) and 1,0,- = 0 to 21r, for each i. The associated stability of each solution was determined by numerically evaluating the eigenvalues of the associated J acobian matrix. Numerical and simulation studies of many Sp x SN_p solutions were carried out. It was found that in the post-bifurcation stage, for absorber amplitudes below the cusp level, the only stable solution branch is the S; x SN_; branch predicted by equations (4.14), (4.18) and (4.19). Equations (3.2a), (3.2b) and (3.3) were used to directly simulate the system dynamics, using Gear’s BDF method [59]. It was found that by utilizing a wide range of initial conditions and the ranges of system parameters described above, the system dynamics always converged to a stable S; x SN._; response in the post- bifurcation parameter range. Figure 4.1 shows a typical set of post-bifurcation absorber responses for N = 4, [1,, = 0.0026 and F9 = 0.048. (Note that different values of [4,, show qualitatively the same system dynamics as the value chosen here, although for higher damping levels 0.15 0.1 0.05 3(0) 0 —0.05 56 I I I I I H I I I I Simulation — Truncated ''''' Nonr'Ii'uncated --- No Distribution an of Absorber Mass (Unison) 0/71’ Figure 4.1: Post-bifurcation steady-state responses of the absorbers for N = 4 (four absorbers), [4,, = 0.0026 and F9 = 0.048. Solid lines: Simulation; Dotted lines: Truncated; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response. 57 the bifurcation of the unison response occurs nearer the cusp point.) In Figure 4.1, the solid lines represent the simulated response. The dotted lines are derived by estimating the response by truncated equations (4.7), (4.18), (4.19) and transforma- tions (4.2). The dashed lines are obtained by assuming the stable S; x SN_; solution (:8; x S; here) and numerically solving the non-truncated averaged equations (3.12) for the absorber responses. The coarse dotted lines represent the simulated absorbers’ responses if they are locked into a unison motion (that is, the absorber inertia is a single lumped mass). This shows that the non-truncated equations are very accurate and that the truncated equations are quite satisfactory. Note that the system re- sponse, as compared with the corresponding unison motion, has N —1(= 3) absorbers with a slight phase shift and little amplitude difference, while one absorber undergoes a motion with drastically different amplitude and phase. It is the localized response of this absorber that will limit the applied torque range. (Initial conditions determine which absorber goes to the large amplitude, but in practice small symmetry-breaking discrepancies may favor localization in a particular absorber.) Figure 4.2 shows various estimates and simulations of the rotor acceleration for the same case as Figure 4.1. The 2nd-order approximation is derived by the trun- cated equations and the estimate given in equation (4.22), while the 3rd-order ap- proximation is derived by the truncated equations and the estimate given in equa- tion (4.23). It is seen that the 2nd-order approximation roughly represents the main harmonic component of the simulated acceleration, but offers a poor prediction for Ilyy’llss. This is due to the fact that in the post-bifurcation stage, the terms up to 0(p2) in equation (3.9) saturate and the higher-order harmonics begin to dominate IIyy'Hss. One remedy to this problem is to use the 3rd-order approximation, from equation (4.23), to estimate Ilyy'Hss, which offers a significant improvement over the 2nd-order results. As expected, the numerically-obtained, non-truncated solution is in excellent agreement with the simulated acceleration in all regards. 58 I I I I I I I I 0.01 - 0.005 3131(9) 0 *- - —0.005 - i‘ ' 2nd-order Approx. ------ 3rd-order Approx. ~— Non-Ti'uncated --- _0.01 _ No Distribution of Absorber Mass (Unison) ---- l l I J l l l l 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0/11' Figure 4.2: Post-bifurcation steady-state responses of the rotor ac- celeration for N = 4 (four absorbers), 42,, = 0.0026 and Pa = 0.048. Solid lines: Simulation; Dotted lines: The 2nd-order approximation; Triangles: The 3rd-order ap- proximation; Dashed lines: Non-truncated; Coarsely dot- ted lines: Imposed unison response. 1.8 llsllss 59 0.25 I I I I I I I 0.2 - Simulation — 'IYuncated ---- 0 15 _ Non-Truncated --- ' No Distribution .... of Absorber Mass (Unison) 0.1 - 0.05 I l l l l l 0 1 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 fa Figure 4.3: The Ilsllss’s derived by different approximations versus the applied torque level. The system parameters used are N = 4 (four absorbers) and [2,, = 0.0026. Solid lines: Simulation; Dotted lines: Truncated; Dashed lines: Non-truncated; Coarsely dotted lines: Imposed unison response. 0.08 60 Figure 4.3 shows the peak absorber amplitude ||s||ss versus the applied torque level. The maximum amplitude, which fixes the range of the applied torque, is set by the restriction in equation (3.4) and is marked as “Cusp” in the figure. From this figure, one observes that the truncated equations give a conservative prediction of the feasible torque range while the non-truncated equations give a very accurate estimate. Also, by comparing the unison and non-unison ||s||83’s, one can see that the distribution of the total absorber mass into several smaller masses significantly decreases the operating torque range. Figure 4.4 shows the percent reduction in this range relative to the unison response for different numbers of absorbers. It is seen that as N increases, the feasible range is dramatically decreased by the bifurcation. Figure 4.5 shows the rotor angular acceleration [lyy'Hss versus the applied torque level. In this figure, the 2nd-order approximation completely saturates after the bifurcation, which is not observed in the simulations. The 3rd-order results are much improved, and the non-truncated equations again give a very accurate result. By comparing the [Iyy'llss’s for the unison and non-unison responses in the post- bifurcation range, one can see that the distribution of absorber mass slightly improves absorber system performance by decreasing the ||yy'||ss levels. Figure 4.6 shows the ratio of the resulting ||yy'||ss to that for the unison response for various numbers of absorbers with F9 = 0.0555 and [1,, = 0.0026. It is seen that the ||yy’||ss’s obtained from simulations are well approximated by the non-truncated equations. However, the second and third order results significantly under and over estimate this ratio, respectively. Also, it is seen that the actual ratio approaches unity as N increases. 4.7 Concluding Remarks This study considered the dynamic effects of using several masses to compose the required inertia for a system of tuned absorbers. For usual sizing calculations, one implicitly assumes that these masses move in a unison manner. In chapter 3, it was .A.X~V ... ...,..v..-7-- {5.36.7}???- Percentage Reduction (‘70) . 61 80 I I I I I I I 75 '- . . O O 70 ~ . .3. 65 '- . 3 El- _ 60 - O + .. El + 55 " D -( + 501 D Simulation + - 'Ii'uncated I Non-'Duncated D 45-]: .. I? 40 - - 35 l l l l l l l 2 3 4 5 6 7 8 9 The number of absorbers, N. Figure 4.4: The percent reduction in torque range, relative to the unison motion, versus the number of absorbers for [4,, = 0.0026. “+”: Simulation; “0”: Truncated; “Cl”: Non- truncated. 10 llnyllss 62 0.025 . , , , . T . 0.02 I Simulation — 'Duncated, 2nd-order Approx. ------ 0 015 _ Truncated, 3rd-order Approx. —¢— ’ Non-'Ii'uncated --- No Distribution ---- of Absorber Mass (Unison) 0.01 - - 0.005 - ........................................................ .. 0 , - i h l i 1 l 1 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0,07 F9 Figure 4.5: The Ilyy'llss’s derived by different approximations versus the applied torque level, for system parameters N = 4 (four absorbers) and [4,, = 0.0026. Solid lines: Simula- tion; Dotted lines: The 2nd-order approximation; Trian- gles: The 3rd-order approximation; Dashed lines: Non- truncated; Coarsely dotted lines: Imposed unison re- sponse. 0.08 The ratios of “yy'Ilss 63 I I I I j I I A A 1.2 r- A _ A A A 16> or g c o c o c 41 A a 3 3 11! 10’ 0.8: 4 1h . Simulation '1' Truncated, 2nd-order Approx. 0 Truncated, 3rd—order Approx. A 0.6 L- Non-Truncated D A No Distribution of the Absorber Mass (Unison) 0 $ 0 C C C C C C . 0.4 - - l l l l I l J 2 3 4 5 6 7 10 The number of absorbers, N. Figure 4.6: The ratio of Ilyy'llss to that for the unison response versus the number of absorbers for F9 = 0.0555 and [1,, = 0.0026. “+”: Simulation; “0”: The 2nd-order ap- proximation; “A”: The 3rd-order approximation; “Cl”: Non-truncated; “o”: Imposed unison response. 64 determined that this motion can become dynamically unstable as the torque level is increased. In the present work the post-bifurcation dynamics are investigated. The results were obtained and verified by employing three methods: (1) low-order trunca- tions of the averaged equations, (2) numerically solving the non-truncated averaged equations, and (3) simulations. The truncated equations offer reliable qualitative results in terms of the dependence on system parameters, but are not very accurate in some respects. In contrast, the non-truncated results, while requiring numerical solutions of the steady-state equations, are very accurate in all respects. It was found that the post-bifurcation dynamics are dominated by a stable S; x SN-; steady-state solution branch. This is very reminiscent of mode localization, in that one absorber undergoes a much larger amplitude of motion relative to the others (see [62] for relevant work on nonlinear localization). It was also found that this S; x SN_; branch leads to the maximum ||s||ss and it results in a mild saturation of [Iyy'llss after bifurcation. Combining the results shown in Figures 4.4 and 4.6 indicates that one does not gain a significant reduction in the level of torsional oscillations by the distribution of the total absorber mass into N masses, but the feasible torque range is drastically reduced. Designers of absorber systems can refer to the information provided herein in order to obtain refined estimates of system performance before testing. However, it is recognized that other effects may have comparable influence on the overall system behavior. Of particular importance is the level of absorber damping; while generally small in practice, it is difficult to measure and may vary during operation (due to wear, temperature differences, etc.). It is interesting to note that when designing an absorber system, it is desirable to keep this damping as small as possible in order to keep the absorber oscillating in an out-of-phase manner relative to the disturbing torque. This offers optimal torque counteraction if the absorbers move in unison. 65 However, for a multiple absorber system, a smaller damping level will cause the bifurcation to non-unison response at a smaller level of the disturbing torque level, causing a potentially dramatic decrease in the applicable torque range. As stated in the introduction, this investigation is only the first step in the study of unison absorber motions. To be of any practical use, the results must be extended to include: other absorber paths, including the widely—used, intentionally mis-tuned circular path; the effects of multiple harmonics in the torque; rotor flexibility and the distribution of torque along the axis of rotation; and mistuning, i.e., symmetry breaking, to name a few. Preliminary simulations that include small mistunings among the absorbers indicate that the individual absorber dynamics can be drasti- cally altered by mistunings on the order of 1%. However, it is also observed that the overall ||s||ss’s and llyy'llss’s are quite robust to such changes. An investigation of these effects will bring the research squarely into the active realm of mode 10- calization (Happawana et al [22]; Hodges [25]; Pierre and Dowell [45]; Vakakis and Cetinkaya [62]). CHAPTER 5 THE EFFECTS OF IMPERFECTIONS AND MISTUNING ON THE PERFORMANCE OF THE PAIRED, SUBHARMONIC CPVA SYSTEM A recent study by Lee et al. [30] has demonstrated a new configuration of cen- trifugal pendulum vibration absorbers (CPVA’s) that is very effective at reducing torsional vibration levels in rotating systems that are subjected to harmonic exter- nal torques. This system is composed of a pair of absorbers riding on epicycloidal paths that are tuned to one-half-order relative to the frequency of the applied torque. Such a configuration is referred to as the subharmonic absorber system. It was shown in [30] that the restoring torque generated by an ideal, perfectly tuned, undamped pair of subharmonic ”absorbers is exactly a pure harmonic over a wide range of amplitudes. This has significant potential advantages over conventional designs, since it generates no higher-harmonic torques, even when accounting for nonlinear effects. The aforementioned results are based the assumption that the absorber paths are perfectly tuned and are manufactured exactly as desired. In practice, however, due to manufacturing tolerances, wear, thermal effects, etc., the absorber paths are never perfect. It was the initial goal of this study to determine the sensitivity of the system response to such small imperfections. During the course of the investigation it was found that a slight mistuning of the linear natural frequencies of the absorbers can actually help to improve some aspects of the absorber performance, although at a price in terms of other measures. In order to account for these effects and to predict the corresponding performance of the absorber system, an extensive analysis 66 67 is conducted herein that includes imperfections and intentional mistuning in the mathematical system model. An evaluation of absorber performance is accomplished by evaluating two perfor- mance measures: the angular acceleration of the rotor and the range of the applied torque. The former is used to quantify the level of vibration reduction, which is desired to be as small as possible, while the latter is imposed by the size of the absorbers’ masses and their limited range of travel. To calculate these two perfor- mance measures, the system dynamic response is approximated using the method of averaging for a particular scaling of the system parameters. The solutions of the averaged equations are derived and considered in light of the system performance goals. Bifurcation diagrams are used to evaluate absorber performance and to distill some guidelines for the design of absorber paths. 5.1 The Subharmonic Absorber System 5.1.1 The Perfectly-Tuned Absorber System A subharmonic absorber system was proposed by Lee et al. [30] which is composed of a pair of identical absorbers with individual masses m, = £20 and identical damping coefficients [2,, = [16, i = 1, 2. These absorbers ride on identical paths specified by n 2 x;2(s,-)=1— (2) 3?, 2': 1,2 (5.1) which is equivalent to R,(S,-) = \/R3 _ (g)? 5?. This path can be shown to be a particular epicycloid, resulting in absorbers whose natural frequency in the constant rotation rate case is n9/ 2, that is, one-half that of the applied torque. The equations of motion (2.12a) and (2.12b) for N = 2 and the identical paths given by equation (5.1) have an exact solution when the absorber damping is zero, 68 [4,, = 0, and condition (2.9) (for constant rotor speed) is satisfied. It is given by y(0) = 11 (5.2a) 31(0) = —82(0) i%(/%cos (g0) , (5.2b) where V = mg“: is the ratio of the total nominal moment inertia of both absorbers about point 0 to that of the rotor. It is seen from equation (5.2a) that in this response the rotor runs at a constant speed and the absorbers move in an exactly out-of-phase (s; = —32) subharmonic response of order two relative to the disturbing torque. In this response the absorbers exactly counteract the applied torque, hence the designation of the subharmonic absorber system. The physics of this absorber response can be seen by observing equation (2.12b), which describes the balance of the torques acting on the rotor. It is seen that the motions of the individual absorbers generate torque harmonics of all odd orders, which, due to their out—of—phase nature, cancel each other in the summation. However, each absorber also generates a single even-order torque, of harmonic 72, through the Coriolis term fiszyz. Since even-order torques add together in an out-of-phase motion, these add, creating a purely order n torque that exactly cancels the disturbing torque. This steady-state operating condition corresponds to a perfectly constant rotor speed, which is the ultimate design goal of such an absorber system. Note also that this solution, while not absolutely global due to the limited range of absorber motion imposed by the cusps, is valid and exact over a wide range of torque amplitudes (described in more detail below). When the system possesses small, nonzero absorber damping, it was shown in [30] that this pair of subharmonic absorbers is able to limit the rotor acceleration 0 to a level that is of the same order as the absorber damping, and that this acceleration saturates at a fixed, small level as the torque amplitude is increased over a wide torque range. 69 5.1.2 Absorber Imperfections, Mistuning and Limitations The dynamically favorable property described in the previous section can only be approximated in practice. Several effects will come into play that limit the ideal solution, including tolerances in the cutting process used for generating the absorber paths, the presence of rollers in the bifilar configuration (whose dynamics do not follow the absorbers’ motions [16]), and deformations due to wear, elasticity or ther- mal effects. In order to account for these imperfections, the absorber path functions (5.1) are generalized to the following, 11436941): 1’ (3)23?“ 29:5st i=1,2 (5-3) .i where the 9’s are used to quantify the the deviations from the ideal path. Note that all g’s are assumed to be small in magnitude in the following analysis. In order to have control on the system dynamics, it is also worthwhile to examine the effect of intentionally mistuning the linear frequencies of the absorbers relative to their design value of n/2. To incorporate this mistuning, the absorber path describing functions are reformulated as 2 . x.(8.;eaj,Awe) = 1 - [(g) + M] S? - 29:58?» i: 1’2 (5-4) ,- where the Aw’s represent the intentional mistuning for each path. For simplification equation (5.4) is re-expressed as . n 2 . . $;‘(3;';6;'j) = 1 — (2) s? — $6,533, i=1,2,. (5.5) where 6,,- = 9;,- for all i, j, except 6.2 = 9.2 + Aw,- for i=1, 2. This split may appear to be artificial, but the idea is that the parameters Aw,- are to be designed into the path, whereas the ggj’s are small and generally unknown perturbations in the path. From equation (2.8), it should be noted that the value of the function g,(s,-) must be kept real during absorber motions. This leads to a restriction on the amplitudes 70 of the absorber motions. For the case when all mistunings and imperfections are small, 6,, << 1, the aforementioned restriction is approximated by 4 nVn2+4’ This restriction, derived by maintaining the g(s)’s real, keeps the absorbers from s,(9) S 3m + 0(6), V 0 and i, where sum, = (5.6) passing the cusp points of the epicycloidal paths. This also imposes a finite operating range on the disturbing torque level F9. For the case of perfect absorber paths it is given by . 7 212V F < P = ——-, 5.7 o .. 0.0 n2 + 4 ( ) over which the desired system response given in equations (5.2) can be maintained. (Note that these explicit forms can be given for the subharmonic absorber since the desired ideal steady state response is known exactly.) 5.2 Measures of Performance Two performace measures will be used to quantify the effectiveness of an ab- sorber system. The first is the amplitude of torsional oscillations here represented by its peak angular acceleration at steady state, denoted by ||yy'||ss. The second performance measure is the range of the applied torque amplitude over which the absorber can operate, denoted by F0. A complete description of these two perfor- mace measures was provided in section 2.6. Note that for the perfect, undamped subharmonic absorber, ||yy'||ss = 0 and the corresponding torque range is given by I‘m in equation (5.7). One of the main goals of this work is to determine ||yy'||ss and the generalization for condition (5.7) for the damped, imperfect system. These results will point out some limitations that are imposed on the subharmonic absorber system by parameter uncertainties, but it will also offer the designer some flexibility in designing the path to achieve certain goals. 71 5.3 Scaling and Reduction of the Equations of Motion Approximate solutions are sought for the damped and imperfect system by mak- ing some scaling assumptions and employing asymptotic analysis techniques. This is accomplished by first utilizing the definition of the small parameter c in equa- tion (3.5) and the scaling (3.6) used for a different case in section 3.1.1. With the definition of e E V, the small imperfections and mistunings can be scaled by 8,3 = 63,5 Vj, and 2 = 1,2. (5.8) Note that typical values of the 5,3’3 are less than one percent, whereas V may range from one to ten percent. The conservative assumption (5.8) is made in order to incorporate the effects of imperfections and mistunings in the first order analysis. The unperturbed system dynamics for this scaling are determined by considering equation (2.12b) with c = 0, that is, V = 0, which yields y = 1. Using this in equation (2.12a) with [1,, = 0 yields a linear oscillator with frequency 72/2 for the absorber motion. Thus, the steady-state solution of the unperturbed system is simply a constant rotor speed, y = 1, and the absorber motion is harmonic with frequency 11 / 2 and arbitrary amplitude. The rotor acceleration can be derived by following the same procedure used for the case with N tautochronic absorbers described in section 3.1.2. This gives the following expression for the acceleration, I 1 2 n2 I n 2 dg°(s-), ~ , 3131(9) = -€{§§(—'2—3431‘ (‘2') 90(3j)31+'—d;J—3j2)—I‘osm(n0)} +0(e")- (5.9) where . 4 2 4 90(86) = 96(3650' = 0) = \]1 - (115:1) 342, i=1,2. 72 Likewise, a set of weakly coupled, weakly nonlinear oscillators for the absorber dy- namics can be obtained, given by 2 s,- + (:21) s,- = ef,(s;,sg,sl,sz,6) + 0(62), i=1,2 (5.10) where f£(31,82;3i,3;a9) = -fla8;-hi(34) I 1 2 n2 I n 2 6190(3) I o 0 +134+9 (3i)l[§j§(--Z-Sjsj - (5) g (5031' + 'Tjj'sj? —I‘gsin(n0)], h,‘(S,-) = lzjgng;j-l. 2 j Remarks: 0 This system has two degrees of freedom with a 1:1 internal resonance. In addition, the excitation is in a 2:1 resonance with respect to the absorbers, and it is of parametric form. In this regard, the system is very similar to that considered by Yang and Sethna [67]. o The effects of the imperfections and intentional mistunings are present in the function h’s which results from the term fi—ffy in the equation of motion (2.12a). 5.4 The Averaged Equations In this section some standard coordinate changes are first carried out which put the equations in the desired form. Averaging is then applied, and this followed by a discussion of the system parameters which appear in the averaged equations and by a presentation of a modified form of the equations for a special scaling of the imperfections. With these forms of the averaged equations in hand, the search for approximate steady-state solutions is carried out in the following sections, the results of which are used for performance evaluation. 73 5.4.1 The Periodic Standard Form A linear coordinate transformation between absorber displacements is first used to simplify the ensuing analysis. This transformation splits the leading order system dynamics into two invariant subspaces, representing the unison motion and its com- plement. A subsequent transformation to amplitude/ phase coordinates will render the desired form. The first transformation is given by 5:81:32 and 0:31;”. (5.11) Substituting transformation (5.11) into equations (5.10) yields the following trans- formed equations of motion 2 “ I I 4"+(-’5) 4 6fe(E.€.n.nI0)+0(€"). 2 n” + (§)2n = 4.2.6.444) + 0(4), 2 s .- s N, (5.12) where f}(€I€'InIn'I9) = 4246' — $121“ + n) - $412 (5 - 77) + [6' + %QO(€+'7)+ gym-77)] Y(€+17I£ -nI9). fII(€I€'InI0'I9) = -flan' - %h1(€+ 7)) + $142 (5 - n) + [n'+ 59°(€+n) - $9005 -n)] Y(€ +2.5 - m9), Y(31I32I9) = %:(—%23j3; — (%)290(si)31+ (19:?)3?) —f‘-gsin(n(9). (5.134) Next, the polar coordinate transformation given by 110 I , 0 4: r. «>sz — 7). 4 = m slum — ”7). n0 I . n0 1] = r,, cos((,0,, - —2—), 7} = my, sm(<,o,, - 3). (5.14) 74 is applied. Substituting the above transformations into equations (5.12) yields a set of first-order differential equations which describe the dynamics of 7'6, (p5, r,7 and cpn in the periodic standard form [39], as follows, I 26 A n9 1‘5 = 7F€(r5799€arm‘19mg) Sil’l((p§ — _2—) + 0(62)7_ (515a) . 2e . n0 2 1’60: = ;F£(réa9°£wrm$0m6) 6030?: - "2") + 0(6 ), (5°15b) I 26 e , 120 r77 = :Ffl(r€a ‘péa rm ‘Pm 6) SH“??? - '2_) + 0(52), (5.15C) . 2e . n0 2 rfl‘pn = :Fiio‘é? (P5, rm (pm 0) COS“??? - 7) + 0(6 )a (515d) where the functions F5 and i}, are simply f5 and fl, expressed, respectively, in terms of coordinates re, 90:, r,, and (pm as obtained by incorporating transformation (5.14) into f: and f". Equations (5.15a) to (5.15d) are in the desired form for averaging. 5.4.2 Application of Averaging Considering only the first order terms in e in equations (5.15), averaging is per- formed in 9 over one period of the excitation, 5;”. The resulting averaged equations are expressed in terms of the first-order averaged variables Pg, 955, F" and 95". Due to the complicated nature of the system, this process results in many terms in the form of integrals which do not yield closed-form expressions. In order to obtain simplified, approximate estimates of the rotor acceleration and the operating torque range, it is assumed that the oscillation amplitudes of the absorbers, that is, 1": and F", are small and of the same order, denoted by 0(F). The averaged equations are then expanded in terms of fig and F". This yields the following set of truncated, averaged equations, where each has been expanded to the desired order, 0(F3), _ ~ - _ g n . _ _ _ . ._ dé — 2 parf ( + 2n ’7 + 2n 8111(CPE — ‘1077) + Zror£ $111280: +cnlfy": sin(2g55 — 2957,) + 0(1‘5), (5.16a) 75 _ ~ ‘ - 38 ‘3 95 ‘2" _ 61905 _ 6&2 n 7": _ 6’72“? + iii + Jig-L" cos(c,5€ — 95,7) 2n 2n 1~ _ _ _ _ _ _ - +ZI‘gFg cos 2956 + Cn1T£T3COS(2(,9€ — 2%,) + c1127“;3 + 61.372732, +065), (5.16b) df -1,_ 5 F 35,?3 354527} , _ _ 1~_, _ :51 = 7““ ‘ ( "2 i + 2'23 + "2.." SW» - w) + 1mm... +cnlfn1"? sin(2,a,, — 2%) + 0(55), (5.16c) _d‘ 5 _ 5 F 35.?3 95,525: _ _ 1~ _ _ ”-28%! = ——’:—2r,, — ( ":1: + 2"”5 + fi— cos(<,c>,7 — (pg) + ngr" cos2 0, the stability of the SMl and SM2 solutions can be determined entirely by the signs of DA and DB. Furthermore, due to the fact that [1,, > 0 no Hopf bifurcations occur from SMl or SM2. The stability for each branch on SMl and SM2 is now determined. Utilizing transformations (5.18) when necessary, DA and DB can be derived. For SMl, Dm = :tcngfgfl/Fg—4fig. (5.28) Since D41- is negative on the branch SMl', this leads to one positive eigenvalue, and thus SM]. is always unstable. For the branch SM1+, DA” is positive and this leads to negative eigenvalues. Thus, the stability of SM1+ must be determined by the sign of D3”, which is given in Appendix I. It can be shown that for 5: small, D3” is positive. Hence, the branch SM1+ is stable. 85 For SM2, DB... = twig/1534,13. (5.29) Applying the same approach used for SMl yields the following results. The branch SM2" is always unstable and the stability of the branch SM2+ is determined by the sign of DA“, which is given in Appendix J. It can be shown that for 55 small, DA“ becomes negative at a level of F9 denoted by F3, at which point a secondary bifurcation occurs. An example of this is shown in Figure 5.1, where SM2+ is unstable for F9 > F3. 5.5.3.3 Range of Validity Based on condition (5.6), only a finite torque range is valid for each branch, in order to keep the motions of the absorbers below the cusps. Only stable solution branches are considered, since these will dictate the steady-state system behavior. For the solutions on SMl, only the stable branch SM1+ is of interest. Using equations (5.23), a condition can be determined such that point A in Figure 5.1 is above the cusp amplitude, thus violating condition (5.6). This condition is given by ~ 2 652 > 2726112 - 94—, (5.30) which, if satisfied, implies that no stable SMl solutions are valid. For the case with n = 2 and small 864’ the R.H.S. of the above equation is approximately -3/4, and thus the condition is satisfied for any realistic value of 59. The same argument sustains for different values of n. It is thus concluded that for small 55, the stable solutions on the branch SMl do not correspond to legitimate steady-state responses for the equations of motion (2.12). This result is largely due to the “internal mistuning” mentioned in section 5, since the term “—n2/4” in the R.H.S of inequality (5.30) results from the effect of internal mistuning. 86 On the other hand, it is seen from equations (5.25), representing the SM2 single- mode solutions, that internal mistuning has no effect (to leading order) on the out- of-phase responses. This fact actually allows the stable solution SM2+ to be valid up to a torque level denoted by F 9, at which the absorbers hit the cusps. (Note that F9 herein is a rescaled version of F 9 defined in equation (5.7); i.e., F9 = 6F9). Based on the solutions given in equations (5.25) and the restriction on the the absorber motions given by the approximation in equation (5.6), F 9 can be approximated by .. ~ 2 % 2n 96554 4652 4 1’ —— 4 . . 9 [(17.2 + 4 + n3(n2 + 4) + n ) + ”a (5 31) "ju This limit is now compared against the secondary bifurcation torque amplitude, ~3, described in the previous section. Utilizing the information given in Appendix J, this torque can be numerically computed and compared with equation (5.31). It is determined that F; > F 9 over the following ranges of the mistuning parameters: 592 E {-0.03, 0.03] and 594 E {-0.03, 0.03]. Therefore, the important conclusion is reached that the SM2+ responses are stable all the way out to the cusp amplitude for realistic values of imperfections. The stable SM2+ branch is central to the effectiveness of the subharmonic vi- bration absorber system, as described in section 5.6. 5.5.4 Coupled-Mode Solutions The existence, stability and range of validity of the coupled-mode solutions are now considered. 5.5.4.1 Solution Branches and Their Stability Observing the averaged equations (5.20), one can first classify all possible steady- state solutions into two distinct groups: the first satisfies sin(2<,5£ - 2%,) = 0 and 87 the other does not. Solutions in the group with the property sin(2¢£ — 2%,) = 0 are sought first. This property implies cos(2¢£ — 2%,) = :tl, which enables one to solve the averaged equations (5.20) for steady-state solutions. As a result, eight steady-state solutions are found, given by 1I515 = .4(cn2cn4-:(|C‘n3icm)2)l " + gal:l + W552 + (Long—cm + 1) (f3 — Milli] ’ Fr2215 = _4(cn2cm+cfi::3:tcn1)2)l [-1652 + b%1(n+ 1%!) + ($155“ + 1) (P3 — 4fl<21’)%] 5526 = [4(c.2c..+i(‘c‘.3:tcm)2)l _" + 4‘53 + W562 + (2‘53“ + 1) (f3 " 4’13) i] . = [........:z*:.......:.] + ‘C"3::"” (2fla)+, due to the effect of the internal mistuning. In the following, “Fl :2 0” will be applied to find possible post-bifurcation solutions. In the post-bifurcation stage, the system might converge to any steady-state solution with non-zero components of fig, 2 S 2' S 4. To classify these solutions, the following sets of indices are defined 73 E {ilolimF,-(6)=0,2SiS4},and A7 a {ilolimf,(0)¢0,2SiS4}, (6.12) which contains those indices corresponding to zero and nonzero steady-state ampli- tudes, respectively. For those F,- with i in Z, the solution for the steady-state phase (,5.- is arbitrary. For the remaining Pg’s, that is, those with i in A7, it can be assumed 119 that the corresponding phases are identical; i.e., 95,- = 95,-, Vi, j 6 A7 (one can utilize a procedure similar to that given in Appendix E to justify this assumption). Apply- ing the above results and “F1 2 0” to equations (6.11c) and (6.11d) yields that the post-bifurcation solutions must satisfy —1 1~ . 0 = —2—~O+Zl"gsm2<,5, (6.13a) 0 — —&12+1F 02' —1-—\II(7=--f 1" F ‘ ) ‘e/V (613b) — n 4 0C 5 99 3272. :12, 3) 4,014 a 2 a ° where cfi = ,a,, ieN, and (6.14a) ‘I’(F5;F2,F3,F4,6’14) = 3n4F§+3n4F§+3n4F3—2n47"21"3-2n47“3i‘4—2n41‘2i‘4 +192&,,(4r~3 — 3522*.- - 373.7".- — 3m.) +144614(i~§ + r; + F3 + 25213 + 213?. + 2mm). (6.14b) Equations (6.13) lead to \I’(fi; f2, 713,714, 514) = ‘1»!(7-‘1'; 7:2, F31 F4, 614), zaj 6 JV. (6.15) Note that equation (6.15) is automatically satisfied for a system with zero fourth- order imperfection (that is, 514 = 0) due to the invariance of the function \II(F,-; F2, F3, F4, 0) under arbitrary exchanges of [$2, F3,F4]. In this case, there exist an infinite number of steady state solutions (at this level of approximation) which lie on an ellipsoid prescribed by 8° = {[f2,f3,F.]I(f.-; F2,1"‘3,F4.0) = 0}, (6-16) where (f,-;1‘~2,1‘-3,F4,6r14) = —32612 + 8n(f‘g — 4,03% — won; 52, 13,174,614). (6.17) However, in practice, the fourth-order imperfection 614 is always a small, nonzero quantity. In the following, the possible steady-state solutions are sought by solving 120 equations (6.13) with assistance from a graphical interpretation in the phase space of the dynamical system. For each i, equations (6.13) are satisfied for any solutions on the ellipsoid 8‘ = {[7‘2,7_‘3,7‘4]|‘1’(fi; 62.63.27... 61.) -- 0}- (6-18) One should note that for any steady-state solution, it must satisfy equations (6.13) for all i 6 A7 simultaneously. Hence, all possible steady-state solutions are the intersection points of the 8‘,i 6 A7; i.e., the set 5 = fl 8" (6.19) ilef contains all possible steady-state solutions. Figure 6.1 interprets the graphical re- lationship among the aforementioned ellipsoids, where the case with JV = {2, 3} is used for the sake of a clear presentation. It is seen from this figure that with a small, nonzero 514, each ellipsoid 8‘ is slightly distorted away form 8° but in a different preferable direction for different i. This results in only finite number of steady-state solutions, which lie at the intersection points of the E‘,i 6 A7, denoted by points Ij, lSj S 4, in the figure. Based on equations (6.18) and (6.19), the aforementioned intersections, i.e., the steady-state solutions, can be found by solving @(fiW-‘zafaaflfiul = (DU—"j; 772.53.54.51“) = 0, 2',j 6 A7, (6.20) which automatically satisfies equation (6.15). It can be shown that the mode shapes for all solutions in the set 8 can be found by examining equation (6.15). They are listed in Table 6.1, where the corresponding isotropy subgroup is used for classifica- tion. It is seen from this table that there exist only three distinct types of solutions: S; x 83, 51 x 33, S; x 81 x 82. The existence of two different mode shapes for the 81 x S; and 81 x S; x 82 solution branches is due to different choices of .51. In fact, Figure 6.1: The graphical interpretation of the distorted ellipsoids. Isotropy Subgroup Mode Shapes of [F2, F3, F4] S; x S, [5,5,0] S, x S; [5,5,5] or [5,0,0] 81 x 31 x S; [F, —1",0] or [7'3 1", 21“] Table 6.1: The solutions branches classified by their isotropy sub- groups and their mode shapes. they are dynamically equivalent. Figure 6.2 depicts the typical responses in the time domain for these solutions. With the mode shapes in hand, the steady-state solutions can be obtained by solving equations (6.20). Furthermore, by numerically evaluating the Jacobian of the truncated, averaged equations (6.11) numerically at these solutions, one can 122 Time Time Time 02='3=N 8 T-1ll 03 = —J‘ SgXSz 51X83 51XS1X53 Figure 6.2: The mode shapes of the steady-state solutions with var- ious isotropy groups detect the corresponding stabilities for each steady-state solution. 6.4.1.2 Absorber Performance and Design Guidelines In this section, the absorber performance will be evaluated by computing two performance measures based on the solution branches and their stabilities. Fur- thermore, generic design guidelines for the absorber paths are given based on the predicted performance. However, due to the high multiplicity of the post-bifurcation solutions and the complexity of the corresponding stability boundaries, closed-form representations of the two performance measures are not pursued in detail. The de- sign guidelines are distilled through a general discussion based on a representative case study. Based on a number of simulations, the design guidelines presented later are robust over a wide range of system parameters. Figures 6.3 to 6.5 show the stability and feasibility boundaries for the representa- tive case, for the solutions with the three different mode shapes, as functions of the two imperfection parameters, 612 and 614 (612 and 614 denote the unscaled quanti- 123 ties of 6;; and 6M; i.e., 6;,- = 66,5). The common system parameters used for this study are T; = 0.035, [1,, = 0.005, n = 2 and u = 0.1662 (The latter two parameters are taken from the 2.5 liter, in-line, four-stroke, four cylinder engine considered by Denman [16]). In this figure, “S” and “U” denote stable and unstable regions re- spectively. Also, the dashed line divides the feasible and infeasible regions, where the corresponding absorber motions hit or not hit the cusps. It is seen that among the three types of solutions, a large set of the S; x S; solutions survive as stable and feasible; a small set of the S; X S; solutions lead to stable and feasible motions; only a tiny set of the S; x S; x S; solutions are stable and feasible. Based on simulations, all the stable and feasible steady-state solutions in areas “abc” and “def” in Figure 6.4 and 6.5, respectively, are unrealistic since the absorber motions will most likely hit the cusps during the transient responses. Thus, they will not be considered for absorber performance evaluation. As results, only stable, feasible S; x S; and S; x S; solutions in Figure 6.3 and 6.4 are the candidates for performance evaluation in the following. Figure 6.6 show the contours of the rotor accelerations with a fixed F9 = 0.035, for all stable and feasible solutions, except for those in areas “abc” and “def” in Figure 6.4 and 6.5. Figure 6.7 and 6.8 show the feasible ranges of the disturbing torque for the S; x S; and S; x S; solution branches respectively. Note that the expression of the rotor acceleration in equation (6.5) and the limitation of absorber motions in inequality (6.4) are used to here generate these figures. It is seen from Figure 6.6 to 6.8 that small, positive &;2’s and &;4’s, leading to the S; x S; branches, have good balance between achieving small rotor accelerations and rendering larger feasible ranges of the disturbing torque, while small, positive 632’s and small, negative &;4’s, leading to the stable, feasible S; x S; branches, render larger rotor accelerations, smaller torque ranges and vulnerability for absorbers to hit the cusps during the transient response. 124 0.02 I I j I 0.015 0.01 - U S 0.005 Feasible -0.005 -0.01 -0.015 Feasible \\.|/ _- _ -0.02 -0.01 O 0.01 012 Figure 6.3: The stability and feasibility boundaries of the solutions with isotropy subgroup S; x S; for Pa = 0.035, it; = 0.005, n = 2 and u = 0.1662. 125 Feasible 0.02 1 I 0.015 (I) 0.01 0.005 -0.005 r I -0.01 -0.01 5 -0.02 J Feasible / .032 -0.01 0'12 Figure 6.4: The stability and feasibility boundaries of the solutions with isotropy subgroup S; x S; for Pa = 0.035, [to = 0.005, n = 2 and V = 0.1662. 126 Feasible 0.02 / I / L 0.015 \‘d; 0.01 - - Feasible \ l x -0.02 -0.01 O 0.01 Figure 6.5: The stability and feasibility boundaries of the solutions with isotropy subgroup S; x S; x S; for I}; = 0.035, [1,, = 0.005, n = 2 and u = 0.1662. 127 \1 Figure 6.6: The contours of the rotor acclerations for F; = 0.035, it; = 0.005, n = 2 and u = 0.1662. 128 Figure 6.7: The feasible disturbing torque range of the S; x S; solu- tion branch for [1,, = 0.005, n = 2 and u = 0.1662. 129 0.02 0.015 0.01 0.005 —0.005 -0.01 -0.015 -0.02 -0.01 0 0.01 Figure 6.8: The feasible disturbing torque range of the S; X S; solu- tion branch for [1,, = 0.005, n = 2 and u = 0.1662. 130 6.4.2 Arbitrary Pairs of Absorbers The similar analysis can be conducted for a system with an arbitrary number of pairs of absorbers. The following conclusions are drawn from considerations of results obtained from analysis and simulations of systems with one, two and three pairs of absorbers (the later results are not presented here). 0 As the number of absorbers increases, the number of existing solution branches increases; e.g., there exist five distinct mode shapes of solution branches for N = 6. This fact complicates the analysis. However, only the solutions with isotro'py subgroups SN]; x SN/2 and S; x SN_; survive as stable and feasible solutions without vulnerability of violating the cusp condition. 0 The dependence of the absorber performance, in terms of the two measures, on the imperfection parameters 6;; and 6;.; is generically similar to that for N = 4. Positive, small 6;; and 6“, leading to a stable SN); x SN); response, are suggested to render smaller rotor accelerations and keep the absorber motions within the cusp levels. 6.5 Remarks and Design Guidelines The results given above are based on scaling assumption (6.10), which says that the differences in the paths are even smaller than the general levels of imperfections and mistunings. Based on simulations, similar to the results obtained for the case N = 2 in the last chapter, the existence of small nonzero 6,5,2 S i S N, slightly decreases the applicable torque range and increases the rotor acceleration. F urther- more, odd (6,5, j = 1,3, ..... ) and higher (6,,-, j Z 4) order imperfections have no distinguishable effects on the absorber performance. 131 In summary, the above results indicate that the following general guidelines be followed when designing the paths for a multi-pair subharmonic absorber system: 0 The absorber paths should be kept as identical as possible. 0 The imperfection parameters 6;; and 6;; should be selected to be small and positive. 0 One can refer to the predicted dynamics and performance evaluations (for example, represented by Figure 6.3 to Figure 6.8, for N = 4) in order to choose values of 6;; and 6;; for a particular specification in terms of vibration level or torque range. CHAPTER 7 CONCLUSIONS AND FUTURE WORK This study focused primarily on investigating the nonlinear dynamics of a rotat- ing system with multiple centrifugal pendulum vibration absorbers (CPVA’s). It is motivated by the fact that in practical implementations the total absorber inertia needs to be divided into several absorber masses that are stationed about and/or along the axis of rotation, due to spatial and balancing considerations. If all the absorbers move in exact unison, the absorber designs suggested by researchers in the past sustain. However, in chapter 3, through a proposed methodology including proper trans- formations and the method of averaging, it was shown that for certain absorber de- signs the unison motion of N identical absorbers may become unstable at a moderate level of the disturbing torque. The corresponding stability criterion was derived. it was found that the critical disturbing torque level is proportional to the square root of the absorber damping when viscous damping is assumed. In chapter 4, the post-bifurcation performance of the absorbers was evaluated by solving for the resultant system response via symmetric bifurcation theory. With the ability to compute two performance measures, the rotor acceleration and the applicable range of the disturbing torque, the absorber absorber performance was re-assessed. It was found that the distribution of the absorber inertia results in a drastic decrease of the disturbing torque range and a slight decrease of the rotor acceleration. Utilizing the same methodology, in chapter 5, the next effort was given to an investigation of the effects of path imperfections and mistuning on the dynamics 132 133 of a system with a pair of subharmonic absorbers. Based on the analytical results obtained, it was found that the effects of path mistuning and symmetric imperfections dominate non-symmetric imperfections, due to resonance effects. It was also found that differences between the two paths of the absorbers have a generally deleterious effect on system performance. Furthermore, by neglecting higher-order dynamics, the average mistunings and imperfections at 2nd and 4th order can be used to design systems that trade off between the operating range of the system and the level torsional vibrations. In chapter 6, the study in the previous chapter was extended to the case with multiple pairs of subharmonic absorbers. As found in the case with a single pair of absorbers, the average mistunings and imperfections dominate the absorber perfor- mance due to resonances. It was also found that the response in which the absorbers move in two unison groups, half in each and exactly out-of-phase, is the ideal re- sponse of the system in terms of absorber performance. Specific design guidelines for absorber paths were distilled based on some case studies and simulations. The analytical work presented in this dissertation is part of a larger framework for absorber dynamics analysis. Listed below are the additional specific problems to be investigated in the future, which include analytical and experimental studies. 0 Circular Paths The intentionally-mistuned circular paths are easily manufactured and widely used in industry. A recently completed perturbation analysis and simulations [55] have shown that for the single-absorber and damped system, a rotor fit- ted with absorbers riding on “mistuned” circular paths can exhibit excellent performance in terms of vibration reduction in the large torque range, even though perfectly tuned tautochronic absorbers are more effective in the low and moderate torque operating range. A systematic analysis on the damped and multi-absorber system is needed for further generalization of the aforemen- 134 tioned results. Multi-Harmonic Torque Inputs In most applications, the applied torque on the rotor is not a pure harmonic. For IC engines, the torque acting on the crankshaft is generated by the gas pressure in the cylinders and through the inertial affects of pistons and other moving components as they transmit torque through crank throws and con- necting rods. This torque is periodic in the rotating angle of the crankshaft, and enters the system equations as a complicated combination of external and parametric excitation which is dependent on the velocity and the acceleration of the crankshaft. However, it can be approximated well by its first several harmonics. To reflect this fact an analysis for systems under general types of multi-harmonic excitation is needed. Nonstationary Conditions There are no reports in the literature dealing with the potential problems as- sociated with transitions in rotational speed; all analyses have been carried out for the case in which the average rotor speed remains fixed. The results are applicable to the aircraft and helicopter applications, but for other appli- cations, e.g., automotive engines, speeds vary in many different ways. It is of interest to consider the effects of such nonstationary operating conditions. Relevant analytical work should be possible since the time scale for the speed changes is slow compared to that of rotation. Herein, asymptotic methods for nonstationary problems may be applied [37]. Flexible Shafts In all aforementioned analyses, the rotor is assumed to be perfectly rigid. How- ever, a system with large but finite rotational flexibility may exhibit different dynamics, especially when the absorbers and the applied torques are distributed 135 along the axis of the rotor. This design strategy is actually used in powertrains to maintain balance and distribute stress along the crankshaft. To investigate the related dynamics, the natural oscillating frequencies of the rotor would be first assumed much larger than that of each absorber subsystem; i.e., the crankshaft is much more “stiff” than each absorber. The equations of motion can then be re-arranged into a singular perturbation form in which the global dynamics is decomposed into fast and slow components, captured by invari- ant manifolds in the phase space subsystem [19, 8]. Our task is to determine the parameter ranges for which the long-time behavior of the system can be simply described by the slow manifold, in which case the crankshaft torsional dynamics are negligible when compared to the “soft” dynamics of each absorber system. However, it is expected that the effects of finite flexibility will have a similar role to that of imperfections, thereby leading to the realistic possibility of localization of absorber response. Experiments In all aforementioned theoretical developments of CPVA systems, the system dynamics is idealized in several aspects in order to obtain analytical estimates of system behavior. For example, the damping is taken to be small and of viscous type and the dynamic effects of rollers are ignored in modeling, to name just two of several. An experimental device needs to be built in the laboratory to verify the validity of the designs offered by the analytical models, and also to provide a measuring stick for the discrepancies between desired / predicted absorber performance and reality. Borowski et al. [3] conducted an experimental study in which it was demonstrated that attaching a CPVA system to an automotive engine crankshaft can actually decrease noise and vibration levels inside a car, but their conclusions were based on qualitative measures from the passengers’ feelings. To evaluate the absorber performance on more solid ground, we need 136 to acquire quantitative measures of vibration levels by building an experimental model in the laboratory and carrying out systematic, controlled experiments. The challenging parts of the experimental buildup would be: (1) precision control of the manufacture of the absorber paths; (2) dynamical measurement of absorber motions; and (3) measurement and quantification of dissipation mechanisms. APPENDICES ““"F‘T‘E‘I L1 [5" THE EXPRESSIONS FOR 111 The terms a—Hl 3H; 5;? 8H; 0% 0H; '5; 0H; 31‘,- ’0’. 8.8. 8.8. 8% 1 8.8. 3 8H; 8r. 1 APPENDIX A 8111 @111 and __20H 8“" 5.5., 8r,- s.s., 8501' 5.5. Brj 8.8. 8H; 8H; . . 3.3., 8ng 8.,_ and Br: [8.8. In (316) are given by g + (n2 + n4)2r4 _ 6H; 690.‘ 1 8(n2 + n4)2r5 __ 6H; 81‘; 8.8. mo [(1 — (n2 + n4)r2) — (3(n2 + 11.“)21"4 — 12(n2 + n4)r2 + 8)] mIH [(n2 + n“)2r4 + 4(n2 + n“)r2 — 8 + 8\/1 — (n2 + n4)r2] The above results were obtained using contour integrals and the residue theorem. 137 APPENDIX B ON THE EIGENVALUES OF C In section 3.3, we claim that if C2Nx2N is a block matrix of the form r . A2x2 82x2 B2x2 B2X2 A2x2 82x2 CZNX2N = a B2x2 LBzxz B2x2 32x2 A2x2 ] then an eigenvalue of [A — B] is an (N — 1) times repeated eigenvalue of C. Further- more, an eigenvalue of [A + (N — 1)B] is an eigenvalue of C. Since the proofs for different N’s are similar, we provide only the proof for N = 4 here. Let A; be an eigenvalue of [A — B] and the associated eigenvector be u, and thus, [A — B]u = /\;u. (B.l) We further define u u u —u 0 0 v; E v; E v; _=_ 0 —u 0 . 0 . 0 . _“ J Based on (B.1), one can verify that CD] = A101, 002 = A102, and 0’03 = A;v3. 138 139 Since 12;, v; and v; are independent, an eigenvalue of [A— B], A;, is a 3 times repeated eigenvalue for C. Let A; be an eigenvalue of [A + (N — 1)B] and the associated eigenvector be w, and thus, [A + 3B]w = A2w. (B.2) We further define F . w w 124 E w 1 w 1 Similarly, one can verify that 004 = A204 based on (B.2). Thus, an eigenvalue of [A + 38] is an eigenvalue of C. Note that 1);, 12;, v; and v; are independent regardless of the choices of the eigenvectors u and w. Furthermore, we know that [A — B] and [A + 3B] have two sets of independent eigenvectors (11;, 11;) and (w;, w;), respectively, and each pair of eigenvectors spans R2. Hence, the following eight eigenvectors spans R8: . 1 . 1 . . .. . . . - . U1 112 111 112 U] 112 w; F 102 .111 —u2 0 0 0 0 w1 “’2 0 0 -u; —u; 0 0 w; w; 0 0 0 0 —‘U1 —U2 “’1 “’2 . .. . . L . l . - . .l . .J .1 We can conclude that two eigenvalues of C are the eigenvalues of [A + 33] and the other six eigenvalues of C are the thrice-repeated eigenvalues of [A — B]. The proof is similar for an arbitrary N. APPENDIX C ON THE EIGENVALUES or [A + (N — 1)B] In section 3.3, we claim that all the real parts of the eigenvalues of [A+ (N —1)B] are negative. To Show this is equivalent to prove that its trace is negative and its determinant is positive. From (3.16), they can be determined in series form as follows Trace[A+ (N —1)B] _-_ _fia, (Cl) 122 +112 n4 n6 2 Det[A+(N—1)B] - ——4—+ (T— 71—) r 4 4 + [15n2 +10n4 + 7126 — [13(1 + n2)“ 716—: 6 6 + [(1+ n2)(40n2 +10n4 — 6n6 — 3113.0 + n2)2)] :5; + 0(1‘7). Obviously the trace is negative. (Since 8;; = B;; = 0, this trace is the same as the trace of [A — B]. See Appendix D for the proof of (C.1).) The general proof that Det[A + (N — 1)B] is positive appears to be quite difficult, so we satisfy ourselves here by proving that the sum of the first two terms in the series is positive. Since the function g,(s,~) in (3.3) is required to be real, 7‘ must satisfy r S m. Under this condition, we can derive that the sum of the first two terms in Det[A + (N — 1)B] is positive if 1 n2+fi§ 140 141 This inequality can be proved by the fact 6 4 4 Tl — Tl 2 —2n ~2 n2 + n4 — n — n2 + n4 < 0 < ”a, for ”—1, 2, 3,.” Thus, for sufficiently small 7', it follows that Det[A + (N - 1)B] is positive. APPENDIX D PROOF OF EQUATION (3.17) In section 3.3, we claim that Trace[A — B] = —fi,,. To prove this, we first show that 211’ 1 F;(r)—F;(r) = 51;] cos2x[1—(n2+n4)r2coszx]5d:r o 1 2" 2 4 2 2 l - = 4—/ [1—(n +n )r cos x]2d(sm2z) 7r 0 —(n’+n“) 211’ _1 = —— [r cos(x) sin(x)]2[1 -- (n2 + n“)r2 cos2 1:]de o 27r r613; 3r ' Note that the third step is completed using integration by parts. Incorporating the above result into the expressions for A;; and A;; in Jacobian (3.16), we can show that Trace[A — B] = A11 + A;; = —[1,,. 142 APPENDIX E JUSTIFICATION OF {2,. 9: 21,-, v 2 31,; _<_ N In order to justify the assumption {6,- 2 12,, \7’ 2 S i, j S N, in the post—bifurcation stage (cf. equations (4.8)), the transformation with 17; capturing the dynamics in V and the remaining 77,-( 2 S i S N) capturing the dynamics in W, where all 175’s are orthogonal to each other, is employed in place of transformation (4.1). Then, by also introducing the angular transformation 77,- : g;cos('r,- — n6) and 17:- : ng,sin(‘r,- — n0), 2 S i S N, (El) and proceeding along the usual lines for the application Of averaging, one arrives at the following steady-state conditions, in place of equations (4.8), ~ _ _flaéi F35; - 1, 0 — —2 +—4n srn(2*r,), (E.2a) 1“”. - N—l 3- N - 0 = _flécos(27",-)—(——)2—§,- :52. . (B.2b) 4n 4 i=2 3 where :5 and ‘7' are the approximate (averaged and truncated) versions Of g and T. The above equations give Tg=7:'j, (mod 71') V2,jEN (13.3) By the definitions Of the {g’s and the 1),-’3, each 5.- with i 6 N is a linear combination Of the ng’s with i 6 N. Hence, 1;,- = 16,-, (mod 7r) V i,j 6 N. Now, choose an arbitrary 2’; E N. For all jo E N with 12,-0 = {61-0 + 7r (mod 27f), replace (FEAR-o) I43 144 by (—]6,-0,1,f2,-0) to equivalently represent the signal sic, and then proceed with the analysis in section 4. One finds that the results are the same as those Obtained if $1=§Z1Vidé N is assumed. APPENDIX F PROOF OF Trace[A + (N — 2)B] < 0 AND Det[A+ (N — 2)B] > 0 AS 6 —) 0+ 4 7 In section 4.4.3, it is claimed that on the S; X SN-; branch with 26 2 '3” Trace[A+ (N— 2)B] < 0 and Det[A+ (N— 2)B] > 0 as 6 -+ 0+ near the bifurcation point. Through a nontrivial computation it can be shown that Trace[A + (N — 2)B] = —fl,, < 0, Det[A + (N — 2)B] = 2—56 1{461(11/ )12N2;3‘ + 763(N— 2) (n + 7.82626; +16(N — 2);2:(n2 + n‘)i361cos(1Z-17)1) —8(N — 2)2,2§;(n2 + n4)2;32f6';’ cos(216 — 2.2,) -16(N — 2)(N —1)Np,n3(n'-’ + n4)636lsin(16—161)}. (F.1) On the S; X SN_; solution branch with 16 2 L21, cos(26— 161 )2 %, cos(216 — 216,) 2 0, and sin(16—161) ~ —1 (F2) -7? Thus, Det[A + (N — 2)B] > 0 on this branch as 6 —> 0+ near the bifurcation point. 145 APPENDIX C. THE LOW-ORDER APPROXIMATION OF 99(6) To Obtain the expressions for yy'(9) in equation (4.22), simplification is carried out in two steps. First, it can be shown that n25; + F9 sin(n0) 2 0 (G.1) by incorporating the approximate steady-state solutions for p; and d); in equations (4.7). Second, the remaining term is reduced based on the corresponding truncated steady-state equations (4.8). It can be shown that before the bifurcation the absorber motions undergo unison motion, which yields 277.2 N I I T2311,- = 27126161 i=1 = 7236: sin(2f61 — 2120). ((3.2) After the bifurcation, using the transformations given in equations (4.2) and (4.4) yields 2712 N ’ 3 ~2 . 2 :2 1 N 2 3,6, = n 6; Sln(2¢1 - 2119) — (N " 1) Z P19111910 — 2719) j=l j?“ + Z 2fij5k Sln('¢_)j + #3): — 277.0) . (G.3) M9“ 3?- j?“ Utilizing some trigonometric identities and the approximate solutions given in equa- tions (4.7), one can Show R.H.S of (G.3) = c08(2IZ,- - 2n0){n3isisin(2zZ.-) 146 147 ~ ~ +7.3 2 {2i3.i3,-sin($.- — 227,-) - (N —1>5§sin[2( .-— an} .191” +713 2 26j6ksin(2tb,- - d),- - 11%)} j,k¢1,i & #1: + sin(2gf); — 2n9){n3)~5jC03(2151) + (N " ”"313? +152125.1,cos($.—$,)—(N—is,1) cosINzZ 2%,-)1} 155” +713 Z 2BjBkCOS(2l/;i — 15;“ — {NJ}: 2 S i S N- j,k¢1.i a; #1: (GA) Incorporating the truncated averaged equations in equation (3.12) yields R.H.S of ((1.4) = 2,1,, cos(216, — 21.9), 2 g i g N. (G.5) Based on the results in Appendix B, one finds 26,, cos(216; -— 2710) = 26,, cos(21l—J — 2n0), 2 S i S N, after the bifurcation. APPENDIX H AVERAGED EQUATIONS IN CARTESIAN COORDINATES The truncated, averaged equations expressed in Cartesian coordinates are given by dA 1 6 6, 76$ = 7B6Ae+(—+ £2 3'12)B€+ _Bn +ZPOB£ —anB{(Ag + B?) + CnsAchBn - Cn6B£A727 + 61173ng +3621 2 3 +2” (A, 3, + A23, +33 ,3: + 3, + 2A ,,A,3,) (H.1a) dB —1 6 6 ~ 71076 = 3.2—6,3, - (7:3 + Z) A,E — -1:—A, + ZRA, +Cn2A{(Ag + B?) "' C115861417817 + CnGAfB: _ Cn7A£A72p _3_6fl4 2 2 2 3 2n (3, A, +3,A, +3A, A +A, +23 ,,A,3,) (H.1b) dA -1 6 6, d6" = 2 —,1,A, + ;-—2B, + 72-3—B,+:1~‘98, +Cn4Bn (A927 + 8,2,) + CnSAnAé-Bé — 0116877142 + Cn7BnBza 36"“ A 3 A33 33 32 33 2A 3 +2n,( 5+ ,+ g + ,+ , ,,A) (H.1c) dB —1, 6, 6 702 = 711.8» — 7/15— -5-2A, + -I‘aA -Cn4An (A?7 + Bi) - 0115/153an + 6116/4nt — 0717/19/13 ‘36,; (8,515 + 32A, + 3A,A2 + A2 + 23,A, 3 ,.) (H.1d) 148 149 where 42— 5 6 n n +3“, —— and 6,17: C25: 128 n C26 256 2n 256 Applying assumption (5.19), the above equations become 53% = ‘B—l'fiaAe + (€513 + :11) Be + iffiBE ‘ @1234": + 8:) +cn5A,A,B, — enngAf, + 61173633, 552; = €1,103, _ (Sufi +21) A£+ifoAg + Caz/4:04: + 852) _c,53,A,B, + cue/1:83 - Gav/1:243, d—(gfl = :ZI—flaA, + 3238, + £118, + MBA/4,2, + B3,) +c,5A,A,3, - 6,63,A§ + 6,73,33, 2% = ’71,,3, _ $4, + film, — 664,114: + 83) — Cn5A§B€Bn + (311614nt — 9171417142- _ 12n3 + n5 3654 —4n3 — 3715 + 2n (H.2a) (H.2b) (H.2c) (H.2d) APPENDIX I STABILITY OF SOLUTIONS ON SMl+ Incorporating the SM1+ solutions given by equation (5.25) into the sub-block matrix B in the corresponding J acobian, one can obtain the determinant of B, DB”. It is given by 031+ = _f_§_+%_2+i5:2 (2:61?’)<1 8::>1~>—J m 6“] [<2 22* + 6] 150 APPENDIX J STABILITY OF SOLUTIONS ON SM2+ Incorporating the SM2+ solutions given by equation (5.26) into the sub-block matrix A in the corresponding Jacobian, one can obtain the determinant of A, DA“. It is given by 6 f2 62 n DA2+ = 66- Tg+gf+$+fi C116 + ___Cn-cn("7)g ..2 6' "2 ~2 '6' 4562 —4 I‘ — 4 — — + + 128cm, ~ 2 8%: ~2 ~2 6' 4662 12862,4 (1 fg)[(P9 4") ’T H-2Cn60n7—C 01:6" 562 2_ ~2 1_4_3£3 2% +:)[