Friction-induced Vibration in Lead Screw Systems

Transcription

1 Frition-indued Vibration in Lead Srew Sytem by Orang Vahid Araghi A thei preented to the Univerity of Waterloo in fulfillment of the thei requirement for the degree of Dotor of Philoophy in Mehanial Engineering Waterloo, Ontario, Canada, 9 Orang Vahid Araghi 9

2 Author Delaration I hereby delare that I am the ole author of thi thei. Thi i a true opy of the thei, inluding any required final reviion, a aepted by my examiner. I undertand that my thei may be made eletronially available to the publi. ii

3 Abtrat Lead rew drive are ued in variou motion delivery ytem ranging from manufaturing to high preiion medial devie. Lead rew ome in many different hape and ize; they may be big enough to move a 4 ton theatre tage or mall enough to be ued in a ml liquid dipening miro-pump. Diproportionate to the popularity of lead rew and their wide range of appliation, very little attention ha been paid to their dynamial behavior. Only a few wor an be found in the literature that touh on the ubjet of lead rew dynami and the intabilitie aued by frition. The urrent wor aim to fill thi gap by preenting a omprehenive tudy of lead rew dynami fouing on the frition-indued intability in uh ytem. In thi thei, a number of mathematial model are developed for lead rew drive ytem. Starting from the bai inemati model of lead rew and nut, dynami model are developed with varying number of degree of freedom to reflet different omponent of a real lead rew drive from the rotary driver (motor to the tranlating payload. In thee model, veloity-dependent frition between mehing lead rew and nut thread ontitute the main oure nonlinearity. A pratial ae tudy i preented where frition-indued vibration in a lead rew drive i the aue of exeive audible noie. Uing a omplete dynamial model of thi drive, a two-tage ytem parameter identifiation and fine-tuning method i developed to etimate parameter of the veloitydependent oeffiient of frition. In thi approah the oupling tiffne and damping in the lead rew upport are alo etimated. The numerial imulation reult uing the identified parameter how the appliability of the developed method in reproduing the atual ytem behavior when ompared with the meaurement. The verified mathematial model i then ued to tudy the role of variou ytem parameter on the tability of the ytem and the amplitude of vibration. Thee tudie lead to poible deign modifiation that olve the ytem exeive noie problem. Frition an aue intability in a dynamial ytem through different mehanim. In thi wor, the three mehanim relevant to the lead rew ytem are onidered. Thee mehanim are:. negative damping;. inemati ontraint, and;. mode oupling. The negative damping intability, whih i aued by the negative gradient of frition with repet to liding veloity, i tudied thorough linear eigenvalue analyi of a -DOF lead rew drive model. The firt order averaging method i applied to thi model to gain deeper inight into the role of veloity-dependent oeffiient of frition and to analyze the tability of poible periodi olution. Thi analyi alo i extended to a -DOF model. It i alo hown that higher order averaging iii

4 method an be ued to predit the amplitude of vibration with improved auray. Unlie the negative damping intability mehanim, inemati ontraint and mode oupling intability mehanim an affet a ytem even when the oeffiient of frition i ontant. Parametri ondition for thee intability mehanim are found through linear eigenvalue analyi. It i hown that inemati ontraint and mode oupling intability mehanim an only our in elf-loing lead rew. The experimental ae tudy preented in thi wor demontrate the need for ative vibration ontrol when eliminating vibration by deign fail or when it i not feaible. Uing the liding mode ontrol method, two peed regulator are developed for -DOF and -DOF lead rew drive ytem model where torque generated by the motor i the ontrolled input. In thee robut ontroller, no nowledge of the atual value of any of the ytem parameter i required and only the upper and lower bound of parameter are aumed to be available. Simulation reult how the appliability and performane of thee ontroller. The urrent wor provide a detailed treatment of the dynami of lead rew drive and the topi of frition-indued vibration in uh ytem. The reported finding regarding the three intability mehanim and the frition parameter identifiation approah an improve the deign proe of lead rew drive. Furthermore, the developed robut vibration ontroller an be ued to extend the appliability of lead rew to ae where peritent vibration aued by negative damping annot be eliminated by deign modifiation due to ontraint. iv

5 Anowledgement I wih to expre my gratitude to my upervior, Dr. Golnaraghi. I have benefited greatly from hi guidane and inightful uggetion throughout my reearh. The experimental part of thi wor would not have been poible by without the generou upport of Dr. Erormaz and hi team in the Preiion Control Laboratory. I am epeially thanful to Amin Kamalzadeh and Wilon Wong for their help in the preparation of the hardware and oftware ued in the tet. I would lie to expre my than to my olleague and friend Nima Elaminaab for hi help in the preparation of the tet etup and for hi aitane during many hour of teting. I am alo grateful to him for letting me boune idea off him whenever I needed. I am forever indebted to my wife, Maryam. Without her undying upport and enouragement, thi journey would not have been poible. v

6 To Maryam vi

7 Table of Content Lit of Figure... xi Lit of Table... xix Chapter Introdution.... Lead Srew.... The Audible Noie Problem Thei Overview Contribution... 8 Chapter Literature Review.... Frition-indued Vibration..... Negative Damping..... Kinemati ontraint intability Mode Coupling...5. Lead Srew Drive... 8 Chapter Mathematial Modeling of Lead Srew Drive.... Lead Srew and Nut A Kinemati Pair.... Veloity-dependent Coeffiient of Frition Bai -DOF Model Inverted Bai Model Bai Model with Fixed Nut Bai Model with Fixed Lead Srew Anti-balah Nut....5 Compliane in Lead Srew and Nut Thread....6 Axial Compliane in Lead Srew Support....7 Compliane in Thread and Lead Srew Support A Complete Sytem Model... 5 vii

8 .9 Some Remar Regarding the Sytem Model... 7 Chapter 4 An Experimental Cae Study Preliminary Obervation Step : Frition and Sytem Parameter Identifiation Experiment Reult DC Motor and Gearbox Identifiation Reult Step : Fine-tuning Parameter Studie Effet of Input Angular Veloity Effet of Damping Effet of Stiffne Conluion Chapter 5 Negative Damping Equation of Motion Eigenvalue Analyi Numerial Example Firt-order Averaging Aumption Equation of Motion in Standard Form Firt Order Averaging Steady State Solution Numerial Simulation Reult Conluion Chapter 6 Kinemati Contraint and Mode Coupling Intability Mehanim Kinemati Contraint Intability in -DOF Lead Srew Model A Note Regarding the Solution of the Equation of Motion...87 viii

9 6.. Example Region of Attration of the Stable Steady-liding Fixed Point Stability Analyi of the -DOF Lead Srew Model with Compliant Thread Undamped Sytem Damped Sytem Stability Analyi of a -DOF Model with Axially Compliant Lead Srew Support Undamped Sytem Damped Sytem Mode Coupling in a -DOF Sytem Conluion... 8 Chapter 7 Vibration Control Mathematial Model A Note on the Frition Model Parameter Unertainty Sliding Mode Control for Rigid Drive Numerial Example Sliding Mode Controller for Flexible Drive The Sliding Phae The Reahing Phae Feedforward Input Continuou Sliding Mode Controller Variable Veloity Set Point Numerial Example Conluion... 8 Chapter 8 Reult Summary and Future Wor Reult Summary Future Wor Appendie... 9 Appendix A Tet Setup... 9 ix

10 Appendix B Firt Order Averaging Theorem Periodi Cae Appendix C A Definite Integral Ued in Averaging... Appendix D Steady-tate Solution of the Averaged -DOF Lead Srew Model... 4 Appendix E Higher-order Averaging... 6 Appendix F Firt-order Averaging Applied to the -DOF Lead Srew Model with Axially Compliant Support... 5 Appendix G Similaritie in the Condition for Loal Stability of the Steady-liding Fixed Point Between the -DOF Model of Setion.5 and the -DOF model of Setion.6... Appendix H Further Obervation on the Mode Coupling Intability in Lead Srew Drive... 6 Appendix I Theorem Cited in Chapter Bibliography... 5 x

11 Lit of Figure Figure -: Lead rew deign fator... Figure -: Mehing Stub Ame lead rew and nut (ut view. Detail: radial and axial learane. Figure -: A powered eat adjuter...5 Figure -4: Time-frequeny ignature of the ound reorded from the powered eat adjuter...5 Figure -5: Component of the lead rew drive mehanim found in a type of variable volume pump...6 Figure -6: Time-frequeny ignature of the ound reorded from the pump...6 Figure -: Stribe urve [5].... Figure -: -DOF ma-on-a-onveyer model... Figure -: Simple model to demontrate inemati ontraint intability [9]....4 Figure -4: A imple -DOF model apable of exhibiting mode oupling intability [47]....6 Figure -: Lead rew drive ytem... Figure -: Sign onvention for ontat fore between nut and lead rew... Figure -: Effet of lead angle on the meaurement of thread angle... Figure -4: Geometry of the thread on two different etion plane... Figure -5: Fore ating on a thread...4 Figure -6: Veloity dependent oeffiient of frition...5 Figure -7: -DOF model of a lead rew ytem...7 Figure -8: Inverted bai -DOF model...8 Figure -9: Bai -DOF model with fixed nut...9 Figure -: Bai -DOF model with fixed lead rew... Figure -: Lead rew model with anti-balah nut... Figure -: -DOF lead rew drive model inluding thread ompliane... Figure -: -DOF lead rew drive model inluding ompliane in the upport...4 Figure -4: -DOF lead rew drive model inluding ompliane in the upport and ompliane in the lead rew and nut thread...5 Figure -5: A 4-DOF lead rew drive ytem model...6 Figure 4-: Tet etup for omplete eat adjuter...4 Figure 4-: Sample tet reult from omplete eat adjuter tet...4 xi

12 Figure 4-: Audible noie frequeny ontent for the tet reult hown in Figure 4- (at 8. Pea amplitude at 6Hz...4 Figure 4-4: Single-tra tet etup...4 Figure 4-5: Sample tet reult from ingle-tra tet...4 Figure 4-6: Audible noie frequeny ontent for the tet reult hown in Figure 4-5 (at. Pea amplitude at 5Hz...44 Figure 4-7: Shemati view of the tet etup...45 Figure 4-8: Reitive torque of the motor and the gearbox. Dot: meaurement, dahed line: fitted line to the data point...5 Figure 4-9: Colletion of data point howing Torque/Speed/Fore...5 Figure 4-: Sample meaurement reult. Variation of motor torque with applied axial load at ontant peed. Dot: meaurement, olid line: fitted line to the data point...5 Figure 4-: Variation of β with motor angular veloity...5 Figure 4-: Variation of β with motor angular veloity...5 Figure 4-: Identified veloity dependent oeffiient of frition...54 Figure 4-4: Experimentally obtained variation of limit yle vibration amplitude with input angular veloity (gearbox output and axial fore...55 Figure 4-5: a A ample of tet reult howing ti-lip in open-loop tet, b zoomed view. Bla: lead rew angular veloity; grey: DC motor angular veloity...55 Figure 4-6: Meaurement v. imulation example. Input: R 5.9( N, ω G 5.6( rad / - Parameter: 4.8( Nm / rad,.86 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...57 Figure 4-7: Meaurement v. imulation example. Input: R 7.9( N, ωg 4.( rad / - Parameter: 4.( Nm / rad,.7 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...57 Figure 4-8: Meaurement v. imulation example. Input: R 4.( N, ωg 4.( rad / - Parameter: 4.65( Nm / rad,.4 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...58 Figure 4-9: Meaurement v. imulation example. Input: R 7.5( N, ωg 8.7( rad / - Parameter: 4.75( Nm / rad, 4.7 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...58 xii

13 Figure 4-: Meaurement v. imulation example. Input: R.8( N, ω G 7.( rad / - Parameter: 4.88( Nm / rad, 5.79 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...59 Figure 4-: Meaurement v. imulation example. Input: R 6.7( N, ω G 7.9( rad / - Parameter: 4.67( Nm / rad, 4. ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation...59 Figure 4-: Variation of oupling tiffne,, with gearbox output veloity and axial fore...6 Figure 4-: Variation of frition boundary effet, /r, with gearbox output veloity and axial fore...6 Figure 4-4: Variation of lead rew upport damping, ( -, with gearbox output veloity and axial fore...6 Figure 4-5: Variation of frition aling, µ, with gearbox output veloity and axial fore...6 Figure 4-6: Contour plot of the teady tate vibration amplitude v. applied axial fore and gearbox output peed...6 Figure 4-7: Effet of lead rew rotational damping on the threhold of intabilitie. The thi bla line orrepond to the intability threhold in Figure Figure 4-8: The effet of oupling tiffne and axial loading on the dynami behavior of the lead rew - Gearbox output angular veloity 4 (rad/...64 Figure 5-: Loal tability of fixed point of the -DOF lead rew ytem Figure 5-: ytem trajetorie for -4 < r ; untable teady-liding fixed point (,...7 Figure 5-: ytem trajetorie for -4 > r ; table teady-liding fixed point (,...7 Figure 5-4: Firt order averaging reult. -4 Grey: trunated equation of motion; Bla: amplitude of vibration from firt order averaging...8 Figure 5-5: Firt order averaging reult. -4 Grey: trunated equation of motion; Bla: amplitude of vibration from firt order averaging...8 Figure 5-6: Firt order averaging reult. -4 Grey: original equation of motion; Bla: amplitude of vibration from firt order averaging...8 Figure 5-7: Firt order averaging reult. -4 Grey: original equation of motion; Bla: amplitude of vibration from firt order averaging...8 Figure 5-8: Bifuration diagram of the averaged amplitude equation. Stable; Untable...8 Figure 5-9: Effet of Stribe frition on bifuration...84 xiii

14 Figure 5-: Effet of negative damping on bifuration Figure 6-: Evolution of the eigenvalue a the tranlating ma, m, i varied...89 Figure 6-: Sytem trajetorie for ontant µ. (a m < m r (b m > m r...9 Figure 6-: Intability aued by inemati ontraint - DOF model with very high ontat tiffne and damping. (a phae-plane, (b ontat normal fore...9 Figure 6-4: Effet of damping on the untable ytem...9 Figure 6-5: Effet of damping on the teady-tate vibration of the lead rew ytem under inemati ontraint intability...9 Figure 6-6: Effet ontat parameter on the repone of the ytem under inemati ontraint intability...9 Figure 6-7: Sytem trajetorie for R- (N, ω4 (rad/, and r...94 Figure 6-8: Stability of the -DOF ytem a ontat tiffne and oeffiient of frition are varied. m 5 and Rω>. Hathed region: untable... Figure 6-9: Variation of the real and imaginary part of the eigenvalue a the ontat tiffne i varied. µ., m Figure 6-: Stability of the -DOF ytem a ontat tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region... Figure 6-: Variation of the real part (a and imaginary part (b of the eigenvalue a the ontat tiffne i varied. µ.8, m Figure 6-: Region of tability of the -DOF model with damping. Bla: table, white: untable. Rω>, m5,, and Figure 6-: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. µ.5, m 5,, and Figure 6-4: Region of tability of the -DOF model with damping. Bla: table, white: untable. Rω>, m5,, and Figure 6-5: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. µ.5, m 5,, and Figure 6-6: Region of tability of the -DOF model with damping. Bla: table, white: untable, Grey: region of intability of the undamped ytem Rω>, m5,, and Figure 6-7: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. m 5,, and 4-4. Solid: µ.5; dahed µ...7 xiv

15 Figure 6-8: Region of tability of the -DOF model with damping. Bla: table, white: untable, Grey: region of intability of the undamped ytem. Rω>, m5, 4 4, and Figure 6-9: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. m 5, 4 4, 8 -, and µ Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied. m 5 and Rω>. Hathed region: untable...4 Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region...5 Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region, and; the hathed area: primary inemati ontraint intability region....6 Figure 6-: Variation of the real part (a and imaginary part (b of the eigenvalue a the ontat tiffne i varied. 4 6, m 5, and m Figure 6-4: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5,, and Figure 6-5: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m. g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5,, and Figure 6-6: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and, hathed: undamped intability region. Rω>, m5,, and Figure 6-7: Variation of real part of the eigenvalue (a and damped natural frequenie (b a a funtion of lead rew upport axial tiffne,. µ.,.,... Figure 6-8: Variation of the real part of eigenvalue (a and the damped natural frequenie (b a a funtion of the oeffiient of frition, µ Figure 6-9: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5, 4, and xv

16 Figure 6-: Variation of real part of the eigenvalue (a and damped natural frequenie (b a a funtion of lead rew upport axial tiffne,. µ...4 Figure 6-: Variation of the real part of eigenvalue (a and the damped natural frequenie (b a a funtion of the oeffiient of frition, µ ( Cloe-up view of the real part of eigenvalue...4 Figure 6-: a Evolution of the three natural frequenie of the undamped -DOF ytem (with ontant oeffiient of frition a a funtion of and. b Stability map...7 Figure 6-: Loal tability of fixed point of the -DOF lead rew ytem with ontant oeffiient of frition. Bla: table, white: untable...8 Figure 7-: lead rew drive model... Figure 7-: Sytem trajetorie under the ation of the ontinuou liding mode ontroller. Dahed: x ( (, olid: ( (, x ; dahed-dot: boundary layer...4 Figure 7-: Controlled input. Dahed: x ( (, olid: ( (, x...4 Figure 7-4: Sytem trajetorie under the ation of the ontinuou liding mode ontroller. Dahed: x ( (, olid: ( (, x ; dahed-dot: boundary layer...4 Figure 7-5: Controlled input. Dahed: x ( (, olid: ( (, x...44 Figure 7-6: Effet of β on the performane of the ontroller...45 Figure 7-7: Sytem trajetorie under the ation of the ontinuou liding mode ontroller. Dahed: x ( (, olid: x ( (, ; hathed region: bound on the olution given by the tability ondition Figure 7-8: Cloe-up view of ytem trajetorie howing limit-yle behavior. Dahed: x ( (, olid: ( (, x...46 Figure 7-9: A portion of veloity error time hitory...47 Figure 7-: Smoothed abolute funtion...66 Figure 7-: Controller performane of example #; (a lead rew veloity error, (b motor veloity error...7 Figure 7-: (a Variation of the torional defletion of the oupling, (b variation of normal ontat fore...74 Figure 7-: Controlled input of example #...74 Figure 7-4: Sytem trajetory for example #...75 xvi

17 Figure 7-5: Lead rew (a and motor (b angular veloitie of the loed loop ytem. Bla: ytem repone; dahed gray: veloity et point...76 Figure 7-6: Controlled input of example #...76 Figure 7-7: Effet of d on the performane of the ontroller Figure 7-8: Effet of feedforward part on the performane of the ontroller. Gray: without feedforward; bla: with feedforward...79 Figure 7-9: Controlled input of example #4. Bla: with feedforward input; gray: without feedforward input Figure 7-: Performane under the ation of feedforward input alone...8 Figure 7-: Smoothed oeffiient of frition...8 Figure 7-: Lead rew (a and motor (b angular veloitie of the loed loop ytem. Bla: ytem repone; dahed gray: veloity et point...8 Figure 7-: Cloe-up view of the tranient vibration of the lead rew...8 Figure 7-4: Controlled input for example #5...8 Figure A-: Experimental etup for preliminary tet on the omplete powered eat adjuter...9 Figure A-: Data aquiition in Matlab/Simulin environment...9 Figure A-: Experimental etup for the lead rew frition identifiation tet...9 Figure A-4: Intrumentation ued in the frition identifiation tet etup...94 Figure A-5: Sample tet reult. (a Lead rew angular diplaement, (b Lead rew angular veloity Figure A-6: Near teady-tate portion of a ample tet reult. (a Lead rew angular veloity, (b Axial load, and ( Motor torque...96 Figure D-: The veloity-dependent oeffiient of frition....5 Figure D-: Shemati plot of amplitude equation for Cae I...7 Figure D-: Shemati plot of amplitude equation for Cae II...8 Figure D-4: Shemati plot of amplitude equation for Cae III - Senario.... Figure D-5: The ae of two poitive zero of (D... Figure D-6: Shemati plot of amplitude equation for Cae III - Senario... Figure D-7: Firt example, µ, -4. Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity...4 Figure D-8: Seond example, µ., µ Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity...4 xvii

18 Figure D-9: Third example, µ., µ 5-4. Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity...5 Figure E-: Firt, eond, and third order averaging reult. (a Numerial averaging reult; gray olid: nonlinear ytem equation; dotted bla: firt order averaging; dahed-dot: eond order averaging; olid bla: third order averaging, (b bla: meaurement; gray: imulation reult...4 Figure H-: Averaged amplitude of vibration, y (rad, a lead rew upport damping,, and ontat damping are varied...7 Figure H-: Bifuration of Poinare etion. a Along the horizontal dahed line in Figure H- b Along the vertial dahed line in Figure H Figure H-: Effet of ontat tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue...9 Figure H-4: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at 4x Figure H-5: Effet of ontat tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue...4 Figure H-6: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at 9x Figure H-7: Effet of oupling tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue...4 Figure H-8: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at Figure H-9: Effet of oupling tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue...44 Figure H-: Bla line: Evolution of pea lead rew vibration frequenie, dotted grey line: eigenfrequenie...44 Figure H-: Simulation reult for lead rew at 4; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion...45 Figure H-: Simulation reult for lead rew at 7; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion...45 Figure H-: Simulation reult for lead rew at ; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion...46 xviii

19 Lit of Table Table 4-: Known or aumed ytem parameter value...5 Table 4-: Identified parameter...5 Table 4-: Numerial value of the identified parameter...5 Table 5-: Parameter value ued in the imulation...7 Table 5-: Parameter value ued in the imulation...8 Table 6-: Parameter value ued in the imulation...89 Table 6-: Element of the Jaobian matrix for the -DOF model...6 Table 7-: Parameter value ued in the imulation...6 Table 7-: Controller parameter for example #...7 Table 7-: Controller parameter for example #...77 Table 7-4: Controller parameter for example # Table 7-5: Controller parameter for example #5...8 Table A-: Partial lit of omponent of the lead rew tet etup...95 Table C-: Parameter value ued in the imulation... Table E-: Parameter value ued in the higher order averaging example... xix

20 Chapter Introdution Lead rew are ued in variou motion delivery ytem where power i tranmitted by onverting rotary to linear motion. Paaging indutrie, indutrial automation, manufaturing, medial devie, and automotive appliation are ome of the area where lead rew an be found. Lead rew ome in many different hape and ize; they may be large enough to upport and move a 4-ton theatre tage [], lightweight enough to be onidered for wearable roboti appliation [], or even mall enough to derive miro-pump ued in medial appliation for dipening fluid volume of le the ml with great preiion []. The liding nature of ontat in lead rew put great importane on the role of frition on the performane of thee ytem. In addition to effiieny onern, driving torque requirement, or wear, frition an be the aue of dynami intabilitie in the lead rew drive, reulting in elfexited vibration whih deteriorate the performane of the ytem and may aue unaeptable level of audible noie. In Setion., an overview of the lead rew feature and deign apet i preented. Appliation area, benefit, and drawba of lead rew are diued in thi etion. Two real-world example, where the lead rew drive generate unaeptable level of audible noie due to frition-indued vibration, are preented in Setion.. The preent reearh i motivated by thee two example.. Lead Srew A very intereting hitorial aount of the development of rew from Arhimede water nail to the wor of Leonardo da Vini and up to the twentieth entury i given in [4]. In thi referene, geometrial peifiation of tranlating rew a oppoed to fatening rew are preented. In appliation where tranmitting power (rather than poitioning i of primary importane, tranlating lead rew are alo nown a power rew [5,6]. When ued in vertial appliation, thee

21 ytem are ometime alled rew ja []. Set forth by the general ondition under whih lead rew are traditionally ued, the mehani of the lead rew i limited to the fator affeting their tati or quai-tati performane, uh a effiieny, driving torque requirement, and load apaity [4-6]. There are, however, numerou other important apet involved in the ueful deign of a lead rew drive ytem. Some of thee iue are ummarized in Figure -. It i important to mention that, to ome degree, almot all of thee iue influene the other apet of the lead rew deign. Figure -: Lead rew deign fator

22 Starting from the appliation of the lead rew, there are two general area: appliation where tranmiion of power i of primary importane, and appliation where aurate poitioning of the tranlating part i the prime objetive. The former i the fou of thi reearh. In either appliation area, the onverion of rotary to linear motion an be ahieved by either rotating the lead rew or rotating the nut. The two example preented in the next etion demontrate eah of thee two onfiguration. There are a number of thread geometrie available for lead rew that are deigned to addre variou requirement uh a eae of manufaturing, load arrying apaity, and the quality of fit [4]. The mot popular of thee geometrie are the Ame and tub-ame type. See Figure - for the hemati view of a mehing tub-ame lead rew and nut. 4.5 o Axial learane Balah Radial learane Figure -: Mehing Stub Ame lead rew and nut (ut view. Detail: radial and axial learane Manufaturer offer a wide range of produt in repone to divere appliation where lead rew are utilized. For poitioning tage, high preiion ground lead rew with or without anti-balah nut are offered a an alternative to the more otly but muh more effiient ballrew driven tage [7,8]. There are variou deign developed by the manufaturer for the anti-balah nut. Thee nut eentially have two halve onneted with preloaded pring that move with repet to one another to ompenate for balah and wear [7,9-,6]. The drawba of uing thee nut i in the inreaed frition fore, whih lower the effiieny and inreae the required driving torque. In addition to their lower ot ompared to ballrew, there are a number of ditint feature that mae a lead rew drive the favorable hoie in many appliation. Thee feature inlude [-5]: Quieter operation due to the abene of re-irulating ball ued in ballrew, The feature of thi deign are further diued in Setion.

23 Smaller moving ma and maller paaging, Availability of high helix angle reulting in very fat lead, Availability of very fine thread for high reolution appliation, Poibility of elf-loing to prevent the drive from being badrivable thu eliminating the need for a eparate brae ytem, Lower average partiulate generation over the life of the ytem, Elimination of the need for periodi lubriation with the ue of elf-lubriating polymer nut, and Poibility to wor in wahed-down environment. Deign fator given in Figure - are diued by the manufaturer a part of their publi tehnial information or produt eletion guideline (ee for example [9-,6,7]. There i, however, a major exeption: frition-indued vibration. Thi important fator the ubjet of thi reearh ha been barely touhed by the ientifi ommunity, a will be diued in Chapter.. The Audible Noie Problem Thi reearh wa motivated by two real-world example where lead rew are ued to onvert rotary motion into tranlation. In both of thee example, the ytem produe unaeptable level of audible noie under normal operating ondition. The firt example involve the horizontal motion drive of an automotive powered eat adjuter. The omplete automotive powered eat adjuter i hown in Figure -. The horizontal drive i ontruted of two parallel lead rew lider ytem. Torque i tranmitted from a DC motor to the lead rew through two worm gearboxe. Two flexible oupling onnet the gearboxe to the motor and to the two lead rew. The nut are tationary and are onneted to the eat frame. The lead rew lider together with the motor and gearboxe move with the eat a lead rew advane in the nut. In many ae, an extra fore applied (by the paenger in the diretion of motion aue the ytem to generate audible noie, whih i unaeptable to the ar manufaturer. In laboratory tet, under ertain load and travel peed ondition, the eat adjuter produed a ignifiant audible noie with the dominant frequeny of around 5Hz. A ample of thee tet reult i hown in Figure -4. Thi ytem i tudied in detail in Chapter 4. 4

24 Figure -: A powered eat adjuter Figure -4: Time-frequeny ignature of the ound reorded from the powered eat adjuter The eond example i a miro-pump for medial appliation. The lead rew drive omponent are hown in Figure -5. A tepper motor rotate the nut, whih i integrated into the rotor. The rotary motion i onverted by the lead rew to tranlation, whih drive the piton. By moving the piton ba and fort, the pump apirate and then dipene predetermined volume of fluid. Figure -6 how reult of a tet performed on the pump when the fluid line were not onneted. It i intereting to ee that the pump generated noie with different frequeny ontent in apirate and dipene phae. Similar to the ae of the powered eat adjuter, preliminary obervation ugget 5

25 that the oure of vibration i the frition-indued vibration of the lead rew drive. Further experimental and theoretial tudy of thi ytem, however, i not undertaen in thi reearh. Figure -5: Component of the lead rew drive mehanim found in a type of variable volume pump Figure -6: Time-frequeny ignature of the ound reorded from the pump. Thei Overview Thi thei onit of eight hapter and nine appendie. After the preent introdutory hapter, a review of the relevant previou wor i preented, in Chapter. Thi review i divided into two part. In the firt part, Setion., the general ubjet of frition-indued vibration in dynamial ytem i 6

26 reviewed and the three major intability mehanim relevant to the lead rew ytem are tudied. In the eond part, Setion., publiation on the lead rew dynami are reviewed. The mathematial model of lead rew ytem ued in thi tudy are developed in Chapter. Thee model over a wide range of onfiguration. In Chapter 4, an experimental ae tudy i preented. In thi tudy, a horizontal drive of an automotive powered eat adjuter i onidered. A two-tep ytem and frition parameter identifiation approah i developed and applied to the experimental data. The identified parameter are ued in a mathematial model of the ytem to perform parameter tudie through numerial imulation. The frition-indued intability aued by the negative lope in the frition-veloity urve i the ubjet of Chapter 5. In Setion 5., eigenvalue analyi i performed on a one degree-of-freedom (DOF lead rew model to etablih ondition for the negative damping intability mehanim to our. The method of averaging i ued in Setion 5. to further tudy the ytem behavior. In thi analyi, a more omplete piture of the ytem tability propertie i obtained. The inemati ontraint and mode oupling intability mehanim are tudied in detail in Chapter 6. In Setion 6., the parametri ondition for intability due to inemati ontraint mehanim for a -DOF model are derived. Uing the eigenvalue analyi of the linearized ytem, tability ondition for the teady-liding fixed point are derived for two different -DOF model in Setion 6. and 6.. Thee etion over both inemati ontraint and mode oupling intability mehanim. Mode oupling intability in a -DOF ytem model i briefly diued in Setion 6.. Chapter 7 diue way to atively attenuate vibration in lead rew drive aued by the negative damping intability mehanim. In Setion 7., a robut liding mode ontroller i developed for a - DOF implified ytem model that aume all of the rotating part (from driver, i.e. motor, to lead rew are all rigidly onneted. Thi aumption i relaxed in Setion 7. and another liding mode ontroller i developed to regulate lead rew angular veloity in the fae of unertainty in frition and other ytem parameter. In the proof of tability of thi novel ontroller, the firt order averaging method i ued extenively. Conluion drawn in thi thei are ummarized in Chapter 8. Alo in that hapter, area for future reearh are identified and preented. For the ae of the ontinuity of the main reult, ome additional material and ontribution are relegated to the appendie. In Appendix A, detail of the tet etup ued in the tudy of the powered eat adjuter (diued in thi hapter and Chapter 4 are preented. A modified firt order averaging theorem whih i ued in Chapter 5 and 7 i tated and proven in Appendix B. In Appendix C, tep 7

27 taen to evaluate a definite integral enountered in the firt order averaged equation are preented. The exitene and the tability of the teady-tate olution of the firt order averaged equation of - DOF lead rew drive (Chapter 5, are diued in Appendix D. Two extenion of the firt order averaging reult of Chapter 5 are given in Appendie E and F. In Appendix E, general expreion for the eond and third order averaging method are preented. Alo in thi appendix, a numerial example i preented to demontrate the improved auray of the higher order averaged equation ompared to the firt order averaged equation in the predition of the amplitude of teady-tate vibration. In appendix F, the firt order averaging method i applied to the equation of motion of a -DOF lead rew ytem to tudy the effet of the negative damping intability mehanim. The inemati ontraint and mode oupling intability mehanim are explored further in Appendie G and H. In Appendix G, the ondition for the loal tability of the teady-liding fixed point of two different -DOF lead rew model are ompared and important imilaritie and differene are pointed out. In Appendix H, everal numerial imulation reult of a -DOF lead rew model are preented and the effet of variou ytem parameter on the nonlinear behavior of the ytem and the mode oupling intability mehanim are diued. Finally, in Appendix I, theorem ited in the proof of tability of the eond liding mode ontroller developed in Chapter 7 (Setion 7.4 are inluded for referene..4 Contribution The urrent reearh wa motivated by real-world problem and wa aimed to preent a omprehenive tudy overing the field of frition-indued vibration in lead rew drive. The ignifiane of thi wor i further emphaized by the notieable la of previou tudie on the dynami behavior of lead rew drive that adequately aount for the frition-indued intability mehanim. The major ontribution of thi wor an be ummarized a follow: Mathematial model: The mathematial model developed in Chapter preent a unified framewor for the tudy of lead rew dynami taing into aount rotation/tranlation and loading diretion a well a important ytem element. Frition parameter identifiation: A novel two-tep identifiation approah i developed that i apable of identifying frition a well a damping and tiffne parameter. The preented experimental reult onfirm the appliability of the developed method a well a the utility of the mathematial model in aurately prediting the vibratory behavior of the 8

28 lead rew drive ytem. Frition-indued intability mehanim: Parametri ondition for the three type of frition-indued intability mehanim (i.e. negative damping, mode oupling, and inemati ontraint are etablihed in a unified framewor. Comprehenive tudy of the negative damping frition-indued intability mehanim utilizing the averaging method: In addition to preenting a omplete piture of the tability propertie of the nonlinear ytem, the obtained reult highlight the poibility of uing the perturbation tehnique (peially, higher order averaging a an aurate and effiient method to predit the teady-tate amplitude of vibration in parameter tudie. Comprehenive treatment of the inemati ontraint intability mehanim in lead rew drive: The poibility of frition-indued intability due to inemati ontraint i tudied in detail for -DOF and -DOF model. Exluive to Multi-DOF model, a eondary inemati ontraint mehanim i identified and it role on the expanion of the untable domain in the pae of ytem parameter i explored. Comprehenive treatment of the mode oupling intability mehanim in lead rew drive: The mode oupling intability, a one of the major mehanim of frition-indued vibration in lead rew ytem, i tudied and parametri tability ondition are derived. Ative vibration ontrol: Two robut liding mode ontroller that effetively regulate the lead rew drive veloity in the fae of parameter unertainty, are developed. 9

29 Chapter Literature Review In thi hapter, the major previou wor publihed on the frition-indued vibration of the dynamial ytem inluding lead rew are reviewed. The frition-indued intability mehanim are reviewed in Setion.. Literature on the effet of frition on lead rew ytem are reviewed in Setion... Frition-indued Vibration A hitorial review of trutural and mehanial ytem with frition i given by Feeny et al. [8]. Their paper tart from the firt human experiene in fire-maing and early invention of the anient ulture to the early wor of Leonardo da Vini, and expand to the modern-day ientifi advane in frition utilization and prevention. An eential part of any tudy on the behavior of a dynamial ytem with frition i to appropriately aount for the frition effet uing a uffiiently aurate frition model. There are numerou wor found in the literature on the variou frition model for imulation and analyi of dynamial ytem. In one of the firt urvey paper on frition modeling by Armtrong-Helouvry et al. [9], variou frition model are tudied. Thee model an be divided into the following two ategorie: Model that are baed on the miro-mehanial interation between rough urfae and aim to explain the frition fore. Model that inorporate variou time or ytem dependent parameter to reprodue the effet of frition in a dynamial ytem. The latter ategory i the ubjet of numerou wor a an be een in review paper by Ibrahim [], Awrejewiz and Olejni [], and Berger []. A reported in thee wor, frition an be onidered dependent on any of the following fator: relative liding veloity, aeleration, frition

30 memory, pre-lip diplaement, normal load, dwell time, temperature, et. The frition model ued in the dynami modeling of ytem an be further divided into tati model and dynami model. In the dynami frition model uh a the o-alled LuGre model [], the frition fore i dependent on additional tate variable that are governed by nonlinear differential equation temming from the model for the average defletion of the ontating urfae. At the prie of inreaed omplexity of the ytem dynami, thee model are apable of reproduing variou feature of frition uh a veloity and aeleration dependene, pre-lip diplaement, and hyterei effet. Depending on the peifi problem being invetigated, appropriate frition model hould be hoen to reflet the relevant feature of the phyial ytem. The implet approximate frition model may be given by (ee for example [9,] F f ( v N gn( v µ (. where F i the frition fore, v i the relative liding veloity, ( v f µ i the veloity-dependent oeffiient of frition, and N i the normal fore preing the two liding urfae together. Thi model i extenively ued in the tudy of frition-indued vibration. When ome form of lubriation i preent between the liding bodie, the variation of frition with veloity i typially explained by the Stribe urve [4]. A hown in Figure -, four different regime are identified in thi model [5]. Figure -: Stribe urve [5]. The firt regime i the tati frition where lubriant doe not prevent the ontat of the aperitie

31 of the two urfae and frition at imilar to the no lubriant ituation. In the eond regime, the liding veloity i not enough to build a fluid film between the urfae and lubriation ha inignifiant effet. In the third regime with the inreae of veloity, lubriant enter the load-bearing region, whih reult in partial lubriation. In thi regime, inreaing the liding veloity dereae frition. Finally in the fourth regime, the olid-to-olid ontat i eliminated and the load i fully upported by the fluid. In thi regime, the frition fore i the reult of the hear reitane in the fluid and inreae linearly with veloity. Different model have been propoed to reprodue thi type of veloity-dependent frition (ee for example [6,7]. The important feature of thee model i the exitene of a region of negative lope in the frition-veloity urve, whih may lead to elf-exited vibration. Thi type of intability i diued in Setion.. below. Wherever liding motion exit in mahine and mehanim, frition-indued vibration may our, and when it doe, it an have evere effet on the funtion of the ytem. Exeive noie, diminihed auray, and redued life are ome of the advere onequene of frition-indued vibration. To thi end, lead rew ytem are no exeption, ine the lead rew thread lide againt mehing nut thread a the ytem operate. Numerou reearher have tudied elf-exited vibration phenomena in variety of fritional mehanim [-,8]. Poibly the loet mehanim to a lead rew drive in term of dynami and frition-indued intabilitie, i a di brae. Fortunately, there are innumerable publiation found in the literature that are dediated to variou apet of di brae noie and vibration. Major experimental and theoretial wor on the automotive di brae queal problem are reviewed in a paper by Kinaid et al. [9]. Major elf-exited vibration mehanim in the ytem with frition relevant to the lead rew drive an be ategorized into three type [8-]:. Dereaing frition fore with relative veloity or negative damping,. Kinemati ontraint intability,. Mode oupling... Negative Damping The negative lope in the frition/liding veloity urve or the differene between tati and inemati oeffiient of frition an lead to the o-alled ti-lip vibration (ee for example [4,]. In mot intane, reearher adopted the well-nown ma-on-a-onveyer model to tudy the ti-lip vibration (See for example [,4,7]. Thi imple model i hown in Figure -. Here, for

32 impliity, the oeffiient of frition i onidered to dereae linearly with relative veloity. Figure -: -DOF ma-on-a-onveyer model The equation of motion for thi model an be written a ( v x& ( v x& m& x + x& + x Nµ gn (. where N > i the normal fore between the ma and the onveyer and v > i the onveyer ontant veloity. Tranferring the teady-liding tate to the origin give where y x x and ( µ b [ µ ( v y& ( v y ( v ] m & y + y& + y N gn & µ (. N. x µ v b b Conidering mall perturbation around the teady-liding fixed point where v b y& >, linearized equation of motion i found from (. a ( Nµ y& + b b m & y + y (.4 It i obviou that when < Nµ, the ytem (.4 i untable. In thi ituation, the vibration amplitude grow until it reahe the ti-lip boundary, i.e. v b y&. Uing an exponentially dereaing model for the oeffiient of frition, Hetzler et al. [4] ued the method of averaging ([5] to tudy the teady-tate olution of a ytem imilar to the one hown in Figure -. They howed that a damping i inreaed, the untable teady-liding fixed point goe through a ubritial Hopf-bifuration ([6], reulting in an untable limit yle that define the region of attration of the table fixed point. Thomen and Fidlin [7] alo ued averaging tehnique to derive approximate expreion for the amplitude of ti-lip and pure-lip (when no tiing our vibration in a model imilar to Figure -. They ued a third-order polynomial to deribe b b

33 the veloity-dependent oeffiient of frition. In ae where the oeffiient of frition i a nonlinear funtion of liding veloity (e.g. humped frition model, the preene of one or more etion of negative lope in the frition-liding veloity urve an lead to negative damping and elf-exited vibration. In thi type of frition intability, no tiing our between the two rubbing urfae (ee for example [,8,9]... Kinemati ontraint intability When frition i preent, the ontraint equation ued to model inemati pair in dynamial ytem an lead to intabilitie even when the oeffiient of frition i aumed to be ontant (ee for example [8] and referene therein. In the ontext of ontrained multi-body ytem dynami with frition, the ame mehanim i the aue of jamming or wedging [4]. In the tudy of di brae ytem, thi type of intability i ometime referred to a prag-lip vibration (ee for example [9] and referene therein. Thi type of intability i uually haraterized by violation of the olution exitene or uniquene ondition of the ytem equation of motion [4]. The implet example to demontrate the inemati intability i hown in Figure - [9]. In the model hown, a male rigid rod pivoted at point O i ontating a rigid moving plane. A fore L i preing the free end of the rod againt the moving plane. The normal and frition fore applied to the rod are given by N and hown that at equilibrium F N where µ i the ontant ineti oeffiient of frition. It an be f µ L N (.5 µ tan θ Figure -: Simple model to demontrate inemati ontraint intability [9]. 4

34 From (.5 it i evident that if θ ( tan, then N and further motion beome µ impoible. In a more realiti etting where ome flexibility i aumed, the motion ontinue by defletion of the part (ee for example Hoffmann and Gaul [4]. After uffiient deformation of the ontating bodie, lippage our whih allow the bodie to aume their original onfiguration and the yle ontinue. Thi ituation i nown a the prag-lip limit yle... Mode Coupling In the ontext of the linear dynamial ytem, the effet of non-onervative fore on tability are well undertood (ee for example [4]. Conider the equation of motion of a eond order undamped multi-dof linear autonomou ytem a M q& + ( K + S( η q & (.6 where q i the vetor of generalized oordinate, M i a poitive-definite ymmetri inertia matrix, K if the ymmetri tiffne matrix, and S ( η i an aymmetri matrix originating from the nononervative fore, and η i a parameter of interet. The natural frequenie of thi ytem are found from the olution of the harateriti equation given by ( ω, η det( K + S( η ω M (.7 Auming the initial ytem ( η η i table, the tability may be lot by divergene or flutter a the parameter η i varied. The divergene intability boundary ( η ηd i found from ([4] ( + S( η det K (.8 At thi ritial value, one of the root of (.7 vanihe. The flutter intability boundary an be found by etting (, η ω ω (.9 where i given by (.7. The flutter boundary ( η ηf i haraterized by the oaleene of two of the ytem natural frequenie. By further variation of the parameter beyond it ritial value, two Thee ytem are alo nown a irulatory ytem. 5

35 root beome omplex onjugate. In ae where S ( η i ew-ymmetri, ytem (.6 an only beome untable through flutter intability a divergene i not poible [4]. The addition of veloity-dependent fore to thi ytem yield where C i poitive emi-definite matrix and ( C + G q + ( K + S q M q& + & (. T G G define the gyroopi fore. It ha been hown that the addition of damping an have a omplex effet on the tability of the ytem and it may even detabilize the otherwie table ytem [4]. For further detail on thi ubjet, ee alo [44]. The role of frition a a follower fore in detroying the ymmetry of the tiffne matrix reulting in flutter intability wa firt ued to explain brae queal [9]. Ono et al. [45] and Motterhead and Chan [46] tudied hard di drive intability uing a imilar onept. Conider the -DOF ytem hown in Figure -4 tudied by Hoffman and Gaul [47]. Thi model onit of a point ma liding on a onveyer. The ma i upended uing vertial and horizontal linear pring and damper. An additional pring plaed at 45 o angle i alo onidered whih at a the oupling between vertial and horizontal motion. The frition fore i modeled uing Coulomb frition law; F µ F where µ i the ontant oeffiient of frition. Alo the onveyer belt i t n moving with ontant veloity of v >. The downward fore R i aumed large enough to enure ontat between ma and onveyer belt i not lot. b Figure -4: A imple -DOF model apable of exhibiting mode oupling intability [47]. 6

36 7 The equation of motion for thi ytem an be written in matrix form a ( µ R x v z z x z x z x m m b z z x z x & & & && & & gn (. Shifting the equilibrium point (teady liding tate to the origin by etting ~ x x x and ~ z z z, where + + µ + R z x z z x (. give ( ( [ ] + µ + + µ ~ gn ~ ~ ~ ~ ~ ~ ~ x v z z z x z x z x m m b z z z x z x & & & && && (. The ymmetry-breaing role of frition i learly hown in (.. Note that the right-hand-ide of thi equation non-zero only when b x v &~. In a mall neighborhood of the origin, (. implifie to a linear homogeneou differential equation + + µ ~ ~ ~ ~ ~ ~ z x z x z x m m z z x z x & & && && (.4 Comparing (.4 with (., it an be onluded that m m M, z x C, G, + + z x K, and ( µ µ z S Negleting damping, from (.9 the flutter intability threhold i alulated a ( m z x z z x z x f + + µ (.5 If f µ µ the two natural frequeny beome idential, given by For further diuion inluding the effet of damping and numerial example refer to the original paper [47] and alo [48].

37 x + z + ω ω (.6 Note that (.6 an be obtained from (.9. Inreaing the oeffiient of frition beyond it flutter ritial value, µ > µ f, reult in the a pair of omplex onjugate quared natural frequenie, whih indiate intability of the teady liding fixed point. Reently, a great number of paper were publihed on the ytem exhibiting mode oupling intability due to frition and the omplex effet damping on uh ytem. See paper by Hoffmann and hi oworer [49-5] and Jézéquel and hi oworer [5-59]. Other reent wor on thi ubjet inlude [6-64].. Lead Srew Drive A mentioned earlier, when it ome to tranlation lead rew, very few publihed wor are found that diu the dynami of thee ytem and the effet of frition on their vibratory behavior. Olofon and Eerfor [65] invetigated the frition-indued noie of rew-nut mehanim. They diued the tribologial apet of lubriated interation between lead rew and nut thread, whih aount for the Stribe frition. Baed on experimental reult, they have onluded that: a the queaing noie i the reult of elf-exited vibration between lead rew and nut thread; b in the ytem tudied (oniting of a long and lender rew, thee vibration exite bending mode hape of the lead rew, and; the quea noie i generated only when the nut i in the viinity of one of the node of the bending mode hape of the lead rew. In a tudy of the effet of frition on the exitene and uniquene of the olution of the equation of motion of dynamial ytem, Dupont [4] onidered a -DOF model of a lead rew ytem. He invetigated the ituation under whih no olution exited and learly identified one of the oure of intability in the lead rew ytem; i.e. the inemati ontraint intability mehanim. For the elfloing rew, he found that there i a ertain limiting ratio between the lead rew inertia (rotating part and the ma of the tranlating part, beyond whih no olution exit. Baed on a ae tudy, Gallina and Giovagnoni [] diued the deign of rew ja mehanim to avoid elf-exited vibration. They developed a -DOF model of a lead rew ytem whih inluded lead rew rotation (oupled with the nut tranlation and lead rew axial diplaement. Uing eigenvalue analyi of the linearized equation, they found relationhip that define the tability domain in term of the parameter of the ytem. They onluded that to avoid vibration in elf- 8

38 loing derive, lead rew hould have low axial and high torional tiffne. Gallina [66] further expanded thi tudy and, uing both eigenvalue analyi and experiment, howed that by inreaing lead rew inertia it i poible to avoid intability under ertain ondition. Oledzi [67] tudied elf-loing mehanim. He laified all type of mehanial drive, inluding worm gear and lead rew, with the emphai on the poibility of elf-loing. A unified notation wa ued to preent geometrial feature of the drive and to derive the equation of motion of a general inemati pair. He alo modeled the inemati pair uing elati ontat intead of rigid ontat. The imulation reult preented howed the poibility of ti-lip vibration. Generally, in high-auray linear poitioning appliation, ball-rew are ued beaue of their low frition, high lead auray, and balah-free operation [68]. Conequently, the majority of wor in the literature regarding poition ontrol and dynami of rew drive fou on ball-rew [69-76]. Lead rew are alo ued for imilar poitioning appliation. For example, Otua [69] ompared a high-preiion lead rew drive equipped with an anti-balah nut with two type of ball-rew drive for nanometer poitioning appliation. The experimental reult obtained howed the poibility of ahieving nanometer auray with all three ytem. Partiular to the lead rew, the nonlinear behavior of the drive due to the ti-lip phenomenon wa tudied. The anti-balah nut were found to have an advere effet due to preloading of the thread and inreaed frition. Sato, et al. [77] onidered the dynami of a lead rew poitioning ytem with balah. They et up an experiment uing a liding table, a lead rew, and a DC motor. In their experiment the table poition, rew rotation angle, and DC motor urrent were meaured. Although they did not undertae detailed modeling of lead rew and nut interation, they were able to etimate lead rew/nut frition uing a diturbane oberver under the ation of a linear proportional plu derivative feedba ontroller. It i worth mentioning that lead rew drive were alo ued in redundant poitioning ytem for only oare table motion [7,7]. In thee ytem, a high-preiion parallel poitioning ytem uh a a piezo atuator i ued for fine-tuning. Another example i the wor by Sato, et al. [78], where they introdued an ative lead rew mehanim. By uing two nut onneted together by a piezoeletri atuator, they were able to atively ontrol balah to ahieve poition auray of better than nm. 9

39 Chapter Mathematial Modeling of Lead Srew Drive In thi hapter, a olletion of mathematial model are developed whih are ued throughout thi thei to tudy the dynami behavior of lead rew ytem. Depending on the ytem element onidered and the type of analyi undertaen, different model are developed with varying number of degree of freedom. Figure - how a typial lead rew drive ytem. A motor poibly through a gearbox rotate the lead rew via a oupling. The rotational motion i onverted to tranlation at the lead rew-nut interfae and tranferred to the moving ma. The weight of the moving ma i upported by bearing. The lead rew i held in plae by upport bearing at it either end. Lead rew Moving ma Coupling DC Motor - Gearbox Lead rew upport Nut Lead rew upport Figure -: Lead rew drive ytem In Setion., a et of mehing lead rew and nut i onidered a a inemati pair and related inemati and ineti relationhip are preented. The veloity-dependent frition model ued in thi thei i diued in Setion.. The bai -DOF lead rew drive model i developed in Setion.. Thi model will be ued in Chapter 5 and 6 to tudy negative frition gradient and inemati ontraint intability mehanim, repetively. A model of lead rew with anti-balah nut i

40 preented in Setion.4 and the role of preloaded nut on the inreaed frition i highlighted. Additional DOF are introdued to the model in Setion.5,.6, and.7 to aount for the flexibility of thread and the axial flexibility of lead rew upport. Thee model are ued in Chapter 6 to invetigate the mode oupling and inemati ontraint intability mehanim. The - DOF model of Setion.5 i alo ued in Chapter 4 to obtain imulation reult baed on the identified ytem parameter. By ombining the DOF of the model of Setion.5 and.6, a - DOF model of lead rew drive i developed in Setion.7. Thi model i ued in Chapter 6 to tudy the mode oupling intability mehanim. In Setion.8, a omplete ytem model i preented that inlude all of the element of a typial linear drive ytem. Thi model i ued in Chapter 4 a the bai of the developed frition parameter identifiation method. Robut ontroller are developed for implified verion of thi model in Chapter 7.. Lead Srew and Nut A Kinemati Pair The rotary motion i onverted to linear tranlation at the interfae of lead rew and nut thread. The inemati relationhip defining a lead rew i imply x r m tan λ θ (. where θ i the lead rew rotation, x i the nut tranlation, λ i the pith angle, and r m i the pith irle radiu. The interation between ontating lead rew and nut thread an be eaily viualized by onidering unrolled lead rew and nut thread [5]. Thi way, the rotation of lead rew i replaed by an equivalent tranlation. Auming one thread pair to be in ontat at any given intant, Figure - how the interation of the lead rew and nut thread for both left-handed and right-handed rew. The ign onvention ued for the ontat fore, N, i hown in thi figure. In the onfiguration hown, when the right-handed lead rew i rotated lowie/moved up (rotated ounterlowie/moved down the nut move baward/right (forward/left. For the left-handed rew, the diretion of motion of the nut i revered. Alo, when the nut thread are in ontat with the leading (trailing lead rew thread, the normal omponent of ontat fore, N, i onidered to be poitive (negative. The frition fore i given by By properly orienting the x-axi, thi relationhip applie to both left-hand and right-hand thread.

41 F ( µ N gn (. f v where µ i the oeffiient of frition and v i the relative liding veloity. The frition fore at tangent to the ontating thread urfae and alway oppoe the diretion of motion but doe not hange diretion when normal fore, N, hange diretion. Right-handed (RH Lead Srew x θ Left-handed (LH Lead Srew x θ Trailing nut thread in ontat, N> Trailing nut thread in ontat, N> F f N λ λ N F f x Nut thread N F f θ Lead rew thread θ F f Lead rew thread N x Nut thread Leading nut thread in ontat, N< F f N x x Leading nut thread in ontat, N< N F f θ N N θ λ F f F f λ Lead rew thread Nut thread Nut thread Lead rew thread Figure -: Sign onvention for ontat fore between nut and lead rew Before moving on to the dynami model of lead rew ytem, the effet of thread geometry on See Setion..

42 the ontat fore are onidered here. The fore interation hown in Figure - i eentially orret for the quare thread where the normal fore i parallel to the lead rew axi. For Ame or other type of thread, a light modifiation i needed to tae into aount the thread angle. Figure - how the thread emi-angle a meaured on a etion through the axi of a rew, and a meaured on a etion perpendiular to the helix, Figure -4, one an write [79] ψ a ψ n. Uing the geometri relationhip in xn xa tan ψn, tan ψa (. y y x n x a oλ (.4 Figure -: Effet of lead angle on the meaurement of thread angle Figure -4: Geometry of the thread on two different etion plane

43 Combining (. and (.4 give tan ψn tan ψa oλ (.5 Figure -5 how a portion of the lead rew with loalized ontat fore Nˆ (perpendiular to the thread urfae and frition fore F f. The X-axi of XYZ oordinate ytem i parallel to the lead rew axi. The x-z plane i perpendiular to the helix. The projetion of ontat fore on the x-y (or X-Y plane i alulated a, N N ˆ oψ (.6 n Figure -5: Fore ating on a thread F f Sine Nˆ i the normal fore, uing (. the frition fore for trapezoid thread i alulated by ( θ µ ˆ Nˆ gn &, where µˆ i the true oeffiient of frition. One an define the apparent oeffiient of frition a ( tan ψ o λ + µ ˆ o ψn µ (.7 µ a Uing (.6 and (.7, the frition fore i written onveniently a µ N gn( θ& ame a (. and will be ued throughout thi wor. F f, whih i the 4

44 . Veloity-dependent Coeffiient of Frition The rubbing ation of ontating lead rew thread againt nut thread i aumed to be the main oure of frition in the ytem onidered in thi thei. A mentioned in Chapter, numerou veloity-dependent oeffiient of frition model an be found in the literature [9,,]. Thee model generally inlude the following three part:. Coulomb or ontant frition. Stribe frition. Viou or linear frition In thi wor, the following model for the frition oeffiient i onidered v v µ µ ~ + µ ~ ( e ~ + µ v (.8 where ~µ, ~µ, and ~µ repreent Coulomb, Stribe, and viou frition oeffiient, repetively. v i the relative liding veloity between ontating nut thread and lead rew thread. Alo, v ontrol the veloity range of the Stribe effet. See Figure -6 for a hemati view of the veloity dependent oeffiient of frition given by (.8. Figure -6: Veloity dependent oeffiient of frition The reaon for hooing thi frition model are twofold. The model truture allow for the three above-mentioned omponent of frition to be eaily eparated for the purpoe of foued analyi. In Thi model i ometime nown a the Tutin model [6]. 5

45 addition, it wa found that thi partiular formula lend itelf very well to the experimental obervation reported in Chapter 4. Baed on Figure -, the liding veloity an be written a Subtituting (.9 into (.8 and rearranging, where r rm, µ µ ~ ~ µ, µ µ ~, and µ oλv rm v θ& (.9 oλ r θ& µ µ + µ e + µ θ& (. µ ~ (. i ued a the bai model for the veloity dependent oeffiient of frition.. Bai -DOF Model r m. In the remainder of thi thei oλ Figure -7 how -DOF lead rew drive model with both right-handed (RH and left-handed (LH lead rew. Note that, with the hoen x-axi diretion, the inemati relationhip (. hold for both LH and RH lead rew. In thee model, θ i i the input rotational diplaement applied to the lead rew through a flexible oupling (torional pring. R i the axial fore applied to the nut and i the linear damping oeffiient of the lead rew upport. I and m deignate inertia of the lead rew and ma of the tranlating part, repetively. Baed on the fore diagram hown in Figure -, and irrepetive of the hand of the lead rew, it I& θ an be written ( θ θ θ & + r ( N λ F oλ T gn( θ& i m in f (. and ( x m& x N oλ Ff in λ + R F gn & (. Baed on the eleted onvention for the axe and fore in Figure -7, the equation of motion for the drive with left-handed rew are idential to thoe with right-handed rew. A a reult, from thi point on, the handedne of the lead rew i aumed to be nown but i not inluded in the diuion. 6

46 where the term gn( θ & T and F gn( x& bearing of the tranlating ma, repetively. repreent the frition in lead rew upport and the Figure -7: -DOF model of a lead rew ytem Eliminating N between (. and (., yield ( I r λξm & θ + θ + θ & θ ξ( R F gn( θ& gn( θ& T where (. and (. were ued and m tan i (. µ tan λ ξ r m (.4 + µ tan λ where ( µ µ gn θn & (.5 wa ued for abbreviation. The normal ontat fore i alulated a 7

47 N ( R F ( θ& I + mr tan λ[ ( θ θ + θ & + T gn( θ& ] gn m i ( oλ + µ in λ( I r tanλξm m (.6 Note that due to the appearane of gn ( N, through µ, in the denominator of (.6, thi equation an only be olved iteratively for N. The equation of motion derived in thi etion an alo deribe other variation to the bai model. Thee model reflet other poible onfiguration that may be found in variou appliation. In Setion.., the inverted bai model i introdued where unlie the model of thi etion, nut i rotated auing the lead rew tranlate. A onfiguration with fixed nut and another with fixed lead rew are preented in Setion.. and.., repetively. In thee two model, the rotating part alo tranlate... Inverted Bai Model In ome appliation, the nut i rotated whih aue the lead rew to tranlate. Figure -8 how thi onfiguration for a imple -DOF model. It an be hown that for thi onfiguration, the equation of motion i idential to (.. Figure -8: Inverted bai -DOF model See, for example, the miro-pump hown in Figure -5. 8

48 .. Bai Model with Fixed Nut In another poible onfiguration, the nut may be fixed and the lead rew rotation i onverted to it tranlation together with other onneted part (i.e. motor, frame, payload, et.. Thi onfiguration i hown in Figure -9. A mentioned before, the equation of motion of thi ytem i alo given by (.. Figure -9: Bai -DOF model with fixed nut.. Bai Model with Fixed Lead Srew The lat variation of the bai lead rew drive model onidered here i hown in Figure -. In thi onfiguration, the lead rew i fixed in plae and the nut rotate, auing it to tranlate along the lead rew together with other moving part (i.e. motor, gearbox, payload, et.. The equation of motion of thi ytem i alo given by (.. See, for example, the powered eat adjuter in Figure -. 9

49 Figure -: Bai -DOF model with fixed lead rew.4 Anti-balah Nut A mentioned in Chapter, anti-balah nut are ommonly ued to ounter the effet of balah and wear in a lead rew drive. An anti-balah nut i uually made of two part that are onneted together through a preloaded pring. Figure - how a hemati model of a lead rew drive with a two-part nut. The pring n i preloaded uh that a fore the nut and where Newton eond law give where P δ at between the two halve of δ n i the initial ompreion of the pring. Negleting the ma of the nut, the I&& θ + r ( θ θ θ & + ( λ µ ( θ& i rm N in gn N oλ ( N in λ µ gn( θ& N oλ T gn( θ& m n ( θ& N in λ + R F gn( x P n (.7 m & x N oλ µ gn & + (.8 ( θ λ P N oλ µ gn & N in (.9 where N > and N > are the thread ontat fore orreponding to left and right part of the nut, repetively. Combining (.7, (.8, and (.9 and uing (., give

50 ( I r tan λξ m θ & + θ + θ & θ ξ ( R F gn( θ& T ( θ& ( ξ + ξ P & gn (. m i where ξ ξ r m r m µ gn + µ gn µ gn µ gn ( θ& tan λ ( θ& tan λ ( θ& + tan λ ( θ & tan λ (. Figure -: Lead rew model with anti-balah nut Compared with (., ( ξ + P term i the additional reitive torque aued by the preloaded ξ nut. The ontat fore N (for left thread in Figure - i found a N ( R F gn( θ& I + mrm tan λ[ ( θ θi + θ & + T gn( θ& ] + ( I + rm tan λξ ( oλ + µ gn( θ& in λ( I r tan λξ m m m P (. where, again, ompared with (.6, the ontat fore i inreaed due to the preload P. Note that the above implified formulation i valid a long a N >. If thi ondition i violated, (i.e. the left ontat i broen for the duration of uh motion, the number of DOF i inreaed to

51 two. In uh ae, whih may be aued by large R >, the dynami of the ytem i more ompliated ine the impat of the thread and repeated lo of ontat hould be onidered..5 Compliane in Lead Srew and Nut Thread In Setion., the lead rew and nut were modeled a a inemati pair leading to an iterative equation for determining the ign of the ontat fore. The analyi may be greatly implified by auming ome degree of ompliane in the lead rew and/or nut thread. Figure - how the ame ytem a in Figure -7(b exept for the ontat between thread whih i now modeled by pring and damper. With thi hange, the number of DOF i inreaed to two. Figure -: -DOF lead rew drive model inluding thread ompliane Conforming to the ign onvention defined in Figure -, the defletion (or interferene of thread an be alulated a δ x oλ r θin λ (. The implet way to approximate the ontat fore i by modeling the fore/defletion relationhip m Refer to Setion 6...

52 of the thread a that of linear pring and damper. Thu N δ + δ& (.4 Subtituting (.4 into (. and (. and uing (. yield I&& θ + r ( θ θ θ & + r ( x oλ r θin λ( in λ µ oλ m i mx && ( x& o λ r θ& in λ( in λ µ o λ gn( θ& m m ( xoλ rm θin λ( oλ + µ in λ ( x& oλ r θ& in λ( oλ + µ in λ + R F gn( x& m m T (.5 (.6 where µ i defined by (.5..6 Axial Compliane in Lead Srew Support Another important oure of flexibility in the ytem may be the ompliane in the lead rew upport. To model thi feature, a hown in Figure -, pring and damping are added to the bai model of Setion., whih allow the lead rew to move axially. Similar to Setion., (. and (. give fore/aeleration relationhip for lead rew rotation and nut tranlation, repetively. Moreover, the lead rew tranlation DOF i governed by m & x x x& + N oλ + F in λ (.7 f The inemati relationhip between θ, x, and x i given a x x r tan λθ (.8 m Eliminating N between (. and (. and alo between (. and (.7 and uing (.8 and (. yield ( I r λξm & θ ξmx && ( θ θ θ& ξ R F gn( x& ( gn( θ& m tan i T (.9 For the model of thi etion, the relative veloity given by (.9 hould be hanged to. However, in pratial ituation where the lead angle ( λ i mall and thread are almot rigid the term negligible and it i omitted. i

53 where ξ i given by (.4. ( m m & x + mrm λθ && x x& + R F gn( x& + tan (. Figure -: -DOF lead rew drive model inluding ompliane in the upport The normal ontat fore i alulated a N I m ( I I ( R F gn( x& + x + x& r tan λ( θ θ + r tan λ θ & + T gn( θ& m m I I ( oλ + µ in λ + r tan λξ m m i m m m (..7 Compliane in Thread and Lead Srew Support By ombining the two model preented in Setion.5 and.6, a -DOF model of the lead rew drive reult, whih i hown in Figure -4. The equation of motion of thi ytem are defined by (., (., and (.7. The only hange i in the alulation of ontat fore N given by (.4; the thread defletion, intead of (., i alulated by ( x oλ r θin λ δ x (. m 4

54 Figure -4: -DOF lead rew drive model inluding ompliane in the upport and ompliane in the lead rew and nut thread.8 A Complete Sytem Model In thi etion, another model i developed whih inlude other element in the power tranmiion hain, namely a DC motor and a worm gearbox. Figure -5 how the 4-DOF onfiguration onidered. For the motor, Newton eond law give I M && θ M T M T fm M ( θw θm Mθ& M gn θ& + (. where torque, θ M i the rotor angular diplaement, I M i the inertia of the rotor, T M i generated (input T fm and M are the internal frition and damping of the motor, repetively. Alo, i the torional tiffne of the oupling onneting the motor to the gearbox, and angular diplaement of the worm. θ W deignate the 5

55 Figure -5: A 4-DOF lead rew drive ytem model For lead rew and nut, imilar to (. and (., one an write I& θ ( θ θ θ & + r ( N λ F o λ T gn( θ& G m in f (.4 ( x m& x N oλ Ff in λ + R F gn & (.5 where N i given by (.4. For the worm and worm gear, Newton eond law give I W & θ W d ( θ W M θw W θ& W ( W λw + µ WG W θ& W λw T fw θ& in gn o gn W (.6 I G & θ G d ( θ θ G G Gθ& G + ( W oλw µ WG W gn θ& G in λw T fg gn θ& G (.7 where I W and I G are the inertia of the worm and worm-gear, repetively. θ G i the angular diplaement of the gear. d W and d G are the pith diameter of worm and worm gear, repetively. 6

56 W i the normal omponent of the ontat fore between mehing worm thread and gear teeth. and T fg are the internal frition torque of the worm and the worm gear, repetively. λ W i the pith angle of the worm and µ WG i the oeffiient of frition of the mehing worm and worm gear. Eliminating W between (.6 and (.7 give T fw ( I + α ξ I && θ + ( + α ξ θ& + ( + α ξ W α W W θ M W G α W ξ G W W θ + ( α T + α ξ T gn θ& W W fw W G W G W fg W G W θ G (.8 where d W αw tan λw (.9 dg i the gearbox ratio (i.e. θ G α W ξ W θ W and d d W G ( tan λw + µ WG gn( θ& WW ( µ gn( θ& W tan λ WG G W (.4 The equation of motion for the 4-DOF model in Figure -5 are given by (., (.4, (.5, and (.8..9 Some Remar Regarding the Sytem Model Depending on the onfiguration of an atual lead rew drive, one or more of model preented in thi hapter may be uitable to aurately apture the mot prominent and/or relevant feature of the ytem dynamial behavior. Thi i ertainly the ae in the ubequent hapter of thi thei. However, many other feature are not inluded in thi wor. The feature inlude: Dependene of frition on poition: A the lead rew turn, the nut progree along the lead rew thread reating the poibility of a poition-dependent oeffiient of frition. In thi wor, the mathematial frition model i aumed to be independent of poition for impliity of mathematial modeling. From an experimental point view, a the ae of For the worm gearbox onidered here, α W n W / n G where n G i the number of gear teeth and n W i the number of worm tart. 7

57 Chapter 4, the identified frition (and other poible poition-dependent parameter may be onidered a an averaged value for the woring portion of the lead rew. Nonlinearity: The only nonlinear effet onidered in thi wor ome from frition. However, many other oure of nonlinearity are poible in a lead rew drive, whih are exluded here to implify the tudy of the frition-indued vibration. Mot notable fator are: preene of nonlinearity in the ontat fore of thread aued by defletion, nonlinear torional tiffne of the oupling, and diontinuity due to balah. Torional defletion of lead rew: For a long and/or lender lead rew, the frequeny of the firt few torional mode of vibration may be low enough to influene the ytem dynami. Moreover, the winding/unwinding ation of torional defletion may affet the thread learane and the overall load ditribution auing further deviation from the model onidered here. In thi wor, the lead rew are onidered uffiiently tiff and modeled a rigid bodie. Axial defletion of lead rew: Similar to the previou point, thi effet may influene the lead rew-nut interation in two way: by introduing new mode of vibration and by affeting the thread learane and load ditribution. Lateral defletion of lead rew: Three ituation may lead to thi type of vibration: lateral loading, exeive axial loading leading to buling (a fator for long lender lead rew, and finally whirling (for very high rotation peed. All thee ituation are onidered to be outide the ope of thi wor. Mialignment: Deign and/or aembly problem may lead to axial offet of the enterline of lead rew and nut. The mialignment may alo our in the form of a ewed nut. In both of thee ae (whih are exluded from the urrent reearh, thread ontat and load ditribution may be effeted everely. Manufaturing iue: Depending on the manufaturing method and quality of the produt, lead rew an uffer from lead error (partiularly in longer deign. There may be external ontaminant or urfae defet on lead rew or nut. Although thee and other imilar iue may have ignifiant impat on the funtion of a lead rew drive, they are 8

58 exluded from thi fundamental tudy of the frition-indued vibration. Additional element: The tudy of lead rew drive, or any other mehanial ytem, an be augmented by other mehanial element (e.g. a vibrating omponent on the moving part, additional DOF due to the flexibility of the moving part, external time-dependent foring, et.. Thee ae are outide the ope of thi wor and depending on the problem they repreent, may warrant a eparate tudy. Balah: Lead rew drive generally uffer from balah. Here, balah i not onidered ine the fou i on the effet of frition on power rew where the reiting load i onidered to be ontant and the ytem i onidered to be moving with a ontant input veloity. Balah ertainly will play a major role in poitioning appliation of lead rew, a ubjet that i outide the ope of thi wor. Wear: Throughout the operating life of lead rew drive, wear aue hange to the ontating urfae, thereby affeting the load ditribution aro the thread. Thi effet i onidered to be outide of the urrent tudy. 9

59 Chapter 4 An Experimental Cae Study A mentioned in the Setion., one of the motivation behind thi reearh wa the exeive audible noie generated from the horizontal drive of an automotive eat adjuter. In thi hapter, the experimental wor, theoretial modeling, and ytem parameter identifiation of the horizontal drive mehanim of thi eat adjuter are preented. In Setion 4. ome preliminary obervation are made regarding the audible noie generated by the ytem. Thee obervation are followed by a foued tudy on the role of frition a the oure of vibration and the generation of audible noie. The detail of the parameter identifiation approah, whih onit of two tep, are preented in Setion 4. and 4.4. In the firt tep, parameter identifiation formulation are formed baed on the teady-tate pure-lip behavior of the ytem. In thi tep, external etting (i.e. applied axial fore and preet motor angular veloity are related to the frition and damping element of the ytem, maing it poible to identify variou parameter from experimental reult uing the leat quare tehnique. Identified reult for the lead rew drive of the eat adjuter are preented in Setion 4.. In the eond tep, Setion 4.4, identified parameter are fine-tuned baed on the open-loop vibrating behavior of the ytem. In thi tep, an optimization tehnique i utilized to math the model repone to the meaurement when the ytem trajetory follow a limit yle. Parameter tudie baed on the fine-tuned model are given in Setion 4.5. Conluion are ummarized in Setion Preliminary Obervation The firt tep in finding the aue of exeive audible noie in an operating mehanial ytem may be to analyze the noie ignal and the external ondition under whih uh a noie i generated. In thi etion, ome of the preliminary obervation made on the powered eat adjuter invetigated 4

60 here are preented. In the firt tep, a erie of tet were performed on the omplete eat adjuter. Figure 4- how the tet etup and intrumentation ued to tudy the omplete eat adjuter. A pneumati ylinder mounted on the wall wa ued to apply axial fore. Load Cell Poition Tranduer Pneumati Cylinder Mirophone Sound Level Meter Figure 4-: Tet etup for omplete eat adjuter A ample thee meaurement are preented in Figure 4-. A hown in Figure 4-(, the applied fore of approximately 8N aued the eat adjuter to generated audible noie. Figure 4-(d how that eat wa traveling at a veloity of approximately 7mm/. The ound level meter meaurement in Figure 4-(b how an approximately db jump ourred in the noie level (from the baground noie level during a portion of the eat travel. During the ame interval, the audible noie timefrequeny plot in Figure 4-(a learly how the utained preene of noie with a dominant frequeny of approximately 6Hz. The frequeny ontent of the noie ignal i alo hown in Figure 4- at t 8. See Setion. for the detail of the eat adjuter mehanim. For the intrumentation detail ee Appendix A 4

61 Figure 4-: Sample tet reult from omplete eat adjuter tet. Figure 4-: Audible noie frequeny ontent for the tet reult hown in Figure 4- (at 8. Pea amplitude at 6Hz The invetigation wa then ontinued by a eond erie of tet fouing only on the lider ompriing the horizontal motion ytem of the eat adjuter. A mentioned earlier, the two lider are equipped with lead rew ytem driven by a ingle DC motor. The tet etup for thi erie of tet i hown in Figure 4-4, whih i imilar to the etup ued for the omplete eat adjuter tet. 4

62 Pneumati Cylinder Load Cell Sample Lead Srew/Nut Aemblie One of the Two Slider DC Motor Poition Tranduer Gearbox Sound Level Meter Mirophone Figure 4-4: Single-tra tet etup Sample meaurement reult from one of the ingle lider tet are preented in Figure 4-5. In thi tet, a horizontal fore of about N (Figure 4-5( wa needed to indue the noie at a traveling veloity of approximately mm/ (Figure 4-5(d. The audible noie ontinued for about 4 with a dominant frequeny of about 5Hz (Figure 4-5(a aompanied by an almot db inreae in the noie level (Figure 4-5(b. Figure 4-5: Sample tet reult from ingle-tra tet. 4

63 The frequeny petrum of the reorded noie at how the dominant ignal frequeny of 5Hz. t i plotted in Figure 4-6. Thi plot learly Figure 4-6: Audible noie frequeny ontent for the tet reult hown in Figure 4-5 (at. Pea amplitude at 5Hz Thee tet and many imilar other under different fore and veloity etting revealed a trong orrelation between the two tet etup, onfirming the initial gue a to the oure of audible noie: frition-indued vibration in the lead rew drive. 4. Step : Frition and Sytem Parameter Identifiation The firt tep in the frition identifiation proe i formulated baed on the ytem repone under teady-tate pure-lip ondition (i.e. no rotational vibration and ontant lead rew angular veloity. The aim of thi formulation i to relate the meaurable ytem input and tate to the internal frition and damping parameter through the mathematial model deribed in Setion.8. Thi model orrepond to the third tet etup developed for thee tudie, whih i hown in Figure 4-7. For the intrumentation detail ee Appendix A 44

64 Figure 4-7: Shemati view of the tet etup Similar to the tet etup hown in Figure 4-4, only one of the two lider wa inluded in the etup. The woring part of the tet etup were taen from an atual eat adjuter. Two enoder were ued to meaure the angular diplaement of the lead rew and the motor. A load ell wa ued to meaure the fore exerted by the pneumati ylinder. The input voltage and urrent to the DC motor were alo meaured. With the help of a ontroller regulating the urrent input to the DC motor [8,8], the lider wa et to move at ontant preet veloitie in the appliable range. At eah tet, the following quantitie beome available a averaged value over the onidered travel troe of the nut: Motor angular veloity, ω M Motor torque, T M Axial Fore, R The angular veloity of the motor i alulated by numerial differentiation of it meaured angular diplaement. The motor torque i alulated from the meaured input urrent and the nown motor torque ontant. The teady tate relationhip are derived from (., (.4, (.5, and (.8 by etting all aeleration to zero and auming poitive angular veloitie. The teady-tate equation are found a ( α θ θ ω T T + (4. M fm W G M M M ( θ θ ω T + r ( N in λ F oλ (4. G m f 45

65 N oλ Ff in λ F + R (4. ( + α ξ ω + ( + α ξ W α W ξ W W G W θ + ( α T + α ξ T W G fw W W W W fg θ G α W θ M (4.4 where ω ω α ω are the ontant angular veloitie and G W M Eliminating N between (4. and (4. give ξ d ( tan λw + µ WG ( µ tan λ W W (4.5 dg WG W ( θ θ ω T ξ ( R F G (4.6 where ξ ( N tan λ ( N tan λ µ gn r m (4.7 + µ gn Sine the aim of thi tudy i to invetigate ae where the axial load i applied in the diretion of motion, N i aumed to be poitive. Conequently, at teady-tate for R > F, (4.7 i implified to ξ µ tan λ r m r ( µ tan λ (4.8 + µ tan λ m where the approximation i obtained by auming µ tan λ <<. Combining (4., (4.4, and (4.6 yield where T M ( R Cω + T + ξ (4.9 M f W ξ F C C + α (4. WξW T f T (4. f + ξw T and C + W + αwξw G M (4. 46

66 Tf TfM + TfW + ξw T (4. fg To eparate fore effet from the veloity effet, (4.9 i rearranged a T M ( ω + ( R F β ( ω β M M (4.4 where For eah motor peed etting, experimental data point R ω ( ωm Tf + CωM β (4.5 ( ω ξ ξ ( ω β M W M (4.6 ( i, the traight line deribed by (4.4 an be fitted to the M ( j ( i, j ( i ( i, T to obtain β and β a funtion of motor angular veloity. Baed on (4.5, another traight line an be fitted to M ( i ( i ω M,β data point to obtain T f and C. Expanding the eond veloity dependent oeffiient, β ( ω M, uing (4.8, give β d m ( ω ξ ( µ ( ω tan λ W (4.7 At teady-tate veloity of ω α W ωm µ µ, the oeffiient of frition defined by (. beome + µ e + µ α ω (4.8 r αw ωm Subtituting (4.8 into (4.7 and rearranging give where β rαw ωm ( ω [ ω ] M γ W M γ e γ (4.9 γ M ( µ tan λ r ξ (4. m W γ mξw µ r (4. γ mξw αwµ r (4. 47

67 Now the urve deribed by (4.9 an be fitted to the previouly obtained data point (i.e. ( i ( i ω M,β to etimate the three new parameter; one find where and ˆγ, ˆγ, and ˆγ. Uing the leat quare tehnique, T T ( A A A B Γˆ (4. [ γˆ γˆ γ ] T Γ ˆ (4.4 ˆ ( r α ω W W ( r α ω W W ( ( e ωw A (4.5 M e e M ( n r α ω W W ω W M ( n ωw β ( ( β B (4.6 M ( n β where n i the total number of data point available. Note that A given by (4.5 i dependent on r, whih i one of the unnown parameter deribing the Stribe effet in the aumed model of the veloity dependent oeffiient of frition. To retify thi problem, a imple optimization routine i ued to find the bet value for r uh that the urve fitting error of (4. i minimized. Define e e e ( ( r ( ( r M ( n( r B A ( r γˆ γˆ γˆ ( r ( r ( r (4.7 where dependene on r i made expliit. The optimized value of r given by ˆr i now found imply a 48

68 n rmin r r max i e( i( r rˆ arg min (4.8 Thi ta may be performed uing a number of numerial optimization tehnique. In the next etion, e( i ( r n i the minimum i found graphially. 4. Experiment Reult i imply omputed and plotted over a range of appropriate value, r >, and In thi etion, the parameter identifiation approah deribed in Setion 4. i applied to the meaurement performed uing the tet etup hown in Figure 4-7. Before exploring the frition torque produed at the ontat between lead rew and nut thread, a preliminary tep i required to etimate and iolate internal damping (4. and frition (4. of the DC motor and the gearbox. Thi tep i preent firt in Setion 4.. and then the lead rew frition and damping identifiation reult are preented in Setion DC Motor and Gearbox In a erie of preliminary tet, DC motor and gearbox were dionneted from the lead rew, and the input urrent of the DC motor wa meaured at different level of preet ontant angular veloitie. Figure 4-8 how the reult of thee tet. By fitting a traight line to thee data point uing the leat quare tehnique, the overall damping, C, and reidual frition torque, T f, were etimated. Thee reult together with other nown ytem parameter are given in Table

69 Figure 4-8: Reitive torque of the motor and the gearbox. Dot: meaurement, dahed line: fitted line to the data point Table 4-: Known or aumed ytem parameter value Parameter Value Lead rew pith diameter, d m.66 mm Lead rew lead angle, λ 5.57 Ma of tranlating part, m.8 g Average reitane of the lider, F < N Aumed ontat tiffne lead rew and nut, 8 N/m Aumed ontat damping lead rew and nut, 6 g/ Lead rew inertia, I 6. g.m Worm pith diameter, d W 9.44 mm Worm gear pith diameter, d G.4 mm Worm lead angle, λ W 8.5 Gearbox ratio, α W 9 Nominal torional tiffne of the oupling,. N.m/rad Aumed oeffiient of frition of gearbox meh, µ WG. Overall DC motor and the gearbox internal damping, C. N.m Overall DC motor and the gearbox internal frition, f T N.m.rad/ 4.. Identifiation Reult Figure 4-9 how data point olleted from all of the meaurement performed. In thi figure, motor 5

70 torque (meaured from motor input urrent i plotted againt meaured fore and meaured peed. Flutuation in the upply air preure to the ylinder, together with the peed-dependent internal frition of the piton rod, aued variation in the applied fore from one experiment to the next. Figure 4-9: Colletion of data point howing Torque/Speed/Fore A deribed in the previou etion, a traight line i fitted to the data point at eah veloity etting, whih give variation of motor torque veru applied axial fore aording to (4.4 for eah of the available veloity et point. Figure 4- how a few ample of thee urve fitting. The urve fitting reult aording to (4.5 and (4.9 are hown in Figure 4- and Figure 4-, repetively. The etimated parameter are lited in Table 4-. Table 4-: Identified parameter Parameter Value Unit T.46 N.m f C 7.4e-5 N.m./rad r.8 /rad γ.54e- m γ.59e-5 m γ -8.99e-8 m In thee alulation, the effet of F wa negleted ine preliminary obervation howed that the lider frition fore i onitently le than N, whih i le than % of the applied fore, R. 5

71 Figure 4-: Sample meaurement reult. Variation of motor torque with applied axial load at ontant peed. Dot: meaurement, olid line: fitted line to the data point Figure 4-: Variation of β with motor angular veloity Baed on (4. and uing value of T f from Table 4- and T f from Table 4-, the reidual frition of the lead rew upport, T i found to be T. N. m where baed on the parameter value in Table 4- and (4.5, ξ W.7 wa ued. 5

72 Figure 4-: Variation of β with motor angular veloity Alo uing (4. and C from Table 4- and C from Table 4-, the damping oeffiient of the end upport i found to be 4 4 N. m. rad Thi value i adjuted in Setion 4.4, ine the ytem tability (in imulation depend heavily on the damping of the lead rew, and in the experimental reult, there wa quite a bit of variability. Uing (4., (4., and (4. and their identified value in Table 4-, the three frition parameter defined by (4.8 an be alulated. Thee value are lited in Table 4- and the reulting veloity-dependent oeffiient of frition i plotted in Figure 4-. Table 4-: Numerial value of the identified parameter Parameter Value Unit µ.8e- - µ.e- - µ -4.47e-4 /rad 5

73 Figure 4-: Identified veloity dependent oeffiient of frition 4.4 Step : Fine-tuning The identifiation formulation in the Setion 4. depend on the nowledge of the liding oeffiient of frition of the gearbox ( µ i given in Table 4- through the appearane of ξ W in (4., WG (4., (4., (4., and (4.. Unertainty in the value of thi parameter, together with the unnown nonlinearity of the oupling tiffne, neeitate a further tep of parameter identifiation and fine-tuning. The approah in Setion 4. wa baed on teady tate (no vibration ondition. Aordingly, in Setion 4., tet were performed while eeping the angular veloity of the lead rew nearly ontant under the ation of a peed ontroller. In thi etion, reult from a erie of tet on the ytem without the ontroller (normal operating ondition are ued to fine-tune the model through identifying variation in damping, tiffne, and frition. Figure 4-4 how the hange in the vibration amplitude a the applied axial fore and the input angular veloity of lead rew i hanged. Eah point in Figure 4-4 repreent the averaged experimental value of amplitude of vibration over a m interval, where a limit yle wa deteted. Thee reult how that the amplitude of vibration inreae with gearbox output angular veloity. Figure 4-5(a how a ample of the tet reult. In the loe-up view in Figure 4-5 (b, the tilip rotational vibration of lead rew an be een learly. Note that in the atual lead rew ytem, variou parameter (e.g. lubriation, urfae ondition, 54

74 and load ditribution on lead rew and nut thread hange a the nut tranlate along the lead rew. Thi mean that the etimated parameter obtained from the meaurement are effetively the averaged value over the ditane that the nut wa et to tranlate along the lead rew in the tet. ω G Figure 4-4: Experimentally obtained variation of limit yle vibration amplitude with input angular veloity (gearbox output and axial fore (a (b Stiing Figure 4-5: a A ample of tet reult howing ti-lip in open-loop tet, b zoomed view. Bla: lead rew angular veloity; grey: DC motor angular veloity Conider the following ot funtion 55

75 Ψ (,,, r µ n M S ( θ& ( i θ& ( i M ( maxθ& ( i i i ~ i + n ~ M ~ S ( θ ( i θ ( i M ( max θ ( i i (4.9 whih alulate the weighted um of quared differene between meaured (uperript M and ~ M ~ imulated (uperript S angular diplaement ( θ, S θ and angular veloity ( M θ & S, θ &. The ~ ignifie that the mean value i removed. Alo, n i the number of data point inluded in the time window during whih the meaured ytem trajetory follow a limit-yle. Additional parameter in the ot funtion Ψ in (4.9 are defined by the following modifiation to the veloity-dependent oeffiient of frition defined by (. r θ& θ& µ µ + µ + µ ~ r θ& µ e e (4. where.9 < µ <. i a aling added to the identified frition to aount for any variation in µ WG from one experiment to the next. In addition, the frition oeffiient i moothed over near zero relative veloitie to failitate numerial integration and improve onformity of the imulation reult to the tet data a the trajetorie approah the zero liding veloity boundary. Beaue of the high gear ratio of the worm gear, frition-indued vibration of the lead rew do not aue oniderable flutuation in the angular veloity of the DC motor (See Figure 4-5, for example. Hene, from thi point on, in the numerial imulation, it i aumed that the angular veloity output of gearbox, θ & G, i ontant. The reulting implified model i given in Setion.5. The parameter minimizing Eq. (4. are found for every inidene where a limit yle i found in the meaured repone (Figure 4-4 over a wide range of gearbox output angular veloitie and applied axial fore. In Figure 4-6 to 4-, ample reult are hown that ompare meaurement with the imulation reult obtained from the -DOF model of Setion.5 with the fine-tuned parameter. 56

76 Figure 4-6: Meaurement v. imulation example. 4 Input: R 5.9( N, ωg 5.6( rad / - Parameter:.8( Nm / rad,.86 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation Figure 4-7: Meaurement v. imulation example. 4 Input: R 7.9( N, ωg 4.( rad / - Parameter:.( Nm / rad,.7 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation 57

77 Figure 4-8: Meaurement v. imulation example. 4 Input: R 4.( N, ωg 4.( rad / - Parameter:.65( Nm / rad,.4 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation Figure 4-9: Meaurement v. imulation example. 4 Input: R 7.5( N, ωg 8.7( rad / - Parameter:.75( Nm / rad, 4.7 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation 58

78 Figure 4-: Meaurement v. imulation example. 4 Input: R.8( N, ωg 7.( rad / - Parameter:.88( Nm / rad, 5.79 ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation Figure 4-: Meaurement v. imulation example. 4 Input: R 6.7( N, ωg 7.9( rad / - Parameter:.67( Nm / rad, 4. ( Nm / rad a phae plot, b frequeny repone. Gray: meaurement, bla: imulation Thee reult how the effetivene of the fine-tuning tep in mathing the dynamial behavior of the model with that of the real ytem. In order to perform parameter tudie through imulation, two-variable bilinear fitting wa 59

79 performed for eah of the four parameter in (4.9 with repet to the gearbox output angular veloity, thee fitting. ω G, and the applied axial fore, R. Contour plot in Figure 4- to 4-5 how the reult of Figure 4-: Variation of oupling tiffne,, with gearbox output veloity and axial fore Figure 4-: Variation of frition boundary effet, /r, with gearbox output veloity and axial fore 6

80 Figure 4-4: Variation of lead rew upport damping, ( -, with gearbox output veloity and axial fore Figure 4-5: Variation of frition aling, µ, with gearbox output veloity and axial fore It i intereting to note that, a expeted, oupling tiffne varie mainly with the axial fore and exhibit a wor-hardening behavior. Alo, the frition aling ( µ wa found to be very loe to one whih how that the initial etimate for µ WG wa quite aurate. 6

81 4.5 Parameter Studie Uing the identified and fine-tuned model parameter, variou performane map an be obtained to tudy the effet of the variation of ytem parameter on the initiation of limit yle and the amplitude of teady-tate vibration. The amplitude of vibration i diretly related to the generated audible noie from the ytem. In the following, the effet of input angular veloity, damping of the lead rew upport, and oupling tiffne are invetigated Effet of Input Angular Veloity Figure 4-6 how the ontour of teady-tate vibration amplitude a a funtion of gearbox output angular veloity and the applied axial fore. It an be een from thi figure that, beyond a ertain value of applied axial fore, inreaing the angular veloity inreae the amplitude of vibration. Thi finding i well orrelated with the ubjetive tet on the audible noie intenity level from the lead rew ytem and the experimental reult in Figure 4-4 Figure 4-6: Contour plot of the teady tate vibration amplitude v. applied axial fore and gearbox output peed 4.5. Effet of Damping To invetigate the effet of damping on the threhold of intabilitie, the above imulation were repeated for three value of the ontant damping oeffiient. Figure 4-7 how the reult of thee 6

82 imulation where a vibration amplitude of. (rad ha been taen a the approximate threhold of table/untable region. It an be een that, by inreaing the damping, intabilitie our at higher level of axial fore. Thi phenomenon will be analyzed further in Chapter 5 uing the eigenvalue analyi method and the method of averaging. Figure 4-7: Effet of lead rew rotational damping on the threhold of intabilitie. The thi bla line orrepond to the intability threhold in Figure Effet of Stiffne The effet of the torional tiffne of the oupling wa alo onidered. Figure 4-8 how the variation of the vibration amplitude a a funtion of applied axial fore and the torional tiffne of the oupling. Thi map wa obtained from numerial imulation auming ω G 4 (rad/. Similar plot an be obtained for other preet angular veloitie. Figure 4-8 how that if the applied fore i below N, by either inreaing or dereaing the tiffne of the oupling from it urrent deign value, rotational vibration leading to exeive noie may be eliminated. Note that the horizontal axi i a aling parameter applied to the oupling tiffne of the ytem. At higher axial load, by uing tiffer oupling beyond time the urrent value muh lower vibration amplitude are obtained. However, other deign requirement may prevent the ue of a high tiffne oupling in the ytem. Thi finding open the door for two poible deign modifiation, whih an either eliminate the teady-tate rotational vibration or 6

83 redue it amplitude in uh a way that no audible noie i emitted from the ytem. The former i aomplihed by inorporating a very oft flexible haft and the latter by replaing the flexible haft with a olid haft, whih guarantee high torional tiffne. Experimental reult howed that by implementing eah of the above olution, in idential ituation, the audible noie i eliminated from the originally noiy ytem. Figure 4-8: The effet of oupling tiffne and axial loading on the dynami behavior of the lead rew - Gearbox output angular veloity 4 (rad/ 4.6 Conluion In thi hapter, a two-tep identifiation/fine-tuning approah i developed to etimate variou frition and damping parameter of the ytem. In the firt tep, uing the teady-liding tet reult, the veloity effet (i.e. damping and veloity-dependent part of frition were eparated from the fore effet (i.e. oulomb oeffiient of frition and appropriate parameter were etimated uing the leat quare tehnique. Then, uing the open-loop tet reult in whih limit yle were oberved, effetive load-dependent torional tiffne of the lead rew ytem wa etimated by minimizing a ot funtion that quantified the differene between meaured and alulated diplaement and veloitie. In thi tep, frition and damping parameter were alo adjuted o that maximum onformity between meaurement and imulation reult wa ahieved. The preented numerial imulation howed the auray of the identified mathematial model of the lead rew ytem under a wide range axial loading and input peed etting. 64

84 Parameter tudie were performed to ae the effet of lead rew rotational damping and oupling torional tiffne on the onet of intabilitie. Simulation reult howed that by inreaing the damping, intabilitie our at higher level of applied fore. In addition, it wa hown that the torional tiffne of the oupling ould hange the axial loading range (dependent on the input peed where the ytem generate ignifiant audible noie. 65

85 Chapter 5 Negative Damping The onverion of rotary to tranlational motion in a lead rew ytem our at the mehing lead rew and nut thread. The ontating thread lide againt eah other reating a frition fore oppoing the diretion of motion. The frition-indued intability mehanim in dynamial ytem were introdued in Chapter. In thi hapter, the role of the veloity-dependent frition oeffiient on the tability of lead rew ytem i tudied. A oberved and reported in numerou previou wor found in the literature, a dereaing oeffiient of frition with relative liding veloity an effetively at a a oure of negative damping auing intabilitie that lead to elf-exited vibration. In thi tudy, the -DOF model of Setion. i ued, whih apture all of the eential feature of the ytem dynami pertaining to the negative damping intability mehanim. The equation of motion of the -DOF lead rew model i preented in Setion 5.. In Setion 5., eigenvalue analyi method i ued to tudy the loal tability of the teady-liding tate and to drive parametri ondition for the negative damping intability mehanim. Thi tudy i expanded in Setion 5., uing the method of firt order averaging. A omplete piture of the tability propertie of the ytem i obtained in thi etion. The reult from the averaging analyi an alo be ued to predit the amplitude of vibration when intability our and to tudy the effet of variou ytem parameter on the teady tate vibration. Thi onept i important in undertanding the role of frition-indued vibration on the generation of audible noie from a lead rew drive mehanim. A ummary of reult and onluion i given in Setion Equation of Motion To tudy the negative damping intability mehanim, the -DOF model of Setion. i hoen. See Setion... 66

86 Negleting F and T for impliity, (. beome where Γ & θ + θ + θ & θi ξr (5. Γ I rm tan λξm (5. and ξ i given by (.4. Alo, the veloity-dependent oeffiient of frition i defined by (.. Let z θ θ then &z θ & ω and &z & & θ where ω θ dt i a ontant repreenting the input i angular veloity. Subtituting thi hange of variable into (5., give d i Γ & z + z& + z ω ξr (5. At teady-liding we have & z&, z&, and z z. Subtituting thee value in (5. yield ω + ξ R (5.4 z where and ξ ( Rω tan λ ( Rω tan λ µ gn r m (5.5 + µ gn µ µ r ω + µ e + µ ω (5.6 The hange of variable u z z onvert (5. to Alo, (. beome u& + u& + u ( ξ ξr (5.7 Γ µ r u& +ω ( u& µ + µ e + µ u& + ω (5.8 Furthermore, the equation for the ontat fore, whih wa given by (.6, i implified to ( u + u& Γ R + mrm tan λ N ( oλ + µ in λγ (5.9 67

87 where Γ i found from (5. by replaing ξ with ξ and the abbreviation (.5 i now written a ( u u& µ ( u& gn( N( u, u& gn( & + ω µ, u (5. 5. Eigenvalue Analyi The equation of motion in tate-pae form an be formulated by the following hange of variable y u, y u& (5. Auming Γ, (5.7 i repreented by two firt-order differential equation y& y& y y y Γ ( ξ ξ R (5. The tability of the ytem fixed point (i.e. the origin an be loally evaluated by alulating the eigenvalue of the Jaobian matrix of (5.. Rewriting (5. a Y & f ( Y where Y [ y ] T and [ f ( Y f ( Y ] T f, the Jaobian matrix i written a Y Y y f A (5. Auming ω and R to be away from zero, arrying out the differentiation yield A + ˆ, Γ (5.4 Γ Γ where Γ I rm tan λξm and ( + tan λ rm R ˆ d (5.5 ( + µ gn( Rω tan λ µ where d µ i the gradient of the oeffiient of frition urve v. relative veloity and i given by d r µ e + µ (5.6 µ r ω Note that ĉ i the equivalent damping oeffiient due to the veloity dependent frition and 68

88 beome negative if d <. The eigenvalue of the Jaobian matrix evaluated at the fixed point (i.e. A are given by µ + ˆ e, ± Γ (5.7 ( + ˆ e 4 Γ Γ When Γ >, the teady liding fixed point beome untable if + ˆ < (5.8 The above intability threhold an be tated alternatively in term of the applied axial fore, R. The ytem i untable if R > ( + µ gn( Rω tan λ r m ( + tan λ d µ and d < (5.9 µ Stable/untable region aording to (5.9 for varying value of R and µ i hown hematially in Figure 5-. Figure 5-: Loal tability of fixed point of the -DOF lead rew ytem. Violation of thi inequality may alo lead to intability. In Chapter 6, thi type of intability i diued in detail. 69

89 Expetedly, when negative frition damping i preent ( d <, there i a limiting value of axial fore, beyond whih the ytem beome untable. Thi limit proportionally inreae by inreaing the damping in the lead rew upport. It i intereting to note that thee finding are in agreement with the experimental reult reported in Chapter Numerial Example The parameter value ued in the numerial example preented here are given in Table 5-. Mot of thee value are taen from experimental tudy of Chapter 4 given in Table 4- and Table 4-. µ Table 5-: Parameter value ued in the imulation Parameter Value Parameter Value d m.7 mm µ. 8 λ 5.57 µ. I 6. g.m µ /rad N.m/rad r.8 rad/ 5 Nm/rad R ± N m.8 g ω ±4 rad/ For an axial fore of R ± N and input angular veloity of ω ± 4 damping oeffiient, r, aording to (5.9 are found a rad/, the ritial gn R ω + then If ( r.5 4 R ω then If gn( r.4 Figure 5- and Figure 5- how the ytem trajetorie for eah imulation the initial ondition wa y (, ω 4 < r and r >, repetively. In. A expeted, for the damping level below (above the ritial value, the fixed point i untable (table. In the untable ae, ytem trajetorie are attrated to a limit yle. Uing the method of averaging, in the next etion the periodi olution of the nonlinear equation of motion (limit yle are tudied and the amplitude of teady-tate vibration are etimated. It i intereting to note that reult in Figure 5- how that, in ae where fore and angular veloity have the ame ign (i.e. fore aiting the motion, the diplaement amplitude i oniderably maller than ae where the axial fore reit the motion (i.e. R ω <. 7

90 Figure 5-: ytem trajetorie for -4 < r; untable teady-liding fixed point (, Figure 5-: ytem trajetorie for -4 > r; table teady-liding fixed point (, 5. Firt-order Averaging The eigenvalue analyi of the previou etion doe not reveal any information regarding the behavior of the nonlinear ytem one intability our. The exitene of periodi olution (limit yle, region of attration of the table trivial olution, and the effet of ytem parameter on thee feature a well a the ize of the limit yle (amplitude of teady-tate vibration are important 7

91 iue that are addreed in thi etion. The powerful method of averaging i the perturbation tehnique utilized here to tudy the behavior of the -DOF lead rew model a a wealy nonlinear ytem. For the lead rew equation of motion to be onidered a a wealy nonlinear ytem, the frition and damping oeffiient mut be mall. The relative mallne requirement of thee parameter will be put into a more onrete etting later in the etion. Before performing the averaging, (5.7 mut be tranformed to tandard form [5]. To that end, ome implifiation are neeary. In the following etion, firt the equation of motion i implified and then onverted into a non-dimenionalized form. In the next tep, a mall parameter, ε, i introdued and the new dimenionle parameter are ordered to reah an approximate wealy nonlinear equation of motion aurate up to O ( ε. 5.. Aumption A mentioned earlier, the urrent tudy i only onerned with the intability aued by negative damping. Thu, it i aumed that Γ > for all u&. From (5., it i eay to ee that the equation of motion of the -DOF lead rew ha a diontinuity whenever θ & roe zero. To deal with thi ituation, the oeffiient of frition i moothed at zero relative veloity (i.e. u& + ω aording to µ r u& +ω r u& +ω ( u& ( µ + µ e + µ u& + ω ( e (5. where r > i a relatively large number. Note that in Chapter 4, a imilar frition-veloity relationhip wa ued. Subtituting (5. into (5. yield µ r u& +ω r u& +ω ( u, u& ( µ + µ e + µ u& + ω ( e gn( u& + gn N( u, u& ω ( (5. It mut be noted that, although (5. i diontinuou at ( u u N,&, the differential equation of the ytem, given by (5.7, i ontinuou, ine in it original form, given by (. and (., only the produt µ N appear. From (5.9, we have Similar moothed frition model were ued by other, ee for example [4,8,8]. 7

92 Γ R gn N gn + u + u& mrm tan λ ( Equation of Motion in Standard Form The firt tep toward tranforming the equation of motion to a proper form for averaging i to nondimenionalize the parameter. Thi i an important tep to appropriately order eah parameter aording to it ize. Expanding (5.7, yield I r m ( Rω ( Rω µ tan λ µ m u u u rm gn tan λ µ tan λ tan λ & + & + R (5. + µ tan λ + µ gn tan λ + µ tan λ Introdue the dimenionle time τ Ωt, where Ω I (5.4 The derivative with repet to τ i given a ( d( d dt Ω dτ (5.5 Alo, define non-dimenional parameter m m r m tan λ I (5.6 ~ (5.7 I R Ω ω rm R, ω (5.8 Uing thee new parameter, (5. i tranformed to µ tan λ m u + u ~ + u + µ tan λ ω µ gn Ω + µ gn ( Rω ( Rω tan λ µ tan λ R tan λ + µ tan λ (5.9 where prime denote derivate with repet to τ. Now that the equation of motion i in it non- 7

93 dimenionalized form, baed on phyial inight, parameter are ordered uing the mall poitive parameter ε. The new parameter, together with m and R are all aumed to be ( mall parameter. Auming, µ µ µ, µ (5. tan λ tan λ ~ (5. tan λ O with repet to ε where ε tan λ i taen a the ω Ω ρε where ρ i O ( and aling u a u ερv (5. give where [ εξ ( v, v, ε m] v + εv + v εr[ Ξ ( ε Ξ (, ε ] v ( ε gn + ε ( Rω µ gn( Rω µ Ξ (5. (5.4 Ξ ( v v, ε µ + ε, ( v, v, ε µ ( v, ε Alo, the expreion for the igned veloity-dependent oeffiient of frition, µ ( v, ε new dimenionle parameter, i µ r ω v +ω r ω v +ω ( v, v, ε ( µ ˆ + µ ˆ e + µ ˆ ωv + ω ( e gn R ( ωv + ω gn εξ ( ε R + v + εv m where µ ˆ µ tan λ, i,,. After rearranging, (5. beome; where, i i ( v,v ε (5.5, in term (5.6 v + v εf, (5.7 74

94 f [ ] ( v, v, ε ( εξ m v + mξ v + R( Ξ Ξ It i important to notie that, depite the preene of the two ign funtion (i.e. gn ( + R µ ε R + v + ε v m + ε µ gn in (5.8, ( v,v,ε repet to it argument for ( v, v, ε D [, ε ] ome ontant. To how thi, we only need to invetigate ( v,v,ε (5.8 v and f i bounded and Liphitz ontinuou with, and D i any ompat ubet of R and ε i f at intane where v + and N (whih i equivalent to R m εξ ( ε R + v + εv. For the firt ae, notie that µ v,, ε and µ i ontinuou at v, provided that N. Furthermore ( uniformly in ε. Alo, N. µ (,, (, v v ε µ v v, ε lim lim + v v v v µ ( v, v, ε v and µ ( v, v, ε ε ( µ ˆ + µ ˆ rωgn( N, N For the eond ae, let ( v R m ( R v v for all ( v v >, in the domain D and δ v,, ε εξ ε + + ε. Subtituting thi relationhip into f v v, ε ε δ εξ v, v, ε m R m + v Ξ v,, ε i bounded and { [ ] ( } { v, v, ε δ v,, ε } (5.8 give (, (. Sine ( ontinuou on [, ε ] ( v v D, and alo ine εξ ( v v,, εm i away from zero, f ( v,v,ε i ontinuou on D [, ε ]. Alo it i eay to ee that lim ( v, v, ε f lim δ v f v ( v, v, ε, lim ( v, v, ε, lim ( v, v, ε, lim ( v, v, ε, and lim ( v, v, ε δ + δ f v are bounded, thu onfirming the Liphitz ontinuity of (5.8 with repet to it argument. To tranform (5.7 into the tandard form, the following hange of variable i ued Thi lead to δ + f ε δ f ε δ + f v, exit and v a oϕ, v ain ϕ (5.9 Thi i the onequene of initial aumption Γ > (ee etion

95 ( aoϕ, ain ϕ, ε ϕ a εf in (5.4 ε ϕ f a o a ( aoϕ, in ϕ, ε ϕ (5.4 Sine ϕ i away from zero, dividing (5.4 by (5.4 yield da f ε dϕ ε f a ( aoϕ, ain ϕ, ε ( aoϕ, ain ϕ, ε in ϕ εg oϕ ( ϕ, a, ε ( Firt Order Averaging In thi etion, the averaging method ([5] i applied to (5.4. To obtain the firt order averaged equation, the right-hand ide of (5.4 mut be averaged over a period (i.e. T ontant. Thi give π while eeping a a ε π π ε π g π ( ϕ, a, f dϕ ( a oϕ, ain ϕ, in ϕdϕ (5.4 There are a few variation on the bai theorem for the periodi firt order averaging [5,44, 84]. In Appendix B, a lightly modified verion of the theorem proven in [44] i preented and proved whih etablihe the error etimate of the averaged ytem, (5.4, with repet to the original differential equation, (5.4. Subtituting (5.9 into (5.8 and then ubtituting the reult into (5.4 give Note that the hange of variable (5.9 i only allowed in ituation where the RHS of (5.4 remain bounded a a approahe zero [84]. Here, thi hange of variable i allowed when R i away from zero, ine after ~ f a o ϕ, a in ϕ, ε af a, ϕ, ε for ome bounded funtion expanding (5.8 uing power erie, we get ( ( ~ ( a,ϕ f for a < a and for uffiiently mall a < uh that N. For impliity of notation, from thi point on, prime denote differentiation with repet to ϕ. 76

96 a + ε π ε π π ( ain ϕ + main ϕoϕ + µ Rin ϕ π ( main ϕoϕ + Rin ϕ µ dϕ dϕ (5.44 After arrying out the integration of the firt term, beome where µ a ε a ε + π π ( main ϕoϕ + Rin ϕ µ ( ϕ, a dϕ r ω ω ain ϕ r ω ω ain ϕ ( ϕ, a ( µ ˆ + µ ˆ + µ ˆ e ω ω ain ϕ ( e gn R ( ω ω ain ϕ gn + aoϕ m (5.45 (5.46 The averaged differential equation given by (5.45 i too ompliated to be approahed analytially. Limiting our tudy to the ituation where ω > and alo where R > i large enough uh that N remain poitive over the domain of interet, µ implifie to µ r ω in ϕ ω in ϕ ( ϕ, ( µ ˆ + µ ˆ + µ ˆ ω in ϕ ( gn( in ϕ a r a a e a e a (5.47 Subtituting (5.47 into (5.45 and implifying yield a ε a ε + R π π in ϕµ ( ϕ, a dϕ (5.48 In addition to the aumption of N >, if the maximum amplitude i limited to, i.e. a, (5.47 further implifie to µ rωa in ϕ r ωain ϕ ( ϕ, a ( µ + µ e µ ain ϕ( r e (5.49 where r and alo r ω e µ ˆ + µ ω µ ˆ (5.5 e rω µ µ ˆ (5.5 77

97 µ ˆ µ ω (5.5 Subtituting (5.49 into (5.48 and implifying give a ε + ε π R a + ( + µ r µ ar π in ε µ π ϕe R π r ωain ϕ in ϕe dϕ Carrying out the ret of the integration, one find + µ R a ε a + εµ RΛ + εr µ arλ, rωa in ϕ ε π dϕ r µ R π ε π in ϕe r µ R π ( r + r in ϕe ωain ϕ dϕ ( r ωa εr µ RΛ ( rωa ( rωa εr µ RΛ (( r + r ωa,,, r ωain ϕ dϕ (5.5 (5.54 where π n m ζ in φ Λ n, m ( ζ in φo φe dφ (5.55 π General formulae for (5.55 are derived in Appendix C. For the integral in (5.54, one find Λ Λ,, ( ζ ( ζ n n n n ( n! ( n n ( n! ζ n ζ n n (5.56 (5.57 In the next etion, teady tate olution of (5.54 are tudied. It mut be noted that, in ae where table (untable non-trivial olution exit and a a, the above averaging proe guarantee that the original ytem, (5.9, ha table (untable limit yle in an O ( ε neighborhood of the irle r ωa Ω, with a period O ( ε to π for uffiiently mall ε > [84] Steady State Solution Subtituting (5.56 and (5.57 into (5.54 and rearranging 78

98 + µ R a ε a nω + εr n n n ( n! n n n n ( r µ ( n r + µ ωr r µ ωr r µ ω( r + r It i obviou that a i the trivial olution. To determine it tability, d a da i derived and evaluated at a. From (5.58 a n (5.58 da dd ε a [ ( + µ R + R( r µ + µ ωr r µ ωr r µ ω( r + r ] (5.59 da dd From (5.59, one an find a parametri ondition for the tability of the trivial olution (i.e. a <, whih i found to be > R + (( r µ + µ ωr r µ ωr r µ ω( r r By ubtituting the original ytem parameter, (5.6 i implified to > r r m µ R u& It i intereting to note that, (5.6 i aurate to ( ε linear eigenvalue analyi ((5.5 and (5.6: u& (5.6 (5.6 O when ompared to what wa found from + tan λ µ > r R, R >, ω > (5.6 m ( + µ tan λ y y Unfortunately, the other poible olution (i.e. table or untable limit yle an only be found numerially due to the omplexity of the averaged equation. However, ome important inight an be gained be examining (5.58. In Appendix D, it i hown that, depending on the ytem parameter one of the following three ae define the dynami behavior of the averaged ytem:. The trivial olution i table and no other olution exit.. The trivial olution i table and i urrounded by an untable limit yle, whih define the region of attration of the trivial olution. The untable limit yle i inide a table limit yle. 79

99 . The trivial olution i untable and i urrounded by a table limit yle. Two natural extenion of the averaging reult of thi etion are inluded a appendie. In Appendix E, the poibility of uing higher-order averaging to improve the auray of the predited vibration amplitude of the -DOF i hown. The method of firt-order averaging i applied to a - DOF model of Setion.6 in Appendix F Numerial Simulation Reult In thi etion, a few numerial example are preented. In thee example the ytem parameter value, unle otherwie peified, are thoe lited in Table 5-. For the parameter value and the initial ondition eleted, all imulation reult atify v and N > ondition. A a reult, the implified averaged ytem equation given by (5.5 or (5.58 i ued. Computationally, it i muh more effiient to ue (5.5 intead of the infinite um of (5.58. Table 5-: Parameter value ued in the imulation Parameter Value Parameter Value d m.7 mm µ. 8 λ 5.57 µ. I 6. g.m µ /rad N.m/rad r.8 rad/ 5 Nm/rad r rad/ m g ω 4 rad/ R N Figure 5-4 and Figure 5-5 how omparion between numerial integration of the approximate (trunated equation of motion given by ( v,v, v + v εf (5.6 and the fixed point of the averaged amplitude equation, (5.5, for two value of lead rew damping; < 4 4 r and > r, repetively. Note that in thee figure, both amplitude are aled by ω Ω to reflet the phyial ytem vibration level. Reult how very aurate predition of the teady-tate amplitude of vibration by the firt order averaging method. However, when ompared with the original (untrunated equation of motion, 8

100 (5.7, the averaging reult have ome differene a hown in Figure 5-6 and Figure 5-7. Thi deviation i aued by the effet of the higher order term omitted from the firt order averaging proe. It mut be noted that, the teady-liding amplitude of vibration in Figure 5-6 and Figure 5-7 are predited very aurately by the averaged equation for the parameter value given in Table 5-. u, ω/ω a (rad u, ω/ω a (rad Figure 5-4: Firt order averaging reult. -4 Grey: trunated equation of motion; Bla: amplitude of vibration from firt order averaging Figure 5-5: Firt order averaging reult. -4 Grey: trunated equation of motion; Bla: amplitude of vibration from firt order averaging 8

101 u, ω/ω a (rad u, ω/ω a (rad Figure 5-6: Firt order averaging reult. -4 Grey: original equation of motion; Bla: amplitude of vibration from firt order averaging Figure 5-7: Firt order averaging reult. -4 Grey: original equation of motion; Bla: amplitude of vibration from firt order averaging Figure 5-8 how the bifuration diagram of the amplitude equation, (5.5, where the damping oeffiient,, i taen a the ontrol parameter. The trivial olution (i.e. the fixed point of the original ytem undergoe a ubritial pith-for bifuration [6] at approximately 4 r. N. m. / rad. It an be hown that thi bifuration orrepond to a Hopf bifuration of the original ytem [6]. The untable branh, hown by the dotted line, determine the domain of 8

102 attration of the trivial or teady-liding fixed point. Figure 5-8: Bifuration diagram of the averaged amplitude equation. Stable; Untable The limiting value and in Figure 5-8 orrepond to the limit diued in Appendix D, Setion D.. Figure 5-9 and Figure 5- how the effet of Stribe frition ( µ and linear negative frition ( µ parameter on the amplitude bifuration diagram, repetively. In thee figure, bifuration plot are drawn with repet to the applied axial fore, R, a the ontrol parameter. A hown, µ ontrol the domain of attration of the table trivial olution without ignifiant hange to the limiting value of R. The reaon for thi i that the term r ω e i negligible for the onidered value of µ r and ω (ee (5.6. However, µ diretly ontrol the threhold of intability of the trivial or teady-tate olution. Further example and diuion are preented in Appendix D. 8

103 Figure 5-9: Effet of Stribe frition on bifuration Figure 5-: Effet of negative damping on bifuration. 5.4 Conluion In thi hapter, uing the -DOF model of a lead rew drive developed in Setion., the intability aued by the negative gradient of the frition oeffiient with repet to veloity wa tudied. The loal tability of the teady-liding fixed point of the ytem wa tudied by examining the eigenvalue of the Jaobian of the linearized ytem. It wa hown that the teady-liding fixed point of the ytem loe tability if the ondition given by either (5.8 or (5.9 i atified. 84

104 The eigenvalue analyi reult wa extended by the appliation of the method of averaging. It wa hown (ee Appendix D that depending on the ytem parameter, one the following ae define the dynami behavior of the ytem:. The trivial olution i table and no other olution exit.. The trivial olution i table and i urrounded by an untable limit yle that define the region of attration of the trivial olution. The untable limit yle i inide a table limit yle. The preene of Stribe effet i a neeary ondition in thi enario.. The trivial olution i untable and i urrounded by a table limit yle. The numerial imulation reult preented, alo howed the appliability of the averaging reult in approximating the amplitude of periodi vibration. The auray of the approximation a preented in Appendix E an be improved by uing higher order averaging. 85

105 Chapter 6 Kinemati Contraint and Mode Coupling Intability Mehanim In Chapter, three frition-indued intability mehanim were introdued. Negative damping intability wa tudied in the previou hapter. In thi hapter, the two remaining intability mehanim, i.e. inemati ontraint and mode oupling, are tudied. In ontrat to the negative damping intability, thee two mehanim an affet a ytem even when the oeffiient of frition i ontant. The mode oupling intability mehanim i exluive to multi-dof ytem. In Setion 6., the inemati ontraint intability mehanim i tudied uing the bai -DOF lead rew model of Setion.. Mode oupling intability i tudied in Setion 6. and 6. uing the -DOF model of Setion.5 and Setion.6, repetively. In thee etion, the inemati ontraint intability i alo tudied. Mode oupling in the -DOF model of Setion.7 i diued in Setion 6.4. In eah etion, numerial example are given to demontrate the finding. Conluion drawn in thi hapter are reviewed in Setion Kinemati Contraint Intability in -DOF Lead Srew Model To tudy the inemati ontraint intability, the ame -DOF model ued in the previou hapter i onidered here. For impliity, a ontant oeffiient of frition i aumed (i.e. µ µ µ. The eigenvalue of the Jaobian matrix given by (5.7 are implified to, e Γ (6. e ± 4 Γ Γ From (6., it i evident that regardle of the linear damping (, divergene intability our 86

106 whenever Γ <. In term of ytem parameter, the fixed point i untable whenever the following inequalitie hold imultaneouly: I II III µ > tan λ Rω > Γ I rm tan λξm < (6. The firt ondition in (6. i the elf-loing ondition in lead rew drive [5,67]. The eond ondition tate that intability an only our if the axial fore (load i applied in the diretion of motion (aiting. The third ondition etablihe a limiting ratio between lead rew inertia and the tranlating ma. 6.. A Note Regarding the Solution of the Equation of Motion Before preenting a numerial example, it i worthwhile to tudy the untable behavior of the ytem when ondition (6. are fulfilled. Setting F and T to zero for impliity, the ontat fore given by (.6 beome N ( I mrm tan λξ R + mrm tan λ( u + u& ( oλ + µ in λ( I mr tan λξ m (6. where the hange of variable (5. wa ued. Limiting our tudy to a ae where R > and ω >, (5.5 implifie to ξ µ λ r tan + µ tanλ (6.4 m Alo, in the ae of a ontant oeffiient of frition with u& + ω >, ξ given by (.4 redue to ( N tan ( N tan λ µ gn λ ξ r m (6.5 + µ It an be een that, under the inemati ontraint intability ondition, if See Setion 6.. below for further diuion. Thi relationhip hold for other ytem onfiguration (See Setion. where I and m deignate the inertia of the rotating part and tranlating part of the lead rew drive, repetively. 87

107 mr ( u + u > ( I mr tan λξ R tan λ & (6.6 m m then (6. ha no olution; etting gn ( N in the RHS of (6. reult in a negative ontat fore, and etting gn( N reult in a poitive ontat fore. If uh a ituation happen, further motion i impoible and the lead rew i to be onidered tationary (i.e. u& + ω. The intantaneou lead rew eizure i aompanied by an infinite impule-lie normal fore. On the other hand, if under the ame ondition a above mr ( u + u < ( I mr tan λξ R tan λ & (6.7 m m then (6. ha two ditint olution: N ( I mrm λξ R + mrm tan λ( u + u& ( oλ + µ in λ( I mr tan λξ tan, N > m + (6.8 and N ( I mrm λξ R + mrm tan λ( u + u& ( o λ + µ in λ( I mr tan λξ tan, N < m (6.9 where ξ + ξ µ λ r tan + µ tan λ and µ + tan λ ξ r m. µ tan λ m Thee two olution in turn lead to two different poible olution for the equation of motion given in tate-pae form by (5.. One find ( u + u& N > u& and I mr m tan λξ + ( u + u& N < u& (6. I mr m tan λξ Dupont [4] ha tudied the problem of exitene and uniquene in the forward dynami equation of fritional ytem and reported imilar reult for a lead rew. A poible way to reolve thi problem i to give the ontating bodie flexibility [85, 86]. Thi i done in the model preented in Setion.5, where nut and lead rew thread an deform. In the numerial example preented next, both rigid and flexible model are ued to imulate the prag-lip behavior of the lead rew. 6.. Example The parameter value ued in the numerial imulation of thi etion are lited in Table 6-. Firt, 88

108 notie that the elf-loing ondition i atified for the eleted value of the ontant oeffiient of frition (i.e..8 > tan( µ. Table 6-: Parameter value ued in the imulation Parameter Value Parameter Value d 5 m.7 mm Nm/rad λ 5.57 R N 6 I. g.m µ. 8 N.m/rad ω 4 rad/ Uing (6., the ritial tranlating ma i found to be; m r. (g. Figure 6- how the evolution of the real and imaginary part of the two eigenvalue given by (6. a the tranlating ma, m, i varied. For m > mr, the ytem loe tability due to divergene. Figure 6-: Evolution of the eigenvalue a the tranlating ma, m, i varied. and Figure 6-(a and Figure 6-(b how the phae plane plot of the -DOF model for m (g m (g, repetively. It an be een that, by roing the inemati intability threhold, the ytem beome untable. Thi intability i haraterized by a violent motion aompanied by very high deeleration and ontat fore. To ee what happen during the prag phae, the ame ytem parameter are ued in the numerial imulation of the -DOF model of Setion.5. In thi example, very high ontat tiffne 89

109 and damping value are eleted; 8. The y y projetion of the trajetorie hown in Figure 6-(a i almot inditinguihable from the -DOF ytem trajetorie plotted in Figure 6-(b. The impule-lie pea in the ontat fore a the ytem goe through the prag phae i hown in Figure 6-(b. For the eleted value of the ontat tiffne and damping, thi fore pea to about N. Figure 6-: Sytem trajetorie for ontant µ. (a m < m r (b m > m r Figure 6-: Intability aued by inemati ontraint - DOF model with very high ontat tiffne and damping. (a phae-plane, (b ontat normal fore A mentioned earlier, damping doe not affet the tability of the -DOF model when the inemati ontraint intability mehanim i ative. However, damping ha a oniderable effet on 9

110 the behavior of the nonlinear ytem. Figure 6-4 how phae plot of the -DOF model with three level of lead rew upport damping. Figure 6-4: Effet of damping on the untable ytem For eah of thee three imulation reult, the line ( u, u N & i alo drawn. In agreement with the diuion of Setion 6.., the onet of lead rew eizure i the point where the trajetory reahe thi line. Note that from (6., the line ( u, u N & i given by I mrm tan λξ u& u R mr (6. m tan λ A damping i inreaed, the amplitude of vibration i lightly redued. A hown in Figure 6-5, inreaing damping inreae the mean defletion of oupling element, whih inreae the mean thread normal fore. The reult preented in Figure 6- were obtained uing a -DOF with very high ontat tiffne and damping. A hown in Figure 6-6, by dereaing the ontat parameter (i.e. and the trajetorie beome moother and the defletion of the oupling element (i.e. torional pring, beome poitive during the prag phae. In the next etion, the tability of thi -DOF model i tudied in detail. 9

111 Figure 6-5: Effet of damping on the teady-tate vibration of the lead rew ytem under inemati ontraint intability Figure 6-6: Effet ontat parameter on the repone of the ytem under inemati ontraint intability 6.. Region of Attration of the Stable Steady-liding Fixed Point The linear eigenvalue analyi of Setion 6. howed that when the ondition given by (6. are not atified and >, the trivial fixed point of the ytem i aymptotially table. However, there an be ituation where the region of attration of the table fixed point i quite mall, leading to intabilitie even when (6. doe not hold. 9

112 Conider the ae where µ > tan λ, R <, and ω >. It i obviou that only the firt ondition of (6. i atified and hene the teady-liding fixed point i table. Further, aume that µ tan λ I rm tan λ m < (thi i the third ondition of (6. if R >. Firt, notie that for + µ tan λ any initial ondition uh that ( u(, u( > + reet to zero (i.e. ( ω N &, no motion i poible and veloity intantaneouly u&. Thi behavior onform to the ame argument a in Setion 6.., when no olution exit (i.e. etting gn ( N in (6. reult in a negative ontat fore and etting gn ( N yield a poitive ontat fore. At thi point (i.e. ( u (, ω ytem trajetory follow a path below the, if N i negative, the u& ω line (i.e. revered rotation of the lead rew and reahe the N line again. Thi yle ontinue until the point ( I mr λξ m tan R ω +, ω mrm tan λ, where the N line interet the horizontal u& ω line. Alo note that initial motion from ondition where N ( u(, u& ( < and ( < ω u& i not poible (thi alo follow a imilar argument a Setion 6.., and the ytem trajetory intantaneouly tranfer to ( (, ω u from whih the motion ontinue toward to origin. The olution that tart from initial ondition, atifying N ( u(, u& ( < and ( ω u& are attrated toward the origin and, if they do not touh the N line, reah it aymptotially. Beaue of the aumption of ontant oeffiient of frition, (5.7 i implified and thee olution atify Γ u & + u& + u (6. If any of thee trajetorie reah the N line ay at t t intantaneouly and tart from the ret at ( u (, ω t, then the motion top. Thi pattern ontinue and may even reult in a limit yle at teady tate. Otherwie, the olution reahe the origin aymptotially. and Figure 6-7(a how two trajetorie tarting well away from the equilibrium point for N. m. rad R 5. Other ytem parameter are taen from Table 6-. Although trajetorie ro the N line, the origin i table. In Figure 6-7(b, the applied axial fore i inreaed to R N while the other parameter are unhanged. In thi ae, the ytem trajetorie are attrated to a limit yle. N 9

113 Figure 6-7: Sytem trajetorie for ω4 (rad/. (a R-5 (N; (b R- (N From the above diuion, one an onlude that, if trajetory tarting from ( u, ω where ( I mr λξ R i large enough uh that every m tan R ω u + aymptotially reah the mr tan λ origin, the teady-liding fixed point i globally table. Otherwie, the region of attration i only a ubet of R. 6. Stability Analyi of the -DOF Lead Srew Model with Compliant Thread In thi etion, the -DOF model of Setion.5 with ontant oeffiient of frition i onidered. In what follow, firt, the equation of motion of the ytem are implified and the teady-liding fixed point i tranferred to the origin by introduing a uitable hange of variable. Then, the loal tability of the teady-liding fixed point i tudied by evaluating the eigenvalue of the Jaobian matrix of the linearized ytem. In Setion 6.., the ae of an undamped ytem i analyzed and neeary and uffiient ondition for intability are derived. The ae of a damped ytem, whih i omewhat more ompliated, i treated in Setion 6... The analye preented involve both analytial and numerial approahe. The equation of motion given are given by (.5 and (.6 m 94

114 I&& θ + r ( θi θ θ & + rm ( x oλ rm θin λ( in λ µ oλ ( x& o λ r θ& in λ( in λ µ o λ m mx && m ( xoλ rm θin λ( oλ + µ in λ ( x& oλ r θ& in λ( oλ + µ in λ + R m (6. (6.4 where µ i defined by (.5. Alo, here F and T are negleted for impliity. To implify the ubequent analyi, the teady-liding fixed point of the ytem i tranferred to the origin. Toward that end, let u u θ θ i x r m tan λθ i (6.5 Subtituting (6.5 into (6. and (6.4 and etting u & u& u&& u& yield u u and u u where ξ ω R (6.6 u R u + r tan λu o λ + µ gn (6.7 m ( Rω tan λ where ξ i given by (5.5. To tranfer the fixed point to the origin, let y y θ u x u θ r tan λ m θ i i (6.8 Subtituting (6.8 into (6. and (6.4 and rearranging yield y y y and where [ ] T ( y y& M & y + Cy& + Ky f, (6.9 I M (6. mˆ 95

115 ( Rω ot λˆ ( ( ( ˆ + µ Rω λˆ gn ot Rω tan λˆ ˆ + µ gn Rω tan λˆ + ˆ µ gn K (6. ˆ µ gn ( Rω ot λˆ ˆ + µ ( Rω λˆ gn ot ( Rω tan λˆ ˆ + µ gn( Rω tan λˆ + ˆ µ gn C (6. ˆ µ gn ot λ f ( y, &y ( µ ( ω µ ( ( + ω ( ˆ ˆ + ˆ ˆ + ˆ gn R gn N y& y y y& y& R (6. tan λ where N i given by (.4 and ˆ r in λ (6.4 m ˆ r in λ (6.5 m mˆ r m tan λm (6.6 Rˆ + µ rm tan λ gn ( Rω R. (6.7 tan λ Note that for a ontant oeffiient of frition ( µ µ, the nonlinear fore vetor given by (6. i non-zero only when the repone trajetory reahe the ti-lip boundary ( v or when the ontat fore hange ign. Setting the fore vetor to zero give the linearized verion of the equation of motion for the ae of ontant oeffiient of frition. One find M & y + Cy& + Ky (6.8 From (6. and (6., the role of frition in breaing the ymmetry of the tiffne matrix and the damping matrix i evident. Beaue of thi effet, either by varying the ontant oeffiient of frition ( µ, the oupling torional tiffne (, or the ontat tiffne (, the linearized ytem (6.8 an have idential undamped natural frequenie, whih define the onet of the flutter intability [4]. 96

116 6.. Undamped Sytem The natural frequenie of the undamped ytem are the root of the following equation whih i a quadrati equation in where det( K ω M (6.9 ω. Expanding (6.9 yield 4 a ω + a ω + a (6. 4 a a a 4 mi ˆ ˆ mˆ mˆ (6. ˆ ot λ( µ gn( Rω tan λ ˆ I( + µ gn( Rω tan λ ( + µ gn( Rω tan λ Sine a 4 > and a >, intability our (i.e. natural frequenie beome omplex number whenever or a > ( < Intability ondition given by (6. an be rearranged a a a a a (6. ( + µ gn( Rω tan λ ˆ ( I ξ r tan λm ˆ m m > (6.4 where ξ i given by (5.5 and alo (6.6 wa ued. Obviouly, if Γ I ξrm tan λm >, inequality (6.4 annot be atified. On the other hand, if gn ( ω R and µ > tan λ then for uitable value of ytem parameter, the inequality Γ < i atified. In thi ae, the ytem i untable if In thi wor, it i aumed that the ondition; µ<ot λ alway hold. Violation of thi ondition would require a very high oeffiient of frition in a lead rew with a helix angle greater than 45 o, whih i not enountered in any pratial ituation [4]. 97

117 o λ( + µ tan λ ξ I r m tan λ > m (6.5 where (6.4 and (6.6 were ued. It i intereting to note that the neeary (but not uffiient ondition for intability to our aording to (6.5 i the ame a the inemati ontrain intability ondition derived in the previou etion for the -DOF model. Here, thi type of intability i alled the eondary inemati ontraint intability mehanim to ditinguih it from the ae of the previou etion, where gn ( ω intability. R, µ > tan λ, and Γ were uffiient ondition for Moreover, ine a atifie (6., the boundary that i defined by a i inide the region defined by (6.. A <, the undamped -DOF model redue to the -DOF model of the previou etion and (6.5 beome the ame a the ondition for the inemati ontraint intability of the -DOF model. The eond intability ondition, given by (6., repreent the mode oupling (flutter intability. The equation for the flutter intability boundary (i.e. oaleene of the two real natural frequenie i found by replaing the le-than ign with the equal ign in (6.. After ome manipulation, one find where ˆ ˆ b + b + b (6.6 b b b [ mˆ ( µ gn( Rω ot λ + I( + µ gn( Rω tan λ ] mˆ [ mˆ ( µ gn( Rω ot λ I( + µ gn( Rω tan λ ] mˆ (6.7 Thi equation i quadrati in and ˆ and an be olved to find parametri relationhip for the onet of the flutter intability. The ondition for the olution to be real poitive number are (either for a a funtion of ˆ or vie vera 4 b b b (6.8 b < (6.9 98

118 Inequality (6.8 an be expreed a ( µ gn( Rω ot λ ( + µ gn( Rω tan λ 6m ˆ I (6.4 whih hold if and only if ξ >. Thi in turn require that The eond inequality (6.9 an be expreed a whih i atified for ξ >. µ tan λ R ω > (6.4 ( + µ gn( ω tan λ( I + r tan λξ m > R m (6.4 Hene, for the undamped ytem, mode oupling an only happen in the elf-loing lead rew drive where R ω > (i.e. where the applied fore i in the diretion of the tranlation Numerial Example and Diuion Figure 6-8 how the tability region of the -DOF model (6.9 for the parameter value in Table 6- and 5 m. Note that ( µ Γ for µ.. A a reult, tability i not affeted by (6.. > The hathed region orrepond to the parameter range, where the two natural frequenie are omplex and the ytem i untable. The boundary of thi region i the flutter intability boundary defined by ω ω. 99

119 Figure 6-8: Stability of the -DOF ytem a ontat tiffne and oeffiient of frition are varied. m 5 and Rω>. Hathed region: untable. 6 It i intereting to note that the flutter boundary i tangent to the µ tan λ line (at.7. A predited in the above, the intability region lie entirely on the right of thi line. Figure 6-9 how the variation of the real and imaginary part of the eigenvalue of the undamped ytem for µ. > tan λ. It an be een that flutter intability our a the two natural frequenie merge. Further inreae of the ontat tiffne unouple the two mode and the tability i retored. Figure 6-9: Variation of the real and imaginary part of the eigenvalue a the ontat tiffne i varied. µ., m 5.

120 For larger value of the tranlating ma, m, inequality (6. an alo beome ative in the onidered range of parameter, thu reating a mixed ituation where both mode oupling and eondary inemati ontraint intability mehanim influene the tability of the ytem. An example of thi ituation i given by the tability map of Figure 6-. The tranlating ma i inreaed to m 5g, reulting in Γ vanihing at µ. 78. The hathed region in thi figure how parameter range where thee two mehanim are ative. Conider the variation of the ytem parameter µ and along the dahed line in Figure 6-. In the table region, the two frequenie are ditint real number. At the flutter intability boundary, the, > two frequenie merge, i.e. ω a, a. By further inreaing the parameter, the frequenie, > beome omplex-valued, i.e. ω a ± ib, a, b. At the eondary inemati ontraint intability boundary the real part of,, > ω vanihe, i.e. ω ± ib, b. If the parameter are inreaed even further, the real part of the quared frequenie beome negative, i.e., ω a ± ib, a, b >. At the eond boundary of flutter intability, the quared frequenie are, > idential and purely imaginary, i.e. ω a, a. Beyond thi threhold, quared frequenie are different but remain purely imaginary. Figure 6-: Stability of the -DOF ytem a ontat tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region.

121 The variation of the real and imaginary part of eigenvalue (i.e. natural frequenie for µ.8 >.78 a i varied i plotted in Figure 6-. Mode oupling intability our at the flutter boundary: It i intereting to note that, due to the eondary inemati ontraint intability, the ytem remain untable even when i large enough that the two mode deouple. (a (b Figure 6-: Variation of the real part (a and imaginary part (b of the eigenvalue a the ontat tiffne i varied. µ.8, m Damped Sytem The eigenvalue of the damped -DOF ytem ( η, i K4 are the olution of the fourth-order equation det( η M + ηc + K (6.4 Auming all of the ytem parameter to be non-negative, the tability ondition baed on the Routh-Hurwitz riterion are found a D i D D + mˆ (6.44 ˆ > ( + µ gn( ω tan λ ˆ ˆ ˆ D + m + R (6.45 >

122 D mi ˆ ( ˆ + ˆ + µ gn( Rω ( tan λ D > (6.46 { ( ˆ ˆ ( gn( tan }( ˆ ˆ ˆ D D mi ˆ + + µ Rω λ + D > (6.47 where ( + µ gn( Rω tan λ + mˆ ( µ gn( ω λ D I ot (6.48 R Thee inequalitie are too ompliated to be ueful in etablihing loed-form parametri tability boundarie. However, ome important peial ae an be proven, whih are lited here. Speial Cae No. : The ytem fixed point i table when the fore i applied oppoite to the nut tranlation diretion, i.e. R ω <. Speial Cae No. : The ytem fixed point i table when the fore i applied in the diretion of the nut tranlation and the lead rew drive i not elf-loing, i.e. R ω > and µ < tan λ. It i lear that elf-loing, a well a appliation of the load onto the nut in the diretion of travel, are the two neeary (but not uffiient ondition for the intability to our. Speial Cae No. : The ytem fixed point i untable when R ω >, µ > tan λ, and. Speial Cae No. 4: The ytem fixed point i untable when R ω >, µ > tan λ, and. The preene of damping only in the rotational DOF (i.e. lead rew upport damping, or in the tranlating DOF (i.e. ontat damping, detabilize the ytem. Similar qualitative obervation are found in the literature involving imple ytem (ee for example [87-9]. Speial Cae No. 5: For very large ontat tiffne and damping (i.e.,, the

123 ytem fixed point i table if D > and untable if D < whih agree with the ondition for the inemati ontrain intability of the -DOF model in Setion 6. ine gn D gn Γ Numerial Example and Diuion In Figure 6-, the two damping oeffiient are hoen a and 5. The other ytem parameter are eleted a before, and m 5. It an be een that the addition of damping, ontrary to ommon experiene, ha redued the tability region. Figure 6-: Region of tability of the -DOF model with damping. Bla: table, white: untable. Rω>, m5,, and 4-5. Variation of the eigenvalue for thi ae are plotted in Figure 6- for µ. 5. The oaleene of the natural frequenie an be een in thi figure. It mut be noted that, due to the preene of damping, the two frequenie do not math exatly, and the intability region doe not neearily orrepond to the range where they are loe. Mathing of the frequeny of the two oupled mode i exat for the peial ae of proportional damping (ee for example [9]. 4

124 (a (b Figure 6-: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. µ.5, m 5,, and 4-5. By inreaing the damping, a hown in Figure 6-4, the table region i expanded beyond the intability region of the undamped ytem. In thi figure the damping oeffiient are and 4 4. Similar to Figure 6-, Figure 6-5 how that the evolution of the real and imaginary part of the eigenvalue a i varied, for µ. 5. The inreaed damping ha reulted in the overdamping of the lower mode of vibration for roughly 5 <.96. More importantly, in thi higher damping level, the range over whih the two natural frequenie are loe ha been almot ompletely eliminated. 5

125 Figure 6-4: Region of tability of the -DOF model with damping. Bla: table, white: untable. Rω>, m5,, and 4-4. (a (b Figure 6-5: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. µ.5, m 5,, and 4-4. Similar to the undamped ae, the effet of eondary inemati ontraint intability on the damped ytem i onidered next. The tranlating ma i now inreaed to m 5g. At thi value, Γ hange ign at approximately µ µ. 78. Figure 6-6 how the tability region of the ytem for and

126 Figure 6-6: Region of tability of the -DOF model with damping. Bla: table, white: untable, Grey: region of intability of the undamped ytem Rω>, m5,, and 4-4 A it wa the ae in the previou example, the addition of damping ha dereaed the table parameter range. The evolution of real and imaginary part of the eigenvalue for two value of ontant oeffiient of frition i plotted in Figure 6-7. (a (b Figure 6-7: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. m 5,, and 4-4. Solid: µ.5; dahed µ.. 7

127 A hown, at µ. 5 the ytem beome untable approximately for ytem i untable over the entire range ontat tiffne value for µ.. 5 > 9.5. The By further inreaing the damping oeffiient, the table region grow. However, due to the preene of eondary inemati ontraint intability (i.e. Γ <, the table region i bounded by the vertial line D where D i defined by (6.44. The bound in term of the frition oeffiient i found from thi equation a + I + mˆ ˆ µ d tan λ (6.49 I ˆ tan λ m It i intereting to note that in (6.49 only the ratio of the two damping oeffiient appear. Conequently, for a fixed ratio of ˆ, the ytem will be untable for ize of damping oeffiient. It mut be mentioned that, for large µ > µ d regardle of the ˆ, the bound obtained by (6.49 will be too high a intability may our at muh lower value of the oeffiient of frition. On the other hand, for ˆ, d µ µ, where ( µ Γ ; thi beome the peial ae No. 4 mentioned above where the ytem i untable for µ > tan λ. Figure 6-8 how the tability map of the ytem for 4 4 and 8, whih are time higher than the value ued in Figure 6-6. The ratio of damping oeffiient here i ˆ.8. The table region i entirely on the left ide of the vertial line defined by µ µ d. In Figure 6-9, the evolution of the real and imaginary part of the eigenvalue are plotted for the ame parameter value a in Figure 6-8 and µ. 8. The lower mode of vibration i overdamped almot over the entire range of. The ytem beome untable for 7 >.65. 8

128 Figure 6-8: Region of tability of the -DOF model with damping. Bla: table, white: untable, Grey: region of intability of the undamped ytem. Rω>, m5, 4 4, and 8 - (a (b Figure 6-9: Variation of the real part of the eigenvalue (a and the natural frequenie (b, a the ontat tiffne i varied. m 5, 4 4, 8 -, and µ.8 6. Stability Analyi of a -DOF Model with Axially Compliant Lead Srew Support In thi etion, the -DOF model of Setion.6 i onidered. A will be diued below, the 9

129 tability ondition of the lead rew model with axially ompliant lead rew upport bear many reemblane to the tudy of previou etion, where the oure of flexibility wa the ompliant thread (ee Appendix G for a omparion of the tability ondition of thee two ytem. There are ome ditint and important differene that will be highlighted in what follow. Firt, the ytem model i implified and onverted to a uitable form for linearization and tability analyi. Then, imilar to the previou etion, the loal tability of the teady liding fixed point i analyzed uing the eigenvalue analyi method. The tability of the undamped ae i diued in Setion 6.. while the ytem with damping i overed in Setion 6... In thee analye, both mode oupling and inemati ontraint intability mehanim are enountered. The equation of motion are given by (.9 and (. ( I ξmr λ && θ ξmx && ( θ θ θ& ξr m tan (6.5 ( m m x& + mrm λθ && x x + R i + & tan & (6.5 where, one again, F and T are negleted for impliity. To bring the teady-liding fixed point to the origin, the following hange of variable i ued θ y x r m + θ + u i tan λy + u (6.5 where ω ξ R (6.5 u Subtituting (6.5 into (6.5 and (6.5 and after ome implifiation where R u (6.54 ( y y& M & y + Cy& + Ky f, (6.55 I ξmrm tan λ ξmrm tan λ M (6.56 m m + m

130 C (6.57 K (6.58 ( ξ ξ f (6.59 R Notie that, unlie the model tudied in Setion 6., here the inertia matrix i aymmetri, and tiffne and damping matrie are ymmetri. For the ae of ontant oeffiient of frition, the linearized equation of motion are given imply by where where ξ i given by (5.5. M & y + Cy& + Ky (6.6 I ξmrm tan λ ξmrm tan λ M (6.6 m m + m Similar to the tep taen in previou etion, the ae of undamped ytem i onidered firt, and then the damped ytem i tudied. 6.. Undamped Sytem The natural frequenie of the undamped ytem are the root of the following equation: whih i a quadrati equation in where ( ω M det K (6.6 ω. Expanding (6.9 yield 4 a ω + a ω + a (6.6 4

131 a a a 4 I ( m + m mmξ rm tan λ ( m + m ( I ξ mr tan λ m (6.64 Sine a >, intability our (i.e. natural frequenie beome omplex number whenever a 4 < (6.65 or a > a ( > or 4 4 < a a a (6.67 In term of ytem parameter, the intability ondition given by (6.65 an be written a where ~ Γ I m~ ξ r tan λ (6.68 m < m~ mm m + m ~ The neeary ondition for Γ i < (6.69 > ξ whih, in turn, require gn ( ω R and µ > tan λ. It i intereting to note that (6.68 tae the form of the inemati ontraint intability ondition given by (6. with m ~ a an equivalent tranlating ma. The intability ondition (eondary inemati ontraint aording to (6.66 an be written a m ( I mξrm tan λ + mi > ( m + m + ( I ξ mr tan λ < m (6.7 For the intability to our aording to (6.7, the following ondition are neeary and uffiient I mξ r tan λ (6.7 m <

132 ξmr m tan λ I > m + m (6.7 It i intereting to note that inequality (6.7 i the ame a the inemati ontraint intability ondition given by (6.. Finally, inequality (6.67 give the neeary and uffiient for the mode oupling intability. Replaing the le-than ign with an equal ign for the intability boundary and after implifiation, one find where b + b + b (6.7 b b b ( I ξmrm tan λ [ m ( ξmrm tan λ I m( ξmrm tan λ + I ] ( m + m (6.74 Thi equation i quadrati in and and an be olved to find parametri relationhip for the onet of the flutter intability. The ondition for the olution to be real poitive number are 4 b b b (6.75 In term of ytem parameter, inequality (6.75 beome whih yield b < (6.76 mm 6ξm r tan ( ξ m λ m + m I rm tan λ (6.77 m + m The eond inequality given by (6.76, yield mm I ξ rm tan λ ξ > (6.78 m + m ( m m m m + m ξrm tan λ I (6.79 <

133 whih i atified whenever (6.78 i atified. From (6.68 and (6.78, it an be onluded the mode oupling and the (primary inemati ontraint intability region have no overlap in the parameter pae Numerial Example and Diuion Figure 6- how the tability region of the undamped -DOF model in the µ parameter pae. Other ytem parameter value not given in the figure are eleted aording to Table 6-. The hathed region orrepond to the parameter range where the two natural frequenie are omplex and the ytem i untable. The boundary of thi region i the flutter intability threhold defined by ω ω. In thi figure, flutter boundary i plotted for two other higher value of the lead rew ma, m. For mall value of lead rew ma, thi plot i almot idential to Figure 6-8. Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied. m 5 and Rω>. Hathed region: untable. By inreaing the tranlating ma to m 5, the ondition (6.7 i atified for approximately µ > µ.78. A hown in Figure 6-, there are two overlapping area in the µ parameter pae that ontitute the untable region. Thee region orrepond to the mode oupling and eondary inemati ontraint intability mehanim. 4

134 Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region. For the above mentioned value of the tranlating ma, if m. 6, then ondition (6.68 i atified for µ > µ. 85. A a reult, the vertial line µ. 85 beome the divergene boundary in the parameter pae, a hown in Figure 6-. The variation of the real and imaginary part of the eigenvalue (i.e. ω are plotted in Figure 6-6 for 4 a a funtion oeffiient of frition. For thi value of the upport tiffne, the ytem beome untable due to mode oupling a the real part of the two eigenfrequenie merge. A eond intability region alo exit for µ >. 85, where the ytem beome untable due to the inemati ontraint. Similar to the reult of Setion 6., at the threhold of the inemati ontrain intability, one eigenvalue i at infinity. 5

135 Figure 6-: Stability of the -DOF ytem a upport tiffne and oeffiient of frition are varied, when m 5 and Rω>. The hathed area: mode oupling intability region; the hathed area: eondary inemati ontraint intability region, and; the hathed area: primary inemati ontraint intability region. (a (b Figure 6-: Variation of the real part (a and imaginary part (b of the eigenvalue a the ontat tiffne i varied. 4 6, m 5, and m.6 6

136 6.. Damped Sytem The eigenvalue of the damped -DOF ytem ( η, i K4 are the olution of the fourth-order equation i ( η + ηc + K det M (6.8 Auming all of the ytem parameter to be non-negative number, the tability ondition baed on the Routh-Hurwitz riterion are found to be ( m + m mm ξ r tan λ D (6.8 I m > ( m + m + ( I ξ r tan λ D m (6.8 m > ( m + m + + ( I ξ mr tan λ D (6.8 m > ( + D D (6.84 D > { D ( + D }( + D D (6.85 > One again, the reulting ondition are too omplex to be ueful for parametri tudy of tability. However, a hown in Appendix G, there are many imilaritie between thee ondition and thoe derived for the -DOF model in Setion 6.. Expetedly, imilar peial ae a in Setion 6.. an be proven for the model of thi etion. Thee ae are lited here. Speial Cae No. and : The ytem fixed point i table if ξ <. From the definition of ξ, (5.5, it i eay to ee that ξ < only if R ω < or µ < tan λ, whih oinide with the firt and eond peial ae of Setion 6... Speial Cae No. : The ytem fixed point i untable when R ω >, µ > tan λ, and. Speial Cae No. 4: The ytem fixed point i untable when R ω >, µ > tan λ, and 7

137 . Speial Cae No. 5: The ytem fixed point beome untable due the inemati ontrain intability mehanim when D < Numerial Example and Diuion Figure 6-4 and Figure 6-5 how ample of the region of tability of the ytem a µ and are varied. The hathed region in thee figure i opied from Figure 6- and orrepond to the region of intability of the undamped ytem. In Figure 6-4 the two damping oeffiient were hoen a and 5. It an be een that the addition of damping ha redued the tability region. However, by inreaing the damping further, a hown in Figure 6-5, the table region i expanded beyond the intability region of the undamped ytem. In thi figure the damping oeffiient were and 4 4. In both erie of example, it an be een that by inreaing the lead rew ma, m, the tability region grow toward the untable region of the undamped ytem. (a (b ( Figure 6-4: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5,, and -5 8

138 (a (b ( Figure 6-5: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m. g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5,, and 4-4 If the tranlating ma i large enough uh the olution of ( µ Γ lie in the onidered range of the oeffiient of frition, then an upper bound for the table region may be found. The olution of ( µ Γ i given by I + rm tan λ µ m (6.86 I rm tan λ m Replaing > by in the eond tability ondition (6.8 give thi limiting value for the oeffiient of frition. Solving D for µ, yield m + I + + rm tan λ m m µ d tan (6.87 m λ + I rm + tan λ m m Similar to (6.49, only the ratio of the two damping oeffiient appear in (6.87. A a reult, for a fixed ratio, no matter how large the damping i, the table region will be on the left ide of the vertial 9

139 line µ µ d in the µ parameter plane. Note that, for large uh that (6.87 doe not yield a poitive olution, other tability ondition mut be heed. If, then µ d µ. However, in thi ae, the ytem tability i defined by peial ae No. 4 above. On the other hand, for value of m uh that (6.8 i violated (inemati ontraint intability, then regardle of damping and tiffne parameter, the ytem i untable for Figure 6-6 how the tability map of the -DOF ytem for µ > µ where + I + rm tan λ m m µ (6.88 rm I + tan λ m m. At thi value, Γ ( µ m 5g at µ µ. 78. The damping value are the ame a the previou example; and 4 4. For m. and m., one ha µ d. 58 and µ d. 7. Vertial line for thee two ae orreponding to µ µ d are hown in Figure 6-6. (a (b ( Figure 6-6: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and, hathed: undamped intability region. Rω>, m5,, and 4-4

140 For m. 6, the ytem i untable due to inemati ontraint for µ > µ. 85 regardle of the damping and tiffne parameter of the ytem. Figure 6-7 and Figure 6-8 how the evolution of the real and imaginary part of the eigenvalue for m. 6 a upport tiffne ( and oeffiient of frition are varied, repetively. In Figure 6-7, the real part of the eigenvalue and the damped natural frequenie are plotted for three loe value of the ontant oeffiient of frition a the upport tiffne i varied. It i intereting to note that, although the variation of the natural frequenie remain motly unhanged, the tability of the ytem i very enitive to the hange in the oeffiient of frition. (a (b Figure 6-7: Variation of real part of the eigenvalue (a and damped natural frequenie (b a a funtion of lead rew upport axial tiffne,. µ.,.,. The plot in Figure 6-8 how the ourrene of the inemati ontraint intability at µ µ.85. Similar to the undamped ae and the -DOF model of Setion 6., at the inemati ontraint intability boundary, one eigenvalue i at infinity. Alo note that, in Figure 6-8 the ytem i untable over a mall range of oeffiient of frition below the inemati ontraint boundary due to mode oupling.

141 (a (b Figure 6-8: Variation of the real part of eigenvalue (a and the damped natural frequenie (b a a funtion of the oeffiient of frition, µ Figure 6-9 how imilar tability map a thoe in Figure 6-6 but with time larger damping oeffiient. A predited, for the m. and m. ae, the proportional inreae of the damping oeffiient expanded the tability region toward the limiting µ µ d line. For m.6, the limiting value of the oeffiient of frition i unhanged ompared to Figure 6-6, ine the inemati ontraint threhold i independent of damping. For thi ae, however, the damping inreae expanded the tability region into the mode oupling intability region of the undamped ytem.

142 (a (b ( Figure 6-9: Region of tability of the -DOF model with damping. (a m. g, (b m. g, and ( m.6 g. Bla: table, white: untable, and hathed: undamped intability region. Rω>, m5, 4, and 4 -. For the ae of m. 6, Figure 6- how the evolution of the real and imaginary part of the eigenvalue a a funtion of upport tiffne for µ.. The inreaed damping ha reulted in the overdamping of the lower vibration mode for mot of the parameter range onidered. The ytem 7 loe tability due to mode oupling at.6. In Figure 6-, the evolution of the real and imaginary part of the eigenvalue i hown a the oeffiient of frition i varied for 6. 5 The ytem loe tability due to inemati ontraint at µ µ. 85, whih i exatly the ame value a in Figure 6-8.

143 (a (b Figure 6-: Variation of real part of the eigenvalue (a and damped natural frequenie (b a a funtion of lead rew upport axial tiffne,. µ.. ( (a (b Figure 6-: Variation of the real part of eigenvalue (a and the damped natural frequenie (b a a funtion of the oeffiient of frition, µ ( Cloe-up view of the real part of eigenvalue 6.4 Mode Coupling in a -DOF Sytem In thi etion, loal tability of the fixed point of the -DOF model deribed in Setion.7 i invetigated. The fou of thi etion i on the mode oupling intability mehanim. Parameter 4

144 tudie and omparion are done numerially. Similar to what wa done in Setion 6. and Setion 6., the equation of motion are firt tranformed to a uitable form orreponding to the teadyliding fixed point. The equation of motion are given by (., (., and (.7. Negleting F and T, thee equation implify to ( θ θ θ & + r ( N in λ F oλ I & θ (6.89 i m f m& x N o λ Ff in λ + R (6.9 Introduing the hange of variable m & x x x& + N oλ + F in λ (6.9 f u u u θ θ x r x i m tan λθ i (6.9 into (6.89, (6.9, and (6.9 and etting all time derivative to zero, the teady-liding fixed point i found a u u u ω+ ξr R + r o λ + µ tan λ R m tan λu + R (6.9 To tranfer thi point to the origin and preent the ytem in tate-pae form, the following hange of variable i applied y y y y y y θ θi θ& ω x r x& r x u x& m m u tan λθ i tan λω u (6.94 5

145 whih reult in a ytem of ix firt order differential equation (, K6 y& i fi y i (6.95 To tudy the tability of the teady-liding fixed point, the eigenvalue of the Jaobian matrix are evaluated at y. The Jaobian matrix i given by ( R, ω f A y i j y γ γ γ 4 6 γ γ γ 4 6 γ γ γ 4 6 γ γ γ γ γ γ γ 6 γ46 γ66 (6.96 where γ ij are given in Table 6-. Table 6-: Element of the Jaobian matrix for the -DOF model γ γ ρ + I γ γ ρ + I γ 4 γ 5 6 rm I in λ rm in λ η o λ I I rmρ rmρ rmρ rmρ oλ I oλ I oλ I oλ I γ 4 γ 4 ρ γ γ ρ m 4 44 γ γ m r in λ m r in λ η in λ m m ρ ρ ρ ρ oλ m oλ m oλ m o λ m γ 6 γ 6 ρ γ γ γ γ ρ r m 6 64 r m in λ m in λ η in λ + m m ρ ρ oλ m oλ m ρ o λ m m ρ o λ m m ρ ( u u5 rm λu dµ r ω η rm o λ in d µ rµ e + µ in λ + µ gn( R ω oλ ρ oλ + µ gn( R ω in λ Figure 6-(a how the variation of the three undamped natural frequenie of the -DOF model 6

146 with a ontant oeffiient of frition a a funtion of lead rew upport tiffne ( and ontat tiffne (. Lead rew parameter are taen a before with m 5 g and m. g. The orreponding tability map, whih i obtained by examining the real part of the eigenvalue, i depited in Figure 6-(b. Thi map how that the ytem beome untable whenever two of the ytem mode merge. Note that the parameter range where oupling between the firt and the eond mode our agree with intability range of the undamped -DOF model of Setion.5 (Figure 6-8 for large value of and alo the undamped -DOF model of etion.6 (Figure 6- for large value of. Figure 6-: a Evolution of the three natural frequenie of the undamped -DOF ytem (with ontant oeffiient of frition a a funtion of and. b Stability map. Figure 6- how the tability map of the -DOF model a the ontat tiffne ( and the upport tiffne ( are varied. In the by erie of plot inluded in thi figure, the ontat damping oeffiient ( and lead upport tranlational damping oeffiient ( tae the value:, 7

147 , and N. m 4 N. m. rad. Alo the lead rew damping (angular oeffiient i hoen a. Other parameter are eleted a before. Thee plot learly how the role of damping in both tabilizing an untable fixed point and detabilizing a table one. It i alo intereting to note that, baed on the ymmetry of the plot and for the eleted value and range of value of parameter, the tiffne and damping of the two tranlational DOF (i.e. x and x have a imilar effet on the tability of the ytem. Figure 6-: Loal tability of fixed point of the -DOF lead rew ytem with ontant oeffiient of frition. Bla: table, white: untable 6.5 Conluion In thi hapter, the inemati ontraint and mode oupling intability mehanim were tudied. Together with negative damping intability tudied in Chapter 5, thee mehanim ontitute the three frition-indued intability mehanim relevant to the lead rew ytem. It wa found that both inemati ontraint and mode oupling intability mehanim hare the ame neeary ondition namely, that the lead rew mut be elf-loing (i.e. µ > tan λ and applied axial fore mut be in the ame diretion a the tranlating part (i.e. R ω >. It wa alo 8

148 found that the quantity Γ µ tan λ I rm tan λ m + µ tan λ, play a entral role in almot all of the intability enario overed in thi hapter. Speifi to the type of intability mehanim and the ytem model, parametri relationhip were found that define the ondition for the loal tability of the teady-liding fixed point. An itemized lit of the major finding i given here.. Kinemati ontraint intability in the -DOF bai lead rew model: Inequality (6., i.e. Γ < define the intability relationhip in term of frition, geometry, and the inertia of the rotating and tranlating part of the lead rew drive.. Mode oupling intability in the undamped -DOF model with ompliant thread: The flutter intability boundary i given by (6.6, whih define the parameter region where the two natural frequenie merge and beome omplex-valued.. Seondary inemati ontraint intability in the undamped -DOF model with ompliant thread: For the parameter value that atify Γ <, inequality (6.5 define the ondition leading to eondary inemati ontrain intability, whih overlap and expand the mode oupling intability region. 4. Mode oupling intability in the damped -DOF model with ompliant thread: The Routh-Hurwitz tability ondition are given by (6.44 to ( Mode oupling intability in the undamped -DOF model with axially ompliant upport: The flutter intability boundary i given by (6.7, whih define the parameter region where the two natural frequenie merge and beome omplex-valued. 6. Seondary inemati ontraint intability in the undamped -DOF model with axially ompliant upport: For the parameter value that atify Γ <, inequality (6.7 define the ondition for the eondary inemati ontrain intability, whih overlap and expand the mode oupling intability region. 7. Kinemati ontraint intability in the undamped -DOF model with axially ompliant upport: For the parameter value that atify both Γ < and inequality (6.68, i.e. ~ ~ Γ <, the ytem i untable and the line Γ define the divergene intability boundary. 9

149 8. Mode oupling intability in the damped -DOF model with axially ompliant upport: The Routh-Hurwitz tability ondition are given by (6.8 to ( Kinemati ontraint intability in the damped -DOF model with axially ompliant upport: Sine damping doe not affet the intability aued by the inemati ontraint ~ tability, ondition are the ame a in the undamped ae, i.e. Γ. < Mode oupling i by far the mot omplex intability mehanim of the three mehanim onidered in thi wor. In Appendix H, variou numerial example are given that how the omplex effet of ytem parameter on the mode oupling intability and the reulting behavior of the ytem (e.g. amplitude and frequeny of teady-tate vibration. In thi hapter, uing a -DOF model, it wa hown that when mode oupling intability mehanim an affet a ytem, all of the relevant DOF mut be inluded in the model. It wa alo hown that the ompliane aued by the thread flexibility ha imilar effet on the tability of the ytem a the axial ompliane in the lead rew upport.

150 Chapter 7 Vibration Control When effort toward deigning a lead rew drive fail to guarantee vibration-free operation due to deign ontraint, material propertie, or variation in the operating ondition, ative vibration attenuation may beome neeary. Other ytem performane requirement uh a aurate traing of ertain poition and veloity profile, may alo neeitate the ue of loed-loop ontrol. In thi hapter, robut ontrol trategie are developed for lead rew drive ytem with the main goal of uppreing vibration aued by the negative damping intability mehanim while regulating the lead rew angular veloity to a ontant referene veloity. Furthermore, the ontroller are deigned to perform thee ta in ituation where there i ignifiant unertainty in the value of variou ytem parameter. The ontroller developed here are baed on the liding mode ontrol method, whih wa pioneered by Utin [9]. The bai idea i to deign a ontrol input that onfine the ytem trajetorie to a predefined liding urfae or manifold. On thi urfae, the ytem dynami are governed by an aymptotially table differential equation. The ontrol law that move ytem trajetory to the liding mode i typially diontinuou and withe between two value baed on the tate of the ytem. The popularity of thi approah in the deign of feedba ontroller tem from it robutne propertie to model unertaintie [84,9]. Numerou publihed wor are found in the literature on the appliation of the liding mode ontrol to variou mehanial ytem (ee for example the urvey paper [9]. The main drawba of the liding mode ontrol approah i what i nown a hattering. In the implementation (or when the otherwie negleted dynami of the atuator are inluded in the model, the delay in the withing ation required by the diontinuou ontroller reult in high frequeny vibration that an deteriorate the performane of the ontroller or lead to intabilitie [84,9].

151 A number of approahe were put forward by many reearhe that aim to redue or eliminate hattering [94]. Thee method inlude; ontinuou approximation of the diontinuou ontroller law [84,9,95,96], oberver-baed liding mode ontrol [94,97], and higher-order liding mode ontrol [98]. The mot popular approah the one adopted here i the ontinuou liding mode ontrol by linear approximation of the ontrol law or boundary layer method [96,99]. In thi approah, in a mall ditane of the liding (withing urfae (i.e. the boundary layer, the ontrol law i replaed by an approximate linear high-gain feedba. Elimination of hattering with thi approah uually lead to redued performane of the ontroller, ine the trajetorie are only guaranteed to tay inide the boundary layer and not on the liding urfae. Integral liding mode ontrol i one of the approahe propoed to ounterat thi effet [99]. In thi hapter, a implified verion of the model preented in Setion.8 i onidered. The mathematial model of thi ytem preented in Setion 7.. The unertainty in the ytem parameter i diued in Setion 7. and ample parameter value ued in the imulation are given. In Setion 7., a liding mode ontroller i deigned with the aumption that the entire drive from power generator (e.g. DC motor to the moving load i rigidly onneted, and thu the ytem i repreented by a ingle DOF. The rigidity ondition i relaxed in Setion 7.4, leading to a -DOF ytem model. Another liding mode ontroller i developed to deal with the added omplexity of thi model. The peial feature of thi ontroller i it ability to tabilize the ytem in the fae of unmathed unertaintie [84]. Numerial imulation reult are inluded that demontrate the effetivene of both of thee ontroller. Finally, the onluion are preented in Setion Mathematial Model A lightly implified verion of the model preented in Setion.8 i onidered here a hown in Figure 7-. For the motor, Newton eond law give; J&& θ M T M T fm M ( θw θm Mθ& M gn θ& + (7. where θ M i the rotor angular diplaement, J i the inertia of the rotor, and T M i the generated (input torque. T fm and M are the internal frition and damping of the motor, repetively. Alo, i the torional tiffne of the oupling onneting the motor to gearbox, and θ W deignate the input angular diplaement of the gearbox. For the lead rew and nut, imilar to (. and (., one an write

152 I& θ ( θ θ θ & + r ( N λ F oλ gn θ& G m in f (7. T m& x N o λ Ff in λ + R F gn x& (7. where the term T gn θ & and F gn x& reflet the frition in lead rew upport and bearing of the tranlating ma, repetively. Alo the normal ontat fore i given by (.6, where θ θ. i G J Figure 7-: lead rew drive model For impliity, that the gearbox i aumed to be ma-le and fritionle. A a reult, for the gearbox, the following inemati relationhip hold: where ( θ θ r ( θ θ G W M (7.4

153 Eliminating N between (7. and (7. yield where θ G rθ W (7.5 ( R F θ& gn θ& Γ & θ + θ + θ & θg ξ gn T (7.6 Γ I rm tan λξm (7.7 and a before, µ gn( u + ω gn N µ &. Alo µ From (7.4 and (7.5 one find µ tan λ ξ r m (7.8 + µ tanλ r u& +ω r u& +ω ( u& ( µ + µ e + µ u& + ω( e (7.9 r r θ + θ + r + M θ G (7. Subtituting (7. into (7. and (7.6 yield the equation of motion of the -DOF model where ( θ rθ + θ & ξ( R F θ& gn θ& T Γ & θ + M gn (7. M M fm M ( θ rθm Mθ& M J& θ T T gn θ& + r (7. (7. + r The ontat fore, N, i given by N IR IF gn θ & + mrm tan λt gn θ & + mrm tan λθ& mrm tan λ Γ ( oλ + µ in λ ( rθ θ M (7.4 4

154 7.. A Note on the Frition Model The frition model (7.9 inlude a linear term µ u& + ω whih ould aue problem in the ubequent global tability analyi when µ <. Although, in pratial ituation, µ i mall enough uh that over the veloity range of interet µ doe not beome too loe to zero, here we need to formally et a limit on the veloity ω max, uh that for ω > ωmax, the oeffiient of frition taper off and beome a ontant. The eaiet way to aomplih thi while eeping the moothne propertie of the frition model i to modify the veloity-dependent oeffiient of frition to µ r u& +ω u& + ω +ω ( r u µ + µ + µ ω & u& e tanh ( e max (7.5 ωmax where ω max > i a large veloity uh thatµ + µ ωmax >. The modified veloity-dependent oeffiient of frition given by (7.5 i ued in the ubequent etion only where global ytem propertie are onidered. In all other ae, it i aumed that u& + ω << ω max, uh that the differene between (7.5 and (7.9 i negligible. 7. Parameter Unertainty A mentioned earlier, ontrol trategie are ought that are robut to parameter unertaintie. The only information aumed here are the bound on eah of the parameter. In thi etion, ome relationhip for the upper and lower bound of ytem parameter are preented whih are ued in the numerial example preented in the ubequent etion. From (7.5, one find where ( ϖ < µ µ (7.6 µ µ µ µ + µ + µ + µ ω max, µ, µ > (7.7 Baed on (7.6, the upper and lower bound for ξ defined by (7.8 are given by ξ ξ ξ (7.8 5

155 where µ + tan λ µ tan λ ξ r m ξ r m (7.9 µ tan λ + µ tan λ Note that ξ < while ξ < ξ an be negative, poitive or zero depending on the diretion of motion. It follow from (7.7 and (7.8 that where γ Γ γ (7. γ I mξ γ I mξ (7. It i aumed γ > to exlude parametri ondition where inemati ontraint intability an our. In (7. it i aumed that I I I and m m. For all of the other ytem parameter in (7. and (7., only the nowledge of upper bound ( and lower bound ( are aumed to be available. Throughout thi hapter, numerial imulation reult are preented that are obtained uing parameter value lited in Table 7-. Thee value are motly taen from the experimental ae tudy of Chapter 4. The Min. and Max. olumn are the aumed available bound on the unertain parameter. In addition, the Nom. olumn lit the nominal value. Table 7-: Parameter value ued in the imulation Parameter True Value Min. Nom. Max. Unit r m mm λ deg r m. g.5. g I gm J gm Nm/rad Nm/rad M 5. 6 Nm/rad R 8 N T fm.. Nm 6

156 Table 7- (ontinued: Parameter value ued in the imulation Parameter True Value Min. Nom. Max. Unit µ µ µ /rad r.8.4. /rad r. 5 /rad T Nm F N 7. Sliding Mode Control for Rigid Drive The rigid drive i obtained be auming to be very large whih yield θ θ rθ rθ (7. Eliminating ( θ rθ M between (7. and (7. and uing (7., yield G ( + Γ & θ ( + θ & + θ& ξ( θ& J r r rt rt r R F gn r T gn θ& W M M M fm gn (7. The normal ontat fore i now alulated from N ( r I + J ( R F gn x& + rm tan λm( r T gn θ & + rtfm gn θ& rtm + M θ & + r θ& ( oλ + µ in λ( r Γ + J (7.4 The eond order differential equation (7. an be written in tate-pae form a x& x x& C ( x x + G( x [ u ( x ] (7.5 where x x θ ωt θ& ω (7.6 and ω i the referene ontant lead rew angular veloity and u TM i the ontrolled input. Alo The approah followed in thi etion i taen from [84]. 7

157 C ( x M + r J + r Γ (7.7 ( r G x (7.8 J + r Γ ( x T ( x + ω + rξ( R F gn( x + ω + rt ( x + ω + ( r + rω fm gn (7.9 gn Note that < C min C( x < Cmax, < G min G( x < Gmax, and ( x max M where C min + r, J + r γ M C max M + r J + r γ (7. r G min, J + r γ r G max (7. J + r γ ( R + F + ( r + r ω max TfM + rt + r ξ M (7. Let ax + x, a > (7. define the liding urfae. Note that when, we have x ax, and if it i ubtituted in the firt equation of (7.5, x& ax i exponentially table. The differential equation governing i found a & ax + x& ax C ( x x + G( x [ u ( x ] (7.4 Selet ρ ( x uh that ax C ( x x G( x ( x G( x ax C ( a + C max ( x x G( x ( x x G G min min + G max max ρ ( x (7.5 8

158 Then the ontrol input i hoen a ( x gn( u β (7.6 where ( ρ( x + β β x (7.7 for ome β >. With thi ontroller, the Lyapunov funtion andidate for (7.4 i hoen a V. The derivative of V along the ytem trajetorie i V& & ρ G β G [ ax C( x x G( x ( x ] + G( x ( x G( x + G( x u ( x [ ρ( x β( x ] min u (7.8 Therefore, the trajetorie tarting away from the liding urfae reah it in finite time and one on the liding urfae, annot leave it. It wa already hown that trajetorie on the liding urfae are exponentially attrated to origin. The ontroller given by (7.6 i diontinuou and uffer from the hattering problem in the implementation. One way to deal with thi problem i to approximate (7.6 uing a aturation funtion where the aturation funtion, at (, i defined a ( x at( u β (7.9 ε at ( u u gn ( u u u > Repeating the previou analyi, one find that the ytem trajetorie tarting outide the boundary layer { ε } x reah it in finite time and one inide, annot leave it. Inide the boundary layer the x equation beome x & ax +, ε (7.4 9

159 The derivative of V x atifie & ε ax + x ax + x ε a( ϑ x, x (7.4 aϑ V where < ϑ <. A a reult, the trajetorie reah the et Ω ε { x ε, ε } (7.4 aϑ in finite time and remain inide it for all future time. Thi reult how ultimate boundedne with an ultimate bound that i a funtion of ε and an be made arbitrarily mall. Further analyi i needed to ee what happen inide ytem equation are x& Ω, where the ontrol i given by ( ε x& x C ax + x u β x ε ax + x ε ( x x G( x ( x G( x β( x Thi ytem ha an equilibrium point at ( x, where ( (. The loed-loop (7.4 x ε (7.44 aβ To hift the equilibrium point to origin, et y x x and y x. The ytem equation beome y& y& y C ( y y G( y ( y G( y β( y ay + ε y β ( ( or y& y& y ag ( y β( y ε y C ( y G + ( y β( y ε y + Θ ( y (7.45 4

160 where ( ( ( ( Θ y G y β y G( y ( y. Note that ( β( Inide Ω ε, we have inf x x Ω ε ε Θ.. Conequently, for < ω ε, ( funtion. From the Liphitz ontinuity of Θ, aume ( y < κ y Θ i a Liphitz ontinuou y Θ for ome κ. Tae y a udu V ( y, y y ε + (7.46 G u β ( ( u a a Lyapunov funtion andidate. Sine G ( u > and ( > β u for u Ωε, it i obviou that V i a poitive definite funtion of ( y, y. The derivative of V along the trajetorie i found a V& a y y + y& ε G( y β( y a y y ε G C G + G( y β( y C( y + ( ( y y β y ε ( y κ + y ( y β( y ε y y ag + G ( y β( y yθ( y ( y β( y ε y C ( y G + ( y β( y ε y + Θ ( y (7.47 From (7.47 it i evident that V & an be made emi-negative definite by hooing ε mall enough, regardle of the value of κ. Uing LaSalle invariane priniple [84], and ine the et {( y y V& ( y, y } ontain only the trivial trajetory (,, ( x, i aymptotially table. 7.. Numerial Example, one onlude that the point The ontroller deigned in thi etion i imply a linear proportional plu integrator ontroller followed by a aturation funtion. To illutrate the effetivene of the developed ontroller, ome numerial reult are preented here. The parameter value are taen from Table Example # Figure 7- how two ytem trajetorie under the ation of the liding mode ontroller (7.9. 4

161 Controller parameter are eleted a a, β, and ε. The boundary layer i hown by two parallel dahed-dotted line (i.e. x ax ± ε. Inide the boundary layer, ytem (7.4 ha an aymptotially attrative equilibrium point ( x,, given by (7.44. In thi example, x 4 4 rad. The ontrolled input time hitorie are hown in Figure 7-. Figure 7-: Sytem trajetorie under the ation of the ontinuou liding mode ontroller., x, ; dahed-dot: boundary layer Dahed: x ( ( olid: ( ( Figure 7-: Controlled input. Dahed: x ( (, olid: x ( (, 4

162 7... Example # Figure 7-4 and Figure 7-5 how imulation reult obtained uing the ame etting a the above exept for ε whih wa inreaed to 5 (reulting in a muh wider boundary layer. Here, the time required to reah teady tate i inreaed but the required input ignal i moother. In addition, the diplaement error ha inreaed to previou example. x rad (about time higher ompared to the Figure 7-4: Sytem trajetorie under the ation of the ontinuou liding mode ontroller., x, ; dahed-dot: boundary layer Dahed: x ( ( olid: ( ( 4

163 Figure 7-5: Controlled input. Dahed: x ( (, olid: x ( (, 7... Example # Figure 7-6 how ytem trajetorie tarting from ( (, ( (, x (dahed line and tarting from x (olid line for two value of β. Other ontrol parameter are a and ε 5. Reult how that by inreaing β, whih inreae the ontrol gain a per (7.7, trajetorie go through a horter reahing phae. Moreover, the teady tate diplaement error dereae a given by (7.44. The teady tate error are 5 and 6 for 5 β and β 5, repetively. 44

164 β 5 β 5 β 5 β 5 Figure 7-6: Effet of β on the performane of the ontroller Example #4 In the foregoing Lyapunov tability analyi, it wa hown that to have aymptoti tability, ε mut be hoen mall enough aording to (7.47. Otherwie, the bet provable reult would be the ultimate boundedne given by (7.4. In thi example, parameter are hoen to demontrate thi ituation. The ontrol parameter are eleted a a 5, ε 5, and β.. Figure 7-7 how ytem trajetorie tarting from ( (, ( (, x (dahed line and tarting from x (olid line. It i lear from thi figure that the two trajetorie remain in the invariant et Ω ε hown by the hathed region. In the loe-up view depited in Figure 7-8, it an be een that the ontrolled ytem trajetorie are attrated to a limit yle. The pulating behavior of the ontrolled ytem i hown in Figure 7-9. The veloity error varie between -.7 to.4 rad/. 45

165 Figure 7-7: Sytem trajetorie under the ation of the ontinuou liding mode ontroller. Dahed: x ( (, olid: x ( (, ; hathed region: bound on the olution given by the tability ondition. Figure 7-8: Cloe-up view of ytem trajetorie howing limit-yle behavior., x, Dahed: x ( ( olid: ( ( 46

166 Figure 7-9: A portion of veloity error time hitory For the eleted parameter, aording to (7.47, the limit yle diappear for ε. 8 and trajetorie onverge aymptotially to the equilibrium point. 7.4 Sliding Mode Controller for Flexible Drive The ontroller preented in Setion 7. wa deigned for a implified ytem model where the ytem element are onidered to be rigid and rigidly onneted. The rigidity of the drive, however, i not a realiti aumption in many ae, and it i natural to aume ome degree of flexibility in the oupling, gearbox, or even the lead rew itelf. Here, a liding mode ontroller i developed for the -DOF model preented in Setion 7. that properly addree the inreaed omplexity of the model and the effet of the veloity-dependent oeffiient of frition. One of the hallenge in deigning a liding mode ontroller (or any robut ontroller for that matter i the appearane of the o-alled unmathed unertaintie in the ytem model. Starting with (7. and (7., and introduing the following hange of variable Unmathed unertaintie do not atify the mathing ondition. Under the mathing ondition, the unertain term appear in the tate equation of the ytem at the ame point a the ontrol input [84]. 47

167 x θ rθm x θ& ω (7.48 x θ& r ω the equation or motion in the tate-pae form with three tate an be written a M x& x x& x& rx x x ξr ω Γ Γ Γ Γ u TfM gn J J r M M ( x + r ω + x x r ω J J J (7.49 where Γ and ξ are defined by (7.7 and (7.8, repetively and u i the ontrolled input. Alo T and F were negleted for impliity. x The firt tep in deigning a liding mode ontroller for (7.49 i to find the liding urfae: ( x ϕ, x. One onfined to thi urfae, the ytem trajetorie mut aymptotially attrat to the origin. Thi proe an be viewed a finding a tate feedba for the firt two of the ytem equation when x i een a the ontrol input. x& x x& rϕ x Γ ( x, x x Γ ξr ω Γ Γ (7.5 One uh a urfae i found, the next tep would be to deign a ontroller ation that bring the ytem trajetory from any initial point in the region of interet to the liding urfae x ϕ( x, x. Here the liding urfae i hoen a where d and d are ontant to be determined. x d d (7.5 r r x x 7.4. The Sliding Phae Upon ubtituting (7.5 into the firt two equation of (7.49, the governing equation of the liding motion are found a 48

168 ( d + x& dx + x x& Γ x Γ x Γ ξr Γ ω (7.5 Converting (7.5 to a eond order differential equation, give ( + Γd x + (( d + + d x ( d + ( ω+ ξr Γ& x + & (7.5 where the lead rew angular veloity error, x (argument of ξ and Γ funtion i given by x x& + dx d + (7.54 Inpired by the reult of Setion 5., one would expet that a imilar analyi uing the averaging ~ method hould reveal that there i a poitive ontant d (dependent on d and other ytem ~ parameter, uh that etting d > d would lead to aymptoti (exponential tability of the equilibrium point of (7.5. The equilibrium point, ( x, x, of (7.5 i the olution of dx + x x ( d + x ξ ω R (7.55 where and ξ ( R gn( x + ω tan λ ( R gn( x + ω tan λ µ gn r m ( µ gn ( x µ µ (7.57 where µ ( i defined by (7.5. and From (7.55, x i found a the unique olution of the following equation d + + ξ + ω x R (7.58 d + 49

169 x d (7.59 d + x Let z x x (7.6 then (7.5 beome ( + Γd z& + (( d + + d z ( d + R( ξ ξ Γ& z + (7.6 The ontat fore, N, i implified from (7.4 to N Γ R + mr m z& + d z tan λ + mr d + Γ ( oλ + µ in λ m tan λz (7.6 Let τ Ωt where ( d + Ω (7.6 I Note that ompared with (5.4, the natural frequeny of the undamped unperturbed ytem i multiplied by d +. Alo, define the non-dimenional parameter m m r m tan λ I (7.64 ~ (7.65 ( d I + R Ω ω rm R (7.66 Note that (7.64 and (7.66 are the ame a (5.6 and (5.8, repetively. (7.65 differ from (5.7 by a fator of d +. Alo, define 5

170 Uing thee new parameter, (7.6 i tranformed to ~ I d d d (7.67 ( d + Ω µ tan λ ~ µ tan λ m z + z ~ + d m z + z + dz ~ ~ + µ tan λ + µ tan λ µ ( Rω λ µ ω gn tan tan λ R ( R + µ gn ω tan λ + µ tan λ Ω (7.68 where prime denote derivate with repet to the dimenionle time τ. Now that the equation of motion i in it non-dimenionalized form, imilar to Setion 5.., parameter are ordered uing the mall poitive parameter ε. The new parameter µ µ µ, µ (7.69 tan λ tan λ together with m and R are all aumed to be ( mall parameter. Finally, z i aled a ~ d d (7.7 tan λ ~ (7.7 tan λ O with repet ε where ε tan λ i taen a the z ερv, where > ρ i ( O and it i aumed ω Ω ρε (7.7 The ytem equation, (7.68, beome m v [ εξ ( ] [ ( v, v, ε m v + εv + εd εξ v, v, ε ] + v + ε dv εr[ Ξ ( ε Ξ ( v, v, ε ] (7.7 where, imilar to (5.4 and (5.5, ( ε gn + ε ( Rω µ gn( Rω µ Ξ (7.74 5

171 Ξ ( v v, ε µ + ε, ( v, v, ε µ ( v, v, ε Alo, the expreion for the igned veloity-dependent oeffiient of frition, µ ( v,v, ε of the new dimenionle parameter i µ, ( v, v, ε µ ( v, v, ε gn( ζ gn( N( v, v ε (7.75, in term (7.76 where r ζ ( ( e r ζ ( v v, ε µ ˆ + µ ˆ e + µ ˆ ω tanh( ζ ω µ, max max (7.77 and µ ˆ µ tan λ, i,,, and i i ω d dx ζ( v, v, ε v + ερ v + + ω (7.78 d + d + + d Alo, from (7.6, one find R gn ( N gn εξ R + εv + ( + ε d v (7.79 m After rearranging, (7.7 beome ( v,v ε v + v εf, (7.8 where f [ ] ( v, v, ε dv ( εξ m v + mξ v + εdv + R( Ξ Ξ (7.8 Comparing (7.8 with (5., we an ee that a damping term, d v, and a higher-order term, ε dv, are added to the ytem equation. Note that (7.76 ha imilar Liphitz ontinuity propertie a (5. with repet to ( v,v,ε. To tranform (7.8 into the tandard form, the following hange of variable i ued Thi lead to v a oϕ, v ain ϕ (7.8 5

172 ( aoϕ, ain ϕ, ε ϕ a εf in (7.8 ε ϕ f a o a ( aoϕ, in ϕ, ε ϕ (7.84 Sine ϕ i away from zero, dividing (7.8 by (7.84 yield da f ε dϕ ε f a ( aoϕ, ain ϕ, ε ( aoϕ, ain ϕ, ε in ϕ εg oϕ ( ϕ, a, ε (7.85 The ytem equation (7.85 i in tandard form with a bounded periodi right-hand ide. The averaged amplitude equation an be found a a ε π π ε π g π ( ϕ, a, f dϕ ( a oϕ, ain ϕ, in ϕdϕ (7.86 Expanding (7.86 give ε a π + ε π π π ( + d ain ϕ + main ϕoϕ + µ Rgn( Rω in ϕ ( main ϕoϕ + Rin ϕ µ ( ϕ, a dϕ dϕ whih immediately an be implified to where and µ a ε π ( + d a εm R + in ϕ aoϕ + ( ϕ, a π m µ dϕ r ϖ ϖa in ϕ r ϖ ϖa in ϕ ( ϕ, a ( µ ˆ + µ ˆ e + µ ˆ ϖ ϖ ain ϕ ( e gn R ( ϖ ϖ ϕ ϕ + ain gn a o m (7.87 (7.88 5

173 ϖ dx + ω d + (7.89 ϖ ω d + (7.9 Note that by etting d d, (7.88 implifie to (5.46. In the following, the ae of R or R i diued. R O( i treated firt and then the ae of O( ε The Cae of R O( For R O(, there i R a N m Alo, ine for < ϖ ϖ R + m m for a < an. R uh that gn aoϕ gn gn( R a we have gn( ϖ in ϕ ( ϖ ϖ a then gn for µ ( in ( ( ( r ϖ αa ϕ e ϖ r ϖ ( αa in ϕ ( ϕ a µ ˆ + µ ˆ e + µ ˆ ϖ ( αain ϕ, gn ϖ a and α ϖ a < a, where min( an, α a ε π ( + d a ε + R in ϕµ ( ϕ, a π. Subtituting (7.9 into (7.87 yield (7.9 dϕ, a < a (7.9 For impliity, here it i aumed ϖ >. Simplifying (7.9, one get µ rϖαa in ϕ r ϖαa in ϕ ( ϕ, a ( µ + µ e µ ain ϕ( r e (7.9 ϖ where r e r and µ ˆ ˆ + µ µ ϖ (7.94 µ µ e ˆ r ϖ µ ˆ µ ϖ α (7.95 (7.96 It i intereting to note that (7.9 i very imilar to (5.49. Subtituting (7.9 into (7.9 and 54

174 arrying out the integration, one find + d + µ R a ε a + εµ RΛ + εr µ arλ, ( r ϖ αa εr µ RΛ ( rϖ αa ( rϖ αa εr µ RΛ (( r + r ϖ αa,,, (7.97 where Λ, ( and Λ, ( are defined by (5.56 and (5.57, repetively. After linearization, one find that the origin i loally exponentially table if (in term of the original ytem parameter, dµ + di > rm R (7.98 dζ ζ Comparing (7.98 with (5.6, it i lear that the only differene i the addition of damping whih i aued by the ontroller. d I to the linear Sytem (7.97 i truturally idential to (5.54 and a uh, reult obtained in Setion 5..4 and Appendix D apply here. Conequently, there exit d, uh that eleting d > d guarantee that the origin i an exponentially table fixed point of the averaged equation (7.97, and no other fixed point exit over a [, a ] with repet to (, ε [, a ] [, ε ]. Sine the right-hand ide of (7.85 i twie ontinuouly differentiable a and ( ϕ,, ε for uffiiently mall ε, the origin of (7.85 i alo exponentially table. For g, then, baed on Theorem of Appendix I, a > a, baed on the argument of Setion 7.. and (7.5, (7.88 i replaed by µ r ϖ ϖ a ϕ ( ϖ ϖ ain ϕ in r ϖ ϖa in ϕ ϕ, a µ ˆ + µ ˆ e + µ ˆ ω tanh ( e gn R ( ϖ ϖ ϕ ϕ + ain gn ao m max ω max (7.99 whih i bounded and periodi with repet to ϕ. Subtituting (7.99 into (7.87 yield the averaged amplitude equation. Due to the boundedne (7.99, the following two definite integral are alo bounded: Replae + d by and ϖ α by ω and (7.97 beome (

175 ψ π ( a in ϕoϕµ ( ϕ a dϕ < ψ, π ψ π ( a in ϕµ ( ϕ a dϕ < ψ, π Uing (7. and (7., from (7.87 we have π ( + d a ε + ( main ϕoϕ + Rin ϕ µ ( ϕ, a a ε π ε ( + d a + εmaψ + εrψ dϕ (7. (7. (7. Let then ( < d max a R ( mψ, mψ + ψ, (7. a a for d > d and a > a. d max d d > guarantee that the origin i an exponentially table fixed point of the Seleting (, averaged ytem for initial ondition a ( a [,. Sine the right-hand ide of (7.87 i ontinuouly differentiable with repet to a, Theorem of Appendix I tate that the olution of (7.87 and (7.85 that tart from the ame initial value a [, remain ( ε [ ϕ O loe for all ϕ,. Conequently, for uffiiently mall ε and after a finite time, the olution of (7.85 enter [,a ] where, a diued above, the olution i exponentially attrated to the origin. Thi property naturally extend to the original ytem, ( The Cae of O( ε Here it i aumed that R or R R εr, where R O( to the ae of R by imply etting R Subtituting R εr into (7.7 give with repet to ε. The ubequent argument apply and R. 56

176 ( v v, ε + ε f ( v, ε v + v εf, (7.4, v where f ( v v dv ( m ( v m v dv, ε εξ + Ξ + ε, (7.5 and f ( v, v, ε ( εξ m ( Ξ Ξ R (7.6 where Ξ and Ξ are given by (7.74 and (7.75, repetively. Furthermore, the aling redue the ize of x, given by (7.58, from O ( to O ( ε R εr,. In term of ordered ytem parameter, (7.58 i rewritten a µ gn + + ερ R + ε ρ d + µ gn ( ε d x ( ε ( R gn( x + ω ( R gn( x + ω ( + gn( ω (7.7 where (7.6, (7.65, (7.66, (7.67, (7.69, (7.7, (7.7, and (7.7 were ued. Auming d O(, it i lear that x O(. The firt order averaged equation (7.87 i redued to where ε a ε π ( + d a εma + in ϕ oϕ µ ( ϕ, a π dϕ (7.8 µ ( ϕ, a µ ˆ + µ ˆ e a r ω ω in ϕ d + a gn ω ω in ϕ d + + µ ˆ a ω ω in ϕ e d + a r ω ω in ϕ d + (7.9 The following two expreion were ued to reah (7.8. Firt, from (7.78 and after ubtituting the olution of (7.7, we have Seond, from (7.79 ζ ω ( v, v, v + ω d + 57

177 gn m R ( N( v, v, gn ε ε ε Ξ R + εv + ( + ε d gn ( N( v, v, gn( v v to, then gn ω ω in ϕ gn( ω If a < + d d a + and onequently, (7.9 i implified µ ( ϕ, a µ ˆ + µ ˆ e e r ω r ω a d+ a in ϕ d+ in ϕ a + µ ˆ ω in ϕ d +, a < + d (7. gn( ω Define ψ π π ( a in ϕ o ϕ µ ( ϕ a, dϕ (7. then there exit ψ, uh that ( < ψ a ψ, a < + d (7. In order to guarantee exponential tability of the origin of the averaged ytem for the initial a, + d d > max, d where ondition atifying ( [ ], it uffie to elet ( d mψ (7. for then, (7.8 yield ( + d a ( ( + d a + εmaψ a < ε + εmaψ < β, β a ε a > where (7. wa ued. Moreover, ine (7. i twie ontinuouly differentiable with repet to a uniformly in ϕ, the right-hand ide of (7.85 i twie ontinuouly differentiable with repet to (, ε [, a ] [, ε ] a. Alo, ine g ( ϕ,, ε, baed on Theorem of Appendix I, for 58

178 uffiiently mall ε, the origin of (7.85 i exponentially table. For a + d, baed on the diuion of Setion 7.., (7.9 i modified to guarantee boundedne a a. (7.9 i replaed by µ ( ϕ, a µ ˆ + µ ˆ e e a r ω ω in ϕ d+ a r ω ω in ϕ d+ + µ ˆ ω max a gn ω ω in ϕ d + a ω ω in ϕ d + tanh ω max (7.4 Due to the boundedne of (7.4, there i a ψ 4, uh that where ( a ( < 4 ψ a ψ, a + d (7.5 ψ i given by (7. after ubtituting (7.4. One again, exponential tability of the origin of the averaged ytem for the initial ondition ( [ + d, (, d d > max where 4 Seleting d max(, d d a i guaranteed, if we elet d4 mψ4 (7.6 >, 4 guarantee that the origin i an exponentially table fixed point of the averaged ytem for the initial ondition a ( a [,. Sine the right-hand ide of (7.8 i ontinuouly differentiable with repet to a, Theorem of Appendix I tate that the olution of (7.8 and (7.85 that tart from the ame initial value a [, remain ( ε [ ϕ O loe for all ϕ,. Conequently, for uffiiently mall ε and after a finite time, the olution of (7.85 enter [, d ] +, where a diued above, the olution i exponentially attrated to the origin. Thi property extend to the original ytem, ( Remar The above argument did not provide a loed form relationhip in term of ytem parameter to determine the two ontrol parameter, d and d. It i, however, poible to alulate thee 59

179 parameter baed on the value of the ytem parameter, if available, or their upper and lower bound Remar The above proof of tability i baed on the method of averaging and i appliable to ytem with wea nonlinearitie. In the preeding derivation, thi requirement on the ize of the nonlinear term i quantified by the introdution of the mall parameter ε. In partiular, Theorem of Appendix I tate that, under the aumption of the theorem, there exit an ε >, uh that for ε [, ε ] the exponential tability of the origin of the unaveraged ytem an be dedued from the exponential tability of the origin of averaged ytem. Conequently, the upper bound on the ize of the nonlinear term i defined by ε. Here, it i aumed that ytem parameter value (or their repetive upper and lower bound are uh that the above ondition i atified. Now that it ha been hown that there exit ontrol parameter d and d uh that the ubytem (7.5 (i.e. the governing equation of the liding motion i exponentially table, we an proeed with the deign of a ontrol input that bring the ytem tate to the liding urfae in finite time The Reahing Phae Baed on (7.5, the equation of the liding urfae i given by where the variable i defined a (7.7 d d x x + x (7.8 r r A ontroller i ought that bring to zero in finite time. Differentiating (7.8 with repet to time, give Conidering the omplexity of the ytem equation and the number of parameter involved, the traightforward alulation of an upper bound on ε (i.e. ε* an be very diffiult. Numerial imulation may be ued a an alternative to etimate thi bound for a partiular lead rew drive. See, for example, [], where a imilar iue i diued in the ontext of the ontrol of the underatuated manipulator. 6

180 6 x r d x r d x & & & & + (7.9 Subtituting (7.49 into (7.9, yield ( ( ω + ξ Γ Γ + + Γ ω+ ω + R r d x r d r d x J d x r d J r Jr r x T J u J M M f gn & (7. Tae the Lyapunov funtion andidate a V ; it derivate along the ytem trajetorie i found a ( ( ( x x x J u J x r d r d x J d x r d J r R r d Jr r x T J u J V M M fm gn + ρ + ρ + ρ ρ + Γ + + Γ + ω + ξ Γ ω+ + ω + & & (7. where,, K ρ i i are aumed to be nown ontant atifying the following relationhip ( ( gn ρ ω + ξ Γ ω+ + ω + R r d J r r x T M fm (7. ρ Γ r d J r (7. ρ Γ + r d J r d J (7.4 ρ M J d (7.5 Uing nown upper and lower bound of the parameter, one an find

181 ( ξ R + ω M J d ρ Tf + ω + (7.6 r Γ r d ρ r + J (7.7 r Γ d d ρ J + (7.8 r r Γ ρ d J + M (7.9 Let ( β( gn( u x + β (7. where ( ρ + ρ x + ρ x + ρ x and β > i ome nown ontant, (7. beome β x (7. V& u + J J J β J ( ρ + ρ x + ρ x + ρ x J ( β( x + β gn( + β( x (7. Thi reult implie that the ytem trajetorie reah the liding urfae in finite time and, one on the urfae, they annot leave it. The liding mode ontroller oniting of a reahing phae and a liding phae i now developed. However, a few iue remain to be addreed before igning off on thi ontroller. Thee iue are: The teady-tate error, Chattering, whih i aued by diontinuou nature of the ontroller and delay in the atual withing ontroller implementation, and Initial high torque demand. In the following three etion, everal approahe are preented to modify the ontroller in order 6

182 to irumvent thee problem without jeopardizing the tability propertie of the original ontroller Feedforward Input From (7.5, we aw that the origin i not an equilibrium point of the governing equation for the liding motion. Thi equilibrium point wa alulated by (7.58 and tranlated to a lead rew veloity error given by (7.59, whih in turn reult in a motor veloity error when onidering the liding urfae equation given by (7.5. In thi etion, a feedforward input i onidered to redue the veloity error a muh a poible, depending on the availability and the auray of the nominal value for the ytem parameter. For the ytem (7.49, the ontant input i required to have the teady tate point at u Tf + M r ω+ rξ R + rω, ω (7. > x x x ξr + ω (7.4 However due to parameter unertainty, both u and x are unnown value. A a ompromie, it i poible to modify the ontroller input with u v + (7.5 û where û i the feedforward input obtained from (7. uing nominal value of ytem parameter. Subtituting (7.5 into (7.49 and etting ~ x x (7.6 x ˆ where ˆx i obtained from (7.4 by uing the nominal ytem parameter, the equation of motion beome after implifiation 6

183 64 ( ( ( ( v J R J r Jr T r x T J R R x x x J rj r x x x M M f f M + ω + ξ + ω ω + Γ ω ξ ξ Γ + Γ Γ ˆ ˆ ˆ ˆ ˆ ˆ gn ˆ ˆ ˆ ˆ ˆ & & & (7.7 where the tilde on x wa dropped to implify the notation. The development of the liding mode ontroller follow the ame tep a deribed above and, exept for pointing out the differene, it i not repeated here. The eond order differential equation governing the liding motion, (7.5, beome ( ( ( ( + ω ξ ξ Γ + Γ R R d x d d x d x ˆ ˆ ˆ ˆ ˆ && & (7.8 where the hat deignate the nominal parameter value. Note that had we nown the true ytem parameter, (7.8 would yield x. However, one would expet that by the addition of the feedforward omponent, a maller x i obtained when the nominal parameter value are reaonably loe to the true value. Thi in turn tranlate to maller teady tate veloity error for motor and lead rew. The addition of the ontant input alo hange the liding mode ontroller law. The equation for &, given by (7. i hanged to + + Γ + Γ + x J d x r d r d x r d J r v J M & (7.9 where

184 65 ( ( ( ( ω + ξ + ω + ω ξ ξ Γ ω + R J r Jr R R r d T r x T J M M f f ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ gn (7.4 The only differene ompared with the previou ontroller i in ρ, whih i now given by ( ( ( ( ( ( R r r r d J R R R R r d J T T T T M M M M f f f f ω + ξ + ω + ω Γ + ξ ξ ξ ξ Γ + ρ ˆ, ˆ max ˆ ˆ ˆ ˆ ˆ, ˆ max ˆ ˆ ˆ, ˆ max ˆ ˆ ˆ ˆ, ˆ ˆ ˆ max ˆ ˆ, ˆ max( ( Continuou Sliding Mode Controller The ontinuou verion of the ontroller given by (7. an be ontruted by replaing ignum funtion with a unit aturation funtion ( ( ε + β β at u x (7.4 For thi ontroller, the previou Lyapunov analyi an only how that the trajetorie tarting outide of the boundary layer (i.e. ε reah it in finite time and, one inide, annot leave it for all future time. Namely, (7. for the derivate of the Lyapunov funtion, now read, ε β J V & (7.4 To enure that the trajetorie inide the boundary layer remain bounded, the boundedne of the olution of the following ytem mut be hown ( ω Γ ξ Γ Γ Γ + + R x x x r x d x d x & &, ε (7.44 where (7.44 i obtained by ubtituting (7.8 into the firt two equation of (7.49. However,

185 ine the aim of the ontroller i to eliminate lead rew vibration, proof of boundedne will not be uffiient. Intead, the analyi ontinue by utilizing the high gain behavior of the liding mode ontroller inide the boundary layer to prove the aymptoti tability of the ytem toward it fixed point (whih i not the origin. Thi ta i aomplihed by onidering the ytem a a ingularly perturbed problem. The tability property i then given by Theorem of Appendix I. To ue thi theorem, the ontrol law given by (7. mut be modified lightly to give it the required moothne propertie. Note that the partial derivative of β ( x, given by (7., with repet to, i,, x i, are not ontinuou. Here, β ( x i replaed by ( x to (7., exept that the abolute funtion are replaed by σ δ ( u β ~, whih i defined imilar u, u δ δ u (7.45 +, u < δ δ where δ > i a parameter deignating the extent of the moothed region. A illutrated in Figure 7-, funtion σ (. i ontinuouly differentiable to any order and ( u u u δ σ δ,. Figure 7-: Smoothed abolute funtion Now the modified ontroller ~ ( ( + β u β x, ε ε (7.46 i ontinuouly differentiable inide the boundary layer. For ε, the equation of motion beome 66

186 x& x& ε x& x rx x Γ J ε T J ξr ω x Γ Γ Γ d d x x + r r ~ ( β( x + β fm x tanh where gn ( ζ wa replaed by ( + r ε ω + r M εr x + x J ω ε J M x (7.47 tanh ζ ε for ome mall ε > a an approximation that ha the required moothne propertie. The ytem deribed by (7.47 reemble a tandard ingular perturbation problem [84]. To onvert it to the tandard form, the fixed point mut be tranferred to the origin. Let where x i given by the olution of the following equation ( ε ( ε z x x z z x x ( (7.48 z x x ε ~ d ( β( x + β ( x + ξ R + ω + ( d + + ε r + ε rt ( x + ξ R + ω fm Alo, x ( x, x x and ξ(, x + ω tanh + rε ξ x. + ε M M x ω x i found from ( x + ξ R + ω x x (7.49 (7.5 alo x x / r (7.5 For ε, (7.49 implifie to 67

187 whih ha a unique olution, ξ ( x R + ω + ( d + x d (7.5 d * x. Sine (7.49 i ontinuouly differentiable with repet to x and it derivative doe not vanih at x and ε (tability requirement et d, baed on * x > the impliit funtion theorem [] it ha a olution for uffiiently mall ε. Subtituting (7.48 into (7.47 yield z& z& ε z& z Γ J rz εr + J ( z + x ( z + x ~ d d ( β( z + x + β z z + z + (( d + x d x ( z + x Γ ε J M z r ξ Γ x + r R ω Γ r ε T J r fm z tanh + r ( x + ω ε + r M whih i in the tandard ingularly perturbed form for the autonomou ytem [84], given by for ( z, z x, y z, and ε ε. x& f εy& g ( x, y, ε ( x, y, ε ω (7.5 Firt, the redued model and the boundary-layer model are defined. Setting ε, the third differential equation degenerate to an algebrai equation. The unique olution of thi equation i given by where the fat that ( ( + β d d z h( z, z z z (7.54 r r β z i ued. Alo, for ε, x and x oinide with the olution of (7.58 and (7.59, whih give ( + x d x d. ε ε Subtituting (7.54 into the firt two equation of (7.5 and etting ε, the redued model i found a 68

188 z& z& d z + z Γ ( d + z Γ z ξ ξ Γ R (7.55 whih an be expreed a a ingle eond-order differential equation, given by ( + Γd z& + (( d + + d z ( d + R( ξ ξ Γ& z + (7.56 Note that (7.56 i exatly the ame a (7.6, and the analyi of Setion 7.4. prove that the origin i an exponentially table equilibrium point of the redued model. To obtain the boundary-layer model, we firt apply the following hange of variable d d y z z + z (7.57 r r Upon ubtituting thi into the third equation of (7.5, one find εy& J ε J ~ εr ( β( z + x + β y + (( d + x d x + ( z + x M x z + r d ε z& + ε r d r z& ε T J r fm z tanh + r ε ( x + ω J + r M ω (7.58 Define the fat time ale a and etting ε to zero, yield τ ε t, whih yield ε y& dy dτ. Subtituting thi into (7.58 dy dτ J ~ ( β( + β y z (7.59 whih an be hown to have an exponentially table origin (i.e. y. Let Lyapunov funtion andidate. The derivative of V along the ytem trajetorie i found a d d V y τ dy d τ J ~ β ( β( + β y y V y be the z (7.6 J Citing Theorem in Appendix I, it i onluded that the ytem (7.5 ha an exponentially table equilibrium point at the origin for uffiiently mall ε. 69

189 7.4.5 Variable Veloity Set Point The error i inverely proportional to d + and by hooing a large enough d, the poition error an be limited to ome allowable maximum thu reduing the lead rew and motor veloity error. Unfortunately, thi approah may inreae the already large torque demand on the motor at the beginning of motion when the differene between ytem initial tate (uually at ret, θ & and the deired tate ( θ & ωd i ignifiant. From a pratial point of view, thi may not be deirable or feaible. An effetive way to redue the input torque demand i to inreae the preet veloity from zero to it deired value, gradually. If ω ω( t then the hange of variable (7.48 give x& x& x& θ& rθ& M && θ ω& && θ r M ω & (7.6 A a reult (7.49 beome x& x& x& x rx x ξ ω ω& x R Γ Γ Γ Γ u Tf gn J J J r M M ( x + r ω + x x r ω r ω& J J (7.6 From (7.6, it i eay to ee that the & equation given by (7.9 hange to & u T J J d d + r r fm gn x Γ ( x + r ω d Γ r M r d ω+ Jr J r d + r ( ξr + ω ω& x + d Γ J M x (7.6 where the term d + ω& r i the only differene ompared to (7.. To ompenate for thi bounded addition, only ρ in (7.6 need to be modified. Let d + ( ξ R + ω + α M J d ρ TfM + ω + (7.64 r Γ r r 7

190 where α and ω are hoen uh that Note that ine ω ( t and ( t ω& are nown funtion of time, ω and α an be hoen to be timevarying too. α up ω& ( t and ω ω( t t up (7.65 t We alo need to now the effet of hanging ω ( t on the tability of the liding phae. Adopting the ame liding urfae a before, the governing equation of the liding motion beome x& x& d x x Γ + ( d + x Γ x ξr ω ω& Γ Γ (7.66 where ompared to (7.5, ω& ha entered into the eond equation. In the following, (7.66 i onverted to a form where the variation of ω ( t appear a a bounded perturbation. The ultimate boundedne of the liding motion i then follow from the global exponential tability of the unperturbed ytem. Let where d ω ( t ωd ϖ( t, ϖ( t ωd ω i the final deired ontant angular veloity value and ( t (7.67 ϖ i the variable part. For the problem onidered here, ϖ ( t i aumed to be a dereaing funtion of t, with ( ωd ( ϖ t, t t for ome > (7.66 yield where t. Alo, ( t x& x& d x x Γ + ϖ and ϖ& i bounded for t. Subtituting (7.67 into ( d + x Γ x ξr ω Γ Γ d + δ (7.68 See for example [84, Lemma 9.]. 7

191 δ, ϖ( t + ϖ& ( t (7.69 Γ ( t x ( x i a bounded term baed on the above aumption for ϖ ( t and ( t ( x Γ i away from zero for x. ϖ& and alo baed on the fat that Setting δ, (7.68 redue to (7.5 with ontant ω ωd whih i globally exponentially table. Thi property, together with the boundedne of δ ( t, x, guarantee that the olution of (7.68 remain bounded [84]. Furthermore, one t > t, the perturbation term vanihe and the exponential tability of the equation of motion of the liding phae i retored Numerial Example In the numerial example preented here, the ontroller law defined by (7.4 i ued. The ytem parameter and their bound are given by Table 7-. In addition, the ytem i onidered to be at ret at the tart of imulation Example # In the firt example, the feedforward input of Setion 7.4. and the variable veloity et point of Setion are not ued. The eleted ontroller parameter are lited in Table 7-. Table 7-: Controller parameter for example # Parameter Value Parameter Value d 5 ω d d β 4 rad ε Figure 7- how the time hitorie of the lead rew and motor veloity error. A an be een, the liding mode ontroller i able to dampen the lead rew vibration in a hort time. The teady tate error i approximately x. rad (about % 5 for the lead rew, whih i equivalent to a x. rad error in the teady tate value of the motor angular veloity, aording to (7.5. The time hitory of the x tate i hown in Figure 7-(a. Thi variable eentially repreent the torional defletion of the element (oupling onneting the motor to the lead rew. The teady 7

192 tate defletion, x, i approximately.54 rad in aordane with (7.5. In Figure 7-(b, the time hitory of the normal ontat fore of the thread i plotted. Note that, during the imulation, ontat fore remain greater than zero and at teady tate reahe approximately, whih an alo be found from (.6 a N N N R oλ + µ gn R ( ω in λ The ontrolled input time hitory i given in Figure 7-. A expeted, at the beginning of motion, the ontroller demand high torque from the motor to bring the ytem tate from 9, 4, 4 toward the boundary layer. The initial torque i about 6. N. m. At teady tate, the torque required to maintain the ytem ontant veloity i with (7.. u. N. m, whih agree Figure 7-: Controller performane of example #; (a lead rew veloity error, (b motor veloity error 7

193 Figure 7-: (a Variation of the torional defletion of the oupling, (b variation of normal ontat fore The ytem trajetory i hown in Figure 7-4. The two plane in thi figure are defined by ( d r x + ( d r ± ε x and are added to viualize the boundary layer. A an be een, the x trajetory enter the boundary layer at point A and, one inide, remain there for all future time. Figure 7-: Controlled input of example # 74

194 O A Figure 7-4: Sytem trajetory for example # Example # In thi example, the veloity et point i gradually varied from to ω d aording to t t t tanh + tanh ( δ δ ω t ω d (7.7 t + tanh δ where t and δ ontrol the onet of peed hange and teepne of it aent, repetively. Note that, baed on (7.7, ω( t and ωd &, t. t δ + tanh δ The ontroller parameter are eleted a the previou example. Alo for thi example, t. 5 δ.. Figure 7-5 how the lead rew and motor angular veloity time hitorie. The plot in thi figure how ueful traing of the variable veloity et point. The ontrolled input for thi example i hown in Figure 7-6. Compared with Figure 7-, it i lear that by varying the veloity et point from zero to the deired final value, the initial high ontrol torque i eliminated. 75

195 Figure 7-5: Lead rew (a and motor (b angular veloitie of the loed loop ytem. Bla: ytem repone; dahed gray: veloity et point Figure 7-6: Controlled input of example # Example # In thi example, the effet of the parameter d on the performane of the ontroller i invetigated. The parameter value ued in thi example are lited in Table

196 Table 7-: Controller parameter for example # Parameter Value Parameter Value d 4 ω d d,, t 4 rad.5. β δ ε Figure 7-7 how the lead rew and motor veloity error time hitorie for three different value of d. It an be een from thi figure that by inreaing the value of d, the teady tate veloity error dereae, a wa hown by (7.5. Figure 7-7: Effet of d on the performane of the ontroller. In Setion 7.4., it wa hown that d ha a tiffne-lie effet on the redued order model of liding phae. It i intereting to note that the imulation reult of Figure 7-7 how thi effet; by inreaing d, the amplitude of the tranient vibration of the lead rew i redued Example #4 In thi example, the feedforward input of Setion 7.4. i added to the ontroller. To implement the feedforward input together with the variable veloity liding mode ontroller of Setion 7.4.5, the 77

197 feedforward input defined by (7. i hanged to ( t + rξˆ Rˆ + rˆ ω( t u Tˆ + ˆ r ω (7.7 fm M Alo, the term ρ given by (7.4 for the feedforward input of Setion 7.4., and the term ρ given by (7.64 for the variable veloity method of Setion 7.4.5, are ombined to give ρ max( J d ( T Tˆ Tˆ f f, f T f + max( ˆ ξr ξˆ Rˆ, ˆ ξr ξˆ Rˆ J d + ω max Γ rˆ r + ˆ Γ rˆ ω ( ˆ ˆ, ˆ ˆ + max( M ˆ M, M ˆ M + ( ξˆ d Rˆ + ω ( ˆ ˆ ˆ max, + α r r (7.7 Finally, the liding mode ontrol law given by (7.4 i hanged to where ~ [ x + x x x ] x ˆ and u ~ x (7.7 ( β( + β at ε ˆx i alulated by ubtituting nominal value into (7.4. The ontroller variable for thi numerial example are given in Table 7-4. Figure 7-8 how the performane of the liding mode ontroller with and without the feedforward input. The addition of the feedforward omponent learly improved the teady tate reult. It mut be noted that, thi improvement i highly dependent on the auray of the nominal value of the ytem parameter. Table 7-4: Controller parameter for example #4 Parameter Value Parameter Value d 5 ω d d t 4 rad.5 β δ. ε 78

198 Figure 7-8: Effet of feedforward part on the performane of the ontroller. Gray: without feedforward; bla: with feedforward The ontroller effort i hown in Figure 7-9 for the two ae onidered in thi example. It an be een that the addition of feedforward input only lightly inreaed the input. The redution of the liding mode ontroller gain (omparing (7.7 with (7.64 i ompenated by the nominal input (7.7, reulting in redued teady-tate error. Figure 7-9: Controlled input of example #4. Bla: with feedforward input; gray: without feedforward input. 79

199 For omparion, Figure 7- how imulation reult when only the feedforward input wa applied (open-loop ytem. The ytem i untable and the lead rew DOF exhibit ti-lip behavior. The vibration amplitude of the motor, on the other hand, i ignifiantly maller whih i due to the high gear ratio, r, between motor and lead rew. The error in the teady tate value of motor veloity error i attributed the differene between nominal value and true value of the ytem parameter. Figure 7-: Performane under the ation of feedforward input alone Example #5 In the development of the liding mode ontroller of thi hapter, the veloity-dependent oeffiient of frition wa moothed to atify the moothne requirement for the averaging proe of Setion 7.4. and the ingular perturbation analyi of Setion However, there i no limit on the ize of r parameter, ine thi parameter doe not appear in the ontrol law. The only differene i that a omewhat larger d value may be needed to guarantee tability due to the inreaed negative lope of the frition urve at the lower relative veloitie. Thi, however, doe not affet the tability of the Similar behavior wa oberved in the open-loop tet reult of Chapter 4. See Figure 4-5(a. 8

200 ytem one the variable referene veloity reahe it deired final value. Thi feature an be ued to effetively reover the ae where the oeffiient of frition doe not vanih near zero relative veloity by eleting very large value for r. In thi example, the atual value for r i et to (ompared with the previou example, where r. The reulting oeffiient of frition i hown in Figure 7-. Controller parameter ued in thi example are given in Table 7-5. Figure 7-: Smoothed oeffiient of frition Table 7-5: Controller parameter for example #5 Parameter Value Parameter Value d 4 ω d d t 4 rad.5 β δ. ε r The performane of the ontroller i hown by the error time hitorie plotted in Figure 7-. The ti-lip tranient vibration in the lead rew angular veloity i learly viible in the loe-up view of Figure 7-. The ontrolled input for thi example i hown in Figure

201 Figure 7-: Lead rew (a and motor (b angular veloitie of the loed loop ytem. Bla: ytem repone; dahed gray: veloity et point Figure 7-: Cloe-up view of the tranient vibration of the lead rew 8

202 Figure 7-4: Controlled input for example #5 7.5 Conluion Ative vibration ontrol for lead rew ytem wa tudied in thi hapter. Baed on the liding mode ontrol method, two robut regulator were deigned that are apable of tabilizing the teady liding fixed point of the ytem to a predefined et point. Thee ontroller atively diminih the vibration aued by the negative damping mehanim that i aumed to be preent in the ytem. To implement thee ontroller, no nowledge of any of the ytem parameter i needed and only the bound of the parameter are aumed to be available. In Setion 7., the model of the lead rew drive wa implified under the aumption that all of the rotating part, from motor to lead rew, are rigidly onneted. The problem of hattering, whih i an inherent onequene of all diontinuou (withing ontroller uh a the liding mode, wa addreed, and a ontinuou verion of the ontroller wa developed to avoid hattering. Stability of thi ontroller wa proven uing an appropriate Lyapunov funtion. Numerial imulation reult alo onfirmed the appliability of the ontroller. Thi aumption of rigidity wa relaxed in Setion 7.4. In thi etion, a more realiti -DOF lead rew drive ytem wa analyzed. Firt, a bai liding mode ontroller wa developed that addreed the preene of unmathed unertaintie in the ytem. The proe of the proof of tability of thi ontroller involved the analyi of the governing equation of liding motion a a wealy nonlinear 8

203 ytem. Drawing heavily on the reult of Setion 5., it wa proven that for a uitable hoie of liding urfae, the liding motion i globally exponentially table. Three area of defiieny were identified regarding the performane of the bai liding mode ontroller. Eah of thee iue wa addreed through an appropriate modifiation to the bai ontroller that retained it tability propertie:. Chatter Thi problem tem from the diontinuou nature of the ontroller. Similar to the ae of Setion 7., boundary layer approximate ontrol law wa introdued. The aymptoti tability of the ytem inide the boundary layer wa proven by reating the equation of motion into a tandard ingular perturbation problem. Steady tate error The origin (i.e. zero teady liding error tate i not the fixed point of the liding phae. A a reult, the table ytem exhibit a teady tate differene between the attained veloity and the deired et point. To redue thi error, an additional feedforward input wa added to the ontroller that utilize the available nowledge of the ytem parameter (i.e. nominal value.. High initial torque demand For initial ondition that are away from the liding urfae, the ontroller require high torque at the tart of the motion. It wa proven that the developed liding mode ontrol i apable of tabilizing the ytem while the veloity et point i varied. The proof of tability only require bounded variation veloity et point with bounded rate of hange. By gradually varying the veloity et point from zero (ytem at ret to the deired value, the initial high torque wa eliminated. The numerial imulation reult howed the effetivene of the modified liding mode ontroller in dampening the vibration aued by the negative damping intability mehanim and regulating the angular veloity of the lead rew to the deired et point. 84

204 Chapter 8 Reult Summary and Future Wor In thi hapter, a ummary of the reult obtained in thi thei i preented and poible area for future wor are diued. 8. Reult Summary Thi thei over a wide range of topi regarding the dynami of lead rew drive with frition. Mathematial modeling, model/frition parameter identifiation, mehanim of frition-indued intability, and robut vibration ontrol are the four area that mae up thi reearh. The reult obtained in thi wor aim to fill a ubtantial gap in the literature regarding the dynami of lead rew ytem with frition. A lead rew drive onit of a mehing lead rew and nut pair and onvert rotary motion to tranlation. Depending on the onfiguration of the ytem, the driver (e.g. eletri motor may rotate either the lead rew or the nut. In both ae, the load may be tranlated by either tranlating the lead rew or tranlating the nut. Clearly, vibratory behavior in any mehanial ytem require ome degree of flexibility in that ytem. In lead rew drive, numerou ompliant element may ontribute to the overall flexibility of the ytem and affet the number of degree of freedom required for modeling the ytem aurately. In thi wor, a unified mathematial framewor i preented for modeling the lead rew drive. Depending on the omplexity of the drive under tudy and the element (flexible or rigid inluded in the power tranmiion hain, the mathematial model developed in thi wor are apable of repreenting the ytem dynami while inorporating the handedne of the rew, diretion of motion, and the diretion of applied fore. All model preented in thi thei are baed on the bai -DOF lead rew drive model. A mathematial model by itelf i an abtrat objet uitable only for qualitative tudie. In order to onvert uh a model to a deign tool, a (quantitative validation proe i needed. In thi thei, the 85

205 lead rew drive mehanim of a powered eat adjuter i invetigated. The preliminary tet reult how that the frition-indued vibration of the two lead rew in the horizontal motion mehanim of thi eat adjuter i the aue of unaeptable level of audible noie. Baed on the developed mathematial model of the lead rew drive of thi eat adjuter, a novel parameter identifiation tehnique i introdued, whih onit of an identifiation tep and a fine-tuning tep. Uing the meaurement data olleted from the drive, ytem parameter uh a veloity-dependent frition parameter, damping, and oupling tiffne are identified. Comparion between the meaurement and imulation reult onfirm the auray of the identified parameter and validate the mathematial model of the lead rew drive. Thi model i then ued to perform parameter tudie, whih reult in the diovery of poible imple deign modifiation to eliminate audible noie problem. Three ditint fritional intability mehanim an affet a lead rew ytem. Thee mehanim are:. Negative damping,. Kinemati ontraint, and. Mode oupling. In thi thei, all of thee mehanim are tudied in detail. Negative damping intability, whih i a onequene of dereaing frition with inreaing liding veloity, i tudied firt. Thi mehanim i reponible for the intabilitie oberved in the lead rew drive of the eat adjuter mentioned above. The negative damping intability mehanim i tudied uing the -DOF bai lead rew model. After linearization, loal tability of the teady-liding fixed point i invetigated by evaluating the eigenvalue of the Jaobian matrix of the ytem, whih reulted in a parametri ondition for tability in term of the oeffiient of frition (more preiely, the rate of deay of frition with liding veloity and rotational damping of the lead rew upport. Thi analyi i expanded by the appliation of the method of averaging. It i hown that, depending on the value of the parameter and beaue of the aumption of exponentially deaying oeffiient of frition, one of the following three enario define the teady tate olution of the averaged ytem:. The origin (teady-liding tate i table and no other periodi olution (limit yle exit;. The origin i untable and it i urrounded by a table limit yle, or;. The origin i table and it i urrounded by an untable limit yle (whih define the region of attration of the table trivial fixed point. The untable limit yle i inide a table limit yle. The inemati ontraint and mode oupling tability mehanim an detabilize a mehanial ytem even with a ontant (veloity-independent oeffiient of frition. The inemati ontraint intability an our in -DOF ytem, wherea mode oupling intability i exluive to multi-dof ytem. 86

206 Parametri ondition for the onet of the inemati ontraint intability are found by analyzing the eigenvalue of the linearized verion of the bai -DOF model. Thi analyi give three ondition that together define the uffiient ondition for the teady liding tate to beome untable. The firt ondition tate that the lead rew mut be elf-loing. The eond ondition require that the fore applied to the nut be in ame diretion a the nut tranlation. The third ondition define a limit ratio between the ma of the tranlating part and the inertia of the rotating part. In other word, there i limiting value for the ma of the tranlating part (depending on the lead rew inertia, oeffiient of frition, and geometry of lead rew, below whih intability doe not our. The mode oupling i by far the mot omplex intability mehanim of the three mehanim tudied in thi thei. In an undamped ytem with two or more DOF mode oupling (flutter, intability our whenever two of the ytem mode merge. The neeary ondition for the mode oupling intability i the preene of fore that are not onervative. In lead rew ytem, frition provide the neeary ondition for the mode oupling intability to our. In thi wor, two -DOF model and a -DOF model of lead rew drive are tudied. By evaluating the eigenvalue of the Jaobian matrix of the linearized equation, parametri tability ondition are derived in eah ae. Thee ondition, epeially in the ae of damped ytem, are very omplex, and numerial analyi i neeary to evaluate the effet of eah ytem parameter on tability. The two -DOF model tudied differ from eah other in the oure of additional ompliane. One model inorporate ompliane in the thread (ontat of the lead rew and the nut thread are modeled with linear pring and damper and the other inorporate axial flexibility in the lead rew upport. It i found that elf-loing i a neeary ondition for intability in both of thee model. Another neeary ondition for intability (imilar to the inemati ontraint intability i the appliation of external fore in the diretion of motion. It i alo hown that introduing damping to only one of the two DOF of the ytem (either tranlational or rotational, detabilize the teady liding fixed point. Deign ontraint, propertie of the available material, or variation in the operating ondition may hinder effort to deign a lead rew drive that operate without frition-indued vibration. In uh ae, ative vibration ontrol may be applied to guarantee vibration free operation. In thi thei, two peed regulator are deigned baed on the liding mode ontrol method. The mot important feature of thee ontroller i their robutne to parameter unertaintie. It i aumed that the atual value of the ytem parameter are not available and only their upper and lower bound are nown. 87

207 The firt ontroller applie to a implified -DOF model, where it i aumed that all of the rotating part are rigid and rigidly onneted. Neeary modifiation are made to thi ontroller to eliminate the hattering problem, whih i an inherent drawba of the diontinuou (withing liding mode ontrol law. Stability propertie of the modified ontroller are proven baed on Lyapunov diret method. The eond ontroller i developed for a more realiti model that inorporate a flexible torional element between motor and lead rew. The preene of unmathed unertaintie (i.e. unertain term and input do not enter the equation at the ame point mae thi model muh more ompliated ompared to the -DOF ae. Three modifiation are applied to thi liding mode ontroller to improve it performane. Firt, to eliminate hattering, an approximate ontinuou verion of the ontroller i developed. Seond, a feedforward input i added to improve the teady tate error. Third, the veloity et point i et to vary with time from zero to the final deired value, thu eliminating the high torque demand at the tart of motion. Uing Lyapunov diret method, the method firt order averaging, and the ingular perturbation approah, tability the ontroller i proven for eah of thee modifiation. The preented numerial imulation reult how the effetivene of the two liding mode ontroller. 8. Future Wor Baed on the reult obtained in thi thei, ome area for further reearh are identified. Thee topi are lited here.. The model preented in Chapter do not inlude lateral, torional, or axial flexibility of the lead rew. In appliation where long and lender lead rew are ued, it may be neeary to inorporate flexibility of the lead rew in to the model.. Although balah an be inluded in the model developed in Chapter, the dynami effet of balah on the frition-indued vibration or ative vibration ontrol are not onidered here. The tudy of the effet of balah i important in poitioning lead rew.. The miro-pump ytem preented in Setion. ue tepper motor, and the additional nonlinearitie introdued by thi driver to the ytem warrant a further foued tudy. 4. In thi thei, the method of averaging wa ued to tudy the negative damping mehanim. Thi method or other perturbation tehnique an be applied to ytem with mode-oupling or inemati ontraint mehanim. Suh analye may provide a more 88

208 omplete piture of the nonlinear ytem a well a approximate and effiient method to predit teady-tate amplitude of vibration for parameter tudie. 5. The ontroller preented in Setion 7.4 aume that all of the tate are meaured. Further reearh may be direted toward the development of output feedba verion of thi ontroller that do not require meaurement of all of the tate. 89

209 Appendie 9

210 Appendix A Tet Setup A. Complete Seat Adjuter Experimental Setup A mentioned in Setion 4., the preliminary phae of the experiment on the powered eat adjuter wa limited to the analyi of the audible noie generated under different operational ondition. Figure A- how the tet etup developed for thee experiment. Figure A-: Experimental etup for preliminary tet on the omplete powered eat adjuter The intrument ued, hown in Figure 4-, were a follow: Fore meaurement: OMEGA panae tyle LCHD lb apaity Sound level (dba: TES 5A Sound Level Meter 9

211 Audible noie (ound wave: A general purpoe PC mirophone Seat diplaement: CELESCO poition tranduer SP- Signal from load ell, poition tranduer, and ound level meter were olleted uing a PC equipped with a Meaurement Computing data aquiition ard model PCI-DAS6/6. A mall Matlab/Simulin program wa written to reord ignal reeived by the data aquiition ard. A reenhot of the data aquiition program during one tet i hown in Figure A-. The ampling frequeny wa et to Hz. The ignal from the mirophone wa reorded by Window tandard ound reorder aeory oftware. The ound ampling frequeny wa 5Hz. Figure A-: Data aquiition in Matlab/Simulin environment A. Single Slider Experimental Setup The experiment performed on the omplete eat adjuter were repeated for a ingle lider. Figure 4-4 how the tet etup developed for thee tet. To implify the tet etup, the lead rew lider mehanim wa intalled upide-down ompared to it onfiguration in the omplete eat adjuter. In thi etup, the DC motor rotate a ingle lead rew, whih i horizontally fixed. A in the ae of the omplete eat experiment, a pneumati ylinder applie the required axial fore to the ytem. A hown in Figure 4-4, fore i applied diretly to the nut parallel to the lead rew axi

212 Intrumentation ued in thee tet were thoe lited in the previou etion. A. Lead Srew Experimental Setup The tet etup ued in the frition identifiation experiment of Chapter 4 i hown in Figure A- and A-4. See Table A- for a lit of intrument and omponent of thi tet etup. Two eparate et of experiment were performed uing thi etup: a loed-loop tet of Setion 4.. and open-loop tet of Setion 4.4. In loed-loop tet, the DC motor i driven through a ervo amplifier operating in the urrent mode (ee Figure A-4. In thi mode, the urrent output of the amplifier i proportional to the input voltage ontrol ignal. Conequently, the motor torque i proportional to the ontrol ignal. The amplifier gain and the DC motor torque ontant are. (A/V and.66 (N.m/A, repetively. Power Supply for the DC Motor Power Supply for the Solenoid Valve Solenoid Valve Load Cell Signal Conditioner dspae Unit Motor Speed Controller Program Figure A-: Experimental etup for the lead rew frition identifiation tet 9

213 The ontrol ignal for the loed-loop tet i generated by the dspae ontroller, whih i programmed in Matlab. The pneumati ylinder i ativated by a olenoid valve whih i alo ommanded by the ontroller. Two idential analog rotary enoder (inuoidal ignal, Vpp are ued to meaure the angular diplaement of the lead rew and the DC motor. Thee enoder have a reolution of 6 ount per revolution, whih i interpolated up to 4 time by the dspae ontroller and reorded. Other meaured ignal in thee tet are the load ell ignal (applied axial fore and the motor urrent, whih are alo aquired by the dspae ytem. Rotary enoder meauring angular diplaement of lead rew Load Cell Solenoid Valve (on the other ide of the lider DC Motor Servo Amplifier Rotary enoder meauring angular diplaement of DC motor Figure A-4: Intrumentation ued in the frition identifiation tet etup Figure A-5 how a ample of meaured angular diplaement and alulated angular veloity (by numerial differentiation of the lead rew. The meaurement data orreponding to the aelerating (tart of motion and deelerating (end of motion portion of eah tet i diarded and the reulting near teady-tate meaurement i averaged and reorded a one data point. See Figure A-6 for a ample of near teady-tate meaurement reult. 94

214 Table A-: Partial lit of omponent of the lead rew tet etup NO. ITEM SPECIFICATION MODEL MANUFACTURER 6 line per revolution, inuoidal Heidenhain Rotary Enoder ERN 8 inremental ignal (Vpp Load Cell lbf Mini Univeral Lin Load Cell LC7- Omega Load Cell Signal Conditioner Strain Gage Amplifier DMD-465 Omega 4 Motor Servo Amplifier Pule width modulation amplifier A8M Advane Motion Control 5 Power Supply DC Regulated Power Supply - BK Preiion 6 Solenoid Valve 4 way, olenoid valve with enter exhaut MVSC 4ER Mindman Pneumati 7 Pneumati Cylinder Double ating with ¾ troe MCQNF - Mindman Pneumati Figure A-5: Sample tet reult. (a Lead rew angular diplaement, (b Lead rew angular veloity. 95

215 Figure A-6: Near teady-tate portion of a ample tet reult. (a Lead rew angular veloity, (b Axial load, and ( Motor torque. In the eond part of the experiment with thi tet etup, the ame onfiguration wa ued but without the motor peed ontroller. The motor ervo amplifier wa withed to voltage mode and the dspae ytem wa only ued to ollet data. A ample of the open-loop experimental reult i hown in Figure 4-4(b. 96

216 Appendix B Firt Order Averaging Theorem Periodi Cae In thi appendix, baed on theorem and proof given in [5,44], a verion of the firt order averaging for the periodi ytem i tated and proven that i ued in Chapter 5. Theorem Conider the following ytem in tandard form Suppoe Then The funtion [ ] n (, x, ε, x( x x & εf t (B. f : R D, ε R i a T-periodi with repet to t for x D. + n D R i an open bounded et and > There exit a ontant > ε i ome number. f. M uh that ( t,x,ε M f ( t,x,ε i Liphitz ontinuou with repet to x and ε with Liphitz ontant λ x and λ ε, repetively. f dτ exit uniformly with repet to x. T The average, ( x f ( τ, x, Conidered the averaged ytem; The olution of (B., ( ;, x T ( z, z( x z & εf (B. z t, belong to interior ubet of D on time ale ε. 97

217 (B. and (B. There exit >, ε, and L > >, uh that the following hold for the olution of ( t, ε z( t, ε ε For ε ε and t L ε. Alo, i independent of ε. x (B. Proof: Let ( t, ε x( t, ε z( t, ε denote the error. From the two differential equation, (B.4 i found a The integrant in (B.5 an be written a E (B.4 t [ ] ( t, ε ε f ( τ, x( τ, ε, ε f ( z( τ, ε E dτ (B.5 [ ] [ f ( τ, x, ε f ( τ, z, ε ] + [ f ( τ, z, ε f ( τ, z, ] + f ( τ, z, f ( z where argument of E, x, and z are omitted for brevity. A a reult, from (B.5 we have t t t [ ] ( τ, x, ε f ( τ, z, ε dτ + ε f ( τ, z, ε f ( τ, z, dτ + ε f ( τ, z, f ( z E ε f dτ (B.6 The firt and eond term on the right-hand ide of (B.6 an be etimated uing the Liphitz ontant λ x and λ ε t t [ f ( τ, z, f ( z ] E ελ E dτ + ε λ t + ε dτ (B.7 x The third term in (B.6 or (B.7 i etimated a follow: ε t N it [ ( τ, z, f ( z ] dτ f ( τ, z, f ( z [ ] ( i t [ f ( τ, z, f ( z ] f dτ + i T NT dτ (B.8 where N i hoen uh that, NT t ( N + T. We have 98

218 ( i it [ f ( τ, z( τ, ε, f ( z( τ, ε ] T it ( i whih hold ine A a reult dτ [ f ( τ, z( τ, ε, f ( z( τ, ε f (( i T, z( ( i T, ε, + f ( z( ( i T, ε ] T it ( i [ f ( τ, z( ( i T, ε, f ( z( ( i T, ε ] T dτ dτ N ( i i T N N it [ f ( τ, z, f ( z ] it ( i i T it ( i i T dτ [ f ( τ, z( τ, ε, f ( z( τ, ε f ( τ, z( ( i T, ε, + f ( z( ( i T, ε ] [ f ( τ, z( τ, ε, f ( τ, z( ( i T, ε,] dτ + N it ( i i T dτ [ f ( z( ( i T, ε f ( z( τ, ε ] dτ (B.9 λ ελ T x N it x i T ( i MN [ z( τ, ε z( ( i T, ε ] dτ f t with The firt inequality in (B.9 hold ine f ( x ha the ame Liphitz ontant a (,x,ε repet to x f ( x f ( x [ f ( τ, x, f ( τ, x,] dτ f ( τ, x, f ( τ, x, T T λ T T T x x x dτ λ x x x The eond inequality in (B.9 hold ine z ( τ,ε i the olution of (B. and a uh i lowly varying: Alo note that z ( τ, ε z( ( i T, ε εtm T dτ 99

219 t NT Uing (B.9 and (B., (B.8 beome t [ ( τ, z, f ( z ] dτ TM f (B. [ ( τ, z, f ( z ] dτ εt Mλ N + TM TM ( λ L + f (B. x x Note that in (B., the inequality (B.7 beome NT t L ε wa ued. Finally, uing (B. the error etimate t ( λ L + E ελ E dτ + ελ L + εtm (B. x Applying the Gronwall lemma [5,44] to (B. yield ε x x ελ xt ( t, ε z( t, ε ε[ λεl + TM ( λ xl + ] e λ x L ε[ λ L + TM ( λ L + ] e ε x (B. λ x L Taing [ λ L + TM ( λ L ] e omplete the proof. ε x +

220 Appendix C A Definite Integral Ued in Averaging A een in Setion 5.., the averaging equation lead to the following integral π n m ξ in φ Λn, m( ξ in φo φe dφ (C. π Uing the power erie expanion for the exponential funtion (C. an be written a n n! x e (C. n x Λ π n+ m n, m( ξ ξ in φo π! φdφ (C. Noting that, π in p φo q φdφ if p or q i odd. The following derivation are due to Moll []. The following two ae are identified: Cae : n and m are both even number. Replaing n and m in (C. by n and m, yield

221 Λ n,m ( ξ π π π π ξ in n ( φo π π n+ φ! ξ! in m in φo ( n+ ξ! m φdφ φo in m φ dφ φdφ (C.4 It an be hown that π in p x o q x dx 4 π in p x o q x dx Β ( p +, q + (C.5 where Β repreent the Beta funtion and i given by ( p Γ( q ( p + q Γ Β ( p, q (C.6 Γ whih i defined uing the Gamma funtion and i given for p, q Z + a ( p! ( q! Β( p, q (C.7 ( p + q! and in the peial ae ( π p +, q + ( p q ( p!( q! Β (C.8 + p! q!( p + q! Subtituting (C.5 into (C.4 and uing (C.8 yield Λ n,m ( ( n +!( m! ξ ( n + m (! ( n +! m! ( n + + m ξ + Cae : n i an odd number and m i an even number. Replaing n and m in (C. by n and m, yield!, n, m,, K (C.9

222 ( ( ( π + π + π φ φ φ ξ π φ φ φ ξ π φ φ ξ φ φ π ξ Λ, o in! o in! in! o in m n m n m n m n d d d (C. One again ubtituting (C.5 into (C. and uing (C.8 yield ( ( ( ( ( ( ( + + ξ ξ Λ,!!!!!! 4 m n m n m n m n m n, K,,, m n (C. Here are ome peial ae: ( ξ ξ Λ,! (C. ( ξ ξ Λ,! (C. ( ( ( ( ( ( + ξ ξ + + ξ Λ,!!!! (C.4

223 Appendix D Steady-tate Solution of the Averaged -DOF Lead Srew Model In thi appendix, the averaged amplitude equation derived for the -DOF lead rew model in Chapter 5 i examined. It will be hown that, depending on the ytem parameter, the averaged equation an have,, or fixed point. The dynami behavior of the ytem i deribed by one of the following enario:. The trivial olution i table and no other olution exit.. The trivial olution i table and i urrounded by an untable limit yle, whih define the region of attration of the trivial olution. The untable limit yle i inide a table limit yle.. The trivial olution i untable and i urrounded by a table limit yle. Before ontinuing, it i important to tae a loer loo at the veloity-dependent oeffiient of frition given by (5.. The oeffiient of frition a a funtion veloity, ω >, an be written a; where r > r. > r ω r ω ( ω ( µ + µ + µ ω( e µ e (D. A hown in Figure D-, depending on the value of µ and µ, four ae an be identified for the variation of the oeffiient of frition with veloity. Thee ae are: Cae I: µ µ. In thi ae, for the entire range of appliable veloitie we have r ω Cae where µ < are not onidered ine the term µ e i added to the frition model only to emulate the Stribe effet (i.e. dereaing of the oeffiient of frition with inreaing relative veloity at low veloitie. 4

224 µ ω >. µ Cae II: µ µ <. In thi ae, ω when µ ω < ωb and < ω when ω > ω b. ω b deignate the loal maximum of the frition urve and define the boundary of the moothed neighborhood of zero veloity. Note that ω b a r. Figure D-: The veloity-dependent oeffiient of frition. Cae III: µ > µ. Thi ae i imilar to Cae II. However, a ditintion i made due to the poibility of additional periodi olution, a hown later in thi etion. µ Cae IV: µ > µ >. In thi ae, imilar to the Cae II and III, ω µ ω < ω b. Alo < ω when µ ω b < ω < ωm and ω when when ω ωm. Here ω b i 5

225 defined a before and ω m deignate the loal minimum of the frition urve. It mut be noted that for the ae where µ and partiularly when µ <, the aumed model of frition implie a limiting value for the oeffiient of frition at high veloitie. Beyond a ertain value of the veloity, whih i typially well outide the range of interet, the frition-veloity urve i aumed to be ontant. In Setion D. to D.4, eah of the above ae i tudied eparately and the exitene of limit yle i invetigated. The tability of eah olution (trivial and non-trivial i alo evaluated. The tudy in thi etion, for impliity, i limited to the ae where R >, N >, ω > ( v +, and θ & u& + ω ω. The lat inequality, limit the averaged amplitude equation to a. Some numerial example are preented in Setion D.5. Auming a (for non-trivial olution and dividing (5.58 by a, an equating the right-hand ide of the reulting equation to zero, the following polynomial equation in a i reahed where a y b a + n b n a n R b + + (( r µ + µ ωr r µ ωr r µ ω( r r (D. (D. b n n ω R n! ( n + n n+ n+ n+ ( r µ ( n + r + µ ωr r µ ωr r µ ω( r r + + n! (D.4 ω where r e r. D. Cae I: µ µ µ A mentioned earlier, in thi ae, > ω for the entire range of appliable veloitie. A a reult, See Setion 7... See hange of variable (5.5 and (5.. 6

226 (5.6 i atified for any and the trivial olution i table in D { a a }. To invetigate the poibility of non-trivial olution, (5.5 i examined. Setting µ in (5.5 and rearranging give a µ Ra r a ε ε π π in π r ωa in ϕ ε dϕ rµ R π ϕe in ϕe r ωa in ϕ From (5.56 and (5.57, we now that the two definite integral in (D.5 are non-negative and alo, a e r π π in r ω ϕe ( a in ϕ r ωa in ϕ dϕ π π in φe r ω ( a in ϕ dφ π π in φdφ dϕ (D.5 whih mean that the eond term in (D.5 i le than or equal to zero for µ. Sine the firt and third term are alo non-poitive, one onlude that ( < a a for a, and there are no other non-trivial fixed point in D. A typial plot of amplitude equation, (D.5, i hown in Figure D-. Figure D-: Shemati plot of amplitude equation for Cae I. D. Cae II: µ µ < Firt, we notie that for mall veloitie atifying where ω i the olution of ( r µ r µ ωr b ω < ω b (D.6 7, (5.6 i atified for any. Thu, the trivial olution i table. Moreover, ine all of the oeffiient of (D. are non-poitive, no other

227 olution exit. When ω > ωb, the trivial olution i table if > R r (( r µ r µ ω > (D.7 and it i untable otherwie. In the ae of table trivial fixed point, one again all of the oeffiient of equation (D. are non-poitive, whih implie that no other olution are poible. However, if (D.7 i not atified, b (given by (D. i poitive while the ret of the oeffiient, b n, (given by (D.4 remain le than or equal to zero. Aording to Dearte Rule of Sign [], (D. ha a poitive olution whih orrepond to a table periodi olution of the original ytem. Going ba to (D.5 with µ < the ondition for the poitive olution, ay the approximation i a ( a, to be inide the region of validity of. In term of the ytem parameter, thi ondition an be written a µ µ B + ω rm R < A + B < (D.8 where, A π π in φe r ω ( + in φ dφ and B π π in φe r ω ( + in φ dφ. Alo < A <. If thi ondition i violated the above firt averaging reult loe it O ( ε auray. To obtain olution with O ( ε auray, (5.45 or (5.48 mut be ued in the averaging proe to obtain reult with O ( ε auray. Typial plot of amplitude equation, (D.5, for thee two ae are hown in Figure D-. Figure D-: Shemati plot of amplitude equation for Cae II. 8

228 C. Cae III: µ > µ Similar to the previou ae, for low veloitie atifying where b ω < ω b (D.9 ω i the olution of ( r µ + µ ωr r µ ωr r µ ω( r + r any. Thu, the trivial olution i table. From (D.9, we have ( µ + r µ ω r r µ ω ( r + r r µ ω <, (5.6 i atified for r (D. whih yield, b <. Multiplying (D. by r n yield, Sine n ( n r n n+ < r < r r, + inequality one find r n n ( r µ r r r µ ω r ( r + r r µ ω+ r µ ω n n n n r < r, ( r r r ( n + r <, and µ <, from the above n n+ n+ n+ ( n + r r r µ ω ( r + r r µ ω+ µ r ω µ < Conequently, from (D.4 it i obviou that b <. Thu (D. ha no other olution. n When ω > ωb, the trivial olution i table if > R + (( r µ + µ ωr r µ ωr r µ ω( r r (D. If (D. hold b < otherwie b > large µ, ay and µ > µ µ, there exit > n+ r > rµ. Note that r ω r e i mall for ω > ωb N uh that for < n N, n n+ n+ ( n + r + r µ r + r µ ( r + r. For uffiiently for n > N µ n+ r < rµ n n+ n+ ( n + r + r µ r + r µ ( r + r. A a reult, the firt few b n are dominated by ω n n+ Rµ ωr + n! ( n +! n and are poitive. A n grow, the other term (all negative will dominate the poitive term and hange the ign of b to 9

229 negative. Hene, depending on the parameter value, one the following three enario may our; Senario : If (D. hold and > µ, then the polynomial equation given by (D. ha two µ ign hange in it oeffiient. Aording to Dearte Rule of Sign, for the firt ae, (D. an have either two or zero poitive root. Typial plot of a a a funtion of a are hown in Figure D-4. Note that imilar to the previou ae, for parameter value uh that a ( > of the table limit yle an be greater than., the amplitude Figure D-4: Shemati plot of amplitude equation for Cae III - Senario. The lead rew damping may at a the deiding fator between the above ituation. A typial plot of (D. i hown in Figure D-5 for the ae where two poitive root exit. Sine the polynomial equation for y (i.e. (D. i ontinuou, aording to the mean-value theorem there i a maximum between the two root and if thi value for b i y max (whih i a finite number, the ondition for non-exitene of non-trivial olution (i.e. no limit yle i b < y max or > + ymax > > where ΩR[ ( r µ + µ r r µ r r µ ( r + ] r. (D.

230 Figure D-5: The ae of two poitive zero of (D. Senario : If (D. hold and µ < µ, then the polynomial equation given by (D. ha no ign hange. A a reult, the trivial olution i table and there are no non-trivial olution. A typial plot of a a a funtion of a are imilar to the one hown in Figure D-. Senario : If (D. doe not hold, then the polynomial equation given by (D. ha only one ign hange. A a reult, there i a non-trivial olution. In thi ae, the trivial olution i untable and there i a table limit yle. A typial plot of a a a funtion of a i hown in Figure D-6. Figure D-6: Shemati plot of amplitude equation for Cae III - Senario. C.4 Cae IV: µ > µ > Similar to the previou ae for veloitie atifying (D.9, the trivial fixed point i table and there are no other olution. In addition to the above ondition, if µ ω > ω ln r rµ (D. m

231 then (5.6 i atified for any, whih implie that the trivial olution i table. For ω > ωb and for uffiiently large µ, N, N, N > N uh that n N N n n+ n+ n+ ( n + r + µ r > r µ r + r µ ( r + r b rµ n > n n+ n+ n+ ( n + r + µ r < r µ r + r µ ( r + r b < n N rµ n < For the firt few term, b n are dominated by ω n n n+ Rµ ωr term whih i poitive. A n grow, + n! ( n +! the negative term grow fater and hange the ign of b n to negative. For even higher value of n more ign hange are poible; however, the polynomial oeffiient beome exeedingly mall for large n and the aoiated root will be well outide the appliable region of the approximation. Baed on the above argument, for ω > ωm or for ω b < ω < ωm when i large enough to atify (5.6, the equation given by (D. ha at leat two ign hange in it oeffiient, reulting in either two or zero valid non-trivial olution. Similar to the argument in Setion C., the damping value may at a a deiding fator between thee two ae. In other word, for uffiiently large damping, no non-trivial olution exit. A typial plot of a a a funtion of a i hown in Figure D-4. Finally for ω b < ω < ω m, if doe not atify (5.6, the equation given by (D. ha at leat one ign hange in it oeffiient, reulting in one non-trivial olution. A typial plot of a a a funtion of a i hown in Figure D-6. C.5 Numerial Example In Chapter 5, numerial example were preented that fou on the auray of the vibration amplitude predition from averaging a well a the effet of frition parameter. The parameter value ued were taen from the experimental ae tudy of Chapter 4 and orrepond to the ae II and III in Figure D- and Setion D. and D. of thi appendix. Here, numerial example are preented that alo over ae IV. In the bifuration plot preented, input ontant angular veloity, ω, i taen a the ontrol parameter. Intead of a, α ( ω Ωa i plotted veru the ontrol parameter to better demontrate the evolution of non-trivial olution in term the original ytem vibration amplitude. Parameter value not given in the example are taen from Table 5- and are lited Table C- for eae of referene.

232 Table C-: Parameter value ued in the imulation Parameter Value Parameter Value d m.7 mm µ. λ 5.57 r.5 rad/ I 6. g.m r rad/ N.m/rad ω -4 rad/ m g R N Reult of the firt example are hown in Figure D-7. In thi example µ and three different value are onidered for 4 µ. The lead rew upport damping i hoen a ( N. m. rad. For eah value of µ, the oeffiient of frition a a funtion of relative angular veloity i plotted in Figure D-7(left. A hown in the teady-tate vibration amplitude plot in Figure D-7(right, for the eleted damping value, a frition reahe it maximum the gradient beome negative ( ω > ωb trivial olution loe it tability and a table limit yle emerge. For the three value of µ eleted,.,.5, and., ω b i found to be ( rad.9,.8, and. 76, repetively. Thee value are in agreement with (D.9. For maller value of µ, the region of intability of the trivial olution i maller. A hown in the loe-up view, larger value of µ reult in table amplitude of vibration loer to the limiting value (for the validity of approximation of a max (or ω Ω, where Ω i the natural frequeny of the α max unperturbed ytem given by (5.4. For even larger value of µ (not hown, the non-trivial olution of the amplitude equation (orreponding to the table limit yle, (5.58, beome greater than and i inadmiible due to violation of the approximating aumption. In thee ae, O ( ε aurate averaging reult an be found by uing (5.45 or (5.48 and arrying on the integration numerially. It i intereting to note that in thi example a well a the two example that follow, a ω i gradually inreaed, the trivial fixed point firt goe through a uperritial pithfor bifuration and then a ubritial pithfor bifuration at a higher veloity. Thee bifuration in the amplitude equation orrepond to Hopf-bifuration of the original equation. Figure D-8 how reult for µ. and 4 µ 4 for three different value of lead rew damping. the

233 Figure D-7: Firt example, µ, -4. Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity Figure D-8: Seond example, µ., µ Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity 4

234 A expeted, the trivial olution beome untable a ω beome greater than ω b. 76. A hown, inreaing the damping dereae the region of intability of the origin. Figure D-9 how reult for µ. and 4 µ 4 for three different value of lead rew damping. A expeted, the trivial olution beome untable a ω beome greater than ω b. 76. Moreover, for the trivial olution beome table again when ω > ωm Note that from (D., one find ω m At higher value of, the trivial olution beome table at lower veloitie. A hown, inreaing the damping dereae the region of intability of the origin and dereae the amplitude of table periodi vibration. Figure D-9: Third example, µ., µ 5-4. Left - variation of the oeffiient of frition with veloity; Right - variation of teady-tate vibration amplitude with input angular veloity 5

235 Appendix E Higher-order Averaging The auray of the reult obtained in Setion 5. depend heavily on the ize of the ytem parameter. In pratial appliation, uh a the experimental example of Chapter 4, the O ( ε error of the firt order averaging may not be uffiiently aurate for the entire range of parameter in the domain of interet. A poible way to improve the auray of the amplitude and frequeny etimate i to extend the averaging to higher order. A lightly different approah to finding the equation of motion in tandard form i ued here. In 4 Setion E., the approximate ytem equation aurate to ( ε O i derived. The tep needed to arry out the averaging proe up to the third order are preented in Setion E.. In Setion E., a numerial example i preented that ompare averaging reult with numerial imulation reult and atual meaurement. E. Equation of Motion in Standard Form Two implifying aumption made in Chapter 5 to reahed the averaged equation (5.54 namely, the ontat fore doe not hange ign and lead rew veloity doe not hange ign. Thee aumption are made here from the tart to enure the equation have the required moothne propertie. For impliity here the analyi i limited to the ae of R > (thu N > and ω > (thu θ & >. Starting from (5. v + ε ε where Ξ ( ε and Ξ (,, ε v implify thee funtion to [ εξ ( ] [ ( v, v, ε m v + εξ v, v, ε m] R[ εξ ( v, v, ε m] [ Ξ ( ε Ξ ( v, v, ε ] v (E. v are given by (5.4 and (5.5, repetively. The above aumption 6

236 ( ε µ Ξ + ε µ (E. where µ ( v i obtained from (5.6 Ξ ( ε and Ξ (, ε µ µ Ξ ( v, ε (E. + ε ( v µ ( v r ω( v + ( ( e r ω( v + ( v µ ˆ + µ ˆ e + µ ˆ ω( v + v an be expanded in power of ε a Ξ 4 ( ε ( µ ( ε µ + O( ε (E.4 (E.5 Ξ 4 ( v, ε ( µ ( ε µ + O( ε (E.6 Alo µ ε m + ε µ + ε ε µ ( µ m + ε ( µ m + ε ( µ 4 ( µ m + O( ε Subtituting (E.5, (E.6, and (E.7 into (E. and eeping term up to ( ε m O give (E.7 ( v v + ε f ( v, v + ε f ( v v v + v εf,, (E.8 where f (E.9 ( v, v v ( µ mv + R( µ µ f, (E. ( v v ( µ mv ( µ m v + R( µ ( µ µ m f ( v, v ( µ m v ( µ m v + R( µ ( µ µ + µ ( µ mv R µ ( µ µ ( µ [ ] m (E. Unfortunately, the expreion involved beome too umberome to be of any pratial ue in loed-form. However, if approahed numerially, thee higher order approximation an be ued to 7

237 etimate the amplitude of the teady-tate vibration, effiiently. E. Higher-order Averaging Formulation In thi appendix, following [4] the general formulation of firt, eond, and third order averaging are derived for a ytem of differential equation in tandard form ( t x + ε X ( t, x + ε X ( x x & εx, (E., t where x a (E. β X i X i (E.4 X i To larify the notation, the following expreion are given here X i X i a β X i( t, ξ ξxi( t, ξ (E.5 X i X i a β and X ij X ij ( a a β X ξ ij t, (E.6 X ij X ij β a β E.. Firt Order Averaging Introduing the following near-identity tranform where ξ i the olution of ( ξ x ξ + εf, (E.7 t 8

238 ( ξ & εp (E.8 ξ where F and P are unnown funtion to be determined. Subtituting (E.7 into (E. (negleting ε and ε term d dt ( ξ + εf ( t ξ εx ( t, ξ + εf (, ξ, t expanding RHS uing Taylor erie expanion and ubtituting (E.8 give F εp + ε + ε t ε + ε F P X X F Negleting ε term ( t, ξ F X, t ( tξ P ( ξ (E.9 The olution to thi equation an be written a Subtituting (E. into (E.9 and integrating T P dt (E. ( ξ X ( tξ, T t (, ξ [ X ( τξ P ( ξ ] dτ + a ( ξ F, where following [5], a ( ξ i hoen uh that F (,ξ t (E. T t t ha a zero mean a dτdt (E. ( ξ [ X ( τξ P ( ξ ], T It an be hown that the olution of (E.8 remain O ( ε loe to the olution of the original differential equation (E. on a time ale of O ( ε, i.e. ( t ξ( t ε x, L t (E. ε 9

239 for ome > and L >. E.. Seond Order Averaging The near-identity tranform i modified to where ξ i the olution of ( tξ + ε F ( ξ x ξ + εf, (E.4, t ( ξ + P ( ξ ξ & εp ε (E.5 where F and P are defined a before, and F and P are unnown funtion to be determined. Subtituting (E.4 into (E. (negleting d dt ε term ( ξ + εf ( t, ξ + ε F ( t, ξ εx t, ξ + εf ( t, ξ + ε F ( t, ξ + ε ( ( t, ξ + εf ( t, ξ + ε F ( t, ξ X Expanding the RHS uing Taylor erie expanion F ξ& + ε + ε F ξ& + ε t F t + ε F ξ& εx + ε X F + ε X F + ε + ε T [ F + εf ] X[ F + εf ] T [ F + εf ] X [ F + εf ] X + ε X F Subtituting (E.5 and negleting ε term F P t F P X F X (E.6 The olution to (E.6 an be written a P or ine ( ξ P ( ξ T + ( ξ [ X ( t, ξ F ( t, ξ F ( t, ξ P ( ξ X ( tξ ], T t ha a zero mean F, dt

240 T P [ X( t, ξ F ( t, ξ + X ( t, ξ ] dt (E.7 T where F ( t,ξ i given by (E.. Subequently, integrating (E.6 give t F, d ( t, ξ [ X ( τ, ξ F ( τ, ξ F ( τ, ξ P ( ξ + X ( τξ P ( ξ ] τ + a ( ξ (E.8 where a ( ξ i hoen uh that F (,ξ T t t ha zero mean a dτdt (E.9 ( ξ [ X ( τ, ξ F ( τ, ξ F ( τ, ξ P ( ξ + X ( τξ P ( ξ ], T It an be hown that given ξ ( t to be the olution of (E.5 for time ( ε O. E.. Third Order Averaging ( t ξ( t εf t ξ( t ( O( x, ε (E. Similar to the previou etion, the near-identity tranform i now defined a where ξ i the olution of ( tξ + ε F ( t, ξ + ε F ( ξ x ξ + εf, (E., t ( ξ + ε P ( ξ + P ( ξ ξ & εp ε (E. where F, F and P, P are defined a Setion E. and F and P are unnown funtion to be determined. Subtituting (E. into (E. d dt ( ξ + εf ( t, ξ + ε F ( t, ξ + ε F ( t, ξ εx t, ξ + εf ( t, ξ + ε F ( t, ξ + ε F ( t, ξ + ε + ε X Expanding the RHS uing Taylor erie expanion ( ( t, ξ + εf ( t, ξ + ε F ( t, ξ + ε F ( t, ξ ( t, ξ + εf ( t, ξ + ε F ( t, ξ + ε F ( t, ξ X

241 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] F X F X F X X F F F F F F F F F F F F F X F X F X X F F F F F F F F F F F F F X F X F X X ξ F F ξ F F ξ F F ξ + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε ε + + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε + ε ε + + ε + ε + ε ε + ε + ε + ε + ε + ε + ε X X X X t t t T T T T & & & & Subtituting nown funtion F, F, P, and P and negleting higher order term, yield F F F F P X P F F X P F F X F X X t T T (E. P i found to be T T T dt X X T F F F F X P F F X P F F X P (E.4 It an be hown that given ( t ξ to be the olution of (E. ( ( ( ( ( ( (,, ε ε ε O t t t t t t ξ F ξ F ξ x (E.5 for time ( ε O. E. A Numerial Example Here a numerial example i preented that i taen from the tet reult of Chapter 4. The parameter value are eleted aording to the ample imulation reult/meaurement depited in Figure 4-6. The identified oupling tiffne, damping oeffiient, and all of the frition parameter are given, for referene, in Table E-. The applied axial fore and input angular veloity value are alo lited in thi table.

242 Table E-: Parameter value ued in the higher order averaging example Parameter Value Frition Parameter Value.8 N.m/rad µ Nm/rad µ. µ /rad Input Value r.8 rad/ ω 5.6 rad/ r.4 rad/ R 5 N µ.97 The reult from the numerial averaging method are preented in Figure E-. In Figure E-(b, the meaurement are ompared with imulation reult howing the auray of modeling and the identified parameter imilar to Figure 4-6. In Figure E-(a, the ame imulation reult are ompared with the firt, eond, and third order averaging. It an be een that for the eleted parameter value, the firt order averaging ha oniderable error in prediting the teady-tate amplitude of vibration (a relative error of approximately %. The eond order averaging reult, on the other hand, how ignifiant improvement in both prediting the teady-tate vibration amplitude (relative error i approximately 5% and onforming to the hape of the oberved limit-yle. The auray of the approximation i further improved, though only lightly, by the third order averaging whih ha approximately 4% relative error in prediting the amplitude of vibration.

243 Figure E-: Firt, eond, and third order averaging reult. (a Numerial averaging reult; gray olid: nonlinear ytem equation; dotted bla: firt order averaging; dahed-dot: eond order averaging; olid bla: third order averaging, (b bla: meaurement; gray: imulation reult 4

244 Appendix F Firt-order Averaging Applied to the -DOF Lead Srew Model with Axially Compliant Support In thi appendix, the method of firt-order averaging i ued to analyze the -DOF model of Setion.6. The equation of motion are given by (.9 and (.. Negleting F and T for impliity and repeating the tranfer of oordinate done in Setion 6.; θ v x r + θ + u m i tan λv + u (F. where u and u are given by (6.5 and (6.54, repetively. After thi hange of variable, equation of motion in matrix form beome where ( v & && v C( v& v& + K( v& v f (& M + (F. v I ξmrm tan λ ξmrm tan λ M (F. m m + m C (F.4 K (F.5 5

245 and ( ξ ξ f (F.6 R Alo ξ ( v, v& ( v, v& tan λ ( v, v& tan λ µ r m (F.7 + µ and where the ontat normal fore i given by (, v& µ ( v& gn( N( v, v& ( v& + ω v (F.8 µ gn N I m + I m r m I tan λξ R + rm tan λ v m I I ( oλ + µ λ + λξ in rm tan m m I + v& m + v + v& (F.9 Auming M to be non-ingular ξmrm tan λ ( I I m + m M ξ tan λ tan λξ ( ( ( (F. mm m I mrm rm I m + m I m + m I m + m Multiplying both ide of (F. by where M, one find mm ( ˆ ( ˆ ( ˆ + r m tan λξ v& && v + C v& & ( ( v + K v& v f v& I m m (F. + ξmr m tan λ ( ˆ I I m + m C ( I ξmr tan λ ( ( (F. m m I m + m I m + m 6

246 ξmrm tan λ ( ˆ I I m + m K ( I ξmr tan λ ( ( (F. m m I m + m I m + m ˆ R f m ( ξ ξ (F.4 ( m + m I The two natural frequenie of the undamped unperturbed ytem are Ω, Ω (F.5 I m + m Let m r m tan λ m I (F.6 Taing ε tan λ a the mall parameter, it i aumed that m and µ µ (F.7 tan λ µ are ( O with repet to ε. We an write the following aymptoti expanion for ξ Alo ξ εr (F.8 m ( µ ε r µ ( µ + K m mm µ mm ε + ε ε m + m + ε µ m + m ( µ + O( (F.9 Define the non-dimenionalized time a τ Ω t ( d( d dt Alo, limiting the analyi to wealy damped ytem, tae. Derivative with repet to τ i given by ( d Ω, ( (F. dτ dτ 7

247 ε IΩ, ε Ω ( m + m (F. Aume ω Ω ρε where ρ i O ( with repet to ε. Let, v ερz (F. Subtituting (F.5, (F.6, (F.7, (F., and (F. into (F. and uing (F.8, (F.9, and (F. and trunating ( ε O term, yield z ~ ~ + K z εcz + εm ~ ( α( µ K z εkz + εf ~ ~ (F. where m α (F.4 m + m ~ K α η (F.5 ~ K η ( µ m (F.6 ~ C α (F.7 ~ f R( µ gn( Rω µ α (F.8 where R Ω ω rm R (F.9 i onidered to be O ( with repet to ε. Alo, η i the ratio of the two natural frequenie 8

248 Ω Ω η (F. Auming the two natural frequenie of the unperturbed ytem are widely apart, let T α (F. η then Ω T K ˆ T (F. η Let z Tw, ytem equation (F. are tranformed to ~ ~ ~ w εt CTw + εm w + Ω ( α( µ Ω w εt KTw + εt f (F. Finally, let y oτ y in τ y in τ y o τ w, w (F.4 y oητ η y4 in ητ ηy in ητ y4 oητ Subtituting (F.4 into (F. and expanding, after ome algebra the following four firt order differential equation in tandard form are reahed y ε ( y in τ + y o τ α εm + η εmη εr in τ ( µ ( y o τ y in τ ( µ ( y oητ η y in ητ ( µ gn( Rω µ in τ 4 in τ in τ (F.5 Note that the more onvenient hoie of amplitude/phae tranform (ee Setion 5.. i not ued here ine the reulting differential equation would have been ingular at the origin. 9

249 y ε ( y in τ + y o τ α εm + η εmη εr o τ ( µ ( y o τ y in τ ( µ ( y oητ η y in ητ ( µ gn( Rω µ o τ 4 o τ o τ (F.6 α y ε η η εηm + εη ( ( ( η y ( in τ + y o τ in ητ ε ηy in ητ + y oητ 4 η ( α( µ ( y oητ η y in ητ in ητ ( µ mα α + ( y o τ y in τ ( µ m α + ( y oητ η y in ητ + εη η + εα R η η η η ( µ gn( Rω µ in ητ 4 4 in ητ in ητ in ητ (F.7 α y ε 4 εηm + εη ( η ( η ( y in τ + y o τ in ητ ε ( ηy in ητ + y o ητ ( α( µ ( ηy oητ y in ητ ( µ mα α + ( y o τ y in τ + εη η + εα R η where ( ε Alo not that argument of µ ( θ & η η η ( µ m α + ( ηy oητ y in ητ ( µ gn( Rω µ oητ 4 4 oητ oητ 4 oητ o ητ (F.8 N y y. O term were negleted. From thi point on, for impliity, we aume that (, > (lead rew angular veloity i given by ( y τ + y o τ θ& ω ω in (F.9 It i more onvenient to expre the above ytem of firt-order differential equation a

250 ( η ψ ψ ε ε y,ψ, g y (F.4 where [ ] T y y y y 4 y. The right-hand-ide of (F.4 i quai-periodi in τ (i.e. it i π periodi with repet to both ψ and ψ. Hene, the firt order averaged equation an be derived from [5] ( π π ψ ψ π ε,, 4 d d ψ y g y (F.4 After ome implifiation, arrying out the integration for (F.5, (F.6, (F.7, and (F.8 yield (dropping the bar ( π π ψ ψ ω ψ ψ µ π η α + ε ψ ψ µ π + ε ε in o in in d y y m d R y y & (F.4 ( π π ψ ψ ψ ω ψ µ π η α + ε ψ ψ µ π + ε ε o in o o d y y m d R y y & (F.4 ( ( ( π π ψ µ π + η α ε ψ µ π α + ε ε d y m d y m y y & (F.44 ( ( ( π π ψ µ π + η α + εη ψ µ π α εη ε d y m d y m y y & (F.45

251 ( where µ µ ω ω( y in ψ + y ψ o. Introduing the polar oordinate the amplitude equation are found a a ε a R + ε π π α a εm + η π y y y y 4 µ in π a a a oβ inβ oβ a ηinβ ( ψ + β dψ ( µ ( ψ + β ( ψ + β o in dψ (F.46 (F.47 a ε (F.48 a Thi equation i imilar to the -DOF ae tudied in Setion 5... In fat, if the ame limitation are onidered a in Setion 5.. (i.e. gn ( θ gn( N where µ µ ( ω( a ψ in & π R a ε a + ε µ in ψdψ π and i defined imilar to (5.49 by, it further implifie to (F.49 µ r ωa in ψ r ωa in ψ ( µ + µ e µ a in ψ ( r e (F.5 and µ, µ, and µ are given by (5.5, (5.5, and (5.5, repetively. Notie that (F.49 i exatly the ame a (5.48 if a i replaed by a. The firt order averaged equation for a, (F.49, and a, (F.48, are deoupled. Furthermore, (F.48 how that, to thi order of approximation, the vibration omponent with the frequeny ω die out exponentially independent of a.

252 Appendix G Similaritie in the Condition for Loal Stability of the Steady-liding Fixed Point Between the -DOF Model of Setion.5 and the -DOF model of Setion.6 The Routh-Hurwitz tability ondition for the -DOF model of Setion.5 and.6, with ontant oeffiient of frition, are given in Chapter 6. Thee ondition were obtained from the harateriti equation of the linearized ytem model. For the -DOF model with ompliant thread, the harateriti equation given by (6.4 i expanded a 4 a η + a η + a η + a η + a (G. 4 where a I mˆ 4 (G. a [( + µ gn( Rω tan λ( I r tan λξ m ] mˆ ˆ m a + (G. [( + µ gn( Rω tan λ( I r tan λξ m ] + mˆ + ˆ ( + µ gn( ω λ ˆ m tan (G.4 R a ( ˆ + ˆ + µ gn( ω ( λ tan (G.5 R ( + µ gn( ω λ a ˆ tan (G.6 R where ˆ, ĉ, and mˆ are given by (6.4, (6.5, and (6.6, repetively. For the -DOF model

253 with axially ompliant lead rew upport, the harateriti equation given by (6.8 i expanded a where 4 b η + b η + b η + b η + b (G.7 4 ( m + m mm ξ r λ b I tan (G.8 4 m ( m + m + ( I ξ r m b m tan λ (G.9 ( m + m + + ( I ξ mr λ b tan (G. m b + (G. r m b (G. To ee the imilaritie between (G. and (G.7, we divide (G. by the tritly poitive quantity ( + µ gn( ω tan λ in λ R where. Alo uing (6.4, (6.5, and (6.6, (G. beome 4 a η + a η + a η + a η + a (G. 4 ( m + a 4 I m (G.4 ( m + m + ( I ξ r λm δ a δ tan (G.5 ( I ξ r λm + ( m + m + + ( I ξ r λm a tan δ tan (G.6 m m a (G.7 + m where m δ i defined a a (G.8 See footnote on page 97. 4

254 m m (G.9 o λ( + µ gn( Rω tan λ δ Note that for µ > tan λ and R ω >, m δ i a mall negative quantity. For example, for µ. 8 o and λ 5.57, m δ m.. The oeffiient given by (G.9, (G., (G., and (G. are truturally idential to (G.5, (G.6, (G.7, and (G.8, repetively. In the two model, and have the ame effet on the eigenvalue a and. The major differene between the two harateriti equation i in (G.8 and (G.4. However, for mall lead rew ma ( m, the differene i mall. A an example, ompare Figure 6- with Figure 6-4. The differene between the tability ondition of the two model beome ignifiant for uffiiently large m and m, ine b 4 an beome negative, leading to inemati ontraint intability (ee Setion 6.., while a 4 i alway poitive. 5

255 Appendix H Further Obervation on the Mode Coupling Intability in Lead Srew Drive Although the linear omplex eigenvalue analyi method i ueful in etablihing the loal tability boundarie of the fixed point of a ytem, it doe not reveal any information regarding the dynami behavior of the original nonlinear ytem. Sine the fou of thi wor i on the lead rew vibration and the poibility of generation of undeirable noie, there i a great deal of interet in the atual behavior of the ytem. Even when the teady-liding fixed point i untable, there ould be ituation where the reulting amplitude of teady-tate vibration i very mall and onequently no audible noie i generated from the ytem. In thi appendix, through numerial imulation, the effet of variou ytem parameter on the dynami behavior of the lead rew drive under the mode oupling intability ondition are invetigated. Firt, in Setion H., variation in the teady-tate vibration amplitude of the lead rew i tudied a the two damping oeffiient in the -DOF model of Setion.5 are varied. The effet of oupling tiffne and ontat tiffne of the ame model are invetigated in Setion H. through five different numerial imulation. In the lat imulation, the poibility of hao i alo briefly mentioned. Unle otherwie peified, numerial value of the ytem parameter are thoe lited in Table 6-. H. Effet of Damping on Mode Coupling Intability in a -DOF Sytem with Contant Coeffiient of Frition In Setion 6.., it wa proven that in the extreme ae where damping i preent in only one of the two ytem DOF, the teady-iding fixed point i untable. In addition, the omplex effet of damping in expanding or reduing the parameter region of tability wa hown by the example in Figure 6- and Figure 6-4. The atual variation in the teady-tate amplitude of vibration an 6

256 have an even more omplex behavior. Figure H- how a map of averaged amplitude of vibration of the lead rew for 7, a lead rew upport damping and ontat damping oeffiient are varied. The two natural frequenie of the undamped ytem are approximately 48. and 94.6 Hz. The initial ondition were hoen loe to the fixed point; u ( u ( and u ( u ( ω & &. For eah pair of damping oeffiient, the numerial imulation were arried out for four eond. The reult for the firt eond were diarded to exlude the tranient. A an be een, the teady-tate amplitude of vibration varie oniderably with the hange of the two damping oeffiient. For the numerial value of the parameter hoen here, the ytem exhibit haoti or multi-period behavior for many of the eleted value of the two damping oeffiient. Figure H-(a and Figure H-(b how the bifuration diagram of Poinare etion ( u&, a the damping parameter i hanged along the horizontal dotted line and the vertial dotted line in Figure H-, repetively. Further example of thi phenomenon are preented in Setion H. below. Figure H-: Averaged amplitude of vibration, y (rad, a lead rew upport damping,, and ontat damping are varied 7

257 Figure H-: Bifuration of Poinare etion. a Along the horizontal dahed line in Figure H- b Along the vertial dahed line in Figure H-. H. Effet of Stiffne on Mode Coupling Intability in a -DOF Sytem with Veloity-dependent Coeffiient of Frition In thi etion, reult from five erie of numerial imulation are preented that how the effet of tiffne (oupling,, and ontat, on the tability of the -DOF model of Setion.5. Thee numerial tudie alo inlude the amplitude and frequeny of the teady-tate vibration of the lead rew. Here the frition oeffiient i aumed aording to (. with parameter taen from Table 4-. Other ytem parameter, not peified in the example, are eleted a before. H.. Example # For thi firt example, the following parameter are onidered; m, and the ontat tiffne,, i varied between 5 4, 4 to 6 (N/m. Figure H-(b how the variation of the real part of the eigenvalue of the ytem Jaobian matrix a the ontat tiffne i varied. The teady-liding fixed point beome untable between 5.9 and A 8

258 hown in Figure H-(a by the two-ided Poinare etion, the amplitude of lead rew vibration varie oniderably in thi range and reahe a maximum of approximately. (rad. Figure H-: Effet of ontat tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue During thi range, the two natural frequenie are loe to eah other a hown by the dahed line in Figure H-4. It an be een in thi figure that the dominant frequeny of vibration follow the higher linear mode at the beginning of the untable range and move toward the lower linear mode at the end of it. 9

259 Figure H-4: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at 4x 5 H.. Example # In thi example, parameter value are hoen the ame a the previou example exept for the lead rew upport damping, whih i lowered to 5 5. A hown in Figure H-5(b, by lowering the lead rew damping, the ytem i untable throughout the imulated range of. The amplitude of vibration, a hown in Figure H-5(a, hit a maximum of approximately. (rad whih i lightly higher than the previou ae. It i intereting to note that at approximately 5 8.7, the ytem undergoe a perioddoubling bifuration [6] whih lat until The evolution of pea vibration frequenie are hown in Figure H-6(a. The dominant vibration frequeny follow the higher linear mode and then hift to the lower mode a the two linear frequenie move loer to eah other. The dominant frequeny of vibration ontinue to follow the lower mode a the two linear frequenie grow apart. The period-doubling bifuration i alo viible in thi frequeny plot and in the etion hown in Figure H-6(b. At the intant plotted, the dominant frequeny i 94Hz and the two pea at 94 either ide of it, aued by the bifuration, are at 94 ± Hz. 4

260 Figure H-5: Effet of ontat tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue Figure H-6: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at 9x 5 4

261 H.. Example # In thi example the ontat tiffne i fixed at varied between. and 8 (N.m/rad. The other parameter are eleted a; m, 6, and in it plae the oupling tiffne,, i 5 4, Similar to example # above, Figure H-7(b how that the teady-liding fixed point i untable in a portion of the parameter range (i.e. approximately between.58 and The Poinare etion in Figure H-7(a how that during the untable parameter range, the amplitude of teady-tate varie oniderably. There i alo a diontinuity at about 5.5 (N.m/rad. The amplitude of vibration remain around.46 (rad before the diontinuity but afterward tart to diminih. Figure H-7: Effet of oupling tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue Parallel to the jump in the amplitude, a ditinguihable jump in the dominant frequeny of lead rew vibration i diernable in Figure H-8(a. At thi point, the two linear frequenie tart to move away from eah other and the frequeny of vibration jump toward the higher mode and follow it on. A etion of thi figure i hown in Figure H-8(b for 4. 4

262 Figure H-8: (a Bla line: Evolution of pea lead rew vibration frequenie, dahed grey line: eigenfrequenie; (b Frequeny ontent of teady-tate lead rew vibration at 4 H..4 Example #4 The parameter hoen for thi example are exatly the ame a the previou example exept for the 5 two damping oeffiient whih are redued to 4 and. A an be een from Figure H-9(b, the ytem fixed point i table throughout the onidered parameter range. Figure H-9(a how the bifuration plot of two-ided Poinare etion. A expeted, the vibration amplitude grow a the oupling tiffne i redued. Alo, it an be een that the ytem exhibit haoti behavior for the eleted value of the parameter. Figure H- how the evolution of dominant frequenie of the lead rew vibration a the oupling tiffne i varied. For 4, 7, and, variou imulation reult are plotted in Figure H-, H-, and H-, repetively. At 4, the petrum plot in Figure H-(a how well eparated pea (thoe pea whih are viible in Figure H- are at 8.6, 57.6, 86., and 5. (Hz. The y -y projetion of the ytem trajetory in Figure H-(b how a periodi orbit (limit-yle. The Poinare etion hown in Figure H-(, onfirm the preene of a 5-period periodi olution at 4. 4

263 Figure H-9: Effet of oupling tiffne (a Two-ided Poinare bifuration diagram, (b Real part of the eigenvalue Figure H-: Bla line: Evolution of pea lead rew vibration frequenie, dotted grey line: eigenfrequenie The petrum plot of lead rew vibration at 7 i hown in Figure H-(a. The two inommenurable dominant frequenie are 6.8 and 7.. The reulting quai-periodi orbit i hown in Figure H-(b, by the filled out region of the phae plane, and in Figure H-(, by the loed-urve Poinare etion. 44

264 Figure H-: Simulation reult for lead rew at 4; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion Figure H-: Simulation reult for lead rew at 7; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion The petrum plot of lead rew vibration at i hown in Figure H-(a. Here, in ontrat to the two previou ae, the petrum ha a broadband harater. The two dominant pea are loated at 95. and 47.Hz. The y -y projetion of the ytem trajetory in Figure H-(b i imilar to Figure H-(b and fill out a portion of the phae plane; however, the poibility of haoti behavior i hown by Poinare etion in Figure H-(. Further invetigation i needed to onfirm that thi 45

265 attrator i indeed haoti, whih i outide the ope of thi wor. Figure H-: Simulation reult for lead rew at ; (a Vibration frequeny ontent, (b y -y projetion of the trajetorie, ( Poinare etion To verify that the attrator i haoti, the enitivity to the initial ondition mut be hown [6]. 46