ABSTRACT. over two times those experienced by a passenger car tire. The results obtained

Transcription

1 BSTCT Title of Thesis: Heavily Loaded Vehicle Tires: nalysis and Characterization Nicholas Garcia, Master of Science, 8 Thesis directed y: Dr. Balakmar Balachandran Department of Mechanical Engineering Viration characterization and eperimental modal analysis were carried ot with a heavily loaded vehicle tire. These tires are typically sjected to loads that are over two times those eperienced y a passenger car tire. The reslts otained throgh eperimental modal analysis were compared to similar reslts otained for standard passenger car tires. The comparisons show that the heavily loaded tire has niqe dynamic characteristics. Comple damping and nonlinear ehavior were considered to eplain these characteristics. fleile ring tire model was sed to investigate these nonlinear characteristics. Comple damping was also eamined to eplain the eperimental oservations. This thesis contains some of the first reslts on viration characterization of heavily loaded vehicle tires. The incorporation of the reslts into a mlti-degree-of-freedom tire model for se with the Effective oad Profile Control scheme sed for vehicle draility simlation stdies was also investigated.

2 HEVILY LODED VEHICLE TIES: NLYSIS ND CHCTEIZTION By Nicholas Garcia Thesis smitted to the Faclty of the Gradate School of the University of Maryland, College Park, in partial flfillment of the reqirements for the degree of Master of Science 8 dvisory Committee: Professor Balakmar Balachandran, Chair Dr. Gregory. Schltz ssociate Professor Hgh Brck

3 Copyright y Nicholas J Garcia 8

4 Dedication To my parents, who have always een there for me. Thank yo for all of the spport and prodding. ii

5 cknowledgements Thanks to: Dr. Balakmar Balachandran, for his gidance, assistance and patience throghot this endeavor. His spervision and nderstanding made this research possile. Dr. Greg Schltz for getting me interested in vehicle dynamics and actal engineering. Withot the opportnity to work with the FSE team, I don t know where I wold e right now. evin efaver from the oadway Simlator grop at erdeen Test Center for his assistance and gidance with rnning the tests and data acqisition. Thanks to the rmy Materiel Systems nalysis ctivity for giving me the time and opportnity to prse a degree while working fll time. iii

6 Tale of Contents Dedication...ii cknowledgements... iii List of Tales... v Chapter : Introdction.... Literatre eview..... Tires: Eperimental Characterization..... Tires: Modeling Efforts Nonlinearities and Modal nalyses..... Effective oad Profile Control (EPC).... Ojectives and Scope Otline of Thesis... 6 Chapter : Eperimental Characterization Eperimental Setp ssmptions Made.... Sorces of Error.... nalysis Procedre....5 Eperimental eslts... 8 Chapter : Modeling Efforts and Comparison with Eperimental eslts.... Fleile ing Model.... Parameters and Nmerical Soltion Model eslts... 5 Chapter : Etension of EPC Framework Transfer Fnction for a Mlti-Degree-of-Freedom Tire Model Connections with Earlier Work Chapter 5: Conclding emarks Contritions to the Field Sggestions for Ftre Work... 8 ppendi and Programs... 8 Biliography iv

7 List of Tales. Data channels Natral freqencies for the first five modes fond throgh eperimental modal analysis Parameters sed in model stdies following Zegelaar (997) Fleile ing Model eslts Smmary... 5 v

8 List of Figres.a Eperimental set-p: the positions and directions of the force ecitations.... Eperimental set-p: the positions and orientations of the accelerometers.... Modes of a standing tire from eperimental modal analysis de to radial implse at point : Fz=N (Zegelaar, 997) Measred freqency response fnctions in the radial direction at point of Figre. de to a radial ecitation at point for two different ondary conditions Mode shapes de to radial ecitation Tire ring model The tire ring and the deformation and the coordinate system sed (Zegelaar, 997) Mass-spring-damper tire model sed in EPC.... Eperimental rrangement FF s for the free and standing tire de to a modal hammer implse FF of the radial acceleration to the inpt tale acceleration. The circles show the areas of potential natral freqencies. Sine sweep test for the standing tire (Fz=9 N) at 5 psi FF of radial acceleration to vertical tire force. Sine sweep test for the standing tire (Fz=9 N) at 5psi FF of vertical tire force to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi FF of spindle acceleration to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi Mode Shapes for the standing tire (Fz=9l). Sine sweep test. The two colors show the etremes of the mode shapes as it epands (one color) and contracts (the other) Comparison of passenger tire modes and heavily loaded vehicle tire modes vi

9 .9 Tangential Mode Shapes for the standing tire (Fz=9l). Sine sweep test FFs of tangential acceleration at different tire locations to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The different accelerometer locations arond the tire are shown y a different color in the figre. ll twelve sensor locations are present in the plot. The figre elow shows the Phase of the FF for only points arond the tire. The noisy channels were removed to give a clearer pictre..... FFs of lateral acceleration at different tire locations to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The different accelerometer locations arond the tire are shown y a different color in the figre. ll twelve sensor locations are present in the plot.... FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire ) epresentative plots at 9 o and 8 o First three modes predicted y the fleile ring model with no damping. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions a) st fleile mode, 97.9 Hz, fleile ring model with no damping ) nd fleile mode,. Hz, fleile ring model with no damping c) rd fleile mode,.8 Hz, fleile ring model with no damping FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with reglar damping, with magnitde e Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire vii

10 ) epresentative plots at 9 o and 8 o FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with reglar damping, with magnitde e Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire ) epresentative plots at 9 o and 8 o FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde e5j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire... 6 ) epresentative plots at 9 o and 8 o First three modes predicted y the fleile ring model with comple damping of magnitde d =e5j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions... 6 a) nd fleile mode, 8. Hz, fleile ring model with comple damping d =e5j ) rd fleile mode, 7.6 Hz, fleile ring model with comple damping d =e5j FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde 5e5j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is viii

11 a) FFs for one hndred and one eqally distrited points arond the tire... 6 ) epresentative plots at 9 o and 8 o First three modes predicted y the fleile ring model with comple damping of magnitde d =5e5j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions... 6 a) nd fleile mode, 8.5 Hz, fleile ring model with comple damping d =5e5j ) rd fleile mode, 9.8 Hz, fleile ring model with comple damping d =5e5j FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde e6j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire... 6 ) epresentative plots at 9 o and 8 o First three modes predicted y the fleile ring model with comple damping of magnitde d =e6j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions a) nd fleile mode,.6 Hz, fleile ring model with comple damping d =e6j ) rd fleile mode, 5.8 Hz, fleile ring model with comple damping d =e6j FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e6 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is i

12 a) FFs for one hndred and one eqally distrited points arond the tire ) epresentative plots at 9 o and 8 o FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e7 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire ) epresentative plots at 9 o and 8 o FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e8 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is a) FFs for one hndred and one eqally distrited points arond the tire ) epresentative plots at 9 o and 8 o Spring and damper tire model sed in EPC Spring and damper tire model sed in EPC in the longitdinal direction... 7

13 Chapter Introdction The stdy of vehicle dynamics starts with the interaction etween the tires of the vehicle and the road. large portion of all of the loads on the vehicle can e tracked ack to the loads on the tires. Since the tires are the only part of the vehicle that is in direct contact with the environment, the loads in the sspension and steering components can all e traced ack to the loads at the tire patch, where the fleile tire contacts the road. In order to predict vehicle dynamics loads it is important to have an accrate tire model. s the performance reqirements for vehicles increase a more precise nderstanding of vehicle dynamics is needed. This reqires a mch etter nderstanding of the dynamic properties of vehicle tires. Modal analysis of passenger vehicle tires started in the 96 s. major contritor to this stdy was Pacejka (99) who proposed a semi-empirical tire model known as the magic tire formla. These eperiments focsed on comparing the ehavior of ias ply with new radial tires. These early tests tilized a fied ale test set p with radial ecitations.. Literatre eview.. Tires: Eperimental Characterization Over the last decade, considerale amont of work has een done y Zegelaar (997) and Yam, Gan, and Zhang () to eamine the three dimensional mode shapes

14 of passenger tires. These eperiments se a free tire set p with the tire sspended from aove with radial and tangential ecitations. Zegelaar (997) eamined the in-plane virations of sch a tire in a free condition and a loaded (standing) condition and compared the eperimental reslts with analytical reslts derived from the fleile ring modal proposed y Gong (99). The loaded condition consisted of a vertical force, F z, of N. By comparison, the heavily loaded tire analyzed in this thesis is sjected to a static loading force of 9 N. This is over two times as mch as the static load a passenger tire is sjected to in Zegelaar and Yam et al. s analyses. Zegelaar s analyses show modes starting arond Hz with the first fleile mode of the free tire to e at 5.9 Hz, see Figre.. In Figres. to., the eperimental arrangement and reslts otained y Zegelaar are shown. Zegelaar (997) performed eperimental modal analysis y placing tri-aial accelerometers arond the tire tread and hitting the tire in varios places with a modal hammer. The inpt force from the hammer is recorded along with the otpts of the accelerometers in order to determine the freqency response fnction etween the inpt and an otpt force. In Figre.a the force ecitations sed are shown and in Figre. the positions and orientations of the accelerometers are shown. The inpt radial force is applied at point, as shown in figre.a. This eperiment is not trly indicative of the response of the tire, since the inpt force is in general applied at the contact patch; that is point. The eperimental modal analysis performed in this thesis will make se of an inpt force at the contact patch of the tire. The eperimental set-p sed y Zegelaar and others is similar to the one that is sed in this thesis work, ecept that a harmonic ecitation prodced y a shaker tale is sed instead of a modal hammer

15 ecitation. Modal hammer strikes cold not impart enogh force to sfficiently ecite the mch stiffer heavily loaded tire stdied in this thesis. Yam, Gan, and Zhang () sed a similar test set p and analyzed the fll three-dimensional motion of the tire to get in-plane and ot-of-plane virations of the tire. Their reslts showed the first fleile mode occrs arond Hz, which agrees with Zegelaar s findings. In the analysis y Yam et al only the free tire modes were eamined. In Figre. the modes fond in Yam et al s analyses in all three dimensions for a radial ecitation are shown. In this thesis the athor eamines the three dimensional virations of a heavily loaded military grade tire de to radial ecitations. The first fleile mode is seen at 5 Hz which is significantly lower than those fond for a standard passenger tire. The natral freqencies of these modes are important since the vehicle can ecite the lower freqencies dring normal operation. Ot-of-plane motion was also oserved in response to a radial in-plane ecitation. The modes for the passenger tire analyzed y Zegelaar are at significantly higher freqencies than those of the mch stiffer and heavier tire stdies in this work. a) ) Fz Figre.: Eperimental set-p: a) positions and directions of the force ecitations and ) positions and orientations of the accelerometers (Zegelaar, 997).

16 Tire tread Spindle Tire rim adial implse inpt location and direction Static load=n Figre.: Modes of a standing tire from eperimental modal analysis de to radial implse at point : Fz=N (Zegelaar, 997).

17 Figre. (contined): Modes of a standing tire from eperimental modal analysis de to radial implse at point : Fz=N (Zegelaar, 997). 5

18 Figre.: Measred freqency response fnctions in the radial direction at point of Figre. de to a radial ecitation at point for two different ondary conditions (Zegelaar, 997). The mode nmering convention sed in Zegelaar s work is shown in Figre.. The peaks in the freqency response fnction (FF) correspond to the natral freqencies of the modes of a particlar system. The modal peaks for the free tire are easy to see, with evenly spaced distinct peaks. These modes are nmered in ascending order. The modal peaks from the free tire (nloaded) are compared to the modes from the standing (loaded) tire. If a modal freqency for the standing tire lines p with that of the free tire, the standing tire mode is given the corresponding integer vale sed for the free tire mode. If a peak from the standing tire freqency response fnction does not line p with a peak in the free tire freqency response fnction, then the standing tire is classified y a real nmer with one half. 6

19 Figre.: Mode shapes de to radial ecitation (Yam et al., ). 7

20 .. Tires: Modeling Efforts The fleile ring model ses a circlar eam that is spported y an elastic spring fondation to descrie the motion of tread of the tire. The eam can end and deform along its ais jst as the tread of the tire wold deform. So this model differs from the classic string model, in that the ring can deform (Pacjeka, 5). The eqations of motion of the fleile ring model are ased on the PhD dissertation of Gong (99). The tire ring model, which is shown in Figre.5, is comprised of a circlar ring that is spported on an elastic fondation. The fleile ring model allows for displacement of the ring elements eyond the standard spindle displacement as it takes into accont the tire tread deformation. In Figre.6, the coordinate system sed to develop this model is shown. Figre.5: Tire ring model (Zegelaar, 997). 8

21 9 The initial position on a ring element is shown as point O, the position reached after the spindle displacement is shown as point, and the position reached after oth spindle displacement and the tire ring deformation is shown y the point B. In the doctoral work of Gong (99) the steps ndertaken are provided. These steps are ased on the strain displacement relations, strain energy, and virtal work for determining the eqations of motion. The reslting eqations have the form cv a a a v s q v t w t v z v c v w p w v F v w E v w EI cos sin Figre.6: Tire ring, deformation field, and coordinate system sed (Zegelaar, 997).

22 cw a a w s q w t v t w z w c v w p w v F v w E v w EI sin cos a v w a w v a a F d v c w c c c t m sin cos (.) az v w a w v a a F d v c w c z c c t z m cos sin ay v a v a ay M d v c c t I where the following notation has een sed. q cv, q cw : eternal distrited force on the ring F a, F az : eternal forces acting on the ring M ay : eternal torqe acting on the ring v, w : tangential and radial displacements a, z a : horizontal and vertical rim displacements : rotational speed a : small deviation of the anglar displacement of the rim de to EI : ending stiffness of ring E: Yong s modls of the ring material I: inertia moment of the cross-section of the ring p F s : pretension in the ring c v : tangential sidewall stiffness

23 c w : tangential sidewall stiffness : radis of the ring : width of the ring p : tire inflation pressre m a : mass of rim I a : moment of inertia of rim (see Figre.6 for the coordinate orientations) In this thesis it is assmed that the distrited forces on the ring, the rim displacements, the anglar displacement of the rim, and the rotational speed are all zero. In addition, the sstittion for the pretension in the ring is carried ot. p F s This leads to the following two eqations of motion: t v v c v w E v w EI v t w w c w w p v w E v w EI w (.7).. Nonlinearities and Modal nalyses There are principal assmptions that form a asis for modal analysis, which is strictly valid for linear systems. One of them is that the system invariants, inclding the natral freqencies, damping ratios, mode shapes, and freqency response fnctions are not affected y the level of inpt ecitation applied dring the test. second assmption

24 is that there is no nonlinear copling etween the modes, that is, the response of one mode does not affect the response of another mode in a nonlinear fashion. Both of these assmptions can reak down when nonlinearities are inherent in the system (e.g.,worden and Tomlinson, ). In nonlinear systems, varios phenomena sch as jmps, nonlinear resonances, and ifrcations can occr that can affect the reslts (Nayfeh and Balachandran, 995). There are a nmer of methods that have een sed for eamining the presence of nonlinearities in eperimental modal testing. Sine sweep and harmonic inpt tests can e particlarly sefl for detecting effects like nonlinear resonances. Eciting the system at one-half, one-third, twice, and three times the linear natral freqency can reveal nonlinear resonances that are common in nonlinear systems (Zavodney, 987). It is qite common for sinsoidal inpts at one freqency to ecite a resonance at a different freqency in a nonlinear system. This does not happen in a linear system, and a slow sine sweep test of a harmonic ecitation is sefl to detect sch an occrrence. Sperposition is only strictly valid for linear systems. The sperposition principle can e sed to detect nonlinearities in a system y oserving deviations from linear sperposition (Nayfeh and Balachandran, 995). Nyqist plots are also a way to detect nonlinearities in a system. Nyqist plot is a polar plot showing the gain and phase of a freqency response. For a linear system ecited close to resonance and ehiiting a response that contains only one mode, Nyqist plots are circlar. For a nonlinear system the Nyqist plots can ecome distorted into ellipses or other shapes (Zavodney, 987). Nonlinear resonances can e a prolem in eperimental modal analysis. The ecitation of one mode at a particlar freqency can lead to a response at another

25 freqency as well as participation of other modes. For a set of given damping and freqency vales, it is possile for a system to ndergo ifrcations in which a fied point of a dynamical system loses staility and the system eperiences a continos echange of energy from one mode to another. These nonlinear phenomena can occr at etremely low ecitation levels (Zavodney, 987)... Effective oad Profile Control (EPC) One important se for accrate tire models is in corrective signal response algorithms for eperimental vehicle dynamics simlations. Effective oad Profile Control (EPC) is a simlation control method developed y MTS Systems Corporation. EPC makes se of the vehicle response with a non vehicle specific control algorithm to determine changes in a vehicle s performance over time. The oadway Simlator grop of the U.S. rmy at the erdeen Testing Center, erdeen Proving Grond, Maryland is interested in this techniqe for carrying ot eperimental vehicle simlations. The principle ehind EPC is that y sing a standard tire model one can determine the effective road profile (EP) de to a random inpt signal. The EP is then sed as a feedack inpt to drive the simlation inpts. Normally, control algorithms reqire for a new vehicle road response to e taken for each vehicle for each test corse eing analyzed. With EPC, this is not necessary. With an accrate tire model, it is epected that the EP can e determined for any vehicle over any test corse. The variance in the EP is determined to find ot when the performance characteristics of a vehicle have changed. This cold occr de to a failre or wear-ot of a sspension component.

26 Figre.7: Mass-spring-damper tire model sed in EPC. The crrent tire models eing sed for EPC are simple single degree of freedom mass-spring-damper models in the vertical direction, see Figre.7. It may e possile to make the process more accrate y sing a three-dimensional tire model ased on the reslts of this thesis, where it is proposed to se oth in plane and ot of plane tire modes to determine the EP. The tire model is sed to predict a specific road profile ased pon a set of spindle dynamic response data. The control algorithm then changes the inpt forces to match an epected road profile. In this way, once one has a tire model in place for each vehicle the same effective road profile can e sed for different vehicle configrations which reslt in different force inpts t correspond to the same road profile for a test corse. The tire

27 model needs to enale oth accracy and efficiency. The control process necessitates that the prediction of the EP from the spindle dynamics with the tire model mst e comptationally fast enogh so that the control feedack can work in real time at high freqencies. nother se for accrate tire modeling efforts is in dynamics modeling efforts. Tire models are reqired for all vehicle dynamics models. These models are created y sing special software packages sch as DDS, VirtalLa, and DMS dynamics modeling software. One of the largest sorces of error in these models is the loads generated at the tire patch of the vehicles. Generic tire models are sed to determine vertical, lateral, and longitdinal forces de to the rolling contact of the tire and the grond. If these tire models are not accrate, the loads and sseqent response of the rest of the vehicle can e significantly affected casing large errors in the overall response of the vehicle. efined tire models are critical for improving the accracy of these models. The reslts from this stdy will hopeflly lead to a etter nderstanding of the tire dynamics and eventally to more accrate tire models for se with these software packages, in particlar, for heavily loaded vehicles.. Ojectives and Scope main ojective of this stdy is to determine the natral freqencies and mode shapes of a heavily loaded tire. These reslts are to eventally e sed to for the creation of an accrate three-dimensional tire model for se in dynamics modeling and eperimental inpt response control algorithms, that is, algorithms like EPC. To 5

28 achieve this ojective, an eperiment was designed and condcted to measre the response of the tire tread to varios inpt signals. second ojective of this stdy is to determine if any non-linear characteristics are present in the response of the heavily loaded tire system and carry ot an attempt to determine the forms and characterize them. Predictions made with the fleile ring model are compared with reslts otained from the eperimental modal analysis in an effort to nderstand the tire viratory response characteristics.. Otline of Thesis The rest of this thesis is organized as follows. In the second chapter, the eperimental modal analysis work is descried. The eperiment setp and procedre are discssed in detail. The reslts are shown, and the mode shape information is eamined. In the third chapter, the analysis is carried ot with the fleile ring model. The predicted modal response from the fleile ring model is compared with the reslts otained from the eperimental modal analyses. Nonlinear terms are added to the fleile ring model and the reslting dynamics is stdied to eplain some of the oservations made in the eperiments. In the forth chapter, the tire model for the Effective oad Profile Control algorithm is epanded into three dimensions. The feasiility of sing tire modes fond in the eperimental modal analysis and the tire model sed in the third chapter for the EPC process is discssed. In the final chapter, conclding remarks are given along with sggestions for ftre work. n appendi containing the programs sed in this work is also inclded 6

29 Chapter Eperimental Characterization. Eperimental Setp To determine the modes and mode shapes of the tire, eperiments were performed y sing a shaker tale at the erdeen Test Center in erdeen Proving Gronds. The tire was monted to the oter frame of the tale and a plate fied to the tale was sed to load the tire and sject it to varios inpts. The fitre was machined ot of steel and was clamped to the oter frame of the tale. plate was olted to the shaker tale y sing a separate machined fitre to apply the loading inpt on the tire. The olts holding the plate to the shaker tale were tightened y sing a hydralic hammer. The plate and fitre were careflly aligned after every installation. The tire was monted with a MTS Spinning Wheel Integrated Force Transdcer (SWIFT) to measre the forces and moments at the center of the tire. This transdcer is attached etween the h and wheel, and it is sed to measre the forces and moments eing inpt into the h and spindle y the tire. ccelerometers were monted on the loading plate, the spindle of the tire, and at varios points on the circmference of the tire. The accelerometers on the tire tread were placed at 5 o, 75 o, 9 o, o, 5 o, 8 o, o, o, 7 o, 85 o, 5 o, and o, as shown in Figre.. s a reference, the loading plate was placed at o. The accelerometer locks (inclding the accelerometers) weighed only a few onces, compared to the tire, which weighed hndreds of ponds. 7

30 The depth of the tire tread and the size of the accelerometer locks hindered attempts to attach the locks to the tire tread in a conventional fashion. The accelerometers were monted to the tire y sing modeling clay that was pressed into the tread of the tire and molded arond the accelerometer locks. This monting procedre took a few tests to get it right, t in the final tests, it was possile to se the clay was ale to attach the accelerometers to the tire tread with only a slight change in the orientation of the locks dring a test. The prolems eperienced while attaching the accelerometers is frther discssed in Section.. Both SoMat and TC s dvanced On-Board Compting System (DOCS) data acqisition systems were sed. The radial tire accelerations, spindle accelerations, and the SWIFT vertical force were recorded on the SoMat while the tale accelerations and three SWIFT forces were acqired with the DOCS system. In Tale., the list of data channels collected dring the eperiments are listed. The inpt signals sed dring the eperiment were a -g implse comprised of a ms saw tooth, a linear sine sweep from 5 Hz to Hz, and a random ecitation inpt. The eperiments were repeated for tire pressres of psi, psi, and 5psi. The static load was kept arond 9 N. The static load is one of the ncontrolled variales in the test that cased some minor inconsistencies throghot the testing. The airag sed to apply the static load for the shaker tale was nreliale and wold lose pressre over the corse of a test. 8

31 Static Load, Fz=9 N Inpt direction Shaker Tale Impact Plate Heavily Loaded Tire ccelerometer Figre.: Eperimental arrangement. SWIFT Transdcer Tale.: Data channels. SOMT DOCS adial cceleration of Pt, SWITFT Force, Vert adial cceleration of Pt, y SWITFT Force, Lat adial cceleration of Pt, z SWITFT Force, Long SWIFT Force, Vert Tale cceleration, Lat 5 adial cceleration of Pt, 5 Tale cceleration, Long 6 adial cceleration of Pt, y 7 adial cceleration of Pt, z 8 adial cceleration of Pt, 9 adial cceleration of Pt, y adial cceleration of Pt, z adial cceleration of Pt, adial cceleration of Pt, y adial cceleration of Pt, z Spindle cceleration, Vertical 5 Spindle cceleration, Lateral 6 Spindle cceleration, Longitdinal 9

32 . ssmptions Made There are some assmptions nderlying the analysis. The first is that the modes are decopled or well separated in terms of their freqency responses. This means that we can ecite each mode separately. It is speclated that this may not actally e the case, and that the analysis will prove or disprove this assmption. The previos work done in the sject of tire dynamics has sed this assmption, and for standard passenger tires this assmption has held p throgh the analysis (Zegelaar, 997). Well separated modes means that the peaks of the freqency response fnctions where the natral freqencies of the modes are located are not too close to each other that the identification of each separate mode ecomes difficlt. If the modes are well separated and decopled they can e solved for separately as discssed in Section.. nother assmption that is made is that the spindle dynamics is negligile in comparison to the radial tire dynamics at the freqencies of interest. Spindle accelerations were taken and the analysis of the reslts is shown in Figre.6. In Figres. to.5, different freqency response fnctions are shown. The spindle dynamics is predominantly relegated to freqencies nder Hz. The modes of interest all have natral freqencies aove Hz, with only a few rigid ody modes even eing close to the spindle freqencies. The amplitde of the FF of the spindle dynamics is also mch lower than the amplitde of the radial tire dynamics for freqencies aove Hz. Ths, it is jstifiale to assme that the spindle dynamics can e neglected in the analysis.

33 . Sorces of Error s mentioned previosly the static load on the tire was another sorce of error in the eperiment. The static load was not constant as it dissipated throghot each test. The load was created y a large airag within the shaker tale apparats. The airag wold lose pressre over every test and the static load wold lose close to one thosand Newtons over the corse of a test. The static load was reconfigred etween each test t it was still a large sorce of inconsistency in the eperiment. To offset the drift in the airag load, the initial static load was increased to arond, N so that the average load wold stay close to 9, N. The triaial accelerometer locks were attached to the tire y sing modeling clay. visal inspection was carried ot efore and after every test rn t the alignment of each lock did change dring and etween each test. The locks cold start to sag as the molding clay deformed dring the test. s the clay deformed dring the test the orientation of the accelerometer locks changed slightly. The changes in the orientations of the locks cased the measred accelerations to e skewed slightly as the directions of the accelerometer aes were not eactly the same for each position arond the circmference of the tire. This variance was not recorded and was visally noticeale after the rest rns, and this is elieved to e the leading case of the variance in the peaks of the FF s associated with a particlar accelerometer location. The changes in the orientations of the accelerometers were elow o over the corse of a test. Dring some tests, one or more accelerometer completed detached from the tire as the clay fitre failed. These tests were repeated, althogh at least one test had a failed accelerometer lock that was not detected in etween tests.

34 The natral freqencies of the system correspond to the peaks in the freqency response fnctions (FF s) etween the radial accelerations measred arond the tire and the tale ecitation acceleration. Dring the eperiments, the data for the inpt acceleration of the tale and the otpt accelerations at different locations on the tire were acqired y sing two different data acqisition systems that had slightly different sampling rates. Withot taking this difference into accont the phase of the FF, etween the tale ecitation, measred from the DOCS, and the tire ecitation, H, measred from the SoMat, grows toward infinity as the freqency increases dring the sine sweep test. H P T SoMat docs r T (.) P docs docs T In eqation (.), r SoMat is the radial acceleration of the tire, docs T is the tale acceleration, P XX is the power spectral density of X, and P YX is the comined power spectral density of X and Y. In order to correct the epression given y (.), the athor mltiplied the FF etween the wheel force and the radial tire ecitation from the SoMat system with the FF etween the tale acceleration and the wheel force from the DOCS system. This reslted in H P SoMatF SoMat P docs docs r F T new (.) PF SoMatF SoMat P T docst docs The vertical tire force was recorded with oth systems. The reslting FF, which is given y eqation., is the FF etween the tale acceleration and a radial tire

35 acceleration. This helped correct the phase prolems that occrred dring the original analysis.

36 . nalysis Procedre To determine the mode shapes of the system, it is assmed that the damping in the system is low and that the modes do not inflence each other (there is no cross copling etween the modes) (Ewins, ). Ths the total response of the system is the sm of all of the modes of the system and their respective amplitdes. This can e written as U, t t (.) i i, In eqation (.), U,t is the response of different points along the circmference of the tire, i,t is the modal response of the system, and i is the contrition of the ith mode to the response. The peaks in the FF s are fitted to individal mass-spring-damper (single degree-of-freedom) systems with natral freqency n and damping factor n. The eqation of motion for a single degree-of-freedom spring-mass-damper system is m c k f. Defining the natral freqency and damping factor as k n (.) m n c km The eqation of motion can e written as g (.5) n n n

37 fter performing a Laplace transform and setting the initial conditions to zero one otains Setting s X s s sx X h (.6) n n j, leads to n X j X X h j (.7) n n n X h j j j Based on this, the contrition of each mode, is given y n n n G n j, where.95 n. 5 n (.8) j n n n The analysis is performed over a small and of freqencies,, arond the natral freqency, n of the considered mode. The total response of the system, H i (j) the FF of the i th point, is the sm of the prodcts of mode shape coefficients, a i,n, and the modal ecitations, G n (j), that is H i nm j a G j (.9) n i, n n where n m is the nmer of modes to e determined. Solving for the mode shape, a i,n, one otains a G G n T j H i j T j G j n i, n, m where n n n n,...,n (.), and G j is the comple conjgate of G j n n. This eqation help otain an average mode shape in the freqency and in qestion. To 5

38 determine the actal mode shape, one mst minimize the error etween the actal FFs, H i, and the calclated FFs, a G j n m n. i, n n, y varying the parameter n The error associated with the system is given y E j a G j, i, n H i i, n n n (.) For twelve freqency locations, the total error for a mode is given y E (.) n E i, n i and, considering 5 modes, the total error for each 5 E i, n n i n n is E. (.) The n vale that gives the lowest total error is determined for all vales of from Hz to Hz. The vales of n n for each n are compared with those assocatied with the peaks from the original FFs. Since it has een assmed that there is no cross copling etween the modes, one can work with each mode separately. The peaks from the FFs are not at eactly the same freqency for every location arond the tire, see Figre.. This is de to inconsistencies in the test set p, and possily, nonlinearities in the system. The vale for n that gives the lowest n near a peak from the FFs is taken to e a mode of the system. The circles in Figre. show the candidate regions for modes of the system. The analysis helps find the optimal vales for n and n to get the smallest error possile. In Tale., the natral freqencies for the modes and their respective damping vales determined from the sine sweep tests are shown. These vales 6

39 are the optimal vales that minimize the error etween the measred and calclated FFs as descried aove. With the natral freqencies determined, one can now look at the modes themselves. The modal amplitdes are comple vales that correspond to the magnitde and the lag of the oscillating point. These can e plotted as t a t i, n i, n sin i, n (.) where n t i, is the radial displacement of the i th point of the n th mode, a i, n is the modal amplitde, and i, n is the phase of a i, n. The sine sweep data was sed to determine the natral freqencies and mode shapes in this analysis. The first five mode shapes fond are shown in the net section. The first mode is a vertical rigid ody mode, while the second mode is a lateral rigid ody mode. 7

40 .5 Eperimental eslts.. FF of the Free and Standing Tire Free tire Standing Tire.8 FF Freqency w (Hz) (Hz) Figre.: FFs for the free and standing tire de to a modal hammer implse In Figre., a representative FF is shown for the free tire for an inpt radial force from a modal hammer. The overall response for the free tire tests was insfficient to perform effective, complete modal analyses, since the inpt force from the modal hammer was too low to ecite the entire tire. The free tire FF has distinct peaks at steady intervals. These peaks correspond to the modes of the system. The standing tire does not have the same shape and it is clear that the peaks for the standing tire FF do not match p with the peaks from the free tire FF. If one ses the mode nmering scheme from Zegelaar (997), the modes for the standing tire wold all e half modes. 8

41 Tale.: Natral freqencies and damping factors associated with the first five modes fond throgh Mode psi 5psi ς (5psi) Hz Hz.656 8Hz Hz. 5 Hz 6 Hz Hz 8 Hz Hz Hz FF, Tale ccel to adial ccel mplitde Phase Freqency (Hz) Figre.: FF of the radial acceleration to the inpt tale acceleration. The circles show the areas of potential natral freqencies. Sine sweep test for the standing tire (Fz=9 N) at 5 psi. Twelve different color plots are shown for the twelve accelerometers sed in the eperiments. 9

42 5 FF, Tire Force to adial ccel mplitde Phase Freqency (Hz) Figre.: FF of radial acceleration to vertical tire force. Sine sweep test for the standing tire (Fz=9 N) at 5psi. Twelve different color plots are shown for the twelve accelerometers sed in the eperiments.

43 mplitde FF, Tale ccel to Tire Force test test test Phase Freqency (Hz) Figre.5: FF of vertical tire force to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The three colors correspond to three different rns.

44 mplitde.8.6. FF, Tale ccel to Spindle ccel (Test) vert lat long Phase Freqency (Hz) Figre.6: FF of spindle acceleration to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The three different color plots correspond to three different accelerometer orientations. In Figres. to.6, the freqency-response fnctions for varios cominations of inpt and otpt measrements are shown. In Figre., the freqency-response fnctions that were otained in the eperimental model analyses are shown. Each FF corresponds to a particlar radial acceleration measrement on the tire and the tale acceleration inpt from the shaker tale. The sampling rate for all of the data channels was set to e Hz. The FFs were calclated sing the tfestimate command in MatLa. This command takes averages and is windowed sing a Hamming window (MatLa, 5).

45 Mode shapes of a heavily loaded tire. psi 5 psi Ecitation direction a) f) 7 7 Hz Hz ) 7 g) 7 8 Hz Hz c) h) 5 Hz 7 6 Hz 7

46 psi 5 psi d) i) Hz 8 Hz d) j) Hz Hz Figre.7: Mode Shapes for the standing tire (Fz=9N). Sine sweep test. The two colors show the etremes of the mode shapes as it epands (one color) and contracts (the other) In Figre.7, the rigid ody modes and the first three fleile modes for the heavily loaded tire that is ecited in the radial direction are shown for two different tire pressres, psi and 5 psi. The arrow shows the direction of the inpt harmonic force corresponding with the vertical orientation of the tire. The first mode shown at Hz is a vertical rigid ody mode while the net at 8 Hz is a horizontal rigid ody mode. The

47 horizontal mode is nepected given that the inpt force is limited to the vertical direction. The net three modes correspond to the first three fleile modes of the tire. The first mode, at 5 Hz, is fairly hard to make ot t it corresponds to a mode (9. Hz) from Zegelaar s work, shown for comparison in Figre.8. The nd mode, 7 Hz, is mch easier to make ot and this corresponds to a mode (5.7 Hz) of Zegelaar (997). The third mode at 9 Hz corresponds to a mode (56.8 Hz) of Zegelaar (997). The modes for the passenger tire are at freqencies nearly dole those seen for the heavily loaded tire. The passenger tire analyses do not pick p the rigid ody modes that this thesis analyses did for the heavily loaded tire. 5

48 The modes for the passenger tire analyzed y Zegelaar (997) are at significantly higher freqencies than those oserved for the mch stiffer and heavier tire considered in this thesis. The modes shown elow are the reslts otained y Zegelaar for a standing passenger tire. Passenger Tire, Fz=N Heavily Loaded Tire, Fz=9N n, f 9. Hz n, f 5Hz n, f 5. 7Hz n, f 7Hz Figre.8: Comparison of passenger tire modes and heavily loaded vehicle tire modes 6

49 Passenger Tire, Fz=N Heavily Loaded Tire, Fz=9N n, f 56. 8Hz n, f 9Hz Figre.8 (contined): Comparison of passenger tire modes and heavily loaded vehicle tire modes 7

50 In Figre.9, the tangential mode shapes are shown. The tangential modes are harder to see than the radial modes. The tangential modes for the passenger car tire fond y Yam et al. (), shown ack in Figre., resemle slightly distorted versions of the radial modes. The modes fond for the heavily loaded vehicle tire follow this trend, althogh they are more distorted. There are no rigid ody modes present and the second modes shown, Figre.9 and h, are discernile leaf modes with elongation in one direction and then in another direction orthogonal to the first. Frther analysis into the mode shapes in the tangential and lateral directions is needed as all of the mode shapes are not clearly discernile. In Figre., the FFs etween the tangential accelerations at different locations arond the tire and the inpt tale accelerations are shown. The FFs for the tangential response contain mch more noise than those for the radial response. This is especially noticeale in the phase plots for the FFs. In Figre., the FFs etween the lateral accelerations at different points arond the tire and the inpt tale accelerations are shown. The freqencies of the first few fleile modes in the tangential and lateral directions start arond 5-6 Hz. This is similar to the freqencies at which the modes for the radial direction were oserved. In comparison to the tangential and lateral modes fond y Yam et al. that started arond Hz, as shown in Figre.. 8

51 Tangential (Longitdinal) Modes psi 5 psi a) g) ) 7 h) c) i) 7.5 Figre.9: Tangential Mode Shapes for the standing tire (Fz=9N). Sine sweep test

52 psi 5 psi d) 7 j) e) k) f) l) 98.7 Figre.9 (contined): Tangential Mode Shapes for the standing tire (Fz=9N). Sine sweep test.

53 FF, Tale ccel to Tangential ccel.8.6 mplitde Phase Freqency (Hz) Phase Freqency (Hz) Figre.: FFs of tangential acceleration at different tire locations to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The different accelerometer locations arond the tire are shown y a different color in the figre. ll twelve sensor locations are present in the plot. The figre elow shows the Phase of the FF for only points arond the tire. The noisy channels were removed to give a clearer pictre.

54 FF, Tale ccel to Lateral ccel.5 mplitde Phase Freqency (Hz) Figre.: FFs of lateral acceleration at different tire locations to tale acceleration. Sine sweep test for the standing tire (Fz=9 N) at 5psi. The different accelerometer locations arond the tire are shown y a different color in the figre. ll twelve sensor locations are present in the plot.

55 Chapter Modeling Efforts and Comparisons with Eperimental eslts In this chapter, a fleile ring model is sed to make predictions and these predictions are compared with the eperimental reslts presented in Chapter. Nonlinear terms are added to the fleile ring model and the reslting dynamics is stdied to eplain some of the oservations made in the eperiments. The comparisons made etween the predicted modal response of the fleile ring model and the eperimental reslts for the heavily loaded vehicle tire are sed to try to identify nonlinear characteristics in the heavily loaded tire dynamics. Fleile ing Model s discssed in Chapter, the fleile ring model eing sed in this analysis was proposed y Gong (99) and it takes the form v v v c v w E v w EI w w w c w w p v w E v w EI (.) where the parameters are as defined in Chapter.

56 For the modal analysis, the fleile ring model is sed along with an added damping term, which in some cases has a comple component. The reslts are otained y solving for the responses of the partial differential eqations descriing the fleile ring when it is sjected to a sine sweep inpt at the ottom edge of the tire. This represents the eperimental setp where the tire is loaded y a plate in the vertical direction. The otpt response is than analyzed with the inpt sine fnction jst as the eperiment data were analyzed, y sing a single degree-of-freedom system to fit a crve to every point arond the tire in order to determine the modal amplitdes. The partial differential eqation from the fleile ring model was solved y sing the MTLB fnction pdepe. In MTLB, the initial-ondary vale prolem for the paraolic PDE is solved in one dimension y sing an iterative solver of the strongform eqation (MTLB, 5). The pdepe solver converts the PDEs to ODEs y sing a second-order accrate spatial discretization method. The time integration is done with the fnction ode5s. fter discretization, the elliptical partial-differential eqations are converted to algeraic eqations. The solver is an iterative solver that ses nmerical time integration to determine the response of the system. The fnction pdepe solves PDEs of the form descried elow: m m c, t,, f, t,, s, t,, (.) t with the initial conditions

57 for t t and all ; that is, t, (.) For all t and either =a or =, the soltion satisfies the ondary conditions of the form: p, t, q, t f, t,, (.) The fnction pdepe solves PDEs involving first derivatives of time and p to second derivatives of space. The eqations of motion for the fleile ring model inclde second derivatives of time and forth derivatives of space. These terms mst e redced to first derivatives of time and, at most, second derivatives of space efore they can e sstitted into the pdepe fnction. In order to redce the order of the eqations, one mst introdce a cople of new variales. The vale of m in eqation (.) is zero in this analysis. For this analysis, five variales will e sed as follows: v (.5 a) v (.5 ) t w (.5 c) w (.5 d) t 5

58 6 w v 5 (.5 e) The eqations of motion (eqation.) in terms of the variales transform to the following five eqations, which are first order in time and at most second order in space in terms of the respective derivatives. t (.6 a) 5 t c E EI v (.6 ) t (.6 c) 5 t c p E EI w (.6 d) 5 (.6 e) The coefficients, initial conditions, and ondary conditions to e sed in eqation (.) for pdepe are as follows: p p c (.7)

59 7 5 p EI E f (.8) 5 5 E c p E c E EI s w v (.9) The ondary conditions corresponding to the eperiments of Chapter read as follows: 55 5 v v (.a) 55 5 v v (.)..* sin 55 5 d t t w w (.c) t d t d t t w w.* cos 55 5 (.d) 55 5 w w (.e) The initial conditions are written as

60 5, 55,. w (.a) w 5, w 55,.* d w (.) For all other v, v, w, w, (.c) For the MTLB code, the initial and ondary conditions are constrcted as shown elow. Initial conditions:, t 55, t 5..* d (.a), t, t (.) Bondary conditions: p 5 t, * sin t t d 5.* cos t t d t d. t (.a) 8

61 9.* cos 55..* sin , 55 t d t d t t d t t t p (.), 55, 5 t q t q (.c) For 5.*cos 5..*sin p EI E t d t d t t d t t (.d) For 55.*cos 55..*sin p EI E t d t d t t d t t (.e) The points arond the circmference of the tire are discretized into a set of one hndred and one eqally distrited points arond the tire starting at 5 from the ottom of the tire and ending at 55 from the ottom of the tire. The gap in the soltion at the ottom of

62 the tire mimics the contact patch where the tire is loaded y the grond (or test fitre). The MTLB file sed to perform these analyses and the coefficients sed are provided in the ppendi. The initial and ondary conditions presented aove descrie a fied sine sweep inpt at the ottom of the tire that takes the form tt d..sin (.) The sine sweep has amplitde of. meters, and this sweep has an initial loaded offset of. meters. The variales and d are varied throghot the test rns in order to ecite the desired range of freqencies. n attempt was also made to analyze the fleile ring model with added damping and nonlinear stiffness terms. Damping terms took the form f w (.5) d where d is the damping coefficient, which cold take real or imaginary vales. Nonlinear cic stiffness terms took the form f n w (.6) where n is the stiffness coefficient and took strictly real vales. These additional terms were added to the Matla fnction in the s coefficient in eqation (.9). The new s coefficient reads as 5

63 5 5 5 E c p E c E EI s w v (.7a) with the damping terms, and as 5 5 E c p E c E EI s w v (.7) with the cic stiffness terms.

64 . Parameters and Nmerical Soltion Tale.: Parameters sed in model stdies following Zegelaar (997) In Tale., the parameters sed for these analyses are shown. These are the same as those sed y Zegelaar (997) for a passenger vehicle tire. The athor was nale to determine accrate vales for many parameters for the heavily loaded tire. Since the parameter vales do not correspond to the heavily loaded tire of this analysis, 5

65 the goal is to carry ot a qalitative comparison with the eperimental reslts showed earlier. The freqencies of the modes are not epected to match p, t the mode shapes and the general profile of the freqency response fnctions may. The analysis was attempted y sing the fleile ring model with only the radial direction terms. These attempts showed reslts that were not reasonale for the system in qestion. This analysis showed that the fleile ring model is not valid for only one variale when the tangential displacement is neglected. If the errors that accmlated dring the eperiment had een handled etter, the spread of the FFs cold have come ot looking more like those of the analytical model. It is easy to pick ot the peaks of the FFs for the fleile ring model since there are no errors stemming from the test setp and the orientations of all measred points arond the tire are identical. The sharp peaks in the fleile ring model also sggest that the actal tire system has a significantly larger amont of damping. In most modal eperiments, the phase vales at the peaks of the FFs and the otpt response tend to e arond 9. The fact that the phase of the FFs for the eperimental data are spread over the entire phase and and was not arond 9 degrees sggest that there may e comple damping in the system that cold spread the phase response (Ewins, ).. Model eslts In this section, the reslts otained from the MTLB PDE solver are shown for the fleile ring model when it is ecited y a harmonic ecitation at the ase of the tire. 5

66 The modal response is otained for the asic fleile ring model as well as for the fleile ring model with added real damping, comple damping, and cic stiffness terms. The FFs etween the radial displacements at mltiple locations arond the tire and the inpt displacements at the ase of the tire are otained and presented. In Figre., the modal response for the fleile ring model with no damping is shown. The fleile modes shapes for the fleile ring model with no damping are shown in Figre.. In Figres.,., and.5 the modal response FFs for the fleile ring model with added comple damping, comined damping, and cic stiffness terms respectively are shown. In Tale., a smmary of the systems solved is shown. The natre and amplitde of the nonlinear terms, the freqency range solved, and the freqencies of the first three modes are presented. The fleile ring model system with reglar damping of magnitde.e+ Ns/m is over damped as seen in Figre.. The fleile ring model systems with comple damping were all missing the first mode from the normal fleile ring model with no damping that corresponds to the rigid mode of the tire. Tale.: Fleile ing Model eslts Smmary Type mplitde, Freqency range solved Freqency of the st mode Freqency of the nd mode Freqency of the rd mode note Normal Damping.E+ Ns/m Damping.E+ Ns/m overdamped Cic Stiffness.E+6 N/m No change Cic Stiffness.E+7 N/m No change Cic Stiffness.E+8 N/m Damping.E+5j Ns/m no rigid mode Damping.E+6j Ns/m no rigid mode Damping 5.E+5j Ns/m no rigid mode 5

67 Fleile ing Model FF mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde Fleile ing Model FF Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 55

68 a) st fleile mode, 97.9 Hz, fleile ring model with no damping ) nd fleile mode,. Hz, fleile ring model with no damping 56

69 c) rd fleile mode,.8 Hz, fleile ring model with no damping Figre.: First three modes predicted y the fleile ring model with no damping. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions. 57

70 Fleile ing Model FF, Damping=e mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde Fleile ing Model FF, Damping=e Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with reglar damping, with magnitde e Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 58

71 Fleile ing Model FF, Damping=e.8 mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde.... Fleile ing Model FF, Damping=e Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with reglar damping, with magnitde e Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 59

72 5 Fleile ing Model FF, Damping=e5j mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde 5 5 Fleile ing Model FF, Damping=e5j Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.5: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde e5j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 6

73 a) nd fleile mode, 8. Hz, fleile ring model with comple damping d =e5j ) rd fleile mode, 7.6 Hz, fleile ring model with comple damping d =e5j Figre.6: First three modes predicted y the fleile ring model with comple damping of magnitde d =e5j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions. 6

74 6 Fleile ing Model FF, Damping=5e5j mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde 6 Fleile ing Model FF, Damping=5e5j Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.7: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde 5e5j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 6

75 a) nd fleile mode, 8.5 Hz, fleile ring model with comple damping d =5e5j ) rd fleile mode, 9.8 Hz, fleile ring model with comple damping d =5e5j Figre.8: First three modes predicted y the fleile ring model with comple damping of magnitde d =5e5j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions. 6

76 Fleile ing Model FF, Damping=e6j mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde Fleile ing Model FF, Damping=e6j Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.9: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with comple damping, with magnitde e6j Ns/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 6

77 a) nd fleile mode,.6 Hz, fleile ring model with comple damping d =e6j ) rd fleile mode, 5.8 Hz, fleile ring model with comple damping d =e6j Figre.: First three modes predicted y the fleile ring model with comple damping of magnitde d =e6j. The dashed line in each figre corresponds to the nominal position, and the lines in le and green correspond to the etremes of the mode shape motions. 65

78 Fleile ing Model FF, Cic Stiffness=e6 mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde Fleile ing Model FF, Cic Stiffness=e Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e6 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 66

79 Fleile ing Model FF, Cic Stiffness=e7 mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde Fleile ing Model FF, Cic Stiffness=e Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e7 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 67

80 5 Fleile ing Model FF, Cic Stiffness=e8 mplitde Phase Freqency a) FFs for one hndred and one eqally distrited points arond the tire. mplitde 5 Fleile ing Model FF, Cic Stiffness=e Phase Freqency ) epresentative plots at 9 o and 8 o. Figre.: FFs of radial displacement at different locations on the tire to inpt displacement. Predictions of the fleile ring model with cic stiffness, with magnitde e8 N/m, are shown. The different locations arond the tire where the fleile ring model predictions were made are shown y a different color. The red line with amplitde of one for all freqencies correspond to the points at the ase of the tire where the inpt was applied, hence, the corresponding FF magnitde is. 68

81 The natral freqencies for the loaded original fleile ring model from this analysis matched the reslts from Zegelaar s work for the passenger tire, in which natral freqencies of 98. Hz,.8 Hz, and.9 Hz were oserved for the first three modes (Zegelaar, 997). The natral freqencies for the fleile ring model with added nonlinear terms did not change significantly nless there was comple damping present in the system. The freqency response fnctions for the fleile ring model with no damping, conventional damping, and nonlinear stiffness terms all have fairly sharp peaks and they do not have the slight peak variations from one point on the tire to the net that the reslts of the eperimental modal analyses for the heavily loaded tire show, see Figres.,.,.,., and.. The addition of these terms does not affect the overall shapes of the freqency response fnctions or the mode shapes. The modes appear at consistent freqency intervals from one another. The cic stiffness terms did not significantly affect the natral freqencies of the modes ntil the magnitde reachede 8N / m. When a larger comple damping term is added to the fleile ring model, a large change in the freqency response fnction is seen. In Figre.8, there is a lot of noise in the response of the system de to the sine sweep inpt. The mode shapes shold e smooth crves and the noise can e attrited to inaccracies in the nmerical soltion provided y the pdepe fnction in MTLB. In Figre.7 and Figre.9, the amplitde and the phase of the freqency-response fnctions are shown for the fleile ring model with a large comple damping term. The modal freqencies are no longer at consistent freqency intervals, and the peaks of the freqency response fnction for all of the points arond the circmference of the tire do not all match p arond the natral 69

82 freqencies as was seen in the previos cases. This phenomenon is very similar to what was oserved in the eperimental modal analyses. The variance was attrited to errors accmlated dring the eperimental setp t it cold also e cased y a large comple damping inherent in the system (Ewins, ). The present analysis is not conclsive aot the inflence of nonlinearities. Frther stdies are needed to nderstand them. 7

83 Chapter Etension of EPC Framework. Transfer Fnction for a Mlti-Degree-of-Freedom Tire Model The crrent EPC tire model ses a single degree-of-freedom model to represent the tire system, as shown in Figre.. The different parameters shown in this figre are as follows: M : M : mass of the sprng mass, M effective tire mass in the vertical direction, M Mtm sp : vertical displacement of the spindle mass : : vertical displacement of the tire mass vertical displacement of the road The goal of the tire model is to determine in terms of the spindle displacement, sp, and the spindle force, F sp. The relationships etween the road displacement and the spindle displacement and forces are sed in the EPC process to predict the road profile for a measred set of spindle parameters. The eqations of motion for the two masses are given y 7

84 C F t M sp sp sp sp (.) M C C (.) sp sp fter carrying ot Laplace transforms of eqations (.) and (.) and setting the initial conditions to zero, the reslt is X sp s M sc F s X sc sp (.) M sc X X sc X sc s (.) Solving for X from (.) leads to s M sc F s X sp sp X (.5) sc On sstitting (.5) into (.), the reslt is sp X X s M sc F s sp sp s M sc X (.6) sp sc This is the formla sed for prediction of the Effective oad Profile in the EPC scheme. sp y sp C y C y Figre.: Spring and damper tire model sed in EPC. 7

85 When an additional degree of freedom is inclded in the longitdinal directions, the system gets a little more comple. The pper portion of the tire model etween the spindle and the tire mass is not directly connected y an aial spring-damper system in the longitdinal direction. It is instead connected y a torsional spring aot the mass M, as shown in Figre.. The location of the tire mass stays the same for the vertical and longitdinal directions; it is etween the spindle and the road contact point. This makes things a little more complicated since the spring and damper vales are not the same etween the spindle and the tire mass and the tire mass and road, in the longitdinal direction. The parameters sed for the longitdinal direction areas follows: M : I : effective tire mass in the longitdinal (y) direction rotational inertia of the sprng mass, M y sp : longitdinal displacement of the spindle y : y : longitdinal displacement of the tire mass longitdinal displacement of the road 7

86 y sp T Î θ θθ, C θθ y M yy, C yy Figre.: Spring and damper tire model for se in EPC in the longitdinal direction. y The eqations of motion for the longitdinal direction are ˆ Cˆ Tt Iˆ (.7) M sp sp sp y y C y y ˆ Cˆ (.8) y yy yy sp sp where sp can e approimated as y ysp sp (.9) and the coefficients ˆ and Ĉ have the appropriate dimensions. fter sstitting (.9) into (.7) and (.8) and altering the coefficients to asor, the respective eqations ecome I y y y y C y y T t sp sp sp (.) M y y C y y y y C y y y yy yy sp sp (.) 7