Parametric and sensitivity analysis of a vibratory automobile model

Transcription

1 Louisiana State University LSU Digita Commons LSU Master's Theses Graduate Shoo Parametri and sensitivity anaysis of a vibratory automobie mode Kania Nioe Vesse Louisiana State University and Agriutura and Mehania Coege, mixon@su.edu Foow this and additiona wors at: Part of the Mehania Engineering Commons Reommended Citation Vesse, Kania Nioe, "Parametri and sensitivity anaysis of a vibratory automobie mode" (). LSU Master's Theses This Thesis is brought to you for free and open aess by the Graduate Shoo at LSU Digita Commons. It has been aepted for inusion in LSU Master's Theses by an authorized graduate shoo editor of LSU Digita Commons. For more information, pease ontat gradetd@su.edu.

2 PARAMETRIC AND SENSITIVITY ANALYSIS OF A VIBRATORY AUTOMOBILE MODEL A Thesis Submitted to the Graduate Fauty of the Louisiana State University and Agriutura and Mehania Coege in partia fufiment of the requirements for the degree of Master of Siene in Mehania Engineering in The Department of Mehania Engineering by Kania Nioe Vesse B.S. in Mehania Engineering Southern University and A & M Coege, 999 May

3 DEDICATION To my husband Bradey D. Vesse, my parents Theophius and Nordia Mixon, Jr., my grandparents Robert and Mary Piere, my une Eddie Piere, my aunty Debra Morgan, and my sibings, LaKenya, Keesha, Damon, Tearai, and Eri ii

4 ACKNOWLEDGMENTS First and foremost I woud ie to give thans and a honor to God. Without Him I woud not be where I am today. I woud ie to anowedge my major professor Dr. Su-Seng Pang for his ontinued guidane and support whie maing sure I stay foused. Thans aso go to Dr. Yitsha M. Ram and Dr. Lynn Lamotte for serving on the graduate ommittee and evauating my thesis. Thans to Dr. Yitsha M. Ram for serving as o-advisor on my graduate ommittee. Furthermore, I woud ie to than Dr. Mihae A. Stubbefied for his advise and support during this advaned degree program. Thans to my oeagues Ashay Nareshraj Singh and Kumar Viram Singh for their inteetua input. Aso, thans to Savador Cupe-Beeha, Mr. Ed Martin and Jim Layton, Jr. for their assistane in onstruting the ab experiment. Thans to my friends Shani Aison, Caroine Stampey, Tamea Wiiamson, Kristian Johnson, Phaedra Be, Shari Jones, and Thereza Shanin for their ontinued support when times were tough. Thans to the Piere, Carter, Mixon and Wiiams famiy for their ontinued support. My gratitude goes to my une Eddie Piere for being ie a father to me and to my aunty Debra Morgan who has aways been there for me. Speia thans to my mom Nordia Mixon for her faith, guidane, ove, support, strength as singe parent, and for being one of my very best friends as we. Thans to my dad Theophius Mixon, Jr. for being stern when I needed disipine, but aso for being my friend when I needed one. Last but ertainy not east, my husband. Thans for a your ove, support, strength and faith when I needed it. iii

5 Thans to NSF/Interative Graduate Eduation Researh Training and NASA/LaSpae for their finania support to mae this possibe. iv

6 TABLE OF CONTENTS DEDICATION... ii ACKNOWLEDGMENTS... iii LIST OF TABLES... vii LIST OF FIGURES.... viii ABSTRACT... xi CHAPTER : INTRODUCTION.. Genera Introdution.... Vibration in Automotive Industry..... Basis of this Researh Organization and Content of the Thesis.5 CHAPTER : LITERATURE REVIEW 7. Introdution 7. Vehie System Modeing..7.3 Vehie Struture Modes...4 Tire Properties..3.5 Vehie Handing..6 CHAPTER 3: VIBRATION, PARAMETRIC AND SENSITIVITY ANALYSIS.8 3. Introdution Vibration Anaysis Free Vibration Fored Vibration (Harmoni Exitation) Parametri Anaysis Sensitivity Anaysis Equations of Sensitivity Sensitivity of Vibration.45 CHAPTER 4: EXPERIMENTAL VERIFICATION Introdution Experimenta Determination Equipment Setup Proedure Resuts Anaytia Determination Vibration Anaysis Parametri Anaysis..68 v

7 4.3.3 Sensitivity Anaysis CHAPTER 5: CONCLUSION AND RECOMMENDATIONS Conusion Reommendations for Future Researh...75 REFERENCES.76 APPENDIX: MATLAB PROGRAMS...78 VITA vi

8 LIST OF TABLES Tabe 3.: Poes of the system... Tabe 3.: Eigenvetors of the system...3 Tabe 3.3: Ranges for eah parameter for parametri anaysis..3 Tabe 3.4: Sensitivity anaysis for Tabe 3.5: Sensitivity anaysis for λ / J λ / J Tabe 4.: Experimenta and anaytia parametri anaysis resuts Tabe 4.: Experimenta and anaytia sensitivity anaysis resuts...73 vii

9 LIST OF FIGURES Figure.: Boune and pith modes of vibration 5 Figure 3.: -degree of freedom system 9 Figure 3.: Poes of the system for free vibration.. Figure 3.3: First mode shape (absoute, rea and imaginary part). 3 Figure 3.4: Seond mode shape (absoute, rea and imaginary part). 4 Figure 3.5: Third mode shape (absoute, rea and imaginary part)....4 Figure 3.6: Fourth mode shape (absoute, rea and imaginary part)..5 Figure 3.7: -degree of freedom system with externa fore.7 Figure 3.8: Magnitude in boune mode for fored vibration Figure 3.9: Phase funtion in boune mode for fored vibration..8 Figure 3.: Magnitude in pith mode for fored vibration..9 Figure 3.: Phase funtion in pith mode for fored vibration....9 Figure 3.: Parametri anaysis for J-magnitude in boune mode...3 Figure 3.3: Parametri anaysis for J-phase funtion in boune mode....3 Figure 3.4: Parametri anaysis for J-magnitude in pith mode..3 Figure 3.5: Parametri anaysis for J-phase funtion in pith mode Figure 3.6: Parametri anaysis for m-magnitude in boune mode.34 Figure 3.7: Parametri anaysis for m-phase funtion in boune mode...34 Figure 3.8: Parametri anaysis for m-magnitude in pith mode.35 Figure 3.9: Parametri anaysis for m-phase funtion in pith mode Figure 3.: Parametri anaysis for -magnitude in boune mode 36 Figure 3.: Parametri anaysis for -phase funtion in boune mode..37 viii

10 Figure 3.: Parametri anaysis for -magnitude in pith mode 38 Figure 3.3: Parametri anaysis for -phase funtion in pith mode Figure 3.4: Parametri anaysis for -magnitude in boune mode 39 Figure 3.5: Parametri anaysis for -phase funtion in boune mode Figure 3.6: Parametri anaysis for -magnitude in pith mode 4 Figure 3.7: Parametri anaysis for -phase funtion in pith mode....4 Figure 3.8: Rea part of sensitivity of the eigenvaues for both DOF( J ) Figure 3.9: Imaginary part of sensitivity of the eigenvaues for both DOF ( J )...48 Figure 3.3: Rea part of sensitivity of the eigenvaues for both DOF( m )...5 Figure 3.3: Imaginary part of sensitivity of the eigenvaues for both DOF ( m )..5 Figure 3.3: Rea part of sensitivity of the eigenvaues for both DOF( )... 5 Figure 3.33: Imaginary part of sensitivity of the eigenvaues for both DOF ( )..5 Figure 3.34: Rea part of sensitivity of the eigenvaues for both DOF( )...53 Figure 3.35: Imaginary part of sensitivity of the eigenvaues for both DOF ( )..53 Figure 4.: Compete experimenta setup..57 Figure 4.: -DOF mass-spring system.58 Figure 4.3: -DOF mass-spring system with added mass.59 Figure 4.4: -DOF mass-spring system with stiffness modifiation Figure 4.5: Cose-up of mass-spring system modifiation Figure 4.6: Resuts for mass-spring system... 6 Figure 4.7: Snap shot resuts for added mass, m =.63 g..63 Figure 4.8: Snap shot resuts for added mass, m =. g..64 ix

11 Figure 4.9: Snap shot resuts for mass-spring system, 3 =9.7 N/m...65 Figure 4.: Snap shot resuts for mass-spring system, 3 =33 N/m Figure 4.: Experiment-imaginary part of sensitivity for both DOF (mass)...7 Figure 4.: Experiment-imaginary part of sensitivity for both DOF ( 3 )....7 x

12 ABSTRACT Vibration response is an important aspet of any engineering probem. This thesis deas with deveoping a method to obtain the vibration, parametri and sensitivity of an automobie anayzed as a -DOF rigid body system with damping. The fous of the vibration anaysis is to obtain the poes, eigenvetors, and natura frequenies of the system. The parametri anaysis gave information on how the natura frequeny is affeted when one of the parameters is perturbed. Information about how muh the system is affeted by the hange in one parameter is what is obtained from the sensitivity anaysis. The theoretia soution ontained in this report uses the parameters of a Honda CR-V to give a genera idea of how an automobie shoud respond when anayzed as a two degree of freedom rigid body. An experiment was designed to impement the theory in pratia appiations. Moda testing was appied to extrat the natura frequenies in order to haraterize the system. From the experiment, the parametri anaysis produed resuts that resuted in error of %. This error is sma and it may seem the experiment is a good design, but that error aused an even greater amount of error in the sensitivity resuts. In that anaysis, the stiffness produed the best resuts, resuting in % error. However, the error for the sensitivity on the mass resuted in % for the first eigenvaue and % for the seond eigenvaue. This high error oud be due to the fat that the mass is a sensitive parameter. For the most part, fairy aurate resuts have been produed from the deveoped theory. Some wor sti needs to be done with the sensitivity to get better resuts. These resuts aso prove that this theory an serve as a vita too in deveoping pratia soutions to vibration ontro probems. xi

13 CHAPTER : INTRODUCTION. Genera Introdution Vibration, whih ours in most mahines, strutures, and mehania omponents, an be desirabe and undesirabe. Vibrations on the strings of a harp are desirabe beause it produes a beautifu sound. However, vibration in a vehie is very undesirabe beause it an ause disomfort to the passengers in the vehie. Vibration is undesirabe, not ony beause of the unpeasant motion, the noise and the dynami stresses, whih may ead to fatigue and faiure of the struture, but aso beause of the energy osses and the redution in performane whih aompany the vibrations. Vibration anaysis shoud be arried out as an inherent part of the design beause of the devastating effets, whih unwanted vibrations oud have on mahines and strutures. Modifiations an most easiy be made in an effort to eiminate vibration, when neessary or to redue it as muh as possibe. There are severa assifiations of vibration, free and fored, undamped and damped, and inear and noninear vibration. Free vibration is when a system, after an initia disturbane, is eft to vibrate on its own. No externa fore ats on the system. If a system is subjeted to an externa fore, the resuting vibration is nown as fored vibration. If the frequeny of the externa fore oinides with one of the natura frequenies of the system, a ondition nown as resonane ours, and the system undergoes dangerousy arge osiations. Undamped vibration is when no energy is ost or dissipated in frition or other resistane during osiations. On the other hand, when energy is ost in this way, it is nown as damped vibration. Linear vibration is when a the variabe fores are direty proportiona to the dispaement, or to the derivatives of

14 the dispaement, with respet to time. On the other hand, if any of the variabe fores are not direty proportiona to the dispaement, or to its derivatives with respet to time noninear vibration ours. In vibration, a number of simpifying assumptions have to be made to mode any rea system. For exampe, a distributed mass may be onsidered as a umped mass, or the effet of damping in the system may be negeted partiuary if ony resonant frequenies are sought, or a non-inear spring may be onsidered inear, or ertain eements may be negeted atogether if their effet is iey to be sma. Furthermore, the diretions of motion of the mass eements are usuay restrained to those of immediate interest to the anayst... Vibration in Automotive Industry An automobie is made up of many omponents. These omponents inude suspension, engine and its omponents, hassis, transmission, braing, et. and represent the many subsystems in a muti-degree of freedom anaysis. Yet, for many of the more eementary anayses appied to it, a omponents move together. The type of anaysis used depends on what omponent is being investigated. For exampe, under braing, the entire vehie sows down as a unit; thus it an be represented as one umped mass oated at its enter of gravity with appropriate mass and inertia properties. One mass is suffiient for aeeration, braing, and most turning anayses. For ride anaysis, it is often neessary to treat the whees as separate umped masses. In that ase, the umped mass representing the body is the sprung mass, and the whees are denoted as unsprung masses. The vehie is treated as a mass onentrated at its enter of gravity for singe mass representation. The point mass at the enter of gravity, with appropriate rotationa

15 moments of inertia, is dynamiay equivaent to the vehie itsef for a motions in whih it is reasonabe to assume the vehie to be rigid... Basis of this Researh Engineering students often understand the theories and prinipes onerning vibration by doing homewor assignments or performing ab experiments. However, it is not simpe to design an experiment to impement these simpe prinipes. A simpe thing as determining the spring fore of a spring or the natura frequeny of a simpe system an be a diffiut experiment to perform. With homewor assignments it is easy to use the nown equations and given information to sove the probem without muh thought. The same goes for ab experiments, the setup is given and the proedure is foowed without muh thought as to why ertain omponents are hosen. A mehania systems have natura frequenies. When anayzing these systems engineers sometimes appy rigid body dynamis. Rigid body dynamis has been used in many industries suh as automotive, agriutura and aerospae. In industry, muti-degree of freedom (DOF) systems is used to obtain measurements and simuate movement under ertain onditions. The purpose of a muti-dof system is to represent the subsystems of the body. When simpifying a mehania system in terms of a rigid body, it is neessary to determine the best method to do that. Tae for instane the researh of Chrstos in []. He was interested in vehie parameters required for vehie dynamis simuation and determined there are three ategories used to deveop the vehie dynamis simuation. The three ategories are ow DOF inear modes, noninear umped parameter modes, and muti-body dynami modes. 3

16 The sophistiation of simuation deveoped has ranged from -DOF inear modes used to study ow severity vehie behavior to ompex muti-body formuations used to investigate imit handing performane and vehie roover. The usage of the -DOF modes is ony a tehnique to extrat meaningfu parameters from measured data. Evauation methods using an unreasonabe number of parameters have no purpose for the deveopment of new ars or for omparison of simiar ars sine drivers preferene vary widey in various handing harateristis. Mimuro et a. showed this in their researh in []. He was interested in parameter evauation of vehie dynami performane. These engineers deveoped a Four Parameter Evauation Method. The atera transient response data was urve fitted with a -DOF mode and the parameters are extrated. The four parameters were expressed in a rhombus to hep reognize vehie performane intuitivey. The area of the rhombus denotes vehie handing potentia, and the distortion denotes handing tendeny. The ey point of this method was maing it possibe to see the parameters that ontradit eah other at a gane []. The use of rigid body dynamis in system anaysis an assist in any objetive the engineer is trying to obtain, it is ie a stepping-stone in system anaysis. The parameters of fous for this experiment are mass ( m ), stiffness ( ), damping ( ), and moment of inertia ( J ). Two types of vibration wi be investigated, free and fored. For both types of vibration, the theoretia resuts wi be ompared to the experimenta resuts. The automobie body or sprung mass has two vibrations that are of interest for this experiment, vertia (boune) and pith. Figure. shows these two modes of vibration. Boune is the transationa omponent of ride vibrations of the sprung mass in the diretion of the vehie z-axis. Pith is the anguar omponent of ride vibration of the 4

17 sprung mass about the vehie y-axis. Therefore, the automobie wi be investigated as a rigid body -DOF system with damping. In addition to the vibration anaysis, a parametri and sensitivity anaysis wi be performed as we. The parametri anaysis wi oo at how perturbations in one parameter affet the natura frequeny. The sensitivity anaysis wi oo at how sensitive a parameter is to hange in one or severa other parameters. Boune y Pith x z Figure.: Boune and pith modes of vibration. Organization and Content of the Thesis This thesis is divided into five setions, the struture and ontent of whih are desribed beow: Setion : Introdution (Chapter ) This setion is a genera introdution to the subjet of vibration, parametri and sensitivity anaysis, foowed by a desription of the organization and ontent of the thesis. 5

18 Setion : Literature Review (Chapter ) This setion is a iterature review to investigate the effet of vibration on the automobie. This setion reviews aboratory studies, whih have investigated the effet of various parameters on automotive vibration. This setion aso onsists of reviews that have investigated vibration on ertain omponents of the automobie, as we as onsidering the automobie with ertain degrees of freedom. Setion 3: Vibration, Parametri and Sensitivity Anaysis (Chapter 3) This setion desribes the theory for modeing free and fored vibration for a -DOF system with damping. Aso inuded are the ontributions of this experiment, whih have not been previousy investigated. The ontributions are the parametri and sensitivity anaysis for eah parameter. Setion 4: Experimenta Verifiation (Chapter 4) Chapter 4 disusses the measurement proess and the measurement instrument used for our experiment, as we as the experimenta setup and resuts of the experiment. Setion 5: Conusion and Reommendations (Chapter 5) This setion ontains a summary of the prinipe onusions reahed on the basis of the iterature review and subsequent experimenta wor. Inuded in this Chapter are reommendations for the diretion of future researh reating to vibration, parametri and sensitivity anaysis. 6

19 CHAPTER : LITERATURE REVIEW. Introdution The iterature reviews that foow are to support the use of rigid body dynami anaysis. Using a -DOF mode annot ony be used to see the frequenies of the system, but aso to oo at the effets of the suspension or the tires. Athough ony the boune and pith modes are being investigated, the automobie has other modes that an be investigated with a -DOF or a muti-dof system. The iterature on the tires is to show that using the stiffness,, is justifiabe in representing the tires. This hapter is divided into 4 setions: Vehie System Modeing, Vehie Struture Modes, Tire Properties and Vehie Handing.. Vehie System Modeing Hovarth was interested in vehie system modeing in order to provide a method of obtaining the dynami response of the fu vehie when the dynami harateristis of eah omponent are nown in [3]. A omparison was made between a Finite Eement (FE) Approah and Moda Modeing. In the FE Approah, the mass and stiffness properties of the eements are representative of the properties of the rea struture at the same oation. The Moda Modeing method was based upon the fat that a strutura omponent an be mathematiay represented by its moda harateristi as we as by distributed mass and stiffness. In other words, moda modes an be used as buiding bos and ouped to mathematia modes of the vehie s other subsystems to simuate the behavior of the ompete vehie system. To obtain the best resuts for moda 7

20 modeing, the vehie was divided into 4 subsystems. The moda modeing tehnique with 4-subsystem division was the best anaysis tehnique beause eah subsystem assured the simpest approah neessary to obtain the required resuts. Therefore, the moda modeing tehnique oud dramatiay redue the probem size and ost with no redution in auray when ompared to the FE approah. Davis further investigated moda modeing by fitting the mode to measure frequeny responses in [4]. The foundation of moda modeing is the foowing prinipe: Vibrationa response of any struture an be onsidered to be the sum of the responses of a set of ouped modes. The foowing 4 vehie struture modes were examined: boune, pith, and beaming modes. He onuded that the dynami behavior of any struture an be ompetey simuated by a moda mode. It is extremey important to predit how the vehie struture interats with its subsystems in order to obtain the tota vehie system. This is why moda modes have beome so usefu. Spehart deveoped a 4-DOF mathematia mode to anayze and predit the handing dynamis of a 4-whee vehie in [5]. There was no aowane for the defetion of any member; therefore, the vehie was treated as an assemby of rigid masses when deriving the equations of motion. This study determined that sma hanges in the suspension design oud mae a notieabe differene in handing. The greatest soures of error in prediting the dynami response were attributed to the unertainty in determining the exat vehie parameter and the performane of tires. Simi and Petronijevi foused on the ar body as an easti struture in [6]. The moda vetor, driving fore, frequeny and vibration harateristis of the easti automobie struture determine the frequeny response. They measured the vibrationa 8

21 harateristis of the automobie body under the harmoni exitation fores at 5 points. The ar body tested was freey supported on four springs. They reaized the input signas oud determine the atua vibration ondition of a partiuar area of a ar body. Possibe hepfu toos in the proess of designing and deveoping the ar body are information about the oation of neutra ines and the shape of vibration modes. Daberow and Kreuzer investigated the integration of Soid Modeing Computer Aided Design (CAD) systems within a dynami simuation environment in [7]. The modeing of a mehania system by means of a muti-body system is based on the omposition of rigid bodies, interonneted by joints, springs, dampers and atuators. The appied fores and torques on the rigid bodies are the fore eements, whih inude springs, dampers and atuators ating in disrete attahment points. Joints with different inemati properties have severa effets on a mehania system. They onstrain the motion of the bodies of the system, determine the DOF of the muti-body system, and resut in onstraint fores and torques. In order to investigate a system without onsidering the geometri mode, modeing method and the soid mode onstrution of the CAD system in use, ertain basi parameters are needed. The basi parameters needed for system dynamis investigations are mass, enter of gravity and moments of inertia. The objet-oriented asses and operations of this new approah are impemented in a system-independent muti-body modeing erne ibrary. Huang and Chao used a quarter-ar -DOF system to design and onstrut the onept of a four-whee independent suspension to simuate the ations of an ative vehie suspension system in [8]. A mode-free fuzzy ogi ontro agorithm is empoyed to design a ontroer for ahieving vibration isoation [8]. To satisfy the 9

22 expetations of ustomers the requirements of ride omfort and driving performane are major deveopment objetives of modern vehies. The suspension system is an important fator to the ride omfort and driving apabiity. The ative suspension onsists of a spring, damper and atuator between the unsprung mass and sprung mass, where the sprung mass is onsidered as a rigid body. The fuzzy ogi ontro agorithm suppressed the sprung mass vibration ampitude due to road variation. The vehie suspension served to suppress the vibration ampitude of the vehie body. This prevented the tire bouning from the road surfae owing to exessive tire deformation. In order to obtain ride omfort the spring and damping fores ating on the body must be ompensated for. The ontro was used to ompensate for these fores ating on the body. The experimenta resuts showed that this ontro strategy has suessfuy redued the osiation ampitude of the vehie body and improved the ride omfort. Varadi et a. foused on a three-dimensiona, omputationay inexpensive vehiuar mode in [9]. In the past the vehie s sprung mass is modeed as a rigid body, but in this researh it is modeed as a nonineary deformabe body. The theory of a Cosserat point was introdued to mode three-dimensiona deformabe bodies with arge deformations. When the theory of a Cosserat point is appied to vehie dynamis, it is assumed that the sprung mass is deformabe. When appied to the sprung mass of a vehie, it resuts in a mathematia mode with reativey few degrees of freedom whose deformation an be used to expain the vents arising during a oision [9]. The methods used for rigid body modes an be appied to the modeing of the unsprung mass of the vehie and the tire/road interation.

23 Zhang et a. investigated how two beam eements suspension/suspension modes with rigid vehie body representation and finite eement tires were studied under proving ground onditions in []. The ony differene between the two modes was that one used fexibe beam eements and the other used rigid beams. One of the obvious advantages of onduting rigid body dynamis anaysis is timing. The main purpose was to show how suspension/suspension omponent fexibiity and the onsequene of sma deformations in these omponents oud affet the noninear dynami anaysis resuts. A the ompiane/stiffness study resuts showed that there were differenes between a suspension/suspension mode with deformabe omponents and one with ony rigid body representation. In onusion, the vehie dynami anaysis resuts an be greaty affeted by the type of suspension/suspension sub-system representation being used. The rigid vehie body struture representation used in the present study had definitey some infuene on the vehie s dynami harateristis and the simuation resuts..3 Vehie Struture Modes Kawagoe et a. were interested in the infuene of ro behavior on drivers evauating handing performane in []. Prototype ars were used for the experiment. In the past, ro behavior of performane has been diffiut to predit. In the past, indies for evauating ro behavior has not orreated we with subjetive evauations. To improve ro behavior, the suspension properties of prototype ars had to be adjusted. The resuts suggested that vehie pith motion is osey reated to the sensation of ro. In onusion, pith motion is an effetive evauation index for vehie ro behavior and it orreates we with subjetive evauation resuts. Controing pith motion during ro to a ower mode at the front end reative to the rear is extremey important. This is the

24 ey to aeptabe ro behavior. Controing of the pith motion an be ahieved by designing suitabe ro enter harateristis, noninear oad hanges and damping fore oeffiients. Mehrabi et a. presented a parametri study of automated vehies using the sensitivity theory in []. A sixth order dynami mode of an axisymmetri vehie was deveoped to represent its 3-DOF motion in the atera, yaw and ro modes. Important parameters of the vehie are grouped into three separate vetors, inertia, stiffness and damping, and geometri-inemati parameter, respetivey. They investigated the effets of various parameters of automated vehies on their dynami performane. The resuts are usefu for the design of both the mehania struture of a vehie and the its steering ontroer. Simuation resuts show that an inrease in the mass of the vehie signifianty dereases the steady-state atera veoity of the vehie. Sprung mass and ro moment of inertia affet the ro transient response. Both transient and steady state responses of atera veoity and yaw rate of the vehie are sensitive to the ornering stiffness of the front tires. Stiffness and damping parameters had virtuay no effet on the steady-state vaue of ro motion. Any inrease in the forward vehie speed inreases the yaw rate onsideraby beause it is the most sensitive state. In genera, the vehie parameters direty reated to ro motion did not ontribute signifianty to the yaw and atera motion. The most signifiant parameter affeting a the motion of the vehie, espeiay yaw rate, was the forward speed of the vehie. The geometri-inemati parameters were the most infuentia on the performane of the vehie, espeiay the yaw rate []. Inertia parameters were the most infuentia when it ame to the transient parts of the various responses.

25 E-Demerdash was interested in the performane of hydro-pneumati imited bandwidth ative suspension system using a haf ar mode in [3]. The boune and pith motions for the sprung mass and vertia DOF for the unsprung mass are onsidered. With the passive suspension systems it is diffiut to ahieve the onfiting requirements. This was beause the prinipe vehie dynami modes: boune, pith and ro, ride omfort and maneuverabiity modes, must have different natura frequenies and damping for optimum suspension design. The haf vehie mode is more onvenient to study their infuene on the interations between the body boune and pith motions and to design the passive as we as the ative suspension systems. The resuts showed that the system with wheebase orreation provides % improvement in pith aeeration and 8% redution in the body vertia aeeration at the rear suspension onnetion ompared with unorreated system. In omparison with passive system, there is 44% redution in the pith aeeration and 9% improvement in the body vertia aeeration at the rear suspension onnetion. This redution in the pith aeeration was usefu to improve the ride omfort in the ongitudina diretion. The advantage of the proposed suspension system was that its hydraui omponents are standard and ommeriay avaiabe in the maret..4 Tire Properties Loeb et a. examined on the reaxation ength of a tire in [4]. The mehania harateristis of the pneumati tire have a arge infuene on the vehie handing performane and diretiona response. Kinemati properties are those that our when the tire is roing on a surfae. The mehania properties onsist of spring-ie properties measured with the tire in a stationary ondition [4]. The spring rate of the 3

26 tire is tested using a Firestone mehania spring rate test rig. An inreasing oad is appied to the tire and the resuting defetions are reorded and then potted. The atera stiffness is determined by auating the initia sope of this pot. Performing a eastsquares urve fit on the measured data and evauating the derivative of the resutant equation at zero defetion an find the sope. They onuded that the ornering stiffness and atera stiffness of a tire might be used to estimate the reaxation ength. Chiesa and Rinonapi ooed at a new mathematia mode deveoped to aount for noninearities between vehies and tires in [5]. Latera and vertia stiffness have basi infuenes on the behavior of the ar in sudden and severe maneuvers; therefore, these two properties of the tire were investigated. They examined the behavior of the ar in various types of maneuvers at a onstant speed. The vertia fexibiity of the tires is not onstant and tends to redue with the atera fore between the tire and the road. They inreased the number of DOF of the mode to inude the ro of the unsuspended masses and the onsequent inrease in body ro in order to tae aount in a rigorous manner of this fexibiity. The inrease in DOF ours beause the ro is not opposed by the rigidity of the suspension aone but by a ower rigidity. The ower rigidity is due to the effet of the fexibiity of the tires being in series with that of the suspensions. The mathematia mode is a system of seven differentia equations. Three of the equations invove atera, yaw, and ro movements of the ar and the other four of the equations invove the reative movements between the tread of the tires and the whee due to atera fexibiity of the tires. An inrease in the atera reation for equivaent vaues of tire sip ange and vertia oad generay produes onsiderabe improvements on the behavior of the vehie. The negative side to this is that the frequeny of osiations to whih the 4

27 vehie is subjeted in the fina part of eah test is in fat notieaby inreased. A positive effet was the inrease in the atera easti rigidity beause it produed effets of more modest dimensions. Ceary, the atera easti rigidity was seen to be a fator that damps the osiations of the vehie. The tas of the driver in hoding the ar on its path is inreased if the osiation of the ar were numerous. By baaning the inrease in the ornering stiffness with a simutaneous inrease in the atera easti stiffness the designer has the possibiity of eiminating or reduing this troube. The auating proedure and the mathematia mode adopted assist in genera to foreasting the effets of any modifiation on the parameters of the ar or of the tires without atuay buiding the vehie [5]. They onuded the evauation of handing harateristis of a vehie must onsider the tota dynami system of the vehie, whih is, the sprung and unsprung masses, the suspension system, and the tires. Tayor et a. investigated the vertia stiffness measured for a radia py agriutura drive tire using five methods: Load-defetion, non-roing vertia free vibration, non-roing equiibrium oad-defetion, roing vertia free vibration, and roing equiibrium oad-defetion in [6]. At a infation pressures, non-roing free vibration resuted in the highest stiffness, and oad-defetion and non-roing equiibrium oad defetion resuts were simiar. The natura frequeny was measured by timing the sma osiations of eah frame whie they were suspended from the pivot point. Osiations were initiated by pushing the frame, whih was then aowed to move freey [6]. The range of motion for the tire inreased when the vertia oad on the tire was inreased. The natura frequeny, the mass moment of inertia of the frame assemby, and the distane from the pivot point of the frame to the axe were required to determine the 5

28 vertia stiffness for a given ondition. The measurement method impats the resuts beause of the notieabe differenes in vertia stiffness, eading the authors to beieve the five methods aused bias in the resuts..5 Vehie Handing Saaani et a. ombined muti-body dynamis with subsystem modes to provide state-of-the-art high fideity vehie handing dynamis for rea-time simuation in [7]. The partiuar formuation hosen to represent the vehie as osey as possibe depends heaviy on how the parameters are obtained from the physia system. The vehie response effets to be modeed must be speified in the muti-body approah. The mutibody formaism in onjuntion with other subsystems is needed to simuate a ompete vehie mode. The muti-body reursive formuations an be used to mode rigid-body noninear motions. They disovered that the vehie subsystems have beome the fundamentas to ompement vehie dynamis simuation. The vehie subsystem modes are as important as rigid body dynamis. Naez and Bindemann foused on the deveopment of four-whee steering systems for passenger vehies in order to enhane their handing properties in [8]. Previous researh on this subjet is usuay based on anaysis of the assi -DOF inear biye mode; however, by using the -DOF biye mode important effets suh as suspension inematis, atera weight transfer and hassis ro are ignored. Therefore, a non-inear 3-DOF vehie mode was used to represent eah suspension as a three dimensiona mehanism. They aso used spatia inematis to simuate suspension motion, giving the simuation the abiity to determine the hange in whee steer and amber anges aused by hassis ro. These two men negeted aerodynamis. By doing 6

29 this, the ony fores and moments, whih at on the vehie, are gravity, and those produed by the tires at the four ontat pathes. The mode is a 3-DOF, three mass mode that onsists of a sprung mass and the unsprung masses of front and rear suspension systems, whih are onneted to the sprung mass through the vehie ro axis. The 3-DOF used for the vehie mode is atera dispaement, vehie yaw and the sprung mass ro ange [8]. They determined that suspension inematis, bump and ro steer infuene vehie response. The resuts showed that suspension and steering systems shoud be designed to either minimize the effets of bump and ro steer, or use these effets to enhane vehie stabiity and response. Mousseau et a. foused on the advent of muti-body system simuations (MSS) programs in [9]. MSS was effetive in prediting the handing performane of vehies; but not as suessfu in prediting the vehie response at frequenies that are of interest in ride harshness and durabiity appiations. For instane, when traveing on fat surfaes a rigid body vehie mode may prove adequate for prediting the handing response. On the other hand, a rigid body vehie may prove inadequate for simuating a vehie traveing over irreguar terrain, sine high-frequeny modes may be exited in the vehie and tire struture. They simuated the vehie with a MSS program. The tires were simuated with noninear FE programs. A simpe -DOF-vehie dynamis mode and a singe-tire FE mode were used for this approah. They assume that rigid bodies an adequatey represent omponents with signifiant masses. The resuts of this study indiate that this approah is a viabe tehnique for simuating vehie dynami response on rough roads [9]. 7

30 CHAPTER 3: VIBRATION, PARAMETRIC AND SENSITIVITY ANALYSIS 3. Introdution This hapter deas with the theoretia anaysis for free and fored vibration. The resuts wi aso inude ases for parametri and sensitivity anaysis. The purpose of the theoretia anaysis using reaisti vaues is to show what happens under rea onditions. 3. Vibration Anaysis 3.. Free Vibration Systems that require two independent oordinates to desribe their motion are aed two DOF systems. Consider the system shown in Figure 3., in whih a mass m is supported on two springs. Assuming that the mass is onstrained to move in a vertia pane, the position of the mass, m, at any time an be speified by a inear oordinate u () t, indiating the vertia dispaement of the enter of gravity of the mass, and an anguar oordinate θ () t, denoting the rotation of the mass m about its enter of gravity. Thus, the system has two degrees of freedom. It is important to note that in this ase the mass m is not treated as a point mass, but as a rigid body having two possibe types of motion. There are two equations of motion for a two DOF system and are generay in the form of ouped differentia equations; that is, eah equation invoves a the oordinates. 8

31 9 From the Free Body Diagram in Figure 3., the equations of motion an be obtained. With the positive vaues of the motion variabes as indiated, the fore equiibrium equation in the vertia diretion an be written as = u u ma F + + = θ θ θ θ & & & & && u u u u u m (3.) or equivaenty, ( ) ( ) ( ) ( ) = θ θ u u u m & & & &. (3.) The moment equation about the C.G. an be expressed as = α G I G M = = θ θ θ θ θ & & & & & & u u u u I G (3.3) or equivaenty, ( ) ( ) ( ) ( ) 4 4 = θ θ θ u u I G & & & &. (3.4) θ u Figure 3.: -degree of freedom system θ u Figure 3.: -degree of freedom system

32 Equations (3.) and (3.4) an be rearranged and written in matrix form as ( ) ( ) ( ) ( ) ( ) ( ) ( ) = θ θ θ u u u I m G & & && &&. (3.5) During free vibration at one of the natura frequenies, the ampitudes of the two DOF (oordinates) are reated in a speifi manner and the onfiguration is aed a norma mode, prinipa mode, or natura mode of vibration. The system has damped natura frequenies instead of natura frequenies beause there is damping in the system. The damped natura frequeny is essentiay, ζ ω ω = n d, where ζ is the damping ratio. Thus a two DOF system has two norma modes of vibration orresponding to the two damped natura frequenies. Now the equations of motion an be used to obtain the poes (eigenvaues) and modes shapes (eigenvetors) for free vibration. A pubished artie on tire parameters states that for a singe tire, the tire spring and damping rates usuay fa into the ranges of 75 to 3 N/m and N-s/m, respetivey []. The vaues for,, m and are taen from the Honda CR-V mode. The vaues for,,, and are pied arbitrariy from the pubished ranges given previousy. Hene, the vaues for the parameters are the foowing: m s N m s N m N m N m m g m /., / `, / 6, / 45,.95,.53, 59 = = = = = = =

33 where m is the mass in iograms, is the distane from the mass enter to end of rigid body, is the distane between front spring to rear spring, and are the spring and damping rate for both front tires, and and are the spring and damping rate for both rear tires. Note in the pages remaining, matries are represented in bod apita etters and vetors are represented as bod owerase etters. For free vibration, the genera equations of motion are of the form M u& + Cu& + Ku =. (3.6) A first order reaization of Equation (3.6) is I u I u& = K u& C M u&& (3.7) where I A =, K (3.8) I B = C M (3.9) and This eads to u z =. (3.) u & Az Bz& =. (3.) The response of this system taes the form λt () t ve z = (3.) where v is a onstant vetor. Substituting (3.) in (3.) eads to the eigenvaue probem

34 ( A B) v = λ. (3.3) In this ontext the vetor v is aed the eigenvetor and λ are the eigenvaues. Further resuts are done using MATLAB. The simuation resuts are shown in Tabe 3. and Figures The system is stabe when the rea part of a poes is negative as indiated in Figure 3.. In Tabe 3., V and V are a ompex onjugate pair and V3 and V4 are aso a ompex onjugate pair of vetors. With ompex numbers in vibration, the rea part is assoiated with the damping in the system and the imaginary part is assoiated with the frequeny in the system. Tabe 3.: Poes of the system Poes s s s 33 s 44 Vaue i i i i Im Re Figure 3.: Poes of the system for free vibration

35 Tabe 3.: Eigenvetors of the system V V V3 V i i i i.4 -.i.4+.i.6-.84i.6+.84i i i i i. +.49i.-.49i i i..8 Mode Shape.6.4 Abs. Imag Rea Index of points Figure 3.3: First mode shape (absoute, rea and imaginary part) 3

36 Abs Mode Shape Rea Imag Index of points Figure 3.4: Seond Mode (absoute, rea and imaginary part).8.6 Mode Shape.4. Abs -. Imag Rea Index of points Figure 3.5: Third Mode (absoute, rea and imaginary part) 4

37 .8.6 Mode Shape Abs Imag Rea Index of points Figure 3.6: Fourth Mode (absoute, rea and imaginary part) 3.. Fored Vibration (Harmoni Exitation) In genera, the dynami of a harmoniay exited n degree-of-freedom system is governed by the equations M u& + Cu& + Ku = g sinωt + h osωt (3.4) () = x, u() v u & = (3.5) where g, h, x and v are onstant vetors. The soution of the probem an be obtained by soving two simper probems. The first assoiated with the homogeneous soution, the other with the partiuar soution, and then superimposing the resuts. This means, where () t u P () t u () t u = + (3.6) u P is the partiuar soution of H Mu& + Cu& + Ku = gsinωt h osωt. (3.7) P P P + 5

38 Trying a soution of the form p () t p osωt qsinωt u = + (3.8) eads to the matrix K ω M ωc ωc p h = K ω M q g (3.9) whih determines p and q giving the soution u () t. The genera soution, u, whih depends on an arbitrary ombination of n ineary independent funtions, of the homogeneous probem H M u& Cu& + Ku =. (3.) H + H H Trying a soution of the form p st u H = re (3.) where s is the eigenvaues and r is the eigenvetor. A more genera soution of Equation (3.) is n j u = a r e (3.) H j= j j s t and is expressed as a ombination of n ineary independent funtions where a j are onstants. The step-by-step formuation to obtain the eigenvaue probem for the homogenous soution is aready given in Equations (3.6)-(3.3). The homogeneous soution is what was overed in Setion 3..; therefore the partiuar soution is of more interest. The equations of motion for fored vibration for the system in Figure 3.7, where h =, are as foows, 6

39 7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = sin 4 4 t I m G ω g θ u θ u θ u & & && &&. (3.3) The soution for Equation (3.3) an aso be written in the form () ( ) Θ U u = t t ω sin (3.4) where U is the magnitude and Θ is the phase funtion. Equation (3.3) is the governing equation and omputer simuations wi be used for further resuts. The resuts are shown in Figures 3.8 to 3.. θ u Figure 3.7: -degree of freedom system with externa fore g sin ωt θ u Figure 3.7: -degree of freedom system with externa fore g sin ωt

40 -4-5 Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.8: Magnitude in boune mode for fored vibration 4 3 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.9: Phase funtion in boune mode for fored vibration 8

41 -5-6 Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Magnitude in pith mode for fored vibration 4 3 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Phase funtion in pith mode for fored vibration By not speifying a speifi vaue for the frequeny of the externa fore aows for a parametri anaysis to see when arge ampitude of vibration ours. Figure 3.8 to 9

42 3. show what happens when arge ampitude of vibration ours. Large ampitude vibration is when the frequeny of the externa fore oinides with one of the damped natura frequenies of the system, and the system undergoes dangerousy arge osiations. 3.3 Parametri Anaysis Parametri studies of dynamia systems are an important tas in a major engineering design. This is espeiay true in the design of ontro systems, when the fous is for ow sensitivity to parameter variations. It is important to now when one parameter is hanged, how that affets the system; if the system beomes unstabe, do the natura frequenies hange, et. These are things to onsider when attempting buiding a system. In the anayses beow, a parameters are hed onstant exept one. Tabe 3.3 shows the initia vaue,, of eah parameter as we as the new vaue after the i th inrementa hange, i. The inrements for eah parameter are hosen arbitrariy at equa intervas. However, eah parameter was not to exeed a ertain imit. For exampe, m was not to exeed the maximum oad arried by the vehie, was not to exeed the tota ength of the vehie, and were not to exeed pubished ranges for stiffness and damping of a vehie. Hene, on eah of the graphs foowing, there wi be six urves to represent the initia vaue and five inrementa hanges to the parameter. 3

43 Tabe 3.3: Ranges for eah parameter for parametri anaysis m (g) (m) (N/m) (N-s/m) The first parametri anaysis is on the mass moment of inertia, J. It is possibe to hange J without hanging the mass by hanging. The simuation resuts are shown in Figures Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Parametri anaysis for J-magnitude in boune mode 3

44 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.3: Parametri anaysis for J phase funtion in boune mode Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.4: Parametri anaysis for J magnitude in pith mode 3

45 4 3 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.5: Parametri anaysis for J phase funtion in pith mode In Figures 3. and 3.4 it shows that when J is inreased, the first damped natura frequeny dereases from approximatey.65 to 9.75 and dereases in magnitude sighty. The orresponding phase funtion for eah inrease is shown in Figures 3.3 and 3.5. For the first mode of vibration, as shown in Figures 3. and 3.3, J does not have any effet on the seond damped natura frequeny. In the pith mode, the seond damped natura frequeny hanges in magnitude ony, whereas the first damped natura frequeny dereases as shown in Figures 3.4 and 3.5. The seond parametri anaysis is on the mass. Changing the mass aso means the mass moment of inertia is affeted as we. However, the effet that hanging the mass wi have on the system wi sti be seen. The simuation resuts are shown in Figures

46 -4 Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.6: Parametri anaysis for m magnitude in boune mode Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.7: Parametri anaysis for m phase funtion in boune mode 34

47 Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.8: Parametri anaysis for m magnitude in pith mode Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.9: Parametri anaysis for m phase funtion in pith mode 35

48 When the mass is hanged, both damped natura frequenies derease but their magnitude inreases. Figures show these hanges. The damped natura frequeny depends on the natura frequeny; therefore, it foows from the parametri anaysis that the damped natura frequeny dereases as the mass inreases. This is what shoud happen sine ω = m. n / The third parametri anaysis is on the stiffness for the front tires,. The simuation resuts are shown in Figures Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Parametri anaysis for magnitude in boune mode 36

49 4 3 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Parametri anaysis for phase funtion in boune mode 37

50 -5-6 Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.: Parametri anaysis for magnitude in pith mode 4 3 Phase, Θ (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.3: Parametri anaysis for phase funtion in pith mode 38

51 There is a sma effet of inreasing. Figure 3. shows the first damped natura frequeny dereases very sighty, amost not notieabe by the human eye. Figures show that both damped natura frequenies hange positions sighty. The inrease in the damped natura frequenies is what shoud happen, but the effet oud be sma beause was inreased by a sma amount, trying not to exeed at the same time. This sma effet is notied in Figures beause at some points a the urves oinide. Athough is ony shown, the same thing happens when a parametri anaysis is performed on. The fourth parametri anaysis is on the damping for the front tires,. The simuation resuts are shown in Figures Magnitude, U (radians) Damped Natura Frequeny, ω d (rad/se) Figure 3.4: Parametri anaysis for magnitude in boune mode 39