Ride Quality and Drivability of a Typical Passenger Car subject to Engine/Driveline and Road Non-uniformities Excitations

Transcription

1 Ride Quality and Drivability of a Typial Passenger Car subjet to Engine/Driveline and Road Non-uniformities Exitations Examensarbete utfört i Fordonssystem vid Tenisa högsolan i Linöping av Neda Nimehr LiTH-ISY-EX--11/4477--SE Linöping 211 I

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3 Ride Quality and Drivability of a Typial Passenger Car subjet to Engine/Driveline and Road Non-uniformities Exitations Examensarbete utfört i Fordonssystem vid Tenisa högsolan i Linöping av Neda Nimehr LiTH-ISY-EX--11/4477--SE Handledare: Examinator: Neda Nimehr ISY, Linöpings universitet Jan Åslund ISY, Linöpings universitet Linöping, 7th June 211 III

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5 Avdelning, Institution Division, Department Division of vehiular system Department of Eletrial Engineering Linöpings universitet SE-58183Linöping, Sweden Datum Date Språ Language Svensa/Swedish Engelsa/English Rapporttyp Report ategory lientiatavhandling Examensarbete C-uppsats D-uppsats Övrigrapport ISBN ISRN LiTH-ISY-EX--11/4477--SE Serietitel oh serienummer ISSN Title of series, numbering URL för eletronis version urn:nbn:se:liu:diva Title Ride Quality and Drivability of a Typial Passenger Car subjet to Engine/Driveline and Road Non-uniformities Exitations Författare Neda Nimehr Author Sammanfattning Abstrat The aim of this wor is to evaluate ride quality of a typial passenger ar. This requires both identifying the exitation resoures, whih result to undesired noise inside the vehile, and studying human reation t applied vibration. Driveline linear torsional vibration will be modelled by a 14-degress of freedom system while engine ylinder pressure torques are onsidered as an input fore for the struture. The results show good agreement with the orresponding referene output responses whih proves the auray of the numerial approah fourth order Runge-utta. An eighteen-degree of freedom model is then used to investigate oupled motion of driveline and the tire/suspension assembly in order to attain vehile body longitudinal aeleration subjet to engine exitations. Road surfae irregularities is simulated as a stationary random proess and further vertial aeleration of the vehile body will be obtained by onsidering the well-nown quarter-ar model inluding suspension/tire mehanisms and road input fore. Finally, ISO diagrams are utilized to ompare RMS vertial and lateral aelerations of the ar body with the fatigue-dereased profiieny boundaries and to determine harmful frequeny regions. Aording to the results, passive suspension system is not funtional enough sine its behaviour depends on frequeny ontent of the input and it provides good isolation only when the ar is subjeted to a high frequeny exitation. Although longitudinal RMS aeleration of the vehile body due to engine fore is not too signifiant, driveline torsional vibration itself has to be studied in order to avoid any dangerous damages for eah omponent by reognizing resonane frequenies of the system. The report will ome to an end by explaining different issues whih are not investigated in this thesis and may be onsidered as future wors. Nyelord Keywords Ride quality, Driveline, Engine exitations, V Road non-uniformities, Suspension System, Torsional vibration, Random proess

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7 Upphovsrätt Detta doument hålls tillgängligt på Internet eller dess framtida ersättare under 25 år från publieringsdatum under förutsättning att inga extraordinära omständigheter uppstår. Tillgång till doumentet innebär tillstånd för var oh en att läsa, ladda ner, sriva ut enstaa opior för ensilt bru oh att använda det oförändrat för ieommersiell forsning oh för undervisning. Överföring av upphovsrätten vid en senare tidpunt an inte upphäva detta tillstånd. All annan användning av doumentet räver upphovsmannens medgivande. För att garantera ätheten, säerheten oh tillgängligheten finns lösningar av tenis oh administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed räver vid användning av doumentet på ovan besrivna sätt samt sydd mot att doumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är ränande för upphovsmannens litterära eller onstnärliga anseende eller egenart. För ytterligare information om Linöping University Eletroni Press se förlagets hemsida Copyright The publishers will eep this doument online on the Internet or its possible replaement for a period of 25 years starting from the date of publiation barring exeptional irumstanes. The online availability of the doument implies permanent permission for anyone to read, to download, or to print out single opies for his/her own use and to use it unhanged for non-ommerial researh and eduational purpose. Subsequent transfers of opyright annot revoe this permission. All other uses of the doument are onditional upon the onsent of the opyright owner. The publisher has taen tehnial and administrative measures to assure authentiity, seurity and aessibility. Aording to intelletual property law the author has the right to be mentioned when his/her wor is aessed as desribed above and to be proteted against infringement. For additional information about Linöping University Eletroni Press and its proedures for publiation and for assurane of doument integrity, please refer to its www home page: Neda Nimehr VII

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9 Abstrat The aim of this wor is to evaluate ride quality of a typial passenger ar. This requires both identifying the exitation resoures, whih result to undesired noise inside the vehile, and studying human reation to applied vibration. Driveline linear torsional vibration will be modeled by a 14-degress of freedom system while engine ylinder pressure torques are onsidered as an input fore for the struture. The results show good agreement with the orresponding referene output responses whih proves the auray of the numerial approah fourth order Runge-utta. An eighteen-degree of freedom model is then used to investigate oupled motion of driveline and the tire/suspension assembly in order to attain vehile body longitudinal aeleration subjet to engine exitations. Road surfae irregularities is simulated as a stationary random proess and further vertial aeleration of the vehile body will be obtained by onsidering the well-nown quarter-ar model inluding suspension/tire mehanisms and road input fore. Finally, ISO diagrams are utilized to ompare RMS vertial and lateral aelerations of the ar body with the fatiguedereased profiieny boundaries and to determine harmful frequeny regions. Aording to the results, passive suspension system is not funtional enough sine its behavior depends on frequeny ontent of the input and it provides good isolation only when the ar is subjeted to a high frequeny exitation. Although longitudinal RMS aeleration of the vehile body due to engine fore is not too signifiant, driveline torsional vibration itself has to be studied in order to avoid any dangerous damages for eah omponent by reognizing resonane frequenies of the system. The report will ome to an end by explaining different issues whih are not investigated in this thesis and may be onsidered as future wors. IX

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11 Anowledgments This wor has been arried out at vehiular system division, ISY department, Linöping University, Sweden. The thesis would not have been possible without the support of many people and division laboratory failities. I wish to express my gratitude to my examiner and supervisor, Dr. Jan Åslund and PhD student Kristoffer Lundahl who were abundantly helpful and offered invaluable assistane, support and guidane. Speial thans also to my bahelor supervisor professor Farshidianfar for sharing the literature and invaluable assistane. I would lie to express my love and gratitude to my beloved parents Maryam and Ahmad for their understanding and endless love, through the duration of my master study. Linöping, May 211 Neda Nimehr XI

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13 Table of Contents 1 Chapter Baground Objetive Assumptions and Limitations Outline Chapter Driveline and vehile Modeling Road Surfae Irregularities Human Response to vibration Chapter Introdution Driveline omponents Engine, flywheel and the main exitation torque Cluth Assembly Gearbox Cardan (propeller) shaft and universal (Hooe s) joints Differential and final drive system Damping in the whole driveline system Overall driveline model Torsional vibration Chapter Introdution Mathematial model and system matries Summary of Modal analysis Natural frequenies Chapter Introdution Mathematial model for fored vibration of the driveline system Time responses of driveline at luth and driving wheels Power spetral densities of time histories Chapter Introdution XIII

14 6.2 Coupled vibration of driveline and the vehile body Tire model and longitudinal fore degrees of freedom system for whole vehile model and its equations of motion Time response of the system Studying the influene of stiffness and damping oeffiients Chapter Introdution Quarter-ar model and performane of suspension system Road roughness lassifiation by ISO and the reommended single-sided vertial amplitude power spetral density Typial passenger ar driver RMS aeleration to an average road roughness Chapter Introdution International Standard ISO : Results and Disussion Thesis onlusion Future wors Referenes Appendix LTI objet Driveline Modeling MATLAB ode Power spetral density funtion Vehile modeling MATLAB odeatlab ode... 7 XIV

15 Figures Figure 1-1, the ride dynami system... 1 Figure 3-1, Front-engine rear-wheel-drive vehile driveline [3]... 7 Figure 3-2, Main engine parts [1]... 8 Figure 3-3, Original System of a ran [3]... 8 Figure 3-4, Equivalent system of ranshaft and its ompat model [3]... 9 Figure 3-5, output torque of a four-stroe single-ylinder engine [1]... 9 Figure 3-6, Line diagram of ylinders arrangement... 1 Figure 3-7, engine torque in the ase of four ylinders... 1 Figure 3-8, 1 seonds pressure reording from ylinder Figure 3-9, 1 seonds pressure reording from ylinder Figure 3-1, Cran mehanism Figure 3-11, Torque output of ylinder 1, total and flutuating part, during 1 seonds and one woring yle Figure 3-12, Torque output of ylinder 2, total and flutuating part, during 1 seonds and one woring yle Figure 3-13, Output torques from 4 ylinders in the same plot Figure 3-14, Compat ranshaft model for a four-ylinder engine Figure 3-15, PSD for output torque from ylinder Figure 3-16, PSD for output torque from ylinder Figure 3-17, Cluth system [22] Figure 3-18, Gearbox model [3] Figure 3-19, Hooe's (ardan) joints [1]... 2 Figure 3-2, propeller shafts and universal joints mathematial model [3]... 2 Figure 3-21, Final drive system and its equivalent model [3] Figure 3-22, Damped torsional vibration mathematial model of driveline system [3] Figure 5-1, Driveline model Figure 5-2, Time response at the luth, using different methods of solution: Bla-> Modal analysis, Blue-> ODE45, Green-> self-written Runge-Kutta ode with nonzero initial onditions and yellow-> with zero initial onditions Figure 5-3, zoomed version of figure 5.2 in order to see the instability of modal analysis and ODE45 solutions Figure 5-4, Time response at the luth with the aid of Runge-Kutta method Figure 5-5, Time response at the driving wheels Figure 5-6, Power spetral density of the time response at the luth Figure 5-7, Power spetral density of the time response at the driving wheels Figure 6-1, tire model [3] Figure 6-2, overall vehile model [3] Figure 6-3, Total engine exitation torques after applying filtration Figure 6-4, torsional veloity vibration of driving wheels due to engine exitation torques by using table 6.1 data values Figure 6-5, zoomed version of Figure 6-4 between seonds 8 to XV

16 Figure 6-6, Longitudinal veloity vibration of the vehile body and axle due to engine exitation torques by using table 6-1 data values Figure 6-7, zoomed version of Figure Figure 6-8, torsional veloity of driving wheels with low damping Figure 6-9, Longitudinal veloities of vehile body and axle with low damping Figure 6-1, Longitudinal veloities of vehile body and axle with low suspension system stiffness Figure 7-1, Two-degrees of freedom model vehile Figure 7-2, Transmissibility as a funtion of frequeny ratio for a single-degree of freedom system Figure 7-3, Modified quarter-ar model inluding seat displaement Figure 7-4, Measured vertial aeleration of a passenger ar seat traveling at 8 Km/hr over an average road Figure 7-5, vehile body vertial aeleration subjet to an average road roughness with 8 Km/hr traveling speed Figure 8-1, ISO :1985 "fatigue-dereased profiieny boundary": vertial aeleration limits as a funtion of frequeny and exposure time [4] Figure 8-2, ISO :1985 "fatigue-dereased profiieny boundary": longitudinal aeleration limits as a funtion of frequeny and exposure time [4] Figure 8-3, vehile body vertial aeleration due to road exitation in omparison with ISO ride omfort boundaries Figure 8-4, Measured longitudinal aeleration of a passenger ar body due to engine exitation torques XVI

17 Tables Table 3-1, Engine properties... 1 Table 4-1, Typial values for equivalent parameters of a vehile driveline [3] Table 4-2, Undamped natural frequenies of whole driveline model using typial parameter values for a passenger ar Table 4-3, first five natural modes of driveline system Table 6-1, Overall vehile properties [3] Table 7-1, Tire/suspension properties [3]... 5 Table 7-2, Classifiation of road roughness proposed by ISO [4] XVII

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19 Chapter1 Introdution 1 Chapter 1 Introdution 1.1 Baground Ride quality is an important parameter for ar manufaturers, whih larifies the transmission level of unwanted noises and vibrations from vehile body to the passengers. The term unwanted is defined aording to human response to vibration whih is different from one person to another and will be desribed more in the next hapters. Inreasing ustomer demands for more omfortable ars and better ride quality, not only requires full understanding of human response to exitation, but also it is needed to study different soures whih may result to vibration of vehile body, and dynami behavior of the automobiles. In order to provide better realization of ride behavior [1], it is useful to show the ride dynami system as follows (Figure 1-1): Figure 1-1, the ride dynami system Aording to Figure 1-1, there are four different exitation soures that may be divided into two ategories: 1) road surfae irregularities and 2) on-board origins whih result from rotating parts (engine, driveline and non-uniformities (imbalanes) of tire/wheel). Sine the days of first vehiles, the attempts have been made to isolate the ar body from road roughness 1, and the ar suspension system is responsible for this duty. Road profile is a random funtion whih ats as an input to suspension system, furthermore theory of stohasti proesses and power spetral densities have been utilized in the literatures to model this random signal. The two degrees of freedom model (2-DOF) nown as a quarterar model is used to simulate suspension system and vehile body [2]. The goal is to optimize the suspension system parameters to derease the undesired effets on the vehile body (Chapter 7) aording to ride omfort riterion that may be seleted. Driveline is one of the onsiderable soures of noise and vibration for any type of automobiles, whih is omposed of everything from the engine to the driven wheels. Driveline torsional osillations fall into two broad ategories: gear rattle and driveline vibration. Idle gear rattle is a onsequene of gear tooth impats, and driveline vibrations are noises whih ome from the driveline system parts suh as engine, luth and universal 1 roughness is desribed by the elevation profile along the wheel tras over whih the vehile passes [1] 1

20 Chapter1 Introdution joints, while the vehile is in motion at different running situations [3], we are interested in the driveline system vibration in three aspets: 1) finding system natural frequenies in order to avoid oinidene with fored frequenies and resonane ourrene, 2) to determine fored response subjet to engine osillatory torque and universal joints and 3) transient response. It should be noted that some omponents of driveline suh as universal (Hooe s) joints result to nonlinear behavior of the system, and in addition the torsional vibration of driveline an be oupled with the horizontal and vertial motions of vehile body and rear axle, these phenomenon may ause the ompliation of the system. Modeling the system (quarter ar model or driveline) and obtaining the response, it s time to evaluate ride quality of the system. Hene it is neessary to speify ride omfort limits. Various methods have been developed over the years for assessing human tolerane to vibration [4] whih will be more explained in hapter Objetive The goal of this thesis is divided into two major parts: 1- To model the driveline and engine flutuating torque in order to find free and fored responses of the system and furthermore studying the sensitivity of driveline behavior by hanging design parameters. The attempt is made to simulate driveline as a 14-degrees-offreedom (DOF) system and the whole vehile as an 18-DOF mehanism and at last determining the horizontal aeleration of sprung mass (vehile body) due to engine torque using Runge-utta numerial method. Aeleration time history is then onverted to frequeny domain using power spetral density tool in order to ompare with ride omfort diagrams. 2- To use quarter ar model and obtain driver response subjet to road random irregularities with the aid of random proess theory. In this part of the report, the importane of the suspension system to derease the undesired motions will be illustrated. 1.3 Assumptions and Limitations In this projet a lumped-parameter model is used for studying the torsional vibration of driveline system whih assumed to be a set of inertia diss lined together by torsional, linear and massless springs [3]. A normal four ylinder rear drive passenger ar will be onsidered and the system parameter values suh as sprung and unsprung masses, all the stiffness and damping oeffiients have been hosen aording to referenes [3] and [4] and different vehile ompanies database. It should also be noted that this wor is based on the PhD thesis by El-Adl Mohammed Aly Rabeih whih is done in The engine flutuating torque (as will be desribed later) onsists of two major parts: gas pressure torque and inertia torque 1 whih are ome from ylinder gas pressure and reiproating omponents of engine, respetively. however in the urrent report, we will only study the effets of the pressure torque sine there is no useful data for the mass of reiproating parts, furthermore the ylinders pressure are measured in the vehiular system engine Laboratory for a four-stroe four-ylinder engine with the firing order of Moreover the nonlinear torque whih is resulted by Hoo s joints has been introdued in this thesis while defining the response of the system subjet to this ouple requires strong nonlinear method whih is beyond the aim of this wor. In order to investigate the road surfae influenes, three assumptions have been inluded: 1 Espeially in high speed vehiles, the inertia torque is very important! 2

21 Chapter1 Introdution The road profile is assumed to be a stationary ergodi random proess, however in reality the road s profiles are non-stationary funtions. the amplitude distribution of the road roughness is assumed to be Gaussian The ar has a onstant speed and travels on a straight line. 1.4 Outline The thesis is omposed of 8 hapters. This introdutory hapter is followed by a short literature review of what has been done so far assoiated to this wor. In hapter 3, driveline system and its different parts have been modeled as well as relations of onverting ylinders pressure to the torques whih are delivered by ranshaft. Chapter 4 inluded of mathematial simulation of driveline, and modal analysis to find natural frequenies of the system. Chapter 5 onsists of introduing different methods of obtaining fored response of 14-degrees of freedom system. In hapter 6 the whole vehile model has been desribed and horizontal vibration of the vehile body and rear axle is obtained subjet to driveline torsional osillations. Besides, the parameter values will be hanged in order to study the sensitivity of the system to stiffness and damping oeffiients. Chapter 7 onsists of using quarter-ar model to define the response of driver to road non-uniformities whih is an input to the system. The report will be ended by hapter 8 whih has inluded international standard ISO for evaluation of human exposure to whole-body vibration [4] and alulating RMS vertial and horizontal aelerations of our model to ompare with ride omfort riteria. Moreover the onlusion setion and future wor suggestions are the last parts of the final hapter. 3

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23 Chapter 2 previous wors 2 Chapter 2 Previous wor 2.1 Driveline and vehile Modeling During the last four deades and from the early days of automotive industry, the attempts have been made to reah the desired ride omfort and quieter vehiles. Sine driveline torsional behavior is one of the major resoures of unwanted vibrations in the ars, signifiant researhes have been done by different ar ompanies and university sientists in order to gain an aeptable model for driveline system and its omponents struture. Syes and Wyman in 1971, and Ergun in 1975 were among the first people who alulated natural frequenies of a onventional automobile driveline system theoretially and experimentally, respetively [3]. Rei in 199 studied the main vibration soures of driveline system and he onsidered the gas pressure torque and its effets [5]. Zhanqi et al in 1992, [6], onstruted a mathematial model inluding torsional, vertial and vehile fore-aft vibrations to study the oupling of those vibrations together, he also has investigated the influenes of different parameter values on the response of the vehile hassis and axle. Signifiant experimental measurements of system behavior and prinipal modes have been done by different researhers in the years [7]. El-Adl Mohammed Aly Rabeih has done a omplete researh in 1997 onerning the driveline and ompelete vehile free and fored vibration modelings, sensitivity analysis of driveline and suspension system parameter values. he used runge-utta numerial method in order to find displaement and veloity time histories at different points of the model [3]. In the last deade, more investigations are foused on the nonlinearity in driveline system suh as gear rattle, balash, and bahvior of the system during luth engagement. 2.2 Road Surfae Irregularities The health, protetion, ride omfort and performane of both driver and passengers in automobiles are influened by the type of the surfae, over whih the vehile moves. Earlier researhes in the automotive industry inluded of subjeting mathematial models to deterministi inputs, however in real situation surfae profiles are rarely simple forms. A signifiant effort has been done between the years in order to find the spetrum of road roughness [1]. Furthermore attempts by various organizations have been done over the years to divide the road surfae irregularities into different lasses, the international organization for standardization (ISO) has presented this lassifiation (A-H) based on the power spetral density [4]. A huge amount of reports during the last four deades have inluded studies about passive, semi-ative and ative suspension systems and vibration isolations in automobiles as well as different optimizations methods to define optimum parameter values of suspension system subjet to random exitation from the road [13]. 2.3 Human Response to vibration Signifiant investigations have been onduted to aquire ride omfort limitations. There are different standards to evaluate drivability and ride quality of a vehile whih are mentioned 5

24 Chapter 2 previous wors in referenes [4] and [17]. Aording to ISO :1985, four parameters have trivial influenes on obtaining human response to vibration whih are intensity, frequeny, diretion and exposure time. In this wor, these riterions will be used to study the ride omfort of the desired model. 6

25 Chapter 3 Desription of driveline and torsional phenomena 3 Chapter 3 Desription of driveline and torsional phenomena 3.1 Introdution In this hapter, torsional models of driveline system and its omponents are defined, sine driveline torsional vibration due to engine torque exitations is one of the main reasons of undesired noise in the vehiles. Further, the neessary relations in order to obtain osillatory engine torques from ylinder pressures time histories are introdued. 3.2 Driveline omponents The driveline funtion is to transmit mehanial energy of the engine to the wheels and that will be ourred through different parts. A lassial front-engine rear-wheel-drive vehile driveline is illustrated in Figure 3-1, Front-engine rear-wheel-drive vehile driveline [3]. The most ommon omponents of the system are engine, flywheel, luth, gearbox, propeller (or Cardan) shaft and universal joints, differential and rear axle assembly and tires. Aording to the goal of the thesis and in order to simplify the analysis of driveline vibration, a lumped parameter model is used for the whole driveline system. However from the vibration theory, it is nown that in the real world the systems are distributed, and therefore annot be modeled as point masses. Although, utilizing this simple simulation has some advantages and the most important one is the ability of estimating natural frequenies and the fored response of the system subjet to different exitations without ompliated mathematis. In the following setions, eah part is desribed in more detail and an appropriate model will be suggested Engine, flywheel and the main exitation torque Engine is the primary power soure on a vehile. Rotating behavior of the engine and disrete stroes during its woring yle result to torsional vibration exitation of the driveline. Figure 3-1, Front-engine rear-wheel-drive vehile driveline [3] 7

26 Chapter 3 Desription of driveline and torsional phenomena The main parts of the engines are: ylinder, piston, onneting rod and ranshaft 1 whih are shown in Figure 3-2, Main engine parts. The ranshaft rotates by pushing the piston up and down in the ylinder area, there are two dead points (at extreme down and up positions) where the pressure on the piston will have no effets to fore the ranshaft to turn, a stroe is alled to the movement of piston from one dead enter to another dead enter. Four-stroe engine onsists of indution, ompression, power and exhaust steps, more desription of engine parts and stroes funtion an be found in the boos of motor vehile tehnology and it is not the aim of this thesis. Figure 3-2, Main engine parts [1] Sine we are interested in torsional vibration of the driveline, the rotational dynamis of the engine will be simulated by taing into aount the ranshaft system whih is shown in Figure 3-3: Figure 3-3, Original System of a ran [3] The ompat ranshaft system an be modeled as follows (Figure 3-4), where the rotational J w (journal+ ran pin+ webs) and reiproating parts J r (piston+ onneting rod+ piston pin) are omposed together as one final inertia dis 2 J 1. 1 The ranshaft (ran) is the part of an engine whih translates reiproating linear piston motion into rotation, ranshaft onnets to flywheel. 2 It should be noted that the engine mounts, whih are important tools to derease the unwanted effets of engine vibration on the other piees and to isolate the engine from the external exitations, will not be onsidered here and their suitable influene an be perused in the next studies. 8

27 Chapter 3 Desription of driveline and torsional phenomena Figure 3-4, Equivalent system of ranshaft and its ompat model [3] In the above figure, T(t) is the engine exitation torque, and is the equivalent torsional stiffness whih is alulated in referene [3]. Beause of the yli operation of a piston engine, the torque whih is delivered at the ranshaft is osillatory and onsists of a steady-state omponent (mean torque) plus superimposed torque flutuations T(t) 1 (Figure 3-5): Figure 3-5, output torque of a four-stroe single-ylinder engine [1] Conerning to pratial issues, there are always more than one ylinder whih are arranged to have their power stroes in suession [2], the most ommon ase is to have four ylinders. the firing order of the engine illustrates the order in whih the ylinders at, in this thesis we will onsider four-stroe four-ylinder engine with firing order , onsequently we will have 72 degrees per yle of operation for this ind of engine and eah stroe taes 18 degrees. Figure 3-6 shows a simplified line-diagram of the ylinders and rans, 1 Moreover, the torsional vibration of the ranshaft due to longitudinal torque on the moving part of the engine, is of partiular importane beause many ranshafts have failed subjet to this torque. 9

28 Chapter 3 Desription of driveline and torsional phenomena (1) (2) (3) (4) Figure 3-6, Line diagram of ylinders arrangement As it is seen in the above figure, pistons move in pairs: 1&4 and 2&3. The measured pressures in engine laboratory are assoiated to ylinders 1 and 2 and the assumption of this wor is that: ylinders 1 & 4, and 2 & 3 have the same pressure distributions, respetively. The following graph presents the expeted torque for a four ylinder engine. Mean torque Num. of stroes Figure 3-7, engine torque in the ase of four ylinders Eah ylinder exitation torque 1 formed from two main parts 2 : gas-pressure torque {Tg(t)} and inertia torque, however as it is also mentioned in setion 1.3, in this report we will study only the influene of the gas-pressure. Gas-pressure torque itself omposed of harmonis 3 and steady-value, while the steady-value (mean value) will not exite torsional vibration; it is omitted in the alulations. Figure 3-8 and Figure 3-9 show the ylinders 1 and 2 pressures time history {p g1 (t) & p g2 (t)} respetively, in addition the harateristis of the LAB engine are given in Table 3-1, Table 3-1, Engine properties Num. of ylinders Piston Diameter, m Cran radius, m Conneting rod length, m Mean Torque 4, N.m *4 1 Engine speed, RPM, rad/se 2 ω= Expeted flutuating torque fundamental frequeny rad/s or Hz 1 whih auses torsional vibration 2 the frition torque is assumed to be small ompared to these two main omponents 3 whih repeats themselves every omplete woring yle, the interval of repetition is two revolutions of the ranshaft (4π) and the period is 4π/ω [3] 4 The useful engine mean torque is the steady part of ylinders net torque whih is measured by a sensor at flywheel point, this value is normally provided by the engine manufaturer 5 Refer to page 12 1

29 Chapter 3 Desription of driveline and torsional phenomena In this stage, the relations of onverting ylinder pressure to delivered torque by the ranshaft are presented. In addition the mean value of alulated torque will be subtrated from the total torque in order to find the flutuating part, Applying fore on the piston (F p ) in Figure 3-1, Cran mehanism = Gas pressure {p(t)} * piston area ( A p ). Original figure Zoomed version Figure 3-8, 1 seonds pressure reording from ylinder 1 Gas torque {Tg(t)}= F p * dx p /dφ where ran angle φ =ωt and ω is the onstant ranshaft speed, furthermore x p denotes the piston displaement Aording to Figure 3-1, Cran mehanism, it is possible to derive the expression for dx p /dθ [21], In the above figure, r is the ran radius and l is the onneting rod length (these values have been provided in Table 3-1). 11

30 Chapter 3 Desription of driveline and torsional phenomena Piston displaement in terms of ran angle an be estimated 1 in the following form: 3.1 Therefore, the differentiation of with respet to is 2 : 3.2 Original figure Zoomed version Figure 3-9, 1 seonds pressure reording from ylinder 2 1 The exat expression is available in referene [23] [21] 2 It should be noted that 12

31 Chapter 3 Desription of driveline and torsional phenomena Figure 3-1, Cran mehanism Finally, the assoiated torque due to ombustion, is Figure 3-11 and Figure 3-12 show the total torque outputs (in 1 seonds and in one woring yle as well) and their flutuating parts from ylinders 1 and 2 respetively. As it was expeted, the omputed torque in one woring yle is similar to what was demonstrated previously (Figure 3-5) for a four-ylinder four-stroe engine. In Figure 3-11 and Figure 3-12, different stroes are learly distinguishable. Cylinders 3 and 4 output torques are equal to ylinders 1 and 2 outputs, aording to the assumption whih was made before. The noises whih are seen in the plots are removable using filter ommands, the neessity of using filtration and the assoiated MATLAB ommands will be desribed more in detail in the next hapters. Finally to he the orretness of presented torque alulations from the measured output pressures of ylinders 1&2, it is funtional to find the mean value for summation of torque 1, torque 2, torque 3 and torque 4 {T g1 (t), T g2 (t), T g3 (t), and T g4 (t)}. We expet that this mean value have to be around the value whih was set during experiment, 1N.m: with the aid of MATLAB ommand mean (torque1+torque2+torque3 +torque 4) = , whih seems aeptable aording to 1 N.m

32 Chapter 3 Desription of driveline and torsional phenomena Figure 3-11, Torque output of ylinder 1, total and flutuating part, during 1 seonds and one woring yle 14

33 Chapter 3 Desription of driveline and torsional phenomena Figure 3-12, Torque output of ylinder 2, total and flutuating part, during 1 seonds and one woring yle It would be useful to plot ylinders output torques in one graph (Figure 3-13): T g1 (t), T g2 (t), T g3 (t), and T g4 (t). 15

34 Chapter 3 Desription of driveline and torsional phenomena Figure 3-13, Output torques from 4 ylinders in the same plot 1 The driveline system is subjeted to these input exitation torques whih are shown in Figure This figure represents ompat ranshaft model of a four-ylinder engine [3]. It should be noted that J d is the torsional damper and J f is the flywheel mass moments of inertia, respetively. Funtion of the flywheel is to derease the magnitude of angular aelerations produed by input exitation torques T g1 (t), T g2 (t), T g3 (t), and T g4 (t). 1 This diagram is similar to Figure

35 Chapter 3 Desription of driveline and torsional phenomena Figure 3-14, Compat ranshaft model for a four-ylinder engine Furthermore, to see the exitation (fored) frequenies 1 whih are assoiated to the above torques, the power spetral densities of the ylinder 1 and ylinder 2 time histories, in Figure 3-13, are obtained using MATLAB ommands and are demonstrated in the following figure. Figure 3-15, PSD for output torque from ylinder 1 1 fundamental frequeny (refer to Table 3-1)and its multipliations 17

36 Chapter 3 Desription of driveline and torsional phenomena Figure 3-16, PSD for output torque from ylinder 2 Aording to Figure 3-15 and Figure 3-16, as it was expeted from Table 3-1, Engine properties, the first (fundamental) frequeny is almost equal to 16.6 Hz (half engine speed 1 ) and the next exitation frequenies are 33.2, 5, 66.9, 83 Hz Cluth Assembly We have the luth system (Figure 3-17) after flywheel whih is made of two different omponents, luth dis and luth mehanism [22]: Figure 3-17, Cluth system [22] 1 Therefore as it will be seen in hapter 5, resonane happens when half engine speed or half multiple of engine speed is equal to one of the natural frequeny of the system 18

37 Chapter 3 Desription of driveline and torsional phenomena The major duties of the luth assembly are to join and disjoin the gearbox with the engine, to transmit engine power to the input shaft, and to supply isolation from the osillatory engine torque osillations. This funtion is ahieved by two mehanisms rotationally onneted by an elasti and dissipative system whih an rotate together (Figure 3-17, Cluth system), The first system is the luth dis and rings onneted to the flywheel, and the seond is the luth hub onneted to the input shaft via spline balash [22]. Two different woring onditions an be onsidered for luth: 1-luth behavior during the steady state running (linear ation) and 2- luth treatment during engagement (nonlinear phenomenon 1 ), however in this wor we simply model the luth system as an inertia dis together with the flywheel whih is onneted to the gearbox via a spring and a damper [3] and it is shown in the driveline overall model in setion Gearbox The third omponent in the driveline system is the gearbox whih onsists of various helial gears in order to provide the ability of hanging the speed ratio between the engine and driving wheels for driver of the ar. We have two major groups of the gearboxes: manual and automati. Sine the dynami model of gearbox mehanism is related to the purpose of the study and onerning the aim of this thesis, whih is analysis of the driveline torsional behavior, therefore modeling of gears and arrying shafts as a simple torsional vibratory system is the primary interest of this step of the report. The following mathematial model is suggested for the torsional vibration of driveline, where the model inluded an equivalent inertia dis for eah of the gear that transmits torque. it should be noted that the inertia of eah dis ontains also the inertia of the idling gears whih results to the redution of the driving gear speed. Figure 3-18, Gearbox model [3] Cardan (propeller) shaft and universal (Hooe s) joints Cardan shaft transmits the engine torque from the differential to the wheels. Sine the engine gearbox shaft, ardan shaft and ba axle are not in line, a universal joint, whih is shown in Figure 3-19, has to be used in order to attah them. The Hooe s joint suffer from one important problem: even when the input shaft has a onstant speed, the output shaft rotates at a variable speed. We now that veloity hange means aeleration and onerning Newton s law, aeleration results to fore. Therefore a seondary ouple will be reated and it is nonlinear. The magnitude of this produed torque is proportional to the torque 1 One of the most important purposes of torsional vibration of the driveline is during luth engagement in manual gear box mehanisms. The study of this topi is too omplex and beyond the goal of this report. 19

38 Chapter 3 Desription of driveline and torsional phenomena whih is applied on the driveline and the Hooe s joint angle, thus the variation of the driveline exitation torque will ause new torque 1 flutuation. Figure 3-19, Hooe's (ardan) joints [1] In order to find a simple model for propeller shaft and universal joints in the whole driveline system, we assume that the mass moment of inertia of the joints is muh larger than the propeller shaft, therefore the system is regarded as an elasti massless shaft (lie a spring) between two inertias [3]. Moreover, the generated torque by the joints will be applied on the two ends of the ardan shaft as it is shown in Figure 3-2, propeller shafts and universal joints mathematial model, Figure 3-2, propeller shafts and universal joints mathematial model [3] Differential and final drive system A differential is a mehanism in automobiles, usually but not neessarily inluding gears, whih has the ability of transmitting torque and rotation through three shafts, normally it reeives one input and provides two outputs. the differential also allows eah of the driving roadwheels to rotate at different speeds. Final drive system onsists of differential and two similar shafts whih are onneted to the wheels, and a simple model for that, is demonstarted in the following figure: 11 The derivation of the vibratory torque whih is generated by universal joints is ompletely desribed in [3]. 2

39 Chapter 3 Desription of driveline and torsional phenomena Figure 3-21, Final drive system and its equivalent model [3] Damping in the whole driveline system Damping is a resisting fore whih ats on the vibrating body and may arise from different soures suh as frition between dry sliding surfaes, frition between lubriated surfaes, air or fluid resistane, eletri damping, and internal frition due to imperfet elastiity. Regarding the damping type, the mathematial model is different and may depend on the veloity of the motion, material, visosity of the lubriant and et... The ases, in whih the frition fores are proportional to veloity, are named as visous damping. In the urrent driveline system, we will only onsider the effets of visous damping in different omponents 1 and other inds of damping are negleted. The equivalent visous torsional damping oeffiients are given in referene [3]. 3.3 Overall driveline model As it was desribed before, the overall torsional model for driveline system is based on disretisation and lumped masses are used. The suggested 14-degrees of freedom linear model for a four-ylinder rear-drive passenger ar is shown in Figure 3-22 whih omposed of inertia diss and massless torsional springs and visous dampers. 1 Cranshaft, engine, luth dis, gearbox, propeller shaft and differential units, and tires. 21

40 Chapter 3 Desription of driveline and torsional phenomena Figure 3-22, Damped torsional vibration mathematial model of driveline system [3] This model is used throughout this thesis in order to study the free and fored vibration of driveline and whole vehile. 3.4 Torsional vibration There are different exitation soures (linear and nonlinear) for the torsional vibration of the driveline model whih is shown in Figure 3-22, Damped torsional vibration mathematial model of driveline system. However as it was mentioned in setion 3.2.1, engine torque osillations 1 (linear behavior), is the main reason of torsional vibration. Studying the nonlinear purposes suh as Hooe s joints is beyond the sope of this thesis. It should be noted that torsional vibration is in primary interest sine firstly, it may ause harmful effets on the different parts of the system and seondly, it will be oupled with the whole body motions of the vehile and results to longitudinal vibration whih is investigated in the next hapters. 1 due to different stroes 22

41 Chapter 4 Undamped natural frequenies of the overall driveline system 4 Chapter 4 Undamped natural frequenies of the overall driveline system 4.1 Introdution This hapter inludes solving driveline differential equations of motion in order to find natural frequenies of the system. To avoid resonane, whih is a harmful phenomenon for mehanial systems, it is neessary to obtain natural frequenies of the struture. Modal analysis is used to study undamped model of the driveline system. 4.2 Mathematial model and system matries The governing differential equation for torsional vibration of the overall driveline system (14-degrees of freedom) whih is shown in Figure 3-22, is 4.1 where,, and are the symmetri mass moment of inertia 1, torsional damping, stiffness, and applying fore (engine flutuating torque) matries, respetively and finally is the 14-dimensional olumn vetor of generalized oordinates. In Table 4-1, parameter values are given for mass moment of inertia, stiffness and damping oeffiients of different omponents in driveline. In order to obtain the system matries for this multidegrees of freedom system, two methods have been desribed in vibration theory boos [21]: Newton s proedure and energy method, the loser one applies Newton s laws on the free body diagram of eah omponent and it is straightforward but time onsuming, while the energy method is based on Lagrange s equations and more pratial for large systems. In this thesis Newton s method is utilized and the inertia, stiffness and damping matries have been determined as follows: 1 Sine we have torsional vibration, M matrix is briefly alled inertia matrix 23

42 24 Chapter 4 Undamped natural frequenies of the overall driveline system Table 4-1, Typial values for equivalent parameters of a vehile driveline [3] Equivalent stiffness oeffiient (N/m or N.m/rad) Equivalent moment of inertia (g.m 2 ) Equivalent system damping oeffiient (N.s/m or N.m.s/rad) Parameter Value Parameter Value Parameter Value 1.2e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e K

43 25 Chapter 4 Undamped natural frequenies of the overall driveline system C J J J J J J J J J J J J J J M Now to find undamped natural frequenies of the system, the right hand side of equation 4.1 and damping matrix are set to be zero. it should be mentioned that, sine the damping matrix is not a linear ombination of the inertia and stiffness matries ( ), it is not possible to use modal analysis 1 to deouple the equations, therefore undamped natural frequenies are to be determined, however they are almost the same with damped frequenies whih are provided in referene [3]. 4.3 Summary of Modal analysis Modal analysis is a proedure to find the natural frequenies of the system by deoupling the system differential equations of the motion whih is given in equation 4.1. The problem is that when the equations are oupled, it is not possible to solve them separately at the same 1 whih is desribed in setion 4.3

44 Chapter 4 Undamped natural frequenies of the overall driveline system time. There are different types of oupling: stati oupling 1 and dynami oupling 2, aording to the mass and stiffness matries whih are already given, we have stati oupling in driveline system of equations. It is also useful to mention that the seletion of oordinate system influenes on the existene or nonexistene of the oupling. As it is nown from vibration theory [21], we an substitute 3 in the equations of motion (equation 4.1), and further we reah to the following expression: By removing the salar value, In order to solve equation 4.3, it is onverted to the form whih is the familiar form for Eigen-value problems. To do so, both sides of relation 4.3 are multiplied by the term from the left as follows: or and finally we obtain: in this ase and. Therefore the natural frequenies of the multidegrees of freedom system are the inverse of the positive square roots of the Eigen-values of matrix. It is possible to find these values by using eig ommand in MATLAB. In order to determine the orresponding mode shape 4 for eah frequeny, the following equation has to be solved: using these mode shapes, we an form the modal vetors matrix that is the base of modal equations 5 for modal analysis 6. however (as it was noted in previous setion) in the urrent system, this method of solution is not usable to attain the fored response of driveline mehanism sine the damping matrix is un-proportional and the above proedure do not deouple the equations whih are oupled by damping oeffiients. Although, in the next When the stiffness matrix is not diagonal 2 When the mass matrix is not diagonal 3 Sine we now that the response of the undamped vibratory system, x will be sinusoidal, therefore it an be shown by exponential form 4 finding the mode shapes of one mehanial system provides the information about the positions at whih large displaement will our and therefore, it would be possible to represent a solution to attenuate the harmful vibration 5 Modal equations are n independent relations for an n-degree of freedom system whih are solvable separately, the new oordinate are alled prinipal oordinates. 6 More desription is available in vibrations boo [21] 26

45 Chapter 4 Undamped natural frequenies of the overall driveline system hapter, fored response of the system is determined using both modal analysis and numerial method, and the preision of the solutions is studied aording to referene [3], in order to see the inauray of modal method and to evaluate the preision of numerial method. 4.4 Natural frequenies Resonane is a harmful phenomenon whih happens in mehanial systems, this results to failure of the mehanism and is very dangerous in the ase of passenger ars. As it is nown, resonane ours when natural frequenies of the system oinident with the fored frequenies, therefore in order to avoid this unwanted situation, it is neessary to now natural frequenies of the system (using the method of previous setion for a multi-degree of freedom system). Moreover we will find the mode shapes of driveline to be aware about the positions whih have onsiderable vibration amplitude. Among the undamped natural frequenies for torsional vibration of the whole driveline model Table 4-2, the first six frequenies are important while they are in operating range of the driveline system. Table 4-2, Undamped natural frequenies of whole driveline model using typial parameter values for a passenger ar Mode number Natural Frequeny (Hz) These values have a very good agreement with the damped natural frequenies for the system, whih are given in [3]- table 5.2. However there are fewer alulations in the ase of finding undamped frequenies for the vibratory systems. Further, the first five mode shapes of the system are obtained as in Table

46 Chapter 4 Undamped natural frequenies of the overall driveline system Table 4-3, first five natural modes of driveline system 1st Mode 2nd Mode 3rd Mode 4th Mode 5th Mode In addition, the seond mode only inludes the driving wheels vibration (with a ommon frequeny) and other omponents of the system are in rest, therefore it an be onluded that the 2nd mode has been never exited by any exitation torques. 28

47 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque 5 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque 5.1 Introdution In this hapter, fored response of driveline model subjet to the engine ylinder pressure input torques is determined, using the same system of differential equations as hapter4, however now exitation fores exist in the right hand side of the equations. Aording to the final solutions, the results of Runge-utta numerial method provide desired auray. 5.2 Mathematial model for fored vibration of the driveline system In order to obtain fored response of the linear damped system, again we onsider the main differential matrix equation for torsional vibration of the driveline mehanism (Figure 5-1): here the fore matrix is a olumn vetor as follows: 5.1 T ( t) T ( t) T ( t) T ( t) 1 where T g ( ), T g ( ), T g ( ) and ( ) 1 t 2 t g 1 g2 g3 g4 3 t T t g are engine flutuation torques from different ylinders whih are given in Figure the applying fores are not sinusoidal but periodi. Figure 5-1, Driveline model 29

48 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque 5.3 Time responses of driveline at luth and driving wheels Steady-state response of the equation 5.1 is found using 3 methods in this setion: Modal analysis as an analytial method, and 2 numerial proedures 1 : ode45 funtion in MATLAB and self-written fourth order Runge-Kutta method. Although the exitation fores are not exatly similar to referene [3] 2, there is a good agreement between the responses from numerial methods that have been utilized here and the results whih are given in [3], however the Modal analysis answer is not satisfatory. In the following paragraphs, a brief desription is provided for eah of the three solution methods: Modal analysis I. Deoupling the system of differential equations 5.1, by using Modal proedure in setion 4.3. In addition, the modal fores have to be obtained. II. Finding the fored response of eah independent equation using lsim 3 funtion in MATLAB, the assoiated ommand is given in Appendix 1.1. m(1,j),(1,j)and (1,j)are read from modal inertia, damping and stiffness matries respetively. The big assumption is that damping matrix will be deoupled through modal proess. III. The obtained solution is represented in Figure 5-2 for luth torsional displaement together with the solutions from the numerial methods. The output result from modal analysis is not orret in the amplitude value, and also it is unstable (whih is more lear in Figure 5-3). All the orresponding matlab odes are attahed in appendia ode 45 funtion I. MATLAB software inludes a series of funtions whih are alled solvers in order to solve ordinary differential equations of eah order, they use Runge-Kutta numerial method with variable time step to find the solution of the equations. ode 23, ode 45 and ode 113 are the most famous funtions of these solvers. In this thesis ode 45 is used to find the fored response of the 14-degrees of freedom driveline system. To do so, first it is neessary to onvert the equations of the motion to a set of first order differential equations (this form is named statevariable form or Cauhy form), this new form 4 is saved in a separate funtion. II. The seond step is to manipulate the fore data in order to use them as an input for the system. III. Finally the ommand [T,Y]=ode45(@(t,z)sde(t,z,t1,F),[,1],[zeros(14,1).zeros(14,1)])is written to find the solution for the system of equations in the desired time interval [,1] with zero intial postions and veloities, there are two problems that this method suffers from: a little unstability whih appears in Figure 5-2 and it is vey time onsuming 5. 1 However the base of both numerial methods is Runge-utta method. 2 He has used sinusoidal terms to simulate the engine torques and the solution is obtained using Modal analysis method. 3 Aording to MATLAB help, lsim is an useful funtion in this software whih is able to determine the vibratory system response to any arbitrary point data input as 4 28 first order differential equations 5 For 1 seonds, it too around 2 days to find the answer 3

49 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque Figure 5-2, Time response at the luth, using different methods of solution: Bla-> Modal analysis, Blue-> ODE45, Green-> self-written Runge-Kutta ode with nonzero initial onditions and yellow-> with zero initial onditions Figure 5-3, zoomed version of figure 5.2 in order to see the instability of modal analysis and ODE45 solutions Self-written Runge-Kutta ode I. In order to improve the alulations time, a MATLAB m-file is written to find the solutions for a system of fourteen seond order differential equations using numerial method whih is alled Runge-Kutta 1, in this wor the fourth order Runge-Kutta will be utilized whih means that the error per step is on the order of h 5, while the total aumulated error has order h 4. 1 From Wiipedia -> In numerial analysis, the Runge Kutta methods are an important family of impliit and expliit iterative methods for the approximation of solutions of ordinary differential equations. These tehniques were developed around 19 by the German mathematiians C. Runge and M.W. Kutta. 31

50 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque II. III. For this method we have to do the same manipulations lie the previous proedure to onvert all the 14 seond order equations to 28 first order equations (State-variable form). The final result is shown in Figure 5-2, whih has not only less instability 1 in omparison with ode 45 funtion solution, but also it taes a few minutes to obtain the answer whih is very valuable regarding the omputer program effiieny. Another point whih is important is that, aording to Figure 5-4, there is no differene in the response of the system, either we onsider zero intial veloities or when a non-zero matrix 2 is set to be as initial veloities. Figure 5-4, Time response at the luth with the aid of Runge-Kutta method 1 displaement vibrates around zero position whih is preferable 2 We are studying steady-state running and the fore data are not from a period of starting and stopping the engine, we have only pied 1 seonds of running engine. 32

51 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque In omparison with the luth response, whih is given in referene [3], smaller amplitude is ahieved here beause the engine exitation fore do not inlude the generated torque due to reiproating omponents, however the FFT plot ontains the same frequenies. Angular vibration of the driving wheels has also signifiant importane sine it may produe longitudinal vibration of the vehile body 1, therefore the displaement plots are presented in Figure 5-5 by using the last method. Figure 5-5, Time response at the driving wheels 1 this will be desribed more in hapter 6 33

52 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque As it was expeted the amplitude is smaller ompare to luth response, sine between luth and driving wheels there are number of omponents that redue and damp the effets of engine exitation torque, moreover the response plot is almost stable. 5.4 Power spetral densities of time histories We are always interested in the responses of the mehanial systems in frequeny domain in order to find harmful frequenies whih are within the operating range of the mahine. Power spetral density is a tool to obtain Fourier transform of random signal time histories [4]. After determining time responses of luth and driving wheels in the previous setion, it would be useful to plot PSD diagrams. we expet to see frequenies of the applying fore whih are different for different engine speeds (in this report for 2 RPM, aording to Figure 3-16, 16.6 Hz is the fundamental frequeny of the engine exitation torque and further we have 33.2, 5, 66.9, 83 Hz) sine the natural frequenies of the driveline are nown, it is obvious that high amplitudes will our if applying fore frequenies oinident with the natural frequenies of the system. There are different built-in funtions in MATLAB to attain power spetral densities of random signals. PSD funtion is used in this thesis whih is attahed in appendix Figure 5-6 and Figure 5-7 show the PSDs of luth response and driving wheels 1, respetively. Figure 5-6, Power spetral density of the time response at the luth As it was expeted all the foring frequenies have reated sharp peas in the frequeny domain of the system response. 1 In this setion only the PSD diagrams have been obtained for the time domain results from the Runge- Kutta numerial method. 34

53 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque Figure 5-7, Power spetral density of the time response at the driving wheels There are two important points that have to be mentioned at the end of this hapter: I. Sine the engine speed is not onstant in a ar, therefore the foring frequenies are not fixed and in addition, even at the onstant engine speed, there exist more than one exitation frequeny. as a result, the problem of ontrolling driveline system beomes ompliated and it is not possible to use passive ontrol method. II. On the other hand, while the response of the system is preditable, it is feasible to find an optimization tehnique to derease the unwanted effets as well as ahieving ative algorithm to ontrol the output vibrations. 35

54 Chapter 5 Steady-state Response of linear driveline model due to engine flutuating torque 36

55 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the 6 Chapter 6 Vibration of the whole vehile system due to torsional vibration of the driveline 6.1 Introdution The main goal of this thesis, as it was mentioned in setion 1.1, is to redue the effets of unwanted vibrations on the automobiles passengers. So far, we have studied the engine flutuation torques from different ylinders, as the main exitation soure of driveline, while failure of eah omponent of driveline system an result to a dangerous ourrene. furthermore it was noted that driveline vibration beside road surfae irregularities is the main reason of undesired noises inside the passenger ars, aordingly it is neessary to obtain the vehile body response due to driveline vibration in order to use the RMS aeleration of the vehile body (sprung mass of the ar) through ride omfort diagrams and study the drivability of a speifi ar. 6.2 Coupled vibration of driveline and the vehile body Torque flutuations at the driving axle will result to variation of the drive fores at the ground and therefore may generate longitudinal vibrations in the vehile. Hene torsional vibration of the driveline is oupled with the vibrations of the vehile body. In order to ahieve the response of sprung mass subjet to engine exitations, first the whole vehile should be modeled whih is the aim of the following setions. 6.3 Tire model and longitudinal fore The whole vehile model is divided into two main parts: driveline part whih onsists of all the omponents between engine and differential and tire-suspension-body system part. This setion is devoted to tire modeling. One powertrain omponent whih is, most of the time, simplified in both old and new torsional models of the automobiles, is the influene of the tires. However tires are the most important element in the quest to get a ar to handle well, sine they are the only lin between the vehile and the ground. By the efforts of Paeja [23], Delft University of Tehnology has signifiantly ontributed to tire researh. In this thesis, Paeja priniples are used to gain a model for tire whih onsists of longitudinal and vertial stiffness and damping oeffiients, two different inertias are taen into aount in order to have more aurate model: one for the wheel and the other for the tire tread bands [3], all the neessary properties for the alulations are given in Table 6-1. It should be noted that, an ideal tire/wheel assembly is onsidered here, although pratially the imperfetions in the manufaturing of tires, wheels, hubs, braes and other parts of the rotating assembly, may ause to three main groups of irregularities: mass imbalane, dimensional variations and stiffness variations whih are more desribed in referene [1]. Using the 2-degrees of freedom model for tire in Figure 6-1, aording to Paeja formula, the total longitudinal fore relation is found to be: 37

56 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the 6.1 where is the oeffiient of tire rolling resistane whih is available in the database, and all other parameters are demonstrated in Figure 6-1, the orresponding values are provided in Table 6-1. The above expression shows that if the tire has an osillatory rotational/vertial movement, then the longitudinal tire/road ontat fore will be various whih result to longitudinal vibration of the vehile body and this is not desired in a large sale aording to ride omfort riteria. Figure 6-1, tire model [3] degrees of freedom system for whole vehile model and its equations of motion In order to ahieve an overall model inluding torsional vibration of driveline oupling with tire, suspension and vehile body motion, first we onsider the substruture whih inludes sprung (vehile body) and unsprung masses, and respetively (Figure 6-2), moreover it is assumed that vehile body motion is limited in longitudinal and vertial diretions by suspension system harateristis. The seond substruture as it was already explained is driveline from torsional damper to differential (12-degrees of freedom) and it is shown in Figure 6-2 as well. 38

57 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Driveline substruture (12-degrees of freedom) Tire, suspension and vehile body substruture (6-degrees of freedom) Figure 6-2, overall vehile model [3] Aording to the above figure, two substrutures are related to eah other by a torque named whih is equal to: furthermore, while it is supposed that two wheels have the same movements in Figure 6-2, it would be reasonable to regard them as single wheel. Now we follow the same Newton proedure as setion 4.2 in order to attain differential equations of motion for the overall 18-degrees of freedom system. Mass, stiffness and damping matries in equation 6.3 are given in equations 6.4, 6.5 and 6.6 respetively. 6.2 where

58 4 Chapter 6 Vibration of the whole vehile system due to torsional vibration of the driveline = A R R A R A R r r r, where =

59 41 Chapter 6 Vibration of the whole vehile system due to torsional vibration of the driveline = = A R R A R A R r r r and finally the stiffness matrix is defined as follows: where = J J J J J J J J J J J J M 6.6

60 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the = = J t J J t 13 J J t t m u m u m b mb It is useful to denote that and matries in the above expression ontain terms assoiated to tire/road ontat fore whih has been obtained already. 6.5 Time response of the system This setion is devoted to solve the differential equations of 18-degrees of freedom system subjet to engine flutuation torques. In order to avoid nonlinearity we ignore the effets of Hoo s joints and we only onsider the exitation torques from the engine. Also, steadystate running is regarded, and studying the system behavior during luth engagement (transient running) is beyond the aim of this report. In ontrast with the differential equations for the 14-degrees of freedom driveline mehanism whih was solved in setion 5.3, the urrent stiffness and damping matries are not symmetri, as a result finding the time response of the system is more diffiult and time onsumable. Considering the exitation torques data from Figure 3-13, Output torques from 4 ylinders in the same plot, again the solution of the system has been obtained via fourth order Runge-utta method and by substituting the given data in Table 6-1 [3] in equation 6.1 and in its assoiated matries. 42

61 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Equivalent stiffness oeffiient (N/m or N.m/rad) Table 6-1, Overall vehile properties [3] Equivalent moment of inertia (g or g.m 2 ) Equivalent system damping oeffiient (N.s/m or N.m.s/rad) Parameter Value Parameter Value Parameter Value 1.2e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e6 J e4 J e6 J e6 J t e8 m u 18 (for both wheels) e4 m b Typial tire radius in a normal passenger ar 3 m rolling frition oeffiient The fore resisting the motion when a body rolls on a surfae is alled the rolling resistane or rolling frition, some typial rolling oeffiient are provided in In order to redue the omputer efforts and upoming diffiulties due to instability for solving the seond order differential equations of 18-degrees of freedom vehile system, the ylinders pressures embedded noises are filtered 1 before onverting to torques and onsequently applying as an input to the system, the assoiated MATLAB ommands are attahed in appendix , ut-off frequeny is hosen depending on the degree of filtration and 1 Hz is suitable for our study, the filtered version of the torques time histories, whih were shown in Figure 3-13, are demonstrated in Figure 6-3, 1 We used an order n low-pass digital Butterworth filter with normalized utoff frequeny ω n. Butterworth filters sarifie roll-off steepness for monotoniity in the pass- and stop-bands. 43

62 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Figure 6-3, Total engine exitation torques after applying filtration It is lear that the mean value of the torques summation should not hange after filtration and remain around the experiment set value (1 N.m). Here we have N.m whih is meaningful. In the following plots, torsional vibration of driving wheels and longitudinal osillations of the vehile body and axle are represented for the nominal values whih are given in Table 6-1. Zero initial displaements and veloities are used to determine the solution of equations. Figure 6-4, torsional veloity vibration of driving wheels due to engine exitation torques by using table 6.1 data values 44

63 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Figure 6-5, zoomed version of Figure 6-4 between seonds 8 to 9 The result beame stable in the seond five seonds. To have a better loo one seond has pied out and zoomed in Figure 6-5. Aording to Figure 6-5, there is a good agreement between the amplitude of torsional veloity in our result and referene [3] (Fig 8.5), however due to different input fore, the frequeny ontent is dissimilar. As it was earlier denoted number of times, vehile body vibration in every diretion (sprung-mass) is very important in the level of ride omfort and we need vehile body aleration to go through drivability diagrams, hene firstly it is neessary to obtain the plots in Figure 6-6: Figure 6-6, Longitudinal veloity vibration of the vehile body and axle due to engine exitation torques by using table 6-1 data values 45

64 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the From the above figure, it taes onsiderable seonds for ahieving stable results. Huge amount of input data to the system and thirty six differential equations that have to be proessed at the same time is one reason, the last three seonds is almost stable whih is zoomed in Figure 6-7. Aordingly fourth order Runge-utta approah is not that powerful to solve this system and as a future wor another preditor orretor numerial method has to be used to study this ind of systems. Again the results agree with referene [3]. Figure 6-7, zoomed version of Figure Studying the influene of stiffness and damping oeffiients From manufaturing view, sensitivity analysis whih studies the effets of different parameter values has a signifiant importane. Moreover there are a variety of strutural optimization tehniques to find the optimum value of eah parameter in any ind of system, this an be regarded as a future wor for this thesis, here we will only hange stiffness and damping oeffiients and resolve the equations of the vehile system to see the influene on the output results. In Figure 6-8 and Figure 6-9, damping oeffiients of the system are dereased notably whih result to great higher amplitude in torsional and longitudinal veloities of driving wheels and vehile body, respetively. Low damping may also generate instability in system output response sine the dissipated energy is not enough in omparison to the added nonlinear longitudinal fore, 46

65 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Figure 6-8, torsional veloity of driving wheels with low damping 47

66 Chapter 6 driveline Vibration of the whole vehile system due to torsional vibration of the Figure 6-9, Longitudinal veloities of vehile body and axle with low damping Figure 6-1, Longitudinal veloities of vehile body and axle with low suspension system stiffness It should be noted that the solution method may not be enough powerful to show aurate sensitivity to the data value hange. In Figure 6-1, Longitudinal veloities of vehile body and axle with low suspension system stiffness the suspension system stiffness value has been redued to see the influene on the longitudinal vibration of the vehile body and axle. Aording to Figure 6-1, dereasing suspension system longitudinal stiffness may ause to vibration inrease of the vehile body (sprung mass) subjet to engine exitation torques, whih is not desired. On the other hand, appropriate suspension system vertial stiffness oeffiient, as it will be seen in the next hapter, when the input exitation is from the ground 1, depends on the range of fore frequeny as well, and onsequently high or low vertial stiffness may be required. 1 road surfae non-uniformities 48

67 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model 7 Chapter 7 Vehile suspension system response due to ground input using quarter-ar model 7.1 Introdution The results of previous hapter and this one will be used in ride omfort diagrams to study the level of indued disomfort due to engine osillatory torques and road irregularities. 7.2 Quarter-ar model and performane of suspension system In order to evaluate ride quality of normal passenger ars, it is neessary to onsider the possibility for different inds of vibration whih may our in vehile body (sprung mass) 1 system as well as front and rear wheels (unsprung mass) 2 mehanisms. it is useful to note that aerodynami, driveline and engine fores are applied to the sprung mass, however ground non-uniformities input exitation is applied to the tire and onsequently suspension system. For the 18-degrees of freedom overall vehile model in setion 6.4, longitudinal and vertial vibrations of the sprung an un-sprung masses were studied due to engine exitations torques sine the goal was investigating the oupled behavior of the driveline torsional vibration and tire/suspension system. The aim of this hapter is looing more speifially into suspension system response subjet to the ground input and finding the effets of raising/reduing stiffness and damping oeffiients to have better performane of suspension mehanisms in automobiles. a simple two-degrees of freedom quarter ar model whih is suitable for our study is represented in. In this model, sprung and unsprung masses are denoted respetively by m 2 and m 1 while all the orresponding parameter values are given in Table 7-1, using Table 6-1, Overall vehile properties, Figure 7-1, Two-degrees of freedom model vehile 1 pith, boune and roll 2 boune and roll 49

68 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model Table 7-1, Tire/suspension properties [3] Vehile body mass (m 2 ) : sprung mass = 18/4 g (quarter of the whole body mass) Wheel (axle) mass : Unsprung mass = 9 g Suspension Tire Stiffness : (N/m) 9e6 Stiffness : (N/m) 4e5 Damping : (N.s/m) 4 Damping : (N.s/m) 6 the equations of motion for the above system in vertial diretion are found aording to Newton s law and free body diagram of the separate masses (Figure 7-1): The undamped natural frequenies of the quarter ar model for sprung (vehile body) and un-sprung masses (wheels) are as follows (Table 7-1): as it is seen, there is a wide differene between natural frequenies of the vehile body and tire/wheels assembly, therefore in the ase of high exitation frequeny (suh as the input impulse by a bumpy road), aording to Figure 7-2, sine the frequenies ratio (f exitation /f natural ) sprung mass is high, transmissibility will be very low and onsequently we will ahieve desired vibration isolation for vehile body with the aid of suspension mehanism. On the other hand, when the exitation frequeny is low and near to vehile body natural frequeny (in the situation of traveling over an undulating surfae), the transmitted fore is equal to the input or even ould be amplified, and hene a onventional suspension system with fixed properties has not good performane in this ase. This is the reason for a huge amount of researh ativities in the field of ative suspension system 1 (in ontrast to passive system) whih has variable stiffness and damping 2 regarding to the input frequeny. 1 Hydrauli systems whih is now available in the new vehiles to have ative suspension system 2 In order to attain desired high or low natural frequeny, stiff or soft properties are required for the suspension mehanism 5

69 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model Figure 7-2, Transmissibility as a funtion of frequeny ratio for a single-degree of freedom system As it was already nown, vertial aeleration of vehile body is important in ride omfort diagrams and it is related to the level of isolation by the suspension system whih was desribed preisely in previous paragraph, however the passenger will not feel vehile body motion but rather the displaement of his seat, thus we use the model suggested in referene [2] in this thesis and it is shown in Figure 7-3, the neessary steps to find orresponding transfer funtions for obtaining output responses are explained as follows: Figure 7-3, Modified quarter-ar model inluding seat displaement first, it is neessary to onvert the equations 7.1 and 7.2 to Laplae domain: now the only unnowns are and, by applying Cramer s rule and substituting s=i in equations 7.5 and 7.6, we have: 51

70 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model there are different versions of transfer funtions :Reeptane, Mobility and Aelerane, expressions 7.7 and 7.8 are reeptanes, now to determine the mobilities and aeleranes we have the following onverting relations: obility 7.9 elerane 7.1 now aording to random proess theory [24], for a stationary 1 random input (suh as Gaussian funtion whih will be later onsidered as ground exitation for the suspension mehanism in ) in the ase of linear system, there is a well-nown formula in order to attain power spetral density of output response by using power spetral density (PSD) of input and the appropriate transfer funtion among the above three: where is the PSD of desired system point displaement, veloity or aeleration, is the orresponding transfer funtion, and is the PSD of input fore, Using the same approah, the omplex transfer funtion between and is: 7.11 where from referene [2] for a normal passenger ar, we have rad se and onsequently we arrive into: and One important point is that: we annot hange the suspension system stiffness and damping oeffiients, to reah desired vibration isolation, without taing into onsideration two other aspets of suspension mehanism whih are important in its performane: road holding 2 and suspension woring spae 3, these parameters are to be onsidered as Stationary random proess: for this type of random data, mean value and variane are onstant and independent of time 2 it is important for a safe ride that the ontat fores between the wheels and the road are so large that horizontal fores on the vehile an be balaned by fritional fores at the wheels [2] 3 pratially, the woring spae of vehile suspension system is limited [2] 52

71 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model onstraints in the problem of finding optimum values of suspension damping and stiffness whih an be studied in a future wor. 7.3 Road roughness lassifiation by ISO and the reommended singlesided vertial amplitude power spetral density As it was already explained, in setion 1.3, by assuming the availability of three onditions, we an use the road surfae profile whih is reommended by ISO in a form of single-sided power spetral density as follows [4]: where is a fixed datum spatial frequeny equal to 1 (2π) yles m and is attained aording to road roughness lassifiation proposed by ISO whih is also represented in Table 7-2 using referene [4]. In the analysis of vehile vibration, it is more pratial to wor with temporal frequeny in Hz rather than spatial frequeny in yles/m. if the ar travels with a onstant speed v (m/s), then where, aordingly we reah to: Table 7-2, Classifiation of road roughness proposed by ISO [4] Road Class Degree of roughness, 1-6 m 2 /yles/m Range Geometri mean A (very good) <8 4 B (good) C (average) D (poor) E (very poor) F G H >

72 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model 7.4 Typial passenger ar driver RMS aeleration to an average road roughness We are interested in RMS vertial aeleration of the ar seat whih is obtainable by nowing power spetral density of the seat aeleration and using the following formula 1 : 7.1 where is found from equation 7.13 by substituting, we assume that the travelling veloity of the vehile is 8 Km/hr (22.22 m/s) and thus the value of the f =vn = Hz. Figure 7-4 demonstaretes RMS aeleration (m/s 2 ) graph of a typial passenger ar seat in a ertain frequeny range, an average road roughness lass is pied, therefore *1-6 m 2 /yles/m. Figure 7-4, Measured vertial aeleration of a passenger ar seat traveling at 8 Km/hr over an average road As it is seen in Figure 7-4, the pea value of the ar seat RMS aeleration plot happens around frequeny 2 Hz whih is the natural frequeny of the sprung/vehile body mass (first mode frequeny), hene it an be onluded that one of the road surfae input exitation frequenies is in the region of 2 Hz. further in order to ontrol the system response amplitude for this lass of road roughness and avoid high aeleration, the natural frequeny of the suspension system have to be regularized 2 with respet to input fore frequeny by taing into aount road holding and suspension woring spae limitations at the same time. Now, for heing the solution method 3 auray, the effets of passenger seat is ignored in equation 7.13 and the represented model in is applied to obtain the vehile body vertial aeleration power spetral density subjet to surfae irregularities. In this ase, it is possible to ompare the result with the plot whih is given in referene [4]. Figure 7-5 illustrates the 1 f is the enter frequeny, and the RMS value is alulated in one-third otave band, it is neessary to aumulate the spetrum between the lower and upper bands [24]. 2 by hanging stiffness or damping oeffiient in an ative suspension system 3 the proedure of finding power spetral density of the output 54

73 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model sprung/vehile body mass RMS vertial aeleration in similar onditions with Figure 7-4. The plot shape has a good agreement with Fig in referene [4], however the aeleration amplitude is somehow greater due to different road surfae onditions and suspension system harateristis. In ontrast to Figure 7-4, in Figure 7-5 two peas exist: one at the sprung mass natural frequeny around 2 Hz (1st modal frequeny) and the seond one at unsprung mass natural frequeny around 1 Hz (2nd modal frequeny). The reason for this phenomenon is that if we inlude passenger seat mass in the model, it will perform as a absorber whih is a very important subjet in vibration theory, in other words passenger seat mass has absorbed the movement of unsprung mass in the region of resonane frequeny and onsequently no pea will our in the 2nd modal frequeny around 1 Hz 1.. Figure 7-5, vehile body vertial aeleration subjet to an average road roughness with 8 Km/hr traveling speed 1 Although this funtion for the absorber is strongly related to the mass ratio and if it was not in the appropriate region, then absorber has negative influene whih happened in the urrent system for the 1st modal frequeny (2Hz) and the amplitude of aeleration is greater in Figure 7-4 ompare to Figure

74 Vehile suspension system response due to ground input using quarter- Chapter 7 ar model 56

75 Chapter 8 riteria Evaluation of typial passenger ar omfort with respet to ride quality 8 Chapter 8 Evaluation of typial passenger ar omfort with respet to ride quality riteria 8.1 Introdution This hapter is devoted to use the proposed results in Figure 7-4 and Figure 6-6 in ride quality diagrams for evaluating drivability of the typial passenger ar with the given overall vehile properties in Table 6-1, Overall vehile properties. 8.2 International Standard ISO :1985 Aording to ISO :1985 [19], four physial fators have signifiant effets on human response to applied vibration: the strength (power), frequeny, diretion and interval of exposure, there are also three different issues whih are important to evaluate the human reation to the vibratory displaements [3]: preservation of woring effiieny preservation of health or safety preservation of omfort In Figure 8-1 and Figure 8-2, the fatigue-dereased profiieny boundaries for various exposure times are represented in vertial (along the z 2 - axis in ) and longitudinal (along the x 2 - axis in ) diretions, respetively. As it is lear in the proposed diagrams, the omfort boundary will derease by rising the vibration duration. It should be mentioned that also the human body is more sensitive to vibration in some frequeny ranges than in others, for example aording to the following figures, for vertial vibration the ritial frequeny region is 4 to 8 Hz while for longitudinal vibration this frequeny area is less than 2 Hz. Center frequeny of one-third otave band Figure 8-1, ISO :1985 "fatigue-dereased profiieny boundary": vertial aeleration limits as a funtion of frequeny and exposure time [4] 57

76 Chapter 8 riteria Evaluation of typial passenger ar omfort with respet to ride quality Center frequeny of one-third otave band Figure 8-2, ISO :1985 "fatigue-dereased profiieny boundary": longitudinal aeleration limits as a funtion of frequeny and exposure time [4] 8.3 Results and Disussion This setion of the report ontains appropriate figures in order to evaluate ride omfort of a typial passenger ar 1 at a ertain frequeny interval by using the sprung mass responses 2 and the ISO riteria whih are shown in the previous setion. Measured vertial RMS aeleration of the vehile body, with 8 Km/hr traveling speed on an average road roughness, is shown again in Figure 8-3 but together with ISO fatiguedereased boundaries to investigate the level of omfort for this speifi passenger ar, Figure 8-3, vehile body vertial aeleration due to road exitation in omparison with ISO ride omfort boundaries 1 with the given properties in Table whih have been obtained already for different diretions in hapters 6 and 7 (Figure 6-6, Figure 7-4 and Figure 7-5) subjet to engine and road irregularities exitations. 58