ON THE FREQUENCY DOMAIN ANALYSIS OF TIRE RELAXATION EFFECTS ON TRANSIENT ON-CENTER VEHICLE HANDLING PERFORMANCE

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1 Poceedings o the ASME 9 Intenational Mechanical Engineeing ongess & Exposition IMEE9 Novembe 3-9, Lake Buena ista, Floida, USA Poceedings o the ASME 9 Intenational Mechanical Engineeing ongess & Exposition IMEE9-643 Novembe 3-9, Lake Buena ista, Floida, USA IMEE9-643 ON THE FREQUENY DOMAIN ANALYSIS OF TIRE RELAXATION EFFETS ON TRANSIENT ON-ENTER EHILE HANDLING PERFORMANE Justin Sill lemson Univesity- Intenational ente o Automotive Reseach Geenville, S, US ABSTRAT This pape pesents an elegant equency domain appoach that can be used to analyze lateal vehicle dynamics o tansient undestee and ovestee peomance. ommonly used steady-state undestee analysis techniques ae not able to expose some eects, such as tie elaxation, in on-cente tansient maneuves. The appoach pesented hee addesses such tansient issues using a simple two degee o eedom handling model coupled to a model o tie lateal dynamics. In addition to the usual yaw ate and lateal acceleation tanse unctions, this pape poposes using an undestee angle tanse unction as an easy-to-intepet metic to evaluate tansient oncente handling. Using the appoach, it is shown that at low vehicle velocities, the inclusion o tie elaxation intoduces damatically dieent system dynamics by intoducing highly undamped poles into the coupled system o both an undesteeing and an ovesteeing vehicle. ey Wods: tansient undestee, tie elaxation, on-cente handling, tansient handling, vehicle tanse unctions, undestee angle NOMENLATURE Font Tie Slip Angle Rea Tie Slip Angle a Distance om G to Font Axle b Distance om G to Rea Axle A y Lateal Acceleation β Side Slip L ehicle Wheelbase Font Tie Relaxation Length Rea Tie Relaxation Length Font Axle oneing Stiness Rea Axle oneing Stiness J z ehicle Yaw Inetia y Font Tie Lateal Stiness Rea Tie Lateal Stiness Beshah Ayalew lemson Univesity-Intenational ente o Automotive Reseach Geenville, S, US m ehicle Mass Yaw Rate ehicle Fowad elocity Road Wheel Stee Angle Undestee Angle INTRODUTION The lateal stability o vehicles has long been analyzed though simple analytical methods. The dominating appoach has been an undestee analysis which can chaacteize the vehicle s steady-state tendency to undestee o ovestee within the linea handling ange [-6]. The taditional steady-state undestee analysis techniques can aide in the design o the vehicle, the selection o ties, and design o suspension geomety and components. Howeve, these methods exclude tansient eects typical o on-cente diving, i.e, those chaacteized by small steeing petubations om staight ahead diving. The ealm o on-cente handling has become inceasingly impotant because it typiies the most common diving conditions that have low lateal acceleation levels (lowe than. Gs). In such on-cente maneuves, tie elaxation eects play an impotant ole [7], paticulaly o lowe vehicle velocities. To chaacteize the on-cente tansient peomance o a vehicle thee have been some expeimental studies elating the yaw ate and lateal acceleation esponses to steeing input in ode to detemine elative undestee/ ovestee changes [8, 9]. Such expeimental methods have been successully used to study the behavio o the entie system, but cannot be used to isolate the eect o tie dynamics on the vehicle peomance. This pape analyzes the tansient behavio o on-cente handling including tie dynamics. This is done using an elegant equency domain appoach applied to a educed linea single tack model o a two-axle vehicle. In paticula, the pape consides the eect o dieent pevailing lateal tie-elaxation lengths on the ont and ea ties o a vehicle on its tansient undestee peomance. opyight 9 by ASME opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

2 The est o the pape is oganized as ollows. The ist section gives a backgound on the simpliied vehicle handling model without tie-elaxation eects. This is then ollowed by the deivation o the equations o the case with tie-elaxation eects. The models ae then simulated mainly in the equency domain and a discussion is given on two dieent cases o a nominally undesteeing and ovesteeing vehicle. Finally, the obsevations o the study ae summaized in the concluding session o the pape. BAGROUND The Handling Model. The ee-body diagam o the widely used handling (bicycle) model is shown in Figue [,, 4-6, -3]. Figue. Model Fee-Body Diagam The tie oces ae deined by the coneing stiness and the espective tie slip angles, and,which in tun ae unctions o vehicle side slip angle, β, yaw ate,, and oad wheel steeing angle, (see, o example, [4]). F F () y () y F y F y a β + (3) b β (4) With the side slip angle and the yaw ate as the states o vehicle and with the ont oad wheel steeing angle as input, the equations o motion educe to: + a + b β m m β m () + a a + b a + b J J z J z z The chaacteistic equation o this system is given by: a b L s mj z s ( ) z ( ) + m a b J s + L + m b a The lateal acceleation and yaw ate tanse unctions ae detemined to be: Ay [ J z ] s + [ bl] s + [ L] (7) s [ am ] s + [ L] ( s ) ( s) These tanse unctions can be analyzed to detemine a vehicle s handling stability and peomance. Undestee Gadient. In steady-state coneing maneuves, the above simple model can uthe be educed to the detemination o the undestee coeicient o gadient, us. This metic is widely used in detemining the ovesteeing and undesteeing tendency o vehicles as well as the calculation o citical and chaacteistic velocities [, 4-6, ]. c us m b a L Ay gl () us The deinition o an undesteeing and ovesteeing vehicle can be detemined om Equation (9). An undesteeing vehicle has a lage ont tie slip angle than that o the ea leading to a positive value o the undestee gadient. onvesely, an ovesteeing vehicle has a lage ea tie slip angle than that o the ont leading to a negative value o undestee gadient. Expeimental Methods. Thee exist seveal test methods to quantiy tansient handling peomance o a vehicle. These methods include maneuves with stee inputs chaacteized by a step, a sinusoid [4], as well as andom o sine sweeps whee all equency anges ae exploed [8, 9]. The most used expeimental method involves the analysis o a sine sweep maneuve. This maneuve is completed at constant speed and stee amplitude ove a ange o possible dive input equencies (appoximately to 4 Hz). The time histoies o stee angle, yaw ate, and lateal acceleation ae pocessed though FFTs (Fast Fouie Tansoms) to yield the vehicle s equency domain esponse. This pocessed data is used to identiy ou metics including the vehicle s steady-state yaw ate gain, yaw ate natual equency, yaw ate damping, and lateal acceleation phase delay at one hetz o stee excitation. These metics ae plotted in a hombus plot (also known as spide chat) as shown in Figue. This is then used to compae elative dieences o vehicles o coniguations. The tendency o a vehicle, as compaed to some baseline (Q), to (6) (8) (9) opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

3 be moe esponsive (P) o moe stable (R) can easily be visualized. Yaw SS Gain (heading easiness) Yaw ate SS gain (heading easiness) P Q R ovestee (moe esponsive) Ovestee (moe esponsive) Yaw NF (heading esponsiveness) Yaw Damping (diectional damping) undestee (moe stable) Lat Acc Hz (ollowing contollability) Figue. Fou Paamete Evaluation Method o Lateal Tansient Response [8] The hombus plot technique is a poweul tool o analyzing objective test data to detemine dieences in tansient undestee peomance o on-cente handling and has inspied the equency domain analysis consideed in this pape. The method adopted hee enhances the above appoach by demonstating the use o a thid tanse unction, the undestee angle tanse unction. As will be shown below, tends in this tanse unction appea easie to intepet. DERIATION OF EQUATIONS The above handling model (lateal & yaw vehicle motions) can be used in conjunction with a ist ode tansient tie model. Fom Figue, the basic equations o motion can be witten in the om: ( β + ) y + y m F F J af bf z y y () () The tie tansient behavio is modeled by a ist ode dynamic system as []: τ Fy + Fy τ Fy + Fy (3) (4) Whee, the tie time constants τ and τ ae deined by the elaxation length and velocity as [6]: Yaw ate natual equency (Heading esponsiveness) Yaw Damping (diectional Undestee (moe stable) Lat. Acc. Hz (ollowing contollability) τ () τ (6) This deinition o tie tansient behavio was used o simplicity though it should be noted that thee ae many ways in which to account o tie dynamics including highe ode models [4, 7, 7-9]. It should also be noted that when the tie time constants ae zeo the ties ae epesented by only coneing stinesses and slip angles thus educing this vehicle model to the taditional steady-state coneing handling model descibed above. Ate substitutions o the time constants and slip angles, the tie oce equations can be expessed as shown below: a Fy F y β + (7) b Fy F y β (8) The complete model can then be epesented in state space om with the side slip angle, yaw ate, ont axle lateal oce, and ea axle lateal oce as state vaiables. m m β a b β J z J z (9) F y a + Fy F y F y b The outputs o inteest include the vehicle s side slip, yaw ate, lateal acceleation, and the undestee angle. The lateal acceleation and undestee angle ae deined by: Fy + Fy Ay m () a b L UA β + β () The undestee angle was deived om the subtaction o the ont and ea tie slips angles which in tun ae unctions o the vehicle side slip angle and location o the vehicle cente o gavity as given by Equations (3) and (4). It is inteesting to note that the deived undestee angle is elated to the yaw ate by the constant atio o wheelbase to velocity and a shit elated to the stee input. The above output equations as well as the side slip and yaw ate ae expessed in tems o state-space output matices as: β β + () A F y m m y UA L F y 3 opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

4 The tanse unctions o the vehicle s lateal acceleation, yaw ate, and undestee angle with espect to stee angle input can be ound om this state-space epesentation o the model. The chaacteistic equation o the system in Equation (9) is: 4 3 ( s) J zm s J zm( ) + + s ( ) z ( ) z ( ( ) ( )) z ( ) ( ) + m a b J mj s (3) + m a a + b b + + J + s + L + m b a The yaw ate, lateal acceleation, and undestee angle tanse unctions ae given, espectively, by: [ am ] s + [ am ] s + [ L] ( s ) (4) s [ 3 ] + [ ] + [ ] + [ ] ( s) A J s J s bl s L ( s) y z z UA s () 4 3 z z ( ) ( ) z ( ) ( ( ) ( )) z ( ) mj s + mj + s + mb a b J + + m s m a b ( s) + m a + b b + b J + s + (6) Fo compaison, the undestee angle with espect to stee angle tanse unction o the model without elaxation (without tie dynamics) is deined hee by: mj z s UA s ( ) z ( ) + mb a b J + s + m a b ( s) (7) It should be noted that when the elaxation length o the ont and ea ties ae set to zeo the model with elaxation lengths will educe to the vehicle model without elaxation. RESULTS & DISUSSION In this section, the tanse unctions deived above ae used to study the tansient esponse o the vehicle. An undesteeing as well as an ovesteeing vehicle with dieent ea tie coneing stinesses ae analyzed. The elevant values ae given in Table. Table. ehicle Paametes Paamete alue Units Undestee Ovestee ehicle Mass 8 kg Yaw Inetia 686 kg-m Wheelbase.7 m Weight Distibution 63/37 % Tie Lateal Stiness N/mm Font Tie oneing Stiness 4 N/deg Font Relaxation Length.7 m Rea Tie oneing Stiness N/deg Rea Relaxation Length m Undestee Gadient deg/g haacteistic/itical elocity 4 kph The elaxation lengths o the ont and ea ties wee calculated based on a tie lateal stiness o the ont and ea ties as well as the espective coneing stinesses. (8) y (9) y Note that the tie stinesses, y & y, ae the same values o both ont and ea axles. This assumption o the insensitivity o lateal stiness to load as compaed to coneing stiness (due to the dominant eect o stuctual tie components [6]), yields dieent elaxation lengths o the ont and ea axles o both vehicles, because o its dependency on coneing stiness in Equations 8 & 9. Undesteeing ehicle. The undestee vehicle, as deined in Table, was analyzed using the models developed above with and without tie elaxation. The yaw ate and lateal acceleation tanse unctions, deined in Equations (7, 8, 4, & ), o vaious vehicle speeds, including 3, 6, 9,, & kph, ae shown in Figues 3&4. Yawate/Stee (deg/s/deg) With Tie Relaxation Without Tie Relaxation Figue 3. Yaw Rate Tanse Function o Undesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) 4 opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

5 Lat. Accel/Stee (Gs/deg) With Tie Relaxation Without Tie Relaxation Figue 4. Lateal Acceleation Tanse Function o Undesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) As can be seen om the igues, the magnitude and phase delay o the model with and without tie elaxation does show some dieences. Fo low equencies, below. Hz, the models show simila esponse, howeve o highe equencies the phasing is damatically dieent. It is diicult to intepet these dieences and thei signiicance om the pespective o a dive. So it is poposed to use the unconventional undestee angle tanse unction, deined in Equations (6 & 7), to investigate the model peomance (See Figue ). Undestee Angle/Stee (deg/deg).. With Tie Relaxation Without Tie Relaxation vehicle side slip angle vanishes. The ea tie slip angle also appoaches zeo. This is evident om Equations (3) and (4). The eect o tie elaxation on the undestee angle o the vehicle is moe evident at low vehicle speeds, as can be seen by the dieence in the models at the lowest speed o 3 kph. As the velocity is inceased, the undestee esponse o the model including tie elaxation appoaches that o the model excluding tie elaxation. It can also be obseved that the undestee tanse unction shows a peak equency at low velocities that could be signiicant. This peak is located at 3-3. Hz at 3 kph, which lies in a equency ange in which a dive maybe sensitive. It can also be obseved that the model with elaxation length has values o undestee angle geate than one o equencies above Hz. This indicates an inceased measue o undestee when tie elaxation is included. The eect o velocity on tie elaxation can be uthe analyzed by computing the poles o the chaacteistic equations (6) and (3) o inceasing velocity. The loci o the poles o the model with and without tie elaxation ae shown in Figue 6. As expected o an undesteeing vehicle, the model without tie elaxation includes only two imaginay poles [4]. Howeve, the model with elaxation tansitions om ou imaginay poles to two imaginay and two eal ones as the vehicle velocity is inceased. Imaginay - - Model without Tie Relaxation Imaginay - - Model with Tie Relaxation Figue. Undestee Angle Tanse Function o Undesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) It can be obseved that the undestee angle appoaches a magnitude o one degee pe degee o steeing o high equencies. The low o nea zeo values o yaw ate at such equencies causes the undestee angle to appoach the steeing input. This is because, at high equencies, the undestee angle becomes a unction o pimaily the ont tie slip angle which in tun appoaches the stee angle as both the yaw ate and the Real Real Figue 6. Loci o the Poles o Undesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) The eect o elaxation length is educed as velocity inceases as shown by the convegence o the let and ight plots to simila imaginay poles. This is consistent with pevious studies showing that the eect o tie elaxation on vehicle handling is most impotant at low velocities [7]. The initial imaginay poles (nea zeo velocity) that ae on the imaginay axis o low speeds ae quite dieent om the poles o the model without tie elaxation. The chaacteistic opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

6 equation (3) with tie elaxation as the velocity appoaches zeo educes to: 4 ( ) J zm s + m( a b ) J z ( ) s (3) + L It can be shown that the oots o this equation ae pue imaginay o the data in Table veiying the above obsevation that at low speeds, tie-elaxation intoduces dominant oscillations. The impact o the system poles can uthe be seen in the vehicle s esponse to a step stee maneuve o one degee oad wheel angle at inceasing velocities o 3, 6, 9, & kph (Figue 7). Yaw ate (deg/s/deg) Lateal Acceleation (Gs) Undestee Angle (deg) Sideslip Angle (deg) W/ Relaxation W/O Relaxation -.. Figue 7. Undesteeing ehicle Response to Step Stee Input o Inceasing elocities (Aows) As can be seen om the esponse, the model with elaxation exhibits moe oscillation at low speeds compaed to the model without elaxation. Inteestingly, this oscillation causes the undestee angle to become negative o the speed o 3 kph, as the yaw ate oveshoots its steady-state value. Ovesteeing ehicle. The ovesteeing vehicle, as deined by the paametes in Table, was analyzed similaly. The yaw ate and lateal acceleation tanse unctions o a ew selected velocities (3, 6, 9, & kph) ae shown in Figues 8 & 9, espectively. Yawate/Stee (deg/s/deg) With Tie Relaxation Without Tie Relaxation Figue 8. Yaw ate Tanse Function o Ovesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) Lat. Accel/Stee (Gs/deg) With Tie Relaxation Without Tie Relaxation Figue 9. Lateal Acceleation Tanse Function o Ovesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) The yaw ate and lateal acceleation esponse looks simila to the pevious case with the exception o steady-state peomance. The yaw ate and lateal acceleation gain incease geatly as the velocity is inceased as expected o an ovesteeing vehicle [, 4, ]. The undestee angle tanse unction o the ovesteeing vehicle is shown in Figue. The steady-state undestee angle is 8 degees out-o-phase with espect to the steeing input. This is consistent with the expected negative value o undestee angle o an ovesteeing vehicle. 6 opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

7 Undestee Angle/Stee (deg/deg) With Tie Relaxation Without Tie Relaxation Figue. Undestee Angle Tanse Function o Ovesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) It is also notewothy that the asymptotes o magnitude and phase o the ovesteeing vehicle ae identical to that o the undesteeing vehicle in Figue. Theeoe, o the ovesteeing and undesteeing vehicles the undestee peomance o inceasing equencies o excitation convege to pue undestee as seen by an undestee angle o one degee pe degee o steeing. The loci o the poles o the ovesteeing vehicle ove a ange o inceasing velocities ae shown in Figue. Imaginay Model without Tie Relaxation Real Imaginay Model with Tie Relaxation Real Figue. Loci o Poles o Ovesteeing ehicle with and without Tie Relaxation o Inceasing elocity (Aows) The model without elaxation, including two negative eal poles, eaches instability when one o the poles cosses the imaginay axis. The loci o the poles o the model with elaxation pogessively move om ou imaginay poles to ou eal poles o inceasing velocity as shown by the aows in Figue. Accodingly, at low vehicle velocities thee exist two distinct equencies o oscillation. Similaly to the model without elaxation, one o these poles cosses the imaginay axis at a citical velocity inducing instability. Note that at high velocities, the two models convege to each othe meaning that tie elaxation does not damatically educe the citical velocity. The step stee esponse o the ovesteeing vehicle, showing the yaw ate, side slip angle, lateal acceleation, and undestee angle is shown in Figue. Yaw ate (deg/s/deg) Lateal Acceleation (Gs) Undestee Angle (deg) Sideslip Angle (deg) W/ Relaxation W/O Relaxation -. Figue. Ovesteeing ehicle Response to Step Stee Input o Inceasing elocities (Aows) The model with elaxation length again shows oscillation at lowe vehicle speeds, paticulaly on the undestee angle. The useulness o the undestee angle as a esponse is shown by how it eveals the most damatic dieences in peomance, especially at lowe vehicle speeds. ONLUSIONS In this wok, tansient undestee peomance o a vehicle was analyzed using a equency domain appoach. The eects o tie elaxation on tansient handling peomance is shown though an analysis o an undesteeing and ovesteeing vehicle using tanse unctions and complex plane locus o the system poles o the models. The ollowing obsevations wee made om the analysis: Relaxation length poduces lage magnitude atios o the undestee angle tanse unction o both vehicles. Fo inceasing velocity, a locus o the poles analysis shows that the poles o the model with tie elaxation convege to poles o the model without elaxation. In othe wods, the most damatic eects o tie elaxation ae appaent at low velocities, whee highly oscillatoy poles ae intoduced by tie elaxation o both ovesteeing and undesteeing vehicles. Relaxation length does not signiicantly aect the stability o citical velocity magnitude o the studied 7 opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use:

8 ovesteeing vehicle, due to the educed eect o elaxation at such a velocity (4 kph). Fo inceasing excitation equencies, an undesteeing and ovesteeing vehicle convege to pue undestee as measued by a one degee undestee angle pe degee o steeing. As expected, the steady-state peomance o a model with and without tie elaxation ae identical, i.e, in steady-state tie elaxation has no eect on the esponse o the vehicle. It is also poposed in this pape that the undestee angle tanse unction makes o an easy to intepet paamete o such on-cente analyses. REFERENES. Gillespie, T., Fundamentals o ehicle Dynamics. 99: Society o Automotive Enginees.. aogal, I. and B. Ayalew. Independent Toque Distibution Stategies o ehicle Stability ontol. in Wold ongess o the Society o Automotive Enginees. 9. SAE Pape No. 9A-7. Detoit, MI: SAE, Inc. 3. aogal, I., B. Ayalew, and H. Law. An Iteative Appoach o Steady State Handling Analysis o ehicles. in Poceedings o the ASME 8 Intenational Design Engineeing Technical oneences & opmputes and Inomation in Engineeing oneence. 8. ASME Pape No. DET8-33. New Yok, NY: ASME. 4. Genta, G., Moto ehicle Dynamics: Modeling and Simulation. Seies on Advances in Mathematics o Applied Sciences. ol , Singapoe: Wold Scientiic Publishing.. Milliken, W.M.a.D., Race a ehicle Dynamics. 99, Waendale, PA: Society o Automotive Enginees, Inc. 6. Wong, J.Y., Theoy o Gound ehicles. 993, New Yok, NY: John Wiley & Sons. 7. Heydinge, G., W.R. Gaott, and J.P. hstos. The Impotance o Tie Lag on Simulated Tansient ehicle Response. in Wold ongess o the Society o Automotive Enginees. 99. SAE Pape No. 93. Detoit, MI: SAE, Inc. 8. Mimuo, T., et al. Fou Paamete Evaluation Method o Lateal Tansient Response. in Wold ongess o the Society o Automotive Enginees. 99. SAE Pape No. 9734: SAE, Inc. 9. Huang, F., J.R. hen, and L.-W. Tsai. The Use o Random Stee Test Data o ehicle Paamete Estimation. in Wold ongess o the Society o Automotive Enginees SAE Pape No Detoit, MI: SAE, Inc.. Sill, J., Modeling, Testing, and Analysis o On-cente Handling o Mid-Size Spot Utility ehicles, Maste Thesis in Mechanical Engineeing. 6, lemson Univesity: lemson, S.. Ghoneim, Y., et al., Integated hassis ontol System to Enhance ehicle Stability. Intenational Jounal o ehicle Design,. ol. 3: p anopp, D., ehicle Stability. 4: R Pess. 3. Moui, H., M. ubota, and N. Hoiguchi. Study on Eects o Tansient Steeing Eots haacteistics on Dive's Steeing Behavio. in Wold ongess o the Society o Automotive Enginees. 7. SAE Pape No Detoit, MI: SAE, Inc. 4. Noman,. Objective Evaluation o On-cente Handling Peomance. in Wold ongess o the Society o Automotive Enginees SAE Pape No Detoit, MI: SAE, Inc.. Rill, G. Fist Ode Tie Dynamics. in Euopean oneence on omputational Mechanics. 6. Lisbon, Potugal. 6. Rhyne, T., lass Notes on Tie Behavio. 9, lemson Univesity - Intenational ente o Automotive Reseach. 7. Pacejka, H., Tye and ehicle Dynamics. : Oxod: Buttewoth-Heinemann. 8. Hou, Y., et al. A Study o Tie Lag Popety. in Wold ongess o the Society o Automotive Enginees.. SAE Pape No Detoit, MI: SAE, Inc. 9. Begman, W. and. Beauegad. Tansient Tie Popeties. in Automotive Engineeing ongess and Exposition SAE Pape No Detoit, MI: SAE, Intenational. 8 opyight 9 by ASME Downloaded Fom: on /3/ Tems o Use: