Yaswanth Siramdasu1,2 and Saied Taheri1
A Tool for Tire Handling Performance Evaluation REFERENCE: Siramdasu, Y. and Taheri, S., ‘‘A Tool for Tire Handling Performance Evaluation,’’ Tire Science and Technology, TSTCA, Vol. 44, No. 2, April–June 2016, pp. 74– 101. ABSTRACT: In the past, handling performance of the tire–vehicle combination has been evaluated using tire models such as the Pacejka Magic Formula. These models usually lack realistic representation of tire–road interaction and are not suitable for combined steering and braking maneuvers that may activate the antilock braking system. The objective of this study is to develop a computationally simple and accurate tire model, which can be used in the development and evaluation of handling performance of the tire on uneven road surfaces. For an emergency obstacle avoidance maneuver at high speeds, transient tire behavior plays a crucial role in the generation of forces between tire and road. Road undulations and steering inputs both induce high-frequency tire belt vibrations, which have detrimental effects on the handling and tractive behavior of the tire. To meet these requirements, a dynamic six degrees of freedom tire model–based rigid ring approach is developed and integrated with a multiple tandem elliptical cam to include enveloping behavior of the tire. The tire model that is developed in this research is partially based on the work of Schmeitz found in the literature. The tire model was parameterized using experimental parameters found in the literature. The tire model is validated using fixed axle high-speed oblique cleat experimental data. The developed tire model is integrated with the vehicle model in CarSimt. From the simulation of successive step steering input, the increasing influence of tire belt vibrations at higher slip angles was observed due to sudden steering wheel inputs. From the simulation of the step steering input on the bad asphalt road surface with an added cleat and on the flat smooth road surface, it was observed that the lateral performance of the tire at higher slip angles is strongly influenced by the vertical load variations. A single lane change maneuver was simulated on the smooth and bad asphalt road surfaces, demonstrating the strong influence of tire lateral and vertical belt vibrations on the lateral performance of the vehicle. Simulation of high-speed emergency obstacle avoidance braking maneuvers on measured rough and smooth roads showed that the influence of highfrequency vibrations due to road undulations and step steering inputs causes large variations of longitudinal and lateral forces at the axle, thus creating large variations in slip and slip angle of the tire with a degraded braking distance on rough roads. KEY WORDS: handling, rigid ring, tire modeling, Pacejka Magic Formula, steering, highspeed lane change
Introduction One of the most important applications of tire and vehicle models is in the development of safety systems such as antilock braking systems (ABS) and vehicle stability control systems. As the control algorithms for these systems become more advanced, there is a requirement for accurate and computationally effective tire models to predict and understand the forces and moments due to 1
2
Mechanical Engineering Department, Center for Tire Research (CenTiRe), Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24060, USA Corresponding author. Email:
[email protected]
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road contact, short wavelength road disturbances, brake and steer torque variations, and the interaction of the tire with these systems for accurate tuning and development of the vehicle, which are essential features in all modern automobiles. Excitation of tire belt vibrations either due to uneven roads, ABS, or steering systems has detrimental effects on the handling and tractive behavior of the tire [1]. Lateral forces are adversely affected due to load variation on uneven roads, which eventually changes cornering stiffness and relaxation length [1]. Understanding the dynamic response of the tire under these short wavelength road profiles at limit handling helps design vehicle controllers with good reliability, so that transition in vehicle behavior from a linear to a nonlinear regime is slow and predictable for the normal driver. The tire model required for these applications should be at least valid up to 75 Hz and able to roll over uneven and relatively sharp roads [1]. The objective of this study is to develop a computationally simple, accurate, and high-frequency dynamic tire model to analyze the effects of tire belt vibrations, road disturbances, and high-frequency brake and steering torque variations on the handling and braking performance of a vehicle. The developed tire model should be fast enough for vehicle dynamic simulation and should contain physical tire parameters that influence performance of the vehicle. The Pacejka Magic Formula (MF), which is widely used in the tire and automotive industries for steady state handling, fails to meet realistic representations of tire– road interactions. Transient tire models, based on relaxation properties of tires, are valid up to 20 Hz [2], because at higher slip angles relaxation length decreases and tire inertial forces are significant. For this study, a dynamic six degrees of freedom (DOF) rigid ring tire model is developed and integrated with a multiple tandem elliptical cam to include enveloping behavior of the tire (on uneven road surfaces) and integrated with first-order relaxation equations with a slip model and contact mass using residual springs. The rigid ring model was developed by Zegelaar [3] and later enhanced by Schmeitz [4]. Zegelaar [3] studied longitudinal behavior of tires under brake torque variations; Maurice [2] studied lateral transient dynamics using step road wheel steer angle; and Schmeitz [4] used the model for ride comfort analysis. In Sivaramakrishnan et al. [5], a rigid ring is used for braking performance evaluation under the influence of ABS and uneven road surface, and the authors concluded that a 1 cm cleat in road profile increases braking distance by 1 m at 65 kph. In this paper, behavior of tire lateral transient dynamics under hard steering, braking, and combined braking and steering with uneven road effects is studied. Previous studies have shown that under combined braking and steering without road effects the influence of lateral belt vibrations on longitudinal behavior is not seen [6]. In the following sections, first the tire model is presented and is parameterized using experimental parameters found in the literature [1,4]. The developed tire model is then validated using fixed axle high-speed oblique cleat experimental data. All rigid modes of tire below 80 Hz are clearly excited.
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The validated tire model is then integrated with the vehicle model in CarSimt with proper axis transformation. An increasing successive step steering input is applied to observe the excitation of the lateral mode (camber) and variations of forces due to sudden steering input. The influence of road undulations and step steering inputs is studied using smooth and rough roads with cleats. It is observed that the vertical force variations increase the influence of lateral belt vibrations and slip angle. A tight lane change maneuver was also simulated on a flat smooth road and a rough road surface to evaluate any deviation in vehicle behavior due to steering and road inputs. To study combined braking and steering under road undulations, an emergency obstacle avoidance maneuver is performed at 75 kph. The braking performance is degraded on rough roads with cleats by 0.8 m. The lateral dynamics (camber mode) and longitudinal dynamics (rotational mode) are clearly excited on both smooth and rough roads. In the case of ABS braking, the influence of the rotational and lateral modes of the tire is observed under smooth and rough road conditions. Mathematical Modeling The detailed tire model, vehicle model, and ABS model form the basis of the braking and limit handling simulation platform. The dynamic tire model used for this study is based on the work by Schmeitz [4]. Tire Modeling The mathematical modeling of tire consists primary of (1) (2) (3) (4)
Enveloping Model—models the enveloping behavior of the tire using a multiple elliptical cams model. Rigid Ring Tire Model—models the tire inertia properties using the rigid ring connected with springs and dampers to the rim. Contact Model—models the slippage of tire contact patch using lumped mass with 3 DOF model connected to the rigid ring. Slip Model—models the longitudinal and lateral relaxation behavior of the tire with first-order ordinary differential equations.
Enveloping Model The enveloping property of the tire is modeled using the three-dimensional tandem elliptical cam model developed by Schmeitz [4]. In its simple form, the semi-empirical model assumes two elliptical cams in tandem laterally (along contact width 2b) and two longitudinally (along contact patch 2a) connected by rods rolling over the uneven road surface and constrained to move only in the vertical direction as shown in Fig. 1. The longitudinal distance ls between the cams is modeled using model parameter pls, ls ¼ pls 3 2a, which is dependent on
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FIG. 1 — Multiple tandem elliptical cam model [4].
load. The lateral distance between the cams is equal to contact width 2b. This model is used for calculation of effective road surface, over which point the contact rigid ring model moves. This model can be used for any short wavelength road profile and can adapt to load variations. The effective road surface is characterized by effective height w 0, effective slope of road by, and effective camber of road bx. These are calculated at each road position X, which is the center of the rectangle formed by ls and 2b: Zf ;left þ Zr;left þ Zf ;right þ Zr;right be 4 Zf ;left Zf ;left þ Zr;right Zr;right tanby ðXÞ ¼ 2ls Zf ;left Zf ;right þ Zr;left Zr;right tanbx ðXÞ ¼ 2ð2bÞ wðXÞ ¼
ð1Þ
where Zf,left is the global height of the front left cam. The vertical distance (ze) between the local X-axis and the ellipse for any local position xf is calculated as
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ð2Þ
Eq. (2) is a maximal relation, because to calculate Zf it is necessary to find out where the ellipse actually makes contact with the road profile. This is done using an iterative procedure where the road profile is sampled at specific intervals around the global position of the front ellipse, Xf, with a maximum limit of the ellipse width ae, where the global height of the front ellipse is calculated at each interval using Eq. (2), and the maximum obtained value at the end of the iteration is taken as the final global height. The same procedure is also repeated for the rear ellipse. Usually a simple four tandem cam model is not sufficient to characterize camber and slope changes in sharp road profiles. Schmeitz [4] suggests that at a minimum, 10 cams in each longitudinal and lateral direction and a total of 36 cams around the contact patch area are needed to characterize any short wavelength road profiles. The modeling parameters of enveloping cam pls, ae, be, and ce are obtained by quasi-static rolling of tires over various obstacles and optimizing measured and simulated responses. The three-dimensional cam requires the same parameters as the two-dimensional cams, and validation of the two-dimensional enveloping model is presented in a previous studies by the authors [5]. Rigid Ring Model The rigid ring model is based on the assumption that the belt remains rigid and inextensible below 75 Hz. The axle is assumed to be a rigid body and is connected to the ring using springs and dampers. Inertias of belt (mb, Ibx, Iby, Ibz) and axle (Iay) are assumed to be those moved with the belt (tread, bead, and outer sidewall) and the axle (inner sidewall and ABS). The outer circumference contact point is connected to the contact model using residual springs and acted upon by the forces from the contact patch, which are transformed into the effective surface plane. The equation of motion of ring using Newton-Euler formulations with an International Organization for Standardization axis system are given by a a b þ x rb Þ þ kbx x rb þ cbx xrb kbz Xzrb ¼ Fbx mb ðV ax xaaz Vay a m ðV xa V a þ xa V a þ y Þ þ k y_ þ c y ¼ F b b
ay
az ax
ax az
rb
by rb
by rb
by
a a b mb ðV az þ xaax Vay þ z rb Þ þ kbz _ zrb þ cbx zrb þ kbx Xxrb ¼ Fbz Ibx ðx ˙ a þ c¨ Þ Iby ðX þ h˙ rb Þðxa þ w˙ Þ ax
rb
az
rb
b þ kbc c˙ rb þ cbc crb kbw Xwrb ¼ Mbx ˙ þ h˙ rb Þ þ kbh h˙ rb þ cbh hrb ¼ M b Iby ðX by
¨ rb Þ þ Iby ðX þ h˙ rb Þðxa þ c˙ rb Þ Ibz ðx ˙ aaz þ w ax b ˙ þ kbw wrb þ cbw wrb kbc Xcrb ¼ Mbz
ð3Þ
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FIG. 2 — Six DOF Rigid ring tire model adapted from Schmeitz [4].
where xrb, yrb, zrb, crb, hrb, and wrb represent relative transitional and rotational a a a motion between ring and axle. X is rotational velocity of axle. Vax , Vay , Vaz , xaay , a a xay , and xaz are velocities of axle with respect to axle axis. cbx, cby, cbz, cbc, cbh, cbw and kbx, kby, kbz, kbc, kbh, kbw represent stiffness and damping in longitudinal, lateral, vertical, and corresponding torsional stiffness and damping about the b b b b b b axis. Fbx , Fby , Fbz , Mbx , Mby , and Mbz are external forces and moments acting on the belt. The equation of axle rotation is given by ˙ kbh h˙ rb cbh hrb ¼ M a Iay X ay
ð4Þ
a is the brake torque applied at the axle. where May
Contact Model The contact model consists of a 3 DOFs contact mass with inertia (mc, Ic) moving over the effective road surface as shown in Fig. 2. It also includes a slip model to characterize transient dynamic slippage and delayed force generation of the contact patch (relaxation behavior). Contact mass is attached to the point contact of the belt outer circumference through residual springs (crx, cry, crw) and dampers (krx, kry, krw). The inputs to model are the external forces and moments (Fsx, Fsy, Msy) generated due to slippage of contact patch and internal forces (Fcx, Fcy, Mcy) generated due to deflections of residual springs and dampers as shown in Fig. 3. The output from the model is slip velocity of contact patch Vc,sx and Vc,sy, which acts as input to slip model in Eq. (9).
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FIG. 3 — Contact model with residual spring and dampers [1].
mc ðV c;sx w˙ c Vc;sy Þ þ Fcx ¼ Fsx mc ðV c;sy þ w˙ c Vc;sx Þ þ Fcy ¼ Fsy Fy;NL ¨c þ M ¼ M Ic w cz sz
ð5Þ
where Fy,NL is nonlagging (NL) lateral force due the camber of wheel or road. Fcx ¼ krx x rc þ crx xrc F ¼ k y þ c y cy
ry rc
ry rc
Mcz ¼ krw w˙ rc þ crw wrc
ð6Þ
The residual deflections xrc, yrc, and wrc are calculated based on deflections of the belt, and additional slip velocity due to effective rolling radius (curvature of road surface) is given by dby x rc ¼ Vc;sx Vb;cx ðwrb þ wrc ÞVb;cy þ re h˙ b qz dt y ¼ V V þ ðw þ w ÞV rc
c;sy
b;cy
rb
rc
b;cx
w˙ rc ¼ w˙ c w˙ rb xaaz
ð7Þ
The velocities of point contact of belt (Vb,sx and Vb,sy) are given by a a Vb;cx ¼ cosby ðVax þ x rb Þ sinby ðVaz þ z rb Þ a þ y rb þ rlb ðxaax þ c˙ rb Þcosby rlb ðxaaz þ w˙ rb Þsinby Vb;cy ¼ cosbx Vay
ð8Þ
where h˙ b ¼ X þ h˙ rb, re is effective rolling radius, and qz is vertical deflection of tire.
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Slip Model The forces acting in the contact patch (Fsx, Fsy, Msy) are generated due to longitudinal slip j 0 and lateral slip angle a 0 . The dynamic interactions between the tread elements of the rolling tire and the ground lead to variations in the slip. The force generation in the contact patch after rolling a finite distance is called relaxation behavior of the tire. Both the slips are also influenced by the deformation of the belt, and longitudinal slip is also influenced due to variations in the effective rolling radius as seen in Eq. (7). Accurate modeling of this process of force and slip generations plays a crucial role in the analysis of extreme braking and cornering maneuvers. Owing to the viscoelastic nature of the tire, the force generation in the contact patch occurs after rolling a finite distance, which is called relaxation behavior of the tire. Hence, a slip model is required for the tread elements to simulate the transient slippage. The overall relaxation behavior of the tire changes due to loading conditions and the operating slip of the tire. This behavior is modeled using an analytical transient slip model with a delayed longitudinal slip j, slip angle a, and camber c values. It consists of first-order approximations of respective relaxation behaviors of the tire, with a relaxation length constant for slip rc, a relaxation length for self-aligning moment with trail slip r2, and a relaxation behavior of camber force with r3. The inputs to the model are slip velocities calculated from contact model Eq. (5). dj 0 þ jVcx jj 0 dt da 0 þ jVcx ja 0 rc dt da 0 r2 t þ jVcx jat0 dt dc 0 þ jVax jc 0 r3 dt rc
¼ Vc;sx ¼ Vc;sy þ jVcx jwst ¼ jVcx ja 0 ¼ jVax jcea
ð9Þ
Since the lateral slip angle in Eq. (9) is at the contact patch, a yaw deflection of the contact patch is generated: Wst ¼ Msz[1/cbW þ 1/crW], which is taken into account. Maurice [2] suggests that the calculation of the aligning moment of the transient trail ‘t’ of the tire is complex, such that the first delay of slip angle is not enough. For this reason, a lead filter is added with an additional trail slip relaxation length r2, and the first-ordered transient lateral slip angle is used as an input to this equation, which increases the order to two. The transient response of the aligning moment due to step change in camber angle shows an additional short relaxation behavior r3 [1,4], so an additional first-order equation is added. Finally, slip quantities are used with the steady state MF equation to calculate forces and moments due to slippage. Since Mz,MF in Eq. (10) is moment at axle, so a lateral static deflection (yrst) is added to calculate moments at the contact patch.
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TIRE SCIENCE AND TECHNOLOGY Fsx ¼ Fx;MF ðj 0 ; a 0 ; Fcz Þ Fsy ¼ Fy;MF ðj 0 ; a 0 ; Fcz ; cea Þ t ¼ Ft;MF ðj 0 ; at0 ; Fcz ; cea Þ Mzr ¼ Mzr;MF ðj 0 ; a 0 ; Fcz ; c 0 Þ Mz;MF ¼ tFy;MF þ Mzr þ sFx;MF Msx ¼ Msx ¼ Mx;MF yrst FcN Msz ¼ Mz;MF þ yrst Fx;MF
The lateral static deflection is given by 1 1 rl þ þ yrst ¼ Fy;MF cby cry cbc
ð10Þ
ð11Þ
The external forces acting on the belt are calculated based on deformations of residual springs and dampers [Eq. (6)] and transformed into the plane of the effective road surface using Eq. (12). This results in calculation of the forces at axle using Eq. (1). Fbx ¼ Fcx þ by Fcz Fby ¼ Fcy þ Fy;NL þ ðbx þ cb ÞFcz Fbz ¼ by Fcx þ ðbx ca ÞðFcy þ Fy;NL Þ þ Fcz Mbx ¼ Mcx þ by Mcz þ rlb Fby Mby ¼ Mcy þ ðbx þ ca ÞMcz re Fbx þ rlb Fbz by Mbz ¼ by Mcz þ ðbx ca ÞMcy þ Mcz rib by Fby
ð12Þ
where rlb is the distance from the point contact of belt to axle and ca is initial camber of the wheel. Fcz is the vertical force acting on the tire in the effective surface plane, Fcz ¼
rad Fcz þ ðFcy þ Fy;NL Þsincea coscea
ð13Þ
where cea ¼ ca bx is the camber angle with respect to effective plane, and the rad is calculated based on residual spring overall radial force in contact patch Fcz deflections qrz. Constructive Relations Constructive relations consist of empirically modeled relations describing contact patch length and width (a, b), relaxation length (rc), load-deformation rad , qrz), velocity dependent stiffness and damping, rolling resistance Mcy, (Fcz effective rolling radius (re), loaded radius (rl), and steady state slip characteristics.
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Since this paper is not intended for modeling of the tire, for details of these equations and parameters of the model refer to Pacejka and Schmeitz [1,4]. Inputs and Outputs The rigid ring model requires inputs from other models to compute forces at the axle, namely: Axle motion. Axle deflections represented by xa, ya, and za and velocities of a a a axle with respect to axle axis are Vax , Vay , and Vaz and angular velocities are, a a a a xax , xay , and xaz , where xay equals the rotational velocity of wheel X. These deflections are generated due to external forces acting from the axle and reaction forces from the vehicle. When the tire is loaded, the axle is given an initial vertical displacement (which is negative) to represent the deflection in the tire. In the case of experiments with a fixed axle, za is kept constant and Vaz ¼ 0. Effective road surface. This is filtered road surface generated from road undulations using the enveloping model. This consists of the effective road height w, road slope by, and road camber bx. These values are calculated for each time step. This surface acts as an input to the point contact rigid ring model with the contact model. Camber of wheel. This is the initial camber angle of the wheel axis ca with respect to the vehicle. b b b External forces on belt. These are external forces Fbx , Fby , and Fzx acting on the belt from the contact model. These forces are generated from the deflection of residual springs in the contact model, which are transformed into the effective surface plane. a Braking moment. This is denoted by May , and this input is provided by the braking module, which calculates the applied torque at the axle during a braking event. This can either be due to ABS or manual braking. Steering torque at kingpin. Along with the rotational deflections due to suspension and axle motion, in the case of handling studies inputs from steering also induce additional deflections at the axle due to steering wheel input. Based on the above inputs, the rigid ring model calculates the following outputs, which serve as inputs to the vehicle, ABS model, active slip controller, and electronic stability controller (ESC) models: Reaction forces and moments at axle. These are the forces and moments acting on the vehicle from the tire due to internal deflection of the spring and dampers representing the sidewall, given in Eq. (3). Wheel slip ratio. The dynamics of the tire rolling over uneven surfaces causes fluctuations in the angular velocity of the wheel due to the deformation of the belt in the rotational direction. The practical slip ratio j is calculated from the slip model.
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Wheel slip angle. The dynamics of the tire rolling over uneven surfaces and steering inputs causes fluctuations in the lateral deformations at the wheel due to the deformation of the belt in the lateral direction. The slip angle a, which acts an input to the active slip controller, is calculated from the slip model. Contact patch area. This is an effective contact patch area, calculated either based on the net load acting on the contact patch or the vertical deflection of the tire due to road undulations. The length of the contact patch is denoted by 2a and width by 2b. These serve as an input to the enveloping model to calculate the shift between the tandem elliptical cams in the longitudinal and lateral directions. Vehicle Modeling Since the tire model used for this study is validated at least up to 75 Hz, an appropriate nonlinear vehicle model with the long range of validity has to be selected for fidelity of the vehicle dynamics simulation results. Commercially available vehicle dynamics software, CarSim, is used for vehicle modeling. The equations of motion in the CarSim math models are valid for full nonlinear three-dimensional motions of rigid bodies. These nonlinear dynamics have to be considered for vehicle dynamics studies operating under transient conditions. CarSim requires fewer parameters and its solutions are computed more quickly and accurately. This makes it preferable for vehicle dynamic studies and control systems designs. The components that have the greatest effect on handling, braking, and acceleration are suspension, steering kinematics and compliance, center of gravity (CG) location, inertia, wheelbase, etc. These are assembled from data sets that define the whole vehicle. The main advantage of CarSim comes from its solutions (faster than real time) and interfacing with Simulinkt for detailed modeling of components or design of controllers. The major kinematic and compliance effects of the suspension and steering system are specified using data that are obtained from real or simulated kinematics and compliance tests. Details of the linkages and gears in the suspension and steering system are not needed, reducing the amount of information needed to obtain accurate predictions and decrease the computation time significantly. The vehicle mass, inertia, and CG location are obtained from the Inertia Test rig. Tire models are based on the measured data or external models. Since the tire size used for this study is 205/60R15 91V, a B-Class Hatchback is used, and Fig. 4 shows the different components of the vehicle. Aerodynamics effects are neglected, and a standard six-speed automatic transmission model is used in this study. Since these effects have a minimum influence on vehicle dynamics pursued in this research (braking, handling, ride), details of these models are not explained. The vehicle mass, inertia, and CG coordinates are shown in Fig. 4. The front and rear independent suspension and standard power steering for a B Class hatchback are selected. Figure 5 shows the geometry and linear compliance of a steering system with a 17:1 steering ratio. Only front linear compliance is considered, not speed sensitive friction
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FIG. 4 — B-Class hatchback sprung mass in CarSim.
steer, and a look up table with small elastokinematics for rack displacement to steer is considered. Figure 6 shows the model in Simulink with the tire and ABS model integrated with CarSim using the necessary coordinate transformation. The forces at the axle are imported into CarSim, and velocity and deflection at axle are exported to Matlab/Simulink. The braking studies presented in
FIG. 5 — Steering topology in CarSim.
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FIG. 6 — Integration of CarSim with tire and ABS model in Simulink.
simulation are performed using an in-house build-rule based slip threshold ABS model, which is implemented on a Volkswagen Jetta; for detailed modeling and validation of the ABS model with commercial systems refer to Sivaramakrishnan et al. and Ding et al. [5,7]. Tire Model Validation Model Validation using Fixed Axle Oblique Cleat Testing The dynamic tire model is validated based on the measured response of the rolling tire over an oblique cleat of 10 mm at an angle of 43.58 with axle conditions fixed at a particular initial load and constant velocity. The measured change in longitudinal, lateral, and vertical axle forces DFx, DFy, and DFz; change in wheel rotational velocity DX; and change in overturning moment DMx and aligning moment DMz both in time and frequency domains are compared with simulation results. Figures 7 and 8 show responses of tires with 39 kph velocity and 4000 N of vertical load. The time and frequency domain simulated response matched the experimental response. In Fig. 7, the change in rotational velocity (slip) and longitudinal force response are almost the same, which is clearly seen in the corresponding frequency response (power spectral density [PSD]), so if the inphase rotational mode of the tire is excited due to uneven road surface or due to ABS oscillations, it directly affects the longitudinal behavior of the tire. When compared with relative change in DFx and DFy, the former is four times higher. In Fig. 8 all rigid modes of vibration of the tire [1] in-phase rotational mode (35 Hz), camber mode (43 Hz), yaw mode (51 Hz), vertical mode (78 Hz), and anti-
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FIG. 7 — Time domain response of tire rolling over oblique cleat 43.58 at 39 kph and Fz ¼ 4000 N.
phase rotational mode (86 Hz) are excited clearly. The vertical and anti-phase rotational mode is excited but with a magnitude difference from experimental data; this is due to the range of validity of the rigid ring model. It is difficult to see the yaw mode and camber mode separately; the split in yaw and camber occurs with the increase in velocity due to gyroscopic coupling. The response of DMz is at the compromise of complex modeling of aligning torque, and other effects due to elliptical cam model parameters are not optimized for each obstacle.
FIG. 8 — Frequency domain response of tire rolling over oblique cleat 43.58 at 39 kph and Fz ¼ 4000 N.
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FIG. 9 — Increasing step steering wheel input in CarSim.
Simulation With the validation of the tire model, vehicle dynamic simulation using tire model and CarSim are performed to study the importance of tire lateral modes (camber and yaw modes) on handling performance. In this section, the effects of the steering wheel and short wavelength road surface inputs on lateral dynamics of the tire are evaluated. Four cases are considered: in the first case, successive increasing step steering inputs are applied to observe the excitation of the lateral mode (camber) of the tire and the variation of lateral forces due to these sudden steering inputs. In the second case, the influence of the road undulations and steering wheel inputs are studied with a step steering input on a flat smooth road surface and on a bad asphalt road surface with artificially added cleats. This is shown in Figure 13. This serves to determine the vertical force variations due to short wavelength road inputs and to determine whether they cause any increasing influence on the lateral belt vibrations and slip angle variations. In the third case, a tight lane change maneuver was simulated on a flat smooth road surface and on a bad asphalt road surface to evaluate any deviation in vehicle behavior due to steering and road inputs. In the fourth case, replicating an emergency steering and braking maneuver on rough and smooth roads is simulated to study the combined dynamic effects of lateral and longitudinal tire vibrations. Successive Increasing Step Steering As shown in Figure 9, an increasing step steering input of 258 is given starting from t ¼ 2 s for 2 s, and vehicle velocity is at 75 kph. This type of input is used to see how a sudden change in steering input at higher slip angles excites the camber
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FIG. 10 — Vertical, lateral, slip angle response of four tires with increasing step steering wheel input at Vx ¼ 75 kph.
mode of the tire. Figure 10 shows the time history of the lateral force and slip angle for all four tires. Since the vehicle has front wheel steering, the camber mode can only be seen in the front tires. From the lateral force vehicle stability (VS) slip angle curves, the response of the front tires has a cyclic oscillation around the typical steady state curves. With no steering input at the rear tires, its response is a typical steady state behavior with transient relaxation effects. From the time history of the lateral force and slip angle variations for all four tires, it is observed that initially at lower slip angles with the step increase in steering input, the force
FIG. 11 — Frequency response (PSD) of lateral, overturning, aligning moments, and vertical force for four tires with increasing step steering wheel input at Vx ¼ 75 kph.
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FIG. 12 — Step steering input in CarSim.
and slip angle response resembles a first-order delayed system. Owing to the dominance of relaxation effects, the force and slip angle values reach the steady state condition by damping the initial oscillations. However, at the higher slip angles with the increasing steering angle input, relaxation effects of the tire are decreasing and the dominance of the tire inertial belt vibrations is increased. The excitation of tire inertial vibrations is dependent on the vertical load on the tire. It is observed that with the decrease of load on the front left tire, the saturation of lateral forces occurs at lower slip angles. This causes the influence of tire inertial belt vibrations to be visible at a simulation time of 4 s. Figure 11 shows the frequency response (PSD) of forces and moments at the axle. For the front tires, a clear excitation of camber mode of the tire is seen. Since road surface inputs are not considered, the vertical mode of the tire is not excited. Although these sudden steering inputs with a magnitude of 258 in 0.01 s are extreme steering maneuvers, under the influence of uneven road surfaces, the vertical load variation can also excite the lateral dynamics of the tire. This shows the importance of understanding the tire belt vibrations at higher slip angles on uneven road surfaces. Step Steering on Road with Cleat After the study of lateral modes of vibration in the above section, this section is focused on influence of load changes on lateral forces and interaction of combined lateral and longitudinal modes of tires due to uneven road profiles. For emergency obstacle avoidance maneuvers, a study of interactions of lateral and longitudinal modes of tires paves the way for design of ABS-based advanced ESCs, to see if rotational slip is corrupted due to lateral oscillations
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FIG. 13 — Rough road with 1 cm cleat.
and vice versa. In this study, simulation of step steering input as shown in Fig. 12 is given on a smooth road (z ¼ 0) and on a rough road with a cleat of 1 cm shown in Fig. 13. Through iteration, a vehicle velocity of 45 kph is selected such that once a state steering wheel angle of 1808 is reached and before lateral force oscillations reach a steady state value, longitudinal dynamics and vertical dynamics are excited with 1 cm on the right side wheels, to see interaction of longitudinal, vertical modes with lateral modes. The previous validation results show that longitudinal slip is mainly influenced by change in effective rolling radius; here influence of cleat on the slip angle of the tire is studied. In Figs. 14–16, left side plots show response on a rough road and right side plots show response on a smooth road. From Fig. 14, on a smooth road, after zero steering wheel velocity, oscillations in lateral and
FIG. 14 — Slip angle and lateral forces response on rough roads and smooth roads at Vx ¼ 45 kph for step steering input.
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FIG. 15 — Vertical and longitudinal forces response on rough roads and smooth roads at Vx ¼ 45 kph for step steering input.
slip angle are seen. Since the velocity is less with slower step steering input than in the previous section, relaxation effect and damping brings forces to a steady state value. The small cycles in between t ¼ 3.1 and 3.75 s, also seen in vertical forces in Fig. 15, show the roll mode of the vehicle, which is around 5–8 Hz; the roll mode on the rough road with a cleat on the right tires is also excited with a higher amplitude in vertical and lateral forces in this case. The response of slip angles and lateral forces on rough roads seems to have higher amplitude oscillations than around curves on the smooth road; this is due to
FIG. 16 — Frequency response (PSD) of lateral and longitudinal forces response on rough roads and smooth roads at Vx ¼ 45 kph for step steering input.
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variation of vertical load, and the increasing slip angle reduces cornering stiffness, which is predominant for lower load front and rear left tires. In Fig. 16, frequency response (PSD) on a rough road has relative increases in peaks for camber mode. Variation of slip angle and lateral force due to cleats causes significant changes in the value of slip angle and lateral forces. This is due to vertical load variation. For the rear right tire on smooth roads in Fig. 16 no lateral mode is excited, but due to cleat a peak at 42 Hz with a magnitude higher than any tire is seen. This confirms the importance of load variation on the camber mode of the tire. For analyzing interactions of longitudinal and lateral modes, frequency response (PSD) on rough and smooth roads is analyzed. It is interesting to see in-phase and anti-phase modes excited on smooth roads (rightmost plot in Fig. 16). This is due to sudden changes in steering wheel angle at the start and end of step steer, as shown in Fig. 15 for longitudinal response, causing the wheel to slip, which excited these modes. In longitudinal frequency response (PSD), oscillation due to lateral mode (46 Hz) is not seen, and there are no peaks in the lateral response (23 Hz or 78 Hz), which confirms experimental studies in the literature [6] stating that lateral oscillations will not influence the oscillation in rotational speed. Frequency response (PSD) on rough roads also shows no influence of lateral oscillations on longitudinal slip and longitudinal slip on lateral oscillations. Two peaks in the in-phase mode for rear right tires are seen due to shifts in mode as a result of load variations. The second peak in frequency response (PSD) of lateral force, due to the vertical mode of the tire (82 Hz), again emphasizes the importance of road undulations (change in load) on lateral force generation, which is not the case for longitudinal force. From Fig. 17, a sharp spike in lateral acceleration signals measured at the vehicle CG is observed when the front and rear right tires roll over the cleat. Clearly the excitation of roll mode of the vehicle on the uneven road is observed as a result of cleats being placed only on the right side of the vehicle. When the front tire rolls over the cleat, since the lateral forces generated by the other three tires are not saturated, not much variation in yaw velocity is observed. When the rear tire rolls over the cleat, the lateral forces of the other three tires are saturated, so a change in yaw velocity of the vehicle is observed. From this maneuver, the influence and interaction of lateral force with the vertical force variations due to road undulations at higher slip angles is studied. The third case study demonstrates that these effects should not be neglected for tight lane change maneuvers, especially at higher velocities. Lane Change on Bad Asphalt without Cleats A lane change maneuver at a velocity of 80 kph is simulated on a flat smooth road and on a bad asphalt road surface. This is shown in Fig. 13. A preview based closed loop driver controller in CarSim is used with a preview time of 0.5 s. Figure 18 shows the desired trajectory with a lane shifted by 3.5 m. Maximum limits on the applicable steering wheel show the time history of lateral forces and slip angle variations on the flat smooth road and the bad
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FIG. 17 — Vehicle behavior for step steering input on smooth and cleat roads.
asphalt road surfaces. Although the slip angles in this case are less than the previous case, more load transfer occurs due to the high velocity. As shown in Figs. 19 and 20, this causes saturation of lateral forces at smaller slip angles. When the vehicle is about to change lanes by steering right, load on the right tires decreases causing saturation in slip angle, as shown in the corresponding positive region of lateral forces VS slip angle curve. The same decrease in load on left tires occurs when the vehicle is steered left to bring the vehicle straight. It is interesting to observe that the oscillations in the time history of slip angle and lateral force on bad asphalt roads occurs only when the tire is operating in a nonlinear region of the lateral force VS slip angle curve. During the other parts of the maneuver, i.e., when slip angle is not saturated, behavior on smooth and bad asphalt surfaces is nearly the same. This is due to the relaxation length of the tire decreasing just before the saturation of the slip angle of the tire. This causes the excitation of the lateral belt vibrations to any changes in road undulations. Figure 21 shows the frequency response (PSD) of lateral and vertical forces on both road surfaces. In the case of a flat smooth road surface, no signs of excitation of tire modes are seen, but on a bad asphalt road, clearly camber and vertical modes of the tire are excited at their respective frequencies. The importance of relaxation effects grows at higher slip angles. This is clearly evident from the body roll angle response, which had a strong influence on subjective safety feelings for the driver [8]. In conclusion, the tire belt vibrations due to short wavelength disturbances have a significant effect on the lateral dynamics of the vehicle just before the saturation of slip angles is reached. This is in agreement with the previous studies done by Maurice [2] and Pacejka [1]. Since the ESC activates only at the physical limits of the vehicle, these effects should be considered in the design of ESCs.
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FIG. 18 — Single lane change desired trajectory in CarSim.
Figure 22 shows the vehicle response during the lane change maneuver. It is observed that the lateral acceleration signal measured at vehicle CG is influenced directly by the excitation of the lateral camber mode of the tire. This is seen from the lateral acceleration curve with oscillations at higher lateral accelerations. The roll angle of sprung mass of the vehicle on a bad asphalt road surface is increased by 0.58. The preview based driver controller is applying more steering wheel angle on the bad asphalt road as seen from the steering wheel angle curve. From the time history of yaw rate, on both road profiles the vehicle is stable throughout the maneuver. Emergency Steering and Braking Maneuver on Rough and Smooth Roads From the study of road undulations effects on lateral dynamics, in this section effect of ABS and road undulations on lateral dynamics is studied. An emergency obstacle avoidance maneuver is performed with steering wheel angle as shown in Fig. 23 starting at t ¼ 1.6 s, applying braking at t ¼ 2 s, and finally decreasing steering wheel angle to zero at t ¼ 2.4 s, with an initial vehicle
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FIG. 19 — Slip angle and lateral forces response for single lane change maneuver on smooth and bad asphalt roads without cleat.
velocity of 65 kph. Braking distance on rough roads and on smooth roads is calculated to see whether there is any influence of lateral forces on longitudinal behavior and ABS cycling and road unevenness on lateral forces. In Figs. 24–27 left side plots show the response on rough roads and right side plots show the response on smooth roads. From Figs. 24 and 25, it is observed from the time response of the slip angle and lateral forces that after the sudden steering input the slip angles and lateral forces increased on smooth and rough roads. Since the vehicle is operating in the nonlinear region with high slip angles as shown in
FIG. 20 — Vertical force and lateral force vs slip angle response for single lane change maneuver on smooth and bad asphalt roads without cleat.
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FIG. 21 — Frequency response of lateral and vertical forces response for single lane change maneuver on smooth (dotted lines) and bad asphalt roads without cleat (solid lines).
Fig. 26, lateral dynamics of the tire with a disturbance in the form of a cleat or highfrequency cycling of ABS braking torque input is studied. After the application of braking input at t ¼ 2 s, a decrease in lateral force is observed with an increase in slip angle on both smooth and rough roads, thus driving the tire into the sliding region. The ABS controller adapts to these conditions by decreasing the braking torques, as seen from decreased longitudinal force. In case of a rough road after hitting the cleat at t ¼ 2.4 s, a 68 change in front right tire slip angle is observed. This eventually causes large lateral belt oscillations as shown in the lateral force response, thus degrading the lateral stability of the vehicle.
FIG. 22 — Vehicle behavior for single lane change maneuver on smooth and bad asphalt roads.
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FIG. 23 — Steering input in case of emergency obstacle avoidance maneuver.
FIG. 24 — Slip angle and slip ratio response on rough roads and smooth roads from Vx ¼ 65 kph for combined steering and braking.
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FIG. 25 — Longitudinal VS slip and lateral VS slip angle curves on rough roads and smooth roads from Vx ¼ 65 kph for combined steering and braking.
From Fig. 25, since the vertical load on the rear inside tire (rear left) is almost equal to zero, instability at the front end transfers to the rear end also. As a result of a large vertical load, a small slip angle change is observed from a cleat impact on the rear right tire when compared with the right front tire. Figure 26 shows the curve’s slip and slip angle VS longitudinal and lateral forces; the loops are due to nonlinear complex behavior of tire with ABS and evasive steering input. The bigger loops are observed on rough roads due to disturbance from both cleats and ABS, whereas on smooth roads nonlinearities are due to ABS pressure pulse
FIG. 26 — Longitudinal, lateral, and vertical forces response on rough roads and smooth roads from Vx ¼ 65 kph for combined steering and braking.
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FIG. 27 — Frequency response (PSD) of longitudinal, lateral, and vertical forces response on rough roads and smooth roads from Vx ¼ 65 kph for combined steering and braking.
inputs. Fig. 27 shows the frequency response (PSD) of longitudinal and lateral forces on rough and smooth roads. In case of a rough road with ABS, large amplitudes in lateral mode oscillations for front right, rear right, and rear left tires are seen in lateral frequency response (PSD) (43 Hz) on rough roads. Additional investigation is needed, with multiple case studies of hitting cleats at different velocities and different points in steering input. As expected, braking performance on rough roads is increased by 0.8 m with a more degraded lateral stability than on smooth roads, as shown in Fig. 28. Conclusion Handling performance of the tire–vehicle combination has been evaluated using a new developed rigid ring based tire model. The tire model is validated using fixed axle high-speed oblique cleat experimental data. With the integration of vehicle models in CarSim and with tire models in Matlab/Simulink, various vehicle dynamics maneuvers are performed. Simulation of successive step steering input emphasizes the increasing influence of tire belt vibrations at higher slip angles, and simulation of step steering input on measured rough and smooth road profiles shows handling performance of tires is strongly influenced by vertical load variations. Simulation of a single lane change maneuver on the smooth and bad asphalt road surfaces demonstrates the strong influence of tire belt vibrations on the lateral performance of the vehicle. Simulation of emergency obstacle avoidance braking maneuvers is performed on rough and smooth roads with degraded braking performance on rough roads.
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FIG. 28 — Braking performance on rough roads and smooth roads from Vx ¼ 65 kph for combined steering and braking.
Acknowledgments The authors would like to thank the industry members of the Center for Tire Research for their support and valuable input during the course of this study.
References [1] Pacejka, H. B., Tire and Vehicle Dynamics, Butterworth-Heinemann, Oxford, UK, 2012, 3rd ed. [2] Maurice, J., ‘‘Short Wavelength and Dynamic Tyre Behavior under Lateral and Combined Slip Conditions,’’ Ph.D. Dissertation, TUDelft, 2000. [3] Zegelaar, P., ‘‘The Dynamic Response of Tyres to Brake Torque Variations and Road Unevennesses,’’ Ph.D. Dissertation, TUDelft, 1998. [4] Schmeitz, A., ‘‘A Semi-Empirical Three Dimensional Model of the Pneumatic Tyre Rolling over Arbitrarily Uneven Road Surfaces,’’ Ph.D. Dissertation, Delft University of Technology, 2004. [5] Sivaramakrishnan, S., Siramdasu, Y., and Taheri, S., ‘‘A New Design Tool for Tire Braking Performance Evaluations,’’ Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, Vol. 7, 2015, 7p. [6] Zanten, A., Erhardt, R., and Lutz, A., ‘‘Measurement and Simulation of Transients in Longitudinal and Lateral Tire Forces,’’ SAE Technical Paper 900210, 1990. [7] Ding, N., Wang, W., Yu, G., Zhang, W., Xu, X., Nenggen, D., Weida, W., Guizhen, Y., and Wei, Z., 2009. ‘‘Research and Validation of the Adaptive Control Strategy for ABS Based on Experimental Knowledge’’. Automotive Engineering, Vol. 31(1), 2009, pp. 28–32. [8] Dixon, J. C., Tires, Suspension, and Handling, Society of Automotive Engineers, Warrendale, PA, 1996, 2nd ed.
