understanding the limitations of different vehicle models for roll dynamics studies

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Understanding the limitations ofdifferent vehicle models for rolldynamics studiesTaehyun Shim a & Chinar Ghike aa Department of Mechanical Engineering, The University ofMichigan-Dearborn, 4901 Evergreen Road, Dearborn, MI, 48128,USAVersion of record first published: 30 Sep 2010.

To cite this article: Taehyun Shim & Chinar Ghike (2007): Understanding the limitations of differentvehicle models for roll dynamics studies, Vehicle System Dynamics: International Journal of VehicleMechanics and Mobility, 45:3, 191-216

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Vehicle System DynamicsVol. 45, No. 3, March 2007, 191–216

Understanding the limitations of different vehicle models forroll dynamics studies

TAEHYUN SHIM* and CHINAR GHIKE

Department of Mechanical Engineering, The University of Michigan-Dearborn,4901 Evergreen Road, Dearborn, MI 48128, USA

An accurate and realistic vehicle model is essential for the development of effective vehicle controlsystems. Many different vehicle models have been developed for use in various vehicle control systems.The complexity of these models and the assumptions made in their development depend on theirapplication. This article looks into the development and validity of vehicle models for prediction ofroll behavior and their suitability for application in roll control systems. A 14 DOF vehicle model thatincludes dynamics of roll center and nonlinear effects due to vehicle geometry changes is developed.The limitations, validity of simplified equations, and various modeling assumptions are discussed byanalyzing their effect on the model roll responses in various vehicle maneuvers. A formulation of thepopular 8 DOF vehicle model that gives good correlation with the 14 DOF model is presented. Thepossible limitations of the 14 DOF model compared with an actual vehicle are also discussed.

Keywords: Vehicle dynamics; Rollover; Roll center; Suspension

Nomenclature

Vehiclea distance of c.g. from front axle (m)b distance of c.g. from rear axle (m)c track width (m)h c.g. height (m)hrcf front roll center distance below sprung mass c.g. (m)hrcr rear roll center distance below sprung mass c.g. (m)Jx roll inertia (kg m2)Jy pitch inertia (kg m2)Jz yaw inertia (kg m2)m vehicle sprung mass (kg)u/v/w longitudinal/lateral/vertical velocities of c.g. in

body-fixed coordinate (m/s)

θ pitch angle (rad)φ roll angle (rad)

*Corresponding author. Email: [email protected]

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online © 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00423110600882449

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192 T. Shim and C. Ghike

ψ yaw angle (rad)ωx/ωy/ωz roll rate/pitch rate/yaw rate of c.g. in body-fixed coordinate (rad/s)

Suspension/tirebs suspension damping coefficient (Ns/m)Fxt/Fyt/Fzt tire longitudinal/lateral/vertical forces (N)Fxg/Fyg/Fzg longitudinal/lateral/vertical forces at tire contact patch in coordinate

frame 2 (N)Fxgs/Fygs/Fzgs longitudinal/lateral/vertical forces at tire contact patch in coordinate

frame 1(N)Fxs/Fys/Fzs longitudinal/lateral/vertical forces transferred to body in coordinate

frame 1(N)Jw rotational inertia of each wheel (kg m2)ks suspension spring stiffness (N/m)kt tire stiffness (N/m)ls instantaneous length of strut (m), ls = lsi − xs + xsi

lsi initial length of strut (m), lsi = h − (r − xti)

mu unsprung mass (kg)r instantaneous tire radius (m)ro nominal radius of tire (m)s tire longitudinal slipT external torque applied at wheel (N m)ug/vg longitudinal/lateral velocities at tire contact patch in

coordinate frame 2 (m/s)us/vs/ws longitudinal/lateral/vertical velocities at the suspension corner in

coordinate frame 1 (m/s)uu/vu/wu unsprung mass longitudinal/lateral/vertical velocities in coordinate

frame 1 (m/s)xs suspension spring compression (m)xt tire spring compression (m)xsi initial suspension spring compression (m)xti initial tire spring compression (m)

α tire lateral slip (rad)ω angular velocity of wheel rotation (rad/s)δ road wheel steer angle (rad)

1. Introduction

With the growing popularity of sport utility vehicles (SUVs) and the increasing number ofrollover accidents in SUVs, as well as trucks and buses, the reduction of vehicle rolloverpropensity has been an area of active research in recent years. Owing to the high fatalityrate of rollover crashes, National Highway Traffic Safety Administration initiated a rolloverresistance rating test as part of its New Car Assessment Program for passenger vehicles,including SUVs [1, 2].

A number of methods have been proposed and explored to prevent vehicle rollovers.Research on such rollover prevention systems can be categorized into two basic types: passivesystems (e.g. curve speed warning systems) and active systems (e.g. roll stability control).Many passive warning systems use a prediction algorithm to determine the risk of impending

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Limitations of different vehicle models 193

rollover using vehicle or axle roll angles, lateral load transfer, and/or lateral acceleration asinputs [3–10]. They provide some type of warning so that the driver can take corrective action.The active roll control systems reported in the literature [11–16] can be categorized into aboutfour different types the basis of on its actuation schemes: four wheel steering [11], activesuspension [12, 13], active roll bar [14, 15], and differential braking [16–18]. The active rollbar and active suspension are designed to directly control the vehicle roll motion; the fourwheel steering and differential braking are designed to reduce the vehicle rollover tendencyby controlling vehicle yaw motion.

In the development of active/passive roll control systems, a vehicle model that can representrealistic roll behavior is essential to predict impending rollover as well as accurately applyingthe actuation force to avoid vehicle rollover. In the literature, a number of vehicle models havebeen introduced for the application of active/passive roll control systems [8–10, 19–24]. Themajority of these models are low-order roll-plane models derived with various assumptions andhave limitations for their application. Figure 1 shows some roll-plane models such as a rigidvehicle model and a suspended vehicle model. The rigid vehicle model is the most commonroll-plane model and is widely used due to its ease of understanding and parameterization.It assumes that there are no compliances (i.e. frame, suspension, or tire) and no degrees offreedom (prior to lift-off). It represents a theoretical upper bound on vehicle stability [21, 22].The suspended vehicle model [8–10, 19–21, 23, 24] differs from the rigid vehicle model inthat it has a roll degree of freedom for the suspension that connects the sprung and unsprungmasses. The sprung mass is typically assumed to rotate about a kinematic roll center axis [19],which connects the front and rear suspension roll centers. This model also assumes that thesprung mass is much greater than the unsprung mass and that the chassis and tires are muchstiffer than the suspension. These models are based on a vehicle model which has a decoupledyaw-roll motion and one equivalent axle. In order to better represent the vehicle lateral andyaw dynamics as well as coupling of yaw-roll motion due to the transient lateral load transferduring extreme maneuvers, higher order models such as 8 DOF [25–27] and 14 DOF [28–31]are also used in rollover studies. A 14 DOF vehicle model, which considers the suspension ateach corner, has the same benefits of an 8 DOF vehicle model, with the additional capabilitiesof predicting vehicle pitch and heave motions. It also offers the flexibility of modeling non-linear springs and dampers and can simulate the vehicle responses to normal force inputs in thecase of an active suspension system. Moreover, the 14 DOF model, unlike the 8 DOF model,

Figure 1. Schematic of simple rigid and suspended vehicle models used in the rollover study.

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can predict vehicle behavior even after wheel lift-off and thus can be used in developing ortesting the validity of rollover prediction/prevention strategies. In other words, even thoughthe 14 DOF model has low degrees of freedom and consequently certain limitations whencompared with a multi-body dynamics model, it can sufficiently express the vehicle motionsthat are important in most active chassis control systems.

This article looks into the development and validity of vehicle models for prediction of rollbehavior and their suitability for application in roll control systems. A 14 DOF vehicle modelthat includes dynamics of roll center and nonlinear effects due to vehicle geometry changesis developed. The limitations, validity of simplified equations, and various modeling assump-tions are discussed by analyzing their effect on the model roll responses in various vehiclemaneuvers. A formulation of the popular 8 DOF vehicle model that gives good correlationwith the 14 DOF model is presented. The possible limitations of the 14 DOF model comparedwith an actual vehicle are also discussed.

2. Development of vehicle model (model 1)

Figure 2 exhibits the schematic of the two axle, 14 DOF, vehicle model used to investigatevehicle roll response to steering and torque inputs. This schematic includes 6 DOF at the vehi-cle lumped mass center of gravity and 2 DOF at each of the four wheels, including verticalsuspension travel and wheel spin. The body is modeled as being rigid, with body-fixed coor-dinates, xyz, attached at the center of gravity and aligned in principal directions (coordinateframe 1). u, v, and w indicate forward, lateral, and vertical velocities, respectively, of thesprung mass. There is roll angular velocity, ωx , pitch angular velocity, ωy , and yaw angularvelocity, ωz. The attitude and position of body with respect to the inertial frame (XYZ) canbe determined through successive coordinate transformations through the cardan angles (i.e.the roll angle φ, the pitch angle θ , and the yaw angle ψ) as shown in figure 2. The coordinateframe 2 is obtained by rotating the inertial coordinate frame through the yaw angle ψ . In otherwords, the body-fixed coordinate frame 1 is obtained by rotating the coordinate frame 2 firstthrough the pitch angle θ and then through the roll angle φ.

Figure 3 shows the force and velocity components in the right front corner of a vehicle. Thevelocities usrf , vsrf , and wsrf are the velocities of the right front strut mounting point in thelongitudinal, lateral, and vertical directions, respectively, in the body-fixed coordinate frame,

Figure 2. Schematic of 14 DOF full vehicle model with one-dimensional suspension and coordinate frames.

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Figure 3. Description of forces and velocities at the right front corner of a vehicle.

which is attached to the sprung mass center of gravity (coordinate frame 1). These velocitiescan be obtained by transforming the c.g. velocities as

⎛⎝usrf

vsrf

wsrf

⎞⎠ =

⎛⎜⎜⎜⎝

0 0c

20 0 a

−c

2−a 0

⎞⎟⎟⎟⎠

⎛⎝ωx

ωy

ωz

⎞⎠ +

⎛⎝u

v

w

⎞⎠ (1)

The velocities uurf , vurf , and wurf represent the velocities of the unsprung mass mu in thebody-fixed coordinate frame 1, and ugrf , vgrf , and wgrf are the lateral, longitudinal, and verticalvelocities at the tire contact patch in coordinate frame 2. The forces Fxsrf , Fysrf , and Fzsrf are theforces transmitted to the sprung mass along the longitudinal, lateral, and vertical directions,respectively, of coordinate frame 1. The forces Fxgsrf , Fygsrf , and Fzgsrf are the forces acting atthe tire ground contact patch in the same coordinate frame 1. These forces can be written interms of the tire forces Fxgrf , Fygrf , and Fzgrf by projecting its components along coordinateframe 2 as ⎛

⎝Fxgsrf

Fygsrf

Fzgsrf

⎞⎠ =

⎛⎝1 0 0

0 cos φ sin φ

0 − sin φ cos φ

⎞⎠

⎛⎝cos θ 0 − sin θ

0 1 0sin θ 0 cos θ

⎞⎠

⎛⎝Fxgrf

Fygrf

Fzgrf

⎞⎠ (2)

The forces Fxgrf and Fygrf are obtained by resolving the longitudinal (Fxtrf ) and cornering(Fytrf ) forces at the tire contact patch as

Fxgrf = Fxtrf · cos δ − Fytrf · sin δ (3)

Fygrf = Fytrf · cos δ + Fxtrf · sin δ (4)

where δ is the steering angle at the road wheels.The Magic Formula [32] tire model has been used in the development of the tire forces Fxtrf

and Fytrf . The longitudinal and lateral slips used in the tire model were calculated as follows:

srf = (rrfωrf − (ugrf cos δ + vgrf sin δ))

|(ugrf cos δ + vgrf sin δ)| (5)

αrf = tan−1

(vgrf

ugrf

)− δ (6)

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Figure 4. Bond graph representation of right front suspension.

Figure 4 shows the bond graph representation of the right front suspension shown in figure 3.The longitudinal and lateral velocities at the tire contact patch, ugrf and vgrf , can be determinedfrom figure 4 as

ugrf = cos θ(uurf − ωy · rrf) + sin θ(wurf cos φ + sin φ(ωx · rrf + vurf)) (7)

vgrf = cos φ(vurf + ωx · rrf) − wurf sin φ (8)

where rrf is the instantaneous radius of the tire.The longitudinal (uurf ) and lateral (vurf ) velocities of the unsprung mass in the body-fixed

coordinate frame used in the earlier equations are simply written as

uurf = usrf − lsrfωy (9)

vurf = vsrf + lsrfωx (10)

where lsrf is the instantaneous length of the strut as indicated in figure 3.The unsprung mass vertical velocity wurf represents the degree of freedom corresponding

to the suspension deflection and can be expressed by applying Newton’s law for the verticalmotion of the unsprung mass as

muwurf = cos φ(cos θ(Fzgrf − mug) + sin θFxgrf) − sin φFygrf − Fdzrf

− xsrf · ksf − xsrf · bsf − mu(vurf · ωx − uurf · ωy)(11)

where ksf is the suspension stiffness, bsf the suspension damping coefficient, and xsrf theinstantaneous compression of the right front suspension spring. The force, Fdzrf , represents

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the additional load transfer that occurs at the wheels through the suspension links because ofthe reaction force to the force transmitted to the sprung mass through the roll center. This willbe considered in detail at a later stage.

The instantaneous suspension spring deflection xsrf is given as

xsrf = −wsrf + wurf (12)

The vertical force Fzgrf acting at the tire ground contact patch in coordinate frame 2 can bewritten in terms of the tire stiffness (ktf ) and the instantaneous tire deflection (xtrf ) as

Fzgrf = Fztrf = xtrfktf (13)

The instantaneous tire deflection xtrf in equation (13) is given as

xtrf = wgrf − wuirf = wgrf − (cos θ(wurf cos φ + vurf sin φ) − uurf sin θ) (14)

where wuirf is the vertical velocity of the wheel center in the inertial coordinate frame. For thesimulations in this article, it is assumed that the vertical velocity wgrf at the tire contact patch iszero (smooth road). It should be noted that even though the tire is assumed to remain at a fixedangle with the strut, the vertical stiffness of the tire, ktf , is always considered to be normalto the ground between the ground and the wheel center ( point C). Thus, even though thecompliance element 1/ktf is located between the one junctions of the ground vertical velocity(wgrf = 0) and the velocity w1rf in figure 4, the vertical velocity of the wheel center in theinertial frame, wuirf , is used in place of the velocity w1rf in equation (14). The instantaneoustire radius is then determined as

rrf = ro − xtrf

cos θ cos φ(15)

To account for the wheel lift-off, when the tire radial compression becomes less than zero,the tire normal force Fzgrf is set equal to zero. In addition, the instantaneous tire radius isconsidered equal to the nominal tire radius until the tire returns to the road surface.

If xtrf < 0 then Fzgrf = 0 and rrf = r0 (16)

As the tire normal force becomes zero, no lateral (Fygrf) and longitudinal (Fxgrf) tire forcesare developed at that contact patch. Thus, the only forces acting at that suspension corner arethe weight and inertia forces of the unsprung mass, which are very small in magnitude. As thecardan angles and appropriate coordinate transformations between the body-fixed coordinateframe 1 and the coordinate frame 2 at the tire–ground contact patch (figure 2) are consideredfor all the forces and velocities in the system, the model is able to simulate vehicle behaviorafter wheel lift-off and during the rollover event with just the modification mentioned inequation (16).

The instantaneous length of the strut lsrf used in equations (9) and (10) is given as

lsrf = lsif − (xsrf − xsif) (17)

where lsif is the initial length of strut and xsif is the initial suspension spring deflection. Theinitial length of the strut lsif is taken such that

lsif = h − (ro − xtif) (18)

where xtif is the initial tire compression.

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The initial spring compression xsif and the initial tire compression xtif are determined fromthe static conditions as

xsif = mb

2(a + b)ksf(19)

xtif = (mb/2(a + b)) + muf

ktf(20)

The forces Fxsrf and Fysrf transmitted to the sprung mass along the u- and v-axes of the body-fixed coordinate frame are obtained after subtracting the components of the unsprung massweight and inertia forces from the corresponding forces Fxgsrf and Fygsrf acting at the tirecontact patch as

Fxsrf = Fxgsrf + mug sin θ − muuurf + muωzvurf − muωywurf (21)

Fysrf = Fygsrf − mug sin φ cos θ − muvurf + muωxwurf − muωzuurf (22)

The vertical force Fzsrf transmitted to the sprung mass through the strut is given as

Fzsrf = xsrfksrf + xsrfbsrf (23)

Figure 5 shows the forces and velocities in the roll plane of, for example, the front suspension.Generally, the roll center height is defined with reference to the ground. However, for thedevelopment of this model, the front and rear roll centers are assumed to be fixed at distanceshrcf and hrcf , respectively, below the sprung mass c.g. along the negative w-axis of the body-fixed coordinate frame 1. Moreover the roll center is simply considered to be a point ofapplication of the forces transmitted to the sprung mass through the suspension links and notas a kinematic constraint.

In figure 5, Fzslf and Fzsrf are the forces transmitted to the sprung mass through the struts.Fyslf and Fysrf represent the lateral forces transmitted to the sprung mass through the suspensionlinks. In the absence of a roll center, i.e. when the roll center is assumed to be in the groundplane, the total roll moment transmitted to the sprung mass at, for example, the right front

Figure 5. Forces and velocities in the front suspension roll plane.

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corner along the ωx direction is given as

Mxrf = Fygsrf(lsrf + rrf) − (mug sin φ cos θ + muvurf − muωxwurf + muωzuurf) · lsrf

= Fygsrf · rrf + Fysrf · lsrf

(24)

and the force due to the lateral load transfer through suspension links, Fdzrf = 0,

(−mug sin θ + muuurf − muωzvurf + muωywurf)

When a roll center is modeled as shown in figure 5, the roll moment Mxrf transmitted to thesprung mass by the right front corner suspension is given as

Mxrf = Fysrfhrcf (25)

Thus, the inclusion of a roll center reduces the total roll moment transferred to the sprung massby the front suspension. The difference between the roll moments in the absence of the rollcenter (equation (24)) and when the roll center is considered (equation (25)) acts directly onthe unsprung mass and is responsible for the link load transfer forces ( jacking forces), Fdzlf

and Fdzrf . These forces can be estimated as

Fdzrf = −Fdzlf = Fygsrfrrf + Fysrf lsrf + Fygslfrlf + Fyslf lslf − (Fysrf + Fyslf)hrcf

cf(26)

The moments Myrf and Mzrf transmitted to the sprung mass at, for example, the right frontcorner by the suspension along the ωy and ωz directions can be given as

Myrf = −(Fxsgrf(lsrf + rrf) − (−mug sin θ + muuurf − muωzvurf + muωywurf) · lsrf)

= −(Fxsgrf · rrf + Fxsrf · lsrf) (27)

Mzrf = 0 (28)

The equations of motion for the 6 DOF of the sprung mass model can now be derived fromthe direct application of Newton’s laws for the system as

m(u + ωyw − ωzv) =∑

(Fxsij) + mg sin θ (29)

m(v + ωzu − ωxw) =∑

(Fysij) − mg sin φ cos θ (30)

m(w + ωxv − ωyu) =∑

(Fzsij + Fdzij) − mg cos φ cos θ (31)

Jxωx =∑

(Mxij) + (Fzslf + Fzslr − Fzsrf − Fzsrr)c

2(32)

Jyωy =∑

(Myij) + (Fzslr + Fzsrr)b − (Fzslf + Fzsrf)a (33)

Jzωz =∑

(Mzij) + (Fyslf + Fysrf)a − (Fyslr + Fysrr)b

+ (−Fxslf + Fxsrf − Fxslr + Fxsrr)c

2(34)

where m is the sprung mass and the subscript ‘ij’ denotes left front (lf), right front (rf), leftrear (lr), and right rear (rr).

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The cardan angles θ , ψ , φ needed in the aforementioned equations are obtained byperforming the integration of the following equations,

θ = ωy cos φ − ωz sin φ (35)

ψ = ωy sin φ

cos θ+ ωz cos φ

cos θ(36)

φ = ωx + ωy sin φ tan θ + ωz cos φ tan θ (37)

The model developed is a highly complex, non-linear full vehicle model with one-dimensionalsuspension. The next section shows the validation of this model with CARSIM. In the subse-quent sections, we will consider the effect of various modeling assumptions and simplificationson the responses of the vehicle model and understand their applicability and validity.

3. Vehicle model validation

The 14 DOF full vehicle model is validated with CARSIM [33] and ADAMS/Car [34] for aJ-turn maneuver at 50 kph. Figure 6 shows the steering wheel angle input for the maneuver andthe comparative roll angles, lateral acceleration, and yaw rates. Figure 7 shows the comparativenormal forces at the four tires. It can be seen that the developed model correlates very well withCARSIM until wheel lift-off. The slight difference in the responses prior to wheel lift-off andthe deviation that occurs later may be due to the fact that the roll center in CARSIM is assumedto be fixed with respect to the ground, whereas in the model developed here, the roll center isassumed to be at a fixed distance from the sprung mass center of gravity. At extreme roll anglesand certainly after wheel lift-off, this will have an impact on the responses. The difference

Figure 6. Comparison of vehicle responses among the models during the J-turn maneuver at a speed of 50 kph.

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Figure 7. Comparison of tire normal forces among the models during the J-turn maneuver at a speed of 50 kph.

in the responses when compared with ADAMS/Car can be attributed to the variation in rollcenter height that occurs due to the suspension geometry in the multi-dynamics model.

4. Effect of linearization of trigonometric terms (Model 1a)

A small angle assumption is widely employed in the development of vehicle models for variousvehicle control systems. In this section, a small angle assumption is made for the cardan anglesθ and φ, and its effect on the vehicle roll dynamics is investigated. No small angle assumptionis made for the steering angle δ as this could cause differences at high steering inputs whichare not truly representative of the limitations of the vehicle model as such.

Several equations are simplified due to the linearization of the trigonometric terms with thesmall angle assumption. Equation (2), which represents the coordinate transformation of theground forces, can be simplified to equation (38). Here it is assumed that sin θ sin φ = θφ = 0.

⎛⎝Fxgsrf

Fygsrf

Fzgsrf

⎞⎠ =

⎛⎝1 0 −θ

0 1 φ

θ −φ 1

⎞⎠

⎛⎝Fxgrf

Fygrf

Fzgrf

⎞⎠ (38)

Equations (7) and (8) for the longitudinal and lateral velocities at the tire ground contact patchare, respectively, modified to equations (39) and (40).

ugrf = usrf − ωy(lsrf + rrf) + θ(wurf) (39)

vgrf = vsrf + ωx(lsrf + rrf) − wurfφ (40)

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The terms containing θ and φ in the above equations are very small and can be dropped withoutintroducing any significant errors to yield equations (41) and (42) subsequently.

ugrf = usrf − ωy(lsrf + rrf) (41)

vgrf = vsrf + ωx(lsrf + rrf) (42)

Equation (11) for the vertical motion of the unsprung mass is simplified to equation (43)subsequently.

muwurf = xtrf · ktrf − mug + θFxgrf − φFygrf − Fdzrf − (xsrf · ksrf + xsrf · bsrf)

− mu(vurfωx − uurfωy)(43)

Equation (14) for the tire deflection xtrf is simplified to equation (44).

xtrf = −wuirf = −(wurf + vurfφ − uurfθ) (44)

The instantaneous tire radius is then simply given as

rrf = ro − xtrf (45)

The forces Fxsrf and Fysrf transmitted to the body by the suspension are now given byequations (46) and (47).

Fxsrf = Fxgsrf + mugθ − mu(uurf − ωzvurf + ωywurf) (46)

Fysrf = Fygsrf − mugφ − mu(vurf − ωxwurf + ωzuurf) (47)

The equations of motion for the three translational degrees of freedom of the sprung massmodel are modified as

m(u + ωyw − ωzv) =∑

(Fxsij) + mgθ (48)

m(v + ωzu − ωxw) =∑

(Fysij) − mgφ (49)

m(w + ωxv − ωyu) =∑

(Fzsij + Fdzij) − mg (50)

The angles θ , ψ , φ can be obtained as

θ = ωy − ωzφ (51)

ψ = ωyφ + ωz (52)

φ = ωx + ωzθ (53)

Figures 8 and 9 show the effect of the small angle assumption on the vehicle responses. Itcan be seen that for the step steer input in figure 8, the small angle assumption makes nodifference. In the ramp steer responses in figure 9, the small angle assumption seems to affectthe responses only when the roll angle exceeds 8◦–10◦. In general, the small angle assumptionwill not affect the vehicle responses while the concerned angles, θ and φ, remain less than8◦–10◦. The pitch angle θ is unlikely to reach such values except in very hard braking scenariosand, in most cases, the vehicle would already have seen wheel lift-off by the time the roll angleφ reaches 8◦–10◦. Thus, we can conclude that the small angle assumption is significant onlyfor studies of vehicle behavior after wheel lift-off has occurred.

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Figure 8. Comparative responses – step steer – effect of small angle assumption and ignoring lateral and longitu-dinal inertia forces of unsprung masses.

Figure 9. Comparative responses – ramp steer – effect of small angle assumption and ignoring lateral and longitu-dinal inertia forces of unsprung masses.

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5. Effect of the unsprung masses: ignoring the lateral andlongitudinal inertia (Model 1b)

The terms uurf and vurf in equations (46) and (47), which are required to compute the inertiaforces of the unsprung masses, can be determined either by differentiation of the velocitiesuurf and vurf , respectively, or by the equations given subsequently

uurf = u − lsrfωy − ωylsrf (54)

vurf = v + aωz + lsrfωx + ωxlsrf (55)

Differentiation is not desirable in simulations as it slows down the computation time, and it isalso not suitable for the development of control logic. Using the equations above will also addseveral off-diagonal terms to theA-matrix in the formAX = BX + C.Therefore, for the sake ofsimplicity, in addition to the small angle assumption, the terms mu(uurf − ωzvurf + ωywurf) andmu(vurf − ωxwurf + ωzuurf) representing the inertia forces of the unsprung mass are droppedfrom equations (46) and (47), respectively, to yield the simplified equations subsequently.

Fxsrf = Fxgsrf + mugθ (56)

Fysrf = Fygsrf − mugφ (57)

Also, equation (43) for the unsprung mass vertical velocity can be simplified to equation (58)by dropping the term mu(vurfωx − uurfωy) without any significant effect prior to wheel lift-off.

muwurf = xtrf · ktrf − mug + θFxgrf − φFygrf − Fdzrf − (xsrf · ksrf + xsrf · bsrf) (58)

It can be seen from the step steer responses in figure 8 that ignoring the unsprung mass inertiaforces increases the net transient roll moment acting on the sprung mass and thus slightlyincreases the peak transient roll angles and lateral load transfers. This increase in the rollmoment would also result in significantly quicker rollover of the vehicle after two wheel lift-off if the changes in equations (56) and (57) are applied without modifying equation (58) asshown. However, the modification in equation (58) opposes this tendency for quicker rolloverand, as seen in figure 9, the vehicle actually rolls over at almost the same time. It should benoted here that the unsprung mass is NOT lumped with the sprung mass in equations (48)and (49). Doing so causes deviation in the responses.

6. Effect of further simplifications to the 14 DOF full vehicle model

Even after the small angle assumption and ignoring the inertia forces of the unsprung masses,the equations for the 14 DOF model are fairly complex. In this section, certain additionalsimplifications and their effect on the vehicle roll response are studied.

6.1 Simplifying the equations for the forces transmitted to the sprung mass (Model 1c-i)

The equations ((38), (56) and (57)) for the lateral and longitudinal forces transmitted to thesprung mass can be simplified by dropping the terms containing φ and θ as(

Fxsrf

Fysrf

)=

(Fxgsrf

Fygsrf

)=

(Fxgrf

Fygrf

)(59)

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This simplification reduces the roll moment on the sprung mass, which reduces the predictedtransient as well as steady-state roll angles as seen in the step steer responses in figure 10.There is also a consequent reduction in the lateral load transfer at the tires. For small rollangles, this effect is not significant and may be deemed acceptable. However, as seen in theramp steer responses in figure 11, at higher roll angles, there are considerable errors, whichin this particular case make the difference between vehicle rollover and stability.

6.2 Simplifying the equation for the unsprung mass vertical velocity (Model 1c-ii)

The terms containing θ and φ are dropped from the equation for the vertical motion of theunsprung mass to yield equation (60).

muwurf = xtrf · ktrf − mug − Fdzrf − (xsrf · ksrf + xsrf · bsrf) (60)

Dropping the term −φFygrf causes an increase in the vertical acceleration of the unsprungmass and a consequent decrease in the tire compression. This reduces the predicted transientand steady-state tire normal forces, as is clearly seen in both the step steer and ramp steerresponses in figures 10 and 11, respectively. The reduced tire normal forces result in lowerlateral force generated at the tires and consequently reduced roll moment acting on the sprungmass. However, as the normal forces on the loaded tire are more affected by the simplificationin equation (60), the effective roll stiffness of the model is also reduced. Thus, the change inroll angle is not as significant as the change in the tire normal forces. However, it should benoted that this simplification will also affect the response of the vehicle to any normal forceinputs at the suspension corners from an active suspension system.

Figure 10. Comparative responses – step steer – effect of simplifications indicated in Model 1c-i and Model 1c-ii.

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Figure 11. Comparative responses – ramp steer – effect of simplifications indicated in Model 1c-i and Model 1c-ii.

6.3 Simplifying the equation for the tire deflection (Model 1c-iii)

Equation (44) for the tire deflection is simplified to equation (61) subsequently.

xtrf = −wuirf = −wurf (61)

Along with this simplification, it is necessary to change the equations for the angles of thesprung mass as

θ = ωy (62)

ψ = ωz (63)

φ = ωx (64)

The removal of the term −vurfφ from equation (44) results in increased tire compressionprediction. As can be seen in the step steer and ramp steer responses in figures 12 and 13, thisresults in significantly greater tire normal forces at high roll angles. Moreover, the model canno longer predict wheel lift-off and rollover as is apparent from the ramp steer outputs.

6.4 Simplifying the equations for velocities at the tire–groundcontact patch (Model 1c-iv)

The longitudinal and lateral velocities at the tire are taken to be the same as those at thecorresponding sprung mass corner. Thus, equations (41) and (42) are simplified as shownsubsequently.

ugrf = usrf (65)

vgrf = vsrf (66)

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Figure 12. Comparative responses – step steer – effect of simplifications indicated in Model 1c-iii and Model 1c-iv.

As can be seen from the step steer responses in figure 12, this simplification changes the rollfrequency of the model entirely. However, there is no change in the steady-state roll anglesor tire forces and the ramp steer responses in figure 13 are identical. The change in the rollfrequency is very important when considering the response of the vehicle in non-steady state

Figure 13. Comparative responses – ramp steer – effect of simplifications indicated in Model 1c-iii and Model 1c-iv.

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maneuvers such as a double lane change test and also in the development of control logic foractive steer or active normal force controls.

However, equations (41) and (42) can be simplified by using the static c.g. height of thesprung mass h in place of the instantaneous lengths lsrf + rrf without introducing any errorseither in the steady state or transient responses prior to wheel lift-off. Similarly, the initial strutlength lsif can be used in place of the instantaneous strut length lsrf in equations (9) and (10) forthe unsprung mass lateral and longitudinal velocities. The unsprung mass velocities can alsobe replaced by the corner velocities to further simplify the equations. However, this results ina slight difference in the roll frequency of the model. This error is very small and can generallybe ignored.

7. Some other modeling considerations

7.1 Effect of the roll center

The inclusion of the roll center is very important in accurately predicting the roll behavior ofthe vehicle. As seen in the step steer and ramp steer responses in figures 14 and 15, the rollangles predicted are very different if no roll center is considered in the model. The steady-state tire normal forces are not substantially different but the transient peaks of the tire normalforces differ considerably. This can make the critical difference in wheel lift-off/rollover inthe fishhook maneuver, for example, and can also affect the predicted trajectory. The rollfrequency does not change significantly. Naturally, the difference in the responses will begreater if the roll centers are further away from the ground. Thus, it is recommended that theroll center height be always considered even in cases where roll angle is not calculated orimportant, such as in the lower order models (7 DOF/8 DOF) used in yaw control sytems.

Figure 14. Comparative responses – step steer – effect of roll center and tire inclination angle.

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Figure 15. Comparative responses – ramp steer – effect of roll center and tire inclination angle.

In figure 14, even though the model without roll center rolls more and predicts one wheellift-off slightly earlier than the model with roll center, it actually rolls over at a later stage.This discrepancy exists because the roll center considered in the 14 DOF model is fixed withrespect to the sprung mass c.g. and not the ground.

7.2 Effect of tire inclination angle

In the vehicle model developed, the tire was considered to remain normal to the sprung mass. Inthe case of a McPherson-type suspension, for example, this may be a reasonable assumption.However, for a double wishbone or multilink suspension, the camber angle changes are verysmall when compared with the roll angle of the body, and it may be more accurate to considerthe tire remaining normal to the ground, rather than normal to the sprung mass, until wheellift-off. Once wheel lift-off occurs, the tire may be assumed to remain normal to the line joiningthe right and left tire bottoms.

To incorporate the above assumption certain equations need to be modified. For example,the longitudinal and lateral velocities at the tire contact patch are now given as

ugrf = cos θ(usrf − (ωy · lsrf + (ωy cos φtf − ωz sin φtf)rrf))

+ sin θ(wurf cos φ + (ωx · lsrf + vsrf) sin φ + φtfrrf sin φtf)(67)

vgrf = cos φ(vsrf + ωx · lsrf) + φtfrrf cos φt − wurf sin φ (68)

and the instantaneous tire radius is calculated from the tire spring deflection as

rrf = ro − xtrf

cos θ cos φtf(69)

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The equations for the moments acting on the sprung mass and for the lateral load transferforces are also modified as shown subsequently

Mxrf = F ′ygsrf · rrf + Fysrf · lsrf (70)

Fdzlf = −Fdzrf = F ′ygsrf · rrf + Fysrf · lsrf − Fysrfhrcf

cf/2(71)

Myrf = −(Fxsgrf · rrf cos(φ − φtf) + Fxsrf · lsrf) (72)

Mzrf = Fxgsrf rrf sin(φ − φtf) (73)

where

F ′ygsrf = (cos φtfFygrf + sin φtf(Fzgrf cos θ + Fxgrf sin θ)) (74)

The angle φtf in these equations is the inclination of the tire with respect to the ground in theroll plane and can be approximated as

φtf = tan−1

(min(xtrf , 0) − min(xtlf , 0)

cf cos φ

)(75)

and the angular velocity φtf can be approximated as

if φtf = 0, φtf = 0 else φtf = ωx (76)

As seen in the step steer response in figure 14, this change in the tire inclination angle changesthe roll frequency of the model. This effect is similar to that seen in the Model 1c-iii.Also, eventhough the steady-state roll angle does not change, the tire normal forces at this roll angle aredifferent. There is less lateral load transfer for the same roll angle when the tire is consideredto remain normal to the ground. This effect can also be seen in the ramp steer responses infigure 15. Owing to this, the wheel lift-off and rollover occur later if the tire is considered toremain normal to the ground. In these simulations, the camber angle input to the tire model(needed for calculating the tire lateral forces) was kept equal to zero in both cases. Thus, thecomparison indicates how the camber angle variation can affect the vehicle rollover limit inaddition to changing the lateral forces at the tires.

If it is found suitable or desirable to consider the tire to remain normal to the ground ratherthan normal to sprung mass, or if the given camber angle variation is desired to be incorporatedinto the model, the aforementioned equations can be used. The equations can also be simplifiedalong similar lines as the original vehicle model.

7.3 Effect of link load transfer equations

In the vehicle model developed, the link load transfer forces are considered symmetrical(equation (26)). It can be argued correctly that these forces will actually not be symmetrical.The tire developing the greater lateral force will have greater jacking force component. How-ever, it is seen in our simulations that it is not the distribution of the left and right jackingforces, but the sum of their magnitudes, |Fdzrf − Fdzlf |, which has a significant impact on thevehicle responses. If the net moment transferred by the jacking forces, (Fdzrf − Fdzlf)cf , is thesame, then the actual distribution of the left and right forces does not have a significant impact.Therefore, for sake of simplicity, the link load transfer forces can be considered symmetrical.

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8. The 8 DOF full vehicle model

An 8 DOF vehicle model is often used as a simplified lower order model for studying vehi-cle handling in scenarios which do not involve significant longitudinal accelerations. In thissection, a formulation for the 8 DOF model, adapted from various references, that can matchthe 14 DOF model reasonably accurately is presented and its limitations are studied.

Figure 16 shows the schematic of the 8 DOF full vehicle model. The model has four degreesof freedom for the chassis velocities and one degree of freedom at each of the four wheelsrepresenting the wheel spin dynamics. The chassis velocities include the longitudinal velocity,u, the lateral velocity, v, the roll angular velocity, ωx , and the yaw angular velocity, ωz. Thepitch and heave motions are not modeled and the front and rear suspensions are representedsimply by their respective equivalent roll stiffness (kφf/kφr) and roll damping coefficients(bφf/bφr).

The equations of motion for the chassis velocities are obtained as given subsequently:

mt(u − ωzv) =∑

Fxgij + (mufa − murb)ω2z − 2hrcmωzωx (77)

mt(v + ωzu) =∑

Fygij + (murb − mufa)ωz + hrcmωx (78)

Jzωz + Jxzωx = (Fyglf + Fygrf)a − (Fyglr + Fygrr)b + (Fxgrf − Fxglf)cf

2

+ (Fxgrr − Fxglr)cr

2+ (murb − mufa)(v + ωzu) (79)

Figure 16. Schematic of 8 DOF full vehicle model.

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(Jx + mh2rc)ωx + Jxzωz = mghrcφ − (kφf + kφr)φ − (bφf + bφr)φ + hrcm(v + ωzu) (80)

where

hrc = hrcfb + hrcra

a + b(81)

In these equations, the forces Fxgij and Fygij are the longitudinal and lateral forces at the fourtire contact patches and the subscript ‘ij’ denotes lf, rf, lr, and rr. As before, hrcf and hrcr arethe vertical distances of the front and rear roll centers below the sprung mass c.g., and thus hrc

is the vertical distance from the sprung mass c.g. to the vehicle roll center. It should be notedthat as equation (80) for the roll degree of freedom is written by considering moments actingabout the vehicle roll center rather than the sprung mass c.g., the roll inertia of the sprungmass about the vehicle roll center (Jx + mh2

rc) is considered in equation (80).The equations for the wheel dynamics and the longitudinal and lateral tire forces are the

same as those used in the 14 DOF model (equations (3)–(6)). The longitudinal and lateralvelocities at, for example, the right front tire contact patch required in these equations aregiven as

ugrf = u + ωzcf

2(82)

vgrf = u + ωza (83)

The normal forces at the four tires are determined as

Fzglf = mgb

2(a + b)+ mufg

2−

(mufhuf

cf+ mb(hcg − hrcf)

cf(a + b)

)(v + ωzu) − (kφfφ + bφf φ)

cf

− (mhcg + mufhuf + murhur)(u − ωzv)

2(a + b)(84)

Fzgrf = mgb

2(a + b)+ mufg

2+

(mufhuf

cf+ mb(hcg − hrcf)

cf(a + b)

)(v + ωzu) + (kφfφ + bφf φ)

cf

− (mhcg + mufhuf + murhur)(u − ωzv)

2(a + b)(85)

Fzglr = mga

2(a + b)+ murg

2−

(murhur

cr+ ma(hcg − hrcr)

cr(a + b)

)(v + ωzu) − (kφrφ + bφrφ)

cr

+ (mhcg + mufhuf + murhur)(u − ωzv)

2(a + b)(86)

Fzgrr = mga

2(a + b)+ murg

2+

(murhur

cr+ ma(hcg − hrcr)

cr(a + b)

)(v + ωzu) + (kφrφ + bφrφ)

cr

+ (mhcg + mufhuf + murhur)(u − ωzv)

2(a + b)(87)

These equations are fairly simple and linearized. It is possible to include several additionalterms in the equations for the chassis velocities as well as the tire forces. However, in ourexperience, these terms have very little effect on the vehicle responses and can be ignored.As seen in the step steer responses in figure 17, the 8 DOF model as described is able to match

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Figure 17. Comparative responses – step steer – 14 DOF model and 8 DOF model.

the responses of the 14 DOF fairly accurately. However, there is no provision in the 8 DOFmodel to simulate wheel lift-off as was possible in the 14 DOF model through equation (16).Thus, as can be seen from the ramp steer responses in figure 18, the 8 DOF model cannot sim-ulate vehicle behavior beyond wheel lift-off. Nevertheless, the model is valid for applications

Figure 18. Comparative responses – ramp steer – 14 DOF model and 8 DOF model.

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which do not involve wheel lift-off such as active steering and active throttle/brake systems foryaw control.

9. Limitations of the 14 DOF model

There can be differences between the responses of the 14 DOF vehicle model developed whencompared with an actual vehicle or a multi-body dynamics based vehicle model. These errorsare due to several effects that exist in an actual vehicle suspension but are not represented inthe vehicle model.

Compliance elements are not considered in the vehicle model. The tire and suspension strutsare considered to remain normal to the sprung mass. In an actual vehicle, the camber anglevariation with wheel travel will affect the tire force generation and roll behavior. Similarly,the orientation of the struts with respect to the sprung mass also changes in a real suspension.This affects the equivalent spring/roll stiffness.

The lateral and vertical movements of the roll center that occur in an actual vehicle whenthe suspension deflects are also not considered in the 14 DOF model. These affect the net rollmoment acting on the sprung mass as well as the load transfer at the tires. This effect can beincluded in the 14 DOF vehicle model developed by including the roll center movement asan additional input into the model or by expressing the roll center location as a function ofthe wheel travel. However, the roll center is also the instantaneous center of rotation of thesprung mass with respect to the ground. If in the actual suspension there is substantial lateralmovement of the roll center, then it will affect the location, velocity, and acceleration (lateraland more importantly vertical) of the sprung mass c.g. For example, if the roll center movesin the positive v-direction during a positive roll, then it will result in a drop in the verticalposition of the sprung mass c.g. which is not modeled in the 14 DOF model mentioned earlier.This may significantly affect the vehicle roll response and the tire normal forces in extrememaneuvers. A vehicle model that considers the roll center not just as a point of application offorces but also as a kinematic link would be more accurate in such cases.

In spite of these limitations, the 14 DOF model developed is a fairly good representa-tion of the vehicle roll and handling behavior in most cases. It can be used in anticipatingvehicle rollover and for the development of integrated chassis controls that include activesuspension/active roll control systems and active steering and active throttle/brake control.

10. Conclusions

A 14 DOF full vehicle model with one-dimensional suspension and fixed roll center, whichcan simulate vehicle rollover, was developed and validated with CARSIM. The effect ofsimplifications of certain key equations on the vehicle responses was studied to understandthe limitations and extent of validity. Similarly, the effects of including the roll center, thetire inclination angle, and the symmetry/asymmetry of jacking forces were also analyzed. Aformulation for a lower order 8 DOF model is presented, which is able to match the 14 DOFmodel fairly accurately until wheel lift-off. Finally, the possible limitations of the 14 DOFmodel when compared with an actual suspension are discussed.

It is expected that the results and analyses presented here prove useful in answering questionsregarding the validity and applicability of various vehicle models and the effect of certain keymodeling assumptions. The limitations of the 14 DOF model with one-dimensional suspen-sion developed also indicate a need for a vehicle model that can incorporate the effects of a

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moving roll center which acts both as a point of application of forces as well as a kinematicconstraint.

Vehicle parametersSprung mass m = 1440 kgSprung mass roll inertia Jx = 900 kg m2

Sprung mass yaw inertia Jy = 2000 kg m2

Sprung mass pitch inertia Jz = 2000 kg m2

Distance of sprung mass c.g. from front axle a = 1.016 mDistance of sprung mass c.g. from rear axle b = 1.524 mSprung mass c.g. height h = 0.75 mFront/rear track width cf = cr = 1.5 mFront suspension stiffness ksf = 35,000 N/mFront suspension damping coefficient bsf = 2500 Ns/mRear suspension stiffness ksr = 30,000 N/mRear suspension damping coefficient bsr = 2000 Ns/mFront/rear unsprung mass muf = mur = 80 kgFront/rear tire stiffness ktf = ktr = 200,000 N/mNominal tire radius ro = 0.285 mTire/wheel roll inertia Jw = 1 kg m2

Front roll center distance below sprung mass c.g. hrcf = 0.65 mRear roll center distance below sprung mass c.g. hrcr = 0.6 m

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understanding the limitations of different vehicle models for roll dynamics studies

Documents

actual vehicle

validity of vehicle

realistic vehicle model

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various vehicle maneuvers

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dof model

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