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Investigation into untripped rollover of light vehicles in the modifiedfishhook and the sine maneuvers. Part I: Vehicle modelling, roll and yawinstabilityNong Zhang a; Guang-Ming Dong b; Hai-Ping Du a

a Mechatronics and Intelligent Systems, University of Technology, Sydney, Australia b State KeyLaboratory of Vibration, Shock&Noise, Shanghai Jiao Tong University, Shanghai, P.R. China

To cite this Article Zhang, Nong, Dong, Guang-Ming and Du, Hai-Ping(2008) 'Investigation into untripped rollover of lightvehicles in the modified fishhook and the sine maneuvers. Part I: Vehicle modelling, roll and yaw instability', VehicleSystem Dynamics, 46: 4, 271 — 293To link to this Article: DOI: 10.1080/00423110701344752URL: http://dx.doi.org/10.1080/00423110701344752

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Vehicle System DynamicsVol. 46, No. 4, April 2008, 271–293

Investigation into untripped rollover of light vehicles in themodified fishhook and the sine maneuvers. Part I: Vehicle

modelling, roll and yaw instability

NONG ZHANG*†, GUANG-MING DONG‡ and HAI-PING DU†

†Mechatronics and Intelligent Systems, University of Technology, Sydney, P.O. Box 123,Broadway, NSW 2007, Australia

‡State Key Laboratory of Vibration, Shock&Noise, Shanghai Jiao Tong University,Shanghai, P.R. China

Vehicle rollovers may occur under steering-only maneuvers because of roll or yaw instability. In thispaper, the modified fishhook and the sine maneuvers are used to investigate a vehicle’s rollover resis-tance capability through simulation. A 9-degrees of freedom (DOF) vehicle model is first developedfor the investigation. The vehicle model includes the roll, yaw, pitch, and bounce modes and passiveindependent suspensions. It is verified with the existing 3-DOF roll–yaw model. A rollover criticalfactor (RCF) quantifying a vehicle’s rollover resistance capability is then constructed based on thestatic stability factor (SSF) and taking into account the influence of other key dynamic factors.

Simulation results show that the vehicle with certain parameters will rollover during the fishhookmaneuver because of roll instability; however, the vehicle with increased suspension stiffness, whichdoes not rollover during the fishhook maneuver, may exceed its rollover resistance limit because ofyaw instability during the sine maneuver. Typically, rollover in the sine maneuver happens after severalcycles.

It has been found that the proposed RCF well quantifies the rollover resistance capability of a vehiclefor the two specified maneuvers. In general, the larger the RCF, the more kinetically stable is a vehicle.A vehicle becomes unstable when its RCF is less than zero. Detailed discussion on the effects of keyvehicle system parameters and drive conditions on the RCF in the fishhook and the sine maneuver ispresented in Part II of this study.

Keywords: Untripped vehicle rollover; Light vehicle; Vehicle model; Roll instability; Yaw instability

Nomenclature

u0 Constant longitudinal velocity of the sprung mass centre of gravity (CG)v Lateral velocity of the sprung mass CGzs Vertical displacement of the sprung mass CGzui Vertical displacement of the unsprung masses i = 1, 2, 3, 41, 2, 3, 4 Left-front, Right-front, Left-rear, and Right-rear, respectivelyS∗ The earth-fixed inertial reference frameSv The vehicle-fixed non-inertial reference frameφ Roll angle of the sprung mass around x-axis of Sv

θ Pitch angle of the sprung mass around y-axis of Sv

*Corresponding author. Email: [email protected]

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online © 2008 Taylor & Francis

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272 N. Zhang et al.

� Yaw angle of the vehicle around z-axis of Sv or S∗p Roll rate of the sprung mass around x axis of Sv

q Pitch rate of the sprung mass around y axis of Sv

r Yaw rate of the vehicle around z axis of Sv or S∗ms Vehicle sprung massmui Vehicle unsprung mass(Ixx)s Moment of inertia about the x-axis of the vehicle sprung mass(Iyy)s Moment of inertia about the y-axis of the vehicle sprung mass(Izz)s Moment of inertia about the z-axis of the vehicle sprung mass(Ixz)s Product of inertia about the x–z axes of the vehicle sprung mass(Izz)ui Moment of inertia about the z-axis of the vehicle unsprung massa Distance from the sprung mass CG to the front axleb Distance from the sprung mass CG to the rear axletf 1/2 width of the front axletr 1/2 width of the rear axlehs Height of the sprung mass CG above the roll axishra Height of the roll axis above the groundKsi Spring stiffness for each suspension i = 1, 2, 3, 4Csi Damping coefficient for each suspension i = 1, 2, 3, 4zsui Change of spring length for each suspension i = 1, 2, 3, 4Kt Vertical tyre stiffnesszgi Road disturbance input for each tyre i = 1, 2, 3, 4δf Front tyre steer anglecδφ Partial derivative of the roll induced steer at the rear axleδr Roll induced steer angle at rearcγφ Inclination angle coefficientγ Inclination angleαf Front tyre slip angleαr Rear tyre slip angle

1. Introduction

Rollover accidents are dangerous events. According to the National Highway Traffic SafetyAdministration (NHTSA) of USA [1], although only 8% of light vehicles (passenger cars,pick-ups, vans and sport utility vehicles) in crashes roll over, 21% of seriously injured occu-pants and 31% of occupant fatalities are involved in rollovers. Rollover accidents can bedivided into off-road and on-road rollover, where on-road rollover can be further subdividedinto tripped and untripped rollover depending on the mechanism that initiated the on-roadrollover. Rollover crash data show that approximately two-thirds of on-road rollovers areuntripped [2].

Rollover incidents involve a variety of factors and some of which may be beyond the controlof vehicle designers. The on-road untripped rollovers are directly connected with vehiclehandling behaviour that is mainly influenced by vehicle design variables, thus researcheson this subject for light vehicles are highly demanding. Experimental evaluations of certainmaneuvers on selected vehicle types have been made and published by NHTSA [2–4], whichtried to quantify on-road, untripped rollover propensity of light vehicles and incorporate the testresults into the New CarAssessment Program. Baumann and Eckstein [5] performed extensivefield testing and used the recorded wheel inputs in their simulations, and they discovered


Vehicle modelling, roll and yaw instability 273

that the main conditions necessary for untripped rollover were to create high roll rates andmaximum side forces at both axles simultaneously.

Experimental investigation of vehicle rollover crashes are not only expensive and time-consuming but also limited to those maneuvers that can be physically reconstructed. Therefore,more attention has been paid to the investigations into rollover crashes based on computersimulations. Many researchers [6–15] generally studied the effects of vehicle parameters anddriving conditions on stability and handling, which is helpful in the preliminary phases ofvehicle design. Others [16–20] researched on the control of untripped vehicle rollovers byadding electronic equipment, which is beyond the main concern of this paper. In general,vehicle models used by researchers for rollover study can be classified into four categories.Brief descriptions are given for each type of model: (1) yaw-plane model [6–8], which includesall the axles of the vehicle but no roll degree of freedom (DOF) and is often used in vehicle yawstability analysis; (2) roll-plane model [7–11], which consists of a single axle with associatedsuspensions and body mass free to roll only. This model is often used in heavy vehicle rolloveranalysis, tripped rollover analysis, or simple static analysis for untripped rollover; (3) roll–yaw model [12, 13, 16–18], which was initially proposed by Segel [21] , includes both rolland yaw dynamics and is often used by researchers for vehicle rollover analysis and controlfor its simplicity and connectivity between wheel input and vehicle’s roll–yaw responses; (4)full-car models used in [10, 14, 15, 19, 20], which include all of the bounce, pitch, roll, and yawdynamics, are often used in commercial vehicle dynamics programs. The mathematical modelsused in these full-car commercial programs are not publicly and easily accessible. This makesit difficult for researchers to conduct detailed parametric study and in-depth investigation. Inthis paper, a 9-DOF full-car model is developed. The model includes the roll, yaw, pitch, andbounce modes of vehicle body and passive independent suspensions. This full-car model isconstructed under the assumption of constant forward velocity for simplicity and is discussedin detail in section 2.

To assess the on-road, untripped rollover propensity of a vehicle, NHTSA uses a set ofrollover evaluation maneuvers and examines the severity and frequency of vehicle two-wheellift (TWL) under these maneuvers [2]. In the simulation of this paper, two types of maneuversof a light vehicle on a flat dry road surface with various parameters are investigated. One isthe modified fishhook maneuver and the other is the sine maneuver for multiple cycles withconstant frequency.

NHTSA [2] described their type-1a fishhook maneuver in an experiment as follows: ‘Tobegin this maneuver, the vehicle was driven in a straight line at a speed slightly greater thanthe desired entrance speed. The driver released the throttle, and when at target speed, initiatedthe hand wheel commands described in figure 1. Following completion of the counter steer,hand wheel position was maintained for 3 s. The hand wheel was then returned to zero’.Clearly the vehicle speed will decrease to some extent during the maneuver because of tyre-rolling resistance, scrub during turning, and no supplied drive torque. However, for simplicity,constant speed in the fishhook maneuver is assumed in this paper to obtain some aggravatedsimulation results from a viewpoint of safety for real applications. The second modificationon the maneuver is made to the steering amplitude A, which is varied for the purpose ofparametric study.

The sine maneuver for multiple cycles with constant frequency shown in figure 2 is modifiedfrom the single sine steer maneuver and sine sweep steer maneuver used in [18]. In thesingle sine maneuver, vehicle is run for one cycle at resonant yaw rate frequency, whichmay neglect something as cycle number increases; in the sine sweep maneuver, results canonly be analysed in frequency domain. In the simulations on the sine maneuver presentedin this paper, the vehicle is run under the sine maneuver for multiple cycles with constantfrequency ranging from 0.1–1.1 Hz, from which results can be evaluated both in time domain


274 N. Zhang et al.

0 0.5 1 1.5 2 2.5 3

-A

0

A

time [s]

stee

ring

ampl

itude

[deg

ree]

A = steering amplitude

T1 = commanded dwell time

Initial steer and conter steerperformed at 720 deg/seg

T1

Figure 1. Fishhook maneuver description.

Figure 2. Description for sine maneuver.

for considering multi-cycle effect and in frequency domain for enhanced resolution. Further-more, it is somewhat like the slalom vehicle test, which aims to evaluate the vehicle handlingcharacteristics.

The presentation of this paper is organized as follows: section 2 provides the derivationdetail of a 9-DOF vehicle model, which includes the roll, yaw, pitch, and bounce modes andpassive independent suspensions. In section 3, the proposed vehicle model is verified with theexisting 3-DOF roll–yaw model using the same vehicle data and steering inputs. A rollovercritical factor is constructed for assessing the rollover resistance capability of a light vehicle. Insection 4, vehicle rollovers due to roll instability in the fishhook maneuver and yaw stabilityin the sine maneuver are investigated. Conclusion drawn from Part I of this research is givenin section 5.

2. Vehicle model description

The vehicle model shown in figure 3 consists of a rigid sprung mass supported by four inde-pendent suspensions, which includes wheel assemblies as four unsprung masses. Due to theconstraints between the rigid bodies, the unsprung masses translate laterally together withthe sprung mass, which means that the unsprung masses have no independent lateral DOFcompared with the sprung mass. In yaw rotation, the unsprung masses rotate together with thesprung mass relative to the earth-fixed inertial reference; however, the roll and pitch rotationsare restricted only to the sprung mass.

Under the assumption of a constant forward speed and the above constraints, the numberof DOF of the vehicle model is set to nine: sprung mass lateral and vertical centre-of-mass



Figure 3. Nine-DOF vehicle model.

motions, roll and pitch rotations of the sprung mass about the axis at its CG, yaw rotation forthe total vehicle, and four centre-of-mass vertical motions of the unsprung masses.

Denote with S∗ the earth-fixed inertial reference frame and with Sv a vehicle fixed non-inertial reference frame rotating with the angular velocity ω⇀v = [0 0 r]T and translatingwith the velocity v⇀o = [u0 v 0]T. Equations of motion of the 9-DOF vehicle model arederived as follows, under the assumption of the fixed roll axis concept.

2.1 Kinematics

2.1.1 Translation of CG of sprung mass. The position of CG of sprung mass in thereference coordinate Sv is:

l⇀os = Ry(θ)Rx(φ)

⎡⎢⎣

0

0

−hs

⎤⎥⎦ +

⎡⎢⎣

0

0

zs

⎤⎥⎦ =

⎡⎢⎣

−hs cos φ sin θ

hs sin φ

−hs cos φ cos θ + zs

⎤⎥⎦ (1)

where

Rx(φ) =⎡⎢⎣

1 0 0

0 cos φ − sin φ

0 sin φ cos φ

⎤⎥⎦ (2)

Ry(θ) =⎡⎢⎣

cos θ 0 sin θ

0 1 0

− sin θ 0 cos θ

⎤⎥⎦ (3)

Then: (d

dt

)∗l⇀

os =(

d

dt

)v

l⇀

os + ω⇀v × l⇀

os (4)

where ‘×’ denotes the cross-product operator. The velocity of vehicle sprung mass CG pointwith respect to S∗ is expressed as the sum of the translational velocity v⇀o of the vehicle reference


276 N. Zhang et al.

system Sv and the time derivative of the position vector l⇀

os :

v⇀s = v⇀o +(

d

dt

)∗l⇀

os = v⇀o +(

d

dt

)v

l⇀

os + ω⇀v × r⇀os = v⇀o + l⇀

os + ω⇀v × l⇀

os (5)

The acceleration of vehicle sprung mass CG point with respect to S∗ is computed as:

a⇀s =(

d

dt

)∗v⇀s =

(d

dt

)v

v⇀s + ω⇀v × v⇀s = v⇀s + ω⇀v × v⇀s (6)

2.1.2 Translations of CGs of unsprung masses. The positions of CGs of unsprung massesin the reference coordinate Sv are:

l⇀

ou1 =⎡⎢⎣

a

−tf

−hu1 + zu1

⎤⎥⎦, l

⇀

ou2 =⎡⎢⎣

a

tf

−hu2 + zu2

⎤⎥⎦,

l⇀

ou3 =⎡⎢⎣

−b

−tr

−hu3 + zu3

⎤⎥⎦, l

⇀

ou4 =⎡⎢⎣

−b

−tr

−hu4 + zu4

⎤⎥⎦ (7)

Here, we denote with 1 the left-front unsprung mass, 2 the right-front, 3 the left-rear, and 4the right-rear unsprung mass.

Then the velocities and accelerations of vehicle unsprung mass CG points with respect toS∗, v⇀ui and a⇀ui , i = 1, 2, 3, 4, can be obtained using equations (5) and (6).

2.1.3 Rotations of sprung mass and unsprung masses. The angular momentum of thesprung mass can be defined as:

H⇀

s = msl⇀

s l⇀

os × v⇀s + Isω⇀

s = (H⇀

CG)s + (H⇀

ω)s (8)

where (H⇀

CG)s is the angular momentum of the sprung mass due to the motion of the CG, (H⇀

ω)s

is the angular momentum of the mass due to its rotation about the CG, and ω⇀s = [p q r]T.The angular momentum of unsprung mass can be defined as:

H⇀

u =4∑

i=1

[mui (r⇀

oui × v⇀ui) + Iuiω⇀

u] (9)

As unsprung masses only have yaw rotation, then

H⇀

u = [0 0 Huz

]T(10)

The angular momentum of total vehicle can be written as:

H⇀ = H

⇀

s + H⇀

u (11)



2.2 Kinetics

2.2.1 Lateral equation for the entire vehicle mass.

∑Fty = msasy +

4∑i=1

muiauyi (12)

∑Fty = (Fy1 + Fy2) cos(δf) + (Fy3 + Fy4) cos(δr) (13)

where Fyi, i = 1, 2, 3, 4, is the tyre lateral force.

2.2.2 Vertical equation for the sprung mass.

−4∑

i=1

Fsi = msasz (14)

where Fsi is the suspension force, we denote with 1 the left-front suspension, 2 the right-front,3 the left-rear, and 4 the right-rear suspension. Fsi can be computed as:

Fsi = Ksizsui + Csi zsui i = 1, 2, 3, 4 (15)

where Ksi is the spring stiffness, Csi the damping coefficient, and zsui the change of springlength for each suspension. Zsui can be computed as:

left-front : zsu1 = (zs − tfφ − aθ) − zu1 (16)

right-front : zsu2 = (zs − tfφ − aθ) − zu2 (17)

left-rear : zsu3 = (zs − trφ + bθ) − zu3 (18)

right-rear : zsu4 = (zs + tfφ + bθ) − zu4 (19)

In an effort to maintain generalized applicability to different types of suspension, suspensionforces can be modified to have their equivalent forces acting at the track width (i.e., at eachwheel) as above.

2.2.3 Vertical equations for the unsprung masses.

Fsi − Kt(zui − zgi ) = muzui i = 1, 2, 3, 4 (20)

where Kt is the vertical tyre stiffness and zgi the road disturbance input for each tyre.

2.2.4 Roll equation for the sprung mass.

∑Mx = Hx − rHy = (Fs1 − Fs2)tf + (Fs3 − Fs4)tr + msghsφ (21)

2.2.5 Pitch equation for the sprung mass.

∑My = Hy + rHx = (Fs1 − Fs2)a − (Fs3 + Fs4)b + msghsθ (22)


278 N. Zhang et al.

2.2.6 Yaw equation for the entire vehicle mass.∑Mz = Hz = a(Fy1 + Fy2) cos(δf) − b(Fy3 + Fy4) cos(δr)

+ tf(Fy2 − Fy1) sin(δf) + tr(Fy4 − Fy3) sin(δr)

+ Mz1 + Mz2 + Mz3 + Mz4 (23)

where Mzi, i = 1, 2, 3, 4 is the tyre self-aligning moment.

2.3 Tyre force calculation

Magic-Formula (MF) tyre models are considered the state-of-the-art for modeling tyre–roadinteraction forces and moments in vehicle dynamics. Since 1987, Pacejka and others havepublished several versions of this type of tyre model. The latest version of MF published byPacejka [22] is used in this paper.

The input for the MF tyre model consists of the wheel load (Fz), the longitudinal and lateralslip (κ, α), and inclination angle (γ ). The output are the forces (Fx , Fy) and self-aligningmoment Mz in the contact point between the tyre and the road. To calculate these forcesand moments, the MF equations use a set of MF parameters, which are derived from tyretesting data.

For pure slip conditions, the MF equations can be described as:

Y (x) = D sin[C arctan{Bx − E(Bx − arctan(Bx))}] (24)

where Y (x) is either Fx with x the longitudinal slip κ or Fy with x the lateral slip α. Thecoefficient B, C, D, and E depend on the wheel load Fz and the inclination angle γ .

The self-aligning moment Mz is due to the offset of lateral force Fy , called pneumatic trailt , from the contact point. A cosine version the MF equation is used to calculate the pneumatictrail t :

Y (x) = D cos[C arctan{Bx − E(Bx − arctan(Bx))}] (25)

in which Y (x) is the pneumatic trail t as a function of lateral slip angle α.In combined slip conditions, the lateral force Fy will decrease due to the longitudinal slip κ ,

and the longitudinal force Fx will decrease due to the lateral slip α. The forces and momentsin the combined slip conditions are based on the pure slip characteristics multiplied by someweighing functions.

The front and rear tyre slip angle can be computed as follows from figure 4:

front tyre slip angle : αf = v + ar

u0− δf (26)

rear tyre slip angle : αr = v − br

u0− δr (27)

where δf is the front tyre steer angle and δr = φ, Cδφ is the roll-induced steer angle at the reartyre, Cδφ is determined by geometric properties of the rear suspension.

The tyre longitudinal slip κ is set to zero because of the constant vehicle longitudinal velocityassumption, and the inclination angle γ is computed as the multiplication of roll angle by acoefficient.

A set of scaling factors is available in the MF tyre model to examine the influence ofchanging tyre properties without the need to change one of the real MF coefficients. γFz0 isthe scaling factor for tyre nominal load. As can be seen in figure 5, tyre lateral stiffness andmaximum tyre lateral force will increase for a given tyre normal load if γFz0 is increased.



Figure 4. Top view of the model.

–20 –10 0 10 20

–4000

–2000

0

2000

4000

Tyre Slip Angle (Degree)

Tyre

Lat

eral

For

ce (N

)

Fz = 4000Nγ = 0

increase of λFz0

increase of λFz0

Figure 5. Influence of scaling factor λFz0 on tyre lateral force.

3. Model comparison and description of rollover critical factor

A light passenger vehicle, of which parameters are listed in Appendix A, is used to conduct thesimulation of 9-DOF vehicle model. The 3-DOF roll–yaw model presented by Segel [21] isused for cross-checking the accuracy of the 9-DOF model. For both the 3- and 9-DOF models,the MF tyre model is used. Transformed vehicle parameters from 9- to 3-DOF model are alsolisted in Appendix A.

The two maneuvers described in section 1 are used for the comparison simulations betweenthe existing 3- and 9-DOF models proposed in this paper. Common parameters used for thetwo models are:

• front and rear tyre – 195/65 R15 car tyre;• constant vehicle longitudinal velocity – u0 = 75 km/h;• steer ratio from hand wheel to road tyre is 20;• maximum wheel angle for two maneuvers is 270◦;• steering frequency for sine steer is 0.3 Hz.


280 N. Zhang et al.

Figure 6. Comparison of vehicle lateral acceleration between the two vehicle models under fishhook maneuver.

The vehicle lateral acceleration, roll angle, roll rate, and yaw rate for the fishhook maneu-ver are obtained from the two vehicle models, respectively, and are shown in figures 6–9 forcomparison. Differences of vehicle lateral accelerations and yaw rates between the two mod-els are small. However, roll angle and roll rate of the 9-DOF model are obviously larger thanthat of the 3-DOF model. By analysing the suspension force equations (15)–(19) and rollequation (21), it can be deduced that the differences between vertical displacements of theunsprung masses zui , i = 1, 2, 3, 4, cause the roll angle and roll rate differences between thetwo models. If zui , i = 1, 2, 3, 4, are set to zero in the 9-DOF model, the differences becomevery small, as shown in figures 10 and 11.

The vehicle lateral acceleration, roll angle, roll rate, and yaw rate for the sine maneuverare obtained from the two vehicle models, respectively, and are shown in figures 12–15 forcomparison. Differences of vehicle lateral acceleration and yaw rate between the two modelsare small. However, roll angle and roll rate of the 9-DOF model are obviously larger thanthat of the 3-DOF model, as shown in figures 13 and 14. If zui , i = 1, 2, 3, 4, are set to zero

0 2 4 6 8

–6

–4

–2

0

2

4

6

Time(s); [maxmum error: 10.4218%]

Rol

l Ang

le (D

egre

e)

9d of vehicle model3d of vehicle model

Figure 7. Comparison of roll angle between the two vehicle models under fishhook maneuver.



0 2 4 6 8

–30

–20

–10

0

10

20


Rol

l Rat

e (D

egre

e/s)


Figure 8. Comparison of roll rate between the two vehicle models under fishhook maneuver.

in the 9-DOF model, the roll angle and roll rate differences become very small, as shown infigures 16 and 17.

To evaluate a vehicle’s rollover resistance capability, static stability factor (SSF) is a basicindicator, which has been incorporated into NHTSA’s rollover rating program [23].A vehicle’sSSF is calculated using the formula SSF = T/2H, where T is the ‘track width’ of the vehicleand H the ‘height of the CG’ of the vehicle. The lower the SSF number, the more likely thevehicle is to roll over.

In SSF, vehicle lateral acceleration level represents the only, or the first order, characteristicsfor vehicle rollover stability. Hac [8] suggested many modifications to SSF. These include theeffects of lateral movement of vehicle CG during body roll, suspension jacking forces, changesin track width due to suspension kinematics, tyre lateral compliance, tyre gyroscopic forces,and dynamic overshoot in the roll angle. In [24], Dynamic Stability Index (DSI) is proposedand discussed. The DSI is based on the SSF model with the addition of the roll moment of

0 2 4 6 8

–30

–20

–10

0

10

20

30

40


Yaw

Rat

e (D

egre

e/s)


Figure 9. Comparison of yaw rate between the two vehicle models under fishhook maneuver.


282 N. Zhang et al.

0 2 4 6 8–6

–4

–2

0

2

4

6


Rol

l Ang

le (D

egre

e)


Figure 10. Comparison of roll angle between the two vehicle models under fishhook maneuver.

0 2 4 6 8

–30

–20

–10

0

10

20


Rol

l Rat

e (D

egre

e/s)


Figure 11. Comparison of roll rate between the two vehicle models under fishhook maneuver.

0 2 4 6 8 10–10

–5

0

5

10


Late

ral a

ccel

erat

ion

in g

loba

l coo

rdin

ates

(m/s

2 )


Figure 12. Comparison of vehicle lateral acceleration between the two vehicle models under sine maneuver.



0 2 4 6 8 10

–6

–4

–2

0

2

4

6

8

10


Rol

l Ang

le (D

egre

e)


Figure 13. Comparison of roll angle between the two vehicle models under sine maneuver.

the sprung mass (Ixx)sφ, where φ denotes the roll acceleration. In addition to DSI, vehiclehalf-track width is modified to account for the lateral CG offset of a vehicle.

The rollover critical factor (RCF) that quantifies the rollover resistance capability of a lightvehicle is defined as follows:

Factor = g

[(tf + tr)

2− hs|φ|

]− |ay |[(hs + hra) − zs] − (Ixx)s

∣∣φ∣∣ms

(28)

where ay is vehicle lateral acceleration and g the gravity acceleration. RCF accounts for thedecrease of vehicle half-track width due to roll angle, the increase of vehicle CG height dueto the CG vertical displacement, and the influence of roll moment of the sprung mass. Asdescribed by equation (28), smaller RCF indicates low rollover resistance capability of avehicle and rollover will occur when RCF becomes negative.

0 2 4 6 8 10–30

–20

–10

0

10

20

30

40


Rol

l Rat

e (D

egre

e/s)


Figure 14. Comparison of roll rate between the two vehicle models under sine maneuver.


284 N. Zhang et al.

0 2 4 6 8 10

–30

–20

–10

0

10

20

30

40

50


Yaw

Rat

e (D

egre

e/s)


Figure 15. Comparison of yaw rate between the two vehicle models under sine maneuver.

0 2 4 6 8 10–6

–4

–2

0

2

4

6

8


Rol

l Ang

le (D

egre

e)


Figure 16. Comparison of roll angle between the two vehicle models under sine maneuver.

0 2 4 6 8 10–30

–20

–10

0

10

20

30

40


Rol

l Rat

e (D

egre

e/s)


Figure 17. Comparison of roll rate between the two vehicle models under sine maneuver.



4. Vehicle rollover due to roll instability or yaw instability

In some cases, vehicle rollovers take place due to roll instability. This instability is demon-strated in the fishhook maneuver simulation, of which the vehicle velocity is set to 75 km/hand the maximum steering angle is specified as 270◦. Three sets of vehicle parameters areused in the simulation: in addition to the normal vehicle parameters in Appendix A, suspensionstiffness is reduced to 60% of its original value for the second simulation case and increasedto 140% of its original value for the third case. To enhance the comparison effect, the sprungmass height hs is increased to 0.4 m with hra reduced to 0.2 m for a constant SSF value in thedecreased stiffness case.

The RCF for the three cases under the fishhook maneuver are computed and shown infigure 18. For the reduced suspension stiffness case, the RCF value is obviously smaller thanthat of the normal and increased stiffness cases. Figures 19–22 show that the major effect

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Time (s)

Rol

love

r Crit

ical

Fac

tor

Ks DecreaseNormalKs Increase

Figure 18. Comparison of RCF for different suspension stiffness under fishhook maneuver.

0 1 2 3 4 5 6 7-10

-5

0

5

10

Time (s)

Veh

icle

Lat

eral

Acc

eler

atio

n (m

/s2 )


Figure 19. Comparison of vehicle lateral acceleration for different suspension stiffness under fishhook maneuver.


286 N. Zhang et al.

0 1 2 3 4 5 6 7–0.015

–0.01

–0.005

0

0.005

0.01

0.015

Time (s)

CG

Ver

tical

Dis

plac

emen

t (m

)


Figure 20. Comparison of CG vertical displacement for different suspension stiffness under fishhook maneuver.

0 1 2 3 4 5 6 7–15

–10

–5

0

5

10

15

Time (s)

Rol

l Ang

le (D

egre

e)


Figure 21. Comparison of roll angle for different suspension stiffness under fishhook maneuver.

0 1 2 3 4 5 6 7–200

–150

–100

–50

0

50

100

150

200

Time (s)

Rol

l Acc

eler

atio

n (D

egre

e/s2 )


Figure 22. Comparison of roll acceleration for different suspension stiffness under fishhook maneuver.



0 1 2 3 4 5 6 7

–30

–20

–10

0

10

20

30

Time (s)

Yaw

Rat

e (D

egre

e/s)


Figure 23. Comparison of yaw rate for different suspension stiffness under fishhook maneuver.

5 10 15 20 250

2

4

6

8

Time (s)

RC

F

2 4 6 8 10–0.2

0

0.5

Cycles

Min

per

cyc

le

Figure 24. Comparison of RCF between the increased and the normal suspension stiffness under sine maneuver.

5 10 15 20–10

0

10

Time (s)

Acc

eler

atio

n (m

/s2 )

2 4 6 8 10

9.6

9.7

9.8

Cycles

Max

per

Cyc

le

Figure 25. Comparison of vehicle lateral acceleration between the increased and the normal suspension stiffnessunder sine maneuver.


288 N. Zhang et al.

5 10 15 20

–1

0

1

2

Time (s)Zs

(mm

)

2 4 6 8 10

–1.6

–1.4

–1.2

–1

Cycles

Min

per

Cyc

le

Figure 26. Comparison of CG vertical displacement between the increased and the normal suspension stiffnessunder sine maneuver.

5 10 15 20 25

-5

0

5

Time (s)

Rol

l Ang

le (D

eg)

2 4 6 8 105

6

7

Cycles

Max

per

Cyc

le

Figure 27. Comparison of roll angle between the increased and the normal suspension stiffness under sine maneuver.

5 10 15 20–200

0

200

Time (s)

Rol

l Acc

eler

atio

n

2 4 6 8 10

160

200

Cycles

Max

per

Cyc

le

Figure 28. Comparison of roll acceleration between the increased and the normal suspension stiffness under sinemaneuver.



5 10 15 20–60

–40

–20

0

20

40

60

Time (s)

Yaw

Rat

e (D

egre

e/s)

Figure 29. Comparison of yaw rate between the increased and the normal suspension stiffness under sine maneuver.

on RCF comes from roll angle variation shown in figure 21; influences of other factors arepositive but insignificant. Figure 23 shows that the differences of vehicle yaw rates for thesethree cases are small.

The results of the simulation case with increased suspension stiffness in the fishhook maneu-ver show higher rollover resistance capability than that of the normal stiffness case. However,the same vehicle with increased suspension stiffness rolls over due to yaw instability in thesine maneuver simulation, in which the vehicle runs under the sinusoidal steering input withmaximum angle 270◦ at a frequency of 0.46 Hz for 11 cycles, maintaining a constant vehiclevelocity 75 km/h.

In figures 24–31, the results of the vehicle, with which the suspension stiffness is increasedto 140% of its normal value, are represented in solid lines and that of the normal vehicle

0 5 10 15 20 25

–10

0

10

Ang

le (D

egre

e)

Front Tyre Slip Angle ( ); Rear Tyre Slip Angle (---)

0 5 10 15 20 25

–10

0

10

Time (s)

Ang

le (D

egre

e)

__

Ks

1.4 Ks

Figure 30. Comparison of tyre slip angle.


290 N. Zhang et al.

2 4 6 8 100.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (s)

Tim

e La

g pe

r Cyc

le(s

)

Figure 31. Comparison of time lag for each cycle.

are in dashed lines. Figure 24 shows that the minimum RCF values per cycle for the vehiclewith normal value remain constant and positive after the second cycle; however, the RCFfor the vehicle with increased suspension stiffness decreases while the number of steeringcycle increases and it becomes negative after the seventh cycle steering. Factors influencingthe RCF of a vehicle are studied, and the obtained results are given in figures 25–27. For thevehicle with increased stiffness, its maximum lateral acceleration, highest vehicle CG verticalposition, maximum roll angle, and roll acceleration per cycles all increase until certain cyclesand then remain constant, which result in the reduced RCF. For the normal vehicle in dashedlines, these values generally remain unchanged after the second cycle. Figure 29 shows thatthe yaw rates for the vehicle with increased suspension stiffness have an obvious ascendingtrend with respect to time, and this indicates the vehicle’s yaw instability. The lower partof figure 30 shows that the rear tyre slip angle is larger than the front tyre slip angle after15 s for the vehicle with increased stiffness, which indicates the over steering characteristics.At the same time, time lag per cycle between steer input and vehicle lateral accelerationbecomes larger and then remains constant after a certain number of cycles, which is shownin figure 31.

The above presented simulation results may partly explain the rollover accident reportedin [25], which described that a sports utility vehicle suddenly rolled over (untripped) afterhaving crossed several cones in a slalom test.

5. Conclusions

To evaluate a vehicle’s rollover propensity through simulation, a 9-DOF vehicle model hasbeen developed. The model includes the roll, yaw, pitch, and bounce modes of a vehiclebody and passive independent suspensions. Model verification against the existing 3-DOFroll–yaw vehicle model has been carried out, and the validity of the proposed 9-DOF vehiclemodel has been confirmed. The RCF has been constructed for evaluating a vehicle’s rolloverpropensity through simulation based on the SSF and taking into account the influence of otherkey dynamic factors.



The RCF depends explicitly on those key dynamic state variables of a vehicle dynamicsystem such as the vertical displacement, the roll angle, the lateral and the roll angular accel-erations of sprung mass. In comparison to the 3-DOF vehicle model, the proposed 9-DOFmodel provides the more accurate prediction of the outboard shift of the sprung mass CG,increase in CG height, and roll acceleration in extreme fishhook and sine maneuvers, whichaffects the vehicle RCF significantly.

The obtained simulation results show that the fishhook maneuver mainly tests the vehicle’sroll instability, whereas the sine maneuver evaluates the vehicle’s yaw instability. A vehiclewith increased suspension stiffness can improve its rollover resistance capability in the fish-hook maneuver. However, rollover may occur to the same vehicle due to its increased yawinstability in the sine maneuver.

It has been found that the proposed RCF well quantifies the rollover resistance capabilityof a vehicle for the two specified maneuvers. The RCF takes into account both roll and yawinstabilities. In general, the larger the RCF, the more kinetically stable is a vehicle. A vehiclebecomes unstable when its RCF is less than zero. Detailed discussion on the effects of keyvehicle system parameters and drive conditions on the RCF is presented in Part II of this study.

Acknowledgements

Financial support for this research was provided jointly by the Australian Research Council(DP0560077) and the University of Technology, Sydney, Australia.

References

[1] National Highway Traffic Safety Administration, 2003, Initiative to address the mitigation of vehicle rollover.Docket No. NHTSA-2003-14622-1.

[2] Forkenbrock, G.J., Garrott, W.R., Heitz, M. and O’Harra, B.C., 2002, A comprehensive experimental examina-tion of test maneuvers that may induce on-road, untripped, light vehicle rollover - phase IV of NHTSA’s lightvehicle rollover research program. DOT HS 809 513, National Highway Traffic Safety Administration.

[3] Forkenbrock, G.J., O’Harra, B.C. and Elsasser, D., 2003, An experimental evaluation of 26 light vehicles usingtest maneuvers that may induce on-road, untripped rollover and a discussion of NHTSA’s refined test procedures- phases VI and VII of NHTSA’s light vehicle rollover research program. DOT HS 809 704, National HighwayTraffic Safety Administration.

[4] Forkenbrock, G.J., O’Harra, B.C. and Elsasser, D., 2004, A demonstration of the dynamic tests developed forNHTSA’s NCAP rollover rating system - phase VIII of NHTSA’s light vehicle rollover research program. DOTHS 809 705, National Highway Traffic Safety Administration.

[5] Baumann, F.W. and Eckstein, L., 2004, Effects causing untripped rollover of light passenger vehicles in evasivemaneuvers. SAE 2004-01-1057.

[6] Nguyen, V., 2005, Vehicle handling, stability, and bifurcation analysis for nonlinear vehicle models. Master’sthesis, Department of Mechanical Engineering, University of Maryland, College Park.

[7] Allen, R.W., Myers, T.T., Rosenthal, T.J. and Klyde, D.H., 2000, The effect of tire characteristics on vehiclehandling and stability. SAE 2000-01-0698.

[8] Allen, R.W., Klyde, D.H., Rosenthal, T.J. and Smith, D.M., 2003, Estimation of passenger vehicle inertialproperties and their effect on stability and handling. SAE 2003-01-0966.

[9] Hac, A., 2002, Rollover stability index including effects of suspension design. SAE 2002-01-0965.[10] Hac, A., 2005, Influence of chassis characteristics on sustained roll, heave and yaw oscillations in dynamic

rollover testing. SAE 2005-01-0398.[11] Eger, R. and Kiencke, U., 2003, Modeling of rollover sequences. Control Engineering Practice, 11, pp. 209–216.[12] Takano, S., Nagai, M., Nagai, M., Taniguchi, T., and Hatano, T., 2003, Study on a vehicle dynamics model for

improving roll stability. JSAE Review, 24, pp. 149–156.[13] Whitehead, R., Travis, W., Bevly, D.M. and Flowers, G., 2004, A study of the effect of various vehicle properties

on rollover propensity. SAE 2004-01-2094.[14] Garrott, W.R. and Heydinger, G.J., 1992, An investigation, via simulation, of vehicle characteristics that

contribute to steering maneuver induced rollover. SAE 92085.[15] Frimberger, M., Wolf, F., Scholpp, G. and Schmidt, J., 2000, Influences of parameters at vehicle rollover. SAE

2000-01-2669.[16] Chen, B.C. and Peng, H., 2001, Differential-braking-based rollover prevention for sport utility vehicles with

human-in-the-loop evaluations. Vehicle System Dynamics, 36(4), pp. 359–389.


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[17] Lee, A.Y., 2002, Coordinated control of steering and anti-roll bars to alter vehicle rollover tendencies. Journalof Dynamic Systems, Measurement, and Control, 124(1), pp. 127–132.

[18] Johansson, B. and Gafvert, M., 2004, Untripped SUV rollover detection and prevention. 43rd IEEE Conferenceon Decision and Control, Atlantis, Paradise Island, Bahamas.

[19] Hac, A, 2002, Influence of active chassis systems on vehicle propensity to maneuver-induced rollovers. SAE2002-01-0967.

[20] Ungoren, A.Y. and Peng, H., 2004, Evaluation of vehicle dynamic control for rollover prevention. InternationalJournal of Automotive Technology, 5(2), pp. 115–122.

[21] Segel, L., 1956, Theoretical prediction and experimental substantiation of the response of the automobile tosteering control. The Institution of Mechanical Engineers, Proceedings of the Automobile Division, No. 7,pp. 310–330.

[22] Pacejka, H.B., 2002, Tyre and vehicle dynamics. SAE, ISBN 0 7680 1126 4.[23] National Highway Traffic Safety Administration, 2000, DOT announces proposal to add rollover ratings to auto

safety consumer information program. NHTSA Now 6(7).[24] Dorohoff, M.D., 2003, A study of vehicle response asymmetries during severe driving maneuvers. Master’s

thesis, Department of Mechanical Engineering, The Ohio State University.[25] Wilson, K.A. A jeep rollover proves hard to understand. www.autoweek.com, 11/19/2001.

Appendix

Table A1. Basic vehicle parameters for 9-DOF yaw–roll–pitch–bounce model.

Name and value Expression

Ms = 1350 kg Vehicle sprung massMu1 = 42 kg Vehicle unsprung mass of the left-front suspensionMu2 = 42 kg Vehicle unsprung mass of the right-front suspensionMu3 = 42 kg Vehicle unsprung mass of the left-rear suspensionMu4 = 42 kg Vehicle unsprung mass of the right-rear suspension

Ixx−s = 380 kg m2 Moment of inertia about the x-axis of the vehicle sprung mass (about CG)Iyy−m = 1000 kg m2 Moment of inertia about the y-axis of the vehicle sprung mass (about CG)Izz−s = 2240 kg m2 Moment of inertia about the z-axis of the vehicle sprung mass (about CG)Ixz−s = 0 kg m2 Product of inertia about the x–z axes of the vehicle sprung mass (about CG)Izz−u = 1 kg m2 Moment of inertia about the z-axis of the 1/4 vehicle unsprung mass (about CG)

a = 1.253 m Distance from the sprung mass CG to the front axleb = 1.508 m Distance from the sprung mass CG to the rear axletf = 0.75 m 1/2 width of the front axletr = 0.75 m 1/2 width of the rear axlehs = 0.3 m Height of the sprung mass CG above the roll axis

k−basic = 20000 N/mKs1 = k−basic Spring stiffness of the left-front suspensionKs2 = k−basic Spring stiffness of the right-front suspensionKs3 = k−basic Spring stiffness of the left-rear suspensionKs4 = k−basic Spring stiffness of the right-rear suspensionc−basic = 1500 N.s/mCs1 = c−basic Damping coefficient of the left-front suspensionCs2 = c−basic Damping coefficient of the right-front suspensionCs3 = c−basic Damping coefficient of the left-rear suspensionCs4 = c−basic Damping coefficient of the right-rear suspension

Kt = 200000 N/m Vertical tire stiffnessPdr−Pfai = 0.07 rad Roll steer angle at the rear axlePdf−Pfai = 0.8 Relation between roll angle and camber angle



Table A2. Basic vehicle parameters for 3-DOF yaw–roll model.

Name and value Expression

Ms = 1350 kg Rolling sprung massMu = 168 kg Non-rolling unsprung mass

I−xx−s = 380 kg m2 Moment of inertia about the x-axis of the rolling sprung mass (about CG)I−xz−s = 0 kg m2 Product of inertia about the x–z axes of the rolling sprung mass (about CG)I−zz−s = 2240 kg m2 Moment of inertia about the z-axis of the rolling sprung mass (about CG)I−zz−u = 416.83 kg m2 Moment of inertia about the z-axis of the non-rolling unsprung mass (about CG)

THITA−R = 0 rad Inclination angle of the roll axis pointing downa = 1.26711 m Distance from the vehicle CG to the front axleb = 1.49389 m Distance from the vehicle CG to the rear axlec = 0.01411 m Distance from the CG of Ms to the vehicle CGe = 0.1134 m Distance from the CG of Mu to the vehicle CGTw = 1.5 m Average vehicle track widthhs = 0.3 m Distance from the CG of Ms to the roll axis

K−R = 45000 Nm/rad Roll stiffnessc−R = 3375 Roll damping coefficient Nm.s/rad

Pdr−Pfai = 0.07 rad Roll steer angle at the rear axlePdf−Pfai = 0.8 Relation between roll angle and camber angle


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