# Development of a Refined Rollover Model That Recognizes the Effects of Suspension and Tire Deformation

Authors: Xiaowen Song; Jie Li; Aiqun Xu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Vehicle rollover represents one of the most dangerous traffic accidents in the world. To improve the antirollover capability of a vehicle, we established an improved rollover model with a particular focus on the effects of independent suspensions and the lateral deformation of the tire. Based on this model, we further developed a new method to mitigate the rollover occurrence by adjusting the stiffness of the spring and the damping coefficient of the damper. Through simulation tests with a brand of SUV, we demonstrated that these adjustments improved the mitigation control as evidenced by better confined steady value and decreased overshoot of the roll angle.


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## Body

## 1. Introduction

Rollover incidents are severe safety problems that occur in all classes of vehicles. According to the USA National Highway Traffic Safety Administration’s rollover research program [1], rollovers account for 8% of total traffic accidents, and, notably, rollover-related severe and fatal injuries are as high as 31% and 21%, respectively. Vehicle rollover can be caused by multiple factors, including vehicle parameters, excessive steering, and soft terrain. Thus, studies aiming to enhance vehicle rollover resistance have been extensively pursued in recent years by numerous researchers from academic institutions as well as automobile industries. The reported studies can be classified into two categories: passive safety and active safety. The former one includes two aspects: (1) configuration of vehicle parameters that are sensitive to the static stability factor (SSF) and (2) detection of rollover signs in advance to signal the vehicle operators to take necessary reactions. Hac proposed a simple and insightful model to predict the propensity of vehicle rollover and recommended a few guidelines for passive suspension design [2]. Bouton et al. designed a rollover risk indicator specifically for off-road vehicles by incorporating the environmental properties and grip conditions into the design [3]. Yi et al. described a new methodology to design model-based estimators to detect possible rollovers. Moreover, the proposed estimator can give useful estimates on the roll angle and roll rate, two critical measurements for the risk of vehicle rollover [4]. Seongjin Yim designed a preview controller with the help of a GPS or an inertial measurement unit which can predict a driver’s steering input [5]. Mathieu et al. developed an algorithm to predict the rollover risk for all-terrain vehicle by means of an on-line estimation of a stability criterion [6]. In addition to the previous passive safety control, there are also four kinds of active safety control techniques, namely, four-wheel steering, active suspension, antiroll bar, and differential braking, which are independent of the operators’ routine handling and are triggered only in response to emergency conditions. Chen and Peng derived an antirollover control algorithm based on time-to-rollover (TTR) metric and combined with human-in-the-loop evaluations [7]. Lu et al. developed a two-layer hierarchical strategy for global integrated control of chassis subsystems associated with controlled suspension, braking system, and steering system [8]. Solmaz et al. utilized differential-braking-based rollover controllers to keep the values of vehicle outputs below certain level, and by doing so, the controllers were shown to be stable under varied vehicle speeds [9]. Cherian et al. described a methodology aiming to apply model-based design to the development and optimization of the vehicle stability control systems using CarSim and Simulink [10]. Vasilio Tsourapas et al. proposed a new method of identifying real-time predictive lateral load transfer ratio for rollover prevention [11]. Hajishafieiha Peiman proposed an active roll control system with a fuzzy logic controller [12, 13].Since light-duty vehicles are featured with relatively high center of mass and considerable deformations of suspensions and tires, the SSF of light-duty vehicles should be significantly reduced by 25% according to Alexksander Hac [2]. In this paper, we established a vehicle rollover model dealing with the changes of suspensions and tires, which has never been reported before. Using this model, we further studied the transient and steady performance of a vehicle with input of lateral acceleration and first uncovered that an optimal relationship between spring stiffness and damper damping coefficient is critical for a proper vehicle dynamic response. With this finding, we then developed a control logic to optimize this relationship in order to limit the vehicle roll state in both transient and steady phases. The effect of the proposed control logic is demonstrated by a set of simulation tests carried out according to the Fishhook 1a test regulations recommended by NHTSA [14].


## 2. Rollover Model

Following the traditional approach, we utilize the simplified three-freedom vehicle model as depicted in Figure1.Simplified vehicle rollover model.
(a)
x-y of the simplified model
(b)
y
-
z of the simplified model
(c)
the vehicle body modelIn Figure1(a), x-y plane, the lateral forces over the front and rear tires are denoted by Fy1 and Fy2. The sideslip angle β and the roll angle ϕ are assumed to be minor. The total forces and torques over the whole vehicle are given by
(1)∑Fy=Fy1cosδ+Fy2=may,∑Mz=aFy1cosδ-bFy2=Izω˙r,
where m is the total mass of the vehicle, the lengths a and b are defined as shown in Figure 1(a), ay is the lateral acceleration, ω˙r is the yaw acceleration, and Iz is the yaw moment of the vehicle inertia.In addition, the lateral forcesFy1=k1α1, Fy2=k2α2, where k1 and k2 are the cornering stiffness of front and rear tires, respectively, α1 and α2 are the slip angles of front and rear tires, α1=β+aωr/vx-δ, α2=β-bωr/vx, vx is the longitudinal velocity, and ωr is the yaw rate. Consequently, the following equation is derived:
(2)ay=(k1+k2)βm+(ak1-bk2)ωrmvx-k1δm,ω˙r=(ak1-bk2)βIz+(a2k1+b2k2)ωrIzvx-ak1δIz.In Figure1(b), y-z plane, the total forces and torques over the whole vehicle are given by
(3)∑Fz=Fn1+Fn2-mg=0,∑Mx=mghsinϕ+may(h0+Δh-h)+Fn2(T2+ΔT)-Fn1(T2-ΔT)=Ixϕ¨,
where ϕ¨ is the body roll acceleration, h0 is the height of the vehicle’s center of gravity (CG) standing above the ground level, h is the distance between the vehicle CG and the assumed roll axis, T is the width of the vehicle track, and Δh and ΔT are the deformations of suspensions and tires. Fn1 and Fn2 are normal forces over the left and right wheels, and Ix is the roll moment of vehicle inertia.The restoring forceFa in Figure 1(c) is associated with the roll movement, Fa=(kϕϕ+cϕϕ˙)/h, where kϕ and cϕ are the total roll stiffness and roll damping of suspensions and ϕ˙ is the roll rate.The total torques over the body with respect to the roll axis are given by(4)∑Mxs=msayh+msghsinϕ-Fah=Ixsϕ¨,
where ms is the sprung mass that represents the total vehicle mass m excluding the suspension and tire and Ixs is the roll moment for the inertia of the sprung mass.Compared with the sprung mass, the unsprung mass is much smaller. Therefore, we can suppose thatms=m and Ixs=Ix, and establish (5) as follows:
(5)ϕ¨=mh(ay+gsinϕ)Ix-kϕϕIx-cϕϕ˙Ix,Fn1-Fn2=2may(h0+Δh-2h)T+2mgΔTT-2(kϕϕ+cϕϕ˙)T,Fn1+Fn2=mg,LTR=(Fn1-Fn2Fn1+Fn2),
where LTR stands for the load transfer ratio, an important criteria to evaluate the vehicle status [15].Sinceay=v˙y+vxωr, β=vy/vx, after introducing the state vector x=[vyωrϕ˙ϕ]T, the motions of this model can be described by
(6)x˙=Ax+Bδ,
where
(7)A=[k1+k2mvxak1-bk2mvx-vx00ak1-bk2Izvxa2k1+b2k2Izvx00k1+k2Ixvxak1-bk2Izvx-cϕIxmgh-kϕIx0010],B=[-k1m-ak1Iz-k1Ix0]T.


## 3. Mitigation Control

The running vehicle intends to overshoot in its response to a suddenly applied lateral acceleration. Therefore, the inner wheels may have lifted off during the steering when the applied acceleration is lower than the nominal permit value that is in compliance with the steady-state laws [16].From (5), we can see
(8)ϕ¨=mh(ay+gsinϕ)Ix-kϕϕIx-cϕϕ˙Ix,
where the roll angle ϕ is small enough to get sinϕ≈ϕ, and after applying Laplace law to the aforementioned differential equation, we established the following new equation:
(9)ϕ(s)ay(s)=(kϕ-mgh)/Ixs2+(cϕ/Ix)s+(kϕ-mgh)/Ix·mh/Ix(kϕ-mgh)/Ix,
where the angular frequency is
(10)ωn=[(kϕ-mgh)Ix]1/2
and the damping ratio is
(11)ξ=cϕ2[(kϕ-mgh)Ix]1/2.When the inputay is set to be the unit step function, the overshoot is
(12)Mp=exp(-ξπ(1-ξ2)1/2).At this point, we found that vehicle dynamics is determined by the first part of (9), whereas the second part defines the vehicle steady state. These findings form the base of the mitigation control referred to in the rest of this section.A specific brand of midsize SUV was used as the test vehicle in this study, of which the relevant parameters are shown in Table1.Table 1
Parameters of the simulated SUV.
Parameter
Value
Unit
m
1460
(kg)
v
x
60
(km/h)
h
0
0.628
(m)
h
0.35
(m)
T
1.68
(m)
a
1.233
(m)
b
1.327
(m)
k
ϕ
45000
(N·m/rad)
c
ϕ
3600
(N·m·s/rad)
a
1
0.13
Null
a
2
0.225
Null
c
1
80200
(N/rad)
c
2
160000
(N/rad)
I
x
550
(kg·m2)
I
z
1400
(kg·m2)With the rollover model and vehicle parameters, we achieved the damping ratio of the original vehicle,ξ=0.38, and the overshoot value of the roll angle at 36% as shown in Figure 2. Thus, the damping ratio needs to be raised in order to reduce the overshoot value. In other words, we can improve the restoring force Fa by increasing the roll stiffness kϕ or the roll damping cϕ. However, it needs to bear in mind that the roll stiffness kϕ and the roll damping cϕ must be increased simultaneously. Otherwise, the damping ratio ξ may be decreased.Figure 2
ϕ
-
a
y transient response.When the gain in roll stiffnessΔkϕ and the gain in roll damping Δcϕ are 1500 and 2150, respectively, the damping ratio ξ will be 0.6, and the overshoot will be much smaller than the original one as shown in Figure 2. As indicated by the calculated values of Δkϕ/kϕ=3.3% and Δcϕ/cϕ=59.7%, a small gain in the roll stiffness results in substantial changes in roll damping in a disproportional manner. This disproportion will become more severe if the damping ratio is higher. Therefore, it is necessary to establish an adjusting relationship between the roll stiffness and the roll damping to ensure a proper match between them.In theory, the overshoot is nearby zero if the damping ratio equals one. However, as we know, high roll stiffness and roll damping cost vehicle ride comfort. For a balance purpose, we assigned the compromised damping ratioξ at a normal factory-set value.There are many factors that can influence the roll stiffnesskϕ and the roll damping cϕ, such as springs, dampers, and bushings. Here, we adopted two ratio factors, a1 and a2. a1 couples the roll stiffness kϕ and the spring stiffness k together in the form of kϕ=a1k.Similarly,a2 serves a link between the roll damping cϕ and damper damping coefficient c in the form of cϕ=a2c. With the compromised damping ratio value, by substituting the original values of k0 and c0 and the gains of Δk and Δc in the spring and the damper into (11), we obtained the following equation:
(13)Δk=a221.44a1IxΔc2+a22c00.72a1IxΔc+(a22c021.44a1Ix+mgha1-k0).We then attempted to identify the value ofΔk (or Δc) for danger prevention. As mentioned previously, the second part of (9) determines the ultimate roll angle value in the steady state, and the prospective control on the roll angle is shown in Figure 3, where Δt is a time period proceeding to the vehicle rollover, ϕth is the desired threshold value of the roll angle, and ϕe(t+Δt) is the estimated value of the roll angle. If ϕe(t+Δt)>ϕth, mitigation control will be activated in the buffer period time of Δt, and if we keep ϕ(t+Δt) the same as ϕ(t), then
(14)ϕ(t)=ϕ(t+Δt),ϕ(t)=mha1k-mghay(t),ϕ(t+Δt)=mha1k*-mghaye(t+Δt),Δk=k*-k,
where aye is the estimated value of the lateral acceleration. Δt is the value in relevance to the system control frequency of a given active suspension.Figure 3
The prospective control on the roll angle.As deduced from (14), the value of Δk is:
(15)Δk=aye(t+Δt)-ay(t)a1ay(t)(a1k-mgh)Finally, (13) and (15) on {Δk,Δc} give rise to the mitigation control that stabilizes the vehicles.


## 4. Simulation Results

In order to verify the proposed mitigation control, simulation tests are conducted according to NHTSA’s Fishhook 1a test description, and the steering maneuver patterns are shown in Figure4.Figure 4
Fishhook 1a maneuver.Using the vehicle parameters in Table1 and the multibody simulation test model, the normal forces on the four wheels are revealed as shown in Figure 5.Normal forces on four wheels of the original vehicle.
(a)
(b)Note that the wild fluctuations occur when the handwheel angle rate changes rapidly, especially within the time ranges of[1.5s,1.75s] and [4.25s,4.5s]. The curves in Figure 5 are roughly in mirrorimage, in which the crests of left-wheel normal force are corresponding to the troughs of the right-wheel normal force. This pattern indicates that the load on the wheels shifts from one side to another. Also, the test result showed that the minimum force over the right-front wheel comes close to only 400 N, a sign that the vehicle is about to lift off the ground.The roll angle and lateral acceleration of the original vehicle in response to the steering input are shown in Figure6, respectively. As shown in Figure 6, the peak value of the roll angle is roughly 7.5 degree; meanwhile, the lateral acceleration is about 10 m/s2. We will try to reduce the maximum roll angle to 5 degrees as our next goal for testing the control model.Roll angle and lateral acceleration of the original one.
(a)
(b)Since the roll angle and lateral acceleration of the vehicle are already built in the simulations, we simplified the mitigation control for passive suspension, in which the changes in the spring stiffness and the damper damping coefficient of the vehicle have already been incorporated into the simulation at the beginning. Using the mitigation control described in the previous section, the adjustments can be easily determined asΔk/k0=0.27, Δc/c0=0.75.The response of the modified prototype to the same steering input is illustrated in Figure7, which shows that the transient state of the roll angle is much smoother and the maximum steady value is controlled below the desired degree.Figure 7
Roll angle of the modified.LTR is another important index used to evaluate vehicle status, and it ranges from −1 to 1. As indicated by R.C.Lin, whenLTR equals −1 or 1 either the left or the right side tires lift off the ground and a smaller value stands for a more stable vehicle [16]. Figure 8 shows the LTRs of both the modified and original vehicles, and it is evident that the modified vehicle features a smaller LTR than the original one does. Moreover, the normal forces over the four wheels are also improved as shown in Figure 9.Figure 8
LTRs of the two prototypes.Normal forces on four wheels of the modified vehicle.
(a)
(b)In summary, simulation results strongly supported the proposed mitigation control, of which the contribution to vehicle stabilization is evident.


## 5. Conclusion

Vehicle rollover is a complex process. To investigate the vehicle transient and steady states, an improved rollover model was established in this study. Based on the model, we further develop a method to limit the overshoot by adjusting the relationship between the spring stiffness and the damper damping coefficient. Additional findings include the mitigation control on vehicle stabilization, and the control logic was verified by a series of simulation tests using a midsize SUV model. Simulation results indicated a decreased overshoot of the roll angle and a better confined steady value. The adjusting method proposed in this paper may help the design of both passive and active suspension controls to increase vehicle stability. We are confident that, with additional studies, the proposed model will be applicable for a real vehicle in the near future.



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*Source: 750896-2013-02-13.xml*