Research method of vehicle rollover mechanism under critical instability condition

Abstract

In this article, a novel rollover prediction algorithm is developed for application on vehicles with large lateral velocity and high center of gravity. Lateral energy is the direct cause of rollover. Rollover prediction model is proposed by taking full account of the impact of the pavement, tire, and suspension and realizes the estimation of the vehicle lateral energy. By calculating the ratio of real-time lateral energy reserve and rollover threshold, the degree of rollover risk is obtained. The double-shift experiment and the Fishhook experiment are performed to verify the accuracy and suitability of the proposed model, and the proposed prediction is 0.2 s ahead of the actual liftoff situation and 0.45 s ahead of the actual rollover situation; therefore, the proposed rollover model can be regarded as an effective method.

Keywords

Vehicle rollover tire suspension

Introduction

According to statistics from the Traffic Management Bureau of the Ministry of Public Security, by the end of 2016, the number of motor vehicles in China reached 290 million, of which 194 million were cars and 360 million motorists were drivers, of which more than 310 million were motorists.¹ However, the number of casualties caused by traffic accidents continues to increase. The number of people killed and injured in traffic accidents in China has been ranked first in the world for 10 years. The number of accidents accounts for only 1.9% of the world’s total, whereas the death number accounts for 15% of the world’s total. However, the accident of rollover has not received due attention. According to statistics, the degree of harm caused by vehicle rollover accidents is only seconds to the crash accident. The number of casualties caused by rollover accidents in the United States accounted for 35% of vehicle accidents.²

In the rollover mode, the lateral acceleration of the vehicle increases sharply. The tire is the only contact medium between the road surface and the vehicle. It has longitudinal, lateral, and vertical coupling motions and has complex nonlinear characteristics. The accuracy of tire model and suspension model is the core of the rollover warning. After thorough in-depth theoretical and experimental research, Professor Pacejka reduced the tires to a “string” structure and established a very famous “Magic Formula” with Bakker.³ Svendenius and Gäfvert⁴ of Lund Science and Technology studied the mechanical properties of tires under compound sliding conditions based on brush model, in consideration of the anisotropy of the tire, the difference in stiffness and friction coefficient, as well as the deformation of the carcass and the roll of the tire; the expressions of the longitudinal force, the lateral force, and the positive moment were finally established and verified by a large number of experiments.

J Preston-Thomas and JHF Woodrooffe developed a set of stability control and alarm systems that use the lateral-load transfer ratio (LTR) as the vehicle’s rollover criteria. The most significant contribution of this system is to use LTR as the rollover judgment indicator for the first time, laying the theoretical foundation for future rollover warnings.⁵ Stavroff and Schubert⁶ proposed in their patent an adaptive algorithm that takes into account changes in the height of the vehicle’s center of mass, which algorithm estimates the center-of-mass height of the vehicle based on the real-time measured yaw rate, vehicle speed, and lateral acceleration. This method improves the accuracy of the rollover warning. Sverrisson⁷ applied for a patent about rollover warning method that uses the lateral acceleration as a percentage of the ultimate lateral acceleration to determine the degree of rollover hazard.

Sadri and Wu⁸ used Lyapunov’s function to analyze the lateral stability and studied the roll stabilization region in the phase diagrams of lateral velocity and yaw rate. Nam et al.⁹ constructed an observer to predict the roll angle through tire lateral force and proposed a four-wheel steering vehicle’s direct roll couple controller. Du et al.¹⁰ developed a roll couple controller, but the controller’s parameters depend on the vehicle’s longitudinal velocity and can only solve finite linear matrix inequalities. The methods used by Nam and Du can obtain lateral slip angle and yaw rate. Lu and colleagues^11,12 discussed the controllability and effective working range of rollover prevention for different key chassis subsystems and used magneto-rheological (MR) semi-empirical suspension as the key actuator to perform overall control.

J Wang et al.¹³ proposed a real-time vehicle roll state observer and rollover warning algorithm. TJ Zhu et al.¹⁴ designed a heavy vehicle rollover warning system based on the improved true three-dimensional reconstruction (TTR) algorithm and linear matrix inequality. Z Jin et al.¹⁵ proposed a rollover warning algorithm based on dynamic stability. X Liu et al.¹⁶ proposed an on-demand active interconnection suspension vehicle rollover control algorithm, and Chu DF¹⁷ proposed an early warning method for large-vehicle rollover under vehicle and road coordination. Zhao Zhiguo proposed the rollover of heavy-duty vehicles based on autoregressive hidden Markov model (AR-HMM). Based on the early warning model and algorithm, Hu XM¹⁸ proposed a rollover warning system based on the roll stability mechanism. Li SQ¹⁹ studied the effect of the lateral and vertical coupling characteristics of tires and pavements on the rollover of the vehicle. J Chiu²⁰ developed the lateral energy model for rollover prediction and prevention, which was verified in Carsim with different vehicles on a lot of driving conditions. H Imine et al.²¹ presented rollover risk prediction of heavy vehicle using high-order sliding mode observer. These studies have elaborated in depth on various rollover prediction methods, mechanisms, models, and key parameters, but lacked analysis of the influence of the complex mechanical properties of flexible carcass on vehicle lateral dynamics model under critical destabilizing conditions.

As the vehicle is a highly nonlinear vibration system, its dynamic rollover process is very complicated. Vehicle lateral speed, steering angle, ground adhesion coefficient, roll angle, and lateral acceleration are vehicle rollover factors. Therefore, we first establish a nonlinear vehicle model with high fidelity to accurately reflect the characteristics of the rollover factors. The novelty of our research is to focus on the tire mechanics model under extreme conditions, the lateral energy rollover model is the subject, and the related research has certain effects on improving liftoff and rollover prediction.

Modeling

Vehicle-rigid-body model

Assume that the vehicle is a rigid body and the lateral speed is 0, when the roll angle γ is 90°, the vehicle is in the rollover position; if γ < 90°, the vehicle is in liftoff or sliding; if γ > 90°, the vehicle is in the rollover state. As shown in Figure 1, according to the lateral energy principle, the conditions under which the vehicle does not rollover are

\frac{1}{2} J_{L} γ'^{2} + mgl \sin γ \leq mgl

(1)

where $J_{L}$ is the moment of inertia of the vehicle relative to point L, and $mgl$ is the gravitational potential energy of the vehicle when γ = 90°

Figure 1.

Vehicle-rigid-body model.

The relationship between $γ'^{2}$ and $\sin γ$ is shown in Figure 2, the shaded area is the area without rollover, it is the slipping or liftoff area, and the blank is the rollover area. The dividing line between these two areas is a straight line $(1 / 2) J_{L} γ'^{2} + mgl \sin γ = mgl$ , and the intersection with the axis sin γ is (1,0).

Figure 2.

Rollover/non-rollover boundary.

First of all, analyze the lateral force of the vehicle, as shown in Figure 3; vehicle is subjected to centrifugal force F_cf in the lateral direction, tire lateral friction force F_yR and F_yL, vertical force G, and support force F_ZR and F_ZL; when the car is driven at high speed, the lateral centrifugal force can cause liftoff or rollover. If F_cf < F_yR + F_yL, then the vehicle will not slide sideways or liftoff; if F_cf ≥ F_yR + F_yL, then the vehicle will slide sideways or liftoff

{\begin{matrix} F_{cf} < (F_{yR} + F_{yL}) = F_{y} \\ F_{yR} = μ F_{ZR} \\ F_{yL} = μ F_{ZL} \\ F_{ZR} + F_{ZL} = G \\ F_{yL} + F_{yR} = F_{y} = μ G \end{matrix}

(2)

where μ is the sliding friction coefficient of the tire and road, and F_y is the lateral resultant force of the tire and road

{\begin{matrix} F_{cf} < (F_{yR} + F_{yL}) = (F_{yR}) = F_{y} \\ F_{yR} = μ F_{ZR} \\ F_{yL} = 0 \\ F_{ZL} = 0 \\ F_{ZR} = G \\ F_{yR} = F_{y} = μ G \end{matrix}

(3)

Figure 3.

Force situation of sideslip and liftoff.

When one side wheel is just off the road, F_cf = F_yR + F_yL = F_yR. Solve the torque around the contact point R between the right tire and the road

{\begin{matrix} F_{cf} h_{g} - G \frac{B}{2} = 0 \\ F_{cf} = m v^{2} / r = m a_{y} \\ G = mg \\ v^{2} / rg = \frac{a_{y}}{g} = \frac{B}{2 h_{g}} \end{matrix}

(4)

If $v^{2} / rg \geq B / 2 h_{g}$ , the vehicle will lift off the ground, where r is the vehicle turning radius. As shown in Figure 4, the horizontal axis is the vehicle driving speed v, the vertical axis is the steering radius of the vehicle, and the dividing line between blank and shaded areas is r = (2v²h_g)/(Bg).

Figure 4.

Sideslip/liftoff boundary.

Vehicle-rigid-body model with suspension model

According to equation (4), $v^{2} / rg$ is defined as a rollover threshold. On a straight level road, when it reaches 0.8, the vehicle will slide sideways. However, according to the experiment data in Table 1, it can be seen that medium-duty truck and heavy-duty truck have started to rollover when they have not reached the rollover threshold. For cars and light-duty truck, they only have slipping without rollover. Accident statistics show that the vehicle does not slip when it rolls over, so it is necessary to further explore the mechanisms of rollover.

Table 1.

Rollover threshold range of different types of vehicles.

Type	Centroid height (mm)	Track distance (mm)	Rollover threshold
Sports car	460–510	1270–1540	1.2–1.7
Mini car	510–580	1270–1540	1.1–1.5
Luxury car	510–610	1540–1650	1.2–1.6
Light passenger car	760–890	1650–1780	0.9–1.1
Passenger and cargo vehicle	760–1020	1650–1780	0.8–1.1
Medium truck	1140–1400	1650–1900	0.6–0.8
Heavy truck	1540–2160	1780–1830	0.4–0.6

Due to the presence of the suspension, the vehicle cannot simply be defined as a rigid body, and a rigid–flexible multi-body analysis is required. As shown in Figure 5, M₁ is the elastic moment of the suspension to point C, k is the roll stiffness of the suspension, $φ$ is the roll angle of the center of mass with respect to the vertical position, $\overset{\cdot}{φ}$ is the roll angle velocity of the center of mass, and $β$ is the angle between the straight line CR and the ground. Define $\sqrt{e_{1 b}^{2} + d_{1}^{2}} = l_{1}$ and $\sqrt{{(B / 2)}^{2} + h_{2}^{2}} = l_{2}$ . Take a moment for the center of mass m

{\begin{matrix} J \overset{\cdot\cdot}{φ} = - (l_{2} \cos β + l_{1} \cos φ) F_{ZR} + (d_{1} + l_{2} \sin β) F_{y} \\ F_{ZL} = 0 \\ F_{Z} = F_{ZR} \end{matrix}

(5)

Figure 5.

Rigid–flexible multi-body analysis.

Take a moment to point C, we can get

{\begin{matrix} - F_{y} l_{2} \sin β - M_{1} + F_{ZR} \frac{B}{2} = 0 \\ M_{1} = \sqrt{e_{1 b}^{2} + d_{1}^{2}} [k (π / 2 - φ) + c \overset{\cdot}{φ}] \end{matrix}

(6)

According to formulas (5) and (6)

\begin{matrix} J \overset{\cdot\cdot}{φ} + (l_{2} \cos β + l_{1} \cos φ) \frac{(2 M_{1} + 2 F_{y} l_{2} \sin β)}{B} \\ - (d_{1} + l_{2} \sin β) F_{y} = 0 \end{matrix}

(7)

Hence

\begin{matrix} J \overset{\cdot\cdot}{φ} + (l_{2} \cos β + l_{1} \cos φ) \\ (\frac{2 l_{1} [k (π / 2 - φ) + c \overset{\cdot}{φ}]}{B} + \frac{2 F_{y} l_{2} \sin β}{B}) \\ - (d_{1} + l_{2} \sin β) F_{y} = 0 \end{matrix}

(8)

Using Newton’s second law to solve $F_{y}$ , define the acceleration of centroid as

{\vec{a}}_{m} = {\vec{a}}_{R} + {\overset{\cdot\cdot}{\vec{r}}}_{Rm}

Assume that the acceleration at point R has only lateral components

{\vec{a}}_{R} = {\vec{a}}_{Ry} = - {\overset{\cdot}{\vec{v}}}_{Ry}

Since

\begin{matrix} {\vec{r}}_{Rm} = - (l_{2} \cos β + l_{1} \sin φ) \vec{y} + (l_{2} \sin β + l_{1} \cos φ) \vec{z} \\ {\overset{\cdot}{\vec{r}}}_{Rm} = - (- l_{2} \sin β \overset{\cdot}{β} + l_{1} \cos φ \overset{\cdot}{φ}) \vec{y} \\ - (l_{2} \cos β \overset{\cdot}{β} + l_{1} \sin φ \overset{\cdot}{φ}) \vec{z} \\ {\overset{\cdot\cdot}{\vec{r}}}_{Rm} = [l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})] \vec{y} \\ - [l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})] \vec{z} \\ {\vec{v}}_{m} = {\vec{v}}_{Ry} + {\overset{\cdot}{\vec{r}}}_{Rm} s \end{matrix}

Hence

\begin{matrix} {\vec{a}}_{m} & = {\vec{a}}_{R} + {\overset{\cdot\cdot}{\vec{r}}}_{Rm} \\ = - {\overset{\cdot}{\vec{v}}}_{Ry} + [l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})] \vec{y} \\ - [l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})] \vec{z} \\ = [- {\overset{\cdot}{\vec{v}}}_{Ry} + l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})] \vec{y} \\ - [l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})] \vec{z} \end{matrix}

(9)

Analyze the vertical and lateral forces of the centroid

{\begin{matrix} \frac{m {v_{m}}^{2}}{r} - F_{y} = m (- {\overset{\cdot}{\vec{v}}}_{Ry} + l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})) \\ F_{Z} - G = m (l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})) \end{matrix}

(10)

The sum of the lateral kinetic energy and potential energy of the vehicle is

{\begin{matrix} E_{all} = mg (l_{2} \sin β + l_{1} \sin φ) + 1 / 2 {mv}_{m}^{2} + 1 / 2 J {\overset{\cdot\cdot}{φ}}_{Rm}^{2} + 1 / 2 k_{s} φ^{2} \\ v_{m}^{2} = {| (- l_{2} \sin β \overset{\cdot}{β} + l_{1} \cos φ \overset{\cdot}{φ}) |}^{2} + {| (l_{2} \cos β \overset{\cdot}{β} + l_{1} \sin φ \overset{\cdot}{φ}) |}^{2} \\ φ_{Rm} = \arctan \frac{| [l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})] |}{| [- {\overset{\cdot}{\vec{v}}}_{Ry} + l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})] |} \end{matrix}

(11)

Tire dynamic model

In rollover conditions, the tire stiffness has complex nonlinear characteristics, especially in the case of considering the deformation of the carcass, the tire mechanical model will become very complex, as shown in Figure 6, and the actual sideslip angle of the tire is much larger than the traditional estimated sideslip angle. To analyze the lateral deformation energy of the tire during rollover, it is necessary to consider the lateral deformation of the carcass, and to analyze the lateral deformation and longitudinal deformation of the tread, as shown in Figures 7 and 8, respectively.

Figure 6.

Actual sideslip angle and the traditional estimated sideslip angle.

Figure 7.

Carcass deformation.

Figure 8.

Longitudinal and lateral deformation of the tread.

The longitudinal length of contact area is $2 a$ , and cut-off point of slip area and adhesion area is $x_{s}$ . As the role of longitudinal and lateral forces, in the point with coordinates x of longitudinal direction, brush deformations are $δ_{xb (x)}$ and $δ_{yb (x)}$ , the longitudinal force $F_{x}$ and lateral force are $F_{x}$ and .

The elastic deformation of the tire includes: longitudinal deformation $δ_{xb (x)}$ , lateral deformation $δ_{yb (x)}$ , and vertical deformation $δ_{z b (x)}$ of the tread, and lateral deformation $δ_{ycar (x)}$ and vertical deformation $δ_{zcar (x)}$ from the carcass. Analyze the lateral energy model of the vehicle, ignoring the longitudinal deformation $δ_{xb (x)}$ , vertical deformation $δ_{z b (x)}$ of the tread, and the vertical deformation $δ_{z car (x)}$ of the carcass, that is, only analyze the lateral deformation $δ_{yb (x)}$ of the tread and lateral deformation $δ_{ycar (x)}$ of the carcass. As shown in Figure 9, $γ_{t} = \arctan \frac{δ_{yb (x)}}{r_{0}}$ and $γ_{tru} = \arctan \frac{δ_{yb (x)} + δ_{ycar (x)}}{r_{0}}$ , obviously $γ_{tru} > γ_{t}$ .

Figure 9.

Lateral deformation relationship between the carcass and the tread.

Define the carcass lateral stiffness as C_ycar and the tread lateral stiffness as C_yb. Analyzing the lateral stiffness of the tire C_ytall, we can obtain the connection between C_ycar and C_yb is in series. Hence

\begin{matrix} C_{yall} = \frac{C_{yar} C_{yb}}{C_{yar} + C_{yb}} \\ F_{y} = \frac{C_{yar} C_{yb}}{C_{yar} + C_{yb}} (δ_{ycar} + δ_{yb}) \end{matrix}

At the same time, it is necessary to consider the effect of tire sideslip on wheelbase

B \to (B - δ_{ycar} - δ_{ycar})

As the rollover threshold is reduced, rollover will become easier and the results of Table 1 will also change

\begin{matrix} \frac{B}{2 h_{g}} \to \frac{(B - δ_{ycar} - δ_{ycar})}{2 h_{g}} = \frac{B}{2 h_{g}} - \frac{δ_{ycar} + δ_{ycar}}{2 h_{g}} \\ = \frac{B}{2 h_{g}} - C_{y 0} \end{matrix}

According to experience analysis

C_{y 0} \approx (4 % - 5 %) \frac{B}{2 h_{g}}

The overall lateral energy of the car becomes

{\begin{matrix} E_{all} = mg (l_{2} \sin β + l_{1} \sin φ) + 1 / 2 {mv}_{m}^{2} \\ + 1 / 2 J {\overset{\cdot\cdot}{φ}}_{Rm}^{2} + 1 / 2 k_{s} φ^{2} + 1 / 2 C_{yall} {γ_{tru}}^{2} \\ v_{m}^{2} = {| (- l_{2} \sin β \overset{\cdot}{β} + l_{1} \cos φ \overset{\cdot}{φ}) |}^{2} + {| (l_{2} \cos β \overset{\cdot}{β} + l_{1} \sin φ \overset{\cdot}{φ}) |}^{2} \\ φ_{Rm} = \arctan \frac{| [l_{2} (\cos β \overset{\cdot\cdot}{β} - \sin β {\overset{\cdot}{β}}^{2}) + l_{1} (\sin φ \overset{\cdot\cdot}{φ} + \cos φ {\overset{\cdot}{φ}}^{2})] |}{| [- {\overset{\cdot}{\vec{v}}}_{Ry} + l_{2} (\cos β {\overset{\cdot}{β}}^{2} + \sin β \overset{\cdot\cdot}{β}) + l_{1} (\sin φ {\overset{\cdot}{φ}}^{2} - \cos φ \overset{\cdot\cdot}{φ})] |} \\ B \to (B - δ_{ycar} - δ_{ycar}) \end{matrix}

Theory verification

Real vehicle test

In order to verify the actual effectiveness of the proposed energy model, we apply this model to the experiment, specifically including the double-shift experiment of 60 km/h, as shown in Figure 10, and the parameters of test system, as shown in Table 2. Due to the great danger of the rollover, the speed of the vehicle is low and the steering speed is also small, there is no rollover, and the parameters of the vehicle are shown in Table 3.

Figure 10.

Real vehicle test.

Table 2.

Test system.

Name	Precision
Speed sensor	Speed: 0.1 km; distance: 0.05%
Fiber optic gyroscope	3%
Centroid test platform	0.1%
Steering force, angle tester	1%
Wind speed probe	Wind speed: 0.03 m/s, humidity:1.8% RH, temperature: 0.5°C
Tape measure	1 mm
Data collector	0.05%

Table 3.

Test vehicle parameters.

Parameter name	Dimension	Value
The total heightof no-load vehicle	mm	2880
The distance from thecentroid to the front axle	mm	4291
The distance from the centroid to the rear axle	mm	2806
Front track	mm	1847
Rear track	mm	1650
The total mass of no-load vehicle	kg	5240
Moment of inertiaaround the z axis	kg m²	12,300
Front effective corneringstiffness, k₁	N/rad	−1.521 × 10⁵
Rear effective cornering stiffness, k₂	N/rad	−3.022 × 10⁵
Tire model	none	215/75 R17.5

As shown in Figure 11, the controlled vehicle was driven at the speed of 60 km/h, and the double-shift test was performed to measure the speed, steering angle, and roll angle. As the existence of steering angle, the longitudinal driving speed changed from 57 to 63 km/h.

Figure 11.

Driving speed and setting speed.

As shown in Figure 12, the left front wheel turns to a little shift at the 0.74th s, reaching −8°; then turns in reverse from 0° to 50° for 0.98 s; then turns to the other direction from 50° to −32°; and then turns from −32° to 50°.

Figure 12.

Front wheel steering angle.

From Figures 12 and 13, it can be seen that the steering angle has reached the maximum value, but the maximum angle of tire roll angle is only ±3° and no rollover occurs. We also find that the change of roll angle is consistent with the change of steering angle, which shows that the root cause of rollover is from the steering angle at high speed.

Figure 13.

Tire roll angle over time.

According to the lateral energy method, the double-shift experiment at 60 km/h was analyzed in Figure 14. The vehicle steering at the 0.74th second resulted in an increase in the lateral speed, the roll angle, and the lateral energy which increases to 56% of the liftoff energy. In the 2.0th second, the vehicle steering from 0° to 50° caused the lateral energy to drop to 0, then increases to 62% of the liftoff energy, that is, about 2 s, the lateral energy changed between 40% and 62% of the liftoff energy. In the 4.0th second, the vehicle steering caused the lateral energy to rapidly decrease to 0, then increased to 60% of the liftoff energy, that is, about 3.2 s, the lateral energy varied between 60% and 48% of the liftoff energy. In the 7.2th second, the vehicle turned to the reverse direction, which caused the lateral energy to decrease to 0, then increases to 60% of the liftoff energy. In general, the sharp steering of the vehicle during driving causes an increase in the lateral speed, and roll angle of vehicle and tire. When the lateral energy increases to the liftoff threshold, liftoff occurs, and continues to increase to the rollover threshold and rollover will occur.

Figure 14.

Roll rate based on the lateral energy method.

Simulation experiment verification

In order to further verify the validity of the proposed energy method, in Carsim, the Fishhook experiment (40, 50, and 60 km/h) has been conducted, and the friction coefficient of the road surface is 0.85. Vehicle speed and steering angle are large to reach the degree of liftoff and rollover, and test vehicle parameters are shown in Table 4.

Table 4.

Test vehicle parameters in Carsim.

Parameter name	Dimension	Value
The total height ofno-load vehicle	mm	2230
The distance from the centroid to the front axle	mm	1180
The distance from the centroid to the rear axle	mm	2620
Front track	mm	1575
Rear track	mm	1575
The total mass of no-load vehicle	kg	1720
Yaw inertia	kg m²	2488
Roll inertia	kg m²	614
Tire model	None	255/75 R16

The changes of steering angle are shown in Figure 15. In the 1.1th second, the vehicle turned leftwards to 15.3°; in the 1.6th second, the vehicle suddenly turned to the right and the steering angle changed from +15.3° to −12.4°. No rollover occurred in the conditions of V = 50 km/h and V = 40km/h; however, in the 3.0th second with V = 60km/h, rollover occurred.

Figure 15.

Steering angle.

The changes of driving speed under three conditions are shown in Figure 16.

Figure 16.

Speeds under three conditions.

Figure 17 is a roll estimation algorithm based on the LTR (load translation rate). In these three cases, only when V = 60 km/h, liftoff occurred in the 1.6th and 2.4th second with LTR = 100%; however, rollover cannot be predicted, in fact, rollover occurred in the 3.15th second. Because the LTR theory is based on the vertical load transfer rate of two left wheels and two right wheels, rollover case cannot be predicted.

Figure 17.

Liftoff prediction with LTR.

According to the lateral energy theory prediction: in Figure 18, V = 60 km/h, it is predicted that the vehicle has enough liftoff energy in the 1.9th second and the vehicle lifted off the ground in the 2.1th second, that is, the proposed prediction is 0.2 s ahead of the actual situation; it is predicted that the vehicle has enough rollover energy in the 2.7th second and the vehicle rolled over in the 3.15th second, that is, the proposed prediction is 0.45 s ahead of the actual situation; under the conditions V = 50 km/h and V = 40 km/h, it is predicted that the vehicle will not rollover, which is consistent with the actual case.

Figure 18.

Lateral energy prediction method.

If the lateral energy model does not include the effect of tire lateral deformation, the theoretical prediction of liftoff time will be delayed by 15% and the theoretical prediction of rollover time will be delayed by 22%; the changes in the tire lateral deflection angle and wheelbase are shown in Figures 19 and 20, respectively; and Table 5 shows the comparison between LTR and lateral energy method for estimating the rollover.

Figure 19.

Tire lateral deflection angle with V = 60 km/h.

Figure 20.

Wheelbase B changes with V = 60 km/h.

Table 5.

Comparative analysis between LTR and lateral energy method.

	LTR			Lateral energy method
	Liftoff	Sideslip	Rollover	Liftoff	Head time	Sideslip	Rollover	Head time
V = 40 km/h	×	×	×	×		×	×
V = 50 km/h	×	×	×	×		×	×
V = 60 km/h	√	×	√	√	200 ms	×	√	450 ms

From the experiments of the real vehicle and simulation vehicle under different maneuvers, we have applied the proposed rollover prediction algorithm to several different conditions, which range from non-liftoff, non-rollover, to liftoff or rollover. The improved rollover prediction algorithm experiment provides a consequence which predicts every case of liftoff and rollover. With the existing LTR model, we can find that only liftoff can be predicted, but not rollover, whereas the proposed rollover prediction algorithm can give the prediction of pre-liftoff and pre-rollover, significantly increasing the effectiveness of active prevention countermeasures.

For the Fishhook case at V = 40 km/h and V = 50 km/h with μ = 0.85, even though we find that the vehicle does not lift off, shown in Figure 20, the individual wheel loads show that the rear inside wheel does lift off momentarily; however, the proposed rollover prediction algorithm does not predict the liftoff, since our algorithm predicts liftoff and rollover of an entire side, at least two wheels, not a single wheel.

With the LTR method, we cannot predict the liftoff until liftoff has occurred, also cannot predict rollover, whereas the proposed rollover prediction method has the ability to predict liftoff and rollover in advance, which significantly increases the effectiveness of vehicle active safety.

Conclusion

Based on the energy method, the rollover theory of the vehicle is developed from nonlinear vehicle system dynamic. The effect of the suspension and the tire on liftoff and rollover is taken into account, and the double-line test and Fishhook test are used to verify the proposed model. Each test of the different vehicles applies the predictive algorithm to different cases (liftoff, rollover, and no liftoff or no rollover), and especially in the test case of V = 60km/h, it is predicted that the vehicle has enough liftoff energy in the 1.9th second and the vehicle lifted off the ground in the 2.1th second, that is, the proposed prediction is 0.2 s ahead of the actual situation; it is predicted that the vehicle has enough rollover energy in the 2.7th second and the vehicle rolled over in the 3.15th second, that is, the proposed prediction is 0.45 s ahead of the actual situation. The rollover predictive method can be proved to be effective and conservative.

Footnotes

Handling Editor: Marianna Imprialou

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was supported by the National Natural Science Foundation of China (grant no. 51705220), the Jiangsu Province Higher Education Natural Science Research Project (grant no. 17KJD580001), the Jiangsu Provincial Higher Education Natural Science Research Major Project (grant no. 17KJA580003), and Foundation for Jiangsu Province “333 Project” Training Funded Project (grant no. BRA2015365).

ORCID iD

Bo Li

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