# Investigate the Relationship between the Vehicle Roll Angle and Other Factors When Steering

Authors: Duc Ngoc Nguyen; Tuan Anh Nguyen
Journal: Modelling and Simulation in Engineering
 (2023)

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## Abstract

Rollover is a dangerous phenomenon. It is closely related to the vehicle roll angle. The greater the roll angle, the greater the risk of rollover. The vehicle roll angle when steering depends on many factors, such as the size of the vehicle, speed of movement, steering angle, etc. In this paper, the author has simulated the oscillation of a car when steering using MATLAB® software with three specific cases. The purpose of the study is to evaluate the dependence of the roll angle on other factors. Each case handles two scenarios: vehicle speed change (fixed height) and vehicle height change (fixed speed). The model of a complex dynamic, a combination of many nonlinear components, is used to simulate vehicle oscillations. According to the study’s results, the roll angle will increase if the speed or the distance from the center of gravity (CG) to roll axis (RA) increases, respectively. Once the roll angle’s value rises, the roll index also increases, which causes the dynamic force at the wheel to decrease. If the vertical force at the wheel approaches zero, a rollover may occur. The rollover phenomenon occurred in the second case, corresponding to speedsv=80 (km/h) and v=85 (km/h). The peak values of the roll angle are 7.77° and 7.63°, respectively. This result helps to identify the factors affecting the rollover phenomenon more clearly.


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## Body

## 1. Introduction

Since more than 130 years ago, the automotive sector has seen significant growth. Today’s automobiles are far more extensive and faster compared to the past. Therefore, the stability and safety of the vehicle are crucial concerns that require complete attention. Several phenomena may be used to illustrate the instability of a moving car on the road, with vehicle rollover being the most hazardous. Rollover incidents can have severe repercussions for individuals and items [1].The phenomenon of rolling over happens when the wheels are completely pushed off the road, i.e., when the dynamic force at the wheels approaches zero [2–4]. Regarding the problem of rolling, the wheel cannot return to its previous position after being lifted off the road. Once the vertical force on the wheel is zero, the roll angle may reach its highest limit [5]. Rollovers typically occur when a vehicle steer to avoid an obstruction or enters a roundabout [6]. Then, centrifugal force may appear. This force will create a moment that tilts the vehicle body [7]. There are four typical reasons for rollovers. Initially, external elements such as wind force might influence a vehicle’s performance in inclement weather [8]. Nevertheless, the value of the lateral wind force is typically insignificant. A few instances of heavy truck rollover are due to the lateral wind force. The second factor contributing to this occurrence is road conditions [9]. Besides, the size of the vehicle also affects this problem. The roll moment of a car will grow according to the trackwidth and height, and this was emphasized by Nguyen et al. [10]. Because adjusting the size of the vehicle may impact its performance, it is not simple to do so. The third factor relating to this phenomenon is the vehicle’s speed. This is considered the primary source of the occurrence of rollover. If the velocity is more, the centrifugal force will grow according to the velocity’s square. Consequently, the risk of a rollover will increase. Lastly, the value of the steering acceleration and steering angle significantly impact the issue of vehicle instability. If these values rise, the rollover limitation will reduce fast, making the vehicle more susceptible to rolling.Li and Bei developed the notion of a static roll index in reference [11]. This index is only dependent on the size of the vehicle, according to Xin et al. [12]. However, this indicator is not significant. The dynamic roll index thereby substitutes the static roll index, which describes the rollover phenomenon more clearly, commonly known as load transfer ratio (LTR) [13]. This index is a result of the differential in dynamic force between the wheels, according to Jin et al. [14]. As soon as the value of this indicator exceeds 1, the car rolls over. This indicator is utilized in a large number of rollover research [15, 16].In the study of roll dynamics, several dynamical models are routinely employed. Zhang et al. merged a 1/2 model and a single-track dynamics model [17]; this is a typical design. However, it is relatively straightforward, and the accuracy is typically low. This model handles the tire as a linear deformation [18]. The vehicle’s motion should be evaluated completely using a spatial dynamics model. This model accounts for the impact of all wheels [19]. It may be utilized for both commercial and specialized vehicles [20]. Nonlinear double-track dynamics must be included in the model of spatial dynamics. In reference [21], Nguyen performed this combo. However, constructing this model is quite difficult. Today, multiple methods have been implemented to improve the vehicle’s stability at high speeds. According to Nguyen, the stabilizer bar can decrease the body roll angle during steering [22]. Moreover, the adaptive suspension system enhances the vehicle’s stability [23–26]. In addition, redistributing power to the driving wheels helps lower the incidence of rollover [27]. If the rollover limits are predefined, the phenomena of rollover may be accurately predicted. Several methods are used to determine the roll limit, such as predicting the vehicle’s roll angle [28–30], the roll index [3, 4], etc. In general, they are quite complex.Additionally, there are many solutions used to avoid rollovers. In reference [31], Bencatel et al. propose the application of the reference governors (RG) to control the car steering system. In reference [32], Edlund et al. introduce the use of the rollover protection systems (ROPS) anti-roll system for quad bikes. The sharing of the active steering (AS) and active anti-roll bar (AARB) to reduce the rollover phenomenon has been shown in reference [33] by Liu et al. Nguyen et al. also show in reference [34] how to use an active stabilizer bar to reduce the roll angle of the vehicle body. In addition, many other control algorithms that help limit the rollover phenomenon have also been introduced by many authors [35, 36]. In addition, many studies examining the dependence of rollover on other factors have also been published in the past few years. For instance, in reference [37], Alrejjal et al. indicate variables related to external factors, such as weather, roadways, etc. In reference [38], Hassan et al. state the influence of tires on lateral stability and rollover when steering. Besides, the vehicle’s mass also dramatically influences the vehicle’s rolling stability, according to Alrejjal and Ksaibati [39]. However, studies on the influence of vehicle speed and height on rollover are rarely done, although this is a significant issue. Therefore, it is necessary to investigate the dependence between height, speed, and the rollover phenomenon.As mentioned above, the rollover phenomenon is closely related to the vehicle roll angle. If the roll angle is too large, this phenomenon may occur. The roll angle depends on many factors, such as the size of the vehicle and the use of the vehicle by the driver. Therefore, it is necessary to determine the relationship between them. This paper focuses on surveying and evaluating the relationship between the vehicle roll angle and the vehicle’s height, travel speed, and steering angle. A complex dynamic model is used to simulate car oscillations under specific circumstances. This is completely new and practical content. The main content of this paper consists of four parts, including an introduction section, materials section, results section, and conclusions section. The specific contents will be presented in turn in this order.


## 2. Materials

To simulate car oscillations, a complex dynamic model needs to be built. This model includes smaller components, such as the spatial oscillation model, the tire model, and the motion model. These models are referenced in [3, 4, 21].The spatial oscillation model is described in Figure1. Equations (1) to (11) are used to describe this model.Figure 1
A model of the oscillation.Total forces acting on the sprung mass:(1)Fims=∑i,j=12FKij+FCij.Total moments acting on the vehicle body (roll):(2)Mϕ=∑i,j=12−1j−1FKij+FCijtwi+gsinϕ+v̇y+β̇+ψ̇vxcosϕmshϕ.Total moments acting on the vehicle body (pitch):(3)Mθ=∑i,j=12−1i−1FKij+FCijai.Total forces acting on the unsprung masses:(4)Fimuij=FKTij−FKij−FCiji,j=1,2¯.Where:Force of inertia (sprung mass),Fims:
(5)Fims=msz¨si,j=1,2¯.Force of inertia (unsprung mass),Fimuij:
(6)Fimuij=muijz¨uiji,j=1,2¯.Force of spring,FKij:
(7)FKij=Kijzs−zuij+−1j+1twiϕ+−1i+1aiθi,j=1,2¯.Force of damper,FCij:
(8)FCij=Cijżs−żuij+−1j+1twiϕ̇+−1i+1aiθ̇i,j=1,2¯.Force of tire,FKTij:
(9)FKTij=KTijzrij−zuiji,j=1,2¯.Moment of inertia (roll),Mϕ:
(10)Mϕ=Jϕ+mshϕ2ϕ¨.Moment of inertia (pitch),Mθ:
(11)Mθ=Jθ+mshθ2θ¨.The components related to motion, such as lateral velocityvy, longitudinal velocity vx, and yaw angle ψ, are unknowns. They can be calculated through a motion model with three degrees of freedom (Figure 2).Figure 2
A model of the motion.Total forces acting on the vehicle (longitudinal):(12)Fix=∑i,j=12Fxijcosδij−Fyijsinδij−F1+Fcex.Total forces acting on the vehicle (lateral):(13)Fiy=∑i,j=12Fxijsinδij+Fyijcosδij−F2−Fcey.Total moments acting on the vehicle (yaw):(14)Mψ=∑i,j=12−1jFxijcosδij−Fyijsinδijtwi+−1i+1Fxijsinδij+Fyijcosδijai+Fici−Mzij.Where:Force of inertia (longitudinal):(15)Fix=ms+∑i,j=12mijv̇x.Force of inertia (lateral):(16)Fiy=ms+∑i,j=12mijv̇y.Centrifugal force (longitudinal):(17)Fcex=ms+∑i,j=12mijβ̇+ψ̇vy.Centrifugal force (lateral):(18)Fcey=ms+∑i,j=12mijβ̇+ψ̇vx.Moment of inertia (yaw):(19)Mψ=Jψψ¨.Heading angle:(20)β=arctanvyvx.The longitudinal force, lateral force, and moment at the wheel can be determined by the tire model. Tire modelling can be simple or complex, depending on the intended use of the studies. In this work, the author proposes using the Pacejka tire model to determine the above unknowns.Longitudinal force:(21)Fx=DxsinCxartanBxκx.Where:(22)κx=1−Exλ+ExBxarctanBxλ.(23)Cx=1.65.(24)Dx=a1Fz2+a2Fz.(25)BCDx=a3Fz2+a4Fzea5Fz,(26)Bx=BCDxCxDx,(27)Ex=a6Fz2+a7Fz+a8.Lateral force:(28)Fy=DysinCyartanByκy+Svy.Where:(29)κy=1−Eyα+Shy+EyByarctanByα+Shy.(30)Cy=1.30.(31)Dy=a1Fz2+a2Fz,(32)BCDy=a3sina4artana5Fz,(33)By=BCDyCyDy,(34)Ey=a6Fz2+a7Fz+a8,(35)Shy=a9γ,(36)Svy=a10Fz2+a11Fzγ.Aligning moment:(37)Mz=DzsinCzartanBzκz+Svz.Where:(38)κz=1−Ezα+Shz+EzBzarctanBzα+Shz,(39)Cz=2.40,(40)Dz=a1Fz2+a2Fz,(41)BCDz=a3Fz2+a4Fzea5Fz,(42)Bz=BCDzCzDz,(43)Ez=a6Fz2+a7Fz+a8,(44)Shz=a9γ,(45)Svz=a10Fz2+a11Fzγ.After the dynamic modelling process is completed, the simulation process will take place.


## 3. Results

Numerical simulation is performed to evaluate the dependence of the roll angle on other factors. MATLAB® software is used for this purpose. Three cases correspond to the three steering angles used to simulate car oscillations (Figure3). These three steering angles have differences in both magnitude and steering acceleration. Their values are not much different but can be enough to cause a rollover (for the second case and third case) when the vehicle travels at high speed. In each case, the vehicle speed and height will be varied to determine this relationship. The parameters that serve the simulation process are given in Table 1. These values are referenced in the CARSIM® application. CARSIM® software is a specialized application for simulating car oscillations. This software can be used independently or in conjunction with other software, such as MATLAB® [40], Python® [41], and LabVIEW® [42]. Many studies often refer to the specifications of cars using this software. Some parameters can be slightly adjusted to suit the actual conditions.Figure 3
Steering angle.Table 1
The simulation parameters.
No.SymbolDescriptionValueUnit1K11/K12Spring stiffness of the front axle45,200N/m2K21/K22Spring stiffness of the rear axle44,600N/m3KT11/KT12Spring tire of the front axle178,000N/m4KT21/KT22Spring tire of the rear axle178,000N/m5C11/C12Damper coefficient of the front axle3300Ns/m6C21/C22Damper coefficient of the rear axle3220Ns/m7msSprung mass1740kg8muijUnsprung mass41kg9twiHalf of the track width front/rear axle805/815mm10aiThe distance between the center of gravity to the front or rear axle1190/1360mm11JϕMoment of inertia (longitudinal-axis)720kg/m212JθMoment of inertia (lateral-axis)2700kg/m213JψMoment of inertia (vertical-axis)2680kg/m2


### 3.1. The First Case

In the first case, the steering angle is minor (Figure3). The dependence of the roll angle on the velocity is shown in Figure 4. In this case, the distance from center of gravity (CG) to roll axis (RA) is constant, with h=550 (mm). It is clear that the roll angle will increase if the vehicle speed increases. According to this result, the peak of the roll angle reached 4.84°, 5.37°, 5.90°, and 6.43°, respectively, corresponding to the speed range from 60 to 75 (km/h). When the speed increases up to v=80 (km/h), the maximum value of the roll angle will continue to increase, reaching 6.94°. This number can go up to 7.47° once the velocity reaches the limit of the investigation process, v=85 (km/h). After these values peak, they tend to decrease over time, even though the steering angle does not change. This happens because of elastic tire deformation, which is described by a complex nonlinear tire model. These values tend to converge in a specific domain. However, there is still a small difference between the values. The most significant decline is when the vehicle moves at top speed and vice versa.Figure 4
The change of the roll angle (the first case—velocity change).When the vehicle body roll angle changes, the rollover index (RI) value also changes accordingly. The change in RI is depicted in Figure5. The trend of the change in the values in Figure 5 is similar to the values in Figure 4. The peak of the values is still reached at time t≈3.7 (s). This is the time after the steering angle has reached its maximum at t=2 (s). The maximum roll index obtained is 0.93, corresponding to a speed of v=85 (km/h). If the velocity decreases, the value of RI also decreases accordingly. The values of RI obtained are 0.87, 0.80, 0.74, 0.67, and 0.60, respectively, corresponding to the speed range v = {80; 75; 70; 65; 60} (km/h). After reaching the peak, their values will gradually decrease and converge in a certain domain.Figure 5
The change of the roll index (the first case—velocity change).If the vehicle’s speed is fixed and the distance from CG to RA varies, the vehicle’s roll angle also changes according to the value of the distance. Compared with the two graphs above, the changing trend of the values in Figure6 is different. However, this difference is not so significant. According to this finding, the roll angle will increase once the distance from CG to RA increases accordingly. Their peak values are still reached at t≈3.7 (s), then gradually decreasing over time. The difference between the values corresponding to the distances is not large and relatively even. The peak value of the roll angle corresponding to h=550 (mm) is only 4.84°, while this value can reach 5.82° if the distance increases to h=650 (mm).Figure 6
The change of the roll angle (the first case—distance change).Because the difference in roll angles for different distances is not too much, the change of RI, in this case, is not too significant (Figure7). The changing trend of RI still follows the changing direction of the vehicle roll angle, which has been proved by the results obtained above. In this case, the difference between RImin and RImax is only about 0.14, and the gap between the values is quite even.Figure 7
The change of the roll index (the first case—distance change).


### 3.2. Second Case

The steering angle in the first case is relatively small, so it is necessary to use a larger steering angle to investigate the car’s oscillations comprehensively. In the second case, the values of the steering angle and the steering acceleration increase (Figure3). The change of the roll angle at a fixed distance, h=550 (mm), and the difference in speed are shown in Figure 8. According to this result, if the velocity rises, the roll angle increases accordingly. The peak values of the roll angle at speeds from 60 to 75 (km/h) are 5.47°, 6.09°, 6.68°, and 7.28°, respectively. At higher speeds, v=80 (km/h), the rollover occurs at time t≈7.8 (s), i.e., the vertical force at the wheel approaches zero. The main cause of this phenomenon is that the steering angle is too large, and the vehicle is traveling at high speed, which causes the vehicle’s roll angle to increase suddenly and drastically reduce the value of the vertical force at the wheel. Once the car is rolled over, the roll angle will reach its maximum value. In this situation, the peak value of the roll angle goes up to 7.77°. Once the speed increases, v=85 (km/h), the car rolls over earlier at time t≈3.1 (s). So, the maximum roll angle will be smaller, only about 7.63°.Figure 8
The change of the roll angle (the second case—velocity change).Once a rollover occurs, the value of the roll index approaches 1, as demonstrated in Figure9. This index can be reduced if the vehicle uses modern anti-roll systems, decreasing the risk of rolling over.Figure 9
The change of the roll index (the second case—velocity change).Compared with the above situation, if the speed is fixed atv=60 (km/h) and the distance changes, the rollover may not occur (Figure 10). This is not to say that velocity has a more significant influence than the distance in the car’s rollover mechanism. The roll angle variation also highly depends on the range of speeds and distances chosen to perform the simulation. Generally, the changing trend of the roll angle and roll index in the second case is similar to the first. However, their value is larger than in the other case. If the peak of the roll angle is not too large, the value of RI will also not be considerable (Figure 11).Figure 10
The change of the roll angle (the second case—distance change).Figure 11
The change of the roll index (the second case—distance change).


### 3.3. Third Case

In both cases mentioned above, only the J-turn steering angle is used. Therefore, it is necessary to use another steering angle to describe the vehicle’s oscillation. A sine-like steering angle, which describes the double lane change of the vehicle, is shown in the third case. The amplitude of this steering angle is larger than the amplitude of both the first and second cases.During the first oscillation phase, the roll angle of the vehicle body increases rapidly (Figure12). Their peak values reach 5.84°, 6.47°, and 7.11°, respectively, with the speed v = {60, 65, 70} (km/h). If the speed increases to v=75 (km/h), a rollover will occur. The limited roll angle that the vehicle can achieve is 7.82°. Once the speed reaches v=80 (km/h) or v=85 (km/h), rollover occurs sooner. Therefore, the limited roll angle is only 7.74° and 7.68°, respectively, for these two velocity values. In the second phase of the oscillation, the value of the vehicle body roll angle is smaller than that of the first phase. This is because of the elastic deformation of the tire caused when using a nonlinear tire model. In general, as the speed increases, the roll angle also increases accordingly.Figure 12
The change of the roll angle (the third case—velocity change).The rollover phenomenon is more clearly described by the rollover index (RI) (Figure13). Their values have approached 1, corresponding to situations where the vehicle has a speed of v = {75, 80, 85} (km/h). Meanwhile, the value of RI obtained is smaller as the vehicle’s speed decreases.Figure 13
The change of the roll index (the third case—velocity change).Figures14 and 15 depict a situation where the height is changed while the velocity is fixed. According to this finding, the roll angle and rollover index values increase with increasing height. The increase is relatively even. However, the rollover phenomenon does not occur even when the car’s moving speed is up to v=85 (km/h). In this situation, the peak value of the roll angle is only 7.06°, corresponding to RI = 0.90. It can be seen that the influence of velocity is more significant than that of distance (this is only true in the situation under consideration).Figure 14
The change of the roll angle (the third case—distance change).Figure 15
The change of the roll index (the third case—distance change).


## 3.1. The First Case

In the first case, the steering angle is minor (Figure3). The dependence of the roll angle on the velocity is shown in Figure 4. In this case, the distance from center of gravity (CG) to roll axis (RA) is constant, with h=550 (mm). It is clear that the roll angle will increase if the vehicle speed increases. According to this result, the peak of the roll angle reached 4.84°, 5.37°, 5.90°, and 6.43°, respectively, corresponding to the speed range from 60 to 75 (km/h). When the speed increases up to v=80 (km/h), the maximum value of the roll angle will continue to increase, reaching 6.94°. This number can go up to 7.47° once the velocity reaches the limit of the investigation process, v=85 (km/h). After these values peak, they tend to decrease over time, even though the steering angle does not change. This happens because of elastic tire deformation, which is described by a complex nonlinear tire model. These values tend to converge in a specific domain. However, there is still a small difference between the values. The most significant decline is when the vehicle moves at top speed and vice versa.Figure 4
The change of the roll angle (the first case—velocity change).When the vehicle body roll angle changes, the rollover index (RI) value also changes accordingly. The change in RI is depicted in Figure5. The trend of the change in the values in Figure 5 is similar to the values in Figure 4. The peak of the values is still reached at time t≈3.7 (s). This is the time after the steering angle has reached its maximum at t=2 (s). The maximum roll index obtained is 0.93, corresponding to a speed of v=85 (km/h). If the velocity decreases, the value of RI also decreases accordingly. The values of RI obtained are 0.87, 0.80, 0.74, 0.67, and 0.60, respectively, corresponding to the speed range v = {80; 75; 70; 65; 60} (km/h). After reaching the peak, their values will gradually decrease and converge in a certain domain.Figure 5
The change of the roll index (the first case—velocity change).If the vehicle’s speed is fixed and the distance from CG to RA varies, the vehicle’s roll angle also changes according to the value of the distance. Compared with the two graphs above, the changing trend of the values in Figure6 is different. However, this difference is not so significant. According to this finding, the roll angle will increase once the distance from CG to RA increases accordingly. Their peak values are still reached at t≈3.7 (s), then gradually decreasing over time. The difference between the values corresponding to the distances is not large and relatively even. The peak value of the roll angle corresponding to h=550 (mm) is only 4.84°, while this value can reach 5.82° if the distance increases to h=650 (mm).Figure 6
The change of the roll angle (the first case—distance change).Because the difference in roll angles for different distances is not too much, the change of RI, in this case, is not too significant (Figure7). The changing trend of RI still follows the changing direction of the vehicle roll angle, which has been proved by the results obtained above. In this case, the difference between RImin and RImax is only about 0.14, and the gap between the values is quite even.Figure 7
The change of the roll index (the first case—distance change).


## 3.2. Second Case

The steering angle in the first case is relatively small, so it is necessary to use a larger steering angle to investigate the car’s oscillations comprehensively. In the second case, the values of the steering angle and the steering acceleration increase (Figure3). The change of the roll angle at a fixed distance, h=550 (mm), and the difference in speed are shown in Figure 8. According to this result, if the velocity rises, the roll angle increases accordingly. The peak values of the roll angle at speeds from 60 to 75 (km/h) are 5.47°, 6.09°, 6.68°, and 7.28°, respectively. At higher speeds, v=80 (km/h), the rollover occurs at time t≈7.8 (s), i.e., the vertical force at the wheel approaches zero. The main cause of this phenomenon is that the steering angle is too large, and the vehicle is traveling at high speed, which causes the vehicle’s roll angle to increase suddenly and drastically reduce the value of the vertical force at the wheel. Once the car is rolled over, the roll angle will reach its maximum value. In this situation, the peak value of the roll angle goes up to 7.77°. Once the speed increases, v=85 (km/h), the car rolls over earlier at time t≈3.1 (s). So, the maximum roll angle will be smaller, only about 7.63°.Figure 8
The change of the roll angle (the second case—velocity change).Once a rollover occurs, the value of the roll index approaches 1, as demonstrated in Figure9. This index can be reduced if the vehicle uses modern anti-roll systems, decreasing the risk of rolling over.Figure 9
The change of the roll index (the second case—velocity change).Compared with the above situation, if the speed is fixed atv=60 (km/h) and the distance changes, the rollover may not occur (Figure 10). This is not to say that velocity has a more significant influence than the distance in the car’s rollover mechanism. The roll angle variation also highly depends on the range of speeds and distances chosen to perform the simulation. Generally, the changing trend of the roll angle and roll index in the second case is similar to the first. However, their value is larger than in the other case. If the peak of the roll angle is not too large, the value of RI will also not be considerable (Figure 11).Figure 10
The change of the roll angle (the second case—distance change).Figure 11
The change of the roll index (the second case—distance change).


## 3.3. Third Case

In both cases mentioned above, only the J-turn steering angle is used. Therefore, it is necessary to use another steering angle to describe the vehicle’s oscillation. A sine-like steering angle, which describes the double lane change of the vehicle, is shown in the third case. The amplitude of this steering angle is larger than the amplitude of both the first and second cases.During the first oscillation phase, the roll angle of the vehicle body increases rapidly (Figure12). Their peak values reach 5.84°, 6.47°, and 7.11°, respectively, with the speed v = {60, 65, 70} (km/h). If the speed increases to v=75 (km/h), a rollover will occur. The limited roll angle that the vehicle can achieve is 7.82°. Once the speed reaches v=80 (km/h) or v=85 (km/h), rollover occurs sooner. Therefore, the limited roll angle is only 7.74° and 7.68°, respectively, for these two velocity values. In the second phase of the oscillation, the value of the vehicle body roll angle is smaller than that of the first phase. This is because of the elastic deformation of the tire caused when using a nonlinear tire model. In general, as the speed increases, the roll angle also increases accordingly.Figure 12
The change of the roll angle (the third case—velocity change).The rollover phenomenon is more clearly described by the rollover index (RI) (Figure13). Their values have approached 1, corresponding to situations where the vehicle has a speed of v = {75, 80, 85} (km/h). Meanwhile, the value of RI obtained is smaller as the vehicle’s speed decreases.Figure 13
The change of the roll index (the third case—velocity change).Figures14 and 15 depict a situation where the height is changed while the velocity is fixed. According to this finding, the roll angle and rollover index values increase with increasing height. The increase is relatively even. However, the rollover phenomenon does not occur even when the car’s moving speed is up to v=85 (km/h). In this situation, the peak value of the roll angle is only 7.06°, corresponding to RI = 0.90. It can be seen that the influence of velocity is more significant than that of distance (this is only true in the situation under consideration).Figure 14
The change of the roll angle (the third case—distance change).Figure 15
The change of the roll index (the third case—distance change).


## 4. Conclusions

The phenomenon of vehicle rollover usually occurs when the driver makes a sudden turn to avoid obstacles. Therefore, it depends a lot on the steering angle and steering acceleration. Besides, this phenomenon also depends on the speed of travel as well as the size of the vehicle. This phenomenon occurs once the vehicle body’s roll angle reaches the limit state, i.e., the vertical force at the wheel is reduced to zero. The consequences of this phenomenon are often very severe, so it is necessary to identify the factors affecting the rollover and prevent it.In this paper, the author has established a complex dynamic model to simulate vehicle oscillation. This complex model includes component models such as the spatial oscillation model, nonlinear motion model, and nonlinear tire model. The simulation takes place in three specific cases in the MATLAB® environment. In each case, the factors’ values are changed flexibly, so that a comprehensive assessment of the dependence of the roll angle on other factors can be made.According to research findings, the roll angle can increase if the speed or height increases. Besides, if the steering angle is more significant, this angle will also be larger. This results in an increased rollover index and a higher risk of a rollover. Therefore, the driver should not suddenly steer at high speed to prevent the vehicle from rolling over. Besides, the height of the vehicle body should be limited, and the trackwidth should also be extended. The results of this study are entirely novel and unique. These results will be the database for other studies in the future.



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*Source: 6069078-2023-01-28.xml*