Adv Syst Sci Appl 2020; 04; 27-35 Published online at https://ijassa.ipu.ru. Determining the Vertical Force When Steering Nguyen Tuan Anh1*, Hoang Thang Binh2 1) Automotive Engineering Department, Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam E-mail: anhngtu@tlu.edu.vn 2) School of Transportation Engineering, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam E-mail: binh.hoangthang@hust.edu.vn Abstract: When the vehicles move at high velocity and quickly steer, the vehicles can be rollover. The first sign of this phenomenon is that two wheels on the same side are completely separated from the road surface. The typical value for this sign is the vertical force FZ at each wheel. If this value gradually decreases to zero, the wheel runs the risk of separating from the road surface that may lead to rollover. Therefore, the value of the vertical force Fz in different conditions needs to be determined to detect the imminent limit of the rollover phenomenon. This research established the spatial dynamic model of the vehicle to determine the vertical force FZ based on the simulation method. Besides, the equation describing the relationship between vertical force FZ, velocity v, and acceleration steering  corresponding to the value of steering angle  is also established by the calculation process. From this equation, the value of the vertical force FZ can be simply calculated with relatively high accuracy based on determining conditions. The results of this research are the basis for determining and establishing the vehicle's rollover limit. Keywords: dynamic vehicle model, vertical force, function, steering acceleration, rollover 1. INTRODUCTION 1.1. The Instability Problem of the Vehicle Automobiles are a popular vehicle in everyday life. However, the number of vehicle accidents is also very large. When the vehicles move at high velocity and quickly steer, the vehicles often encounter the situation: side slip or rollover. The side slip phenomenon usually occurs when the vehicles go on slippery roads, wheels are not able to contact the road surface. When the side slip occurs, the driver could not control the trajectory and direction of the motion of the vehicles. The sideslip problem is usually less dangerous than the rollover problem. The rollover phenomenon occurs when two wheels on the same side are completely separated from the road surface. If there is only one wheel separated from the road surface, the vehicle will be in an unstable state and at risk of rollover [15, 17]. The main cause of this phenomenon is due to the lateral acceleration ay appeared suddenly when quickly steering, especially in the case of fishhook steering [11, 18]. The parameter that warns the risk of rollover is the vertical force at wheels Fz. If this value is sufficiently close to zero, the wheel runs the risk of separating from the road surface. Therefore, if it is possible to determine the vertical force value FZ, the timing of the rollover of the vehicle can be predicted. Some solutions have been proposed to improve this situation such as the use of the active stabilizer bar, the electronic power steering, the active suspension, the electronic stability * Corresponding author: anhngtu@tlu.edu.vn 28 N.T. ANH, H.T. BINH Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) program, [etc.] [1, 16, 19]. The above systems use parameters of input signals such as the lateral acceleration ay, the roll angle , the vertical force at wheels FZ [5, 20]. 1.2. Literature Review The topic of "Rollover Vehicle" has been researched by many authors. Several authors have introduced a rollover index R [6, 8, 9, 10, 14]. This index is determined based on the difference in the vertical force of the wheels. If |R| <1, the car operates stably. In the case of |R| = 1, both wheels on one side are separated from the road surface, the vehicle rollovers completely. However, this index only really makes sense in case both wheels of the same side are separated from the road surface. If only one of the wheels is lifted off the road surface, the rollover index R cannot be determined. Besides, the value of lateral acceleration ay is often used to determine the rollover threshold of the vehicles [3, 7, 13]. The value of the lateral acceleration ay can be calculated and simulated through link equations or experiments. However, this value is only really meaningful in case of ignoring the influence of the dimensions of the vehicles. Some other studies use the roll angle of the vehicle  to determine the vehicle's rollover limited [2, 4, 12]. The value of the roll angle of the vehicles  is calculated by the lateral acceleration ay, which includes the influence of the dimensions. To accurately determine the limits of the vehicle instability, it is necessary to find the time when the wheels are separated from the road surface (FZ = 0). This research focuses on identifying the limits at which the wheels are separated from the road surface. At the same time, the research also established the equation showing the relationship between vertical force FZ, steering angle , and steering acceleration . Therefore, it is easy to determine the vertical force FZ at different times and conditions. 2. METHODOLOGY 2.1. Nomenclature : Pitch angle, rad : Roll angle, rad : Steering angle, rad : Yaw angle, rad ay: Lateral acceleration, m/s2 b: Half of the base width, m Cij: Damping coefficient, Ns/m FCij: Force of the damper, N FKij: Force of the spring, N FKTij: Force of the tire, N FXij: Longitudinal force, N FYij: Lateral force, N FZij: Vertical force, N h: Distance from center to roll axis, m Iz: Moment of inertia of the z-axis, kgm2 Ix: Moment of inertia of the x-axis, kgm2 Kij: Stiffness of the spring, N/m KTij: Stiffness of the tire, N/m l1: Distance from the center to the front axle, m l2: Distance from the center to the rear axle, m m: Sprung mass, kg mij: Unsprung mass, kg DETERMINING THE VERTICAL FORCE WHEN STEERING 29 Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) uij: Bump on the road, m vx: Longitudinal velocity, m/s vy: Lateral velocity, m/s z: Displacement of the sprung mass, m zij: Displacement of the unsprung mass, m 2.2. Double-track Dynamic Vehicle Model The motion of the vehicle is set based on the model of 10 degrees of freedom. The double- track model is set as below (Fig. 2.1). Assuming that the vehicle is moving at a constant velocity, the steering angle is small. Ignore the influence of other factors. The motion determination matrix is described as follows [2]: 1 2 z ij i; j = 1 y xy1 y2 2 2 z i;j = 1 l1 Im + m (v ψ) = (F F ) - v (ψ 0) l1 - Im + m ij                 (2.1) Where: y1 y11 y12F = F + F y2 y21 y22F = F + F Fig. 2.1. The double-track model 2.3. Spatial Dynamic Model 7 DOF To determine the oscillation of the vehicle, it is necessary to establish the spatial dynamic model with 7 degrees of freedom as Fig. 2.2. 30 N.T. ANH, H.T. BINH Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) Fig. 2.2. Dynamic vehicle model 7 DOF The matrix of vehicle oscillation is given as follows [2]: 22 yx 1 2 2 x 2 x 1 b m (a cos φ + gsinφ)mhI + mh (z φ) = (F F ) + 0 1 b I + mh - m I + mh                    (2.2) Where: 1 C11 K11 C21 K21F = F + F + F + F 2 C12 K12 C22 K22F = F + F + F + F To be able to identify the above matrix, use the corresponding link equations. The vertical displacement of the unsprung mass: ij ij KTij Cij Kij m z = F - F - F (2.3) The elastic force of the suspension system: Kij ij ij F = K (z - z ± bφ) (2.4) The damping force of the suspension system: Cij ij ij F = C (z - z ± bφ) (2.5) The elastic force of the tire: KTij Tij ij ij F = K (u - z ) (2.6) The vertical force FZ is calculated by using the equation below: Zij KTij Cij Kij F = F - F - F (2.7) When the vertical force FZ becomes close to zero, the wheel tends to separate from the road surface, the vehicle is in an unstable state. DETERMINING THE VERTICAL FORCE WHEN STEERING 31 Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) 3. RESULT AND DISCUSSION 3.1. Simulation Conditions The established model is used for most of the common vehicles. To conduct simulation and review, the specifications of the reference vehicles are given in Table 3.1. Table 3.1. Vehicle specifications Symbol Description Value Unit m Sprung mass 2000 kg mij Unsprung mass 45 kg l1 Distance from center of gravity to front axle 1250 mm l2 Distance from center of gravity to rear axle 1550 mm b Half of the base width 750 mm Ix Moment of inertia of the x–axis 800 kgm2 Iz Moment of inertia of the z–axis 2400 kgm2 h Distance from the center to roll axis 650 mm At the velocity v = 70 km/h, the steering angle  = 0–50 (linear increase in 1 second), the graph in Fig. 3.1 shows the value of the vertical force FZ at the wheels at the same time. Fig. 3.1. The value of the vertical force FZ From Fig. 3.1 it can be seen that the value of the FZ21 is the smallest, this wheel tends to separate from the road surface first. Therefore, simulation is concentrated at this position to determine the time of separating the wheels from the road surface. The research will simulate the oscillation of the vehicle with two cases: Case 1: Steering angle  = 0–40 (linear increase), corresponding to the different values of steering acceleration and velocity of the vehicle. Case 2: Steering angle  = 0–50 (linear increase), corresponding to the different values of steering acceleration and velocity of the vehicle. 32 N.T. ANH, H.T. BINH Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) 3.2. Results Using the parameters of the reference vehicle and the two proposed cases, the graph in Fig. 3.2 shows the relationship between steering acceleration , velocity v, and vertical force FZ21 of the wheel at position (21). The graph in Fig. 3.2 indicates that: + At the same value of the velocity, if the steering acceleration increases, the vertical force at the wheel will decrease. In case the velocity value is small, this decrease is not much. Otherwise, if the velocity value is large, this attenuation will be significant. + At the same value of steering acceleration, if the velocity increases, the vertical force will drop sharply. This decrease is almost linear (in case the value of steering acceleration is large). + At the same value of velocity and steering acceleration, if the steering angle is larger, the value of vertical force will be smaller. The relationship between the parameters in Fig. 3.2 has the form of irregular planes. Therefore, it is possible to set up many plane equations to determine the vertical force value FZ when the velocity v and steering acceleration  are known. The equation for determining the value of the vertical force FZ is as follows: Z 1 2 3F = A + A ε + A v (3.1) Where: A1, A2, and A3 are the coefficients of the equation for each specific determination interval. Based on the results obtained from the simulation process, the above coefficients are given in Table 3.2. The coefficients A1, A2, and A3 correspond to different values of steering angle and steering acceleration. Fig. 3.2. The relationship between steering acceleration, velocity, and vertical force Table 3.2. The coefficients of the equation Steering angle (deg) Steering acceleration (deg/s2) The coefficients of the equation A1 A2 A3 40   0.14 4989.07 – 71.94 – 43.73 40 0.07   < 0.14 5064.91 – 913.04 – 43.57 40  < 0.07 5099.01 – 2571.43 – 42.80 50   0.14 5023.14 – 136.69 – 54.07 50 0.07   < 0.14 5127.43 – 1434.78 – 53.63 50  < 0.07 4704.57 – 2714.29 – 48.03 DETERMINING THE VERTICAL FORCE WHEN STEERING 33 Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) The vertical force value FZ can easily be determined through the coefficients of the equation. Table 3.3 shows the difference between the results of calculating vertical force FZ by two different methods (simulation and equation). Table 3.3. The value of the vertical force FZ Velocity (km/h) Steering acceleration (deg/s2) Steering angle  = 0–40 Steering angle  = 0–50 Fz simulation (N) Fz equation (N) Tolerance (%) Fz simulation (N) Fz equation (N) Tolerance (%) 60 0.279 2345 2345 0.0 1741 1741 0.0 60 0.140 2354 2355 0.0 1757 1760 0.2 60 0.093 2366 2366 0.0 1776 1776 0.0 60 0.070 2377 2387 0.4 1784 1810 1.4 60 0.056 2387 2387 0.0 1787 1771 0.9 60 0.047 2388 2411 1.0 1791 1797 0.3 60 0.040 2389 2428 1.6 1795 1815 1.1 60 0.035 2391 2441 2.0 1796 1828 1.8 70 0.279 1877 1908 1.6 1165 1200 2.9 70 0.140 1884 1918 1.8 1179 1219 3.3 70 0.093 1896 1930 1.8 1199 1240 3.3 70 0.070 1909 1951 2.2 1216 1273 4.5 70 0.056 1919 1959 2.0 1224 1191 2.8 70 0.047 1926 1983 2.9 1231 1216 1.2 70 0.040 1931 2000 3.5 1237 1234 0.2 70 0.035 1939 2013 3.7 1241 1248 0.6 80 0.279 1440 1471 2.1 635 659 3.6 80 0.140 1448 1481 2.2 649 678 4.3 80 0.093 1461 1494 2.2 674 704 4.3 80 0.070 1477 1516 2.6 705 737 4.3 80 0.056 1493 1531 2.5 722 711 1.5 80 0.047 1506 1555 3.2 736 736 0.0 80 0.040 1516 1572 3.6 750 754 0.5 80 0.035 1525 1585 3.8 760 767 0.9 90 0.279 1033 1033 0.0 119 119 0.0 90 0.140 1043 1043 0.0 138 138 0.0 90 0.093 1059 1059 0.0 167 167 0.0 90 0.070 1080 1080 0.0 200 201 0.5 90 0.056 1103 1103 0.0 230 230 0.0 90 0.047 1123 1127 0.4 253 256 1.2 90 0.040 1142 1144 0.2 273 274 0.4 90 0.035 1157 1157 0.0 287 287 0.0 34 N.T. ANH, H.T. BINH Copyright ©2020 ASSA. Adv. in Systems Science and Appl. (2020) The data in the Table show that there is a difference between the two methods, but the tolerance is quite small. In the case of steering angle  = 0–40, the biggest tolerance is only 3.8%. In the case of steering angle  = 0–50, this value does not exceed 4.5%. In general, this difference is not much and this equation can be used in many different cases. 4. CONCLUSION The stability and safety of the vehicle are expressed through the value of vertical force at the wheel FZ. The velocity, steering angle, and steering acceleration have a great influence on this problem. When the steering angle increases or the steering acceleration increases or both factors increase, the value of the vertical force FZ will decrease significantly. If the value of FZ gets close to zero, the wheels tend to separate from the road surface. At this time, the vehicle is in an unstable situation and very dangerous. To determine the value of the vertical force FZ at different velocities, steering angle, and steering acceleration, the established dynamic vehicle model can be used. However, the setup and simulation process is complicated and inconvenient. This research gave the function of determining the value of the vertical force FZ at specific conditions. Therefore, it is easy to identify FZ without using the dynamic vehicle model to simulate. The function is not perfectly accurate, there is a difference in the vertical force value FZ when determined by this function and simulation process. However, this difference is quite small (less than or equal to 3.8% when the steering angle is from 00 to 40 and less than or equal to 4.5% when the steering angle is from 00 to 50). The results of the research can be used in many different survey cases. 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Application of an Active Anti-roll Bar System for Enhancing Vehicle Ride and Handling, Proceedings of the 2012 IEEE Colloquium on Humanities, Science and Engineering, Kota Kinabalu, Malaysia, 260–265. DOI: 10.1109/CHUSER.2012.6504321. 1. INTRODUCTION 1.1. The Instability Problem of the Vehicle 1.2. Literature Review 2. METHODOLOGY 2.1. Nomenclature 2.2. Double-track Dynamic Vehicle Model 2.3. Spatial Dynamic Model 7 DOF 3. RESULT AND DISCUSSION 3.1. Simulation Conditions 3.2. Results 4. CONCLUSION