The influence of driving axle location on the lateral force of vehicle

le that affects the directional stability of vehicle. This paper presents some research results of the influence of driving axle location on the lateral force of vehicle with 4x2 wheel formula when vehicle is turning. The single track dynamical models of FWD and RWD vehicle while cornering are created. Based on these two dynamical models, the system equations of motion built are enable to study the influence of driving axle location on the lateral force. Some calculated simulation results are shown for illustration. Key words: lateral force, driving axle, turning, transmission system Cite this Article: Nguyen Khac Tuan, The Influence of Driving Axle Location on the Lateral Force of Vehicle. International Journal of Mechanical Engineering and Technology 10(6), 2019, pp. 11-18. 1. INTRODUCTION The transmission system plays an important role in the overall performance of vehicle. Especially, it has a significant influence on the dynamical characteristic, fuel consumption and stability of the vehicle [14,16]. So far, there have been a number of publications related to transmission system [13-16]. However, these works mainly focus on studying dynamic loads appearing in transmission system [13,15,16] or investigating characteristics of acceleration [1,2], the fuel consumption of vehicles with different powertrain structures [1,2,6,10]. Literature [1-16] showed very few published works that are concerned with the influence of transmission layout on the stability of vehicles. In actual working conditions, the lateral force acting on the tires produces a side slip angle that affects the directional stability of vehicle [1,2,6,9,12]. Lateral forces can be caused by many factors such as the impact of side wind, uphill, cornering... The main purpose of this study is to investigate the influence of the location of driving axle on the lateral force acting on the wheels of the vehicle with 4x2 wheel formula while cornering [4,5,9,11]. Nguyen Khac Tuan 12 editor@iaeme.com 2. MATERIAL AND METHOD 2.1. Dynamical model of vehicle while cornering Consider the single track models shows in Figure 1. The parameters used in the vehicle’s model are defined as below: θ – the steer angle of the front axle; z  - the angular speed of the vehicle about the vertical axis ; Fx, Fy: the external forces in the x and y diretions; VA, VD: the velocity of the front and rear tires, respectively; Vx, Vy: the speed of vehicle in the x and y diretions; Rx1i, Rx1i: the forces resulting from braking or tractive effort and/ or rolling resistance, acting parallel to the plane of the wheel to the front and rear wheels, respectively; Ry1i, Ry2i, : the lateral forces, acting to the front and rear wheels respectively; (i=r or f ), f, r: the index for front wheel drive (FWD) and rear wheel drive (RWD), respectively; a,b: the lengths from the mass center to the front and rear axles, respectively. Rear wheel drive Front wheel drive Figure 1. Single track models of vehicle during cornering The equations of motion are writen by summing forces in the x and y directions and summing moments about a vertical axis through the mass center. For the system shown in figure, 1a can be written as:   2 1 1 1 2 1 1 1 2 cos sin cos sin cos sin jx x r x x r y r jy y r y r x r y jz y r x r y r z F R F R R F R R R F M R R a R b M                   (1) where Rxr1 is in the opposite direction of velocity VA and has the form as: Rx1r=f.Z1 For the system shown in figure 1b, the equations of motion can be written as:   1 1 2 1 2 1 1 1 2. . cos sin cos sin cos sin jx x f y f x f jy y f y f x f y jz y f x f y f z F R F R Rx F R R R F M R R a R b M                   (2) a b A C D vD vA vy     Rx2r vx Rx1r Ry1r Ry2r R  vC Fjx Fjy O Mz Mjz y x a b A C D vD vA vy     Rx2f vx Rx1f Ry1f Ry2f R  vc O Fjx jy Mz Mjz y x F The Influence of Driving Axle Location on the Lateral Force of Vehicle 13 editor@iaeme.com In which, Rxr2 is in the opposite direction of velocity VD and has the form as: Rx2f=f.Z2 In the equation (1) and (2): Fjx, Fjy are the centrifugal forces in the x and y diretions; Mz is the moment of turning resistance; Mjz is the yaw moment. These can be expressed as: jx x xP m j  (3) jy y yP m j  (4) 2.jz z z z zM J m      (5) where: Jz is the moment of inertia of the vehicle about the vertical axis; m is the mass of the vehicle. Solving equation (1) can determine the forces Ry1r, Ry2r and Rx2r acting on the tires of RWD vehicle:       1 1 2 1 2 tan cos tan cos jy y z jz y r jy y z jz y r jy y z jz x r x jx F F b M M R fZ L F F a M M R L F F b M MfZ R F F L                     (6) Following a similar approach, from quation (2), the forces Ry1f, Ry2f and Rx1f can be calculated as:     1 2 2 1 2 ) ( ) ( cos . sin cos sin jy y z jz y f x jx jy y z jz y f jy y z jz x f jx x M F F M F M F F F b M R F f Z L a F a M R L F b M R F fZ L                       (7) 2.2. Forces and moments acting on the vehicle while cornering External forces in the x direction. The external force Fx in the direction x is [1,10]: Fx=Fi + Fw =m.sin + K.F.v 2 x0 (8) where: Fi is the climbing resistance; Fw is the aerodynamic drag force;  is the the angle of uphill gradient; Cd denotes the aerodynamic drag coefficient, ρ is the air density and A is the cross sectional area. Turning resistance moment Mz. During cornering of vehicle, the turning resistance moment Mz is generated by the difference of torque at the left and right of driving wheels. This value depends on the locking coefficient of the differential between the wheels, which is determined by the equation [9]: bt bn bt bn d bt bn k M M M M k M M M      (9) Mk – total torque, supplied to both wheels of driving axle. Moment Mz can be determined as: Nguyen Khac Tuan 14 editor@iaeme.com   . .bt bn k d z d d B M M B M k M r r    (10) Centrifugal forces and yaw moment. In order to determine the centifugutial forces Pjx, Pjy and yaw moment Mjz we have to determine normal and tangential accelerations jx, jy and εz. Figure 2. Scheme to determine the acceleration of the vehicle Figure 3. Simulink model for calculating accelerations jx and jy Figure 2 shows the scheme for determining the acceleration of the vehicle jx, jy. We have: ( ) ( ) y y N DO T CD x Z dV j j j V dt     (11) ( ) ( ) x x T DO N CD y Z dV j j j V dt     (12) where: JN(DO), JT(DO): normal and tangential acceleration of point D relative to point O, respectively; JN(CD), JT(CD): normal and tangential acceleration of point C relative to point D, respectively. Angular speed is expressed as [6,10]:  1 2xx z VV R L         (13) Where: δ1, δ2 slip angles of the front- and rear wheels; Rδ - turning radius. The relationship between velocities Vx and Vy has the form: tany xV V  (14) ψ: the body slip angle After a few transformations we get:  1 2x y V b a V L        (15)    1 2 1 2x x y dV b a V b adV dt dt L                  (16) Substituting equations (15), (16) into equations (11), (12) và (13) and then substituting into (3), (4) and (5), yields: O CD j T(DO) j N(CD) j T(CD) j N(DO) The Influence of Driving Axle Location on the Lateral Force of Vehicle 15 editor@iaeme.com  2 1 2 . . xx jx x V b adV P m j m dt R L                 (17)    2 1 2 1 2 . . x x x jy y dV b a V b a V dtP m j m R L                             (18)    1 2 1 2 2 2. . . . . x x jz z z z z z dV V dtM J m m L                               (19) After detemining Fx, Fy, Fjx, Fjy and subsituting their values into equations (6) and (7) the values of lateral forces Ry1f, Ry2f, and Ry1r, Ry2r of FWD and RWD vehicles with the aid of Matlab-simulink software are obtained 3. RESULTS AND DISCUSSIONS For illustration we show the results of calculating the influence of driving axle location on lateral forces while vehicle cornering with the aid of Matlab-Simulink 2018 software. Parameters for the model are chosen as: m=1290 (kg); a=1.19(m); b=1.37(m); B=1.5(m); jz=1752 (kg/m 2 );kd=0.05;f=0.02. The results of calculation of lateral forces generated in the front axle (Ry1f, Ry1r) and rear axle (Ry2f, Ry2r) of FWD and RWD vehicles while cornering are presented in Figures 4 – 8. Figure 4. The lateral force vs velocity for FWD and RWD vehicle with 4x2 wheel formula From the graph in Figure 4 it can be seen that the lateral forces in the rear wheels are relatively similar for both RWD and FWD layouts (Ry2f= Ry2r). However, the lateral force generated in the front wheel is markedly different in two cases, in which the lateral force acting on front wheel of FWD is smaller than that in RWD vehicle (Ry1f<Ry1r). The analysis of results in Figures 5 and 6 also shows that the values of the lateral forces in both layouts of driving axle depend strongly on the two factors: velocity and the steering angle of front wheel. In the same cornering condition of vehicle, the greater the automobile velocity grows, the greater the lateral force is generated. Also, the greater the steering angle θ, the greater the lateral force acting on the wheels. This can be explained as follows: the lateral forces value generated while vehicle cornering is strongly depend on the centrifugal forces Fjx, Fjy and moment Mjz, especially the centrifugal forces Fjy in y direction; according to equations (18) and (19), these forces and moment are directly proportional to the value of Nguyen Khac Tuan 16 editor@iaeme.com velocity Vx and rotation angle θ. Therefore, when the speed Vx or the steering angle θ of front wheel of cornering vehicle increases, the value of lateral forces acting on the wheels grows. Figure 5. The lateral force acting on rear wheels of RWD vs velocity with different of steering angles Figure 6. The lateral force acting on front wheels of FWD vs steering angle with different of velocities Figure 7. The lateral forces vs velocity of RWD vehicle with θ=0,2 rad Figure 8.The lateral forces vs velocity of FWD vehicle with θ=0,6 rad In addition, when comparing the lateral force acting on the wheels of both case front wheel drive and rear wheel drive vehicles (Figure 7 and 8), it is shown that the lateral force acting on the front wheel is always greater than the lateral forces acting on the rear wheel. The Influence of Driving Axle Location on the Lateral Force of Vehicle 17 editor@iaeme.com This result is due to, beside the effect of centrifugal forces and yaw moment. Since the front wheel also has functions of a steering wheel, it bears an additional lateral force generated by steering angle θ. 4. CONCLUSIONS This study presented a model for calculating the lateral forces acting at the wheels of FWD and RWD vehicles with 4x2 wheel formula when cornering. The main conclusions are: (i) The lateral force acting on the wheel depends on many factors, of which the most important factors are the velocity of vehicle and the steering angle of the steering wheel; (ii) For both FWD and RWD vehicles, the lateral forces acting on front wheel are always of greater than those of rear wheel; (iii) The value of lateral force acting on front wheel of FWD vehicle is smaller than the RWD vehicle with 4x2 wheel formula. In other words, in the same cornering condition, FWD vehicle has better directional stability than RWD vehicle. ACKNOWLEDGEMENTS The work described in this paper was supported by Thai Nguyen University of Technology for a scientific project. REFERENCES [1] J.Y. Wong, Theory of ground vehicles, 4th ed., (John Wiley and Sons, Inc., New York, 2008) [2] Hans Pacejka, Tyre and vehicle dynamic 3rd Edition (Butterworth-Heinemann, 2012) [3] Charles Wayne Johnson Lateral stability of the driver/vehicle system: analytical results, PhD thesis, Iowa State University, (1983) [4] Yi LU et al.: Study on the Stability of High-Speed Turning Braking Based on the Hardware-in-the-Loop Test. https://doi.org/10.17559/TV-20170720161424 [5] Li Li et al.: Integrated Longitudinal and Lateral Tire/Road Friction Modeling and Monitoring for Vehicle Motion Control, IEEE Transactions on Intelligent Transportation Systems, Vol. 7 (2006) March. [6] Rajesh Rajamani, Vehicle dynamic and control, (Springer, 2006). [7] Moustapha Doumiat et al. A method to estimate the lateral tire force and the sideslip angle of a vehicle: Experimental validation, American Control Conference Marriott Waterfront, Baltimore, MD, USA (2010), June 30-July 02. [8] Xianbin Wang and Shuming Shi Analysis of Vehicle Steering and Driving Bifurcation Characteristics, Mathematical Problems in Engineering Volume (2015), [9] Д.Р. Эллис, Управляемость автомобиля, (М. Машиностроение, 1975) [10] Тарасик В.П, Теория движения автомобиля, ( СПб. БХВ-Петербург, 2006). Nguyen Khac Tuan 18 editor@iaeme.com [11] Баулина Е.Е. и др, К вопоросу исследования устойчивости и управляемости гибридного автомобиля с изменямым в прсцессе движения типом привода, Известия МГТУ МАМИ (2012), 29-37. [12] В.Н. Кравец, Р.А. Мусарский, Вляние конструктивных факторов на управляемость и устойчиввость автобуса , Известия ВолгГТУ (2015), 32-38. [13] И. С. Цитович и В. Б. Альгин , Динамика автомобиля, (Мн. Наука и техника, 1981). [14] Cornel Stan, Alternative Propulsion for Automobiles, (Springer 2016). [15] Nguyen Khac Tuan, Le Van Quynh, Modeling and simulation of vehicle vertical vibration from powertrain and road excitation.: International Symposium on Technology for Sustainability, Bangkok, Thailand (2012), 512–515. [16] Nguyen Trong Hoan and Nguyen Khac Tuan, Automotive transmission system, (Vietnam educational Publishing house 2018).