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# Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations

###### Recibido: 14 Feb 2022
Detalles de la revista
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Electric vehicles can alleviate the energy crisis. This paper analyzes the electric vehicle handling stability model. At the same time, we analyze the dynamic interaction between the electric vehicle and the driver's human body. Furthermore, we established a nonlinear dynamic mathematical model of human-vehicle 6-DOF steering driving, including the car's lateral, yaw, and roll motions and the driver's body [1]. In this paper, we analyze the influence of human-vehicle dynamics coupling on the steering stability of electric vehicles through numerical calculation.

Electric vehicle handling dynamics model
Human-vehicle system mechanics model

The wheelbase and wheelbase of the electric vehicles studied in this paper are relatively small, so the driver/occupant is distributed front and rear. Because of the nonlinear dynamic interaction between the driver's human body and the electric vehicle, we established the coordinate system shown in Fig. 1 and the electric vehicle's handling stability [2]. The dynamic effect between the car and the driver's body is mainly realized through the elastic connection between the human body and the seat, seat belt, steering wheel, and underbody. When building the model, we treat the sitting human body as a rigid body. The connection between the human and the vehicle is simplified as the spring-damping force connection between the driver's body and the seat at point q1 of the lower body mass center and q2 point of the upper body mass center. We set the intersection of the vertical line of the center of gravity and the roll axis when the car is stationary as A. At the same time we assume that the reference base fixed at this point is A(X, Y, Z). Its Z axis is vertically upward. The horizontal, vertical axis of the vehicle is the X axis, the forward direction is the positive direction, and the Y axis direction is determined according to the right-hand rule. The following assumptions are made for the man-vehicle system handling stability mechanics model: 1) The vehicle and the driver's body perform lateral, yaw, and roll motions. 2) We take the front wheel angle as an input parameter. And the turning angle is small enough to ensure that the tire works within the linear range. 3) The front and rear suspension roll centers are at the same level. 4) When stationary, the driver's body is symmetrical concerning the XZ plane.

Differential equation of electric vehicle handling stability

Based on the mechanical model in Fig. 1, we establish the differential equation of the steering stability motion of the human-vehicle system for the relative motion of the driver's human body and the electric vehicle [3]. We set m1 as the total mass of the electric vehicle. ms is the sprung mass. u is the speed of driving at a constant speed. γ is the yaw angular velocity. g is the acceleration of gravity. Subscript 1 stands for the electric vehicle. The subscript 2 represents the driver's body.

The force balance equation of the electric vehicle along the Y axis is: $m1u(β1+γ1)−m1sh1ϕ1=Ff+Fr−FY$ {m_1}u\left({{\beta_1} + {\gamma_1}} \right) - {m_{1s}}{h_1}{\phi_1} = {F_f} + {F_r} - {F_Y}

In the formula, FY is the force of the driver's body on the electric vehicle Y in the direction. The torque balance equation about the Z axis of the electric vehicle is: $I1Zγ1=aFf−bFr+MZ$ {I_{1Z}}{\gamma_1} = a{F_f} - b{F_r} + {M_Z}

In the formula, MZ is the acting torque of the driver's body on the Z axis of the electric vehicle. The moment balance equation of the body around the X axis is: $I1xsϕ1−m1sh1u(β1+γ1)=m1sh1ϕ1−Kϕϕ1−Cpϕ1+MX$ {I_{1xs}}{\phi_1} - {m_{1s}}{h_1}u\left({{\beta_1} + {\gamma_1}} \right) = {m_{1s}}{h_1}{\phi_1} - {K_\phi}{\phi_1} - {C_p}{\phi_1} + {M_X}

In the formula, MX is the acting torque of the driver's body on the X axis of the electric vehicle. Since the wheel angle is small and the tire cornering characteristics are in the linear range, the tire cornering force is $Ff=kfαf=kf(β1+auγ1−Efϕ1−δ)Fr=krαr=kr(β1+buγ1−Erϕ1)$ \matrix{{{F_f} = {k_f}{\alpha_f} = {k_f}\left({{\beta_1} + {a \over u}{\gamma_1} - {E_f}{\phi_1} - \delta} \right)} \hfill \cr {{F_r} = {k_r}{\alpha_r} = {k_r}\left({{\beta_1} + {b \over u}{\gamma_1} - {E_r}{\phi_1}} \right)} \hfill \cr}

δ is the nominal front wheel angle. kf, kr are the cornering stiffness of the front and rear tires, respectively. αf, αr are the side slip angles of the front and rear tires, respectively. Ff, Fr are the roll steering coefficients of the front and rear wheels, respectively.

Differential equations of human motion for driving

We denote the reference base fixed to the body as B(x, y, z). Its origin B coincides with the origin of the reference base A and the orthogonal unit vectors of the two reference bases have the following relationship: $[XYZ]=[1000cos ϕ1−sin ϕ10sin ϕ1cos ϕ1][xyz]$ \left[{\matrix{X \cr Y \cr Z \cr}} \right] = \left[{\matrix{1 & 0 & 0 \cr 0 & {\cos \,{\phi_1}} & {- \sin \,{\phi_1}} \cr 0 & {\sin \,{\phi_1}} & {\cos \,{\phi_1}} \cr}} \right]\left[{\matrix{x \cr y \cr z \cr}} \right] . Then there is Y = cos ϕ1 y − sin ϕ1 z. We assume an inertial reference base G(g1, g2, g3). The driver's body has three degrees of freedom, lateral, yaw, and roll, and the general spatial motion in the inertial reference base G is shown in Figure 2.

The first derivative of a vector RBC concerning time in the inertial reference frame is $GdRBCdt=BdRBCdt+ΩGB×RBC$ {{^Gd{R_{BC}}} \over {dt}} = {{^Bd{R_{BC}}} \over {dt}} + {\Omega_{GB}} \times {R_{BC}} . The second derivative is $Gddt(GdRBCdt)=Gddt(BdRBCdt)+Gddt(ΩGB×RBC)$ {{^Gd} \over {dt}}\left({{{^Gd{R_{BC}}} \over {dt}}} \right) = {{^Gd} \over {dt}}\left({{{^Bd{R_{BC}}} \over {dt}}} \right) + {{^Gd} \over {dt}}\left({{\Omega_{GB}} \times {R_{BC}}} \right) . So the absolute $Gddt(GdRBCdt)=R°°BC+Ω˙GB×RBC+ΩGM×(ΩGB×RBC)+2ΩGB×R°BC$ {{^Gd} \over {dt}}\left({{{^Gd{R_{BC}}} \over {dt}}} \right) = {{\mathop R\limits^{\circ\circ}}_{BC}} + {\dot \Omega_{GB}} \times {R_{BC}} + {\Omega_{GM}} \times \left({{\Omega_{GB}} \times {R_{BC}}} \right) + 2{\Omega_{GB}} \times {{\mathop R\limits^{\circ}}_{BC}} acceleration of the driver's center of mass C is $RBC=R°°BC+Ω˙GB×RBC+ΩGM×(ΩGB×RBC)+R¨GB+2ΩGB×R°BC$ {R_{BC}} = {{\mathop R\limits^{\circ\circ }}_{BC}} + {\dot \Omega_{GB}} \times {R_{BC}} + {\Omega_{GM}} \times \left({{\Omega_{GB}} \times {R_{BC}}} \right) + {\ddot R_{GB}} + 2{\Omega_{GB}} \times {{\mathop R\limits^{\circ}}_{BC}} .

Among them: the coordinate of the driver's body in the base B is RBC = [lx Δty hz]. lx, hz is a constant [4]. The translational velocity is $R°BC=[0 Δ˙ty 0]$ {{\mathop R\limits^{\circ}}_{BC}} = \left[{0\,\dot \Delta {t_y}\,0} \right] . Translational acceleration $R°°BC=[0 Δ¨ty 0]$ {{\mathop R\limits^{\circ\circ}}_{BC}} = \left[{0\,\ddot \Delta {t_y}\,0} \right] . The angular velocity of base B relative to base G is $ΩGB=[ϕ˙10 γ1]$ {\Omega_{GB}} = \left[{{{\dot \phi}_1}0\,{\gamma_1}} \right] . So the acceleration of the driver's body along the y axis is $azy=u(β˙1+γ1)+lxγ˙1−hzϕ¨1−Δty(ϕ¨12+γ12)+Δ¨ty$ {a_{zy}} = u\left({{{\dot \beta}_1} + {\gamma_1}} \right) + {l_x}{\dot \gamma_1} - {h_z}{\ddot \phi_1} - \Delta {t_y}\left({\ddot \phi_1^2 + \gamma_1^2} \right) + \ddot \Delta {t_y} . According to Euler's theorem, we can get $M2=I2ω˙2+ω˜2I2ω2$ {M_2} = {I_2}{\dot \omega_2} + {\tilde \omega_2}{I_2}{\omega_2} . where M2 is the inertial moment of the driver's body. ω2 is the angular acceleration matrix. $ω˜2$ {\tilde \omega_2} is the angular velocity antisymmetric matrix. I2 is the inertia tensor in the non-inertial basis A. At this time $ω2=[ϕ˙2 0 γ2]$ {\omega_2} = \left[{{{\dot \phi}_2}\,0\,{\gamma_2}} \right] , $ω˜2=[0−γ20γ20−ϕ20ϕ20]$ {\tilde \omega_2} = \left[{\matrix{0 & {- {\gamma_2}} & 0 \cr {{\gamma_2}} & 0 & {- {\phi_2}} \cr 0 & {{\phi_2}} & 0 \cr}} \right] , I2 = diag[I2X I2Y I2Z].

Assuming that the electric vehicle roll angle is small, consider cos ϕ1 = 1. Then the balance equation of lateral force, roll, and yaw motion moment balance equation of the driver body is as follows: m2a2y = FY, I2Xϕ2 = m2gh2MX + m2a2yhz, I2Zγ2 = − MZ.

Elastic connection force at the human-vehicle interface

The relative displacement ΔSq1y, ΔSq2y, of the driver's body and the electric vehicle along the ΔSq1y, = Δtyhq1 (ϕ2ϕ1) + lq1 (ψ2ψ1); y axis at point q1, q2 is ΔSq2y, = Δtyhq2 (ϕ2ϕ1) + lq2 (ψ2ψ1), respectively, hq1, hq2 is the relative vertical position of the point q1, q2 and the roll center, and lq1, lq2 is the relative horizontal position. ψ1, ψ2 are the yaw angles of the electric vehicle and the human body, respectively [5]. The spring-damping forces along the y axis at the lower and upper body mass centers at the human-vehicle interface are Fq1y = kq1yΔSq1y + cq1yΔSq1y; Fq2y = kq2yΔSq2y + cq2yΔSq2y respectively.

kq1y, kq2y, are the translational stiffness coefficients along the y axis at point q1, q2 between the driver's body and the electric vehicle, respectively. cq1y, cq2y are the damping coefficients, respectively [6]. The lateral force of the driver's body on the electric vehicle along the Y axis can be expressed as FY = (Fq1y + Fq1y) cos ϕ1 = Fq1y + Fq1y.

The acting moment about the X axis is $MX=(kq1x+kq2x)(ϕ2−ϕ1)+(cq1z+cq2z) (ϕ˙2−ϕ˙1+)Fq1yhq1+Fq2yhq2$ {M_X} = \left({{k_{q1x}} + {k_{q2x}}} \right)\left({{\phi_2} - {\phi_1}} \right) + \left({{c_{q1z}} + {c_{q2z}}} \right)\,\left({{{\dot \phi}_2} - {{\dot \phi}_1} +} \right){F_{q1y}}{h_{q1}} + {F_{q2y}}{h_{q2}} . kq1x, kq2x are the torsional stiffness coefficients around the x axis, respectively. cq1x, cq2x are the torsional damping coefficients around the x axis, respectively.

The acting moment about the Z axis is $MZ=(kq1z+kq2z) (ψ2−ψ1)+(cq1z+zq2z) (ψ˙2−ψ˙1)−Fq2yhq2Fq1y, lq1−Fq2y,lq2$ \matrix{{{M_Z} = \left({{k_{q1z}} + {k_{q2z}}} \right)\,\left({{\psi_2} - {\psi_1}} \right) + \left({{c_{q1z}} + {z_{q2z}}} \right)\,\left({{{\dot \psi}_2} - {{\dot \psi}_1}} \right) - {F_{q2y}}{h_{q2}}} \hfill \cr {{F_{q1y}},\,{l_{q1}} - {F_{q2y}},{l_{q2}}} \hfill \cr} . kq1z, kq2z is the torsional stiffness Fq1y, lq1Fq2y, lq2 coefficient around the axis, respectively. cq1z, cq2z is the torsional damping coefficient around the z axis, respectively.

Vibration test and parameter identification

Parameters such as inertia and structural dimensions of electric vehicles and drivers are easy to measure. The stiffness damping element connected between people and vehicles needs to combine vibration test and modeling for parameter identification. The interaction between the human body and the car is complex [7]. The detection method is mainly realized through the human body and the seat system. We analyzed the stiffness damping coefficient between the human body and the seat and built a vibration test bench. We performed low-frequency multidirectional vibration experiments as well as data acquisition and processing.

Vibration test of the person-chair system

The test bench consists of a servo motor, a vibration exciter, a seat, a base plate, and sensors. Each excitation can realize a single direction of motion of the sitting human body. However, the test bench can be disassembled and reassembled to realize the translational vibration test in the y-direction and the rotational vibration test around the x and z axes, respectively.

The inertial sensor we use in the test is a miniature attitude reference system. The system can measure the angular velocity and acceleration signals in the local coordinate system of the sensor [8]. According to the output data, we obtain the direction cosine matrix describing its attitude transformation relative to the inertial coordinate system. Before the test, according to the national standard GB/T17245-2004, we calculated the position of the center of mass of the lower body and the upper body, respectively. Experimentally measure the initial displacements L1 L2 of the lower body mass center to sensor II and the upper body mass center to sensor III. In the test, I collect the acceleration time-domain signal along the y-axis on the seat pan. Sensors II and III measure the y-axis acceleration time-domain signal in their local coordinate system [9]. Based on the response signal, we obtain the time domain signal of acceleration along the y-axis at the position of the lower and upper body centroid. Then we perform the following coordinate transformation: $a1′=(a1A1+d2(L1A1)dt2)A0−1$ a^{'}_1 = \left({{a_1}{A_1} + {{{d^2}\left({{L_1}{A_1}} \right)} \over {d{t^2}}}} \right)A_0^{- 1} ; $a2′=(a2A2+d2(L2A2)dt2)A0−1$ a^{'}_2 = \left({{a_2}{A_2} + {{{d^2}\left({{L_2}{A_2}} \right)} \over {d{t^2}}}} \right)A_0^{- 1} .

Among them: A0, A1, A2 is the direction cosine matrix describing the posture transformation of seat, lower body and upper body, respectively. a1, a2 are the original measured response accelerations of the lower and upper body, respectively. a1, a2, are the accelerations in the local coordinate system of the sensor I, respectively.

In the rotational vibration test around the x, z axis, the input and output signals are the rotational angular velocity. There is an only coordinate transformation in data processing and no vector translation transformation. The transformation process is $ω1′=ω1A1A0−1$ \omega^{'}_1 = {\omega_1}{A_1}A_0^{- 1} ; $ω2′=ω2A2A0−1$ \omega^{'}_2 = {\omega_2}{A_2}A_0^{- 1} .

where ω1, ω2 is the original measured response angular velocity of the lower body and the upper body, respectively. ω1, ω2 are the angular velocity in the local coordinate system of the sensor I, respectively.

Person-chair interface parameter identification

The human body is composed of elastic tissues such as tendons, muscles, ligaments, and bones. The seat surface is also a layer of viscoelastic foam. Therefore the person-chair system has translational motion in the y-axis and rotation around the x- and z-axes. We can use a lumped parameter model including stiffness and damping elements to describe it (Figure 3). The point q1 is the centroid of the lower body [10]. The point q2 is the centroid of the upper body. Figure 3(a) describes the y-axis translation of the sitting human body, and Figure 3(b) can be used to describe the rotation of the sitting human body around the x-axis or the z-axis. The upper and lower bodies are elastically connected at point p.

The differential equations of these three motions have the following unified form: $M1x1+Cq1(x1−x0)+Kq1(x1−x0)=Cp(x2−x1)+Kp(x2−x1)M2x2+Cq2(x2−x0)+Kq2(x2−x0)=Cp(x1−x2)+Kp(x1−x2)$ \matrix{{{M_1}{x_1} + {C_{q1}}\left({{x_1} - {x_0}} \right) + {K_{q1}}\left({{x_1} - {x_0}} \right) = {C_p}\left({{x_2} - {x_1}} \right) + {K_p}\left({{x_2} - {x_1}} \right)} \hfill \cr {{M_2}{x_2} + {C_{q2}}\left({{x_2} - {x_0}} \right) + {K_{q2}}\left({{x_2} - {x_0}} \right) = {C_p}\left({{x_1} - {x_2}} \right) + {K_p}\left({{x_1} - {x_2}} \right)} \hfill \cr}

Among them: M1, M2 is the mass (moment of inertia) of the lower and upper body. x1, x2 is displacement (angular displacement). Kq1, Kq2 and Cq1, Cq2 are stiffness and damping, respectively [11]. The expression in the frequency domain after Fourier transform is performed on both sides of the equation at the same time is: $(−ω2M1+jωCq1+Kq1+jωCp+Kp) X1−(jωCp+Kp)X2−(jωCq1+Kq1) X0=0(−ω2M2+jωCq2+Kq2+jωCp+Kp) X2−(jωCp+Kp)X1−(jωCq2+Kq2) X0=0$ \matrix{{\left({- {\omega^2}{M_1} + j\omega {C_{q1}} + {K_{q1}} + j\omega {C_p} + {K_p}} \right)\,{X_1} - \left({j\omega {C_p} + {K_p}} \right){X_2} - \left({j\omega {C_{q1}} + {K_{q1}}} \right)\,{X_0} = 0} \hfill \cr {\left({- {\omega^2}{M_2} + j\omega {C_{q2}} + {K_{q2}} + j\omega {C_p} + {K_p}} \right)\,{X_2} - \left({j\omega {C_p} + {K_p}} \right){X_1} - \left({j\omega {C_{q2}} + {K_{q2}}} \right)\,{X_0} = 0} \hfill \cr}

We fit the above mathematical model to the experimental data. In this way, the stiffness and damping parameters of the person-chair interface can be identified. The objective function used for parameter identification is as follows: $e=∑f=0.58[H1e(f)−H1m(f)]2/[H1e(f)2+∑f=0.58[H2e(f)−H2m(f)]2/[H2e(f)2$ e = \sum\limits_{f = 0.5}^8 {{{\left[{{H_{1e}}\left(f \right) - {H_{1m}}\left(f \right)} \right]}^2}/\left[{{H_{1e}}{{\left(f \right)}^2} + \sum\limits_{f = 0.5}^8 {{{\left[{{H_{2e}}\left(f \right) - {H_{2m}}\left(f \right)} \right]}^2}/\left[{{H_{2e}}{{\left(f \right)}^2}} \right.}} \right.}

H1e, H2e is the amplitude-frequency response function of the lower and upper body measured experimentally. H1m, X1 / X0, H2m, X2 / X0. We first established a 3D model of a sitting human body in SolidWorks and imported it into ADAMS. A three-dimensional human model enables the calculation of the lower and upper body centers of mass and the moment of inertia relative to their respective centers of mass [12]. The results are shown in Figure 4 and Table 1.

Person-chair model identification parameters

Parameter Mean Standard deviation
kq1ty/kN·m−1) 14.1 1.1
kq2ty/kN·m−1) 12.5 1
cq1ty/(N·s·m−1) 198.1 32.6
cq2ty/(N·s·m−1) 423.3 181.1
Calculation and analysis of electric vehicle handling stability

In equations (1)(3), we let FY = 0, FZ = FX = 0. Therefore, we can obtain the electric vehicle handling stability model without considering the dynamic coupling effect of humans and vehicles. Its differential equation degenerates into the traditional three-degree-of-freedom handling stability equation [13]. We consider the driver's body and sprung mass as a whole. At this time, the changes in the total mass and mass distribution of electric vehicles caused by the model can be approximated as: $IiZ′=I1Z+I2Z, I1X′=I1X+I2X, m1′+m2ms′=ms+m2, h=(msh1+m2h2)/ms′$ \matrix{{I^{'}_{iZ} = {I_{1Z}} + {I_{2Z}},\,I^{'}_{1X} = {I_{1X}} + {I_{2X}},\,m^{'}_1 + {m_2}} \hfill \cr {m^{'}_s = {m_s} + {m_2},\,h = \left({{m_s}{h_1} + {m_2}{h_2}} \right)/m^{'}_s} \hfill \cr}

We take the model car's roll angle of 2.5° under 0.4g lateral acceleration as the standard. We choose the relative damping coefficient of the suspension to be 0.3. The moment of inertia is estimated by the ADAMS virtual prototype model and empirical formula [14]. The parameters required for the calculation are shown in Table 2. We combined the mathematical model and Matlab/Simulink to calculate and analyze the dynamic coupling between the driver's human body and the electric vehicle when the front wheel rotation angle is 5°. We analyze the effect of this model on the handling stability of electric vehicles with different curb weights. The calculation results are shown in Figure 5 and Table 3.

Calculated main parameters

Model A B C Model A B C
m1/kg 450 200 100 m1/kg 270 100 50
a/m 0.8 0.8 0.6 b/m 0.8 0.8 0.6
h1/m 0.35 0.35 0.35 h2/m 0.55 0.55 0.55
lx/m 0.05 0.05 0.05 u/(m·s−1) 10 10 6
Iz1/(kg·m2) 380 174 48 Iz2/(kg·m2) 5.4 5.4 5.4
Ix1/(kg·m2) 120 52 18 Ix2/(kg·m2) 26 26 26
la/m 0.152 0.152 0.152 lb/m −0.153 −0.153 −0.153
ha/m 0.577 0.577 0.577 hb/m 0.767 0.767 0.767

Roll Response Steady-State Values and Overshoot

no coupling Consider coupling no coupling Consider coupling
A 3.58 3.81 11.5 14
B 3.1 3.37 14.2 17.8
C 11.2 1.26 23.1 34.8

The human-vehicle dynamic coupling effect prolongs the roll response time and increases the roll angle's steady-state value and transient peak value. The larger the overshoot is, the more unstable the electric vehicle is. The differences between the steady-state values of roll angles of models A, B, and C are 0.23°, 0.27°, and 0.14°, respectively [15]. The corresponding relative increments were 6.4%, 8.7%, and 13.3%, respectively. The difference of overshoot is 2.5%, 3.6%, and 11.7%, respectively. The structural scale and quality of Model A are relatively close to Model B and Model C of the human body. Human-vehicle dynamic coupling reduces the steady-state value of its response. Under the action, the curve fluctuation becomes larger, and the peak response time is prolonged.

Conclusion

1) The human-vehicle dynamic coupling significantly impacts the roll angle response of electric vehicles. 2) Human-vehicle dynamics coupling increases the tendency of electric vehicles to understeer. Under the influence of this effect, the system response time is prolonged, and the steady-state value and transient peak value of the roll angle are increased. At this time, the steady-state value of the yaw angular velocity is reduced, and the curve fluctuation becomes larger.

#### Calculated main parameters

Model A B C Model A B C
m1/kg 450 200 100 m1/kg 270 100 50
a/m 0.8 0.8 0.6 b/m 0.8 0.8 0.6
h1/m 0.35 0.35 0.35 h2/m 0.55 0.55 0.55
lx/m 0.05 0.05 0.05 u/(m·s−1) 10 10 6
Iz1/(kg·m2) 380 174 48 Iz2/(kg·m2) 5.4 5.4 5.4
Ix1/(kg·m2) 120 52 18 Ix2/(kg·m2) 26 26 26
la/m 0.152 0.152 0.152 lb/m −0.153 −0.153 −0.153
ha/m 0.577 0.577 0.577 hb/m 0.767 0.767 0.767

#### Roll Response Steady-State Values and Overshoot

no coupling Consider coupling no coupling Consider coupling
A 3.58 3.81 11.5 14
B 3.1 3.37 14.2 17.8
C 11.2 1.26 23.1 34.8

#### Person-chair model identification parameters

Parameter Mean Standard deviation
kq1ty/kN·m−1) 14.1 1.1
kq2ty/kN·m−1) 12.5 1
cq1ty/(N·s·m−1) 198.1 32.6
cq2ty/(N·s·m−1) 423.3 181.1

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