*2.2. Dynamic Balance of Vehicle–Road System*

Based on the above description, combined with the vehicle vibration balance equation and the road vibration balance equation, the dynamic balance equation of the vehicle–road system is established. The stress analysis of the vehicle model and the body are shown in Figures 2 and 3, respectively.

**Figure 2.** Stress analysis of vehicle model.

**Figure 3.** Stress analysis of body.

where

$$\begin{aligned} F\_{\rm fc1} &= \mathcal{C}\_{\rm s1} (Z\_s - \lambda\_1 \ddot{\boldsymbol{\varphi}} - \lambda\_3 \ddot{\boldsymbol{\theta}} - Z\_{\rm f1}), \\ F\_{\rm ts1} &= K\_{\rm s1} (Z\_s - \lambda\_1 \ddot{\boldsymbol{\varphi}} - \lambda\_3 \ddot{\boldsymbol{\theta}} - Z\_{\rm f1}), \\ F\_{\rm t2} &= \mathcal{C}\_{\rm s2} (\bar{Z}\_s + \lambda\_2 \ddot{\boldsymbol{\varphi}} - \lambda\_3 \ddot{\boldsymbol{\theta}} - \bar{Z}\_{\rm f2}), \\ F\_{\rm t3} &= K\_{\rm s2} (Z\_s + \lambda\_2 \ddot{\boldsymbol{\varphi}} - \lambda\_3 \ddot{\boldsymbol{\theta}} - Z\_{\rm f2}), \\ F\_{\rm t3} &= C\_{\rm s3} (\bar{Z}\_s - \lambda\_1 \ddot{\boldsymbol{\varphi}} + \lambda\_4 \ddot{\boldsymbol{\theta}} - \bar{Z}\_{\rm f2}), \\ F\_{\rm t5} &= K\_{\rm s3} (Z\_s - \lambda\_1 \ddot{\boldsymbol{\varphi}} + \lambda\_4 \ddot{\boldsymbol{\theta}} - Z\_{\rm f3}), \\ F\_{\rm t4} &= C\_{\rm s4} (\bar{Z}\_s + \lambda\_2 \ddot{\boldsymbol{\varphi}} + \lambda\_4 \ddot{\boldsymbol{\theta}} - \bar{Z}\_{\rm f4}), \\ F\_{\rm t5} &= K\_{\rm s4} (Z\_s + \lambda\_2 \ddot{\boldsymbol{\varphi}} + \lambda\_4 \ddot{\boldsymbol{\theta}} - Z\_{\rm f4}), \end{aligned}$$

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and *Ftc*1, *Fts*1, *Ftc*2, *Fts*2, *Ftc*3, *Fts*3, *Ftc*4, and *Fts*4 represent the interaction force between the unsprung mass part of the vehicle and the body.

According to the force balance conditions in Figures 2 and 3, the balance equations of vertical vibration, pitch vibration, and roll vibration of the body are as follows [21–23]:

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$$\begin{aligned} m\_{\rm s} \ddot{Z}\_{\rm s} &+ \mathbb{C}\_{\rm s1} (\ddot{Z}\_{\rm s} - \lambda\_1 \ddot{\boldsymbol{\upmu}} - \lambda\_3 \ddot{\boldsymbol{\upmu}} - \ddot{Z}\_{\rm t1}) + K\_{\rm s1} (Z\_{\rm s} - \lambda\_1 \ddot{\boldsymbol{\upmu}} - \ddot{Z}\_{\rm t1}) \\\ &\lambda\_1 \ddot{\boldsymbol{\upmu}} - \lambda\_3 \ddot{\boldsymbol{\upmu}} - Z\_{\rm t1}) + \mathbb{C}\_{\rm s2} (\ddot{Z}\_{\rm s} + \lambda\_2 \ddot{\boldsymbol{\upmu}} - \lambda\_3 \ddot{\boldsymbol{\upmu}} - \ddot{Z}\_{\rm t2}) \\\ &+ K\_{\rm s2} (Z\_{\rm s} + \lambda\_2 \ddot{\boldsymbol{\upmu}} - \lambda\_3 \ddot{\boldsymbol{\upmu}} - Z\_{\rm t2}) \mathbb{C}\_{\rm s3} (\ddot{Z}\_{\rm s} - \lambda\_1 \dot{\boldsymbol{\upmu}} + \\\ &\lambda\_4 \ddot{\boldsymbol{\upmu}} - \ddot{Z}\_{\rm t2}) + K\_{\rm s3} (Z\_{\rm s} - \lambda\_1 \ddot{\boldsymbol{\upmu}} + \lambda\_4 \ddot{\boldsymbol{\upmu}} - Z\_{\rm t3}) + \mathbb{C}\_{\rm s4} \\\ &(\ddot{Z}\_{\rm s} + \lambda\_2 \ddot{\boldsymbol{\upmu}} + \lambda\_4 \ddot{\boldsymbol{\upmu}} - \dot{Z}\_{\rm t4}) + K\_{\rm s4} (Z\_{\rm s} + \lambda\_2 \ddot{\boldsymbol{\upmu}} + \lambda\_4 \ddot{\boldsymbol{\upmu}} \\\ &- Z\_{\rm t4}) = 0 \end{aligned} \tag{1}$$

*J* .. *θ* + *λ*1 ⎧⎪⎪⎪⎨⎪⎪⎪⎩ *Cs*1( .*Zs* − *λ*1 .*ψ* − *λ*3 .*θ* − .*Zt*1)+ *Ks*1(*Zs* − *λ*1*ψ* − *λ*3*θ* − *Zt*1)+ *Cs*3( .*Zs* − *λ*1 .*ψ* + *λ*4 .*θ* − .*Zt*2)+ *Ks*3(*Zs* − *λ*1*ψ* + *λ*4*θ* − *Zt*2) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ −*λ*<sup>2</sup> ⎧⎪⎪⎪⎨⎪⎪⎪⎩ *Cs*2( .*Zs* + *λ*2 .*ψ* − *λ*3 .*θ* − .*Zt*2)+ *Ks*2(*Zs* + *λ*2*ψ* − *λ*3*θ* − *Zt*2)+ *Cs*4( .*Zs* + *λ*2 .*ψ* + *λ*4 .*θ* − .*Zt*4)+ *Ks*4(*Zs* + *λ*2*ψ* + *λ*4*θ* − *Zt*4) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ = 0 (2) *J* .. *ψ* + *λ*3 ⎧⎪⎪⎪⎨⎪⎪⎪⎩ *Cs*1( .*Zs* − *λ*1 .*ψ* − *λ*3 .*θ* − .*Zt*1)+ *Ks*1(*Zs* − *λ*1*ψ* − *λ*3*θ* − *Zt*1)+ *Cs*2( .*Zs* + *λ*2 .*ψ* − *λ*3 .*θ* − .*Zt*2)+ *Ks*2(*Zs* + *λ*2*ψ* − *λ*3*θ* − *Zt*2) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ −*λ*<sup>4</sup> ⎧⎪⎪⎪⎨⎪⎪⎪⎩ *Cs*3( .*Zs* − *λ*1 .*ψ* + *λ*4 .*θ* − .*Zt*2)+ *Ks*3(*Zs* − *λ*1*ψ* + *λ*4*θ* − *Zt*3)+ *Cs*4( .*Zs* + *λ*2 .*ψ* + *λ*4 .*θ* − .*Zt*4)+ *Ks*4(*Zs* + *λ*2*ψ* + *λ*4*θ* − *Zt*4) ⎫⎪⎪⎪⎬⎪⎪⎪⎭ = 0 (3)

The force analysis of the vehicle's unsprung mass model is shown in Figure 4.

**Figure 4.** Force analysis of vehicle unsprung mass.

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Where i = 1, 2, 3, and 4 represent the right front unsprung mass part, the right rear unsprung mass part, the left front unsprung mass part, and the left rear unsprung mass part, respectively. The force balance equations of the four unsprung mass parts are as follows:

$$\begin{cases} m\_{t1}Z\_{t1} + \mathbb{C}\_{t1}(Z\_{t1} - Z\_{t1}) + \mathbb{K}\_{t1}(Z\_{t1} - Z\_{t1}) - \mathbb{C}\_{s1}(Z\_{s} \\ -\lambda\_{1}\dot{\psi} - \lambda\_{3}\dot{\theta} - \dot{Z}\_{t1}) - \mathbb{K}\_{s1}(Z\_{s} - \lambda\_{1}\psi - \lambda\_{3}\theta - Z\_{t1}) = 0 \end{cases} \tag{4}$$

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$$\begin{aligned} m\ddot{\chi}\ddot{Z}\_{l3} + \mathbb{C}\_{l3}(\dot{Z}\_{l3} - \dot{Z}\_{l3}) + \mathbb{K}\_{l3}(Z\_{l3} - Z\_{l3}) - \mathbb{C}\_{l3}(\dot{Z}\_{l} \\ -\lambda\_1 \dot{\psi} + \lambda\_4 \dot{\theta} - \dot{Z}\_{l2}) - \mathbb{K}\_{l3}(Z\_5 - \lambda\_1 \psi + \lambda\_4 \theta - Z\_{l3}) &= 0 \end{aligned} \tag{5}$$

$$\begin{aligned} m\_{t2}\ddot{Z}\_{t2} + \mathbb{C}\_{t2}(\dot{Z}\_{t2} - \dot{Z}\_{t2}) + \mathbb{K}\_{t2}(Z\_{t2} - Z\_{t2}) - \mathbb{C}\_{s2}(\dot{Z}\_{s} \\ + \lambda\_{2}\dot{\psi} - \lambda\_{3}\dot{\theta} - \dot{Z}\_{t2}) - \mathbb{K}\_{s2}(Z\_{s} + \lambda\_{2}\psi - \lambda\_{3}\theta - Z\_{t2}) &= 0 \end{aligned} \tag{6}$$

$$\begin{aligned} m\_{t4}\ddot{Z}\_{t4} + \mathbb{C}\_{t4}(\dot{Z}\_{t4} - \dot{Z}\_{l4}) + \mathbb{K}\_{t4}(Z\_{t4} - Z\_{l4}) - \mathbb{C}\_{t4}(\dot{Z}\_{s} \\ + \lambda\_{2}\dot{\psi} + \lambda\_{4}\dot{\theta} - \dot{Z}\_{t4}) - \mathbb{K}\_{t4}(Z\_{s} + \lambda\_{2}\psi + \lambda\_{4}\theta - Z\_{t4}) &= 0 \end{aligned} \tag{7}$$

The stress analysis of the wheel is drawn according to the stress of the vehicle wheel, as shown in Figure 5.

**Figure 5.** Stress analysis of wheel.

Where, i = 1, 2, 3, and 4 represent the four wheels, respectively. According to the force balance, the balance equations are obtained:

$$f\_{t1} = \mathcal{C}\_{t1}(\dot{Z}\_{t1} - \dot{Z}\_{l1}) + \mathcal{K}\_{l1}(Z\_{t1} - Z\_{l1}) - m\_{l1}\ddot{Z}\_{l1} \tag{8}$$

$$f\_{l2} = \mathcal{C}\_{l2}(Z\_{l2} - Z\_{l2}) + \mathcal{K}\_{l2}(Z\_{l2} - Z\_{l2}) - m\_{l2}Z\_{l2} \tag{9}$$

$$f\_{l\Im} = C\_{l\Im}(\dot{Z}\_{l\Im} - \dot{Z}\_{l\Im}) + K\_{l\Im}(Z\_{l\Im} - Z\_{l\Im}) - m\_{l\Im}\ddot{Z}\_{l\Im} \tag{10}$$

$$f\_{t4} = \mathbb{C}\_{t4}(\dot{Z}\_{t4} - \dot{Z}\_{l4}) + \mathbb{K}\_{t4}(Z\_{t4} - Z\_{l4}) - m\_{l4}\ddot{Z}\_{l4} \tag{11}$$

In the above equations and pictures (Equations (1)–(11) and Figures 1–5), *ms* represents the body mass; *θ* is the body nod displacement angle, *ψ* represents the body turnover displacement angle; *mt*i represents the unsprung mass of vehicle front and rear; *ml*i represents wheel mass; *Ct*i represents the damping coefficient of the corresponding wheel before and after the vehicle; *Cs*i represents the damping coefficient of the corresponding suspension system before and after the vehicle; *Kt*i represents the stiffness coefficients of the corresponding wheels before and after the vehicle; *Ks*i represent the stiffness coefficients of the front and rear suspension of the vehicle; *Zs* represents the vertical vibration displacement of the body; *Zt*i represent the vertical vibration displacements of the four unsprung mass parts corresponding to the front and rear of the vehicle; *Zl*i represents the vertical vibration displacement of the front and rear wheels of the vehicle; *J*X represents the rotational inertia of the body around the *x*-axis; *J*Y represents the rotational inertia of the body around the *y*-axis; and *ft*i represents the dynamic loads of the four wheels.

Based on the balance equation of each part of the vehicle system mentioned above, the vector expression can be obtained as:

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$$\text{MZ} + \text{CZ} + \text{KZ} = \text{F}\_l \tag{12}$$

where *M* represents the mass matrix of the vehicle; *C* represents the damping matrix of the vehicle; *K* represents the stiffness matrix of the vehicle; *Z* represents the displacement matrix of the vehicle; *Ft* represents the dynamic load vector acted on the road surface by the vehicle system. The above matrix expressions are as follows:


*C* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *c*11 *c*12 *c*13 −*cs*1 −*cs*2 −*cs*3 −*cs*4 0000 *c*21 *c*22 *c*23 *λ*1*cs*1 −*λ*2*cs*<sup>2</sup> *λ*1*cs*3 −*λ*2*c*<sup>4</sup> 0000 *c*31 *c*32 *c*33 *λ*3*cs*1 *λ*3*cs*2 −*λ*4*cs*<sup>3</sup> −*λ*4*c*<sup>4</sup> 0000 −*cs*1 *λ*1*cs*1 *λ*3*cs*1 *cs*1 + *ct*1 000 −*ct*1 000 −*cs*2 −*λ*2*cs*<sup>2</sup> *λ*3*cs*2 0 *c*2 + *ct*2 0 00 −*ct*2 0 0 −*cs*3 *λ*1*cs*3 −*λ*4*cs*<sup>3</sup> 0 0 *c*3 + *ct*3 0 00 −*ct*3 0 −*cs*4 −*λ*2*cs*<sup>4</sup> −*λ*4*cs*<sup>4</sup> 0 00 *c*4 + *ct*4 000 −*ct*4 00 0 −*ct*1 000 *ct*1 000 00 0 0 −*ct*2 0 00 *ct*2 0 0 00 0 0 0 −*ct*3 0 00 *ct*3 0 00 0 0 0 0 −*ct*4 000 *ct*4 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, *K* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *k*11 *k*12 *k*13 −*ks*<sup>1</sup> −*ks*<sup>2</sup> −*ks*<sup>3</sup> −*ks*<sup>4</sup> 0000 *k*21 *k*22 *k*23 *λ*1*ks*1 −*λ*2*ks*<sup>2</sup> *λ*1*ks*3 −*λ*2*ks*<sup>4</sup> 0000 *k*31 *k*32 *k*33 *λ*3*ks*1 *λ*3*ks*2 −*λ*4*ks*<sup>2</sup> −*λ*4*ks*<sup>4</sup> 0000 −*ks*<sup>1</sup> *λ*1*ks*1 *λ*3*ks*1 *ks*1 + *kt*1 000 −*kt*<sup>1</sup> 000 −*ks*<sup>2</sup> −*λ*2*ks*<sup>2</sup> *λ*3*ks*2 0 *ks*2 + *kt*2 0 00 −*kt*<sup>2</sup> 0 0 −*ks*<sup>3</sup> *λ*1*ks*3 −*λ*4*ks*<sup>3</sup> 0 0 *ks*3 + *kt*3 0 00 −*kt*<sup>3</sup> 0 −*ks*<sup>4</sup> −*λ*2*ks*<sup>4</sup> −*λ*4*ks*<sup>4</sup> 000 *ks*4 + *kt*4 000 −*kt*<sup>4</sup> 00 0 −*kt*<sup>1</sup> 000 *kt*1 000 00 0 0 −*kt*<sup>2</sup> 0 00 *kt*2 0 0 00 0 0 0 −*kt*<sup>3</sup> 0 00 *kt*3 0 00 0 0 0 0 −*kt*<sup>4</sup> 000 *kt*4 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, *Z* = *Zs θ Zt*1 *Zt*2 *Zt*3 *Zt*4 *Zl*1*Zl*2*Zl*3*Zl*4*T*,

*ψ* 

*C*11 = *cs*1 + *cs*2 + *cs*3 + *cs*4, *C*12 = *C*21 = −*λ*1*cs*<sup>1</sup> + *λ*2*cs*2 − *λ*1*cs*3 + *λ*2*cs*4, *C*13 = *C*31 = −*λ*3*cs*<sup>1</sup> − *λ*3*cs*2 + *λ*4*cs*3 + *λ*4*cs*4, *C*22 = *<sup>λ</sup>*12*cs*1 + *<sup>λ</sup>*22*cs*2 − *<sup>λ</sup>*12*cs*3 + *<sup>λ</sup>*22*cs*4, *C*23 = *C*32 = *λ*1*λ*3*cs*1 − *λ*2*λ*3*cs*2 − *λ*1*λ*4*cs*3 + *λ*2*λ*4*cs*2, *C*33 = *<sup>λ</sup>*32*cs*1 + *<sup>λ</sup>*32*cs*2 + *<sup>λ</sup>*42*cs*3 + *<sup>λ</sup>*42*cs*4, *K*11 = *Ks*1 + *Ks*2 + *Ks*3 + *Ks*4, *K*12 = *K*21 = −*λ*1*Ks*<sup>1</sup> + *λ*2*Ks*2 − *λ*1*Ks*3 + *λ*2*Ks*4, *K*13 = *K*31 = −*λ*3*Ks*<sup>1</sup> − *λ*3*Ks*2 + *λ*4*Ks*3 + *λ*4*Ks*4, *K*22 = *<sup>λ</sup>*1<sup>2</sup>*Ks*1 + *<sup>λ</sup>*2<sup>2</sup>*Ks*2 − *<sup>λ</sup>*1<sup>2</sup>*Ks*3 + *<sup>λ</sup>*2<sup>2</sup>*Ks*4, *K*23 = *K*32 = *λ*1*λ*3*Ks*1 − *λ*2*λ*3*Ks*2 − *λ*1*λ*4*Ks*3 + *λ*2*λ*4*Ks*2, *K*33 = *<sup>λ</sup>*3<sup>2</sup>*Ks*1 + *<sup>λ</sup>*3<sup>2</sup>*Ks*2 + *<sup>λ</sup>*4<sup>2</sup>*Ks*3 + *<sup>λ</sup>*4<sup>2</sup>*Ks*4, *Ft* = 0000000 *ft*1 *ft*2 *ft*3 *ft*4 *T*

The boundary constraint conditions of the finite element model of the road multi-layer system need to be simplified according to the force and loading conditions of the road under the actual vehicle traveling conditions. When the vehicle is driving in the center of the road, the response at the far side of the road is negligible. Because of the thick soil foundation, the response at the bottom is too small and negligible. Therefore, it can be known from Saint-Venant's principle that longitudinal constraints are imposed on both sides of the principle and fixed displacement are imposed on the bottom to obtain the finite element model of the road [24–26], as shown in Figure 6.

**Figure 6.** Finite element model of road structure layer.
