*3.1. Pavement Vibration Model*

The roughness of the road surface is called unevenness in highway engineering. It originates from the accidental factors that cannot be controlled artificially in the process of construction and the irregular vertical undulation of the ground caused by repeated loads caused by vehicles. The unevenness of the road will cause the vibration of the vehicle. Based on the establishment of the road finite element model in the previous section, according to the forced vibration equation of the damped system with multiple degrees of freedom, the vibration balance equation of the road structure model can be obtained as follows:

.

$$[m]\{\delta\} + [c]\{\delta\} + [k]\{\delta\} = \{f\} \tag{13}$$

where [*m*] represents the overall mass matrix of the road finite element model, [*c*] represents the overall damping matrix, [*k*] represents the overall stiffness matrix, {*δ*} represents the displacement vector of the road finite element model, { . *δ*} represents the velocity vector, { .. *δ*} represents the acceleration vector, and { *f* } represents the vertical load vector, including the dynamic load and static load.

To solve the common vibration problem caused by vehicles acting on the road, the vehicle vibration balance equation and road vibration balance equation in Equations (12) and (13) are needed to calculate the dynamic response of the whole system. To simplify the calculation without losing accuracy and reliability, the high-order term of vibration mode superposition is discarded, and only the contact between the vehicle and the road is considered; that is, the node between the wheel and the road. The vertical vibration displacement at any cross-section *x* position of the road pavement is obtained by superposition of the vibration mode functions of each order of the road:

$$z\_{\mathcal{I}}(\mathbf{x}) = \sum\_{n=1}^{N} A\_{\mathcal{I}} \phi\_n(\mathbf{x}\_{\bar{i}}) \tag{14}$$

where *An* represents the corresponding generalized coordinates and *φn*(*xi*) is the vibration mode function at the corresponding road horizontal position *x* and the *n*th order.
