*3.2. Model of Road Roughness*

The methods of road unevenness can be divided into static cross-section measurement, dynamic cross-section measurement, reaction level measurement system, and subjective evaluation method. The road smoothness is expressed as the superposition of some sine or cosine waves with random phases. For the superposition of N similar sine waves, the smoothness of a random road surface can be expressed as [24]:

$$Z(\mathbf{x}) = \sum\_{i=1}^{n} \sqrt{2} A\_i \cdot \sin(2\pi \cdot \mathbf{x} \cdot n\_{mid\\_j} + \theta\_i) \tag{15}$$

where, *<sup>Z</sup>*(*x*) represents the unevenness value of the random pavement, *x* represents the displacement of the pavement along the horizontal direction, *θ* represents the random number of [0–2*π*], and *nmid* \_ *i* represents the intermediate value of the spatial frequency of the pavement flatness between each cell.

Assuming the vehicle speed is *v*, let *x* = *vt*, and the first-order and second-order derivatives of the above formula can obtain the speed and acceleration of a random road, respectively.

$$\dot{Z}(\mathbf{x}) = \sum\_{i=1}^{n} 2\sqrt{2}\pi vn\_{\rm mid\\_j} A\_i \cdot \cos(2\pi \cdot \mathbf{x} \cdot n\_{\rm mid\\_j} + \theta\_i) \tag{16}$$

$$\ddot{Z}(\mathbf{x}) = \sum\_{i=1}^{n} -4\sqrt{2}\pi^2 v^2 n\_{\text{mid\\_i}}^2 A\_i \cdot \sin(2\pi \cdot \mathbf{x} \cdot \mathbf{n}\_{\text{mid\\_i}} + \theta\_i) \tag{17}$$
