## 3 Vibration Induced by Road Roughness  
When vehicles traverse a bridge, the weight, speed, and dynamic characteristics of the vehicles, as well as the road roughness, collectively influence bridge vibrations and ride comfort. The vibrations induced by road roughness can have a detrimental effect on both the bridge and the vehicles. In general, when a vehicle moves at a constant speed, the road roughness can be modeled as a stationary random process in the spatial domain. For bridges with uneven surfaces, as shown in Figure 1, the road roughness curve is represented as \(r(x)\). Additional displacements need to be incorporated into the original dynamic system, resulting in the updated bridge dynamic deflection as follows:  
\[y_{r}(x,t)=y(x,t)+r(x) \tag{16}\]  
Equation (16) is substituted into Equations (1) and (2) while ignoring the derivative of the road roughness with respect to time, since the additional displacement \(r(x)\) is time-independent until the vehicle group reaches a specific position. Consequently, the resulting equations are established as the coupling equations described above. After integration, the form of the simultaneous equations for the coupled system considering road roughness takes the same form as Equation (11), and the supplementary equations include Equations (12)-(14). However, the expression of Equation (15) is modified as follows:  
\[F(t)=[\rho_{Fr}\phi_{1},\,\rho_{Fr}\phi_{2},\,...\,,\,\rho_{Fr}\phi_{n},\,\rho_{ r}]^{T} \tag{17}\]  
where \(\rho_{Fr}\) equals \(\frac{4(\overline{\rho}_{0}+\overline{\eta}_{1})g}{n\overline{H}\pi}\sin\frac {n\pi l_{p}}{2\overline{L}}-\rho_{k}r(x)\), and \(\rho_{r}\) equals \(k_{1}r(x)\). If a vehicle travels at a constant speed, road roughness becomes a stationary random process in the time domain due to the relationship between the distance traveled and the time. For a bridge experiencing vibrations from vehicles moving at a constant speed, this stationary displacement can be utilized as an input. Under normal conditions, the surface roughness of the bridge is considered a Gaussian random process with zero mean and ergodic states. The power spectral density function of a given road roughness is used to obtain the bridge deck roughness sequence. There are various methods available to obtain the roughness of the bridge deck according to the given power spectral density. The triangle series superposition method has been widely used due to its simplicity and high simulation accuracy. This method involves representing a stochastic process as the sum of several cosine functions. According to the calculation principle of the triangle series superposition method, the road roughness can be obtained as follows:  
\[r(x)=\sqrt{2}\sum_{k=1}^{N}\sqrt{G_{d}(n_{k})\Delta n}\cos(2\pi n_{k}x+\varphi _{k}) \tag{18}\]  
where \(r(x)\) is the simulated road roughness sample; \(G_{d}(n_{k})\) is the power spectral density function of surface displacement, as defined by the standard; \(n_{k}\) is defined as the discrete spatial frequency in the range of \([n_{l},\,n_{u}]\); \(\Delta n\) is the frequency interval in the frequency space; \(\varphi_{k}\) is a set of random numbers uniformly distributed in the range of \([0,2\pi]\); and N is the total number of sampling points. \(\Delta n\) and \(n_{k}\) can be calculated by Equations (19) and (20):  
\[\Delta n=(n_{u}-n_{l})/N \tag{19}\]  
\[n_{k}=n_{l}+(k-1/2)\Delta n \tag{20}\]  
where \(n_{u}\) and \(n_{l}\) are the upper and lower limits of the effective spatial frequency of the road, respectively. [PERSON] (2003) [25] elucidated the process of generating roughness using the triangular series superposition method. Road roughness is primarily characterized by the road power spectral density. In 1984, the International Standard Organization (ISO) released the Draft Standard for Road Roughness [26], and in 1986, China also promulgated the national standard, GB7031-86, regarding road roughness [27].