## 2 Model and Method
### Probability Density Evolution Method for Nonlinear Stochastic Analysis
#### 2.3.2 Probability Density Evolution Method  
The stochasticity of the nonlinear vehicle-track-bridge vertically coupled dynamical system mainly derives from the stochastic irregularity of the track and the stochasticity of the system parameters; then, the system stochastic dynamical equation can be expressed as  
\[[\mathbf{M}(\Theta_{m})]\ddot{\mathbf{Z}}(t)+[\mathbf{C}(\varTheta_{c})]\dot{\mathbf{Z}}(t)+[ \mathbf{K}(\varTheta_{k})]\mathbf{Z}(t)=\mathbf{F}(\varTheta_{I},t) \tag{4}\]  
where, \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\) are system mass, damping, and stiffness matrices; \(\varTheta_{m},\varTheta_{c},\varTheta_{k}\) are the stochastic parameters in the mass, damping, and stiffness matrices; \(\varTheta_{1}\) is the stochastic parameter of stochastic track irregularity; \(t\) is time; \(\mathbf{Z}(t)\), \(\dot{\mathbf{Z}}(t)\), \(\ddot{\mathbf{Z}}(t)\) are the displacement, velocity, and acceleration vectors, respectively; \(\mathbf{F}\) is the wheel-track excitation caused by stochastic track irregularities.  
A system stochastic vector is defined as \(\mathbf{\Theta}=\varTheta_{m},\varTheta_{c},\varTheta_{k},\varTheta_{I}\) under the condition that without the addition of other system stochastic factors during the vehicle operation, the stochasticity of any dynamic response \(\xi(t)\) in the coupled vehicle-track-bridge system originates from the stochastic vector \(\mathbf{\Theta}\), in which all stochastic factors have been included, and the whole system is a probabilistic conservative stochastic system. According to the principle of probability conservation, the probability of the augmented system composed of \([\xi\left(t\right),\varTheta]\) is conserved. Examine the stochastic event \((\xi(t),\varTheta)\in(\varOmega_{l}\times\varOmega_{\varTheta})\), wherein \(\varTheta\) is the sample realization of the stochastic vector \(\mathbf{\Theta}\), \(\varOmega_{l}\) is the distribution space region of \(\xi\left(t\right)\) at moment \(t\), and \(\varOmega_{\varTheta}\) is the distribution space region of \(\xi\left(t\right)\) in the sample \(\varTheta\). At the moment \(t+\Delta t\) after a small time increment \(\Delta t\), the stochastic event evolves as \((\xi(t+\Delta t),\varTheta)\in(\varOmega_{l+\Delta t}\times\varOmega_{ \varTheta})\); obviously, we have  
\[P\{(\xi(t),\mathbf{\Theta})\in(\varOmega_{t}\times\varOmega_{\varTheta})\}=P\{( \xi(t_{t+\Delta t}),\mathbf{\Theta})\in(\varOmega_{lt+\Delta t}\times\varOmega_{ \varTheta})\} \tag{5}\]  
Figure 2: Uniformity of the two-dimensional distribution of sample points by the GP split-circle domain method.  
where \(P[\cdot]\) denotes the probability of a stochastic event. Referring to the literature [40,41], the probability density evolution equation can be  
\[\frac{\partial p_{\xi_{\Theta}}(\xi,\Theta,t)}{\partial t}+\dot{\xi}(\Theta,t) \frac{\partial p_{\xi_{\Theta}}(\xi,\Theta,t)}{\partial\xi}=0 \tag{6}\]  
\(\dot{\xi}(\Theta,t)\) is the velocity of \(\dot{\xi}(t)\) conditional on \(\{\mathbf{\Theta}=\Theta\}\), namely, \(\dot{\xi}(t)=\partial H(\mathbf{\Theta},t)/\partial t\). Its initial condition is:  
\[P_{\xi_{\Theta}}(\xi,\Theta,t)\big{|}_{t=0}=\delta(\xi-\xi_{0})P_{\Theta}(\Theta) \tag{7}\]  
Solving the above equations to obtain the joint probability density function \(P_{\xi\Theta}(\xi,\Theta,t)\) and further integrating it over \(\dot{\xi}_{\Theta}\) to obtain the probability density function of \(\dot{\xi}(t)\), we have:  
\[p_{\xi}(\xi,t)=\int_{\Omega_{\Theta}}p_{\xi_{\Theta}}(\xi,\Theta,t)d\Theta \tag{8}\]  
Since the generalized probability density evolution equation decouples the probability space from the physical space, it is not difficult to obtain the solution to the stochastic nonlinear response of a complex structural system by combining the physical equation with the generalized probability density evolution equation and solving it by numerical methods in sequence.