# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 2 Modeling of Complex Dynamical Properties of the Railway Track and Ballast
The rail support modulus \(Kr\) can be expressed by the layers of springs with different stiffnesses [2; 3; 7; 8; 18; 25; 26; 27; 44]:  
\[K_{r}=\frac{1}{1/k_{p}+1/k_{t}+1/k_{b}+1/k_{s}} \tag{2}\]  
where \(k_{p}\) represents the rail pad stiffness, \(k_{t}\) represents the sleeper stiffness, \(k_{b}\) represents the ballast stiffness, and \(k_{s}\) is the foundation soil stiffness. The following equation can express the motion of the vehicle subsystem:  
\[M_{\varphi}\ddot{x}_{p}+C_{\varphi}\dot{x}_{\varphi}+K_{\varphi}x_{p}=F_{ \varphi}\big{(}\dot{x}_{\varphi},x_{\varphi},\dot{x}_{t},x_{t}\big{)}+F_{ext} \tag{3}\]  
where \(\ddot{x}_{\varphi}\), \(\dot{x}_{\varphi}\), and \(x_{\varphi}\) represent the accelerations, velocities, and displacements of the vehicle subsystem; \(M_{\varphi}\) represents the mass of the vehicle; \(C_{\varphi}\), \(K_{\varphi}\) are the damping and the stiffness originating from the vehicle subsystem, to describe nonlinearities of the suspension; \(\dot{x}_{t}\), \(x_{t}\) are the velocities and displacements of the track subsystem; and \(F_{\varphi}\big{(}\dot{x}_{\varphi},x_{\varphi},\dot{x}_{t},x_{t}\big{)}\) represents the system load, which represents the nonlinear contact forces between the wheel and rail. The parameter \(\dot{x}_{\varphi}\), \(x_{\varphi}\) of the vehicle and \(\dot{x}_{t}\), \(x_{t}\) of the track decide the model. \(F_{ext}\) represents  
Figure 1: Schematic model of Quart vehicle–track system. the external forces, including gravitational forces and forces resulting from the centripetal acceleration when the vehicle is running through a curve. The equations of the vertical motion of rails have been expressed according to fourth-order partial differential equations:  
\[EI_{y}\frac{\partial^{4}Z_{r}(x,t)}{\partial x^{4}}+m_{r}\frac{\partial^{2}Z_{r} (x,t)}{\partial t^{2}}=-\sum_{i=1}^{N}F_{swi}(t)\delta(x-x_{i})+\sum_{j=1}^{4 }P_{j}(t)\delta\big{(}x-x_{wj}\big{)} \tag{4}\]  
In Equation (4), \(Z_{r}(x,t)\) represents the vertical displacements of the rail; \(m_{r}\) represents the rail mass per unit length; \(\rho\) represents the rail density; \(EI_{y}\) represents the rail bending stiffness to the \(Y\)-axis; \(F_{swi}(t)\) represents the vertical dynamic forces at the \(i\)th rail-supporting point; \(P_{j}(t)\) represents the \(j\)th wheel-rail vertical forces; and \(\delta(x)\) represents the Dirac delta function. Therefore, the final equation of the track subsystem is an equation including sleepers and ballast blocks, the model of which can be represented by individual rigid masses, which can be expressed as  
\[M_{t}\ddot{x}_{t}+C_{t}\dot{x}_{t}+K_{t}x_{t}=F_{t}\big{(}\dot{x}_{v},x_{v}, \dot{x}_{t},x_{t}\big{)} \tag{5}\]  
where \(M_{t}\) represents the mass of the track structure; \(C_{t}\) and \(K_{t}\) are the damping and the stiffness of the track subsystem; \(\ddot{x}_{t}\) represents the accelerations of the system; and \(F_{t}\big{(}\dot{x}_{v},x_{v},\dot{x}_{t},x_{t}\big{)}\) represents the load of the track subsystem, to represent the nonlinear wheel-rail forces obtained by the wheel-rail coupling model. Many existing research studies have investigated various nonlinearities in the track system and classic nonlinearities in the train system. Ballast is vital to track alignment, stability, and sustainability. The mechanical behavior of railroad ballast has been interpreted by modeling based on large-scale experiments. However, the existing basic ballast models have not considered amplitude-dependent stiffness, frequency-dependent stiffness, and complex damping properties, although experimental evidence has proven these properties to be critical. To comprehensively quantify the nonlinear properties of ballast systems in the framework of rheology theory, this paper deploys and extends the general multiple parameters model for characterizing time/space-varying amplitude-dependent stiffness, frequency-dependent stiffness, and hysteresis in ballast systems.