# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 2 Modeling of Complex Dynamical Properties of the Railway Track and Ballast
Rheological theory has been used to quantify the nonlinear elastoplastic-viscoelastic properties of materials with multiple nonlinear spring elements, viscous damping, and frictional elements [86; 87; 88; 89; 90; 91]. In numerical simulations of ballast systems, a generalized Maxwell model has been used to establish constitutive relations on a material level regarding stresses and strains. The proposed model in this paper extends the understanding from the material level to the structural component level. The straightforward extension to a structural level is given in this research in terms of forces and displacements. Moreover, since the experimental data are widely established as forces and displacements, the proposed parameter model can be used for system identification and facilitating efficient, accurate inspection. To quantify the complex ballast stiffness, the proposed model is based on a generalized Maxwell model with multiple parameters [90; 91], which is a combination of nonlinear elastic, friction, and viscous portions. The model is based on a superposition of elastic, friction, and viscous forces, which reflects the time/space-varying amplitude-dependent stiffness, frequency-dependent stiffness, and rate-dependent hysteresis. The detailed model of ballast supporting force characterizing the frequency-dependent, amplitude-dependent, and history-dependent stiffness is given as follows. Figure 2 shows the schematic model of the generalized model for the rheological ballast system. It is a generalized Maxwell model with spring/damper/sliding friction elements, comprehensively characterizing the ballast system's nonlinear and time/space-varying viscous-elastic-plastic properties. \[F\big{(}x,\dot{x}\big{)}=F_{e}+F_{f}+F_{v} \tag{6}\]where \(F_{e}\) is the non-linear elastic force, \(F_{f}\) is the ballast internal interaction or friction force, \(F_{p}\) is viscous force. For steady harmonic excitation \(x=x_{0}sin\omega t\), the amplitude of each force can be expressed as  
\[\left\{ \begin{array}{l}F_{e}=k_{t}\frac{2 dt}{\pi}\tan\frac{\pi x}{2t_{t}}+a(y) \\ F_{f}=\frac{F_{max}}{2x_{2}}\left(\sqrt{x_{2}2^{2}+x_{0}2^{2}+6x_{2}x}-x_{2}- x\right)\\ F_{pRC}=\left(\frac{(\omega c_{1}/k_{1})^{2}}{1+(\omega c_{1}/k_{1})^{2}}k_{1 }+\frac{(\omega c_{2}/k_{2})^{2}}{1+(\omega c_{2}/k_{2})^{2}}k_{2}\right)x\\ F_{vIm}=\left(\frac{c_{1}}{1+(\omega c_{1}/k_{1})^{2}}+\frac{c_{2}}{1+( \omega c_{2}/k_{2})^{2}}\right)\omega x\end{array} \right. \tag{7}\]  
where \(F_{vRe}\) and \(F_{vIm}\) are the real and imaginary force components of complex viscoelastic force, in which \(x_{0}\), \(\omega\), and \(t\) are displacement amplitude, circular excitation frequency, and time. \(F_{fmax}\) is the maximum friction force, whereas the displacement \(x_{2}\) is the displacement required to gain a friction force as large as \(F_{f}=F_{fmax}/2\). \(a(y)\) is the time/space-varying term in the elastic force, dependent on spatially variable \(y(t)\) along rail locations due to the variation of ballast properties. \(k_{t}\) is the stiffness coefficient, \(d_{t}\) is the characteristic thickness representing the location of the elastic curve asymptote. \(c_{1}\), \(c_{2}\) are the viscous damping coefficients, and \(k_{1}\), \(k_{2}\) are the stiffness coefficients. A typical dynamical stiffness versus amplitude and frequency based on the model is plotted in Figure 3, with parameters \(k_{1}=35\) MN/m, \(k_{2}=250\) MN/m, \(c_{1}=300\) kNs/m, \(c_{2}=40\) kNs/m, \(F_{fmax}=140\) kN, and \(x_{2}=2\) mm. Figure 4 shows a typical hysteresis loop of the force versus harmonic displacement based on the model, illustrating the complex stiffness nonlinearity. Figure 3: Amplitude and frequency dependency of stiffness. Figure 2: Schematic model of ballast stiffness. The new nonlinear dynamic stiffness model enables comprehensively characterizing the dynamic properties of displacement, frequency dependence, and hysteresis, which have been widely recorded in previous experimental research [3, 8, 24, 25, 38, 44, 57, 58, 59].