# Railway Substructure Modeling Approach for Modal Analysis Using Multibody Simulation
## 2 Materials and Methods
### Proposed Modeling Solution  
The MS proposed here is a combination of four distinct theoretical models, which together create a deterministic solution for the layered elements' geometry in an MBS model. Table 1 presents a list of the main geometrical and mechanical parameters that relate to these theoretical models. The control parameter of this combined model, to verify the realistic scenario conditions, is the natural frequency of the first vertical vibration mode of the subgrade layer. The four theoretical submodels (SM) are described below. The first submodel (SM-Freq) is the theoretical equation of the natural frequencies of a surface-level soil layer from a natural soil deposit [12; 19]. For each vibration wave type (e.g., longitudinal waves or shear waves), the general form of this equation can estimate the frequency values for the \(n\) vibration modes of a soil layer. The kinematics of each vibration mode depend on the type of vibration wave created by the excitation source and its position relative to the soil layer. The specific form of this equation used here (Equation (1)) gives the frequency value of the first vibration mode (\(f_{1}\)), due to a vertical excitation source, based on the soil's characteristic longitudinal wave velocity (\(c_{p}\)) and the soil layer's height (\(h\)). This corresponds to the first vertical vibration mode that is dynamically excited by the presence of an excitation source above the soil layer, such as a railway vehicle, which is the most relevant situation in the present context [12]. \[f_{1}=\frac{c_{p}}{4h} \tag{1}\]  
The second submodel (SM-Wave) is the equation for the characteristic longitudinal wave velocity of a soil layer (\(c_{p}\)) [12; 19]. This equation (Equation (2)) provides a relationship between relevant mechanical properties of the soil to be simulated, namely, its shear modulus (\(G\)), its density (\(\rho\)), its Young's modulus (\(E\)) and its Poisson's ratio (\(\
u\)). Equations (3) and (4) present the formula for the Lame parameters (\(\lambda\), \(\mu\)), which are relevant for Equation (2). SM-Freq and SM-Wave represent a deterministic pair of equations for the natural frequency of a soil layer, for a given set of the relevant input parameters. This theoretical formulation is well known in the geotechnical field [21]. However, it is not enough to define the geometric shape of the railway's layered elements in an MBS model, \begin{table}
\begin{tabular}{c c} \hline \hline
**Vehicle-Track Parameters** & **Vehicle-Track Parameters** \\ \hline Layered elements diameter (\(\theta\)) & Sleeper spacing (\(L_{sil}\)) \\ Subgrade layer height (\(l_{sil}\)) & Soil longitudinal wave velocity (\(c_{p}\)) \\ Ballast layer height (\(l_{lu}\)) & Subgrade layer vertical stiffness (\(k_{sil}\)) \\ Subgrade density (\(\rho_{su}\)) & Ballast layer vertical stiffness (\(k_{i,lu}\)) \\ Ballast density (\(\rho_{lu}\)) & Sleeper vertical stiffness (\(k_{i,sl}\)) \\ Subgrade layer mass (\(m_{i,su}\)) & Car body mass (\(m_{nc}\)) \\ Ballast layer mass (\(m_{i,lu}\)) & Bogie mass (\(m_{p,b}\)) \\ Sleeper mass (\(m_{i,sl}\)) & Axle mass (\(m_{na}\)) \\ Sleeper length (\(2a\)) & Primary suspension stiffness (\(k_{c\approx 1}\)) \\ Sleeper width (\(2b\)) & Secondary suspension stiffness (\(k_{c\approx 2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters used in the proposed modeling solution. since it only establishes layer height. Therefore, there was a need for additional models to be added to the overall formulation. \[c_{p}=\sqrt{\frac{\lambda+2G}{\rho}} \tag{2}\]  
\[\lambda=\frac{vE}{(1+\
u)(1-2\
u)} \tag{3}\]  
\[\mu=\frac{E}{2(1+\
u)}=G \tag{4}\]  
The third sub-model (SM-Lumped) is a lumped-elements type model that can provide an adequate estimate of the subgrade's resonance frequency of interest obtainable from an MBS simulation [17]. This theoretical model was developed in the context of the new TCMM that was previously mentioned.