# Railway Substructure Modeling Approach for Modal Analysis Using Multibody Simulation
## 2 Materials and Methods
### Proposed Modeling Solution
It is a two Degree-of-Freedom (2-DoF) model, containing an assembly of two spring-mass subsystems in series, which represents the vehicle, per axle, and the railway infrastructure beneath it. The lower mass element represents the unsprung mass of one vehicle axle and the modal mass of the infrastructure section that is being dynamically excited by the passing vehicle (\(m_{1}\)), and the second represents the vehicle's sprung mass that is supported by one axle (\(m_{2}\)). These body elements are linked by a spring that represents the vehicle's suspension system (\(k_{2}\)). The remaining element is a spring that characterizes the modal stiffness of the railway infrastructure (\(k_{1}\)). This last element aggregates all stiffness contributions from the entire railway infrastructure, including from the subgrade layer. The outputs of this model are two natural frequencies (\(\omega_{1}\) and \(\omega_{2}\)) [20]. Frequency \(\omega_{2}\) is mainly influenced by the vehicle's modal characteristics and corresponds to the vertical oscillatory motion of the vehicle's sprung mass, which has a frequency range between 1 and 5 Hz for common passenger carriages [10]. On the other hand, frequency \(\omega_{1}\) is influenced by the modal parameters of the railway infrastructure elements. This is the frequency that represents an adequate estimate of the subgrade's first vertical resonance frequency obtainable from an MBS model (\(f_{sim}\)) in the presented conditions [17]. [PERSON] et al. (2021) provides a more in-depth description of this TCMM and this concept [16]. The formula for frequency \(\omega_{1}\) is the equation associated with SM-Lumped (Equation (5)). In this equation, \(k_{2}\) represents the vertical stiffness of the primary suspension of the vehicle, even for a vehicle with two suspension levels, since it is the spring element that separates the sprung and unsprung mass elements of the vehicle. Equation (5) also shows how \(\omega_{1}\) is calculated with the relevant parameters from Table 1.  
\[\omega_{1}=\sqrt{\frac{k_{2}+k_{1}}{m_{1}}}*\frac{1}{2\pi}=\sqrt{\frac{k_{v,s 1}+k_{i,su}}{m_{v,a}+m_{i,su}+m_{i,ba}+m_{i,sl}}}*\frac{1}{2\pi}\cong f_{ sim} \tag{5}\]  
Having integrated the layered elements' mass (included in \(m_{1}\)) in the overall formulation, the conditions to establish their geometry were still not met. The system of equations was still undetermined due to the vertical stiffness term of the subgrade layer (\(k_{i,su}\)). The fourth sub-model (SM-Stiffness) is an equation that gives an estimate of the vertical stiffness that a platform or foundation would perceive when placed on a single soil layer (\(k_{f}\)). This sub-model (Equation (6)) defines the soil layer as a homogeneous half-space and is solely based on the in-plane dimensions of the platform and the mechanical properties of the soil layer [22]. This equation will be used to provide an estimate for the subgrade layer vertical stiffness (\(k_{i,su}\)), a concept that has already been used in another railway-related context to study vibration propagation from a railway track [12]. In the railway context, the platform's dimensions used are the length (\(2a\)) and width (\(2b\)) of a steeper. The soil'smechanical properties relevant for this equation are its shear modulus (\(G\)) and its Poisson's ratio (\(\
u\)). \[k_{f}=\frac{G*b}{(1-\
u)}\left[3.1\left(\frac{a}{b}\right)^{0.75}+1.6\right]=k_{ isu} \tag{6}\]  
With this additional equation, the combined model now represents a deterministic solution for the geometry of the railway's layered elements to be used in an MBS model.