## 3 Mathematical Model  
The rail structure is well represented by an Euler-Bernoulli beam of infinite length on periodic spring supports [10; 17]. The equation of free motion of an unsupported Euler-Bernoulli beam excited by a point harmonic force in the absence of damping is given by [18]:  
\[EI\frac{\partial^{4}y}{\partial x^{4}}+m_{r}^{\prime}\frac{\partial^{2}y}{ \partial t^{2}}=\text{Fe}^{i\omega t}\delta(x) \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Aluminum Rail** & **Conductor Rail** & **Bracket** & **Sleeper** & **Fasteners** \\ \hline Material & Aluminum alloy 6063 & Stainless steel 430 & Structural steel & Concrete & Stainless Steel 316 \\ Density (kg/m\({}^{3}\)) & 2700 & 7740 & 7850 & 2300 & 8027 \\ Shear modulus (Pa) & \(2.59\times 10^{10}\) & \(7.84\times 10^{10}\) & \(7.69\times 10^{10}\) & \(1.27\times 10^{10}\) & \(8.2\times 10^{10}\) \\ Young’s modulus (Pa) & \(6.89\times 10^{10}\) & \(2\times 10^{11}\) & \(2\times 10^{11}\) & & \(2\times 10^{11}\) \\ Cross-sectional area (m\({}^{2}\)) & \(4.70\times 10^{-3}\) & \(6.59\times 10^{-4}\) & - & - & - \\ \hline \hline \end{tabular}  
* Values are based on third rail used in SMRT.  
\end{table} 
Table 1: Material properties of SMRT third rail components *. Figure 2: Finite element model of third rail system: (**a**) conductor rail model; (**b**) typical third rail arrangement, (**c**) single-span third rail model; and (**d**) multi-span third rail model. where \(E\) is the combined effective Young's modulus of the third rail, \(I\) is the second moment of inertia, and \(EI\) is collectively known as bending stiffness of the beam. \(m_{r}^{\prime}\) is the mass per unit length of the beam, \(y(x,t)\) is the vertical displacement/bending of the beam, \(x\) is the coordinate along the rail, and \(Fe^{i\omega t}\) represents the harmonic force with frequency \(\omega\) rad/s. The combined effective Young's modulus of the third rail was estimated by using the material properties of the aluminum rail and stainless-steel conductor layer using Equation (2) [19]. \[\text{Effective Young/s modulus}\left(E\right)=\frac{E_{AL}A_{AL}+E_{SS}A_{SS}}{ A_{AL}+A_{SS}} \tag{2}\]  
where \(E_{AL},E_{SS},A_{AL}\) and \(A_{SS}\) are [PERSON]'s moduli and the cross-sectional areas of the aluminum rail and stainless-steel conductor, respectively. The free wave solution of Equation (1) considering the harmonic motion of frequency \(\omega\) rad/s is given by:  
\[y(x,t)=Ue^{i\omega t}e^{-kx} \tag{3}\]  
where \(U\) is the amplitude coefficient, and \(k\) is the wavenumber. Substituting Equation (3) in Equation (1), the wavenumbers of the unsupported beam is given by:  
\[k=\left(\frac{m_{r}^{\prime}\omega^{2}}{EI}\right)^{\frac{1}{4}} \tag{4}\]  
Moreover, the third rail can be modelled as an infinite beam that is periodically supported by springs separated by a distance \(d\)[8; 17; 20], as illustrated in Figure 3. Analysis for the periodically supported beam is also described in [10; 18; 21]. A general approach is to account for the reaction forces at each support point that is exerted by individual deflected springs on the beam. If the deflection/bending of the beam at the support point \(x_{n}=nd\) is \(u(x_{n})\), the beam experiences a reaction force equivalent to \(-Ku(x_{n})\), where \(K\) is the stiffness of the supporting springs. The procedure to estimate the equivalent spring stiffness for the third rail support structure was explained in Section 2. In general, the dynamics of the periodically supported structure is analyzed using a receptance function \(\alpha(x,x^{\prime})\) defined as the dynamic response of each structure under unit force excitation [12]. The receptance function for the Euler-Bernoulli beam is given by [10]:  
\[\alpha\big{(}x,x^{\prime}\big{)}=u_{1}e^{-ik_{e}|x-x^{\prime}|}+u_{2}e^{-ik_{ p}|x-x^{\prime}|} \tag{5}\]  
where \(k_{e}\) and \(k_{p}\) are the wavenumbers, which are the negative imaginary and positive real solutions of Equation (4).