## 2 Vertical Dynamic Response of a Simply Supported Double-Beam System  
### Mathematical Model Building and Parameter Solving  
Figure 1 shows a simply supported double-beam system under successive moving loads in arbitrary spaces and sizes; after introducing a constructor \(S(\zeta)\), the successive moving loads \(P(t)\) can be expressed as follows:  
\[P(x,t)=\sum_{i=1}^{N}F_{i}\delta[x-v(t-t_{i})]S\Bigg{[}\frac{v(t-t_{i})}{l} \Bigg{]} \tag{1}\]  
\[S(\zeta)= \begin{cases}1&0\leq\zeta\leq 1\\ 0&\text{else}\end{cases}  \tag{2}\]  
where \(\delta\) is Dirac function; \(l\) is the length of a simply supported double-beam system; \(F_{i}\) is the i\({}^{\text{th}}\) moving load. It is assumed that \(F_{1}\) acts on \(x=0\) at the initial moment (\(t=0\)); \(t_{i}=d_{i}/v\), \(d_{i}\) represents the distance from \(F_{i}\) to \(F_{1}\). The vertical vibration of the double-beam system shown in Figure 1 is governed by the two coupled partial differential equations [35]:  
\[E_{1}I_{1}\frac{\partial^{4}y_{1}}{\partial x^{4}}+m_{1}\frac{\partial^{2}y_{ 1}}{\partial t^{2}}+\kappa(y_{1}-y_{2})=P(t) \tag{3}\]  
\[E_{2}I_{2}\frac{\partial^{4}y_{2}}{\partial x^{4}}+m_{2}\frac{\partial^{2}y_{ 2}}{\partial t^{2}}-\kappa(y_{1}-y_{2})=0 \tag{4}\]  
where \(y_{1}(x,t),y_{2}(x,t)\) are the vertical displacements of primary and secondary beams, respectively; \(E_{1},E_{2}\) are the elastic moduli of primary and secondary beams, respectively; \(I_{1},I_{2}\) are the horizontal moments of inertia of primary and secondary beams, respectively; \(m_{1},m_{2}\) are the masses per unit length of primary and secondary beams, respectively; \(\kappa\) is the spring stiffness between the primary and secondary beams. Figure 1: Double-beam system under successive moving loads. To solve the above vibration partial differential equations, the first step is to perform a finite sin-Fourier transform for space coordinate \(x\); for \(0\leq x\leq l\), the finite sin-Fourier transform can be defined as follows [35]:  
\[\psi[y_{n}(x,t)]=U_{n,k}(t)=\int_{0}^{L}y_{n}(x,t)\sin(\xi_{k}x)\mathrm{d}x \tag{5}\]  
\[\psi^{-1}\Big{[}U_{n,k}(t)\Big{]}=y_{n}(x,t)=\frac{2}{l}\sum_{k=1}^{\infty}U_{ n,k}(t)\sin(\xi_{k}x) \tag{6}\]  
where \(n=1,2\), \(\xi_{k}=\frac{k\pi}{l}\), \(k=1,2,3,...\). Under minor deformation conditions, the boundary condition of a simply supported double-beam system can be written as follows:  
\[y_{n}(x,t)\big{|}_{x=0,l}=0,Ely^{\prime\prime}_{n}(x,t)\big{|}_{x=0,l}=0 \tag{7}\]  
According to the boundary condition,  
\[\psi\Bigg{[}\frac{\mathrm{d}^{4}y_{n}(x,t)}{\mathrm{d}x^{2}}\Bigg{]}=\xi_{k}^ {4}U_{n,k}(t) \tag{8}\]  
By performing finite sin-Fourier transforms for both sides of Equations (3) and (4), double-beam infinite-degree-of-freedom system can be transformed into two-degrees-of-freedom system,  
\[\xi_{k}^{4}E_{1}I_{1}U_{1,k}+m_{1}\tilde{U}_{1,k}+\kappa\Big{(}U_{1,k}-U_{2,k }\Big{)}=P_{k} \tag{9}\]  
\[\xi_{k}^{4}E_{2}I_{2}U_{2,k}+m_{2}\tilde  
By multiplying \(\mathbf{\Gamma}_{k}{}^{\mathrm{T}}\),  
\[\mathbf{\Gamma}_{k}{}^{\mathrm{T}}\mathbf{M}_{k}\mathbf{\Gamma}_{k}\mathbf{ \tilde{q}}_{k}+\mathbf{\Gamma}_{k}{}^{\mathrm{T}}\mathbf{K}_{k}\mathbf{\Gamma} _{k}\mathbf{q}_{k}=\mathbf{\Gamma}_{k}{}^{\mathrm{T}}[P_{k},0]^{\mathrm{T}} \tag{18}\]  
The orthogonality of mode of vibration shows that \(\mathbf{\Gamma}_{k}{}^{\mathrm{T}}\mathbf{M}_{k}\mathbf{\Gamma}_{k}\) and \(\mathbf{\Gamma}_{k}{}^{\mathrm{T}}\mathbf{K}_{k}\mathbf{\Gamma}_{k}\) are diagonal matrices; thus, Equation (18) can be simplified as follows:  
\[M_{n,k}\ddot{q}_{n,k}+K_{n,k}q_{n,k}=P_{n,k} \tag{19}\]  
\[M_{n,k}=\left\{ \begin{array}{c}\phi_{k}^{1,n}\\ \phi_{k}^{2,n}\end{array} \right\}^{\mathrm{T}}\mathbf{M}_{k}\left\{ \begin{array} []{c}\phi_{k}^{1,n}\\ \phi_{k}^{2,n}\end{array} \right\} \tag{20}\]  
\[K_{n,k}=\left\{ \begin{array}{c}\phi_{k}^{1,n}\\ \phi_{k}^{2,n}\end{array} \right\}^{\mathrm{T}}\mathbf{K}_{k}\left\{ \begin{array} []{c}\phi_{k}^{1,n}\\ \phi_{k}^{2,n}\end{array} \right\} \tag{21}\]  
\[P_{n,k}=\left\{