## 2 FE-SEA Hybrid Model for Bending Vibration of a Simply Supported Beam  
### Formulation of the Dynamic Equation  
Consider a simply supported beam structure with a uniform cross-section subjected to a vertical load, as shown in Figure 1. The beam is discretized into multiple-node beam elements, as shown in the figure. Let \(L\) be the length of the beam, \(A\) the cross-sectional area, \(EI\) the flexural rigidity, and \(\rho\) the density. Vertical point load \(F_{0}\) is applied at position \(x_{0}\) along the beam. In the finite element method (FEM), the shape function matrix for the beam element is defined as  
\[[1-\frac{3x^{2}}{L^{2}}+\frac{2x^{3}}{L^{3}},x(1-\frac{2x}{L}+\frac{x^{2}}{L^{ 2}}),\frac{3x^{2}}{L^{2}}-\frac{2x^{3}}{L^{3}},-\frac{x^{2}}{L}+\frac{x^{3}}{ L^{2}}] \tag{1}\]  
For modal superposition, the vibration mode function for the \(n\)th mode of the simply supported beam is expressed as  
\[\sqrt{\frac{2}{\rho AL}}\sin(\frac{n\pi x}{L}) \tag{2}\]  
For Combining these two, the shape function vector is defined as  
\[ \begin{array}{l}N(x)=[\phi_{1}(x),\phi_{2}(x),\phi_{3}(x),\cdots]\\ =[1-\frac{3x^{2}}{L^{2}}+\frac{2x^{3}}{L^{3}},x(1-\frac{2x}{L}+\frac{x^{2}}{L^ {2}}),\frac{3x^{2}}{L^{2}}-\frac{2x^{3}}{L^{3}},-\frac{x^{2}}{L}+\frac{x^{3}}{ L^{2}},\sqrt{\frac{2}{\rho AL}}\sin(\frac{\pi x}{L}),\sqrt{\frac{2}{\rho AL}}\sin( \frac{2\pi x}{L}),\cdots]\end{array}  \tag{3}\]  
The nodal displacement vector is denoted as  
\[Q(t)=[q_{1}(t),q_{2}(t),q_{3}(t),\cdots]^{T} \tag{4}\]  
where \(q_{1}(t),q_{2}(t),q_{3}(t),q_{4}(t)\) represent the vertical displacements and rotational angles at the two ends of the beam. The term \(q_{n}(t)\) refers to the modal amplitude of the \(n\)th mode  
Figure 1: Beam bending vibration element. and represents the displacement of the \(n\)th virtual node, which is a non-physical node. These virtual displacements are treated in the same manner as actual nodal displacements, where \(n=5,6,7,\cdots\). Accordingly, the displacement of the beam at any position \(x\) can be expressed as  
\[ \begin{array}{c}U(x,t)=N(x)Q(t)=\sum\limits_{n=1}^{\infty}q_{n}(t)\varphi_{n} (t)\\ =q_{1}(t)(1-\frac{3x^{2}}{L^{2}}+\frac{2x^{3}}{L^{3}})+q_{2}(t)x(1-\frac{2x}{L }+\frac{x^{2}}{L^{2}})+q_{3}(t)(\frac{3x^{2}}{L^{2}}-\frac{2x^{3}}{L^{3}})+q_{4 }(t)(-\frac{x^{2}}{L}+\frac{x^{3}}{L^{2}})\\ +\sqrt{\frac{2}{\rho AL}\sum\limits_{n=5}^{\infty}q_{n}(t)}\sin(\frac{(n-4)\pi x }{L})\end{array}  \tag{5}\]  
The displacement vector \(Q(t)\) satisfies the dynamic equilibrium equation:  
\[M\ddot{Q}(t)+C\dot{Q}(t)+KQ(t)=F \tag{6}\]  
where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrices of the structure, respectively, and \(F\) is the excitation force vector applied to the structure. Applying the Fourier transform to Equation (6) yields  
\[(-\omega^{2}M+i\omega C+K)Q=DQ=F \tag{7}\]  
where \(D\) is the dynamic stiffness matrix of the structure and \(\omega\) is the angular frequency. In the mid-to-high frequency range, due to the high modal density, a large number of mode shapes are required to accurately capture the structural vibration response. For analytical convenience, the modal basis is decomposed into two parts and discussed separately. According to the excitation frequency and computational constraints, the structural modes are categorized into global modes and local modes. The shape functions \(\varphi_{n}(x)\) are accordingly divided into global and local components. Assume that \(S\) mode shapes belong to the global modes. Let the total number of modes be \(N\), where \(N^{g}(1,2,3,\cdots,s)\) and \(N^{l}(s+1,s+2,\cdots,N)\) represent the number of global and local modes, respectively.