## 3 Case Studies: Application to Continuous and Lumped-Parameters Systems
### Case Study 1: Coupling of Continuous Substructures
#### 3.1.3 Validation of RCFBS Using Modal Method  
To determine the receptance matrix components for system ABC, modeled as a flexural beam with fixed-free boundary conditions, the mode shapes \(\mathrm{Y}_{\mathrm{n}}(\mathrm{x})\) are derived from the Euler-Bernoulli beam theory. The mode shapes are given by the following [1]:  
\[\mathrm{Y}_{\mathrm{n}}(\mathrm{x})=\cos\mathrm{h}(\lambda_{ \mathrm{n}}\,\mathrm{x})-\cos(\lambda_{\mathrm{n}}\,\mathrm{x})+\frac{\cosh( \lambda_{\mathrm{n}}\,\mathrm{L})-\cos(\lambda_{\mathrm{n}}\,\mathrm{L})}{ \sinh(\lambda_{\mathrm{n}}\,\mathrm{L})-\sin(\lambda_{\mathrm{n}}\,\mathrm{L} )}(\sinh(\lambda_{\mathrm{n}}\,\mathrm{x})-\sin(\lambda_{\mathrm{n}}\,\mathrm{ x})). \tag{16}\] \[\theta_{\mathrm{n}}(\mathrm{x})=\frac{\mathrm{d}\mathrm{Y}_{\mathrm{ n}}}{\mathrm{d}\mathrm{x}}=\lambda_{\mathrm{n}}\{\sinh(\lambda_{\mathrm{n}}\, \mathrm{x})+\sin(\lambda_{\mathrm{n}}\,\mathrm{x})+\frac{\cosh(\lambda_{ \mathrm{n}}\,\mathrm{L})-\cos(\lambda_{\mathrm{n}}\,\mathrm{L})}{\sinh(\lambda _{\mathrm{n}}\,\mathrm{L})-\sin(\lambda_{\mathrm{n}}\,\mathrm{L})}(\cosh( \lambda_{\mathrm{n}}\,\mathrm{x})-\cos(\lambda_{\mathrm{n}}\,\mathrm{x})).\]  
where \(\theta_{n}(x)\) is the rotational mode shape (slope) evaluated at position \(\mathrm{x}\), derived from \(Y_{\mathrm{n}}(x)\) and \(\lambda_{\mathrm{n}}{}^{4}=\omega_{\mathrm{n}}{}^{2}\frac{\rho\mathrm{A}}{ \mathrm{E}\mathrm{I}}\), representing the n-th eigenvalue associated with the natural frequencies \(\omega_{\mathrm{n}}\) and mode shapes of the beam. The natural frequencies are obtained by solving the transcendental characteristic equation:  
\[\cos\mathrm{h}(\lambda_{\mathrm{n}}\,\mathrm{L})\cos(\lambda_{ \mathrm{n}}\,\mathrm{L})=1 \tag{17}\]  
Solving this equation yields multiple solutions, with the first four modes given by the following:  
\[\lambda_{1}\mathrm{L}=4.73,\lambda_{2}\mathrm{L}=7.85,\lambda_{ 3}\mathrm{L}=10.99,\lambda_{4}\mathrm{L}=14.13 \tag{18}\]  
This equation yields multiple solutions, with the first four modes corresponding to the natural frequencies of 20.03 Hz, 55.18 Hz, 108.15 Hz, and 178.96 Hz. The receptance is calculated by summing the contributions from each mode, using the mode shapes and natural frequencies to determine the response at a given excitation frequency. Given the assumption of no internal points, the receptance matrix \(\mathrm{H}_{\mathrm{ij}}(\omega)\), which relates the translational or rotational displacement at connection points 1 and 2 (as shown in Figure 2) to the applied force or moment, is computed by summing the contributions from each mode:  
\[\mathrm{H}_{\mathrm{ij}}(\omega)=\sum_{r=1}^{\infty}\frac{\psi_{ \mathrm{ir}}(\mathrm{x}_{i})\psi_{\mathrm{jr}}(\mathrm{x}_{i})}{\mathrm{m}( \omega_{r}{}^{2}-\omega^{2})} \tag{19}\]  
where \(\mathrm{m}\) is the mass per unit length, \(\omega_{\mathrm{r}}\) is the natural frequency of the r-th mode, and \(\psi_{\mathrm{ir}}(\mathrm{x}_{i})\) represents the mode shape at position \(\mathrm{x}_{i}\) for the frequency \(\omega_{\mathrm{r}}\). This receptance function, \(\mathrm{H}_{\mathrm{ij}}(\omega)\), characterizes the translational or rotational receptance at point \(\mathrm{x}_{i}\) due to a force or moment applied at point \(\mathrm{x}_{j}\). It encapsulates the cumulative effect of each mode, with mode shapes evaluated at x\({}_{i}\), and is pivotal in understanding the system's dynamic response to excitations. For example, by considering contributions from the first four modes, the receptance functions H\({}_{11}(\omega)\), H\({}_{14}(\omega)\), and H\({}_{22}(\omega)\), which describe the translational displacement at node 1 due to an excitation force applied at the same node, the translational displacement at node 1 due to an excitation moment applied at node 2, and the rotational displacement at node 1 due to a couple excitation applied at the same node, respectively, can be expressed as in Equation (20):  
\[