## 3 Case Studies
### Case Study 1: Coupling of Continuous Substructures
#### 3.1.1 Determining the Components of Subsystems Acceptance  
To determine the receptance matrix components for subsystem A, which consists of a flexural beam with fixed-free boundary conditions, we begin with the governing equations derived from EBT [29]. The equation for beam displacement is as follows:  
\[EI\frac{\partial^{4}y}{\partial^{4}x}+\rho\frac{\partial^{2}y}{\partial^{2}t} = 0. \tag{20}\]  
In this equation, \(EI\),, and \(\rho\) are flexural rigidity, cross-section area, and density, respectively. This equation assumes that rotary inertia and shear deformations are negligible compared to translational inertia and bending deformations, an approximation valid for slender beams. For free vibration analysis, a steady state (harmonic) solution is assumed where the displacement response in a steady-state is given by \(y(x,t)=Y(x)e^{i\omega t}\). The spatial component \(Y(x)\) can be expressed as follows:  
\[Y(x)=a_{1}\text{sin}(\lambda\,x)+a_{2}\text{cos}(\lambda\,x)+a_{3}\text{sin} \,h(\lambda\,x)+a_{4}\text{cos}\,h(\lambda\,x). \tag{21}\]  
Figure 1: Schematic of system AB for the 1 st case study. Figure 2: Decomposition of system AB into substructures A and B for case study 1.  
where \(\lambda^{4}=\omega^{2}\frac{\rho}{EI}\), and \(a_{1},\ a_{2},\ a_{3},\ \text{and}\ a_{4}\) are coefficients determined by applying the boundary conditions defined in Equation (22) to Equation (21). Here, \(\lambda\) is a parameter related to the natural frequencies and mode shapes of the beam, with \(\omega,\ \rho,\,\) and \(EI\) representing the angular frequency, material density, cross-sectional area, and flexural rigidity, respectively. For subsystem A, with fixed-free boundary conditions and with length L, harmonic force and moment excitation are applied at the free end, as shown in Figure 3. The boundary conditions for subsystem A are fixed-free and are specified as follows:  
\[y_{(x=0)}=0,\ \theta_{(x=0)}=\frac{dy_{(x=0)}}{dx}=0. \tag{22}\]  
The boundary conditions for subsystem B are free-free and are specified as follows:  
\[EI\frac{d^{2}y_{(x=L)}}{d^{2}x}=M_{2}^{A}e^{i\omega t},\ EI\frac{d^{3}y_{(x=L)}}{ d^{3}x}=0.\]  
\[EI\frac{d^{2}y_{(x=L)}}{d^{2}x}=0,\ EI\frac{d^{3}y_{(x=L)}}{d^{3}x}=-F_{y_{2}}^{ A}e^{i\omega t}.\]  
The generalized receptance matrix for subsystem A, with translational and rotational DoFs at the free end, is as follows:  
\[[\mathcal{H}_{22}^{A}]= \begin{bmatrix}H_{y_{2}y_{2}}^{A}&H_{y_{2}\theta_{2}} ^{A}\\ H_{\theta_{2}y_{2}}^{A}&H_{\theta_{2}\theta_{2}}^{A}\end{bmatrix} . \tag{23}\]  
Each component is determined from receptance definitions and the displacement function \(Y(x)\), given boundary conditions [5; 30] as follows:  
\[H_{y_{2}y_{2}}^{A}=\frac{y_{2}^{A}}{F_{y_{2}}^{A}}=\frac{-S_{1}}{EI\lambda^{3 }S_{8}},H_{y_{2}\theta_{2}}^{A}=\frac{y_{2}^{A}}{M_{2}^{A}}=\frac{S_{2}}{EI \lambda^{2}S_{8}},\ H_{\theta_{2}\theta_{2}}^{A}=\frac{\theta_{2}}{M_{2}^{A}}= \frac{S_{2}}{EI\lambda^{2}S_{8}}. \tag{24}\]  
where \(S_{1}\) to \(S_{8}\) are defined in Equation (29). For subsystem B, a free-free beam is subjected to forces and moments at both ends, as shown in Figure 4.  
Figure 3: Excitation force and moment on substructures A for case study 1.