## 3 Theory  
Frequency-Based Substructuring (FBS) systematically analyzes the dynamic behavior of reconfigurable dynamic systems by breaking down the system into smaller, more manageable subsystems. This methodology uses frequency-domain coupling to integrate subsystem transfer functions obtained either numerically or experimentally, enabling the creation of hybrid models. By measuring FRFs at critical points, particularly at interface coordinates, engineers can develop effective reduced-order models that accurately capture the subsystem interactions without requiring detailed geometrical complexity. Figure 1: Flowchart describing the road map for modeling of reconfigurable dynamic systems using generalized coupling method. The FRF or receptance matrix can be measured experimentally through experimental methods such as impulse hammers or shakers for structure-borne vibration analysis, and loudspeakers or volume velocity sources for airborne noise analysis. These measurements are limited to disassembled subsystems, improving access to connection points and reducing the overall measurement burden. In real-world situations where there are lots of interference signals, FRFs are calculated using power spectral density (PSD) and cross-power spectral density (CPSD), which mitigate noise and improve precision. For structural vibrations, a dynamic system comprising interconnected subsystems is governed by the equilibrium between externally applied forces and internal forces. The equation of motion for each subsystem is as follows:  
\[[\mathrm{M}^{(\mathrm{i})}]\Big{\{}\widetilde{\mathrm{x}}^{(\mathrm{i})} \Big{\}}+[\mathrm{C}^{(\mathrm{i})}]\Big{\{}\dot{\mathrm{x}}^{(\mathrm{i})} \Big{\}}+[\mathrm{K}^{(\mathrm{i})}]\Big{\{}\dot{\mathrm{x}}^{(\mathrm{i})} \Big{\}}=\Big{\{}\dot{\mathrm{f}}^{(\mathrm{i})}\Big{\}} \tag{1}\]  
where \([\mathrm{M}^{(\mathrm{i})}]\), \([\mathrm{C}^{(\mathrm{i})}]\), and \([\mathrm{K}^{(\mathrm{i})}]\) represent the mass, damping, and stiffness matrix of i-th subsystem, respectively. The vectors \(\Big{\{}\dot{\mathrm{x}}^{(\mathrm{i})}\Big{\}}\) and \(\Big{\{}\dot{\mathrm{f}}^{(\mathrm{i})}\Big{\}}\) denote the generalized response and external excitation force, respectively. By applying the Fourier transform, the dynamic response in the frequency domain can be expressed as follows:  
\[ \begin{array}{c}\Big{[}Z^{(\mathrm{i})}(\omega)\Big{]}[\mathrm{X}^{(\mathrm{ i})}(\omega)\Big{\}}=\Big{\{}\dot{\mathrm{F}}^{(\mathrm{i})}(\omega)\Big{\}}\\ \Big{\{}\dot{\mathrm{X}}^{(\mathrm{i})}(\omega)\Big{\}}=\Big{[}\mathrm{H}^{( \mathrm{i})}(\omega)\Big{]}\{\dot{\mathrm{F}}^{(\mathrm{i})}(\omega)\Big{\}} \end{array}  \tag{2}\]  
where \([\mathrm{Z}^{(\mathrm{i})}(\omega)]=[\mathrm{K}^{(\mathrm{i})}]+\mathrm{j} \omega[\mathrm{C}^{(\mathrm{i})}]-\omega^{2}[\mathrm{M}^{(\mathrm{i})}]\) is the dynamic stiffness (impedance matrix), and \(\Big{[}\mathrm{H}^{(\mathrm{i})}(\omega)\Big{]}=\Big{[}Z^{(\mathrm{i})}( \omega)\Big{]}^{-1}\) represents the FRF or receptance matrix for the i-th substructure. An element of the FRF matrix, \(\mathrm{H}_{\mathrm{ij}}(\omega)\), is defined as the partial derivative of the response at the i-th generalized coordinate with respect to the excitation force applied at the j-th point:  
\[\mathrm{H}_{\mathrm{ij}}(\omega)=\frac{\partial\mathrm{X}_{\mathrm{i}}}{ \partial\mathrm{F}_{\mathrm{j}}} \tag{3}\]  
Using partitioning method, the equations of motion for the j-th substructure of the system are expressed in matrix notation as follows:  
\[ \begin{array}{c}\left\{\mathrm{X}_{\mathrm{i}}^{(\mathrm{j})} \right\}=[\mathrm{H}^{(\mathrm{i})}(\omega)]\Bigg{\{}\dot{\mathrm{F}}_{\mathrm{ i}}^{(\mathrm{j})}\Bigg{\}}\quad\text{where}\quad\Big{[}\mathrm{H}^{(\mathrm{j})}( \omega)\Big{]}=\begin{bmatrix}\mathrm{H}_{\mathrm{ii}}^{\mathrm{j}}&\mathrm{H}_ {\mathrm{ic}}^{\mathrm{j}}\\ \mathrm{H}_{\mathrm{ci}}^{\mathrm{j}}&\mathrm{H}_{\mathrm{cc}}^{\mathrm{j}} \end{bmatrix}\end{array}  \tag{4}\]  
In this context, subscripts i and c denote internal and connection DoFs, respectively.