## 3 FRF Estimation  
Applying the Laplace transform of Equation (25), we acquire the transfer function \(\mathbf{H}(s)\) from the disturbance to the image motion, as follows:  
\[\left\{ \begin{array}{l}\mathbf{P}(s)=\mathbf{H}(s)\mathbf{F}(s)\\ \mathbf{H}(s)=\mathbf{C}_{\rm p}(s\mathbf{I}-\mathbf{A}_{\rm s})^{-1}\mathbf{B}_{\rm s},\end{array} \right. \tag{26}\]  
where \(\mathbf{P}(s)\) and \(\mathbf{F}(s)\) are the Laplace transform of \(\mathbf{p}(t)\) and \(\mathbf{F}(t)\), respectively. Let \(s=j\omega\), then, we obtain the FRF \(\mathbf{H}(\omega)\). The relationship between the disturbance \(\mathbf{F}(\omega)\) and the image motion \(\mathbf{P}(\omega)\) in the frequency domain is:  
\[\mathbf{P}(\omega)=\mathbf{H}(\omega)\mathbf{F}(\omega), \tag{27}\]  
where \(\mathbf{F}(\omega)=[F_{\rm x}(\omega),F_{\rm y}(\omega),F_{\rm z}(\omega),M_{\rm x }(\omega),M_{\rm y}(\omega),M_{\rm z}(\omega)]^{\rm T}\), and \(\mathbf{H}(\omega)\) is a 2 \(\times\) 6 matrix. In fact, when an RWA is installed on a flexible supporting structure, its disturbances excite the modes of the structure, which, in turn, perturb the RWA, itself, thus, creating coupled dynamics [24]. As a result, the loads that are present at the interface vary from those that would be found in a hard-mounted boundary condition [25; 26; 27]. The forces and moments \(\mathbf{f}_{\rm C}\) that are actually transmitted at the interface between the source and the supporting structure can be described as:  
\[\mathbf{f}_{\rm C}=\mathbf{f}_{\rm B}-\mathbf{D}_{\rm wa}\tilde{\mathbf{x}}_{\rm C}, \tag{28}\]  
where \(\mathbf{f}_{\rm B}\) is the forces and moments measured in the hard-mounted boundary condition, \(\mathbf{D}_{\rm wa}\) is the dynamic characteristics (inertia, stiffness, and damping) of the source and \(\tilde{\mathbf{x}}_{\rm C}\) is the vector of the coupled acceleration at the interface. The acceleration vector \(\tilde{\mathbf{x}}_{\rm C}\) can be determined if the supporting structure response at the interface node point (i.e., the dynamic mass of the supporting structure \(\mathbf{D}_{\rm ss}\)) is provided:  
\[\tilde{\mathbf{x}}_{\rm C}=\mathbf{D}_{\rm ss}^{-1}\mathbf{f}_{\rm C}. \tag{29}\]  
Substituting Equation (29) into Equation (28), Equation (28) can be rewritten as:  
\[\mathbf{f}_{\rm C}=\mathbf{f}_{\rm B}-\mathbf{D}_{\rm wa}\mathbf{D}_{\rm ss}^{-1}\mathbf{f}_{\rm C}. \tag{30}\]  
Rearrange Equation (30) and obtain \(\mathbf{f}_{\rm C}\) as:  
\[\mathbf{f}_{\rm C}=\Big{(}\mathbf{I}+\mathbf{D}_{\rm wa}\mathbf{D}_{\rm ss}^{-1}\Big{)}^{-1} \mathbf{f}_{\rm B}, \tag{31}\]  
where \(\mathbf{I}\) is a 6 \(\times\) 6-unit matrix. Equation (31) can also be reformulated in terms of the Power Spectral Density (PSD) as:  
\[\mathbf{\Phi}_{\rm C}=\Big{(}\mathbf{I}-\mathbf{D}_{\rm wa}\mathbf{D}_{\rm ss}^{-1}\Big{)}^{- 1}\mathbf{\Phi}_{\rm B}\Big{(}\mathbf{I}-\mathbf{D}_{\rm wa}\mathbf{D}_{\rm ss}^{-1}\Big{)}^{- \rm H}, \tag{32}\]  
where \(\mathbf{f}_{\rm C}\) and \(\mathbf{f}_{\rm B}\) become \(\mathbf{\Phi}_{\rm C}\) and \(\mathbf{\Phi}_{\rm B}\), respectively, and \(\rm H\) denotes the Hermitian transpose. The PSD output of the image motion \(\mathbf{\Phi}_{\rm out}\) can be obtained by multiplying the \(\mathbf{\Phi}_{\rm C}\) by the transfer function matrix, which links the interface node point to the image motion, \(\mathbf{T}_{\rm C-out}\), to obtain:  
\[\mathbf{\Phi}_{\rm out}=\mathbf{T}\mathbf{F}_{\rm C-out}\Big{(}\mathbf{I}-\mathbf{D}_{\rm wa}\bm {D}_{\rm ss}^{-1}\Big{)}^{-1}\mathbf{\Phi}_{\rm B}\Big{(}\mathbf{I}-\mathbf{D}_{\rm wa}\bm {D}_{\rm ss}^{-1}\Big{)}^{-\rm H}\mathbf{T}\mathbf{F}_{\rm C-out}^{\rm H}. \tag{33}\]  
Due to the complexity of \(\mathbf{D}_{\rm wa}\) and \(\mathbf{D}_{\rm ss}\), many challenging experiments needed to be planned, similar to reference [25].