## 2 Transmissibility Function  
The equation of motion of a multi degree-of-freedom (DOF) system subjected to forced vibration is governed by the following differential equation:  
\[M\ddot{x}+C\dot{x}+Kx=F(t) \tag{1}\]  
where \(M,C,K\) are the mass, damping and stiffness matrices and \(x,\ \dot{x},\ \ddot{x}\) are the displacement, velocity and acceleration responses, respectively. Assuming harmonic excitation forces, the response \(X(\omega)\) in the frequency domain as a function of the excitation, \(F(\omega)\) can be expressed as:  
\[X(\omega)=H(\omega)F(\omega) \tag{2}\]  
where \(H(\omega)\), the frequency response function is defined as:  
\[H(\omega)=\left[\omega^{2}M+i\omega C+K\right] \tag{3}\]  
Focusing on two sets of coordinates \(i\) and \(j\) and partitioning the respective parts of the above expression, for a single excitation force applied at coordinate \(k\), one can write  
\[X_{i}(\omega)=H_{ik}(\omega)F_{k}(\omega) \tag{4}\]  
\[X_{j}(\omega)=H_{jk}(\omega)F_{k}(\omega) \tag{5}\]  
Eliminating \(F_{k}(\omega)\), one gets  
\[X_{i}(\omega)=\frac{H_{ik}(\omega)}{H_{jk}(\omega)}X_{j}(\omega) \tag{6}\]  
The local transmissibility between the two coordinates \(i\) and \(j\), in reference to input location \(k\), can be defined using two responses directly as:  
\[T_{ij}^{k}(\omega)=\frac{X_{i}\left(\omega\right)}{X_{j}(\omega)} \tag{7}\]  
The most commonly used approach to estimate transmissibility is through power spectral density functions, \(G\):  
\[T_{ij}^{k}(\omega)=\frac{X_{i}(\omega)\ X_{k}\left(\omega\right)}{X_{j}(\omega )\ X_{k}(\omega)}=\frac{G_{ik}(\omega)}{G_{jk}(\omega)} \tag{8}\]  
Equation (8) is the ratio of the response cross spectral densities between coordinates \(i\) and \(j\), defining how vibration is transmitted between these coordinates as a function of frequency. It should be noted that \(G\) becomes an auto spectrum when \(k=i\) or \(k=j\) and in that case, it will be analogous to coherence in frequency response functions. It should also be noted that the applied force does not necessarily need to be harmonic. Although the derivation assumed harmonic forces to relate two response spectra of like variables, the expressions still remain valid for other types of excitation including impulse and random loads [49]. The local TF obtained in this manner depends only on the location of the input excitation and does not require the input to be measured. Any change in the structural properties will clearly be reflected in the TF, which is expected to be quite sensitive to even local damage, as it characterizes the vibration response across-small sections of the structure. Among possible loading (unmeasured) conditions that can be implemented when using TF for damage detection, a potential alternative with practical significance is to use a single input moved to different points on the structure. For each excitation location, a transmissibility matrix of size \(N\times N\), elements of which are functions of frequency can be formed for all combinations of outputs between location pairs. The selection of appropriate features as damage indicators is essential, not only for reducing the dimensionality of the original dataset but also for eliminating the unrelated variables to ensure the success of the proposed methodology. To 'close-in' on the damaged region, typically the two successive pairs of coordinates are selected, and the transmissibilities corresponding to the two adjacent coordinates along the structure are interrogated. This corresponds to the first-upper diagonal of the matrix with entries \(T_{12},\ T_{23},\ldots T_{k,k+1}\). Furthermore, to limit the dependence of the TF outside the domain enclosed by the output coordinates, the input location is also arranged such that it is collocated with one of the output coordinates. In this study, the excitation is always applied at the latter DOF in the selected pair (excitation at DOF \(i+1\) for the transmissibility \(T_{i+1}\)), following a consistent directional pairing.