## 2 The Strain Transmissibility Function (TF)  
### The Strain TF of the Intact Structure under Unknown Ambient Excitation  
Transmissibility function (TF) is defined as the Fourier transform or the spectrum ratio of two measurement responses. For an n-DOFs structure subject to unknown ambient excitation \(\mathbf{P}=[P_{1},P_{2},\dots,P_{m}]^{\mathrm{T}}\), the basic motion equation is as follows:  
\[\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{I}\mathbf{P}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) represent the mass, damping, and stiffness matrices of the structural system, respectively; \(\ddot{\mathbf{x}}(t)\), \(\dot{\mathbf{x}}(t)\), and \(\mathbf{x}(t)\) are acceleration, velocity, and displacement response vectors; \(\mathbf{I}\) is the influence matrix of ambient excitation; \(\mathbf{P}(t)\) is assumed to be white noise to the intact structure.  
In the frequency domain, Equation (1) is transformed as:  
\[\mathbf{x}(\omega)=\mathbf{H}(i\omega)\mathbf{I}\mathbf{P}(\omega) \tag{2}\]  
where \(\mathbf{H}(i\omega)=\left(\omega^{2}\mathbf{M}+i\omega\mathbf{C}+\mathbf{K}\right)^{-1}\). Compared with the displacement response, the strain response as a local feature is more sensitive to small deviations. The strain in this study refers to normal strains because the strain gauges only measure normal strains in a certain direction. According to the derivation results in [38], the relationship between displacement and strain is expressed by Equation (3).  
\[\varepsilon_{e}=\mathbf{a}_{e}\mathbf{x};\ \mathbf{a}_{e}=\mathbf{U}_{e}\mathbf{Y}_{e}\mathbf{W}_{e} \tag{3}\]  
where \(\mathbf{a}_{e}\) is the transformation vector, which is described as a \(1\times n\) vector. Furthermore, \(\mathbf{a}_{e}\) is viewed as the multiplication of the selection matrix \(\mathbf{W}_{e}\), the coordinate system transformation matrix \(\mathbf{Y}_{e}\), and the shape function \(\mathbf{U}_{e}\).  
According to the definition of the TF, the TF is expressed as the spectral ratio of two structural responses of the measurement point \(i\) and the reference point \(j\). Since the power spectral density of white noise is constant, the strain TF under ambient excitation is shown in Equation (4):  
\[T_{i,j}^{e}(\omega)=\frac{G_{i,j}^{e}(\omega)}{G_{j,i}^{e}(\omega)}=\frac{ \varepsilon_{i}(\omega)\varepsilon_{j}^{e}(\omega)}{\varepsilon_{j}(\omega) \varepsilon_{j}^{e}(\omega)}=\frac{S_{0}\sum\limits_{r=1}^{n}\sum\limits_{k=1} ^{n}\sum\limits_{f=1}^{m}\left(a_{i}\right)_{r}H_{rk}(\omega)I_{bf}\sum \limits_{r=1}^{n}\sum\limits_{k=1}^{n}\sum\limits_{f=1}^{m}\left(a_{i}\right) _{r}H_{rk}^{*}(\omega)I_{bf}}{S_{0}\sum\limits_{r=1}^{n}\sum\limits_{k=1}^{n} \sum\limits_{f=1}^{m}\left(a_{j}\right)_{r}H_{rk}(\omega)I_{bf}\sum\limits_{r =1}^{n}\sum\limits_{k=1}^{n}\sum\limits_{f=1}^{m}\left(a_{j}\right)_{r}H_{rk}^ {*}(\omega)I_{bf}} \tag{4}\]  
where \(G_{i,j}^{e}(\omega)\) is the strain cross-power spectrum of the structural responses of point \(i\) and the reference point \(j\); \(G_{j,j}^{e}(\omega)\) is the strain auto-power spectrum of the reference point \(j\); \(\varepsilon_{j}^{e}(\omega)\) means the complex conjugate of \(\varepsilon_{j}(\omega)\); \(S_{0}\) is a constant, which is the power spectral density of the excitation; \(H_{rk}^{*}(\omega)\) is the frequency response function; and \(H_{rk}^{*}(\omega)\) is the complex conjugate of \(H_{rk}(\omega)\). From Equation (4), it is noted that \(T_{i,j}^{e}(\omega)\) is not influenced by the ambient excitations and is only related to the structural characteristics.