## 3 CWT-SSSM Feature-Based Damage Index
### Second-Order Strain Statistical Moments (SSSM)  
For a multi-degree-of-freedom linear structural system, under the action of Gaussian white noise \(F(t)\) with a mean value of \(0\) and power spectral density \(S_{0}\), its motion differential equation can be expressed as:  
\[M\ddot{x}+C\dot{x}+Kx=F(t) \tag{7}\]where \(\tilde{x}\), \(\tilde{x}\), and \(x\) are the acceleration, velocity, and displacement responses of the structure under white noise excitation, respectively. \(M\), \(C\), and, \(K\) are the mass, damping, and stiffness matrices of the structure, respectively. Adopting the assumption of Rayleigh damping, the displacement response \(x(t)\) can be solved by the modal decomposition method:  
\[x(t)=\sum_{i=1}^{n}z_{i}(t)\varphi_{i} \tag{8}\]  
where \(z_{i}(t)\) is the \(i\)th-order modal coordinate and \(\varphi_{i}\) is the \(i\)th-order mode shape. If the excitation force acts only on the \(r\)th degree of freedom, then the displacement frequency response function \(H_{sr}(\omega)\) on the \(s\)th degree of freedom can be expressed as:  
\[H_{sr}(\omega)=\sum_{i=1}^{n}\frac{\varphi_{si}\varphi_{ri}}{\overline{m}_{i} (\omega_{i}^{2}-\omega^{2}+2j\omega\omega_{i}\zeta_{i})} \tag{9}\]  
where \(\overline{m}_{i}\), \(\overline{\zeta}_{i}\), and \(\overline{k}_{i}\) are the \(i\)th-order modal mass, modal damping, and modal stiffness, respectively. \(\omega_{i}\), and \(\zeta_{i}\) are the \(i\)th-order natural frequency and damping ratio, respectively. It is worth mentioning that the strain response is more sensitive to local damage than the displacement response [29; 30; 31; 32] because the strain mode shape is a second-order differentiation of the displacement mode shape, while the differentiation operation further amplifies the abrupt changes in the curve caused by damage. Consequently, the strain response data are selected for the damage identification study. The strain response \(\varepsilon(t)\) can also be solved by the modal decomposition method. \[\varepsilon(t)=\sum_{i=1}^{n}z_{i}^{\prime}(t)\psi_{i} \tag{10}\]  
where \(z_{i}^{\prime}(t)\) is the \(i\)th-order modal coordinate corresponding to the strain, which represents the contribution of the \(i\)th-order strain mode \(\psi_{i}\) to the strain response, as well as \(z_{i}(t)=z_{i}^{\prime}(t)\). The strain frequency response function \(H_{sr}^{\varepsilon}(\omega)\) can be expressed as the following formula:  
\[H_{sr}^{\varepsilon}(\omega)=\sum_{i=1}^{n}\frac{\psi_{si}\varphi_{ri}}{ \overline{m}_{i}(\omega_{i}^{2}-\omega^{2}+2j\omega\omega_{i}\zeta_{i})} \tag{11}\]  
According to the theory related to random vibration, the second-order strain statistical moment \(SSSM_{s}\) (variance) of the \(s\)th degree of freedom can be written as the following formula:  
\[SSSM_{s}=\sigma^{2}=\int_{-\infty}^{+\infty}H_{sr}^{\varepsilon}(\omega) \overline{H_{sr}^{\varepsilon}(\omega)}S_{0}d\omega \tag{12}\]  
In this paper, we focus on the second-order statistical moment because the higher-order statistical moment is unstable [18]. In addition, it is noted that the \(H_{sr}^{\varepsilon}(\omega)\) in Formula (11) takes into account the contribution of all the strain mode shapes.