## 3 Physical Parameter Identification under Random Excitation  
For an \(R\) DOF TV system, the vibration differential equation can be defined as follows:  
\[M(t)\ddot{x}(t)+C(t)\dot{x}(t)+K(t)x(t)=f(t) \tag{38}\]  
in which \(M(t)\), \(C(t)\), and \(K(t)\) are the TV mass, damping, and stiffness matrices; \(\ddot{x}(t)\), \(\dot{x}(t)\) and \(x(t)\) are the system acceleration, velocity, and displacement vectors; and \(f(t)\) are the excitation vectors exerted on the system. On the basis of modal superposition theory, the system acceleration vector can be regarded as the sum of multiple modal components. The acceleration vector of \(r\)-th DOF can be expressed as \(\ddot{x}_{ri}(t)(r=1,2,\ldots,R)\). Under the condition that only the acceleration signal is known, the PCT integral algorithm is employed, and Equation (38) is rewritten as follows:  
\[M(t)\ddot{x}(t)+C(t)\overline{\dot{x}}(t)+K(t)\overline{\dot{x}}(t)=f(t) \tag{39}\]  
Further, Equation (39) can be updated with amplitude \(A_{pq}\) and phase as follows:  
\[[M(t)\;C(t)\;K(t)] \begin{bmatrix}A_{pq}^{M}(t)\\ A_{pq}^{K}(t)\\ A_{pq}^{K}(t)\end{bmatrix}\begin{bmatrix}e^{\mathrm{j}|\omega_{1}(t-t_{0})+ \sum_{i=1}^{m}\frac{\alpha_{1}(t-t_{0})^{2}}{(t-t_{0})^{2}}}\\ e^{\mathrm{j}|\omega_{2}(t-t_{0})+\sum_{i=1}^{m}\frac{\alpha_{2}(t-t_{0})^{3} }{(t-t_{1})}}\\ \vdots\\ e^{\mathrm{j}|\omega_{n}(t-t_{0})+\sum_{i=n}^{m}\frac{\alpha_{n}(t-t_{0})^{2 }}{(t+1)}}\end{bmatrix} =f(t) \tag{40}\]  
The amplitude \(A_{pq}\) is given as follows:  
\[\left\{ \begin{array}{l}A_{pq}^{M}(t)=\frac{\mathrm{PCT}_{\mathrm{j}^{q}}( \ddot{x}_{pq}(t))}{\mathrm{C}_{h}\left[-\dot{x}_{q}-\left(\omega_{q}+\sum_{q =1}^{N}c_{q}(t-t_{0})\right)^{q+1}\right]}\\ A_{pq}^{C}(t)=\frac{\mathrm{PCT}_{\mathrm{j}^{q}}(\ddot{x}_{pq}(t))}{\mathrm{C}_ {h}\left[-\dot{x}_{q}-\left(\omega_{q}+\sum_{q=1}^{N}c_{q}(t-t_{0})\right)^{q+ 1}\right]}\\ A_{pq}^{K}(t)=\frac{\mathrm{PCT}_{\mathrm{j}^{q}}(\ddot{x}_{pq}(t))}{\mathrm{C}_ {h}\left[-\dot{x}_{q}-\left(\omega_{q}+\sum_{q=1}^{N}c_{q}(t-t_{0})\right)^{q+ 1}\right]}\end{array} \right.,(p,q=1,2,\cdots,N) \tag{41}\]Expectation operations are performed on both sides, and Equation (40) can be updated as follows:  
\[ \begin{bmatrix}M(t)\ C(t)\ K(t)\left[\begin{array}{c}A_{pq}^{M}(t)\\ A_{pq}^{K}(t)\\ A_{pq}^{K}(t)\end{array}\right]^{\mathrm{T}}\left[\begin{array}{cc}\left[ \begin{array}{cc}e^{\mathrm{j}[\omega_{1}(t-t_{0})+\sum_{i=1}^{n}\frac{\alpha_ {1}(t-t_{0})^{2}}{i+1}]}\\ e^{\mathrm{j}[\omega_{2}(t-t_{0})+\sum_{i=2}^{n}\frac{\alpha_{2}(t-t_{0})^{3}}{ i+1}]}\\ \vdots\\ e^{\mathrm{j}[\omega_{n}(t-t_{0})+\sum_{i=n}^{n}\frac{\alpha_{1}(t-t_{0})^{2}}{ i+1}]}\end{array}\right]\left[\begin{array}{c}\left[\begin{array}{c}e^{ \mathrm{j}[\omega_{1}(t-t_{0})+\sum_{i=1}^{n}\frac{\alpha_{1}(t-t_{0})^{2}}{ i+1}]}\\ e^{\mathrm{j}[\omega_{2}(t-t_{0})+\sum_{i=n}^{n}\frac{\alpha_{2}(t-t_{0})^{3}}{ i+1}]}\\ \vdots\\ e^{\mathrm{j}[\omega_{n}(t-t_{0})+\sum_{i=n}^{n}\frac{\alpha_{1}(t-t_{0})^{2}}{ i+1}]}\end{array}\right]^{\mathrm{T}}\left[\begin{array}{c}\left[ \begin{array}{cc}e^{\mathrm{j}[\omega_{1}(t-t_{0})+\sum_{i=1}^{n}\frac{\alpha_ {1}(t-t_{0})^{2}}{i+1}]}\\ e^{\mathrm{j}[\omega_{2}(t-t_{0})+\sum_{i=n}^{n}\frac{\alpha_{2}(t-t_{0})^{3}}{ i+1}]}\\ \vdots\\ e^{\mathrm{j}[\omega_{n}(t-t_{0})+\sum_{i=n}^{n}\frac{\alpha_{1}(t-t_{0})^{2}}{ i+1}]}\end{array}\right]\end{bmatrix} ^{\mathrm{T}} \tag{42}\]  
Here, random Gaussian white noise is set as the system excitation force, and its correlation function is defined as follows:  
\[E\big{\{}f_{m}(t)\cdot f_{n}(t+\tau)\big{\}}= \begin{cases}\lambda_{m}\delta( \tau)&m=n\\ 0&m=n\end{cases} ,(m,n=1,2,\ldots,n) \tag{43}\]  
Therefore, the right side of Equation (42) can be rewritten as follows:  
\[E\Big{\{}f(t)\cdot f(t+\tau)^{\mathrm{T}}\Big{\}}= \begin{bmatrix}\lambda_{1} &0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{n}\end{bmatrix} ,\tau=0 \tag{44}\]  
Based on Equations (37) and (43), the correlation function matrix can be obtained as follows:  
\[E\left\{