## 2 Basic Theory  
### Initial Displacement Excitation Method  
The forced vibration motion equation of a damped multi-degree of freedom system is  
\[M\ddot{x}+C\dot{x}+Kx=F(t) \tag{1}\]  
where: \(M\) is a mass matrix, which is a positive definite matrix; \(C\) is a proportional damping matrix; \(K\) is a stiffness matrix, which is a positive definite or semi-positive definite matrix; \(F(t)\) is an external force vector. Through generalized eigenvalue analysis, the first \(n\)-order natural frequencies (\(\omega_{1}\), \(\omega_{2}\),..., \(\omega_{n}\)) and natural modes (\(\varphi_{1}\), \(\varphi_{2}\),..., \(\varphi_{n}\)) of the multi-degree-of-freedom system are obtained. The \(n\)-order mode vector can form the following mode matrix \(\phi\). \[\phi=(\varphi_{1}\ \varphi_{2}\ \cdots\varphi_{n}) \tag{2}\]  
According to the orthogonality of modes, any \(n\)-dimensional vibration of the system can be uniquely expressed as a linear combination of modes. \[x=\phi z=\sum_{r=1}^{n}\varphi_{r}z_{r} \tag{3}\]  
Among them, \(z_{r}(r=1,2,\ldots,n)\) is the generalized coordinate describing the motion of the system in the modal space, called the principal coordinate, and \(r\) represents the modal order. The array \(z\) composed of the principal coordinates of each order is the principal coordinate array. \[z=\left( \begin{array}{cccc}z_{1}&z_{2}&\cdots&z_{n}\end{array} \right)^{T} \tag{4}\]  
Substituting the principal coordinates established by the formula into the equation, and multiplying each item to the left by \(\boldsymbol{\phi}^{-1}\) can be obtained. \[M\ddot{z}+C\dot{z}+Kz=0 \tag{5}\]  
Assuming that \(C\) is proportional damping, \(n\) independent single-degree of freedom systems are obtained after modal decomposition. \[m\ddot{z}_{r}+c\dot{z}_{r}+kz_{r}=0 \tag{6}\]  
Assuming that the initial condition of the structure is an arbitrary overall initial displacement, namely. \[t=0:x(0)\
eq 0,\ \dot{x}(0)=0 \tag{7}\]  
At this time, the solution of Equation (6) is:  
\[z_{r}(t)=\frac{z_{r}(0)}{\sqrt{1-\xi_{r}^{2}}}e^{-\xi_{r}\omega_{r}t}\cos( \omega_{dr}t-\theta_{r}) \tag{8}\]  
Among them, \(\omega_{r}\) is the undamped natural frequency, \(\omega_{dr}\) is the natural frequency of the damping system, and \(\xi_{r}\) is the damping ratio. \(v_{2}\)) in the camera image coordinate system. See formulas (13) and (14) for details. \(Z_{c1}\) and \(Z_{c2}\) are the magnification coefficients in the camera imaging model and are related to the internal parameters of the camera. \[z_{c_{1}}\left[ \begin{array}{c}u_{1}\\ v_{1}\\ 1\end{array} \right]=M1\left[ \begin{array}{c}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{array} \right]=\left[ \begin{array}{cccc}m_{11}^{1}&m_{12}^{1}&m_{13}^{1} &m_{14}^{1}\\ m_{21}^{1}&m_{22}^{1}&m_{23}^{1}&m_{24}^{1}\\ m_{31}^{1}&m_{32}^{1}&m_{33}^{1}&m_{34}^{1}\end{array} \right]\left[ \begin{array} []{c}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{array} \right] \tag{13}\]  
\[z_{c_{2}}\left[ \begin{array}{c}u_{2}\\ v_{2}\\ 1\end{array} \right]=M2\left[ \begin{array}{c}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{array} \right]=\left[ \begin{array}{cccc}m_{11}^{2}&m_{12}^{2}&m_{13}^{2} &m_{14}^{2}\\ m_{21}^{2}&m_{22}^{2}&m_{23}^{2}&m_{24}^{2}\\ m_{31}^{2}&m_{32}^{2}&m_{33}^{2}&m_{34}^{2}\end{array} \right]\left[ \begin{array} []{c}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{array} \right] \tag{14}\]  
Simultaneous formulas (13) and (14), after eliminating the amplification factor \(Z\), four equations are obtained, as shown in formula (15), in which there are three unknowns (\(x_{w}\), \(y_{w}\), \(z_{w}\)), and the least square method is used to solve the statically indeterminate equations to obtain the three-dimensional space coordinate values (\(x_{w}\), \(y_{w}\), \(z_{w}\)) of the target point \(P\). \[\left\{