## 3 Structural Dynamic Equation  
Based on the finite element theory, the elastic wedge can be decomposed into a system with finite degrees of freedom, and the structural dynamic equation can be expressed as:  
\[[M]\Big{\{}\ddot{U}\Big{\}}+[C]\{\dot{U}\}+[K]\{U\}=\{P\} \tag{5}\]  
where \([M]\) is the mass matrix of the structure; \([C]\) is the damping matrix of the structure; \([K]\) is the stiffness matrix of the structure; \(\{U\}\) is the nodal displacement array of the structure; \(\{P\}\) is the distribution force equivalent nodal array, and the distribution force refers to the impact pressure on the wedge in the present study.  
Through the Eigen analysis of the structural dynamic system of Equation (5), modal shapes \(\{D_{r}\}\) of the structure in air (dry modes) can be obtained as follows:  
\[\{D_{r}\}=\{D_{r1},D_{r2},\cdots D_{rN}\}^{T} \tag{6}\]  
where \(N\) is the number of nodes for the discrete structure model; \(r\) is the order of the vibration mode; \(D_{rj}\) is the displacement vector of the \(j\)-th node in the \(r\)-th order mode.  
In the 3D space coordinate system \(o-xyz\), \(\{D_{r}\}\) can be expressed as  
\[D_{rj}=\big{\{}u_{r},v_{r},w_{r},\theta_{xr},\theta_{yr},\theta_{xr}\big{\}}_ {j}^{T} \tag{7}\]  
where \(u_{r},v_{r},w_{r}\) are the linear displacement along the \(ox\), \(oy\) and \(oz\) axes, respectively; \(\theta_{xr},\theta_{yr},\theta_{zr}\) are the angular displacement around the \(ox\), \(oy\) and \(oz\) axes, respectively.  
The modal number of elastic structures is actually infinite, and several lower order modes usually represent the primary contribution to structural responses. When \(m\) modes are selected, the displacement shapes of the discrete node system can be expressed as:\[[D]=[\{D_{1}\},\{D_{2}\},\cdots,\{D_{m}\}] \tag{8}\]  
By using the normal mode method, the node displacement of the system under the impact load can be expressed as:  
\[\{U\}=[D]\{p\}=\sum\limits_{r=1}^{m}\{D_{r}\}p_{r}(t) \tag{9}\]  
where \(\{p\}\) is the principal coordinate array; \(p_{r}(t)\) represents the contribution of the \(r\)-th mode to the structural response.  
Substituting Equation (9) into Equation (5) and multiplying the two sides by \(\left[D\right]^{T}\), the principal coordinate equation of the structural system can be achieved:  
\[[a]\{\ddot{p}\}+[b]\{\dot{p}\}+[c]\{p\}=\{Z\} \tag{10}\]  
where \([a]\) is the generalized mass matrix of the structure, \([a]=\left[D\right]^{T}[M][D]\); \([b]\) is the generalized damping matrix of the structure, \([b]=[D]^{T}[C][D]\); \([c]\) is the generalized stiffness matrix of the structure \([c]=[D]^{T}[K][D]\); \([Z]\) is the generalized distribution force matrix, \([Z]=[D]^{T}\{p\}\).  
The generalized damping matrix of the structure is usually determined by experiments or experience. In the normal mode method, it is assumed diagonal such that \(b_{rr}=2\omega_{r}v_{r}a_{rr}\), where \(\omega_{r}\) is the natural frequency and \(v_{r}\) is the structural damping factor. The effect of structural damping is investigated by changing the structural damping factor in the following study.  
Thus, the structural dynamic equation, which takes the nodal displacement as an unknown variable, has been transformed into the structural dynamic equation with the principal coordinate as the unknown variable. After the computation of the generalized distribution force, the principal coordinate corresponding to each mode can be solved, and the structural response can be solved using the normal mode method.