## 4 Numerical Examples
### A Spring-Mass System  
The second example is the spring-mass system (shown in Figure 2) used by [PERSON] [37], which is employed to further verify the proposed second-order sensitivity formula of eigenvector. The nominal model of the system has the parameters \(k_{1}=k_{4}=2\), \(k_{2}=k_{3}=1\) and \(m_{1}=m_{2}=m_{3}=1\).  
Two parameters will be allowed to vary, namely \(m_{1}\) (equivalent to \(p_{1}\)) and \(k_{2}\) (equivalent to \(p_{2}\)). The mass and stiffness matrices and their nonzero derivatives are  
\[M=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]\text{, }K=\left[\begin{array}{ccc}3&-1&0\\ -1&2&-1\\ 0&-1&3\end{array}\right] \tag{42}\]  
\[\frac{\partial M}{\partial p_{1}}=\left[\begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&0\end{array}\right]\text{, }\frac{\partial K}{\partial p_{2}}=\left[ \begin{array}{ccc}1&-1&0\\ -1&1&0\\ 0&0&0\end{array}\right] \tag{43}\]  
The first eigenvalue and eigenvector can be computed as  
\[\lambda_{1}=1\text{, }\phi_{1}=(0.4082\text{, }0.8165\text{, }0.4082)^{T} \tag{44}\]  
Using Equation (24), \(\frac{\partial\phi_{1}}{\partial p_{1}}\) and \(\frac{\partial\phi_{1}}{\partial p_{2}}\) can be calculated as  
\[\frac{\partial\phi_{1}}{\partial p_{1}}=(0.1134\text{, }-0.1134\text{, }-0.0907)^{T} \tag{45}\]  
\[\frac{\partial\phi_{1}}{\partial p_{2}}=(0.1928\text{, }-0.0907\text{, }-0.0113)^{T} \tag{46}\]  
Figure 2: Spring-mass system for example 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(\frac{\partial\phi_{2}}{\partial p_{1}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) & \(\frac{\partial\phi_{2}}{\partial p_{2}}\) \\
**DOF** & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** \\ & **Proposed** & **Superposition** & **Proposed** & **Superposition** & **Proposed** & **Superposition** & **Proposed** & **Superposition** \\ & **Method** & **Method** & **Method** & **Method** & **Method** & **Method**) & **Method** & **Method**) \\ \hline
1 & 1.5 & 1.5 & –1.1 & –1.1 & –0.5 & –0.5 & 0.8 & 0.8 \\
2 & 0.8 & 0.8 & 1.3 & 1.3 & 2.3 & 2.3 & 3.6 & 3.6 \\
3 & –0.2 & –0.2 & 5.5  
Using Equation (35), the second-order sensitivities of eigenvalue can be obtained as  
\[\frac{\partial^{2}\lambda_{1}}{\partial p_{1}^{2}}=-0.0648,\frac{\partial^{2} \lambda_{1}}{\partial p_{2}^{2}}=-0.2315,\frac{\partial^{2}\lambda_{1}}{ \partial p_{1}\partial p_{2}}=-0.1852 \tag{47}\]  
Using Equation (37), the second-order sensitivities of eigenvector can be obtained as  
\[\frac{\partial^{2}\phi_{1}}{\partial p_{1}^{2}}=(-0.0410,-0.1084,-0.0523)^{T} \tag{48}\]  
\[\frac{\partial^{2}\phi_{1}}{\partial p_{2}^{2}}=(0.2593,-0.0788,0.0098)^{T} \tag{49}\]  
\[\frac{\partial^{2}\phi_{1}}{\partial p_{1}\partial p_{2}}=(0.0611,0.0674,0.078 1)^{T} \tag{50}\]  
The above results are the same as those obtained by [PERSON] in reference [37]. It has been shown that the proposed method is exact and reliable.