## 4 Numerical Examples  
### A Truss Structure  
A truss structure as shown in Figure 1 is used to demonstrate the proposed method and the calculation results are compared with those obtained by the modal superposition method. The structure consists of 58 bar elements with a total of 51 DOFs. The basic physical parameters are: modulus of elasticity E = 200 GPa, density \(\rho\) = 7800 kg/m\({}^{3}\), cross-sectional area A = \(3.14\times 10^{-4}\) m\({}^{2}\). According to the finite element model theory, structural global stiffness matrix \(K\) can be expressed as the sum of all elemental stiffness matrices, that is  
\[K=\sum_{i=1}^{58}p_{i}K_{i} \tag{41}\]  
where \(K_{i}\) and \(p_{i}\) are the \(i\)th elemental stiffness matrix and design parameter. Next, the proposed method (Equation (24)) is employed to calculate the eigenvector sensitivity \(\frac{\partial\phi_{j}}{\partial p_{i}}\). Without loss of generality, Tables 1 and 2 present some results (the data corresponding to the first 20 DOFs) of \(\frac{\partial\phi_{1}}{\partial p_{1}}\), \(\frac{\partial\phi_{1}}{\partial p_{2}}\), \(\frac{\partial\phi_{1}}{\partial p_{3}}\), \(\frac{\partial\phi_{1}}{\partial p_{4}}\), \(\frac{\partial\phi_{2}}{\partial p_{1}}\), \(\frac{\partial\phi_{2}}{\partial p_{2}}\), \(\frac{\partial\phi_{2}}{\partial p_{3}}\) and \(\frac{\partial\phi_{2}}{\partial p_{4}}\). For convenience of comparison, the results obtained by the modal superposition method (Equation (6)) are also given in Tables 1 and 2. One can see from Tables 1 and 2 that the proposed method can obtain the same calculation results as those of the modal superposition method. It has been shown that the proposed method is an exact method and easier to operate. Figure 1: A truss structure. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{3}{*}{**DOF**} & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) & \(\frac{\partial\phi_{1}}{\partial\phi_{1}}\) \\  & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** & **(The** & **(Modal** \\ \multirow{2}{*}{**Number**} & **Proposed** & **Superposition** & **Proposed** & **Superposition** & **Proposed** & **Superposition** & **Proposed** & **Superposition** \\  & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method** & **Method**) \\ \hline
1 & 0.3 & 0.3 & \(-0.5\) & \(-0.5\) & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
2 & 0.1 & 0.1 & 1.1 & 1.1 & 0.4 & 0.4 & 0.5 & 0.5 \\
3 & \(-0.0\) & \(-0.0\) & 3 & 3 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
4 & 0.1 & 0.1 & 0.8 & 0.8 & 0.3 & 0.3 & 0.6 & 0.6 \\
5 & \(-0.0\) & \(-0.0\) & 0.3 & 0.3 & \(-0.1\) & \(-0.1\) & \(0.1\) \\
6 & 0.1 & 0.1 & 2.9 & 2.9 & 0.3 & 0.3 & 0.5 & 0.5 \\
7 & \(-0.0\) & \(-0.0\) & 2.4 & 2.4 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
8 & 0.1 & 0.1 & 2.9 & 2.9 & 0.3 & 0.3 & 0.5 & 0.5 \\
9 & \(-0.0\) & \(-0.0\) & 0.9 & 0.9 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
10 & 0.1 & 0.1 & 3.8 & 3.8 & 0.3 & 0.3 & 0.4 & 0.4 \\
11 & \(-0.0\) & \(-0.0\) & 1.9 & 1.9 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
12 & 0.1 & 0.1 & 3.7 & 3.7 & 0.3 & 0.3 & 0.4 & 0.4 \\
13 & \(-0.0\) & \(-0.0\) & 1.4 & 1.4 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
14 & 0.1 & 0.1 & 3.7 & 3.7 & 0.2 & 0.2 & 0.3 & 0.3 \\
15 & \(-0.0\) & \(-0.0\) & 1.6 & 1.6 & \(-0.1\) & \(-0.1\) & \(0.1\) & \(0.1\) \\
16 & 0.1 & 0.1 & 3.7 & 3.7 & 0.2 & 0.2 & 0.3 & 0.3 \\
17 & \(-0.0\) & \(-0.0\) & 1.8 & 1.8 & \(-0.0\) & \(-0.0\) & \(0.1\) & \(0.1\) \\
18 & 0.1 & 0.1 & 3.0 & 3.0 & 0.2 & 0.2 & 0.2 & 0.2 \\
19 & \(-0.0\) & \(-0.0\) & 1.3 & 1.3 & \(-0.1\) & \(-0.1\) & \(0.0\) & \(0.0\) \\
20 & 0.1 & 0.1 & 3.0 & 3.0 & 0.2 & 0.2 & 0.2 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the calculation results for the first eigenvector sensitivity (\(\times 10^{-3}\)).