## 3 An Improved Objective Function Based on SVDM
### The Improved Objective Function  
The traditional objective function is improved by using the fact that the amplitude of the frequency response reaches its local maxima at the natural frequencies. Given the vector \(\mathbf{\mu}\) of the damage factors, let \(h_{\alpha q}(\mathbf{\mu},\omega)\) denote the frequency response of the \(\alpha\)-th degree of freedom (DOF) under the excitation \(q(t)\). It can be expressed as Equation (3):  
\[h_{\alpha q}(\mathbf{\mu},\omega)=\sum_{i}\frac{\varphi_{\alpha i}(\mathbf{\mu})\mathbf{ \varphi}_{i}^{T}(\mathbf{\mu})\mathbf{B}_{q}}{k_{i}(\mathbf{\mu})-\omega^{2}m_{i}(\mathbf{\mu}) +j\omega c_{i}(\mathbf{\mu})} \tag{3}\]  
where vector \(\varphi_{\alpha i}(\mathbf{\mu})\) is the \(\alpha\)-th DOF of the \(i\)-th mode shape; \(k_{i}(\mathbf{\mu}),m_{i}(\mathbf{\mu}),c_{i}(\mathbf{\mu})\) are respectively the \(i\)-th mode generalized stiffness, mass and damping; and \(\mathbf{B}_{q}\) is the load position matrix of excitation \(q(t)\). Denote the contribution of the \(i\)-th mode to the frequency response by \(h_{\alpha q}^{(i)}(\mathbf{\mu},\omega)=\frac{\varphi_{\alpha i}(\mathbf{\mu})\mathbf{ \varphi}_{i}^{T}(\mathbf{\mu})\mathbf{B}_{q}}{k_{i}(\mathbf{\mu})-\omega^{2}m_{i}(\mathbf{\mu}) +j\omega c_{i}(\mathbf{\mu})}\); the process of summation of these modal contributions to the overall frequency response is illustrated in Figure 1.  
When the frequency \(\omega\) is equal to the \(i\)-th damped frequency \(\omega_{di}(\mathbf{\mu})\), i.e., \(\omega=\omega_{di}(\mathbf{\mu})\) (where \(\omega_{di}(\mathbf{\mu})=\omega_{i}(\mathbf{\mu})\sqrt{1-\xi_{i}^{2}}\), \(\xi_{i}=\frac{c_{i}}{2\sqrt{k_{i}m_{i}}}\)), the amplitude of the frequency response \(h_{\alpha q}(\mathbf{\mu},\omega)\) reaches its local maximum value, and if the damping ratio \(\xi_{i}\) is small, then \(\omega_{di}(\mathbf{\mu})\approx\omega_{i}(\mathbf{\mu})\). Therefore, when \(\omega=\omega_{i}(\mathbf{\mu})\), the frequency response \(h_{\alpha q}(\mathbf{\mu},\omega)\) has two following characteristics:  
\[\text{Involving the mode shape }\mathbf{\varphi}_{i}(\mathbf{\mu})\text{:} \tag{1}\]  
Consider the special case \(\omega=\omega_{i}(\mathbf{\mu})\) and note that \(h_{\alpha q}(\mathbf{\mu},\omega_{i}(\mathbf{\mu}))\approx h_{\alpha q}^{(i)}(\mathbf{\mu},\omega_{i}(\mathbf{\mu}))\), that is, at the \(i\)-th natural frequency the frequency response is dominated by the contribution of the \(i\)-th mode. Thus, Equation (3) can be approximated and expressed as Equation (4). It can be seen from Equation (4) that the structural \(i\)-th mode shape \(\mathbf{\varphi}_{i}(\mathbf{\mu})\) is approximately proportional to \(\mathbf{H}_{q}(\mathbf{\mu},\omega_{i}(\mathbf{\mu}))\), where \(\mathbf{H}_{q}=\left[ \begin{array}{cc}h_{1q}&h_{2q}&\cdots&h_{n_{\alpha}q}\\ \end{array} \right]^{T}\), i.e., \(\mathbf{\varphi}_{i}(\mathbf{\mu})\propto\mathbf{H}_{q}(\mathbf{\mu},\omega_{i}(\mathbf{\mu}))\). When \(\mathbf{\mu}\rightarrow\overline{\mathbf{\mu}}\), then \(\omega_{i}(\mathbf{\mu})\rightarrow\overline{\mathbf{\omega}}_{i}\), and \(\mathbf{H}_{q}(\mathbf{\mu},\overline{\omega}_{i})\rightarrow\mathbf{H}_{q}(\overline{ \mathbf{\mu}},\overline{\omega}_{i})\propto\overline{\mathbf{\varphi}}_{i}\). So if \(\mathbf{\mu}\rightarrow\overline{\mathbf{\mu}}\), then \(\mathbf{\varphi}_{i}(\mathbf{\mu})\propto\mathbf{H}_{q}(\mathbf{\mu},\overline{\omega}_{i})\), and \(\mathbf{\varphi}_{i}(\bmInvolving the natural frequency:  
It can be seen from Figure 2 that \(\left|h_{aq}(\mathbf{\mu},\omega)\right|\) reaches its local maximum value at \(\omega=\omega_{i}(\mathbf{\mu})\), and so \(-\left|h_{aq}(\overline{\mathbf{\mu}},\omega)\right|\) attains the local minimum value at \(\omega=\overline{\omega}_{i}\).