## 2 A Review of Structural Frequency Response Analysis Method  
Frequency response analysis can calculate the steady-state response of structures with dynamic loads at a certain frequency range. It can predict the continuous dynamical behavior of structures and verify whether the design can overcome the detrimental effects of resonance, fatigue, and forced vibrations. The frequency response equation of the structure with the harmonic load is [31]:  
\[\left(K+j\omega_{\mathrm{c}}\mathbf{C}-\omega_{\mathrm{c}}^{2}\mathbf{M}\right)\mathbf{U} (\omega_{\mathrm{c}})=\mathbf{F}, \tag{1}\]where \(j=\sqrt{-1}\), \(\mathbf{K}\), \(\mathbf{C}\) and \(\mathbf{M}\in\mathbf{\Omega}^{n\times n}\) denote the stiffness matrix, damping matrix and mass matrix of the structure, and \(n\) is the degrees of freedom of the FOM. \(\mathbf{F}\) and \(\omega_{\rm c}\) are the amplitude and circular frequency under the harmonic load, respectively. The displacement frequency response \(\mathbf{U}(\omega_{\rm c})\) can be expressed as:  
\[\mathbf{U}(\omega_{\rm c})=\Big{(}\mathbf{K}+j\omega_{\rm c}\mathbf{C}-\omega_{\rm c}^{2} \mathbf{M}\Big{)}^{-1}\mathbf{F}. \tag{2}\]  
This method of solving the equation to calculate the frequency response of the structure is called the direct method, which needs to decompose the dynamic stiffness matrix \(\big{(}\mathbf{K}+j\omega_{\rm c}\mathbf{C}-\omega_{\rm c}^{2}\mathbf{M}\big{)}^{-1}\) at each frequency point.  
When the FOM has a few degrees of freedom or the frequency range of the harmonic load is narrow, the efficiency of the direct method is acceptable. When the number of degrees of freedom of the FOM is particularly large or the frequency range of the harmonic load is wide, the efficiency of the direct method is low. However, it can be improved by model order reduction technique. The conversion equation is normally used to express the displacement frequency response of the FOM as follows:  
\[\mathbf{U}(\omega_{\rm c})=\mathbf{T}\overline{\mathbf{U}}(\omega_{\rm c}), \tag{3}\]  
where \(\mathbf{T}\in\mathbf{\Omega}^{n\times m}\) is the reduced basis matrix, \(m\) is the dimension of the ROM, and \(\overline{\mathbf{U}}\in\mathbf{\Omega}^{m\times 1}\) is the displacement frequency response of the ROM. Substitute Equation (3) into Equation (1) and multiply both sides of Equation (1) by \(\mathbf{T}^{\rm T}\) to obtain the frequency response equation of the ROM:  
\[\Big{(}\overline{\mathbf{K}}+j\omega_{\rm c}\overline{\mathbf{C}}-\omega_{\rm c}^{2} \overline{\mathbf{M}}\Big{)}\overline{\mathbf{U}}(\omega_{\rm c})=\overline{\mathbf{F}}, \tag{4}\]  
where \(\overline{\mathbf{K}}\), \(\overline{\mathbf{C}}\) and \(\overline{\mathbf{M}}\in\mathbf{\Omega}^{m\times m}\) are the stiffness matrix, damping matrix and mass matrix of the ROM, respectively. Their expressions are as follows:  
\[\begin{array}{c}\overline{\mathbf{K}}=\mathbf{T}^{\rm T}\mathbf{K}\mathbf{T},\ \overline{\mathbf{C}}=\mathbf{T}^{\rm T}\mathbf{C}\mathbf{T}\\ \overline{\mathbf{M}}\mathbf{=}\mathbf{T}^{\rm T}\mathbf{M}\mathbf{,}\ \overline{\mathbf{F}}=\mathbf{T}^{\rm T}\mathbf{F} \end{array}. \tag{5}\]  
Then, the displacement frequency response can be solved by the direct method:  
\[\Big{(}\overline{\mathbf{K}}+j\omega_{\rm c}\overline{\mathbf{C}}-\omega_{\rm c}^{2} \overline{\mathbf{M}}\Big{)}\overline{\mathbf{U}}(\omega_{\rm c})=\overline{\mathbf{F}}. \tag{6}\]  
Since the dimension of the matrix \(\big{(}\overline{\mathbf{K}}+j\omega_{\rm c}\overline{\mathbf{C}}-\omega_{\rm c}^{2} \overline{\mathbf{M}}\big{)}^{-1}\) is much smaller than the matrix \(\big{(}\mathbf{K}+j\omega_{\rm c}\mathbf{C}-\omega_{\rm c}^{2}\mathbf{M}\big{)}^{-1}(m \ll n)\), it can be predicted that the efficiency of solving Equation (6) by the direct method is much higher than solving Equation (2).