## 6 The Spectrum Methods for Incoherent Ground Displacement Excitations
### The Power Spectrum Method (Solution)  
The first differential equation of motion in Equation (29) is written as Equation (30)  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\mathbf{K}\mathbf{X}+ \mathbf{K}_{c}\mathbf{X}_{g}=\mathbf{0}. \tag{30}\]  
Also, the solution of Equation (30) can be supposed as the form expressed by Equation (31)  
\[\mathbf{X}=\mathbf{\Phi}\mathbf{u}=\sum_{j=1}^{n}\varphi_{j}\cdot u_{j}, \tag{31}\]  
where \(\mathbf{\Phi}=\left[\varphi_{1},\varphi_{2},\cdots\varphi_{n}\right]\) is the mode matrix; \(\varphi_{j}\) is the column vector of the \(j\)th mode; \(u_{j}\) is the generalized coordinate of the \(j\)th mode. Substituting Equation (31) into Equation (30) and then multiplying \(\varphi_{j}^{T}\), Equations (32) and (33) can be obtained by using the mode composition method  
\[\ddot{u}_{j}+\frac{\varphi_{j}^{T}\mathbf{C}\varphi_{j}}{\varphi_{j}^{T} \mathbf{M}\varphi_{j}}\cdot\dot{u}_{j}+\frac{\varphi_{j}^{T}\mathbf{K}\varphi _{j}}{\varphi_{j}^{T}\mathbf{M}\varphi_{j}}\cdot u_{j}=\frac{-\varphi_{j}^{T} \mathbf{K}_{c}}{\varphi_{j}^{T}\mathbf{M}\varphi_{j}}\cdot\mathbf{X}_{g}, \tag{32}\]  
\[\ddot{u}_{j}+2\zeta_{j}\omega_{j}\dot{u}_{j}+\omega_{j}^{2}u_{j}=\sum_{k=1}^{ m}\beta_{jk}x_{gk}, \tag{33}\]  
where \(x_{gk}(t)\) is the ground motion displacement at support \(k\) at the time \(t\); \(\beta_{jk}\) denotes the participation factor of the \(j\)th mode at support \(k\), as expressed by Equation (34)  
\[\left\{\beta_{j1},\beta_{j2}\cdots\beta_{jm}\right\}=\frac{-\varphi_{j}^{T} \mathbf{K}_{c}}{\varphi_{j}^{T}\mathbf{M}\varphi_{j}}, \tag{34}\]  
where \(m\) is the number of supports. Hence, Equation (33) can be converted to Equation (35) using  
\[u_{j}=\sum_{k=1}^{m}\beta_{jk}\delta_{jk},\ddot{\delta}_{j}+2\zeta_{j}\omega_ {j}\dot{\delta}_{j}+\omega_{j}^{2}\delta_{j}=x_{gk}, \tag{35}\]  
\[\mathbf{X}=\sum_{j=1}^{n}u_{j}\varphi_{j}=\sum_{j=1}^{n}\sum_{k=1}^{m}\beta_{ jk}\delta_{jk}\varphi_{j}, \tag{36}\]where \(\delta_{jk}\) is the generalized displacement at the \(k\)th support in the \(j\)th mode. Now, \(z\) is supposed as a structural response and expressed by Equations (37)-(39) through the nodal displacement vector, \(\mathbf{X}\)  
\[z=\mathbf{q}^{T}\mathbf{X}, \tag{37}\]  
\[z=\sum_{j=1}^{n}\sum_{k=1}^{m}\beta_{jk}\mathbf{q}^{T}\boldsymbol{\varphi}_{j} \delta_{jk}=\sum_{j=1}^{n}\sum_{k=1}^{m}b_{jk}\delta_{jk}, \tag{38}\]  
\[b_{jk}=\beta_{jk}\mathbf{q}^{T}\boldsymbol{\varphi}_{j}, \tag{39}\]  
where \(\mathbf{q}^{T}\) is the vector transferring displacement response to the other response; \(b_{jk}\) is the combination factor. Response \(z\) is a random process, and its self-power spectral density function can be obtained from the theory of random vibration, that is,  
\[\mathbf{X}=\mathbf{\Phi}\mathbf{u}=\sum_{j=1}^{n}\varphi_{j}\cdot u_{j}, \tag{31}\]0  
where \(\omega_{l}\) is the natural frequency to the \(i\)th mode; \(i=(-1)^{-\frac{1}{2}}\) is the unit imaginary number; \(H_{i}(i\omega)=\left[\omega_{l}^{2}-\omega^{2}+2i\zeta_{i}\omega_{l}\omega \right]^{-1}\) is the complex frequency response function of the \(i\)th mode; \(H_{j}(-i\omega)=\left[\omega_{j}^{2}-\omega^{2}-2i\zeta_{j}\omega_{j}\omega \right]^{-1}\) is the conjugation of the complex frequency response function of the \(j\)th mode; \(\zeta_{i}\) is the structural damping ratio of the \(i\)th mode; \(S_{xgk}x_{gl}(i\omega)\) is the cross-power spectrum density function of the earthquake displacements on the ground at points \(k\) and \(l\), which is obtained from the acceleration cross-power spectrum density function, \(S_{\overline{x}gk}\overline{x}_{gl}(i\omega)\).