# A Semi-Analytical Approach for the Linearized Vibration of
## 3 Linearized Vibration of Beams with the Effect of Static and Thermal Load
### Solution Procedures
#### 3.2.2 Linearized Modal Analysis  
Upon completion of the nonlinear static analysis, the equation of free vibration, which takes into account the influence of static load and thermal load, can be derived from Equation (10) as follows. \[\rho A\ddot{w}_{1}+c\dot{w}_{1}-(\frac{3}{2}EAw^{\prime 2}_{0}+N_{xx})w^{ \prime\prime}{}_{1}+EIw_{1}{}^{(4)}=0 \tag{15}\]  
Based on the separation of variables, the modal equation can be derived from Equation (15) expressed as Equation (16). \[-EI\phi^{(4)}+(\frac{3}{2}EAw^{\prime 2}_{0}+N_{xx})\phi^{\prime\prime}+\omega^{ 2}\rho A\phi=0 \tag{16}\]  
in which \(\omega\) is the natural frequency of the beam, and \(\phi\) is the mode of vibration. Equation (16) is a variable coefficient partial differential equation and is difficult to obtain an analytical solution. An approximate solution is performed by the transfer-matrix method and local homogenization techniques. The beam is divided into \(n\) segments and the modal equation of the \(i\)th segment is  
\[EI\phi_{i}{}^{(4)}-\frac{3}{2}EAw^{\prime}{}_{0}{}^{2}(x_{i})\phi^{\prime\prime }{}_{i}-\omega^{2}\rho A\phi_{i}=0 \tag{17}\]  
According to the homogenization theory, let  
\[\delta_{i}^{2}=\frac{1}{x_{i+1}-x_{i}}\int_{x_{i}}^{x_{i+1}}\frac{3 Aw^{\prime 2 }_{0}(x)}{2I}+\frac{N_{x}}{EI}\mathrm{d}x\,\qquad\beta^{4}=\frac{\omega^{2}\rho A}{EI} \tag{18}\]  
where \(x_{i}\) and \(x_{i+1}\) are the left and right \(x\)- coordinates of the \(i\)th segment. The approximate general solution of Equation (17) is  
\[\phi_{i}(x)=A_{i}\sin\beta_{i1}(x-x_{i})+B_{i}\cos\beta_{i1}(x-x_ {i})+C_{i}\cosh\beta_{i2}(x-x_{i})+D_{i}\mathrm{sinh}\beta_{i2}(x-x_{i}),\ x_{i}\leq x\leq x_{i+1} \tag{19}\] \[\beta_{i1}=\sqrt{\sqrt{\beta^{4}+\frac{\delta_{i}{}^{4}}{4}}- \frac{\delta_{i}{}^{2}}{2}}\quad\beta_{i2}=\sqrt{\sqrt{\beta^{4}+\frac{\delta _{i}{}^{4}}{4}}+\frac{\delta_{i}{}^{2}}{2}} \tag{20}\]Among them, \(A_{i}\), \(B_{i}\), \(C_{i}\), and \(D_{i}\) are constants determined by boundary conditions. The displacement and force are continuous between the \(i\)th segment and the \(i\)+1 th segment which can be written as  
\[ \begin{array}{c}\phi_{i}(x_{i+1})=\phi_{i+1}(x_{i+1})\ \phi^{\prime}_{i}(x_{i+1})= \phi^{\prime}_{i+1}(x_{i+1})\\ \phi^{\prime\prime}_{i}(x_{i+1})=\phi^{\prime\prime}_{i+1}(x_{i+1})\ \phi^{\prime\prime}_{i}(x_{i+1})=\phi^{ \prime\prime\prime}_{i+1}(x_{i+1})\end{array}  \tag{21}\]  
The relations can be expressed by the transfer matrix  
\[ \begin{array}{c}\Phi_{i}(x_{i+1})\psi_{i}=\Phi_{i+1}(x_{i+1})\psi_{i+1}\\ \psi_{i+1}=\Phi_{i+1}^{-1}(x_{i+1})\Phi_{i}(x_{i+1})\psi_{i}\end{array}  \tag{22}\]  
in which  
\[ \begin{array}{c}\psi_{i}=\begin{bmatrix}A_{i}&B_{i}&C_{i}&D_{i}\end{bmatrix} ^{\rm T}\\ \theta_{i}(x)=\begin{bmatrix}\sin\beta_{i1}(x-x_{i})&\cos\beta_{i1}(x-x_{i})& \cosh\beta_{i2}(x-x_{i})&\sinh\beta_{i2}(x-x_{i})\end{bmatrix}\\ \Phi_{i}(x_{j})=\begin{bmatrix}\theta_{i}(x_{j})&\theta^{\prime}_{i}(x_{j})& \theta^{\prime\prime}_{i}(x_{j})&\theta^{\prime\prime\prime}_{i}(x_{j})\end{bmatrix} ^{\rm T}\end{array}  \tag{23}\]  
The relationship between the \(n\)th segment and 1 st segment is written as  
\[ \begin{array}{c}\psi_{n}=\Gamma\psi_{1}\\ \Gamma=\prod\limits_{i=1}^{i=n}\Phi_{i+1}^{-1}(x_{i+1})\Phi_{i}(x_{i+1})\end{array}  \tag{24}\]  
To determine the natural frequencies of a clamped beam, the roots of the polynomial can be found using the following approach:  
\[ \begin{array}{c}\det\begin{bmatrix}0&1&1&0\\ \beta_{11}&0&&0&\beta_{12}\\ &&\Gamma_{1}&&\\ &&\Gamma_{2}&&\end{bmatrix}=0\end{array}  \tag{25}\]  
in which  
\[ \begin{array}{c}\begin{bmatrix}\Gamma_{1}\\ \Gamma_{2}\end{bmatrix}=\begin{bmatrix}\sin(\beta_{n1}h)&\cos(\beta_{n1}h)& \cos h(\beta_{n2}h)&\sin h(\beta_{n2}h)\\ \beta_{n1}\!\cos(\beta_{n1}h)&-\beta_{n1}\!\sin(\beta_{n1}h)&\beta_{n2}\!\sin h (\beta_{n2}h)&\beta_{n2}\!\cos h(\beta_{n2}h)\end{bmatrix}\end{array}  \tag{26}\]  
The structural mode frequencies can be obtained by numerically solving Equation (25).