## 2 Modification of the Timoshenko Beam Model
### Solution to Natural Frequencies under Different Boundary Conditions
For a clamped-clamped beam, we have the following:  
\[c_{1}+c_{3}=0 \tag{25}\]  
\[c_{1}coshg_{1}L+c_{2}sinhg_{1}L+c_{3}cosg_{2}L+c_{4}sing_{2}L=0 \tag{26}\]  
\[c_{2}g_{3}+c_{4}g_{4}=0 \tag{27}\]  
\[c_{1}g_{3}sinhg_{1}L+c_{2}g_{3}coshg_{1}L+c_{3}(-g_{4})sing_{2}L+c_{4}g_{4}cosg _{2}L=0 \tag{28}\]  
For a dynamic problem, coefficients \(c_{1},\ c_{2},\ c_{3},\ and\ c_{4}\) cannot be all zeroes; otherwise, it becomes a static problem. Thus, the equation of the natural frequency can be expressed as follows:  
\[\left| \begin{array}{cccc}1&0&1&0\\ \cos\mathrm{h}g_{1}L&\sin\mathrm{h}g_{1}L&\cos g_{2}L&\sin g_{1}L\\ 0&g_{3}&0&g_{3}\\ g_{3}\mathrm{sin}\,\mathrm{h}g_{1}L&g_{3}\mathrm{cos}\,\mathrm{h}g_{1}L&(-g_{ 4})\sin g_{2}L&g_{4}\cos g_{2}L\end{array} \right|=0 \tag{29}\]  
Denote Equation (29) as follows:  
\[f_{2}(\omega)=0 \tag{30}\]  
where \(f_{2}(\omega)\) is the determinant of Equation (29) and frequency \(\omega\) is de facto the eigenvalue of the corresponding matrix of the determinant. Name \(f_{2}(\omega)=0\) the frequency polynomial for a clamped-clamped beam, and every zero of the polynomial is namely every order of natural frequency for the beam. After expansion and simplification of the determinant, we will obtain the following equation:  
\[f_{2}(\omega)=-2g_{3}g_{4}+2g_{3}g_{4}cos(g_{2}L)cosh(g_{1}L)-\Big{(}g_{3}{}^ {2}-g_{4}{}^{2}\Big{)}sin(g_{2}L)sinh(g_{1}L) \tag{31}\]  
from which natural frequencies of a clamped-clamped beam can be calculated. For a clamped-free beam, we have the following:  
\[c_{1}g_{1}g_{3}-c_{3}g_{2}g_{4}=0 \tag{32}\]  
\[c_{2}(g_{3}-g_{1})+c_{4}(g_{4}-g_{2})=0 \tag{33}\]  
\[c_{1}coshg_{1}L+c_{2}sinhg_{1}L+c_{3}cosg_{2}L+c_{4}sing_{2}L=0 \tag{34}\]  
\[c_{1}g_{3}sinhg_{1}L+c_{2}g_{3}coshg_{1}L+c_{3}(-g_{4})sing_{2}L+c_{4}g_{4}cosg _{2}L=0 \tag{35}\]Since coefficients \(c_{1},\ c_{2},\ c_{3},\ and\ c_{4}\) are not all zeroes, the equation of the natural frequency can then be expressed as follows:  
\[\left| \begin{array}{cccc}g_{1}g_{3}&0&-g_{2}g_{4}&0\\ 0&g_{3}-g_{1}&0&g_{4}-g_{2}\\ \cos hg_{1}L&\sin hg_{1}L&\cos g_{2}L&\sin g_{1}L\\ g_{3}\sin hg_{1}L&g_{3}\cos hg_{1}L&(-g_{4})\sin g_{2}L&g_{4}\cos g_{2}L\\ \end{array} \right|=0 \tag{36}\]  
Denote Equation (36) as follows:  
\[f_{3}(\omega)=0 \tag{37}\]  
After expansion and simplification of the determinant, we will obtain the following equation:  
\[ \begin{array}{c}f_{3}(\omega)=\\ (-g_{2}+g_{4})g_{1}{g_{3}}^{2}(-\cos(g_{2}L))\cosh(g_{1}L)\\ -g_{1}g_{3}g_{4}\sin(g_{2}L)\sinh(g_{1}L)-g_{2}g_{3}g_{4}\cos h^{2}(g_{1}L)\\ +g_{2}g_{3}g_{4}\sin h^{2}(g_{1}L)+(-g_{1}+g_{3})g_{1}g_{3}g_{4}\big{(}\cos^{2 }(g_{2}L)\big{)}\\ +g_{1}g_{3}g_{4}\sin^{2}(g_{2}L)-g_{2}g_{3}g_{4}\sin(g_{2}L)\sinh(g_{2}L)\\ +g_{2}g_{4}^{2}\cos(g_{2}L)\cosh(g_{1}L)\\ \end{array}  \tag{38}\]  
Similar to the clamped-clamped beam, name \(f_{3}(\omega)=0\) the frequency polynomial for clamped-free beam, and every zero of the polynomial is every order of natural frequency for the beam. The natural frequencies of the clamped-free beam will also be obtained through the equation.