## 2 Frequencies and Mode Shapes of the Double-Beam System
### Mathematical Formulation
#### 2.1.1 Fixed-Fixed Boundary Conditions  
The boundary stiffness is  
\[\left\{\begin{array}{l}k_{v,1}=k_{v,2}=\infty\\ k_{r,1}=k_{r,2}=\infty\end{array}\right. \tag{14}\]  
The boundary conditions of the fixed-fixed double-beam system are  
\[\left\{\begin{array}{l}\varphi_{1}(0)=0\\ \varphi_{1}^{\prime}(0)=0\\ \varphi_{1}(l)=0\end{array}\right.^{\prime},\left\{\begin{array}{l}\varphi_ {2}(0)=0\\ \varphi_{2}^{\prime}(0)=0\\ \varphi_{2}(l)=0\\ \varphi_{2}^{\prime}(l)=0\end{array}\right. \tag{15}\]  
Substituting Equation (15) into Equations (4) and (13) results in an equation in a matrix form as  
\[\mathbf{\Phi X}=\mathbf{0} \tag{16}\]  
where  
\[\mathbf{X}=\left(C_{1},\ C_{2},\ C_{3},\ C_{4},C_{5},C_{6},C_{7},C_{8}\right)^ {\mathrm{T}} \tag{17}\]  
\[\mathbf{\Phi}=\left(\begin{array}{cccccccc}1&1&1&1&0&0&0&0\\ 0&0&0&0&1&1&1&1\\ 1&-1&i&-i&0&0&0&0\\ 0&0&0&0&1&-1&i&-i\\ e^{r_{1}l}&e^{-r_{1}l}&e^{ir_{1}l}&e^{-ir_{1}l}&0&0&0&0\\ 0&0&0&0&e^{r_{5}l}&e^{-r_{5}l}&e^{ir_{5}l}&e^{-ir_{5}l}\\ e^{r_{1}l}&-e^{-r_{1}l}&ie^{ir_{1}l}&-ie^{-ir_{1}l}&0&0&0&0\\ 0&0&0&0&e^{r_{5}l}&-e^{-r_{5}l}&ie^{ir_{5}l}&-ie^{-ir_{5}l}\end{array}\right) \tag{18}\]  
To make the homogeneous Equation (16) have nontrivial solutions, the determinant of \(\mathbf{\Phi}\) should be zero, which results in  
\[\left\{\begin{array}{c}\cos(r_{1}l)\cosh(r_{1}l)-1=0\\ r_{5}=0\end{array}\right. \tag{19}\]  
or  
\[\left\{\begin{array}{c}r_{1}=0\\ \cos(r_{5}l)\cosh(r_{5}l)-1=0\end{array}\right. \tag{20}\]  
Equations (19) and (20) would result in the same solution for Equation (13). For example, if Equation (19) is chosen, the nonlinear equation of \(r_{1}l\) is solved numerically and shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mode** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(r_{1}l\) & 4.73 & 7.8532 & 10.9956 & 14.1372 & 17.2788 & 20.4204 \\ \hline \hline \end{tabular}
\end{table}
Table 1: First six solutions of \(r_{1}l\) for fixed-fixed double-beam systems.  
It is noticed that the wavenumber equation for \(r_{1}l\) in Equation (19) or for \(r_{5}l\) in Equation (20) are the same as that for a single beam with fixed-fixed boundary conditions.  
Then, the mode shape of the secondary beam is obtained by solving Equation (16)  
\[\varphi_{2,n}(x)=C[(\cos r_{1}^{(n)}x-\cosh r_{1}^{(n)}x)-\frac{\cos r_{1}^{(n) }l-\cosh r_{1}^{(n)}l}{\sin r_{1}^{(n)}l-\sinh r_{1}^{(n)}l}(\sin r_{1}^{(n)}x- \sinh r_{1}^{(n)}x)] \tag{21}\]  
where \(C\) is a coefficient set to be one in the analyses in Section 3 and \(r_{1}^{(n)}l\) is the value of \(r_{1}l\) in the \(n\)th basic mode.  
It is shown in Equation (21) that the mode shapes of the secondary beam are the same as the mode shapes of a single beam with fixed-fixed boundary conditions. Substituting Equation (21) into Equation (4) results in the mode shape amplitude ratios between the primary beam and the secondary beam in the two sub-modes  
\[\varphi_{1,n}^{(i)}=\left(1+\frac{1-\left(\frac{\omega_{1}^{(i)}}{\omega_{2,n} }\right)^{2}}{s_{e}}\Big{(}r_{1}^{(n)}l\Big{)}^{4}\right)\varphi_{2,n}\ \ \ i=1,\ 2 \tag{22}\]  
It is seen from Equation (22) that the mode shape of the primary beam is a multiple of the secondary beam. In fact, Equation (22) is valid for the mode shapes of the primary beam of any double-beam systems with symmetric boundary conditions except that the values of \(r_{1}^{(n)}l\) are different for different boundary conditions.