# On Isospectral Composite Beams
## 4 Extension to Multiple Connected Beams  
Let us consider a supported composite system \(\{\widehat{A},\widehat{k}\}_{N}\) obtained by connecting \(N\) equal beams, where \(N\geq 2\), as considered in (5)-(8), with a cross-sectional area \(\widehat{A}(x)\) and where the shearing stiffness of the \((N-1)\) connections is \(\widehat{k}(x)\). The free axial vibration is governed by the boundary-value differential system  
\[\left\{ \begin{array}{l}(\widehat{A}\mathbf{u}^{\prime})^{\prime}+\lambda \widehat{A}\mathbf{u}=\frac{\widehat{k}(x)}{2}\mathbf{C}\mathbf{u},\quad x\in (0,1),\\ \mathbf{u}(0)=\mathbf{u}(1)=0,\end{array} \right. \tag{41}\]  
where \(\mathbf{u}(x)=(u_{1}(x),\ldots,u_{N}(x))\) and where \(\mathbf{C}\) is an \(N\times N\) Jacobi matrix  
\[\mathbf{C}=\left( \begin{array}{cccccc}1&-1&0&\cdots&0&0\\ -1&2&-1&\cdots&0&0\\ 0&-1&2&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&2&-1\\ 0&0&0&\cdots&-1&1\end{array} \right). \tag{43}\]  
Let us represent \(\mathbf{u}\) on the basis of the eigenvectors of \(\mathbf{C}\), namely  
\[\mathbf{u}(x)=\sum_{j=1}^{N}\widehat{\eta}^{(j)}(x)\mathbf{c}^{(j)},\quad \widehat{\eta}^{(j)}(x)=\mathbf{u}(x)\cdot\mathbf{c}^{(j)}, \tag{44}\]  
where  
\[\mathbf{C}\mathbf{c}^{(j)}=\chi^{(j)}\mathbf{c}^{(j)},\quad\mathbf{c}^{(j)} \cdot\mathbf{c}^{(i)}=\delta_{ij}, \tag{45}\]  
where \(i,j=1,\ldots,N\). A direct calculation shows that, for every \(j=1,\ldots,N\), \begin{table}
\begin{tabular}{l c c c c c} \hline \(\lambda_{exact}\) (rad/s) & 19.74 & 49.35 & 98.70 & 167.78 & 256.61 \\ \hline \(\lambda_{FE}\) & 19.74 & 49.36 & 98.76 & 167.99 & 257.12 \\ Error \(\left(\frac{\lambda_{FE}-\lambda_{exact}}{\lambda_{exact}}\right)\times 10^{ -3}\) & 0.13 & 0.32 & 0.67 & 1.24 & 1.98 \\ \hline \(\lambda_{exact}\) (rad/s) & 365.17 & 493.48 & 641.52 & 809.31 & 996.83 \\ \hline \(\lambda_{FE}\) & 366.23 & 495.43 & 644.86 & 814.65 & 1005.0 \\ Error \(\left(\frac{\lambda_{FE}-\lambda_{exact}}{\lambda_{exact}}\right)\times 10^{ -3}\) & 2.89 & 3.96 & 5.20 & 6.60 & 8.17 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison between exact (\(\lambda_{exact}\)) and FE results (\(\lambda_{FE}\)) of the list \((\lambda_{n})_{n=1}^{10}\). \[\chi^{(i)}k^{(i)}(x)=\chi^{(i)}\widehat{k}(x)-2c\widehat{a}^{2}(x)y_{m}^{(i)}(x) \left(\frac{2(y_{m}^{(i)})^{\prime}(x)}{1+c\int_{0}^{x}(y_{m}^{(i)})^{2}ds}-\frac{ c(y_{m}^{(i)})^{3}(x)}{\left(1+c\int_{0}^{x}(y_{m}^{(i)})^{2}ds\right)^{2}}\right), \tag{56}\]  
For a \(c>-1\) and \(|c|\) small enough, \(m=1,2,\dots,\) where \(y_{m}^{(i)}(x)\) is the \(m\)th eigenfunction of (52) and (53). Note that \(k^{(i)}\) also depends on the index \(i\) since \(y_{m}^{(i)}\) depends on \(i\). We can now construct a composite system \(\{\widehat{A}(x),k^{(i)}(x)\}_{N}\) such that all the eigenvalues \(\{\lambda_{n}^{(i)}\}_{n\geq 1}\) of (54) and (55) belong to its spectrum. Let us multiply (54) by \(\mathbf{c}^{(i)}\), where \((i)\) is a fixed index, not summed, where \(i=2,\ldots,N\). Recalling (45), we have  
\[(\widehat{A}(\eta_{n}^{(i)})^{\prime}\mathbf{c}^{(i)})^{\prime}+\lambda_{n}^{( i)}\widehat{A}\eta_{n}^{(i)}\mathbf{c}^{(i)}=\frac{k^{(i)}}{2}\mathbf{C}(\eta_{n}^ {(i)}\mathbf{c}^{(i)}),\quad x\in(0,1), \tag{57}\]  
such that the function \(\mathbf{u}_{n}^{(i)}=\eta_{n}^{(i)}\mathbf{c}^{(i)}\) is a non-trivial solution to  
\[\left\{ \begin{array}{l}(\widehat{A}(\mathbf{u}_{n}^{(i)})^{\prime})^{\prime }+\lambda_{n}^{(i)}\widehat{A}\mathbf{u}_{n}^{(i)}=\frac{k^{(i)}(x)}{2} \mathbf{C}\mathbf{u}_{n}^{(i)},\quad x\in(0,1),\\ \mathbf{u}_{n}^{(i)}(0)=\mathbf{u}_{n}^{(i)}(1)=0.\end{array} \right. \tag{58}\]  
We have proved that the composite systems \(\{\widehat{A},\widehat{k}\}_{N}\), and \(\{\widehat{A},k^{(i)}\}_{N}\) share the same eigenvalues \(\{\lambda_{n}^{(i)}\}_{n\geq 1}\). However, if an index \(j\geq 2\) is chosen with \(j\
eq i\), then \(k^{(i)}\) and \(k^{(j)}\) are not necessarily equal in \([0,1]\). It follows that, in general, it is not possible by this approach to construct isospectral composite systems unless, of course, \(N=2\).