# On Isospectral Composite Beams
## 2 Theory  
The following proposition establishes a spectral equivalence between the composite beam system (5)-(8) and two one-dimensional Sturm-Liouville problems. Proposition 1: _Let \(A\in C^{2}([0,1])\), \(A(x)\geq A_{0}\) in \([0,1]\), \(k\in C^{0}([0,1])\), and \(k(x)\geq k_{0}\) in \([0,1]\), where \(A_{0}\) and \(k_{0}\) are positive constants._  
_Let \(\{\lambda,(u_{1},u_{2})\}\) be an eigenpair of the composite beam System (5)-(8). Then, either_  
\[u_{2}=u_{1}\quad\text{in }[0,1], \tag{13}\]  
_or, alternatively,_  
\[u_{2}=-u_{1}\quad\text{in }[0,1]. \tag{14}\]  
_If (_13_) holds, then_ \(\{\lambda^{U},U=u_{1}+u_{2}\}\) _is an eigenpair of_  
\[\left\{ \begin{array}{l}(AU^{\prime})^{\prime}+\lambda^{U}AU=0,\quad x\in(0, 1),\\ U(0)=U(1)=0.\end{array} \right. \tag{15}\]  
_If (_14_) holds, then_ \(\{\lambda^{V},V=u_{1}-u_{2}\}\) _is an eigenpair of_\[\left\{ \begin{array}{ll}(AV^{\prime})^{\prime}+\lambda^{V}AV=kV,\quad x\in(0,1), \\ V(0)=V(1)=0.\end{array} \right. \tag{17}\]  
Proof. : The proof of (13) and (14) is by contradiction. Let us assume there exists a constant \(\beta\) (\(\beta\
eq 0\)) such that  
\[u_{2}=\beta u_{1}\quad\text{in}\;[0,1],\quad|\beta|\
eq 1. \tag{19}\]  
Using (19) in (5)-(8), we obtain  
\[(\beta+1)((Au_{1}^{\prime})^{\prime}+\lambda Au_{1})=0\quad\text{in}\;(0,1) \tag{20}\]  
which implies  
\[(Au_{1}^{\prime})^{\prime}+\lambda Au_{1}=0\quad\text{in}\;(0,1). \tag{21}\]  
Comparing (21) with (5), we conclude that  
\[k(u_{2}-u_{1})=0\quad\text{in}\;[0,1], \tag{22}\]  
which implies \(u_{2}=u_{1}\); that is, \(\beta=1\), which is a contradiction. Finally, the eigenvalue problems (15)-(16) and (17)-(18) follow from (5)-(8) by summing and subtracting (5) and (6), respectively, and by taking into account the boundary conditions. **Remark 1**. : _Note that, in order to distinguish the two classes of principal modes, it is enough to know the axial strain \(u_{i}^{\prime}\), where \(i=1,2\), at one end of the beam, say \(u_{1}^{\prime}(0)\), \(u_{2}^{\prime}(0)\). In fact, for in-phase motion, we have \(u_{1}^{\prime}(0)=u_{2}^{\prime}(0)\), whereas for out-of-phase vibration, we have \(u_{1}^{\prime}(0)=-u_{2}^{\prime}(0)\). Note that \(u_{1}^{\prime}(x)u_{2}^{\prime}(x)\
eq 0\) for \(x=0\) and \(x=1\)._  
**Remark 2**. : _The eigenvalues of (5)-(8) may not be simple, with multiplicity at most equal to \(2\). For the uniform composite beam with \(A(x)=1\) and \(k(x)=k_{0}\) constant, where \(k_{0}>0\), the eigenvalues are double if and only if \(k_{0}=(n^{2}-m^{2})\pi^{2}\) for integer numbers \(m,n\) with \(n>m\geq 1\). Clearly, if \(0<k_{0}<3\pi^{2}\), then all the eigenvalues are simple._  
**Remark 3**. : _Proposition 1 can also be extended to other boundary conditions, for example, the cantilever end conditions:_  
\[u_{1}(0)=u_{2}(0)=0,\quad u_{1}^{\prime}(1)=u_{2}^{\prime}(1)=0. \tag{23}\]  
_The eigenvalues \(\{\lambda_{i}^{\prime}\}_{i\geq 1}\) of (5)-(6) and (23) are given by_  
\[\{\lambda_{i}^{\prime}\}_{i\geq 1}=\{\lambda_{l}^{U^{\prime}}\}_{l\geq 1}\cup \{\lambda_{n}^{V^{\prime}}\}_{n\geq 1}, \tag{24}\]  
_where \(\{\lambda_{l}^{U^{\prime}}\}_{l\geq 1}\), \(\{\lambda_{n}^{V^{\prime}}\}_{n\geq 1}\) are, respectively, the eigenvalues of_  
\[(AU^{\prime})^{\prime}+\lambda^{U^{\prime}}AU=0,\quad x\in(0,1), \tag{25}\] \[U(0)=U^{\prime}(1)=0, \tag{26}\]  
_and_  
\[(AV^{\prime})^{\prime}+\lambda^{V^{\prime}}AV=kV,\quad x\in(0,1), \tag{27}\] \[V(0)=V^{\prime}(1)=0. \tag{28}\]  
From Proposition 1, it is seen that, when the composite beam vibrates in the principal modes of (15) and (16), the two beams are subject to in-phase motions \((u_{1},u_{2}=u_{1})\), and the strain energy stored inside the connection vanishes. On the contrary, the two beams oscillate according to out-of-phase motions \((u_{1},u_{2}=-u_{1})\) when the composite beam vibrates in the principal modes of (17) and (18).