# On Isospectral Composite Beams
## 1 Introduction  
Composite beams obtained by connecting two beam elements represent a structural solution commonly used in designing long-span floor beams or bridge decks [1]. The connection is the structural component having to bear the major consequences of stress and fatigue during service, and therefore, the evaluation of its integrity is of great importance for practical purposes. Nondestructive techniques based on dynamic measurements are appealing for assessing damage to composite beams and have attracted much interest in recent years; see, for example, [2, 3] and the references therein. In this paper, we consider a class of composite beams formed by two identical straight elastic beams connected together and subjected to longitudinal vibration. Practical examples of this system are steel beams connected by an adhesive layer on the contact surface and wood beams connected by means of wood studs that hinder sliding at the common interface. A schematic view can be seen in Figure 1. In mathematical terms, the small free undamped longitudinal vibration of such a composite beam of length \(L\) is governed by the differential system in \((z,t)\in(0,L)\times(0,T_{0})\)[2]  
\[\left\{ \begin{array}{l}\frac{\partial}{\partial z}\left(EA(z)\frac{\partial w _{1}}{\partial z}\right)+\frac{\mu(z)}{2}(w_{2}-w_{1})=\gamma A(z)\frac{ \partial^{2}w_{1}}{\partial t^{2}},\\ \frac{\partial}{\partial z}\left(EA(z)\frac{\partial w_{2}}{\partial z}\right) -\frac{\mu(z)}{2}(w_{2}-w_{1})=\gamma A(z)\frac{\partial^{2}w_{2}}{\partial t ^{2}},\end{array} \right. \tag{1}\]  
with supported end conditions at both ends, namely  
\[w_{1}(0,t)=w_{1}(L,t)=0,\quad w_{2}(0,t)=w_{2}(L,t)=0,\quad t\in(0,T_{0}), \tag{3}\]  
where \(w_{1}(z,t)\), \(w_{2}(z,t)\) is the axial displacement of the transversal section of beams 1 and 2, respectively, at the transversal section of abscissa \(z\) at the moment of time \(t\), where \(T_{0}>0\). The area of the cross-section is denoted by \(A(z)\), \(A\in C^{2}([0,L])\), \(A(z)\geq A_{0}>0\) in \([0,L]\). \(E\) is the Young's modulus of the material, and \(\gamma\) is the volume mass density. \(E\) and \(\gamma\) are assumed to be constant and positive. The coefficient \(\frac{\mu(z)}{2}\), \(\mu\in C^{0}([0,L])\), \(\mu(z)\geq\mu_{0}>0\) in \([0,L]\) is the shearing stiffness per unit length of the connection. For free vibration with a radian frequency \(\omega\) and for a normalized abscissa \(x=\frac{\tilde{z}}{L}\), the longitudinal displacement may be assumed as  
\[w_{1}(x,t)=u_{1}(x)\cos(\omega t),\quad w_{2}(x,t)=u_{2}(x)\cos(\omega t), \tag{4}\]  
so that the eigenpair \(\{\lambda=\omega^{2}\frac{7L^{2}}{E},(u_{1}(x),u_{2}(x))\}\) satisfies  
\[\left\{ \begin{array}{ll}(Au_{1}^{\prime})^{\prime}+\frac{k}{2}(u_{2}-u_{1})+ \lambda Au_{1}=0,&x\in(0,1),\\ (Au_{2}^{\prime})^{\prime}-\frac{k}{2}(u_{2}-u_{1})+\lambda Au_{2}=0,&x\in(0,1 ),\\ u_{1}(0)=u_{1}(1)=0,&\\ u_{2}(0)=u_{2}(1)=0,&\end{array} \right. \tag{5}\]  
where \(k(x)=\frac{\mu(x)L^{2}}{E}\), \(A(x)=A\big{(}\frac{\tilde{z}}{L}\big{)}\) and both \(u_{1}(x)\) and \(u_{2}(x)\) are not identically vanishing functions. Assuming the beam profile \(A(x)\) is given, an inverse problem that is interesting for applications consists of determining the shearing stiffness coefficient \(k(x)\) from spectral data, e.g., the eigenvalues belonging to the spectrum under either Dirichlet or other boundary conditions. Putting  
\[A=a^{2},\quad\mathbf{y}=(y_{1}=au_{1},y_{2}=au_{2}), \tag{9}\]  
system (5) and (6) reduces to the canonical Sturm-Liouville vectorial form  
\[\mathbf{y}^{\prime\prime}+\lambda\mathbf{y}=\mathbf{Q}\mathbf{y},\quad x\in(0,1), \tag{10}\]  
with  
\[\mathbf{Q}=\left( \begin{array}{cc}q+\frac{k}{2a^{2}}&-\frac{k}{2a^{2}}\\ -\frac{E}{2a^{2}}&q+\frac{k}{2a^{2}}\end{array} \right), \tag{11}\]  
where the potential  
\[q(x)=\frac{a^{\prime\prime}(x)}{a(x)},\quad q\in C^{0}([0,1]), \tag{12}\]  
is given and \(k(x)\) is the unknown coefficient.