# On Isospectral Composite Beams
## 2 Theory
These latter modes are the only principal modes affected by the shearing stiffness of the connection, and therefore, only the family of eigenvalues \(\{\lambda_{n}^{V}\}_{n\geq 1}\) contains information about the stiffness of the connection. It follows that the problem of determining \(k(x)\) from spectral data in (5)-(8) coincides with the problem of determining \(k(x)\) in the scalar Sturm-Liouville Problem (17) and (18). By the Liouville transformation \((y=aV,A=a^{2})\), Problems (17) and (18) can be reduced to the canonical form  
\[\left\{ \begin{array}{l}y^{\prime\prime}+\lambda y=\Big{(}q+\frac{k}{a^{2}} \Big{)}y,\quad x\in(0,1),\\ y(0)=y(1)=0,\end{array} \right. \tag{29}\]  
with \(q=\frac{\epsilon^{\prime\prime}}{a}\). Inverse problems for (29) and (30) are well-known due to the cornerstone contributions by [PERSON]  
where \(c>-1\) and \(m=1,2,\dots\), and \(y_{m}(x)\) is the \(m\)th Dirichlet eigenfunction of (36) and (37) normalized by \(\int_{0}^{1}y_{m}^{2}(s)ds=1\). Therefore, assuming \(q(x)=\widehat{q}(x)+\frac{k}{\hat{a}^{2}}\), for every \(m=1,2,\dots\), we obtain a composite beam \(\{\widehat{A}(x),k(x)\}_{2}\) with shearing stiffness  
\[k(x)=\widehat{k}(x)-2c\hat{a}^{2}(x)y_{m}(x)\Bigg{(}\frac{2y_{m}^{\prime}(x)}{ 1+c\int_{0}^{x}y_{m}^{2}(s)ds}-\frac{cy_{m}^{3}(x)}{\big{(}1+c\int_{0}^{x}y_{m }^{2}(s)ds\big{)}^{2}}\Bigg{)}, \tag{39}\]  
which has exactly the same eigenvalues of \(\{\widehat{A}(x),\widehat{k}(x)\}_{2}\), e.g., \(\lambda_{n}^{V}(\widehat{k})=\lambda_{n}^{V}(k)\) for every \(n\geq 1\). To conclude the construction of a real \"physical\" isospectral composite beam, we need to show that the shearing stiffness \(k(x)\) is positive in \([0,1]\). Since, by hypothesis, \(\widehat{k}(x)\geq k_{0}>0\) in \([0,1]\), by (39), there exists \(k_{0}^{\prime}>0\), and there exists \(\delta>0\) (possibly depending on \(m\) and \(k_{0}^{\prime}\)) such that \(k(x)\geq k_{0}^{\prime}>0\) in \([0,1]\) for every \(c>-1\) and \(c\in[-\delta,\delta]\). In conclusion, the composite beam \(\{\widehat{A}(x),k(x)\}_{2}\) is isospectral to \(\{\widehat{A}(x),\widehat{k}(x)\}_{2}\) under Dirichlet end conditions. By adapting the above analysis and using the results in [22], it can be shown that the construction of isospectral composite beams also extends to other boundary conditions, such as, for example, the support-free (cantilever) and free-free conditions. Here, we recall the main steps for the determination of the potential \(q(x)\) in (39) isospectral to the potential \((\widehat{q}+\widehat{k}/\hat{a}^{2})\) under Dirichlet boundary conditions. The analysis is based on a double application of the Darboux lemma [25]. Denote by \(L_{r}\) the standard Sturm-Liouville operator with potential \(r\in C([0,1])\), i.e., \(L_{r}u=-u^{\prime\prime}+ru\). Let \(\mu\), \(\lambda\) be two real numbers. In its simpler form, the Darboux lemma enables us to find a non-trivial solution \(z\) of a new equation \(L_{q}z=\lambda z\) if we know a non-trivial solution \(g\), \(f\) of the equation \(L_{r}g=\mu g\), \(L_{r}f=\lambda f\), respectively, corresponding to two different values \(\lambda\), \(\mu\) of the parameter and to a potential \(r\). In particular, it turns out that \(z=\frac{1}{g}[g,f]\), where \([g,f]=gf^{\prime}-g^{\prime}f\), and \(\ \hat{q}=r-2(\ln(g))^{\prime\prime}\). The potential \(\hat{q}\) is singular at those points of \([0,1]\) in which \(g\) has a zero. However, for such cases, we can modify the above analysis by applying the Darboux lemma twice, obtaining Expression (39) of the regular (i.e., continuous) potential \(q\) isospectral to \((\widehat{q}+\widehat{k}/\hat{a}^{2})\). We refer to the book [19] (Chapter 5) for more details.