# On Isospectral Composite Beams
## 1 Introduction
Several results have been established for this class of inverse spectral problems, notably by [PERSON] and [PERSON] [4], [PERSON] and [PERSON] [5; 6], [PERSON] [7], [PERSON] and co-authors [8; 9], [PERSON] [10], [PERSON] [11], [PERSON] [12], and [PERSON] [13], to mention a few. In particular, [PERSON] [14] showed that a general two-by-two real symmetric smooth matrix \(\mathbf{Q}\) can be uniquely determined by the eigenvalues belonging to five spectra corresponding to suitable boundary conditions. [PERSON] and [PERSON] [15] extended the above result to the case of an integrable \(m\times m\) general real matrix-valued symmetric function \(\mathbf{Q}\), for which \(1+m^{2}\) spectral data can determine \(\mathbf{Q}\) uniquely. Figure 1: (**a**) Longitudinal view and cross-section of the composite beam. (**b**) Mechanical model for axial vibration. By exploiting the special structure (11) of the matrix \(\mathbf{Q}\) for our composite beam, we show here that the inverse problem of determining \(k(x)\) in (10) and (11) can be framed within the one-dimensional Sturm-Liouville inverse spectral theory and that, roughly speaking, half of the eigenvalues of the Dirichlet spectrum in (5)-(8) and half of the eigenvalues of the cantilever spectrum (i.e., system (5) and (6) with \(u_{i}(0)=u_{i}^{\prime}(1)=0\), \(i=1,2\)) of the composite beam uniquely determine the shearing stiffness \(k(x)\). The analysis is based on the fact that this class of composite beams is spectrally equivalent to two families of one-dimensional Sturm-Liouville problems, and the eigenvalues of one family do not depend on the coefficient \(k(x)\). We refer to Section 2 for a precise statement and related results. Closely related to the inverse eigenvalue problem is the isospectrality problem for (5)-(8), i.e., the characterization of coefficients \(k(x)\), which have the same spectrum as a given coefficient (with fixed \(A(x)\)) for a particular set of boundary conditions. In the scalar case, i.e., a single Sturm-Liouville equation in canonical form with scalar potential, the isospectrality problem was solved by [PERSON] and co-workers in [16; 17; 18; 19]; see also the contributions by [PERSON] and [PERSON] for an impedance operator [20; 21] and [PERSON] and [PERSON] for applications to longitudinally vibrating beams [22] and for special classes of bending vibrating beams [23]. [PERSON] and [PERSON] [5; 6] proposed a method based on a Gelfand-Levitan integral equation and trasmutation operators for a general real symmetric matrix-valued smooth \(m\times m\) matrix \(\mathbf{Q}\), \(m\geq 2\). [PERSON] and [PERSON] [24] developed a method for a complete parametrization of the isospectral set of matrix-valued \(L^{1}\)-potentials. We refer also to [PERSON] [13] for uniqueness theorems for inverse problems for vectorial Sturm-Liouville equations in which all the eigenvalues of the system are of full multiplicity. The above-mentioned papers deal with a complete characterization of the isospectral potentials \(\mathbf{Q}\) for vector-valued Sturm-Liouville operators. As far as this aspect is concerned, the present paper has a more modest purpose: to show how to determine families of composite beams (with fixed \(A(x)\)) that are isospectral to a given one for a given set of boundary conditions. We will show that, under our assumptions, we can resort to a classical Darboux lemma [25] for an explicit construction of isospectral composite beams. The isospectral shearing stiffness coefficients \(k(x)\) belong to a neighborhood of a given coefficient, and the construction is possible for composite beams with simple eigenvalues. The paper is organized as follows. In Section 2, we show our main theoretical results concerning the unique determination of the shearing stiffness coefficient \(k(x)\) from eigenvalue data and the construction of isospectral composite beams. Some examples are presented in Section 3. A generalization of the above analysis to composite systems formed by \(N\geq 3\) connected beams is attempted in Section 4.