Global classification of isospectral Euler–Bernoulli beams with constant stiffness–mass product

For Euler–Bernoulli beams in the class $\mathcal{C}$ where the product of the flexural rigidity and the linear mass density is constant ($I\gamma\equiv C$), Gladwell and Morassi (2010) obtained closed-form families of beams isospectral to a given beam under any combination of pinned and sliding end conditions, by exploiting an equivalence between this class and a vibrating string and applying a Darboux lemma after reduction of the string equation to Sturm–Liouville canonical form. Their construction, however, yields families of isospectral beams, not a complete characterization of the isospectral set. The complete global description — a parameterization or classification of all coefficient pairs in $\mathcal{C}$ sharing a given vibration spectrum — remains open. Later work constructs further families — buckling spectra [2], Lie-factorization [6], Rayleigh and Timoshenko extensions [9, 10], and matrix-string and Camassa–Holm-type isospectral deformations [13, 14] — and solves the inverse problem for discrete beams, but no published result establishes completeness of the isospectral class in $\mathcal{C}$.

Background

A thin beam vibrating in flexure is classically modeled by the Euler–Bernoulli equation $(\gamma(x)u''(x))''=\omega^{2}I(x)u(x)$ on $[0,1]$, where $\gamma$ is the flexural rigidity (the product of Young's modulus and the second moment of area of the cross section), $I$ is the linear mass density (mass per unit length), and $\omega$ is a natural angular frequency; primes denote derivatives with respect to the axial coordinate. The end conditions studied here are combinations of pinned ends (vanishing deflection and bending moment: $u=0$ and $\gamma u''=0$, i.e. $u=u''=0$ since $\gamma>0$) and sliding ends (vanishing slope and shear force: $u'=0$ and $(\gamma u'')'=0$; for constant $\gamma$ the shear condition reduces to $u'''=0$) — the standard self-adjoint end conditions for the variable-coefficient equation, under which the free-vibration frequencies are real. Two beams are called isospectral, with respect to a fixed configuration of end conditions, if they have exactly the same sequence of free-vibration frequencies. A classical question of inverse vibration theory — a continuous analogue of 'can one hear the shape of a drum?' — is to what extent this spectrum determines the beam: for discrete (finite-dimensional) beam models it has long been known that three spectra under different end conditions are needed for uniqueness, so beams sharing one spectrum form a nontrivial family (Gladwell, 2002, who found such families for discrete beams via shifted QR factorization and Toda lattice flows). Gottlieb (1987) earlier found analytically solvable inhomogeneous beams isospectral to a homogeneous one, and Gottlieb (2002) found transformations generating isospectral strings. Gladwell and Morassi (2010) initiated the present question: they considered the class $\mathcal{C}$ of beams for which the product of bending stiffness and linear mass density is constant, $I(x)\gamma(x)\equiv C$. They showed that, in a precise sense, every beam in this class is equivalent to a vibrating string, reduced the string equation to Sturm–Liouville canonical form via a Liouville-type change of variables (a monotone reparametrization of the spatial variable that removes the leading variable coefficient), and applied the classical Darboux lemma (a transformation that produces a new Sturm–Liouville operator with an appropriately shifted, hence spectrum-matching, eigenvalue sequence) to obtain closed-form expressions for families of beams isospectral to a given beam, valid for pinned–pinned, pinned–sliding, and sliding–sliding end conditions. Their construction produces explicit subfamilies — analogous results for one-dimensional Schrödinger operators parameterize the full isospectral manifold — rather than a complete characterization of the isospectral set; a complete global description of the isospectral set remains open.

Problem Statement

Given a reference Euler–Bernoulli beam in the class $\mathcal{C}$ (i.e. positive coefficient pairs $(I,\gamma)$ with $I,\gamma\in C^{2}([0,1])$ and $I(x)\gamma(x)\equiv C$ on $[0,1]$), provide, for each of the pinned–pinned, pinned–sliding, and sliding–sliding end-condition configurations, a complete global description of its isospectral set under that configuration: a parameterization or classification of all pairs $(I,\gamma)$ in $\mathcal{C}$ whose Euler–Bernoulli vibration spectrum under the same end conditions coincides exactly with that of the reference beam. A complete answer must establish both (i) sufficiency — every member of the described family is isospectral to the reference beam under the stated end conditions, as the Darboux-based closed-form families of Gladwell and Morassi (2010) already do for their subfamilies — and (ii) necessity — every beam in $\mathcal{C}$ isospectral to the reference beam arises from the parameterization, so that no isospectral beam is missed.

References

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