Isospectral Euler-Bernoulli Beams with Continuous Density and Rigidity Functions

Isospectral systems are those that have exactly the same free-vibration frequency spectrum with respect to a given boundary condition configuration. In this paper, seven different classes of inhomogeneous Euler-Bernoulli beams with continuous density and flexural rigidity functions are found that are analytically solvable and that are isospectral with a homogeneous beam in the clamped-clamped configuration. Of these exact solutions, one class is isospectral with the homogeneous case in all ten distinct configurations obtainable from clamped, sliding and two antiresonant end conditions, whereas another class is isospectral in the six configurations with clamped, hinged or free end conditions. A connection with inverse problems is discussed.