Unsteady behaviour of a heterogeneous elastic beam floating on shallow water

The unsteady behaviour of a thin elastic Euler beam with heterogeneous structural properties, floating freely on the surface of an ideal incompressible liquid is investigated using the linear theory. The unsteady behaviour of the beam is due to the incidence of a localized wave on its surface or initial deformation. Two methods of solving the problem are proposed in which the sagging of the beam is sought in the form of an expansion in eigenfunctions of the oscillations of a heterogeneous beam (the first method) or of a homogeneous beam (the second method) in the void. In both methods the problem is reduced to solving an infinite system of ordinary differential equations for the unknown amplitudes. The effect of different actions on a beam having a piecewise-constant distribution of the cylindrical stiffness and the specific mass is investigated. The eigenvalues of the systems of differential equations are determined.