Selection of Physical and Geometrical Properties for the Confinement of Vibrations in Nonhomogeneous Beams

Abstract

Confinement of flexural vibrations in nonhomogeneous beams is formulated as one of two types of an inverse eigenvalue problem. In the first problem, the beam’s geometrical and physical parameters and natural frequencies are determined for a prescribed set of confined mode shapes. In the second problem, the beam’s parameters are approximated for a given set of confined mode shapes and frequencies. In both problems, a set of mode shapes, which satisfy all of the boundary conditions and yield vibration confinement in prespecified spatial subdomains of the beam, are selected. Because closed-form solutions are not available, we discretize the spatial domain using the differential quadrature method. As a result, the eigenvalue problem is replaced by a system of algebraic equations, which incorporates the values of the beam’s parameters at all grid points. These equations constitute a well-posed eigenvalue problem, which can be readily solved to determine an equal number of unknowns characterizing the beam properties. In both confinement problems, the unknown physical and geometrical properties must be positive and are approximated using functions constructed from polynomials. These functions are specified at the beam’s left end, right end, or both. Numerical simulations are conducted to confirm convergence of the solution of the inverse eigenvalue problem. It is shown that the physical and geometrical properties can be reconstructed from a few mode shapes. The approximate parameters are finally substituted in the eigenvalue problem to confirm the confined mode shapes of the beam.