On the Eigenvalue Distribution for a Beam with Attached Masses

TL;DR

This paper analyzes the eigenvalue distribution of a hinged beam with attached masses and actuators, combining analytical spectral analysis with numerical simulations to understand free vibrations in this mechanical system.

Contribution

It provides new asymptotic estimates of eigenvalues for a beam with attached masses, integrating analytical and numerical methods.

Findings

Eigenvalues are asymptotically estimated with analytical methods.

Numerical simulations illustrate the eigenvalue distribution.

The model captures the dynamics of a beam with attached masses and actuators.

Abstract

We study a mathematical model of a hinged flexible beam with piezoelectric actuators and electromagnetic shaker in this paper. The shaker is modelled as a mass and spring system attached to the beam. To analyze free vibrations of this mechanical system, we consider the corresponding spectral problem for a fourth-order differential operator with interface conditions that characterize the shaker dynamics. The characteristic equation is studied analytically, and asymptotic estimates of eigenvalues are obtained. The eigenvalue distribution is also illustrated by numerical simulations under a realistic choice of mechanical parameters.