Eigenvibrations of a beam with two mechanical resonators attached to the ends

L N Koronova,K O Levinskaya,D M Korosteleva,S I Solov’Ev,A Glezer,S Roshchupkin

doi:10.1051/matecconf/202032903009

Abstract

The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

Highlights

We investigate the vertical eigenvibrations of a beam of length l
Assume that the ends x = 0 and x = l of the beam are elastically fixed by springs of stiffness K, at points x = 0 and x = l of the beam loads of mass M are joined
We study limit properties as K → ∞ with fixed

Summary

Introduction

We investigate the vertical eigenvibrations of a beam of length l. The vertical deflection w(x,t) of the beam at a point x at time t satisfies the following system of partial differential equations. M and as M → ∞ with fixed K of eigenvalues and eigenfunctions of the parameter spectral problem (5), (6). Spectral approximations for compact operators are investigated in the papers [1,2,3,4]. Preconditioned iterative methods for solving linear spectral problems are proposed and investigated in the papers [7,8,9,10,11,12,13,14]. Iterative methods for solving spectral problems with nonlinear parameter are proposed and investigated in the papers [15–26]. This paper develops and generalizes results of the papers [1,2,3,4,5,6]

Limit properties of the beam-resonators eigenvalue problem

Numerical experiments