Nonlocal Timoshenko simply supported beam: Second spectrum and modes

Abstract

AbstractWe investigate the spectrum of frequencies of a nonlocal simply supported Timoshenko beam. When the mass matrix term is nonsingular, we can find the amplitudes of free vibrations as solutions of a second‐order matrix differential equation. These solutions are given in terms of a fundamental basis involving an impulsive matrix response and its derivative. This latter is given in closed‐form, involving a scalar function, its derivatives and coupling matrices. Simply supported boundary conditions have a nonclassical nature, but the frequencies are natural due to the influence of the nonlocal parameter. From the characteristic equation, we can find two branches of nonlocal frequencies referred to as the first and second spectrum and a critical frequency. This latter is the same cut‐off frequency for waves solutions of the nonlocal beam without boundary conditions. The nonlocal spectra are bounded, in contrast with the unbounded spectra of a local beam. For classifying eigenvalues as simple or double, the scalar wave function is useful. Simple eigenvalues occur when only one of the basic roots of the nonlocal characteristic polynomial are harmonically spaced, and double when both are harmonic and different. The eigenmodes have been found for both branches of the spectrum, and they are always oscillatory. At the critical frequency, the shape modes are determined by a limit process with the scalar function, resulting in occurrence the one pure‐shear mode. For a singular mass matrix, modes will exist for certain values of the nonlocal parameter but they are not unique.