Nonlocal Timoshenko simply supported beam: Second spectrum and modes

Abstract

We 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.

The Dissipative Bresse-Timoshenko System without a Second Spectrum is Well-Posedness and Exponential Stability

“…In 1921, Tymoshenko [1] optimized the Euler-Bernoulli beam model and the Rayleigh beam model and proposed the following hyperbolic system of two coupled wave equations ρ1ϕtt − k(ϕx + ψ)x = 0, ρ2ψtt − bψxx + k(ϕx + ψ) = 0, (1.1) which is called Timoshenko beam model, where ϕ and ψ are the deflection of the beam from its equilibrium position and the rotation of the neutral axis, respectively, ρ1 = ρA, ρ2 = ρI, b = EI and k = k GA are positive constants with ρ is the density, A is the cross-sectional area, I is the second moment of area of the cross-sectional area, E is the Young modulus of elasticity, G is the modulus of rigidity, k is the transverse shear factor. However, it was later discovered that the Timoshenko beam model admits two wave speeds k/ρ1 and b/ρ2, which contributes to a physical paradox called the second spectrum (see, for example, [2,3,4]). Based on these reasons, Elishakoff [5] proposed the following truncated version model by combining d'Alembert's principle for dynamic equilibrium from Timoshenko hypothesis,…”