Non-stationary vibrations of a viscoelastic structure

Abstract

The effect of the linear and nonlinear properties of the beam's material on the dynamic behavior of a structure under different kinematic influences is investigated. With an account for the viscoelastic properties of the material, the problem under consideration is reduced to a system of small-order linear or nonlinear integro-differential equations by the selection of coordinate functions satisfying geometric boundary conditions; the problem is solved by the averaging method or using quadrature formulas. In the article, the problems are solved using the finite element method and the Newmark method. First, a resolving matrix system of linear or nonlinear differential or integro-differential equations is obtained using the finite element method, and then it is solved by the Newmark method. The advantage of the proposed algorithm is the use in the solution of all possible modes of vibration, which are ignored in conventional methods. Comparisons of the results of the forced vibrations of a beam, taking into account the viscoelastic properties of the material under different kinematic influences, show that at the initial time, the elastic and viscoelastic solutions practically do not differ from each other. Over time, the amplitude of oscillations of the points of the beam, reaching a certain maximum value, remains constant and then begins to decrease gradually. Analysis of the presented results of forced vibrations of the beam shows that the general case, when nonlinear and viscous properties of the material are taken into account, leads to the greatest decrease in the amplitudes of displacements of the beam points compared with all other results obtained.