Refined Modeling and Free Vibration of Inextensional Beams on the Elastic Foundation

Abstract

In this study, the nonlinear equation of motion of the beam on the elastic foundation is obtained via the Newton's second law of motion, and its free vibration nature is investigated. Considering the inextensionai condition, the planar model of the beam accounting for the effects of the rotary inertia is derived. Then, the linear vibration and nonlinear vibration of the beam on the elastic foundation are examined. It is shown that the cut-ojf frequency can be observed in the frequency spectrum of the beam response. The effects of the rotary inertia on the natural frequencies are systematically investigated. Einally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed.

Multimode Dynamics of Inextensional Beams on the Elastic Foundation With Two-to-One Internal Resonances

“…The present paper expands upon a previous one by the same group of authors [5], where the beam-soil model was presented and the nonlinear free vibrations for different boundary conditions were addressed, along with the influence of different foundation models. Owing to consideration of the soil subgrade reaction, the beam-soil model and the ensuing behavior exhibit some differences with respect to classical ones.…”

“…Then the full-basis discretization of the form is performed by letting v{x,t) = J2T=i1kif)'l^k{x) and v{x,t) = J2T=i 2*(O'^/t(•*")' where qi! {t) and Zi(i) are the unknown displacement and velocity coordinates, respectively, and (^^(J:) is the kth in-plane mode shape corresponding to the natural frequency Wk (see Wang et al [5], where the eigenvalue problems of the clamped-free beam, hinged-free beam, and free-free beam were investigated). Next, the Galerkin discretization is performed in order to obtain the infinite set of ordinary-differential equations (ODEs) (3), the ODEs include two kinds of time-dependent terms, (1) i.e., the forced excitation and parametric excitation terms.…”

“…Therefore, due to the conservative character of the system, the effective nonlinear coefficients in a condition of two-to-one internal resonance exhibit symmetry and resonance features which result in a relatively simple dynamics to be possibly investigated via the analytical approach [12,14,15]. However, some nonlinearities of the beam on the elastic foundation may come from the second-order moment of the subgrade reaction [5], which is governed by the soil-structure interaction. The conservative character and the symmetry conditions may be disrupted.…”

“…Many studies have been concerned with the linear vibrations of beams on elastic foundations [1][2][3][4][5]. In these studies, the Winkler model and some two-or three-parameter models were widely applied.…”

Critical load and buckling of the single pile foundation subjected to the vertical load

“…1, a cylindrical coordinate system O −rϕz is chosen, with the origin O placed at the upper end of the undeformed pile and the z direction along the axes of pile. In this paper, the subgrade reaction of the elastic foundation denoted as q(z, t) = k 0 v(z, t) − k 1 v (z, t), where k 0 and k 1 are the Winkler parameter and the shear parameter of the soil medium, respectively [6]. Obviously, the displacements of the pile and the soil medium around the pile can be denoted as the separable functions of the cylindrical coordinate r, ϕ and z.…”