Geometric nonlinear dynamic analysis of curved beams using curved beam element

Abstract

Instead of using the previous straight beam element to approximate the curved beam, in this paper, a curvilinear coordinate is employed to describe the deformations, and a new curved beam element is proposed to model the curved beam. Based on exact nonlinear strain-displacement relation, virtual work principle is used to derive dynamic equations for a rotating curved beam, with the effects of axial extensibility, shear deformation and rotary inertia taken into account. The constant matrices are solved numerically utilizing the Gauss quadrature integration method. Newmark and Newton-Raphson iteration methods are adopted to solve the differential equations of the rigid-flexible coupling system. The present results are compared with those obtained by commercial programs to validate the present finite method. In order to further illustrate the convergence and efficiency characteristics of the present modeling and computation formulation, comparison of the results of the present formulation with those of the ADAMS software are made. Furthermore, the present results obtained from linear formulation are compared with those from nonlinear formulation, and the special dynamic characteristics of the curved beam are concluded by comparison with those of the straight beam.

Experimental proof of emergent subharmonic attenuation zones in a nonlinear locally resonant metamaterial

“…Focusing on resonators exhibiting geometric quadratic nonlinearities, several designs can be mentioned, e.g. nonlinear micromechanical cantilever system integrated with nanotubes proposed by Cho et al 40 , the suspended cables studied by Zhao et al 41 , the M-shape resonator designed by Leadenham et al 42 and the arch resonators widely proposed in the literature [43][44][45][46][47][48] , mostly not in the context of metamaterials. Fig.…”

“…This has been achieved by exploiting the quadratic geometric nonlinearity arising in arched beams. Although the concept of arched resonators has been widely applied in micromechanical systems [43][44][45][46][47][48] , to the best of our knowledge, this approach has not yet been used in a practical design of metamaterials that promote the quadratic non-linearity in a local resonator. Numerical and analytical models are used to support the design process of the resonant unit cell, revealing an adequate agreement with the obtained experimental results.…”

Large displacement static and transient analysis of functionally graded deep curved beams with generalized differential quadrature method

“…For example, Gerges et al [2] used a consistent subparametric curved beam element for geometrically nonlinear static analysis of shallow beams. Pan et al [3] conducted geometric nonlinear dynamic analysis of curved beams using curved beam element with subtended angles varying up to 60°. Bathe et al [4] performed nonlinear static and dynamic analysis of beam structures using continuum-based finite element formulations.…”

A computationally efficient numerical model for a dynamic analysis of beam type structures based on the combined finite‐discrete element method

“…This phenomenon arises due to an inability of the formulations to describe a pure (in-extensional) bending deformation, and this is even more evident in the analysis of deep thin arch structures. The solution to this problem has emerged in the form of a coupled displacements polynomial field, mixed trigonometric polynomial fields and in higher-order of polynomials for both axial and transverse displacements [1][2][3][4][5]. The finite elements proposed so far are mainly dependent on the slenderness of their respective arches and on the rise to span ratio of a structure.…”