Large oscillations of beams and columns including self-weight

Santillan, Sophia T.; Plaut, Raymond H.; Witelski, Thomas P.; Virgin, Lawrence N.

doi:10.1016/j.ijnonlinmec.2008.04.007

Cited by 15 publications

(11 citation statements)

References 36 publications

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“…However, very little investigation has been done to date on the effect of gravity on the nonlinear dynamics of vertical cantilevers, especially at extreme amplitudes of vibration, which is the present aim of this paper. Santillan et al derived an analytical perturbation approximation based on the elastica model for the first nonlinear mode of a standing cantilever including self-weight and compared the results with numerical finite difference results [18]. In [18], the authors also compared their backbone approximations to the analytical expression for the amplitude-vs-frequency relationship of cantilever beams derived by Luongo et al [19], though this work did not include the effect of gravity.…”

Section: Introductionmentioning

confidence: 99%

Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes

2023

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In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.

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Section: Introductionmentioning

confidence: 99%

Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes

2023

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“…Moreover, these two classical systems are often studied in the literature on nonlinear beam dynamics [36,53,54,28,55,56]; some of these works are used in what follows as reference simulations in order to validate the quaternion-based model presented in this work. In particular, given that in 3D the beam has two transverse directions orthogonal to the longitudinal direction, we take a special interest in the one-to-one (1:1) internal resonance (IR) phenomenon that is uncovered for beams of (near) symmetrical cross sections.…”

Section: Test Casesmentioning

confidence: 99%

Quaternion-based finite element computation of nonlinear modes and frequency responses of geometrically exact beam structures in three dimensions

Debeurre,

Grolet,

Thomas

2024

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In this paper, a novel method for computing the nonlinear dynamics of highly flexible slender structures in three dimensions (3D) is proposed. It is the extension to 3D of a previous work restricted to in-plane (2D) deformations. It is based on the geometrically exact beam model, which is discretized with a finite element method and solved entirely in the frequency domain with a harmonic balance method (HBM) coupled to an asymptotic numerical method (ANM) for continuation of periodic solutions. An important consideration is the parametrization of the rotations of the beam's cross sections, much more demanding than in the 2D case. Here, the rotations are parametrized with quaternions, with the advantage of leading naturally to polynomial nonlinearities in the model, well-suited for applying the ANM. Because of the HBM-ANM framework, this numerical strategy is capable of computing both the frequency response of the structure under periodic oscillations and its nonlinear modes (namely its backbone curves and deformed shapes in free conservative oscillations). To illustrate and validate this strategy, it is used to solve two 3D deformations test cases of the literature: a cantilever beam and a clamped-clamped beam subjected to one-to-one (1:1) internal resonance between two companion bending modes in the case of a nearly square cross section.

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“…A large variety of beam structures are used as stiffness elements in above-mentioned applications and theoretical studies, for instance, cantilever beams, buckled beams, nanobeams, as well as beams with different shapes and configurations, which have been extensively studies, e.g. see [37][38][39][40][41][42][43][44][45][46][47][48]. In 1980's, some scholars started to study the nonlinear dynamics of beam, e.g., Moon and Shaw et al [49,50], who reduced a cantilever beam by a Galerkin approximation to the Duffing oscillator.…”

Section: Introductionmentioning

confidence: 99%