Experimentally validated geometrically exact model for extreme nonlinear motions of cantilevers

Abstract

A unique feature of flexible cantilevered beams, which is used in a range of applications from energy harvesting to bio-inspired actuation, is their capability to undergo motions of extremely large amplitudes. The well-known third-order nonlinear cantilever model is not capable of capturing such a behaviour, hence requiring the application of geometrically exact models. This study, for the first time, presents a thorough experimental investigation on nonlinear dynamics of a cantilever under base excitation in order to capture extremely large oscillations to validate a geometrically exact model based on the centreline rotation. To this end, a state-of-the-art in vacuo base excitation experimental set-up is utilised to excite the cantilever in the primary resonance region and drive it to extremely large amplitudes, and a high-speed camera is used to capture the motion. A robust image processing code is developed to extract the deformed state of the cantilever at each frame as well as the tip displacements and rotation. For the theoretical part, a geometrically exact model is developed based on the Euler–Bernoulli beam theory and inextensibility condition, while using Kelvin–Voigt material damping. To ensure accurate predictions, the equation of motion is derived for the centreline rotation and all terms are kept geometrically exact throughout the derivation and discretisation procedures. Thorough comparisons are conducted between experimental and theoretical results in the form of frequency response diagrams, time histories, motion snapshots, and motion videos. It is shown that the predictions of the geometrically exact model are in excellent agreement with the experimental results at both relatively large and extremely large oscillation amplitudes.

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

“…For example, the free end of the beam can be easily subjected to more than half of a complete turn, even reaching a state of bending "backwards" beyond the fixed end (see Fig. 4(a) and [1,2,3,4]). At large amplitudes of vibration, so-called geometrical nonlinearities (trigonometric terms related to the cross-section rotation) become consequential.…”

Finite element computation of nonlinear modes and frequency response of geometrically exact beam structures

“…This method leads to a simple formulation in the form of partial differential equations with transverse degrees of freedom only, which are discretized by an expansion onto a normal mode basis. The strategy produces highly accurate results when compared with reference numerical FE simulations or experiments [49,37,50,51], although it is restricted to the case of a single cantilever beam.…”

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

“…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.…”

“…Frequency domain solving schemes can be particularly advantageous in modeling the nonlinear dynamics of vibrating systems since they directly target the steady state of the periodic solution. The series of recent works by Farokhi et al [27,35,28] on highly flexible beams proposed a frequency domain solving scheme to target periodic motions, but the proposed model was limited in scope to cantilever beams. The topic of the current paper is to propose a frequency domain-based geometrically exact model for the modeling of any general beam structure.…”