Comparison of the Von Kármán and Kirchhoff models for the post-buckling and vibrations of elastic beams

International audience We compare different models describing the buckling, post-buckling and vibrations of elastic beams in the plane. Focus is put on the first buckled equilibrium solution and the first two vibration modes around it. In the incipient post-buckling regime, the classic Woinowsky-Krieger model is known to grasp the behavior of the system. It is based on the von Kármán approximation, a 2nd order expansion in the strains of the buckled beam. But as the curvature of the beam becomes larger, the Woinowsky-Krieger model starts to show limitations and we introduce a 3rd order model, derived from the geometrically-exact Kirchhoff model. We discuss and quantify the shortcomings of the Woinowsky-Krieger model and the contributions of the 3rd order terms in the new model, and we compare them both to the Kirchhoff model. Different ways to nondi-mensionalize the models are compared and we believe that, although this study is performed for specific boundary conditions, the present results have a general scope and can be used as abacuses to estimate the validity range of the simplified models.

Inextensibility and Its Effect on the Number of Equilibria of Shallow Buckled Beams

A buckled beam with shallow rise under lateral constraint is considered. The initial rise results from a prescribed end displacement. The beam is modeled as inextensible, and analytical solutions of the equilibria are obtained from a constrained energy minimization problem. For simplicity, the results are derived for the archetypal beam with pinned ends. It is found that there are an infinite number of zero lateral-load equilibria, each corresponding to an Euler buckling mode. A numerical model is used to verify the accuracy of the model and also to explore the effects of extensibility.

Free Vibration of Bistable Clamped-Clamped Beams: A Preliminary Study

We present a preliminary study on bistable clamped-clamped beams both analytically and experimentally relating the linear post-buckling vibrations to the generated sound. In the analytical study, closed-form natural frequencies and mode shapes around the first buckled configuration are derived from an eigenvalue problem. It is found that as the static deflection of the buckled beam increases, the natural frequencies of the anti-symmetric vibrational modes stay constant, while those of the symmetric vibrational modes increase asymptotically. In the experimental study, a bistable clamped-clamped buckled beam made of steel is switched quasi-statically by hand between the two stable configurations. The generated sound is measured by a microphone and analyzed in both temporal and frequency domains, which agrees well with the analytical results. This work lays the foundation for using bistable beams in a variety of applications such as actuators, resonators, energy harvesters, and vibration reduction.

Bifurcations in a Pre-Stressed, Harmonically Excited, Vibro-Impact Oscillator at Subharmonic Resonances

This study investigates the behavior of a damped, inelastic, sinusoidally forced impact oscillator which has its barrier placed such that the oscillator always vibrates under compression about its subharmonic resonant frequencies. The Poincaré sections at near subharmonic resonance conditions exhibit finger-shaped chaotic attractors, similar to the strange attractor mapping of Hénon and the ones found by Holmes in his study of chaotic resonances of a buckled beam. The number of such fingers are observed to increase as the barrier distance from the equilibrium is decreased. These chaotic states are interspersed with regimes of periodic behavior, with the periodicity being in accordance with well defined period adding laws. This study also focuses on the ordered behavior of the one-impact period-[Formula: see text] orbits around the higher subharmonics of the oscillator.