Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

Abstract

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von K" a arm" a an equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency o kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to o kn ; which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper. #

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

“…Whenw 0 = 0 (no imperfection), one recovers the usual von Kármán equations for perfect plates [40]. A free-edge boundary condition is considered, which enforces the following relationships to be fulfilled, for allt and θ:…”

“…In the above equations, ω i is the i-th non-dimensional natural flexural frequency of the perfect plate. The analytical expressions for Φ i and Ψ i as well numerical values for ω i and ζ i can be found in [40]. The critical parameters in such Galerkin expansions is the number of retained mode in the numerical analysis.…”

“…The critical parameters in such Galerkin expansions is the number of retained mode in the numerical analysis. In this study, the number of inplane modes has been fixed to 12, a sufficient value to ensure a five-digits accuracy for the cubic non-linear coefficients up to the 15 th modes, and four-digits accuracy up to the 26 th modes [40,41]. This accuracy is sufficient for the truncations we will study in the remainder of the paper.…”

“…The cubic coefficients Γ u rsp are those of the perfect case, presented in [40]. The linear and quadratic coupling coefficients α u p and β u pr appearing in Eq.…”

“…The modes are classified, as it is usual for circular plates, with two indexes (k, n), k being the number of nodal diameters and n the number of nodal circles. Modes (0, n) are called axisymmetric, while k 0 implies an asymmetric mode, which have two companion (or preferential) configurations for the same eigenfrequency [40,37]. For asymmetric modes, a binary index is often added in order to distinguish the two configurations, say (2, 0, C) for the cosine mode and (2, 0, S ) for the sine configuration.…”