Non-linear vibrations of imperfect free-edge circular plates and shells

Abstract

Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Kármán equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.

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

“…The amplitude of the imperfection is parameterized by a (0,1) only. Evolution of all the linear and non-linear characteristics of this imperfection has already been studied in [37], and the type of non-linearity of the first modes are reported in [55]. Two amplitudes will be studied, a (0,1) =0.45h (h is the thickness of the plate),…”

“…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.…”

“…A static imperfection associated with zero initial stress is denoted asw 0 . The local equations for the imperfect plate reads [37,38,39]:…”

Statistical energy analysis of nonlinear vibrating systems

“…The presence of nonlinearity giving rise to a cascade of energy into higher frequencies has been studied in detail by Camier, Touze and colleagues [6,7], for the case of structural plate elements. The various categories of behaviour exhibited as the vibration amplitude is varied have been explored in [8], while [6,9] reveal how the spectra of the energy response show self-similar scaling laws for both transient and steady-state responses.…”

“…The various categories of behaviour exhibited as the vibration amplitude is varied have been explored in [8], while [6,9] reveal how the spectra of the energy response show self-similar scaling laws for both transient and steady-state responses. The self-scaling laws of [9] are compared favourably with experimental results for the case of harmonic excitation of isolated plates, at amplitudes that produce frequency content across a range of the order of several kilohertz.…”