Effect of Imperfections and Damping on the Type of Nonlinearity of Circular Plates and Shallow Spherical Shells

Physica D: Nonlinear Phenomena

et al. 2014

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The dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence (WT) regime is investigated with a finite difference, energy-conserving scheme. The numerical method allows the simulation of pointwise forcing together with realistic boundary conditions, a set-up which is close to experimental conditions. In the absence of damping, the framework of non-stationary wave turbulence is used. Numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake. Self-similar dynamics of the spectra are found with and without periodic external forcing. For the periodic forcing, the mean injected power is found to be constant, and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy. For the free turbulence, the energy contained in the cascade remains constant while the frequency front increases as t 1/3 . These self-similar solutions are found to be in accordance with the kinetic equation derived from the von Kármán plate equations. The effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies, as compared to the unforced case. The presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics. The steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added.

Section: Imperfect Undamped Platesmentioning

confidence: 99%

Dynamics of the wave turbulence spectrum in vibrating plates: A numerical investigation using a conservative finite difference scheme

Ducceschi

Cadot

Physica D: Nonlinear Phenomena

et al. 2014

Self Cite

“…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),…”

Section: Lyapunov Exponents and Power Spectramentioning

confidence: 99%

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

International Journal of Non-Linear Mechanics

Thomas

Amabili

2011

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric nonlinearity, are used to model the large amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the structure. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveals the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially adressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, ap- *

“…The convergence is addressed, showing that the present model is reliable for imperfection amplitude being more than ten times the thickness with a limited number of expansion functions. The non-linear coefficients with known imperfections have been addressed in Touzé et al (2007), where the type of non-linearity (hardening/softening behaviour) has been computed.…”

Section: Results On Typical Imperfectionsmentioning

confidence: 99%

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

Camier

European Journal of Mechanics - A/Solids

Thomas

2009

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.