Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments

Abstract

This article is dev oted to an experimental v alidati on of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced v ibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the v ibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driv ing frequency are compared to the theoretical ones, leading to a quantitativ e v alidation of the predictions giv en by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relev ance of the approach.

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

“…However, the lowest harmonic distortion is obtained by adjusting precisely the position at rest of the magnet so that the symmetry plane of the magnet coincides with the side plane of the coil. The harmonic distortion is in this case of the order of 1% for a magnet displacement amplitude of the order of 2.5 mm [14], which is much greater than the maximum amplitudes encountered during the nonlinear experiments (see Section 6: the amplitude of the center of the shell is less than 0.5 mm in Fig. 16).…”

“…A transfer of energy toward the asymmetric companion modes is observed, those modes oscillating at half the driving frequency, thus creating a subharmonic in the shell response. A 1:1 internal resonance between two asymmetric modes has already been studied by the authors in the case of a circular plate, theoretically in [13] and experimentally in [14]. The experimental setup used here shares some common feature with the one used in [14].…”

“…The coil is fed through a power amplifier by either a random noise for the modal analyzes (Section 3) or by a sine signal for the nonlinear experiments (Section 6). This excitation device has already been used in [14], where the interested reader can find a precise description of the calibration procedure that allows to evaluate the force acting on the magnet as a function of the current intensity in the coil. It has been found that the force is proportional to the intensity, under the condition that the magnet has a constant position with respect to the coil.…”

Conservative numerical methods for the Full von Kármán plate equations

“…Mode coupling and energy exchange between internally resonant modes are a second common feature, now well established in the literature [4][5][6][7][8][9][10][11].…”

A family of conservative finite difference schemes for the dynamical von Karman plate equations

“…This is an area which has existed for some time, but only now is the computational power of small commercially available computers great enough to begin to attempt simulations of complex objects in something approaching real time, and nonlinear dynamics plays a critical role in all musical instruments. The present case of the plate under large amplitude vibration conditions is a good approximation to the behavior of various percussion instruments, such as gongs and cymbals [24,25], for which linear models are unable to reproduce such audible effects as crashes (buildup of high-frequency energy), pitch glides (because of variations in modal frequencies with amplitude), and also subharmonic generation. 1 The reader may be interested to hear synthetic sound examples generated using the numerical methods described in this article [26].…”