Experiments on Chaotic Vibrations of a Thin Circular Plate with Outer Clamped Edge and a Circular Center Hole

Abstract

This paper presents experimental results on chaotic vibrations of a thin circular plate with a circular center hole. The plate has an asymmetric curved configuration due to an initial imperfection, an in-plane compressive stress and lateral deformation by the gravity. First, linear natural frequencies and natural modes of vibration are measured. The second and the third modes of vibration have one nodal diameter with different natural frequencies. These nodal diameters are perpendicular to each other. The linear natural frequencies of the lowest and the second modes satisfy the condition of oneto-two internal resonance, closely. Characteristics of restoring force of the plate show the type of a softening-and-hardening spring. Next, the plate is excited with periodic acceleration, laterally. Sweeping the excitation frequency, nonlinear responses of the plate are measured. In a restricted frequency range of the principal resonance of the lowest mode, non-periodic responses with amplitude modulation are generated. The time histories of the response are measured for a long-time interval at four positions, simultaneously. The time histories are inspected by the Fourier spectrum, the Poincaré projection, the maximum Lyapunov exponents and the principal component analysis. The non-periodic response is found to be the chaotic response generated from the oneto-two internal resonance coupled with the lowest and the second modes. The results of the principal component analysis with the time histories of the multiple positions on the plate show that the lowest, second and third modes predominantly contribute to the chaotic response. Applying the calculation with short-time intervals cut out from the time histories, it is found that the nodal patterns of the second and the third modes of vibration are fluctuated along the circumferential direction. Furthermore, the exchange of the contribution ratio is shown between the lowest and the second modes of vibration.