Trapped degenerate modes in an embedded-core flat circular plate

Abstract

Vibrational energy trapping and environmental perturbations reduction are of growing interest in high-precision acoustic sensing applications, but combining these two features in a single structure imposes higher demands on conventional resonators. In this paper, we design an 'embedded-core' circular elastic resonator, of which mode shapes are characterized by cyclic symmetry, stable energy trapping, and modal degeneracy. An embedded inner plate consisting of material with a wave velocity lower than that of its outer plate can trap in-plane vibrations in a flat structure layout, while conventional mono-material substrate design always adopts plate thickness variations as a common configuration. A set of two-dimensional equations of motion of the plate, governing cyclic symmetric vibrations with two different circumferential order numbers nθ, is developed and derived via variational equations and modified trigonometric series expansion. It is found that the mode shapes for both torsional modes (nθ=0) and one of the degenerate modes (nθ=2), shows excellent trapping behavior, where the displacement amplitudes decay exponentially outside the inner-outer plate boundary, in agreement with the results of finite element analysis. In detailed analysis of the degenerate mode, appropriate modifications based on material combinations and geometric parameters of the embedded-core plate are proven to not only determine the modal eigenfrequencies, but enhance the acoustic energy trapping efficiency within the inner plate region, which provides effective guides for the optimal design of this unique resonator.