Natural Frequency Analysis of Two Nonlinear Panels Coupled with a Cavity Using the Approximate Elliptic Integral Solution and the Method of Harmonic Residual Minimization

Abstract

The nonlinear structural acoustic problem considered in this study is the nonlinear natural frequency analysis of flexible double panels using the elliptic integral solution method. There are very limited studies for this nonlinear structural-acoustic problem, although many nonlinear plate or linear double panel problems have been tackled and solved. A multistructural/acoustic modal formulation is derived from two coupled partial differential equations which represent the large amplitude structural vibrations of the flexible panels and acoustic pressure induced within the air gap. One is the von Karman’s plate equation and the other is the homogeneous wave equation. The results obtained from the proposed method approach are verified with those from a numerical method. The effects of vibration amplitude, gap width, aspect ratio, the numbers of acoustic modes and harmonic terms, and so forth on the resonant frequencies of the in-phase and out of phase modes are examined.

Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

“…It is assumed that the panel absorber and flexible wall are simply supported. According to [10,11], the governing equations of the nonlinear symmetric panel and flexible wall are shown in the following,…”

“…Thus, the residual terms, R c and R o , are considered on the right side of Equations ( 3) and (4). ω is the nonlinear resonant frequency of the whole structural acoustic system and given by [11]…”

“…In this section, the derivations of the two normalized acoustic pressures (i.e., P c and P o in Equations ( 1) and ( 2)) are shown and described. Firstly, with reference to the 3-dimensional homogeneous wave equation [10][11][12]20]:…”

“…However, frequency-amplitude analysis was not included in [10] or in any other related works. In the work of [11], two governing equations for two identical panels with same vibration amplitude coupled with a cavity were simplified to become just one governing equation. This is the equivalent of a governing equation for a panel-cavity problem with only one flexible panel (not two).…”