Transverse Vibration and Waves in a Membrane: Frequency Domain Spectral Element Modeling and Analysis

Abstract

Although the spectral element method (SEM) has been well recognized as an exact continuum element method, its application has been limited mostly to one-dimensional (1D) structures, or plates that can be transformed into 1D-like problems by assuming the displacements in one direction of the plate in terms of known functions. We propose a spectral element model for the transverse vibration of a finite membrane subjected to arbitrary boundary conditions. The proposed model is developed by using the boundary splitting method and the waveguide FEM-based spectral super element method in combination. The performance of the proposed spectral element model is numerically validated by comparison with exact solutions and solutions using the standard finite element method (FEM).

A generic type of frequency-domain spectral element model for the dynamics of a laminated composite plate

“…By following the approach introduced in the previous studies of the authors [18][19][20]22], the free wave solutions that satisfy the weak form given by Eq. (12) can be obtained approximately by using the boundary splitting method [21] and the SSEM in which the Kantorovich method-based finite strip element method and the frequency-domain waveguide method are used [16].…”

“…However their spectral element models can be assembled only in one plate direction (e.g., the x-direction). Park and his colleagues [18][19][20] developed spectral element models for membranes and isotropic and orthotropic composite plates by using a combination of the boundary splitting method [21] and SSEM [16]. Their spectral element models can be assembled in two plate directions (i.e., the x-and y-directions).…”

Nonlinear vibration analysis of a membrane based on large deflection theory

“…The passive effect of PVDF on the dynamics of an inflatable space-based rectangular shaped structure has been studied and trends in natural frequencies for various patch areas and thickness have been explored. Park et al. (2014) proposed a spectral element model for the transverse vibration of a finite membrane subjected to arbitrary boundary conditions, the performance of the proposed spectral element model was numerically validated by comparison with exact solutions and solutions using the standard finite element method (FEM).…”

Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions

“…However, as their spectral element models can be assembled only in another direction (the -direction), their applications must be limited to very specific boundary conditions. Recently, Park et al [5,18,19] developed spectral element models for rectangular membrane, isotropic plate, and orthotropic laminated composite plate elements by using two methods in combination: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. They derived frequency-dependent shape functions by applying a Kantorovich method-based finite strip element method in one direction and the SEM in another direction in combination, and vice versa.…”

“…Accordingly, their spectral element models can be assembled in both the -anddirections. However, the spectral element models by Park et al [5,18,19] still have some limitations because they are valid only for finite rectangular membranes or plate elements whose four corner nodes are fixed. To the authors' best knowledge, there have been no reports on a generic type of spectral element model that can be assembled in any direction of a plate subjected to arbitrary boundary conditions.…”