Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM

Abstract

SUMMARYIn this paper, the spurious eigenequations for annular plate eigenproblems by using BIEM and BEM are studied in the continuous and discrete systems. Since any two boundary integral equations in the plate formulation (4 equations) can be chosen, 6 (C 4 2 ) options can be considered instead of only two approaches (single-layer and double-layer methods, or singular and hypersingular equations) which are adopted for the eigenproblems of the membrane and acoustic problems. The occurring mechanism of the spurious eigenequation for annular plates in the complex-valued formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and the Fourier series expansion for the circular boundary density are employed to derive the true and spurious eigenequations analytically. For the discrete system, the degenerate kernels for the fundamental solution and circulants resulting from the circular boundary are employed to determine the true and spurious eigenequations. True eigenequation depends on the specified boundary condition while spurious eigenequation is embedded in each formulation. It is found that the spurious eigenvalue for the annular plate is the true eigenvalue of the associated interior problem with an inner radius of the annular domain. Also, we provide three methods (SVD updating technique, Burton and Miller method and CHIEF method) to suppress the occurrence of the spurious eigenvalues. Several examples were demonstrated to check the validity of the formulations.

A boundary integral algorithm for the Laplace Dirichlet–Neumann mixed eigenvalue problem

“…(12) In view of the regularity result [1, Theorem 2.8] it follows that around the point (y 0 1 , y 0 2 ) the functions ψ + y 0 and ψ − y 0 of equation (8) are given in terms of the distance function (9) by (13) for all N ∈ N. If is smooth at (y 0 1 , y 0 2 ), in turn, the asymptotics of ψ + y 0 and ψ − y 0 around this point are given by (14) for any N ∈ N. Here Q i y 0 , i = 1, 4 are polynomials in the listed arguments. (Note that, while correct, the boundary expressions (13) are less detailed than the corresponding volumetric expression (10).…”

“…In the case of smooth , in turn, the equations in (14) tell us that the asymptotic behavior of ϕ near τ = a q 2 and near τ = b q 2 is given, respectively, by the relations…”

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

“…Free vibration analysis of this kind component is helpful to the work of mechanical design and flight control of the structure. As quoted by Leissa [19]: "the free vibrations of circular plates have been of practical and academic interest for at least a century and a half", most of research works have focused on the free vibration analysis of circular and annular plates [2,3,19,[24][25][26][27][28]. However, only few studies have conducted the problem of plate with an eccentric hole [17] or multiple holes.…”

“…The effect of eccentricity of the hole on the vibration characteristics of such plates subject to several boundary conditions is also addressed. Moreover, the inherent problem of spurious eigenvalues using BEM is investigated and the SVD updating technique [2] is employed to suppress the appearance of spurious eigenvalues.…”

Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs

“…Although the results for circular or annular plates are available in the literature [1,2,3,4,5,6,7,8]; however, all of these publications except [1,6,8] are confined to plates with concentric hole. Although a large amount of papers on the circular membrane vibration with circular holes were published [9], very a few papers on the free vibration of plate with several holes can be found.…”