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Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM. (English) Zbl 1121.74059

Authors’ summary: The spurious eigenequations for annular plate eigenproblems by using BIEM and BEM are studied for continuous and discrete systems. Since any two boundary integral equations in the plate formulation (4 equations) can be chosen, 6 \((C_{2}^{4})\) 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 complex-valued formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and 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 spurious eigenvalues. Several examples are demonstrated to check the validity of the formulations.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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