M M S S V V 0 Free vibration analysis of a circular plate with multiple circular holes by using addition theorem and direct BIEM Wei-Ming Lee 1, Jeng-Tzong Chen 2 1 Department of Mechanical Engineering, China University of Science and Technology, Taipei, Taiwan 2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan Sep. 2-4, 2009 New Forest, UK BEM/MRM 31

M M S S V V 1 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation Outlines

M M S S V V 2 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation Outlines

M M S S V V 3 Numerical method Overview of numerical methods Domain type Boundary type FEM FDM BEM BIEM Meshless method

M M S S V V 4 Motivation Two questions of BIEM OR BEM The improper integral in the boundary integral equation It is difficult to calculate the principal-value of plate problem High order derivative when field point and source point are located on different circular boundaries Approaches to these problems The degenerate kernel, tensor transformation Ref: Lee, W. M. & Chen, J. T., Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes. Computational Mechanics, 42, pp.733 – 747, 2008.

M M S S V V 5 The combination of tensor transformation and the higher order derivative increases the difficulty in computation and then affect the accuracy of its solution. In addition, the method proposed by Lee & Chen [10] belongs to point-matching approach and requires more efforts for computation due to the collocation of boundary nodes. Motivation

M M S S V V 6 Is it possible to have a method which needs not … Motivation Tensor transformation The answer is YES and please to see the next. Collocation points

M M S S V V 7 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation

M M S S V V 8 Vibration of plate Governing Equation: u(x)u(x) is the frequency parameter is the biharmonic operator is the domain of the thin plates u is the lateral displacement ω is the angle frequency ρ is the volume density D is the flexural rigidity h is the plates thickness E is the Young’s modulus μ is the Poisson’s ratio

M M S S V V 9 Problem Statement The eigenproblem of a circular plate with multiple circular holes

M M S S V V 10 The integral representation for the plate problem

M M S S V V 11 Kernel function The kernel function is the fundamental solution, which satisfies

M M S S V V 12 The slope, moment and effective shear operators slope moment effective shear

M M S S V V 13 Kernel functions In the polar coordinates of

M M S S V V 14 Direct boundary integral equations displacement slope with respect to the field point x normal moment effective shear force Among four equations, any two equations can be adopted to solve the problem.

M M S S V V 15 Null-field integral equations By collocating the field point outside the domain, i.e.  C x  If degenerate kernel functions are used C x  B  

M M S S V V 16 Addition theorem Degenerate kernel (Separated kernel)

M M S S V V 17 Complex Fourier series expansions of boundary data Displacement Bending slope Bending moment shear force

M M S S V V 18 Analytical eigensolution for a circular plate with multiple circular holes and the null field near the circular boundary B 0 Considering a clamped circular plate with H circular holes

M M S S V V 19 The shear operator : The moment operator :

M M S S V V 20 x Addition Theorem

M M S S V V 21

M M S S V V 22 The first and second null integral equation where (24) (29)

M M S S V V 23 Considering the null field near the circular boundary B p, P=1, …,H

M M S S V V 24 where (31) (36)

M M S S V V 25 A couple infinite system of simultaneous linear algebraic equations If m=0, ±1, ±2,….±M, a truncated (H+1)(2M+1) system of equations is given. (37)

M M S S V V 26 Direct-searching scheme The eigenvalue can be obtained by applying the SVD technique to the system of Eq.(37).

M M S S V V 27 The SVD updating technique To provide sufficient constrains, UM formulation is considered. (40) By combing eqns. (37) and (40), spurious eigenvalues can be suppressed.

M M S S V V 28 A circular plate with an eccentric hole formulation The SVD updating technique

M M S S V V 29 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation

M M S S V V 30 A circular clamped plate with three circular free holes

M M S S V V 31 Natural frequency parameter versus the number of terms of Fourier series Fast rate of convergence with few terms of Fourier series

M M S S V V 32 The minimum singular value versus the frequency parameter by using three different methods

M M S S V V 33 The former five natural frequency parameters and mode shapes for a circular clamped plate with three circular free holes by using the present method, semi-analytical method and FEM

M M S S V V 34 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation

M M S S V V 35 Concluding remarks 2.Based on the addition theorem, two critical problems of improper integration and the higher derivative in the multiply-connected domain problem were successively treated in a novel way. 4.The SVD updating technique can successfully suppress the appearance of spurious eigenvalues. 5.Numerical results show good accuracy and fast rate of convergence thanks to the analytical method. 3.A couple infinite system of simultaneous equations has been derived as an analytical model for the free vibration of a circular plate with multiple circular holes. 1.Natural frequencies and natural modes of a circular plate with multiple circular holes have been solved theoretically.

M M S S V V 36 Thanks for your kind attention The End