M M S S V V 0 Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes Wei-Ming Lee 1, Jeng-Tzong Chen 2 Ya-Kuei Shiu 1, Wei-Ting Tao 1, Jyun-Chih Kao 1 1 Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan 2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 時 間 : 2007 年 06 月 16 日 地 點 : 中國文化大學

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

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

M M S S V V 3 Motivation most research work has focused on the free vibration analysis of circular or annular plate. Only few studies are available for the plate with an eccentric hole or multiple holes. Principal-value calculation: more difficult to calculate than membrane vibration References: Circular hole: to reduce the weight of the whole structure or to increase the range of inspection Our goal:to develop a systematic, excellent accuracy, fast rate of convergence, high computational efficiency and free of calculating principal value approach

M M S S V V 4 Outlines 5. Conclusions 3. Illustrated examples 2. Methods of solution 1. Motivation 4. Discussion

M M S S V V 5 Vibration of plate Governing Equation: is the lateral displacement is the frequency parameter is the biharmonic operator is the domain of the thin plates ω is the angle frequency ρ is the surface density D is the flexural rigidity h is the plates thickness E is the Young’s modulus ν is the Poisson ratio w(x)w(x)

M M S S V V 6 Problem Statement The eigenproblem of a circular plate with multiple circular holes

M M S S V V 7 The integral representation for the plate problem

M M S S V V 8 Kernel function The kernel function is the fundamental solution which satisfies

M M S S V V 9 The slope, moment and effective shear operators slope moment effective shear

M M S S V V 10 Kernel functions In the polar coordinate of

M M S S V V 11 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 12 x s O x Expansion Degenerate kernel (separate form) Fourier series expansions of boundary data

M M S S V V 13 Boundary contour integration in the adaptive observer system

M M S S V V 14 Vector decomposition

M M S S V V 15 Transformation of tensor components

M M S S V V 16 Linear system where L denotes the number of circular boundaries

M M S S V V 17

M M S S V V 18 Spurious eigenvalues and SVD updating technique formulation

M M S S V V 19 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation

M M S S V V 20 A circular plate with an eccentric hole Geometric data: R 1 =1m R 2 =0.4m e=0.0 ～ 0.5m thickness=0.002m Boundary condition: Inner circle : free Outer circle: clamped, simply - supported and free

M M S S V V 21 Results (e =0.2) Natural frequency parameter versus the number of terms of Fourier series for the clamped-free annular plate (R1=1.0, R2=0.4 and e=0.2) The first minimum singular values versus the frequency parameter for the clamped-free annular plate (R1=1.0, R2=0.4 and e=0.2)

M M S S V V 22 The former seven eigenvalues and eigenmodes (e =0.2)

M M S S V V 23 Effect of the eccentricity e on the frequency parameter

M M S S V V 24 The minimum singular value versus the frequency parameter formulation

M M S S V V 25 The minimum singular value versus the frequency parameter using the SVD technique of updating term Suppress the spurious eigenvalue

M M S S V V 26 Discussions of accuracy

M M S S V V 27 Discussions of accuracy

M M S S V V 28 A circular plate with three circular holes Geometric data: R 1 =1m R 2 =0.4m R 3 =0.2m R 4 =0.2m o 1 =(0.0,0.0) o 2 =(0.5,0.0) o 3 =(-0.3,0.4) o 4 =(-0.3,-0.4) Thickness=0.002m Boundary conditions: Inner circles: free Outer circle: clamped R1R1 R2R2 R3R3 R4R4 o2o2 o1o1 o4o4 o3o3

M M S S V V 29 The former six natural frequency parameters and mode shapes for a circular clamped plate with three circular free holes

M M S S V V 30 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Motivation

M M S S V V 31 Concluding remarks A semi-analytical approach for solving the natural frequencies and modes for the circular plate with an eccentric hole was proposed The present method used the null BIEs in conjugation with the degenerate kernels, and the Fourier series in the adaptive observer system. The improper integrals in the direct BIEs were avoided by employing the degenerate kernels and were easily calculated through the series sum. The SVD updating technique can successfully suppress the appearance of spurious eigenvalue. From the numerical results presented in this paper, the present method provides more accurate semi-analytical eigensolutions for the circular plate with an eccentric circular hole or multiple holes so far

M M S S V V 32 Thanks for your kind attention The End