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Null-field integral equation approach for free vibration analysis of circular

plates with multiple circular holes

Wei-Ming Lee1, Jeng-Tzong Chen2

Ya-Kuei Shiu1, Wei-Ting Tao1, Jyun-Chih Kao1

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

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Motivation

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Motivation

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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

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Outlines

5. Conclusions

3. Illustrated examples

2. Methods of solution

1. Motivation

4. Discussion

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Vibration of plate

xxwxw ),()( 44 Governing Equation:

is the lateral displacement

w is the lateral displacement is the frequency

parameter

4 is the biharmonic operator

is the domain of the thin plates

)1(12

3

24

hED

D

h ω is the angle frequencyρ is the surface density

D is the flexural rigidityh is the plates thickness

E is the Young’s modulusν is the Poisson ratio

w(x)

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Problem Statement

The eigenproblem of a circular plate with multiple circular holes

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The integral representation for the plate problem

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Kernel function

The kernel function is the fundamental solution which satisfies

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The slope, moment and effective shear operators

slope

moment

effective shear

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Kernel functions

In the polar coordinate of

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Direct boundary integral equations

displacement

slope

with respect to the field point x

with respect to the field point x

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.

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x

s

eU

O

iUr

qf

xr

Rf

Expansion

Degenerate kernel (separate form)

Fourier series expansions of boundary data

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Boundary contour integration in the adaptive observer system

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Vector decomposition

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Transformation of tensor components

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Linear system

where L denotes the number of circular boundaries

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Spurious eigenvalues and SVD updating technique

formulation

formulation

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Motivation

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A circular plate with an eccentric hole

Geometric data:R1=1mR2=0.4me=0.0 ～ 0.5mthickness=0.002mBoundary condition:Inner circle : freeOuter circle: clamped, simply -supported and free

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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)

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The former seven eigenvalues and eigenmodes (e =0.2)

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Effect of the eccentricity e on the frequency parameter

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The minimum singular value versus the frequency parameter

formulation formulation

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The minimum singular value versus the frequency parameter using the SVD technique of updating term

Suppress the spurious eigenvalue

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Discussions of accuracy

6.84956.85356.5264

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Discussions of accuracy

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A circular plate with three circular holes

Geometric data:R1=1mR2=0.4mR3=0.2mR4=0.2mo1=(0.0,0.0)o2=(0.5,0.0)o3=(-0.3,0.4)o4=(-0.3,-0.4)Thickness=0.002mBoundary conditions:Inner circles: freeOuter circle: clamped

R1

R2

R3

R4

o2o1

o4

o3

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

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Motivation

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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 degenerat

e 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 plat

e with an eccentric circular hole or multiple holes so far.

1.

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Thanks for your kind attention

The End