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MMSS VV
1
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
MMSS VV
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Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Motivation
MMSS VV
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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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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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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 minimum singular value versus the frequency parameter using the SVD technique of updating term
Suppress the spurious eigenvalue
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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.
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