Outlines

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Free vibration analysis of a circular plate with
multiple circular holes by using the multipole
Trefftz method
Wei-Ming Lee Department of Mechanical
Engineering, China Institute of Technology,
Taipei, Taiwan
2009?06?17???????
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Intrduction

• Circular holes can reduce the weight of the whole
  structure or to increase the range of inspection.
  
• These holes usually cause the change of natural
  frequency as well as the decrease of load
  carrying capacity. .
• Over the past few decades, most of the researches
  have focused on the analytical solutions for
  natural frequencies of the circular or annular
  plates.

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• Laura et al. determined the natural frequencies
  of circular plate with an eccentric hole by using
  the Rayleigh-Ritz variational method.
• Lee et al. proposed a semi-analytical approach to
  the free vibration analysis of a circular plate
  with multiple holes by using the indirect and
  direct boundary integral method.
• Spurious eigenvalues occur when using BEM or
  BIEM.

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• The Trefftz method was first presented by Trefftz
  in 1926 and is categorized as the boundary-type
  solution such as BEM or BIEM.
• The Trefftz formulation is regular and free of
  the problem of improper boundary integrals.
• The concept of multipole method to solve
  multiply-connected domain problems was firstly
  devised by Zaviska.
• The multipole Trefftz method was proposed to
  solve plate problems with the multiply-connected
  domain in an analytical way.

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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Free vibration of plate
Governing Equation
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Problem Statement
Problem statement for an eigenproblem of a
circular plate with multiple circular holes
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The integral representation for the plate problem
The solution of free vibration in the polar
coordinate is
The Bessel equation
The modified Bessel equation
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The solution for
where is defined by
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The slope, moment and effective shear
slope
Moment
Effective shear
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Analytical derivations for the eigensolution
The lateral displacement by the multipole
expansion
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The Graf's addition theorem
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The addition theorem
The displacement field near the circular boundary
B0
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where
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The field of bending moment, m(x), near the
circular boundary Bp (p1,,H)
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The moment operator is defined as
The effective shear operator is
defined as
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The field of effective shear, v(x), near the
circular boundary Bp (p1,,H)
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For an outer clamped circular plate (u ? 0)
containing multiple circular holes with the free
edge (m v 0)
A coupled infinite system of simultaneous linear
algebraic equations
m0, 1, 2, ., M
A (H1)(2M1) system of equations the
direct-searching scheme by SVD
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Case 1 A circular plate with an eccentric hole
Geometric data R01m R10.4m e0.5m thickness0.
002m Boundary condition Inner circle
free Outer circle clamped
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Natural frequency parameter versus the number of
coefficients of the multipole representation
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The minimum singular value versus the frequency
parameter
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The former seven frequency parameters, mode types
and mode shapes
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Case 2 A circular plate with three holes
Geometric data R01m R10.4m R20.2m R30.2m O0(
0.0,0.0) O1(0.5,0.0) O2(-0.3,0.4) O3(-0.3,-0.4)
thickness0.002m Boundary condition Inner
circles free Outer circle clamped
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Natural frequency parameter versus the number of
coefficients of the multipole representation
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The minimum singular value versus the frequency
parameter
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The former six natural frequency parameters and
mode shapes
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Outlines
1. Introduction
2. Methods of solution
3. Illustrated examples
4. Concluding remarks
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Concluding remarks
The multipole Trefftz method has successively
derived an analytical model for a circular plate
containing multiple circular holes.
1.
An exact eigensolution can be derived from a
coupled infinite system of simultaneous linear
algebraic equations.
2.
3.
No spurious eigenvalue occurs in the present
formulation.
The proposed results match well with those
provided by the FEM using many elements to obtain
acceptable data for comparison.
4.
Numerical results show good accuracy and fast
rate of convergence thanks to the analytical
approach.
5.
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The End
Thanks for your kind attention