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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年 06月 17日國立海洋大學

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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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

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Free vibration of plate

Governing Equation:w(x)

4 4( ) ( ) ,w x λ w x x ΩÑ = Î

24

3

12 1

hλ

D

E hD

( )

=

=-

wr

m

ω is the angular frequency

D is the flexural rigidityh is the plates thickness

E is the Young’s modulus

μ is the Poisson’s ratio

ρ is the surface density

w is the out-of-plane displacement is the frequency parameter

4 is the biharmonic operator

is the domain of the thin plates

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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 4 4( ) ( )w x λ w xÑ =

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 (p=1,…,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 (p=1,…,H)

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For an outer clamped circular plate (u = θ = 0) containing multiple circular holes with the free edge (m = v = 0)

A coupled infinite system of simultaneous linear algebraic equations

A (H+1)(2M+1) system of equations+ the direct-searching scheme by SVD

m=0, ±1, ±2, …., ±M

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Case 1: A circular plate with an eccentric hole

Geometric data:R0=1mR1=0.4me=0.5mthickness=0.002mBoundary 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:R0=1mR1=0.4mR2=0.2mR3=0.2mO0=(0.0,0.0)O1=(0.5,0.0)O2=(-0.3,0.4)O3=(-0.3,-0.4)thickness=0.002mBoundary condition:Inner circles: freeOuter 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

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Concluding remarks

The multipole Trefftz method has successively derived an analytical model for a circular plate containing multiple circular holes.

An exact eigensolution can be derived from a coupled infinite system of simultaneous linear algebraic equations.

No spurious eigenvalue occurs in the present formulation.

Numerical results show good accuracy and fast rate of convergence thanks to the analytical approach.

1.

2.

3.

4.

5.

The proposed results match well with those provided by the FEM using many elements to obtain acceptable data for comparison.

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Thanks for your kind attention

The End