M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 1 Plate vibration Part III

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海洋大學力學聲響振動實驗室MSVLAB HRE NTOU

Plate vibration

Part III

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction

3. Derivation by Circulants

4. SVD updating terms

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction



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Vibration of plates

xxuxu ),()( 44 Governing Equation:

u is the lateral displacement is the frequency

parameter

4 is the biharmonic operator

is the domain of the thin plates

)1(12 2

3

24

hED

D

h is the surface density Dis the flexural rigidity

is the angle frequency

h is the plates thicknessE is the Young’s modulus is the Poisson ration

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Field representation using RBF

),()( jij sxcxu

)(),( rsx sj

Bx

To match B.C.Field representation

cu

****

****

****

****

****

0r

0r

xi

s1

s2

s3

s4

ri1

rij

s2N

s1

s2

sj

xis3

s4

rij

x1

x2

x3

x4

s2N

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Data bank of RBFRadial basis function (RB

Fs)Radial basis function (RB

Fs)

Mesh methodMesh method

Globally-supported RBFsGlobally-supported RBFs

DRBEMNardiniBrebbia

1982

DRBEMNardiniBrebbia

1982

Method of particular integral

Ahmad & Banerjee1986

Method of particular integral

Ahmad & Banerjee1986

rr 1)( rCr )(

Meshless methodMeshless method

Globally-supported RBFsGlobally-supported RBFsCompactly-supported RBFs

C.S. Chen

Compactly-supported RBFs

C.S. Chen

Volume potential

Volume potential

Method of fundamental

solution

Method of fundamental

solution

)14()1()( 4 rrr

)(

)(

),( )(

rU

sxU

xsUr

c

c

c

(Potential theory)

The present method

)()()( 00 rIrJr

Imaginary-part fundamental

solution

EquivalencePolyzos & Beskos

1994

EquivalencePolyzos & Beskos

1994

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction



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Displacement field of plate vibration

N

jjj

N

jjj xsQxsPxsu

2

1

2

1

),(),(),(

2N is the number of boundary nodes

s is the source point

x is the collocation point

j and j are the unknown densities

P and Q can be obtained from either two combinations of U, , M and V

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Four kernels

))}()((8

Im{),( )1(0

)2(02

riHrHi

xsU

))()((8

1),( 002

rIrJxsU

Imaginary-part fundamental solution

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Four kernels

)),((),(

)),((),(

)),((),(

xsUKxsV

xsUKxsM

xsUKxs

v

m

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Operators

))(

()1()(

)(

)()1()()(

)()(

22

2

22

tntnK

nK

nK

v

m

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))(()(

))(()(

))(()(

xuKxv

xuKxm

xuKx

v

m

Slope, Moment and Shear

Slope

Moment

Shear

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B.C. True eigenequation

Clamped

plate

u(x)=0

(x)=0

Simply-supported

plate

u(x)=0

m(x)=0

Free

plate

m(x)=0

v(x)=0

True eigenequation of three cases

0)()()()( JIIJ

)1(

2

)(

)(

)(

)( 11

J

J

I

I

0)()()1(4)()(2)(1(

))()()()()(1)(1(2

))()()()()()1)(1((

211

22

1122

1144222

aIaJaIaJaa

aIaJaIaJa

aIaJaIaJa

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0)( xu

0)( x

0 U

0 U

0

0

U

U CSM

Clamped plate

Nontrivial

CSMdet0

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0)( xu

0)( xm

0 U

0 mmU

0

0

mmU

U

Simply-supported plate

Nontrivial

SSM0

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0)( xm

0)( xv

0 mmU

0 vvU

0

0

vv

mm

U

U

Free plate

Nontrivial

FSM0

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Governing Equation▽4u(x) = λ4u(x) , xΩ

Find interior modes

Game over

Flow chart of the present method for clamped case by using U and kernels

N

j

N

jjjjj

j

N

j

N

jjjj

sxssxsUx

sxssxsUxu

2

1

2

1

2

1

2

1

)(),()(),()(

)(),()(),()(

Off-Diagonal term

Direct computation L’Hôpital’s ruleInvariant method

Diagonal term

Constructing the four influence matrices[U], [Θ], [U] and [Θ]

SVD technique[SM] = ΦsΣΨs

T

Plot det[SM] or plot σ1 versusλ.

Find eigenvaluesλ

Construction of [SM]

NN

U

USM

44

Find form s

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Case 1: Circular clamped plate using the present method

2 4 6 8

freq u en cy p a ra m eter

1e-249

1e-234

1e-219

1e-204

1e-189

1e-174

1e-159

1e-144

1e-129

1e-114

1E-099

1E-084

1E-069

1E-054

1E-039

1E-024

1E-009

1000000

1E+021

1E+036

det[SM

]

C la m p ed c ircu la r p la teU sin g U , k ern e ls

T T T

T

TT< 3 .1 9 6 > < 4 .6 11 > < 5 .9 0 6 >

< 6 .3 0 6 >

< 7 .1 4 4 >

< 7 .7 9 9 >

T : T ru e e ig en v a lu es

2 4 6 8


1e-249

1e-234

1e-219

1e-204

1e-189

1e-174

1e-159

1e-144

1e-129

1e-114

1E-099

1E-084

1E-069

1E-054

1E-039

1E-024

1E-009

1000000

1E+021

1E+036

det

[SM

]

C la m p ed c ircu la r p la teU sin g U , k ern e ls

S< 3 .1 9 6 > < 4 .6 11 > < 5 .9 0 6 >

< 6 .3 0 6 >

< 7 .1 4 4 >

< 7 .7 9 9 >

S S

S

SS

S : S p u r io u s e ig en v a lu es

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Case 2: Circular simply-supported plate using the present method

2 4 6 8


1e-231

1e-216

1e-201

1e-186

1e-171

1e-156

1e-141

1e-126

1e-111

1E-096

1E-081

1E-066

1E-051

1E-036

1E-021

1E-006

det

[SM

]

S im p ly -su p p o rted c ircu la r p la teU sin g U , k ern els

< 3 .1 9 6 > < 4 .6 11 > < 5 .9 0 6 >< 6 .3 0 6 >

< 7 .1 4 4 >

< 7 .7 9 9 >

S S S

S

S

S


2 4 6 8


1e-231

1e-216

1e-201

1e-186

1e-171

1e-156

1e-141

1e-126

1e-111

1E-096

1E-081

1E-066

1E-051

1E-036

1E-021

1E-006

det

|SM

|

S im p ly -su p p o rted c ircu la r p la teU sin g U , k ern els

T< 2 .2 2 1 >

< 3 .7 2 8 >< 5 .0 6 1 >

< 5 .4 5 2 >

< 6 .3 2 1 >

< 6 .9 6 3 >

< 7 .5 3 9 >

TT

T

T

T

T

T : T ru e e ig en v a lu es

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Case 3: Circular free plate using the present method

2 4 6 8


1e-193

1e-178

1e-163

1e-148

1e-133

1e-118

1e-103

1E-088

1E-073

1E-058

1E-043

1E-028

1E-013

100

1E+017

1E+032

1E+047

det

[SM

]

F ree c ircu la r p la teU sin g U , k ern els

< 2 .2 9 4 >

< 3 .0 1 1 >

< 3 .4 9 9 >

< 4 .5 2 9 >

< 4 .6 40 >

TT : T ru e e ig en v a lu es

T< 5 .7 5 0 >

< 5 .9 3 7 >

< 6 .2 0 5 >

< 6 .8 4 2 >

< 7 .2 7 5 >

< 7 .7 3 7 >

< 7 .9 2 1 >

T

T

T

T

T

T

T

T

T

T

2 4 6 8


1e-193

1e-178

1e-163

1e-148

1e-133

1e-118

1e-103

1E-088

1E-073

1E-058

1E-043

1E-028

1E-013

100

1E+017

1E+032

1E+047

det[SM

]

F ree c ircu la r p la teU sin g U , k ern els

< 3 .1 9 6 > < 4 .6 1 1> < 5 .9 0 6 >

< 6 .3 0 6 >

< 7 .1 4 4 >

< 7 .7 9 9 >

S


S S

S

S

S

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Mode 1 (λ =2.221)

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

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Mode 2 (λ=3.196)

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

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23

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Mode 3 (λ =3.728)

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction



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The degenerate kernels for interior and exterior problems:

,))(cos()]()()1()()([

8

1),,(

,))(cos()]()()1()()([8

1),,(

),(

2

2

RmRIIRJJRU

RmIRIJRJRUxsU

mmm

mmm

E

mmm

mmm

I

；

；

Degenerate kernels for circular case

X

Y

R

S

X

Exterior problem

X

Y

R

S

X

Interior problem

Blue: field pointsRed: source points

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012321

10432

324201222

22321012

1222210

aaaaa

aaaaa

aaaaa

aaaaa

aaaaa

K

N

NNNN

NNN

NN

Circulants

),( ijij xsKa

12212

222

1210 )()()(][

NNNNN CaCaCaIaK

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NN

NC

22

2

00001

10000

00100

00010

Circulants

0 12

N

)2

2sin()

2

2cos(2

2

Ni

Ne N

i

MMSS

28

VV


NN

aaaa NN

U

),1(,,2,1,0

1212

2210

][

)]()()1()()([4 2

][

IIJJNU

Circulants

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The eigenvalues of matrices

T

NN

UN

UN

UN

U

U

U

TUU

22

][

][)1(

][)1(

][1

][1

][0

00000

00000

00000

00000

00000

00000

MMSS

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VV


][

][

][

][

SVD

UU

SVD

SVD

UU

SVD

U

U

Relationships

Diagonal matrix

Diagonal

element

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VV


T

U

U

NN

TTU

TTUCSM

0

0

0

0

22

Determinant (for clamped)

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NN

JIIJ

JIIJN

SM

N

N

N

N

UU

C

),1(,,2,1,0

,0)}()()()({

)]()()()([16

)1(

)(

]det[

11

11)1(

2

2

)1(

][][][][

Eigenequation for clamped boundary

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T

U

U

TTU

TTUS

mm

mm

SM

0

0

0

0

Determinant (for simply-supported)

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][

][

][

][

m

m

SVDm

UU

SVDm

SVD

UU

SVD

U

U

Relationships

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VV


NN

IJ

IJIJ

JIIJN

SM

N

N

N

N

UU

S

),1(,,2,1,0

0)}()(2

))()()()()(1{(

)]()()()([16

)1(

)(

]det[

11

11)1(

2

2

)1(

][][][][

Eigenequation for simply-supported boundary

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VV


T

U

UU

TTU

TTUF

vv

mm

vv

mmSM

0

0

0

0

Determinant (for simply-supported)

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VV


][

][

][

][

v

v

m

m

SVDv

UU

SVDv

SVDm

UU

SVDm

U

U

Relationships

MMSS

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Eigenequation for free boundary

NN

IJIJ

IJIJ

IJIJ

IJIJN

SM

N

N

N

N

UU

F

),1(,,2,1,0

,0)()()1(4)()(2)(1(

))()()()()(1)(1(2

))()()()()()1)(1({

)]()()()([16

)1(

)(

]det[

211

22

1122

1144222

11)1(

42

2

)1(

][][][][

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Comparisons of the NDIF and present method

Kang Present method

Base

Clamped plate

Simply-supported

plate

Treatment Net approach Dual formulation with SVD updating

)(),( 0 rJxsU

)(),( 0 rIxs

0)()()()( 11 rIrJrIrJ

0)( rJ

)1(

2

)(

)(

)(

)( 11

r

rJ

rJ

rI

rI

0)()()()( 11 rIrJrIrJ

)1(

2

)(

)(

)(

)( 11

r

rJ

rJ

rI

rI

0)()()()( 11 rIrJrIrJ

sn

xsUxs

),(

),(

))()((8

1),( 002

rIrJxsU

0)()()()( 11 rIrJrIrJ

0)( rJ

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Circular clamped plate using different methods

2 4 6 8

F req u en cy p a ra m eter

-300

-200

-100

0

Log

arit

hm

val

ues

of d

et [SM

]

C ircu lan t m e th o dN D IF m e th o dP re se n t m e th o d

T

T

TT

TT

T

TT

TT

T

S

S

SS

SS

T

T

TT

TT

S

< 3 .1 9 6 >

< 4 .6 1 1 >

< 5 .9 0 6 >< 6 .3 0 6 >

< 7 .1 4 4 >< 7 .7 9 9 >

< 7 .1 4 3 >< 7 .7 9 8 >

< 3 .1 9 6 >

< 4 .6 1 1 >

< 5 .9 0 6 >< 6 .3 0 6 >

< 3 .1 9 6 >

< 4 .6 1 1 >

< 5 .9 0 6 >< 6 .3 0 6 >

< 6 .3 0 6 >< 7 .1 4 4 >

T : tru e e ig e n v a lu eS : sp u rio u s e ig en v a lu e

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Comparisons of Leissa and present method

Leissa

(Kitahara)Present method

Clamped plate

Simply-supported

plate

Free plate

0)()()()( 11 rIrJrIrJ

)1(

2

)(

)(

)(

)( 11

r

rJ

rJ

rI

rI

0

)()()1(4)()(2(

)1())()()()((

)1)(1(2))()(

)()()()1)(1({

211

22

11

221

144222

IJIJ

IJIJ

IJ

IJ

0)()()()( 11 rIrJrIrJ

)1(

2

)(

)(

)(

)( 11

r

rJ

rJ

rI

rI

)]()([)1()(

)]()([)1()(

)]()()[1()(

)]()()[1()(

23

23

22

22

III

JJI

III

JJJ

J

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction



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SVD updating terms (clamped case)

0

0

U

USM C

0

01

VM

VMSM C

MMSS

44

VV



1

1

48

1

0

0

000

000

000

000

)(

)(

NNVV

MM

UU

TC

TC

SM

SMC

MMSS

45

VV



1

1

440

0

0

0NN

T DCC

Based on the least squares

])()(

)()(

)()[(

detdet

2][][][][2][][][][

2][][][][2][][][][

2][][][][2][][][][

)1(

VMVMVV

MMVUVU

MUMUUUN

N

T DCC

MMSS

46

VV



.0)(,0)(

,0)(,0)(

,0)(,0)(

][][][][][][][][

][][][][][][][][

][][][][][][][][

VMVMVV

MMVUVU

MUMUUU

The only possibility for zero determinant of [D]

at the same time for the same .The common term is

0)()()()( 11 rIrJrIrJ

True eigenequation

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Outlines

5. Conclusions

2. Plate vibration

1. Introduction



MMSS

48

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Conclusions

True eigenequation can be extract out by using SVD updating term.

1.

2.

3.

Spurious eigenequation only depends on the adopted kernel function, while the true eigenequation is relevant to the specified boundary condition.

Since any two combinations of the four types of potentials, six options( ) were considered.4

2C

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J. T. Chen, M. H. Chang, I. L. Chung and Y. C. Cheng, 2002, Comments on eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point matching, J. Acoust. Soc. Amer., Vol.111, No.1, pp.33-36. (SCI and EI)

J. T. Chen, M. H. Chang, K. H. Chen and S. R. Lin, 2002, Boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, Journal of Sound and Vibration, Vol.257, No.4, pp.667-711 (SCI and EI)

J. T. Chen, M. H. Chang, K. H. Chen, I. L. Chen, 2002,Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Computational Mechanics, Vol.29, pp.392-408. (SCI and EI)

J. T. Chen, S. R. Kuo, K. H. Chen and Y. C. Cheng, 2000, Comments on "vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function“, Journal of Sound and Vibration, Vol.235, No.1, pp.156-171. (SCI and EI)

J. T. Chen, I. L. Chen, K. H. Chen and Y. T. Lee, 2003, Comments on “Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function.”, Journal of Sound and Vibration, Vol.262, pp.370-378. (SCI and EI)

J. T. Chen, I. L. Chen, K. H. Chen, Y. T. Yeh and Y. T. Lee, 2003, A meshless method for free vibration analysis of arbitrarily shaped plates with clamped boundaries using radial basis function, Engineering Analysis with Boundary Elements, Accepted. (SCI and EI)

1.

2.

3.

4.

5.

6.

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歡迎參觀海洋大學力學聲響振動實驗室烘焙雞及捎來伊妹兒

http://ind.ntou.edu.tw/~msvlab/

E-mail: [email protected]

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Journal of the Chinese Institute of Engineers

Call for Papers

Special Issue on Meshless Methods

If you are interested, please obser to the following schedule ：

1. Abstract due ： Mar. 31, 2002

2. Full paper due ： Jun. 30, 2003

3. Initial acceptance/rejection ： Sep. 30, 2003

4. Final, modified manuscript due ： Dec. 31, 2003

5. Editing deadline ： Mar. 31, 2004

Guest Editors ：

Professor D. L. Young, Professor C. S. Chen, Professor A. H. –D. Cheng,

Professor H. -K. Hong, Professor J. T. Chen

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國際邊界元素法電子期刊

Betech 2003

Meshless work shop

ICOME 2003

第七屆邊界元素法會議

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Thanks for your kind attention

There is nothing more practical than the right theory.

Whether the theory is right or not depends on the experiment

Documents

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 1 Plate vibration Part III