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PSSC 1998. 10. 13.

Mode Localization in Multispan Beams with Massive and Stiff Couplers on Supports

Dong-Ok Kim and In-Won Lee

Department of Civil Engineering in

Korea Advance Institute of Science and Technology

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CONTENTS

Introduction Definition of mode localization Literature Survey Objectives

Theoretical Background Multispan Beams Simple Structure Occurrence of Mode Localization Conditions of Significant Mode Localization

Numerical Examples Mode Localization in Two-Span Beam

Conclusions

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INTRODUCTION

Definition of Mode Localization– Under conditions of weak internal coupling, the mode

shapes undergo dramatic changes to become strongly localized when small disorder is introduced into periodic structures. (C. Pierre, 1988, JSV)

• Trouble by Mode Localization– When mode localization occurs, the modal amplitude o

f a global mode becomes confined to a local region of the structure, with serious implication for the control problem. (O. O. Bendiksen, 1987, AIAA)

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Introduction

• Example : Mode Localization of Two-Span Beam

Figure 1. Weakly Coupled Simply Supported Two-Span Beam

Figure 2. Non-Localized First Two Mode Shapes

Figure 3. Localized First Two Mode Shapes

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Introduction

• Example : Mode Localization of Parabola Antenna

Figure 4. Simple Model of Space Parabola Antenna

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Introduction Mode Localization of Parabola Antenna

Figure 5. Non-Localized Mode Shape

Figure 6. Localized Mode Shape

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Introduction

Literature Survey

• Localization of electron eigenstates in a disordered solid

– P. W. Anderson (1958)

• Mode localization in the disordered periodic structures

– C. H. Hodges (1982)

• Localized vibration of disordered multispan beams

– C. Pierre (1987)

• Influences of various effects on mode localization

– S. D. Lust (1993)

• Mode localization up to high frequencies

– R. S. Langley (1995)

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Introduction

Objective:To study influences of the stiffness and mass of coupler on mode localization

• Scope• Theoretical Background:

Qualitative analysis using simple model

• Numerical Examples:Verifications of results of the theoretical background using multispan beams

• Conclusions

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THEORETICAL BAGROUND

Multispan Beams

Figure 7. Simply supported multispan beam with couplers.

- Periodically rib-stiffened plates or

- Rahmen bridges

• Two-span beam : Two substructures and one coupler

Figure 8. Simply supported two-span beam with a coupler.

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Theoretical Background

Simple Structure Analysis

Figure 9. Simple model with two-substructures and a coupler.

• Subject : Qualitative analysis of influences of stiffness and mass of coupler on mode localization using simple model

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Theoretical Background

• Eigenvalue Problem

0

0

0

0

0

3

2

1

354343

4242

3131

y

y

y

mkkkkk

kmkk

kmkk

• Equation for ratio of and , and

2

1

21

2

143

2

24

1

23

3543

11)(

y

y

my

y

mkk

m

k

m

kmkkk d

1

31)1(

m

kk

2

42)2(

m

kk

2y1y )2()1( d

where

The ratio represents degree of mode localization corresponding mode.

(1)

(2)

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Theoretical Background

Occurrence of Mode Localization

where

rsrsr 21

4

31 k

ks

2

1

3

42 m

m

k

ks

dmkkkkk

m )( 354343

1

1

2

y

yr • Equation for Degree of Mode Localization

and

(3)

• In Equation (3)Left-hand side : Parabolic curve

Right-hand side : Line passing origin with slop

(4)

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Theoretical Background

• Graphical Representation

Figure 10. Two curves.

f

r

))(( 21 srsrf p

rfl

1s 2s1r

2r

– Steep line

Significant mode localization

1

1,0 21 rr

– Identical substructures: )2()1(

No mode localization

0

1,1 21 rr

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• Delocalization condition

or

• Classical condition ( ) :

• Becoming and , under the condition of

results in significant mode localization, .

Theoretical Background

Conditions for Significant Mode Localization

1||

43

35

kk

mk

143

5 kk

k

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

1||

03 m

3

5433543 0

m

kkkmkkk

(6)

(5)

(7)

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NUMERICAL EXAMPLES

Mode Localization in Two-Span Beam

• Assumptions– All spans have identical span lengths initially.

– Length disturbances are introduced into the first span only.

Figure 11. Simply Supported Two-Span Beam with a Coupler

• Subjects to Discuss– Influences of length disturbance of a span

– Influences of the stiffness and the mass of coupler

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Mode Localization in Multispan Beams

%41 L

0.20K 0.0J

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

0.0K 02.0J 0.20K 02.0J

Figure 12. First ten mode shapes:

• Selected Mode Shapes of Two-Span Beam

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Mode Localization in Multispan Beams

• Measure of Degree of Mode Localization

Classical Measure– Span response ratio

b

aba y

yr ,

ay

by

a

b

: Maximum amplitude of span

where

Note !

Classical measure is good for analysis but not for practice.

: Maximum amplitude of span

(8)

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Mode Localization in Multispan Beams

: Total number of spans

1

T

CTL N

NND

TN

CN

jy j: Maximum amplitude of span

Note !

Proposed measure is good for practice but not for analysis.

where

: Number of spans in which vibration is confined1

1

2

2

1

TT N

jj

N

jjC yyN

(9)

(10)

Proposed Measure– Normalized number of spans having no vibration

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Mode Localization in Multispan Beams

Figure 13. Influence of the stiffness

0 2 4 6 8 10

M ode N um ber

0.0

0.2

0.4

0.6

0.8

1.0

Deg

ree

of M

ode

Loca

lizat

ion

%5.0,100 1 LKC

%0.1,100 1 LKC

%0.1,1000 1 LKC

%5.0,1000 1 LKC

• Coupler with Stiffness

• Stiffness makes the system sensitive to mode localization

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0 2 4 6 8 10

M ode N um ber

0.0

0.2

0.4

0.6

0.8

1.0

Deg

ree

of M

ode

Loca

lizat

ion

Mode Localization in Multispan Beams

Figure 14. Influence of the mass

%5.0,1.0 1 LJC

%0.1,1.0 1 LJC

%0.1,0.1 1 LJC

%5.0,0.1 1 LJC

• Coupler with Mass

• Mass makes the system sensitive to mode localization in higher modes.

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0 2 4 6 8 10

M ode N um ber

0.0

0.2

0.4

0.6

0.8

1.0

Deg

ree

of M

ode

Loca

lizat

ion

Mode Localization in Multispan Beams

Figure 15. Influences of stiffness and mass

%5.0,1.0,100 1 LJK CC

%0.1,1.0,100 1 LJK CC

%0.1,0.1,1000 1 LJK CC

%5.0,0.1,1000 1 LJK CC

• Coupler with Stiffness and Mass

• Stiffness governs sensitivities of lower modes.

• Mass governs sensitivities of higher modes.

• Delocalized modes can be observed.

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CONCLUSIONS

Influences of the Coupler are Discovered.– The stiffness of coupler makes the structures sensitive

to mode localization.

– The mass of coupler makes the structures sensitive to mode localization in higher modes.

– The coupler with stiffness and mass is a cause of delocalization* in some modes.

* Delocalization is that mode localization does not occur or is very weak in certain modes although structural disturbances are severe.

• The mass as well as the stiffness of coupler give significant influences on mode localization especially in higher modes.