1 efficient mode superposition methods for non-classically damped system sang-won cho, graduate...

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Efficient Mode Superposition Methods Efficient Mode Superposition Methods for Non-Classically Damped Systemfor Non-Classically Damped System

Sang-Won Cho, Graduate Student, KAIST, Korea

Ju-Won Oh, Professor, Hannam University, Korea

In-Won Lee, Professor, KAIST, Korea

12th KKNN Seminar12th KKNN SeminarTaejon, Korea, Aug. 20-22, 1999Taejon, Korea, Aug. 20-22, 1999

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IntroductionIntroduction

Mode Superposition Methods for Mode Superposition Methods for Classically Damped System

Mode Superposition Methods forMode Superposition Methods for Non-Classically Damped System

Numerical ExamplesNumerical Examples

ConclusionsConclusions

CONTENTSCONTENTS

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Dynamic Equations of MotionDynamic Equations of Motion

where where MM :: Mass matrix of order Mass matrix of order nn

CC :: Damping matrix of order Damping matrix of order nn

KK :: Stiffness matrix of order Stiffness matrix of order nn

u(t)u(t) :: Displacement vectorDisplacement vector

RR00 :: Invariant spatial portion of input Invariant spatial portion of input

loadload

r(t)r(t) :: Time varying portion of input loaTime varying portion of input loadd

(1))()()()( 0 trRtuKtuCtuM )()()()( 0 trRtuKtuCtuM

INTRODUTIONINTRODUTION

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Methods of Dynamic Analysis Methods of Dynamic Analysis

Direct integration methodDirect integration method- Short duration loading as an impulseShort duration loading as an impulse

Mode superposition methodMode superposition method

- Long duration loading as an earthquakeLong duration loading as an earthquake


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Improved Mode Superposition MethodsImproved Mode Superposition Methods

Mode acceleration (MA) methodMode acceleration (MA) method

Modal truncation augmentation (MT) methodModal truncation augmentation (MT) method

Limitation of MA and MT methodsLimitation of MA and MT methods

Applicable only to classically damped systemsApplicable only to classically damped systems


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ObjectiveObjective

To expand MA and MT methods to analyze To expand MA and MT methods to analyze

non-classically damped systemsnon-classically damped systems


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Previous Studies:

Mode Superposition Methods for Mode Superposition Methods for Classically Damped SystemClassically Damped System

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Mode Displacement (MD) MethodMode Displacement (MD) Method


Modal TransformationModal Transformation

Modal EquationsModal Equations

(1))()()()( 0 trRtuKtuCtuM

where

(3))()()(2)( 02 trRttt T

iiiiii

)(][ nmm 21

m

iiittu

1

)()( (2)

Classically Damped SystemClassically Damped System

),,1( mi

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MA Method (Williams, 1945)MA Method (Williams, 1945)

DisplacementDisplacement

)()()( tututumats

)()( trRKtu ttma

1

st RRR 0

m

i

Tiis RMR

10

m

iiis ttu

1

)()(


where

(4)

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MT Method (Dickens & Wilson, 1980)MT Method (Dickens & Wilson, 1980)


)()()( tututumtts

)()( tPtu Ptmt

(5)


m

iiis ttu

1

)()( where

P : MT vector

: modal displacement)(tP

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- For - For PP

- For ,- For , solve solve

tRPK

PP 1

1/2PMP T

)()()()( trRPtMPPtKPPtMPP TP

TP

TP

T0

(6)


)(tP

(9)

(7)

(8)

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)(tus

)()( tutumtts

)()( tutumats

Methods Displacement Notes

MD only retained modes

MA statically improved

MT dynamically improved

Summary

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This Study:

Mode Superposition Methods forMode Superposition Methods for

Non-Classically Damped SystemNon-Classically Damped System

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State Space EquationsState Space Equations

(1))()()()( 0 trRtuKtutuM C


)(ˆ)()( trRtyAtyB 0

00

0

00

0

RR

tu

tuty

M

KA

M

MCB ˆ,

)(

)()(,,

(10)

where

Eigenvalue ProblemEigenvalue Problemiii BsA

where and : complex conjugate pairsis i(11)

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MD MethodMD Method

State Space EquationsState Space Equations

Modal TransformationModal Transformation

Modal EquationsModal Equations

(10)

(12)

)(ˆ)()( trRtyAtyB 0

i

q

iis zzty

2

1

)( )( nq

(13)


)(ˆ)( trRzstz Tiiii 0

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MA MethodMA Method


(14))()()( tytytymats

)(ˆ)( trRAty ttma

1

st RRR ˆˆˆ 0

0RBR Tiiiis

ˆ]][[ˆ


i

q

iis zzty

2

1

)( )( nq where

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MT MethodMT Method

DisplacementDisplacement)()()( tytyty

mtts

)(ˆ)( tzPty ptmt

(15)


i

q

iis zzty

2

1

)(where

: MT vector

: modal displacement)(tzP

P̂

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- For For

- For , solveFor , solve

tRPA ˆ

PP1

ˆ

21

)( PBP T

)(ˆˆ)(ˆˆ)(ˆˆ trRPtzPAPtzPBP Tp

Tp

T0


)(tz p

(16)

(19)

(17)

(18)

P̂

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Stability of MT methodStability of MT method

- Modal equationModal equation

- Solution ( Solution ( rr((tt)) = sin = sin ( ( tt)), z, z(0)(0)==0 )0 )

- Stability conditionStability condition

0 APPs TP

(19)

(21)

)(ˆˆ)(ˆˆ)(ˆˆ trRPtzPAPtzPBP Tp

Tp

T0

PAPs TP

ˆˆwhere where

)()(ˆˆ

22tcostsins

s

RPz p

p

tT

P

tPse (20)


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Characteristics of MT SolutionCharacteristics of MT Solution

- SolutionSolution

- Property of Property of

- SimplificationSimplification

(19)

(22)

)(ˆˆ

tsins

RPz

p

tT

p

)()(ˆˆ

22tcostsinse

s

RPz p

s

p

tT

PP

t

|| PsPAPs TP

ˆˆ

(23)


Ps

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Comparison MT Solution with MA SolutionComparison MT Solution with MA Solution

- MT solutionMT solution

- MA solutionMA solution

- Coefficient of MT solutionCoefficient of MT solution

)(ˆ)( 1 tsinRAty ttma

)()( tytymamt tt

(24)

(25)

)(ˆˆ

ˆ)( 1 tsins

RPRAty

P

tT

ttmt


(26)1ˆˆ

ˆˆˆˆˆˆ

PAP

PAP

s

PAP

s

RPT

T

P

T

P

tT

PARtˆˆ where where

(27)

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)()( tytymtts

)()( tytymats

Methods Displacement Notes

MD only retained modes

MA stable

MT conditionally stable

Summary

)(tys

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StructuresStructures

Cantilever Beam with Lumped DampersCantilever Beam with Lumped Dampers

- To compare the MA and MT methods with To compare the MA and MT methods with

MD methodMD method

10-Story Shear Building10-Story Shear Building

- To show the divergent case of MT methodTo show the divergent case of MT method

NUMERICAL EXAMPLESNUMERICAL EXAMPLES

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Cantilever Beam with Lumped DampersCantilever Beam with Lumped Dampers

El-Centro EarthquakeEl-Centro Earthquake

1

2

3

9

10

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100 IN

Fig. 1 Beam ConfigurationFig. 1 Beam Configuration

E = 3.0107

L = 100

A = 4

C = 0.1

I = 1.25

= 7.4110-4

10 Beam Elements

ModeNumber

Eigenvalues

1 – 4.43482 – 39.29620i

2 – 4.43482 + 39.29620i

3 – 88.4454 – 231.3995i

4 – 88.4454 + 231.3995i

5 – 677.3535 – 147.892i

6 – 677.3535 + 147.892i

Table 1 EigenvaluesTable 1 Eigenvalues


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0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Moment at Each NodeMoment at Each Node

MD Method MA & MT Methods

Mm /

Md

1 m ode

2 m odes

3 m odes

1 m ode

2 m odes

3 m odes

1 2 3 9 10 114 5 6 7 8

Node Number Node Number

1 2 3 9 10 114 5 6 7 8


Mm: Moment by mode superposition methodsMd : Moment by direct integration method

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0.2

0.6

1.0

0.0

0.4

0.8

1.2

0.2

0.6

1.0

0.0

0.4

0.8

1.2

Shear Force at Each NodeShear Force at Each Node

1 m ode

2 m odes

3 m odes

1 m ode

2 m odes

3 m odes

MD Method MA & MT Methods

1 2 3 9 10 114 5 6 7 8

Node Number Node Number

1 2 3 9 10 114 5 6 7 8

Numerical ExamplesNumerical ExamplesS

m / S

d

Sm: Shear force by mode superposition methodsSd : Shear force by direct integration method

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10-Story Shear Building10-Story Shear Building

Harmonic Loading ( Harmonic Loading ( = 32.0 rad/sec= 32.0 rad/sec))

m1=1Ksec2/IN

m2=2k1=800 K/IN

k2=1600

m3=2

m4=2

m5=3

m6=3

m7=3

m8=4

m9=4

m10=4

Fig. 2 10-Story Shear Building

ModeNumber

Eigenvalues

1 – 0.0316 – 4.0100i

2 – 0.0316 + 4.0100i

3 – 0.0066 – 10.8381i

4 – 0.0066 + 10.8381i

5 – 0.0058 – 17.421i

6 – 0.0058 + 17.421i

Table 2 EigenvaluesLoad Case 2)( tsin

Load Case 1)( tsin


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Load Case 1Load Case 1

0.0 0.4 0.8 1.2 1.6 2.0-4.0E-4

-2.0E-4

0.0E+0

2.0E-4

4.0E-4

0.0 0.4 0.8 1.2 1.6 2.0-4.0E-4

-2.0E-4

0.0E+0

2.0E-4

4.0E-4

MA MethodMA Method MT MethodMT Method

Dis

pla

cem

ent

01047109 3 .ps MA and MT solutions are sameMA and MT solutions are same

Time (sec ) Time (sec )


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01024436 3 .ps

0.0 0.4 0.8 1.2 1.6 2.0-4.0E-3

-2.0E-3

0.0E+0

2.0E-3

4.0E-3

Dis

pla

cem

ent

Load Case 2Load Case 2

0.0 0.4 0.8 1.2 1.6 2.00.0E+0

1.0E-3

2.0E-3

3.0E-3

4.0E-3

MT method gives no solutionMT method gives no solution

MA MethodMA Method MT MethodMT Method

Time (sec ) Time (sec )


No solutionNo solution

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Expanded MA and MT methods were applied Expanded MA and MT methods were applied

to non-classically damped system.to non-classically damped system.

MA method is stable,MA method is stable,whereas MT method is conditionally stable.whereas MT method is conditionally stable.

MT method gives same results with MA methodMT method gives same results with MA methodwhen MT method is stable.when MT method is stable.

CONCLUSIONSCONCLUSIONS

1 efficient mode superposition methods for non-classically damped system sang-won cho, graduate...

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