1 efficient mode superposition methods for non-classically damped system sang-won cho, graduate...
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1
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
2
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
3
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
4
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
IntroductionIntroduction
5
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
IntroductionIntroduction
6
ObjectiveObjective
To expand MA and MT methods to analyze To expand MA and MT methods to analyze
non-classically damped systemsnon-classically damped systems
IntroductionIntroduction
7
Previous Studies:
Mode Superposition Methods for Mode Superposition Methods for Classically Damped SystemClassically Damped System
8
Mode Displacement (MD) MethodMode Displacement (MD) Method
Dynamic Equations of MotionDynamic Equations of Motion
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
)()(
Classically Damped SystemClassically Damped System
where
(4)
10
MT Method (Dickens & Wilson, 1980)MT Method (Dickens & Wilson, 1980)
DisplacementDisplacement
)()()( tututumtts
)()( tPtu Ptmt
(5)
Classically Damped SystemClassically Damped System
m
iiis ttu
1
)()( where
P : MT vector
: modal displacement)(tP
11
- For - For PP
- For ,- For , solve solve
tRPK
PP 1
1/2PMP T
)()()()( trRPtMPPtKPPtMPP TP
TP
TP
T0
(6)
Classically Damped SystemClassically Damped System
)(tP
(9)
(7)
(8)
12
Classically Damped SystemClassically Damped System
)(tus
)()( tutumtts
)()( tutumats
Methods Displacement Notes
MD only retained modes
MA statically improved
MT dynamically improved
Summary
13
This Study:
Mode Superposition Methods forMode Superposition Methods for
Non-Classically Damped SystemNon-Classically Damped System
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Non-Classically Damped SystemNon-Classically Damped System
Dynamic Equations of MotionDynamic Equations of Motion
State Space EquationsState Space Equations
(1))()()()( 0 trRtuKtutuM C
Non-Classically Damped SystemNon-Classically Damped System
)(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)
Non-Classically Damped SystemNon-Classically Damped System
)(ˆ)( trRzstz Tiiii 0
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MA MethodMA Method
DisplacementDisplacement
(14))()()( tytytymats
)(ˆ)( trRAty ttma
1
st RRR ˆˆˆ 0
0RBR Tiiiis
ˆ]][[ˆ
Non-Classically Damped SystemNon-Classically Damped System
i
q
iis zzty
2
1
)( )( nq where
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MT MethodMT Method
DisplacementDisplacement)()()( tytyty
mtts
)(ˆ)( tzPty ptmt
(15)
Non-Classically Damped SystemNon-Classically Damped System
i
q
iis zzty
2
1
)(where
: MT vector
: modal displacement)(tzP
P̂
18
- For For
- For , solveFor , solve
tRPA ˆ
PP1
ˆ
21
)( PBP T
)(ˆˆ)(ˆˆ)(ˆˆ trRPtzPAPtzPBP Tp
Tp
T0
Non-Classically Damped SystemNon-Classically Damped System
)(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)
Non-Classically Damped SystemNon-Classically Damped System
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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)
Non-Classically Damped SystemNon-Classically Damped System
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
Non-Classically Damped SystemNon-Classically Damped System
(26)1ˆˆ
ˆˆˆˆˆˆ
PAP
PAP
s
PAP
s
RPT
T
P
T
P
tT
PARtˆˆ where where
(27)
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Non-Classically Damped SystemNon-Classically Damped System
)()( 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
11
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
Numerical ExamplesNumerical Examples
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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
Numerical ExamplesNumerical Examples
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
Numerical ExamplesNumerical Examples
28
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 )
Numerical ExamplesNumerical Examples
29
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 )
Numerical ExamplesNumerical Examples
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
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