002 ray modeling dynamic systems
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Modeling Dynamic Systems
• Basic Quantities From Earthquake Records
• Fourier Transform, Frequency Domain
• Single Degree of Freedom Systems (SDOF) Elastic Response Spectra
• Multi-Degree of Freedom Systems, (MDOF) Modal Analysis
• Dynamic Analysis by Modal Methods
• Method of Complex Response
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Earthquake Records
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Numerical Concept
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Acceleration vs. Time
-4.0000E-01
-3.0000E-01
-2.0000E-01
-1.0000E-01
0.0000E+00
1.0000E-01
2.0000E-01
3.0000E-01
4.0000E-01
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
Time (sec)
Acc
el (
g)Acceleration vs. Time
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Acceleration vs. Time, t=16.00 to 20.00 seconds
-4.0000E-01
-3.0000E-01
-2.0000E-01
-1.0000E-01
0.0000E+00
1.0000E-01
2.0000E-01
3.0000E-01
4.0000E-01
16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00
Time (sec)
Acc
el (
g)
Acceleration vs Time t=16 to 20 sec
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Harmonic Motion
SDOF Response
-1.00E-02
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000
time (sec)
Dis
pl. (
m)
Mass = 10.132 kgDamping = 0.00Spring = 1.0 N/mωn=√k/m=0.314 r/s
Drive Freq = 0.0 Drive Force = 0.0 NInitial Vel. = 0.0 m/sInitial Disp. = 0.01 m
Period=1/Frequency
Amplitude
X=A sin(ωt-φ)
sec)/(radiansfrequencyω
waveofamplitudeA
)(radianslagphaseφ
timet
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Fourier Transformti
N
ss
SeXtx
2/
0
Re)( 2
,...,2,1,02 N
stN
sS
1
0
1
0
21,
2
2,0,
1
N
k
tkik
N
k
tkik
SN
sforexN
Nssforex
NX
S
S
)sin()cos( tkωitkωe SStkωi S
22SSS XXXMag
S
S
X
Xφ
1tan
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Fourier Transform of El Centro Accleration Record
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 20 40 60 80 100 120
Circular Frequency, v
Mag
nitu
de
Fourier Transform; El Centro
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Earthquake Elastic Response Spectra
m
k/2c
)sin(0 tωP
k/2
x m
k/2ck/2
x
xg
xt
(a) (b)
)sin(0 tωPkxxcxm critccD / kmccrit
)(0 tPxmkxxcxmorkxxcxmxm earthquakegg
systemsdampedDm
kωsystemsundamped
m
kω dn )1(; 2
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Duhamel's Integral
p(τ)t
)(sin
)()( )()1(
t
m
dpetdx D
D
t
dtepm
tx Dt
t
D
)(sin)(1
)( )(
0
ttBttAtx DD cos)(sin)()(
τdτωe
eτp
ωmtA Dtξω
ξωτt
D
cos)(1
)(0 τdτω
e
etp
ωmtB Dtξω
ξωτt
D
sin)(1
)(0
A
ζD
tζωm
τtA )(
1)(
tωtpτξω
τtωτtpτtt
D
D
AA
cos)()exp(
)(cos)()()(22
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Elastic Response Spectrum
Displacement Response Spectrum El Centro, 1940 E-W
0.00E+00
1.00E-02
2.00E-02
3.00E-02
4.00E-02
5.00E-02
6.00E-02
7.00E-02
1.00E-01 1.00E+00 1.00E+01 1.00E+02
Frequency (rad/sec)
Dis
pla
cem
ent
(m) D=0.0
D=0.02
D=0.05
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Multi-Degree of Freedom
(a) (b)
m1
k1/2c1k1/2
x1
m2
k2/2c2k2/2
x2
k3/2k3 /2
x3 m3
c3
y1
y2 y4
y3
y5
θ1
θ2θ3 θ4
θ5
iiNii
N
N
Si
S
S
x
x
x
kkk
kkk
kkk
f
f
f
2
1
21
22221
11211
2
1
ji
coordinateofntdisplacemeunittoduecoordinatetoingcorrespondforcekij
ji
coordinateofvelocityunittoduecoordinatetoingcorrespondforcecij
ji
coordinateofonacceleratiunittoduecoordinatetoingcorrespondforcemij
p(t)kxxcxm
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Modal Analysis
(t)pkxxm
)(tpXkΦXmΦ p(t)φXkΦφXmΦφ T
nTn
Tn
p(t)φkφφmφφ Tnn
Tnn
Tn nn XX
)(tPXKXM nnnnn
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Modal Damping
)(tPXKXCXM nnnnnnn
n
nnnnnnn M
tPXKXωξX
)(2
)()( ttPKCM nnnnnn pφkφφcφφmφφ Tnn
Tn
Tn
Tn
kmc 10 aa
nbT
nbnbTnnb kmmaC φφφcφ ][ 1
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pUMK
Uu
2
thene ti
tie puKuM
FEM Frequency Domain
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G1,ρ1,ν1
u1
u2
u7
u8
),,( 1111 GfnK
8
7
2
1
8,87,82,81,8
8,77,72,71,7
8,27,22,21,2
8,17,12,11,1
u
u
u
u
kkkk
kkkk
kkkk
kkkk
)( 11 fnm
constant
constant
cybxau iii
Finite Elements
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tωiepuKuM
tie
p
p
p
p
p
u
u
u
u
u
kkk
kkkk
kkkkk
kkkk
kkk
u
u
u
u
u
m
m
m
m
m
5
4
3
2
1
5
4
3
2
1
5,54,53,5
5,44,43,42,4
5,34,33,32,31,3
4,23,22,21,2
3,12,11,1
5
4
3
2
1
5
4
3
2
1
valuedcomplexare
forsolveωgivenω
andeωtheneif tωitωi
UK
UppUMK
UuUu
,
,,2
2
22 1221* DiDDGG
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Method of Complex Response
• Given earthquake acceleration vs. time, ü(t)
• FFT => ω1, ω 2 , ω 3...ωn ; {p}1 ,{p}2 ,{p}3,{p}n
• Recall that
• Solve
• FFT-1 => ü (t)
pUMK 2ω
tiN
ss
SeXtx
2/
0
Re)(
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212,428 nodes, 189,078 brick elements and 1500 shell elements
Circular boundary to reduce reflections
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Finite Element Model of Three-Bent Bridge
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Zoom 1 Zoom 2
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