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

Earthquake Records

Numerical Concept

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)

Accel (g)

Acceleration vs. Time

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)

Accel (g)

Acceleration vs Time t=16 to 20 sec

Harmonic Motion

onacceleratixvelocityxntdisplacemexwhere

φtωAωxφtωAωxφtωAx

===

−−=−−=−=

&&&

&&&

,,

)sin()cos()sin( 2

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)

Displ

. (m

)

Mass = 10.132 kgDamping = 0.00Spring = 1.0 N/mωn=√k/m=0.314 r/sDrive 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

=

=

Fourier Transformti

N

s

sSeXtx

ω∑=

=2/

0

Re)( &&&&2

,...,2,1,02 N

stN

sS =

∆=

πω

<≤

==

=

∑

∑−

=

∆−

−

=

∆−

1

0

1

0

21,

2

2,0,

1

N

k

tki

k

N

k

tki

k

SN

sforexN

Nssforex

NX

S

S

ω

ω

&&

&&

&&

)sin()cos( tkωitkωe SS

tkωi S ∆−∆=∆−

22SSS XXXMag &&&&&& ℑ+ℜ=

ℜ

ℑ= −

S

S

X

Xφ

&&

&&1tan

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, ω

Mag

nitu

deFourier Transform; El Centro

Earthquake Elastic Response Spectra

m

k/2c

)sin(0 tωP

k/2

x m

k/2c

k/2

x

xg

xt

(a)(b)

)sin(0 tωPkxxcxm =++ &&&critccD /= kmccrit =

)(0 tPxmkxxcxmorkxxcxmxm earthquakegg =−=++=+++ &&&&&&&&&&

systemsdampedDm

kωsystemsundamped

m

kω dn )1(; 2−==

Duhamel's Integral

p(τ)t

−= −− )(sin

)()( )()1( τω

ωτττξ

tm

dpetdx D

D

t

ττωτω

τξω dtepm

tx D

t

t

D

)(sin)(1

)( )(

0

−= −−∫ttBttAtx DD ωω 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

+∆−

∆−∆−+∆−= ∑∑

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)

Displacement (m

) D=0.0

D=0.02

D=0.05

Multi-Degree of Freedom

(a) (b)

m1

k1/2c1

k1/2

x1

m2

k2/2c2

k2/2

x2

k3/2k3 /2

x3 m3

c3

y1

y2 y4y3

y5

θ1θ2 θ3 θ4

θ5

=

iiNii

N

N

Si

S

S

x

x

x

kkk

kkk

kkk

f

f

f

M

L

LLLL

L

L

M

2

1

21

22221

11211

2

1

ji

coordinateofntdisplacemeunittoduecoordinatetoingcorrespondforcekij =

ji

coordinateofvelocityunittoduecoordinatetoingcorrespondforcecij =

ji

coordinateofonacceleratiunittoduecoordinatetoingcorrespondforcemij =

p(t)kxxcxm =++ &&&

Modal Analysis

(t)pkxxm =+&&

)(tpXkΦXmΦ =+&&p(t)φXkΦφXmΦφ

T

n

T

n

T

n =+&&

p(t)φkφφmφφT

nn

T

nn

T

n =+ nn XX&&

)(tPXKXM nnnnn =+&&

Modal Damping

)(tPXKXCXM nnnnnnn =++ &&&

n

n

nnnnnnM

tPXKXωξX

)(2 =++ &&&

)()( ttPKCM nnnnnn pφkφφcφφmφφT

nn

T

n

T

n

T

n ≡≡≡≡

kmc 10 aa +=

n

bT

nbnb

T

nnb kmmaC φφφcφ ][ 1−==

{ } { }[ ] [ ]{ }{ } { }pUMK

Uu

=−

=2ω

ω thene ti

[ ]{ } [ ]{ } { } tie ωpuKuM =+&&

FEM Frequency Domain

G1,ρ1,ν1

u1

u2

u7

u8

[ ] ),,( 1111 νρGfnK =

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 ρfnm =

constant

constant

cybxau iii

=

=

++=

σε

Finite Elements

[ ]{ } [ ]{ } { } 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

ω

=

+

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 −+−=

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

s

sSeXtx

ω∑=

=2/

0

Re)( &&&&

212,428 nodes, 189,078 brick elements and 1500 shell elements

Circular boundary to reduce reflections

Finite Element Model of Three-Bent Bridge

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