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)

Acc

el (

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)

Acc

el (

g)

Acceleration vs Time t=16 to 20 sec

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

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

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

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

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

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

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

Modal Analysis

(t)pkxxm

)(tpXkΦXmΦ p(t)φXkΦφXmΦφ T

nTn

Tn

p(t)φkφφmφφ Tnn

Tnn

Tn nn XX

)(tPXKXM nnnnn

Modal Damping

)(tPXKXCXM nnnnnnn

n

nnnnnnn M

tPXKXωξX

)(2

)()( ttPKCM nnnnnn pφkφφcφφmφφ Tnn

Tn

Tn

Tn

kmc 10 aa

nbT

nbnbTnnb kmmaC φφφcφ ][ 1

pUMK

Uu

2

thene ti

tie puKuM

FEM Frequency Domain

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

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

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)(

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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