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Modeling Dynamic Systems
• Basic Quantities From Earthquake Records
• Fourier Transform, Frequency Domain
• Single Degree ...
Earthquake Records
Numerical Concept
Acceleration vs. Time
Acceleration vs. Time

4.0000E-01
3.0000E-01
2.0000E-01

Accel (g)

1.0000E-01
0.0000E+00
-1.0000E-0...
Acceleration vs. Time, t=16.00 tot=16 to 20 sec
vs Time 20.00 seconds
Acceleration
4.0000E-01
3.0000E-01
2.0000E-01

Accel...
Harmonic Motion
t = time

A = amplitude of wave
ω = frequency (radians / sec) SDOF Response
1.00E-02
8.00E-03
6.00E-03

X=...
Fourier Transform
2π s
ωS =
N ∆t

N /2


(t ) = Re ∑ X s e iωS t
x
s =0

1 N −1  e −iωS k∆t

,
∑ xk
 N k =0
 = ...
Fourier Transform; El Centro
Fourier Transform of El Centro Accleration Record

0.008
0.007
0.006

Magnitude

0.005
0.004
...
Earthquake Elastic Response Spectra
P0 sin(ωt )
x

xt

m
c

k/2

m

x
k/2

c

k/2

k/2

xg

(a)

m + cx + kx = P0 sin(ω ...
Duhamel's Integral
t

p(τ)

dx (t ) = e

−ξ (1) ( t −τ )

t

1
x(t ) =
mω D

 p (τ )dτ

sin ω D (t − τ )

 mω D


p(...
Elastic Response Spectrum
7.00E-02

Displacement Response Spectrum
El Centro, 1940 E-W

6.00E-02

Displacement (m)

5.00E-...
Multi-Degree of Freedom
x3

m3
c3

k3 /2

x2

k1/2

k3/2

c2

k2/2

m1
c1

k1/2

y3

y2

m + cx + kx = p(t)
x 

y4

y1
...
Modal Analysis

m + kx = p(t)
x

mΦX + kΦX = p(t )

T
T
T

φ n mΦX + φ n kΦX = φ n p(t)
T
T
T

φ n mφ n X n + φ n ...
Modal Damping


M n X n + C n X n + K n X n = Pn (t )
 + 2ξ ω X + K X = Pn (t )

Xn
n n
n
n
n
Mn
T
M n ≡ φ n mφ n

T...
FEM Frequency Domain


[ M ]{ u} + [ K ]{ u} = { p} e

iωt

{ u} = { U} e then
{[ K ] − ω 2 [ M ]}{ U} = {p}
i ωt
Finite Elements
u1

u7
G1,ρ1,ν1

u2

u8

[ K1 ] = fn(G1 , ρ1 ,ν 1 )
[ m1 ] = fn( ρ1 )
ui = ai x + bi y + c

 k1,1
k
 2,...

[ M ]{ u} + [ K ]{ u} = { p} eiωt
m1

m2







m3
m4


  u1   k1,1 k1, 2
 u   k

k
  2   2,1 2...
Method of Complex Response
• Given earthquake acceleration vs. time, ü(t)
• FFT => ω1, ω 2 , ω 3...ωn ; {p}1 ,{p}2 ,{p}3,{...
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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002 ray modeling dynamic systems

  1. 1. 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
  2. 2. Earthquake Records
  3. 3. Numerical Concept
  4. 4. Acceleration vs. Time Acceleration vs. Time 4.0000E-01 3.0000E-01 2.0000E-01 Accel (g) 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 Time (sec) 60.00 70.00 80.00 90.00
  5. 5. Acceleration vs. Time, t=16.00 tot=16 to 20 sec vs Time 20.00 seconds Acceleration 4.0000E-01 3.0000E-01 2.0000E-01 Accel (g) 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 Time (sec) 18.50 19.00 19.50 20.00
  6. 6. Harmonic Motion t = time A = amplitude of wave ω = frequency (radians / sec) SDOF Response 1.00E-02 8.00E-03 6.00E-03 X=A sin(ωt-φ) Displ. (m) 4.00E-03 Amplitude 2.00E-03 0.00E+00 φ = phase lag (radians ) Mass = 10.132 kg Damping = 0.00 Spring = 1.0 N/m ωn=√k/m=0.314 r/s Drive Freq = 0.0 Drive Force = 0.0 N Initial Vel. = 0.0 m/s Initial Disp. = 0.01 m -2.00E-03 -4.00E-03 -6.00E-03 Period=1/Frequency -8.00E-03 -1.00E-02 0.000 5.000 10.000 15.000 20.000 time (sec) 25.000 30.000 35.000 40.000
  7. 7. Fourier Transform 2π s ωS = N ∆t N /2  (t ) = Re ∑ X s e iωS t x s =0 1 N −1  e −iωS k∆t  , ∑ xk  N k =0  =  X S  N −1 −iω k∆t 2 k e S , x N∑  k =0 e − iωS k∆t N s = 0,1, 2,..., 2 N 2   N  for 1 ≤ s < 2   for s = 0, s = = cos(ωS k∆t ) − i sin(ωS k∆t )    Mag X S = ℜX + ℑX 2 S 2 S   ℑX S φ = tan   ℜX  S  −1    
  8. 8. Fourier Transform; El Centro Fourier Transform of El Centro Accleration Record 0.008 0.007 0.006 Magnitude 0.005 0.004 0.003 0.002 0.001 0 0 20 40 60 Circular Frequency, v 80 100 120
  9. 9. Earthquake Elastic Response Spectra P0 sin(ωt ) x xt m c k/2 m x k/2 c k/2 k/2 xg (a) m + cx + kx = P0 sin(ω t ) x   m + mg + cx + kx = 0 or x x ωn = k m (b) D = c / ccrit c crit = km m + cx + kx = − mg = Pearthquake (t ) x  x undamped systems; ωd = k (1 − D 2 ) damped systems m
  10. 10. Duhamel's Integral t p(τ) dx (t ) = e −ξ (1) ( t −τ ) t 1 x(t ) = mω D  p (τ )dτ  sin ω D (t − τ )   mω D  p(τ ) e −ξω (t −τ ) sin ω D (t − τ ) dτ ∫ 0 x(t ) = A(t ) sin ω D t − B(t ) cos ω D t t t 1 eξωτ 1 eξωτ A(t ) = ∫ p(τ ) eξωt cos ωD τ dτ B(t ) = mωD ∫ p(t ) eξωt sin ωD τ dτ mωD 0 0 A ∆τ 1 A  A  A(t ) =  ∑ (t ) ∑ (t ) = ∑ (t − ∆τ ) + p(t − ∆τ ) cos ωD (t − ∆τ ) mωD ζ ζ 2  2  exp(−ξω∆τ ) + p(t ) cos ωD t
  11. 11. Elastic Response Spectrum 7.00E-02 Displacement Response Spectrum El Centro, 1940 E-W 6.00E-02 Displacement (m) 5.00E-02 D=0.0 4.00E-02 D=0.02 D=0.05 3.00E-02 2.00E-02 1.00E-02 0.00E+00 1.00E-01 1.00E+00 1.00E+01 Frequency (rad/sec) 1.00E+02
  12. 12. Multi-Degree of Freedom x3 m3 c3 k3 /2 x2 k1/2 k3/2 c2 k2/2 m1 c1 k1/2 y3 y2 m + cx + kx = p(t) x  y4 y1 y5 θ1 (a) k12  k1N   x1  k 22  k 2 N   x2              ki 2  kiN   xi  kij = force corresponding to coordinate i due to unit displacement of coordinate j cij = force corresponding to coordinate i due to unit velocity of coordinate j mij = force corresponding to coordinate i due to unit acceleration of coordinate j m2 k2/2 x1  f S 1   k11  f  k  S 2   21  =      f Si   ki1    θ2 θ3 (b) θ4 θ5
  13. 13. Modal Analysis m + kx = p(t) x  mΦX + kΦX = p(t ) T T T  φ n mΦX + φ n kΦX = φ n p(t) T T T  φ n mφ n X n + φ n kφ n X n = φ n p(t)  M n X n + K n X n = Pn (t )
  14. 14. Modal Damping   M n X n + C n X n + K n X n = Pn (t )  + 2ξ ω X + K X = Pn (t )  Xn n n n n n Mn T M n ≡ φ n mφ n T C n ≡ φ n cφ n T K n ≡ φ n kφ n c = a 0 m + a1k C nb = φ T c b φ n = ab φ T m[m −1 k ]b φ n n n T Pn (t ) ≡ φ n p(t )
  15. 15. FEM Frequency Domain  [ M ]{ u} + [ K ]{ u} = { p} e iωt { u} = { U} e then {[ K ] − ω 2 [ M ]}{ U} = {p} i ωt
  16. 16. Finite Elements u1 u7 G1,ρ1,ν1 u2 u8 [ K1 ] = fn(G1 , ρ1 ,ν 1 ) [ m1 ] = fn( ρ1 ) ui = ai x + bi y + c  k1,1 k  2,1   k  7 ,1 k8,1  k1, 2 k 2, 2 k1, 7 k 2, 7 k7, 2 k8, 2 k7,7 k8, 7 k1,8 u1  k 2,8 u2        k7 ,8 u7    k8,8 u8    ε = constant σ = constant
  17. 17.  [ M ]{ u} + [ K ]{ u} = { p} eiωt m1  m2       m3 m4    u1   k1,1 k1, 2  u   k  k   2   2,1 2, 2    u3  +  k3,1 k3, 2      k 4, 2  u4   m5  u5        k1,3 k 2,3 k 2, 4 k 3, 3 k 3, 4 k 4,3 k 4, 4 k 5, 3 k 5, 4   u1   p1       u2   p2      k3,5  u3  =  p3 eiωt  k 4,5  u4   p4      k5,5  u5   p5       if { u} = { U} e iωt then { u} = −ω 2 { U} e iωt and {[ K ] − ω [ M ]}{ U} = {p} given ω, {p}, solve for { U} 2 [ K ], { U} are complex − valued ( G* = G 1 − 2 D 2 + 2iD 1 − D 2 )
  18. 18. 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 N /2 • Recall that  (t ) = Re ∑ X s e iωS t x s =0 { [ K ] − ω [ M ] } { U} = {p} 2 • Solve • FFT-1 => ü (t)
  19. 19. 212,428 nodes, 189,078 brick elements and 1500 shell elements Circular boundary to reduce reflections
  20. 20. Finite Element Model of Three-Bent Bridge
  21. 21. Zoom 1 Zoom 2

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