The document discusses modeling dynamic systems and earthquake response. It covers basic concepts like Fourier transforms, single and multi-degree of freedom systems, modal analysis, and elastic response spectra. Numerical methods are presented for dynamic analysis in the frequency and time domains, including the finite element method and method of complex response. Examples of earthquake records and harmonic motion are shown.

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. Earthquake Records

3. Numerical Concept

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. FEM Frequency Domain

[ M ]{ u} + [ K ]{ u} = { p} e
iωt
{ u} = { U} e then
{[ K ] − ω 2 [ M ]}{ U} = {p}
i ωt

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. 
[ 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. 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. 212,428 nodes, 189,078 brick elements and 1500 shell elements
Circular boundary to reduce reflections

21. Finite Element Model of Three-Bent Bridge

22. Zoom 1
Zoom 2