system dynamics

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

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