O slideshow foi denunciado.
Utilizamos seu perfil e dados de atividades no LinkedIn para personalizar e exibir anúncios mais relevantes. Altere suas preferências de anúncios quando desejar.
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
Zoom 1

Zoom 2
Ray : modeling dynamic systems
Ray : modeling dynamic systems
Ray : modeling dynamic systems
Próximos SlideShares
Carregando em…5
×

Ray : modeling dynamic systems

879 visualizações

Publicada em

system dynamics

Publicada em: Educação
  • I have done a couple of papers through ⇒⇒⇒WRITE-MY-PAPER.net ⇐⇐⇐ they have always been great! They are always in touch with you to let you know the status of paper and always meet the deadline!
       Responder 
    Tem certeza que deseja  Sim  Não
    Insira sua mensagem aqui
  • These are one of the best companies for review articles. High quality with cheap rates. ⇒⇒⇒WRITE-MY-PAPER.net ⇐⇐⇐ I highly recommend it :)
       Responder 
    Tem certeza que deseja  Sim  Não
    Insira sua mensagem aqui
  • Seja a primeira pessoa a gostar disto

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

×