metodos aproximados para resolver sistema de un grado de libertad
DESCRIPTION
dinámica estruturalTRANSCRIPT
![Page 1: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/1.jpg)
METODOS APROXIMADOS PARA RESOLVER SISTEMAS DE UN GRADO DE LIBERTAD PROBLEMA N°1
DATOS:
m 0.849 t*S2/cm
k 16.16 t/cm
PERIODO FUNDAMENTAL
T 1.44 Seg.
tp 0.360 Seg.
tp 0.720 Seg.
tp 1.439 Seg.
condiciones iniciales
x(0) 0 cm
v(0) 0 m/seg.
x(0) 2 cm
v(0) 40 m/seg.
Amplitud:
A 83.261
Se toma como Δt = T/10
1. Método basado en la interpolación de la excitación.
![Page 2: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/2.jpg)
Tabla de coeficientes para calcular:
Condiciones:
x(0) 0 cm
v(0) 0 m/seg.
a) Para un periodo Tp = 0.36 seg.
COEFICIENTES
A 0.813
B 0.131
C 0.008
D 0.004
A' -2.485
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B' 0.756
C' 0.073
D' 0.080
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.31 0.00 0.00 0.00 0.00
0.14 79.15 0.61 0.19 0.83 6.36 0.25 0.31
0.29 49.12 0.38 0.00 1.80 13.79 1.53 1.88
0.43 0.00 0.00 0.00 1.22 9.36 3.01 3.71
0.58 0.00 0.00 0.00 -0.28 -2.13 3.44 4.23
0.72 0.00 0.00 0.31 -1.58 -12.13 2.57 3.17
0.86 78.98 0.60 0.19 -1.40 -10.69 1.06 1.30
1.01 49.56 0.38 0.00 -0.20 -1.54 0.37 0.46
1.15 0.00 0.00 0.00 0.17 1.33 0.45 0.55
1.30 0.00 0.00 0.00 -0.05 -0.37 0.51 0.62
1.44 0.00 0.00 0.00 -0.24 -1.83 0.37 0.46
b) Para un periodo Tp = 0.72 seg.
COEFICIENTES
A 0.813
B 0.131
C 0.008
D 0.004
A' -2.485
B' 0.756
C' 0.073
D' 0.080
![Page 4: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/4.jpg)
Ecuaciones de recurrencia:
SOLUCION NUMERICA USANDO INTERPOLACION DE LA EXITACION
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.19 0.00 0.00 0.00 0.00
0.14 48.90 0.37 0.31 0.51 3.93 0.16 0.19
0.29 79.15 0.61 0.31 1.62 12.44 1.10 1.35
0.43 79.24 0.61 0.19 2.38 18.22 2.96 3.64
0.58 49.12 0.38 0.00 1.89 14.49 4.99 6.14
0.72 0.28 0.00 0.00 -0.09 -0.66 5.90 7.26
0.86 0.00 0.00 0.00 -2.42 -18.52 4.73 5.82
1.01 0.00 0.00 0.00 -3.71 -28.46 1.88 2.31
1.15 0.00 0.00 0.00 -3.56 -27.27 -1.49 -1.83
1.30 0.00 0.00 0.00 -2.10 -16.07 -4.11 -5.05
1.44 0.00 0.00 0.00 0.05 0.40 -5.04 -6.20
c) Para un periodo Tp = 1.439 seg.
COEFICIENTES
A 0.813
B 0.131
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C 0.008
D 0.004
A' -2.485
B' 0.756
C' 0.073
D' 0.080
Ecuaciones de recurrencia:
SOLUCION NUMERICA USANDO INTERPOLACION DE LA EXITACION
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.10 0.00 0.00 0.00 0.00
0.14 25.72 0.20 0.19 0.27 2.07 0.08 0.10
0.29 48.92 0.37 0.26 0.93 7.13 0.60 0.74
0.43 67.34 0.52 0.31 1.64 12.55 1.77 2.17
0.58 79.17 0.61 0.33 2.01 15.40 3.44 4.23
0.72 83.26 0.64 0.31 1.78 13.64 5.19 6.38
0.86 79.20 0.61 0.26 0.90 6.93 6.44 7.92
1.01 67.40 0.52 0.19 -0.42 -3.20 6.68 8.21
1.15 49.00 0.37 0.10 -1.82 -13.95 5.67 6.97
1.30 25.81 0.20 0.00 -2.90 -22.19 3.51 4.32
1.44 0.09 0.00 0.00 -3.34 -25.61 0.66 0.82
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Condiciones:
x(0) 2 cm
v(0) 40 cm/seg.
a) Para un periodo Tp = 0.36 seg.
COEFICIENTES
A 0.813
B 0.131
C 0.008
D 0.004
A' -2.485
B' 0.756
C' 0.073
D' 0.080
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.31 0.52 4.00 1.63 2.00
0.14 79.15 0.61 0.19 0.58 4.42 2.00 2.46
0.29 49.12 0.38 0.00 0.91 6.99 2.74 3.37
0.43 0.00 0.00 0.00 0.07 0.50 3.28 4.03
0.58 0.00 0.00 0.00 -1.26 -9.64 2.72 3.34
0.72 0.00 0.00 0.31 -2.04 -15.59 1.19 1.46
0.86 78.98 0.60 0.19 -1.18 -9.08 -0.44 -0.54
1.01 49.56 0.38 0.00 0.55 4.25 -0.67 -0.82
1.15 0.00 0.00 0.00 1.16 8.90 0.22 0.27
1.30 0.00 0.00 0.00 0.79 6.07 1.12 1.38
1.44 0.00 0.00 0.00 0.15 1.17 1.55 1.91
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b) Para un periodo Tp = 0.72 seg.
COEFICIENTES
A 0.813
B 0.131
C 0.008
D 0.004
A' -2.485
B' 0.756
C' 0.073
D' 0.080
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.19 0.52 4.00 1.63 2.00
0.14 48.90 0.37 0.31 0.26 1.98 1.90 2.34
0.29 79.15 0.61 0.31 0.74 5.63 2.31 2.85
0.43 79.24 0.61 0.19 1.22 9.36 3.23 3.97
0.58 49.12 0.38 0.00 0.91 6.99 4.27 5.25
0.72 0.28 0.00 0.00 -0.54 -4.12 4.52 5.55
0.86 0.00 0.00 0.00 -2.21 -16.90 3.24 3.98
1.01 0.00 0.00 0.00 -2.96 -22.67 0.84 1.03
1.15 0.00 0.00 0.00 -2.57 -19.71 -1.72 -2.12
1.30 0.00 0.00 0.00 -1.26 -9.63 -3.49 -4.30
1.44 0.00 0.00 0.00 0.44 3.39 -3.86 -4.75
![Page 8: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/8.jpg)
c) Para un periodo Tp = 1.439 seg.
COEFICIENTES
A 0.813
B 0.131
C 0.008
D 0.004
A' -2.485
B' 0.756
C' 0.073
D' 0.080
ti Pi C×Pi D×Pi+1 Bůi ůi Aui ui
0.00 0.00 0.00 0.10 0.52 4.00 1.63 2.00
0.14 25.72 0.20 0.19 0.02 0.12 1.83 2.25
0.29 48.92 0.37 0.26 0.04 0.32 1.82 2.23
0.43 67.34 0.52 0.31 0.48 3.70 2.03 2.50
0.58 79.17 0.61 0.33 1.03 7.90 2.71 3.34
0.72 83.26 0.64 0.31 1.33 10.18 3.80 4.68
0.86 79.20 0.61 0.26 1.12 8.55 4.94 6.08
1.01 67.40 0.52 0.19 0.34 2.59 5.63 6.93
1.15 49.00 0.37 0.10 -0.83 -6.38 5.43 6.68
1.30 25.81 0.20 0.00 -2.06 -15.75 4.13 5.08
1.44 0.09 0.00 0.00 -2.95 -22.62 1.84 2.27
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2. Método basado en la Diferencia Central
Condiciones iniciales:
x(0) 0 cm
v(0) 0 cm/seg.
a) Para un periodo Tp = 0.36 seg.
CALCULOS INICALES:
ϋ0 0.00 cm/s2
u -1 0.00 cm
ǩ 42.26
a 39.69
b -65.79
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60
ui
ti
ti vs ui
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ti Pi ui-1 ui ṕ ui+1
0.00 0.00 0.00 0.00 0.00 0.00
0.14 79.17 0.00 0.00 79.17 1.87
0.29 49.03 0.00 1.87 172.28 4.08
0.43 0.00 1.87 4.08 193.84 4.59
0.58 0.00 4.08 4.59 139.97 3.31
0.72 0.00 4.59 3.31 35.86 0.85
0.86 79.08 3.31 0.85 3.46 0.08
1.01 49.26 0.85 0.08 20.97 0.50
1.15 0.00 0.08 0.50 29.39 0.70
1.30 0.00 0.50 0.70 26.07 0.62
1.44 0.00 0.70 0.62 12.98 0.31
b) Para un periodo Tp = 0.72 seg.
CALCULOS INICALES:
ϋ0 0.00 cm/s2
u -1 0.00 cm
ǩ 42.26
a 39.69
b -65.79
ti Pi ui-1 ui ṕ ui+1
0.00 0.00 0.00 0.00 0.00 0.00
0.14 48.92 0.00 0.00 48.92 1.16
0.29 79.17 0.00 1.16 155.32 3.68
0.43 79.21 1.16 3.68 275.06 6.51
0.58 49.03 3.68 6.51 331.37 7.84
0.72 0.14 6.51 7.84 257.68 6.10
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0.86 0.00 7.84 6.10 89.95 2.13
1.01 0.00 6.10 2.13 -101.96 -2.41
1.15 0.00 2.13 -2.41 -243.21 -5.75
1.30 0.00 -2.41 -5.75 -282.85 -6.69
1.44 0.00 -5.75 -6.69 -211.93 -5.01
c) Para un periodo Tp = 1.439 seg.
CALCULOS INICALES:
ϋ0 0.00 cm/s2
u -1 0.00 cm
ǩ 42.26
a 39.69
b -65.79
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ti Pi ui-1 ui ṕ ui+1
0.00 0.00 0.00 0.00 0.00 0.00
0.14 25.73 0.00 0.00 25.73 0.61
0.29 48.95 0.00 0.61 89.01 2.11
0.43 67.37 0.61 2.11 181.76 4.30
0.58 79.19 2.11 4.30 278.55 6.59
0.72 83.26 4.30 6.59 346.20 8.19
0.86 79.18 6.59 8.19 356.53 8.44
1.01 67.34 8.19 8.44 297.25 7.03
1.15 48.91 8.44 7.03 176.82 4.18
1.30 25.70 7.03 4.18 21.81 0.52
1.44 0.00 4.18 0.52 -132.10 -3.13
Condiciones iniciales:
x(0) 2 cm
v(0) 4 cm/seg.
a) Para un periodo Tp = 0.36 seg.
CALCULOS INICALES:
ϋ0 -39.83 cm/s2
u -1 1.01 cm
ǩ 42.26
a 39.69
b -65.79
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ti Pi ui-1 ui ṕ ui+1
0.00 0.00 1.01 2.00 91.42 2.16
0.14 79.17 2.00 2.16 142.11 3.36
0.29 49.03 2.16 3.36 184.40 4.36
0.43 0.00 3.36 4.36 153.61 3.63
0.58 0.00 4.36 3.63 65.96 1.56
0.72 0.00 3.63 1.56 -41.58 -0.98
0.86 79.08 1.56 -0.98 -47.59 -1.13
1.01 49.26 -0.98 -1.13 14.22 0.34
1.15 0.00 -1.13 0.34 66.84 1.58
1.30 0.00 0.34 1.58 90.69 2.15
1.44 0.00 1.58 2.15 78.41 1.86
b) Para un periodo Tp = 0.72 seg.
CALCULOS INICALES:
ϋ0 -39.83 cm/s2
u -1 1.01 cm
ǩ 42.26
a 39.69
b -65.79
ti Pi ui-1 ui ṕ ui+1
0.00 0.00 1.01 2.00 91.42 2.16
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0.14 48.92 2.00 2.16 111.86 2.65
0.29 79.17 2.16 2.65 167.45 3.96
0.43 79.21 2.65 3.96 234.84 5.56
0.58 49.03 3.96 5.56 257.36 6.09
0.72 0.14 5.56 6.09 180.24 4.26
0.86 0.00 6.09 4.26 38.90 0.92
1.01 0.00 4.26 0.92 -108.71 -2.57
1.15 0.00 0.92 -2.57 -205.76 -4.87
1.30 0.00 -2.57 -4.87 -218.23 -5.16
1.44 0.00 -4.87 -5.16 -146.49 -3.47
c) Para un periodo Tp = 1.439 seg.
CALCULOS INICALES:
ϋ0 -39.83 cm/s2
u -1 1.01 cm
ǩ 42.26
a 39.69
b -65.79
ti Pi ui-1 ui ṕ ui+1
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0.00 0.00 1.01 2.00 91.42 2.16
0.14 25.73 2.00 2.16 88.68 2.10
0.29 48.95 2.16 2.10 101.13 2.39
0.43 67.37 2.10 2.39 141.53 3.35
0.58 79.19 2.39 3.35 204.54 4.84
0.72 83.26 3.35 4.84 268.76 6.36
0.86 79.18 4.84 6.36 305.48 7.23
1.01 67.34 6.36 7.23 290.50 6.87
1.15 48.91 7.23 6.87 214.27 5.07
1.30 25.70 6.87 5.07 86.44 2.05
1.44 0.00 5.07 2.05 -66.66 -1.58
3. comparación entre método de interpolación vs diferencia central
Condiciones iniciales:
x(0) 0 cm
v(0) 0 cm/seg.
a) Para un periodo Tp = 0.36 seg.
![Page 16: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/16.jpg)
b) Para un periodo Tp = 0.76 seg.
c) Para un periodo Tp = 1.439 seg.
![Page 17: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/17.jpg)
Condiciones iniciales:
x(0) 2 cm
v(0) 4 cm/seg.
a) Para un periodo Tp = 0.36 seg.
![Page 18: Metodos aproximados para resolver Sistema de Un Grado de Libertad](https://reader036.vdocuments.net/reader036/viewer/2022081722/563db7d6550346aa9a8e6bcc/html5/thumbnails/18.jpg)
b) Para un periodo Tp = 0.72 seg.
c) Para un periodo Tp = 1.439 seg.