taller 3 p3
DESCRIPTION
Dinámica estructuralTRANSCRIPT
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3. a( ) SOLUCION DE DESPLAZAMIENTO
Datos del punto 1
b 0.45m L 3m
h 0.45m
masa 40700kg
I b h
3
12
fc 28
E 3900 fc 1000000 N
m2
E 2.064 1010
N
m2
k 2 12 E IL
3
Matriz de Rigidez del edificio
K ( )
2k
k
0
k
2k
k
0
k
k
1.254 108
6.268 107
0
6.268 107
1.254 108
6.268 107
0
6.268 107
6.268 107
N
m
Matriz de Masa
M
masa
0
0
0
masa
0
0
0
masa
4.07 104
0
0
0
4.07 104
0
0
0
4.07 104
kg
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MODO 1
w1 17.463 rad
s
Ω w1
β Ω
w1
ξ1 5%
Suponemos una carga Po
Po 1000N
Φ1
1
1.80
2.24
M1 Φ1T
M Φ1 3.768 105
kg
α1
1
M1
1.629 10 3
1
kg
Φ1.norm α1 Φ1
1.629 10 3
2.932 10 3
3.649 10 3
1
kg
P1 t( ) Φ1.normT
0
0
Po sin Ω t( )
K 1 Φ1.normT
1.254 10
8
6.268 107
0
6.268 10
7
1.254 108
6.268 107
0
6.268 107
6.268 107
Φ1.norm
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η1 t( )P1 t( )
K 1
1
1 β2
2
2 ξ1 β 2
1 β2
sin Ω t( ) 2 ξ1 β cos Ω t( )
η1 t( ) simplify 0.05972971435146153602 kg N sin 34.926 rad t
s
u1 t( ) Φ1.norm η1 t( )
MODO 2
ξ2 3%
Φ2
1
0.45
0.81
M2 Φ2T
M Φ2 7.565 104
kg
α2
1
M2
3.636 10 3
1
kg
Φ2.norm α2 Φ2
3.636 10 3
1.636 10 3
2.945 10 3
1
kg
P2 t( ) Φ2.norm
0
0
Po sin Ω t( )
K 2 Φ2.normT
1.254 108
6.268 107
0
6.268 107
1.254 108
6.268 107
0
6.268 107
6.268 107
Φ2.norm
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η2 t( )P2 t( )
K 2
1
1 β2
2
2 ξ2 β 2
1 β2
sin Ω t( ) 2 ξ2 β cos Ω t( )
η2 t( ) simplify 0.020491171190223558868 kg N cos 17.463 rad t
s
sin 17.463 rad t
s
u2 t( ) Φ2.norm η2 t( )
MODO 3
ξ3 1%
Φ3
1
1.246
0.554
M3 Φ3T
M Φ3 1.164 105
kg
α3
1
M3
2.931 10 3
1
kg
Φ3.norm α3 Φ3
2.931 10 3
3.652 10 3
1.624 10 3
1
kg
t 0.01s
P3 t( ) Φ3.normT
0
0
Po sin Ω t( )
K 3 Φ3.normT
1.254 10
8
6.268 107
0
6.268 10
7
1.254 108
6.268 107
0
6.268 107
6.268 107
Φ3.norm
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η3 t( )P3 t( )
K 3
1
1 β2
2
2 ξ3 β 2
1 β2
sin Ω t( ) 2 ξ3 β cos Ω t( )
η3 t( ) 0.016235004579380538987 kg N cos 0.17463 rad( ) sin 0.17463 rad( )
u3 t( ) Φ3.norm η3 t( )
b( ) Desplazamiento total mediante superposición modal
modos de vibracion correspondiente son:
Φ Φ1.norm Φ2.norm Φ3.norm
Φ
8.196 10 3
9.161 10 4
2.328 10 3
1
kg
u t( )
1.629 10 3
2.932 10 3
3.649 10 3
1
kgη1 t( )
3.636 10 3
1.636 10 3
2.945 10 3
1
kgη2 t( )
2.931 10 3
3.652 10 3
1.624 10 3
1
kgη3 t( )
u t( ) simplify
0.000074505898447652860044 kg cos 0.17463 rad( ) N sin 0.17463 rad
kg
0.0001210
0.000033523556067205742308 kg cos 0.17463 rad( ) N sin 0.17463 rad
kg
0.0001454
0.00023113655138694014259 kg N sin 0.34926 rad( )
kg
0.000060346499155208380866
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210388961302206 kg N sin 0.34926 rad( )
kg
240411653635942 kg N sin 0.34926 rad( )
kg
kg cos 0.17463 rad( ) N sin 0.17463 rad
kg