8/18/2019 TALLER 7 EJE X v1

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 ANALISIS: EJE X 

Matrices de rigidez y matriz de masa:

≔m1   ――197.70

981

≔m2   ――192.65

981

≔m3 m2   ≔m4 m2   ≔m5 m2

≔m6 m2   ≔m7   ―148.6

981⋅ton   ― s

2

cm

≔k1   4067.4536653≔k2   5263.2244944   ≔k3 k2   ≔k4 k2   ≔k5 k2   ≔k6 k2≔k7   5067.3929296

≔K

+k1 k2   −k2   0 0 0 0 0−k2   +k2 k3   −k3   0 0 0 0

0   −k3   +k3 k4   −k4   0 0 0

0 0   −k4   +k4 k5   −k5   0 00 0 0   −k5   +k5 k6   −k6   00 0 0 0   −k6   +k6 k7   −k70 0 0 0 0   −k7 k7

⎡⎢⎢⎢

⎢⎢⎢⎣

⎤⎥⎥⎥

⎥⎥⎥⎦

=K

9330.678   −5263.224 0 0 0 0 0−5263.224 10526.449   −5263.224 0 0 0 0

0   −5263.224 10526.449   −5263.224 0 0 00 0   −5263.224 10526.449   −5263.224 0 00 0 0   −5263.224 10526.449   −5263.224 00 0 0 0   −5263.224 10330.617   −5067.3930 0 0 0 0   −5067.393 5067.393

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

―ton

cm

≔ M

m1   0 0 0 0 0 0

0   m2   0 0 0 0 0

0 0   m3   0 0 0 0

0 0 0   m4   0 0 0

0 0 0 0   m5   0 0

0 0 0 0 0   m6   0

0 0 0 0 0 0   m7

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

= M

0.202 0 0 0 0 0 0

0 0.196 0 0 0 0 0

0 0 0.196 0 0 0 0

0 0 0 0.196 0 0 0

0 0 0 0 0.196 0 0

0 0 0 0 0 0.196 0

0 0 0 0 0 0 0.151

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

Cálculo de Frecuencias angulares y periodos de vibración

≔K1   ⋅ M−1

K

=K1

46299.42   −26116.46 0 0 0 0 0−26801.05 53602.11   −26801.05 0 0 0 0

0   −26801.05 53602.11   −26801.05 0 0 00 0   −26801.05 53602.11   −26801.05 0 00 0 0   −26801.05 53602.11   −26801.05 00 0 0 0   −26801.05 52604.91   −25803.850 0 0 0 0   −33452.98 33452.98

⎡⎢⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥⎥

⎦

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≔evp   eigenvals   K1

=evp

⋅1.025 105

⋅8.919 104

⋅6.978 104

⋅4.76 104

⋅2.649 104

⋅1.008 104

⋅1.151 103

⎡⎢⎢

⎢⎢⎢⎢⎢⎣

⎤⎥⎥

⎥⎥⎥⎥⎥⎦

≔ Freq   sort   evp   = Freq

⋅1.151 103

⋅1.008 104

⋅2.649 104

⋅4.76 104

⋅6.978 104

⋅8.919 104

⋅1.025 105

⎡⎢⎢

⎢⎢⎢⎢⎢⎣

⎤⎥⎥

⎥⎥⎥⎥⎥⎦

≔i   ‥0 6

≔w+i   1

 ‾‾‾‾ Freqi

≔T +i   1

2 ―w

+i   1

=w1

33.934 =T 1

0.185

=w2

100.418 =T 2

0.063

=w3

162.749 =T 3

0.039

=w4

218.172 =T 4

0.029

=w5

264.153 =T 5

0.024

=w6

298.652 =T 6

0.021

=w7

320.117 =T 7

0.02

Cálculo de Modos de vibración

≔v   eigenvecs   K1

=v

−0.163 −0.311 −0.43 0.498 0.483 −0.357 −0.1320.35 0.511 0.387 −0.025 0.366 −0.495 −0.229

−0.476 −0.368 0.197 −0.504 −0.112 −0.447 −0.3150.517 −0.023 −0.506 −0.088 −0.48 −0.23 −0.388

−0.467 0.398 0.108 0.484 −0.373 0.073 −0.4450.335 −0.506 0.44 0.197 0.102 0.348 −0.482

−0.163 0.303 −0.406 −0.465 0.491 0.498 −0.499

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

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Normalizando los modos de vibración

≔V 1

  ―v

6

v ,0 6

=V 1

11.7292.3832.935

3.3613.6433.773

⎡⎢⎢⎢

⎢⎢

⎣

⎤⎥⎥⎥

⎥⎥

⎦

≔V 2

  ―v

5

v ,0 5

=V 2

11.3871.2520.646

−0.203−0.976−1.397

⎡⎢⎢⎢

⎢⎢

⎣

⎤⎥⎥⎥

⎥⎥

⎦

≔V 3

  ―v

4

v,0 4

=V 3

10.759

−0.233−0.994−0.773

0.212

1.017

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

≔V 4

  ―v

3

v,0 3

=V 4

1−0.05−1.011−0.177

0.9720.394

−0.933

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

≔V 5

  ―v

2

v,0 2

=V 5

1−0.899−0.457

1.175−0.252−1.023

0.942

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

≔V 6

  ―v

1

v,0 1

=V 6

1−1.642

1.1810.074

−1.2791.625

−0.975

⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

≔V 7

  ―v

0

v,0 0

=V 7

1−2.151

2.922−3.178

2.873−2.061

0.999

⎡

⎢⎢⎢⎢⎢

⎣

⎤

⎥⎥⎥⎥⎥

⎦

Matriz de modos de vibración normalizados

≔ϕ

V 1

,0 0

V 2

,0 0

V 3

,0 0

V 4

,0 0

V 5

,0 0

V 6

,0 0

V 7

,0 0

V 1,1 0

V 2,1 0

V 3,1 0

V 4,1 0

V 5,1 0

V 6,1 0

V 7,1 0

⎛V 1⎞

,2 0

⎛V 2⎞

,2 0

⎛V 3⎞

,2 0

⎛V 4⎞

,2 0

⎛V 5⎞

,2 0

⎛V 6⎞

,2 0

⎛V 7⎞

,2 0

V 1

,3 0

V 2

,3 0

V 3

,3 0

V 4

,3 0

V 5

,3 0

V 6

,3 0

V 7

,3 0

⎛V 1⎞

,4 0

⎛V 2⎞

,4 0

⎛V 3⎞

,4 0

⎛V 4⎞

,4 0

⎛V 5⎞

,4 0

⎛V 6⎞

,4 0

⎛V 7⎞

,4 0

V 1

,5 0

V 2

,5 0

V 3

,5 0

V 4

,5 0

V 5

,5 0

V 6

,5 0

V 7

,5 0

⎛V 1

⎞,6 0

⎛V 2

⎞,6 0

⎛V 3

⎞,6 0

⎛V 4

⎞,6 0

⎛V 5

⎞,6 0

⎛V 6

⎞,6 0

⎛V 7

⎞,6 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

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=ϕ

1 1 1 1 1 1 11.729 1.387 0.759 −0.05 −0.899 −1.642 −2.1512.383 1.252 −0.233 −1.011 −0.457 1.181 2.9222.935 0.646 −0.994 −0.177 1.175 0.074 −3.178

3.361 −0.203 −0.773 0.972 −0.252 −1.279 2.8733.643 −0.976 0.212 0.394 −1.023 1.625 −2.0613.773 −1.397 1.017 −0.933 0.942 −0.975 0.999

⎡⎢⎢⎢

⎢⎢

⎣

⎤⎥⎥⎥

⎥⎥

⎦

Cálculo de Me y Ke

≔ Me   ⋅⋅T

ϕ M ϕ

= Me

10.576 0 0 0 0 0 0

0 1.459 0 0 0 0 00 0 0.802 0 0 0 00 0 0 0.757 0 0 00 0 0 0 1.025 0 00 0 0 0 0 1.99 00 0 0 0 0 0 7.376

⎡

⎢⎢⎢⎢⎢

⎣

⎤

⎥⎥⎥⎥⎥

⎦

≔ Ke   ⋅⋅T

ϕ K ϕ

= Ke

12178.086 0 0 0 0 0 00 14714.682 0 0 0 0 00 0 21243.257 0 0 0 00 0 0 36012.716 0 0 00 0 0 0 71520.608 0 00 0 0 0 0 177513.885 00 0 0 0 0 0 755893.806

⎡⎢

⎢⎢⎢⎢

⎣

⎤⎥

⎥⎥⎥⎥

⎦

 Vector de masas participantes

≔ I 

1

111111

⎡

⎢⎢⎢⎢⎢

⎣

⎤

⎥⎥⎥⎥⎥

⎦

≔L   ⋅⋅T

ϕ M I    =L

3.532

0.4030.1540.0850.0580.0460.04

⎡

⎢⎢⎢⎢⎢

⎣

⎤

⎥⎥⎥⎥⎥

⎦

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Masas Efectivas:

≔Le,1 1

  ――

L,0 0

2

 Me,0 0

=Le,1 1

1.18

≔Le,2 1

  ――

⎛L,1 0

⎞2

 Me,1 1

=Le,2 1

0.111

≔Le,3 1

⎛L,2 0

⎞2

 Me,2 2

=Le,3 1

0.029

≔Le ,4 1

⎛L,3 0

⎞2

 Me,3 3

=Le ,4 1 0.01

≔Le,5 1

⎛L,4 0

⎞2

 Me,4 4

=Le,5 1

0.003

≔Le,6 1

⎛L,5 0

⎞2

 Me,5 5

=Le,6 1

0.001

≔Le,7 1

  ――⎛L ,6 0⎞

2

 Me,6 6

=Le,7 1

⋅2.136 10−4

Masas Efectiva Total:

≔ Mtotal   ∑=i   1

7

Le,i   1

= Mtotal   1.335

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Factor de Participación Modal (FPMi)

≔r1

  ――

L,0 0

 Me,0 0

=r1

0.334

≔r2

  ――

L,1 0

 Me,1 1

=r2

0.276

≔r3

L,2 0

 Me,2 2

=r3

0.191

≔r4

L,3 0

 Me,3 3

=r4 0.113

≔r5

L,4 0

 Me,4 4

=r5

0.057

≔r6

L,5 0

 Me,5 5

=r6

0.023

≔r7

  ――L

,6 0

 Me,6 6

=r7

0.005

Sumatoria de participación de modos

=∑=i   1

7

ri

1

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Factor de Participación Masas

≔ P 1

  ―――

⋅⎛Le,1 1

⎞   100

 Mtotal

= P 1

88.379

> %90}≔ P 2   ―――⋅⎛Le ,2 1⎞   100 Mtotal = P 2 8.352≔ P 

3  ―――

⋅⎛Le,3 1

⎞   100

 Mtotal= P 

32.203

≔ P 4

  ―――⋅Le ,4 1 100

 Mtotal= P 

40.723

≔ P 5

  ―――

⋅Le,5 1

100

 Mtotal= P 

50.248

≔ P 6

  ―――

⋅Le,6 1

100

 Mtotal= P 

60.078

Sumatoria de participación efectiva

≔ P 7

  ―――

⋅Le,7 1

100

 Mtotal= P 

70.016 ≔Suma   ∑

=i   1

7

 P i

=Suma   100

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