Sistemas de Vibración Sin Amortiguamiento (1)

7/23/2019 Sistemas de Vibración Sin Amortiguamiento (1)

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SISTEMAS DE VIBRACIÓN SIN AMORTIGUAMIENTO

A. PARA TRES GRADOS DE LIBERTAD

m1 ´ y

1+k

1 y

1−k

2 ( y2− y

1 )=0

m2 ´ y

2+k

2 ( y2− y

1 )−k 2 ( y3− y

2 )=0

m3 ´ y

3+k

2 ( y3− y

2)=0

k

y1(¿¿1+k 2)+ y2 (−k 1 )+0=0

m1 ´ y

1+¿

−k

y1(¿¿2)+ y

2 (k 2+k 2 )+ y3 (−k 2 )=0

m2 ´ y

2+¿

−k k

y2(¿¿2)=0

y2(¿¿ 2)+¿m3

´ y3+0+¿

[ M ] { ´ y }+ [ K ] { y }=0

[ M ]=m

1 0 0

0 m2

0

0 0

m3

{ ´ y }={´ y

1

´ y2

´ y3

}[ K ]=[

k 1+k

2 −k

2 0

−k 2

2k 2 −k

2

0 −k 2 k

2

]



|[ K ]−w2 [ M ]|=0

PÓRTICO 2-2

E (kg/m²)I( m

4

)H k = (12EI)/H³

b (m) h(m)

k'210x10^

7 0.3 0.3 4.10246804.312

2

k210x10^

7 0.3 0.3 3.00 630000

N! "#$#% k TOTA&

k1 3 k'740412.

37

k2 3 k 180000

|[ K ]−w2 [ M ]|=0

|2630412.94−22478.98w2 −1890000 0

−1890000 3780000−19604.72w2 −1890000

0 −1890000 1890000−16103.56w2|=0

−7.09674 x 1012ω

6+3.03168 x 1015ω

4−5.55997 x 1017ω

2+3.49396 x1019

ω2

1=117.162

ω2

2=155.0155151+134.1329585 i

ω2

3=155.0155151−134.1329585 i

T()*)(% #+ "*+(, ,#*+ -# w2

|−3269.31476 −1890000 0

−1890000 1483071.795 −1890000

0 −1890000 3274.70328

| x

{∅

11

∅12

∅13}

=0

m1

22478.8

m2

1604.72

m3

16103.6



∅11=1 ∅

12=−0.0017 ∅

13=−1.001

1

-.0017

-1.001

PÓRTICO 7-7

E (kg/m²)I(m^4)

H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.2 0.2 4.1011022.141

3

k210x10^

7 0.2 0.2 3.0030381.444

4

N! "#$#% k TOTA&

k1 4 k' 476088.6

k2 4 k 121277.78

m1

2706.48

m2

2388.02m3

161.88



|[ K ]−w2 [ M ]|=0

|1691366.34−27506.48w2 −1215277.78 0

−1215277.78 2430555.56−25388.02w2 −1215277.78

0 −1215277.78 1890000−15691.88w2|=0

−1.09582 x1013ω

6+3.04277 x1015ω

4−3.35824 x 1017ω

2+1.3059 x 1019

ω2

1=80.953

ω

2

2=98.3589075

+71.03927627

i

ω2

3=98.3589075−71.03927627 i

T()*)(% #+ "*+(, ,#*+ -# w2

|

−535365.7354 −1215277.78 0

−1215277.78 375319.1769 −1215277.78

0 −1215277.78 619695.2384

| x

{

∅11

∅12

∅13

}=0

∅11=1 ∅

12=−0.4405 ∅

13=−1.1360

1

-0.4405

-1.136



PÓRTICO 8-8

E (kg/m²) I(m^4) H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.2 0.2 4.1011022.141

3

k210x10^

7 0.2 0.2 3.0030381.444

4

N! "#$#% k TOTA&

k1 4 k' 476088.6

k2 4 k 121277.78

m1 1171.30

m2 1207.80

m3 830.60

|[ K ]−w2 [ M ]|=0

|1691366.34−11751.30w2 −1215277.78 0

−1215277.78 2430555.56−12097.80w2 −1215277.78

0 −1215277.78 1890000−8309.60w2|=0

−1.18133 x1012ω

6+6.76061 x1014ω

4−1.56444 x 1017ω

2+1.3059 x 1019

ω2

1=176.928

ω2

2=197.6799382+152.9295788 i

ω2

3=197.6799382−152.9295788 i

T()*)(% #+ "*+(, ,#*+ -# w2



|−387767.6664 −1215277.78 0

−1215277.78 290116.0016 −1215277.78

0 −1215277.78 419799.0912| x {

∅11

∅12

∅13

}=0

∅11=1 ∅

12=−0.3191 ∅

13=−1.0762

B. PARA CUATRO GRADOS DE LIBERTAD

m1 ´ y1+k 1 y1−k 2 ( y2− y1 )=0

m2 ´ y

2+k

2 ( y2− y

1 )−k 2 ( y3− y

2 )=0

m3 ´ y

3+k

2 ( y3− y

2)−k 2 ( y4

− y3 )=0

m4 ´ y

4+k

2 ( y 4− y

3 )=0

k y

1(¿¿1+k 2)+ y2 (−k

1 )+0+0=0

m1 ́y

1+¿

−k

y1(¿¿ 2)+ y

2 (k 2+k 2 )+ y3 (−k 2 )+0=0

m2 ´ y

2+¿

−k

k y3(¿¿ 2)+ y

4 (−k 2 )=0

y2(¿¿2)+¿m

3 ́y

3+0+¿

−k

k y

4(¿¿2)=0

y3(¿¿2)+¿m4

´ y4+0+0+¿



[ M ] { ´ y }+ [ K ] { y }=0

[ M ]=

[m

1 0

0 m2

0 0

0 0

0 0

0 0

m3

0

0 m4]

{ ´ y }={´ y

1

´ y2

´ y3

´ y4

} { y }={ y

1

y2

y3

y4

}[ K ]=[

k 1+k

2 −k

2

−k 2

2k 2

0 0

−k 2

0

0 −k 2

0 0

2k 2 −k

2

−k 2 k

2

]|[ K ]−w

2 [ M ]|=0

1

-0.3191

-1.0762

PÓRTICO 3-3



E (kg/m²)I(m^4)

H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.3 0.4 4.1083264.3

6

k210x10^

7 0.3 0.4 3.00 212620

N! "#$#% k TOTA&

k1 2 k' 1662.11

k2 2 k 42200

)1 2731.84

)2 204.3

)3 23788.12

)4 8610

|5918429.11−27351.84w

2

−4252500

−4252500

8505000−25045.39w2

0 0

−4252500 0

0 −4252500 8505000−23788.12w2 −4252500

0 0 −4252500 4252500−8610w2 |=0

1.4030604 x1017ω

8−1.974672 x1020ω

6+1.0150224 x1023ω

4−2.2552656 x1025ω

2+1.4935116 x 1027

w1

2=109.792

w2

2=595.19

x / 31.20818.864

x / 31.20818.864

*,* w1

2

[2915415.893 −4252500

−4252500 5755216.541

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

5893254.729 −4252500

−4252500 3307190.88]{

∅11

∅12

∅13

∅14

}=0

∅11=1 ∅12=0.6856 ∅13=−0.0721 ∅14=−0.7855



1

1

0.6856

-0.0721

-0.7855

*,* w2

2

[−10361112.54 −4252500

−4252500 −6401765.674

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

−5653451.143 −4252500

−4252500 −872085.9]{

∅11

∅12

∅13

∅14

}=0

∅21=1 ∅

22=−2.4365 ∅

23=2.6679 ∅

24=−1.1103



1

-2.4365

2.6679

-1.1103

PÓRTICO 4-4

E (kg/m²)I(m^4)

H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.3 0.4 4.1083264.3

6

k210x10^

7 0.3 0.4 3.00 212620

N! "#$#% k TOTA&

k1 2 k' 1662.11

k2 2 k 42200

)122768.

0

)2117.2

6

)320277.6

0

)4 12046.3

|5918429.11−22768.50w

2

−4252500

−4252500

8505000−19179.26w2

0 0

−4252500 0

0 −4252500 8505000−20277.60w2 −4252500

0 0 −4252500 4252500−12046.3w2 |=0



1.06669 x 1017ω

8−1.57425 x1020ω

6+8.60451 x1022ω

4−2.06068 x1025ω

2+1.49351 x1027

51 / 121.62

2/ 613.88

3/ 370.04224.68

4 / 370.04224.68

*,* w1

2

[

3148595.548 −4252500

−4252500 6171804.662

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

6038189.405 −4252500

−4252500 2787043.512

]{

∅11

∅12

∅13

∅14

}=0

∅11=1 ∅

12=0.7404 ∅

13=0.0746 ∅

14=−0.6345

1

0.7404

0.0746

-0.6345

*,* w2

2



[−8061156.668 −4252500

−4252500 −3270835.489

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

−3945203.069 −4252500

−4252500 −3143783.644]{

∅11

∅12

∅13

∅14

}=0

∅21=1 ∅

22=1.8956 ∅

23=−2.4580 ∅

24=0.3848

1 1

0.7404

0.0746

-0.6345

1.8956

-2.458

0.3848

PÓRTICO 5-5

E (kg/m²) I(m^4) H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.3 0.4 4.1083264.3

6

k210x10^

7 0.3 0.4 3.00 212620

N! "#$#% k TOTA&

k1 2 k' 1662.11

k2 2 k 42200

)1 24.26

)2 24100.10)3 23321.70



)4 12784.80

|5918429.11−29545.26w

2

−4252500

−4252500

8505000−24100.10w2

0 0

−4252500 0

0 −4252500 8505000−23321.70w2 −4252500

0 0 −4252500 4252500−12784.80 w2

|=0

2.12305 x1017ω

8−2.65493 x 1020ω

6+1.22661 x1023ω

4−2.47124 x 1025ω

2+1.49351 x 1027=0

/100.636

/1.8

x / 31.16187.6

x / 31.16187.6

*,* w1

2

[2945112.325 −4252500

−4252500 6079662.336

000000000 0000000

−4252500 00000000000000 −4252500

000 000 00000000

6157997.399 −4252500

−4252500 2965888.867]{∅

11

∅12

∅13

∅14}

=0

∅11=1 ∅

12=0.6926 ∅

13=−0.0098 ∅

14=−0.7068

1

0.6926

-0.0098

-0.7068



*,* w2

2

[−9432047.085 −4252500

−4252500 −4016399.756

000000000 0000000

−4252500 00000000

000000 −4252500000000 00000000

−3611975.809 −4252500−4252500 −2389945.118

]{∅

11

∅12

∅13

∅14}

=0

∅21=1 ∅

22=−2.218 ∅

23=1.0949 ∅

24=1.288

1 1

0.6926 -2.218

1.0949

1.288

PÓRTICO 6-6

E (kg/m²)I(m^4)

H k = (12EI)/H³ b (m) h(m)

k'210x10^

7 0.3 0.4 4.1083264.3

6

k210x10^

7 0.3 0.4 3.00 212620

N! "#$#% k TOTA&

k1 2 k' 1662.11

k2 2 k 42200

)1 37334.10

)2 310.17

)3 278.14

)4 1048.1



|5918429.11−37334.10w

2

−4252500

−4252500

8505000−35109.17w2

0 0

−4252500 0

0 −4252500 8505000−27985.14 w2 −4252500

0 0 −4252500 4252500−10458.1w2 |=0

3.83625 x 1017ω

8−4.26324 x1020ω

6+1.71381 x1023ω

4−2.9467 x1025ω

2+1.49351 x1027

/83.317

/474.613

x / 276.687147.76

x / 276.687147.76

*,* w1

2

[2807845.233 −4252500

−4252500 5579791.729

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

6173345.599 −4252500

−4252500 3381157.253]{

∅11

∅12

∅13

∅14

}=0

∅11=1 ∅

12=0.6603 ∅

13=−0.1336 ∅

14=−0.8542

1

0.6603

-0.1336

-0.8542

*,* w2

2



[−11800820.09 −4252500 0

−4252500 −8158268.501

000000000 0000000

−4252500 00000000

000000 −4252500

000 000 00000000

−4777125.489 −4252500

−4252500 −711050.2153]{

∅11

∅12

∅13

∅14

}=0

∅21=1 ∅

22=−2.7750 ∅

23=4.3237 ∅

24=−2.0821

1

-2.775

4.3237

-2.0821

Documents

Sistemas de Vibración Sin Amortiguamiento (1)