Ecuaciones Diferenciales Ordinarias

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA
FACULTAD DE INGENIERA CIVIL, ARQUITECTURA Y GEOTECNIA

ESCUELA ACADMICO PROFESIONAL DE INGENIERA GEOLGICA GEOTECNIA / MTODOS NUMRICOS EN INGENIERIA GEOLGICA 3 AO
PRACTICA ENCARGADA # 03
PROBLEMA:
Cuando el cuerpo sujeto al resorte se mueve en un medio que produce friccin sobre el cuerpo,
entonces decimos que el movimiento se efecta con amortiguamiento, supongamos
que el
amortiguamiento es directamente proporcional a la velocidad. Por la segunda ley de Newton

tenemos:
Donde es la constante de friccin o constante de amortiguamiento. La misma que se puede

escribir como:
Dnde:
Resolver la EDO para los siguientes casos y reproducir las funciones mostradas. Asumir las
contantes necesarias para lograr los casos a, b y c.
a) Cuando
, estamos en caso de un movimiento sobre-amortiguado.
Pgina 1

b) Cuando
, estamos en caso de un movimiento crticamente amortiguado.
c) Cuando
, estamos en caso de un movimiento sub-amortiguado.
FORMULACIN DEL PROBLEMA

CASO I: Movimiento Armnico Simple
Consideramos un cuerpo de masa m que se sujeta al extremo de un resorte suspendido de un
soporte rgido y se trata de encontrar el desplazamiento x(t) de la masa como funcin del tiempo.
Para ello se utiliza la segunda ley del movimiento de Newton y la ley de Hooke
Ley de Hooke: Un resorte al ser estirado o contrado opone una fuerza F llamada fuerza de
Restitucin que es proporcional a su alargamiento.
Dnde: k > 0 es una constante llamada mdulo o constante del resorte S es el alargamiento del
resorte despus que la masa ha sido colgada y el sistema est en reposo en la posicin de
equilibrio.
Es decir: Si una masa se sujeta a un resorte lo alargar en una magnitud S y alcanzar la posicin
de equilibrio en la cual su peso w es equilibrado por la fuerza de restitucin k s
Pgina 2

Donde m es la masa y g la aceleracin, es decir la condicin de equilibrio significa que

o(
Cuando el sistema est en movimiento, esto quiere decir que la masa se desplaza de su posicin
de equilibrio en una magnitud x y despus se suelta, la fuerza neta correspondiente est dada por
la segunda ley del movimiento de Newton.
F = m*a
Donde a es la aceleracin
Suponiendo que sobre el sistema no actan otras fuerzas exteriores (movimiento vibratorio libre) y
la masa oscila libre, entonces podemos igualar F a la resultante del peso y la fuerza de restitucin
obteniendo la Ecuacin Diferencial que representa el movimiento.
El signo negativo indica que la fuerza de restitucin del resorte acta en direccin opuesta a la del
movimiento. Los desplazamientos medidos hacia abajo desde la posicin de equilibrio se
considerarn positivos.
Dividiendo por la masa la ecuacin anterior, se obtiene
Donde
, y la ecuacin diferencial describe el Movimiento armnico simple o Movimiento
vibratorio libre no amortiguado.
Pgina 3

Las condiciones iniciales asociadas, son obvias
Interpretacin Fsica
Si analizamos el signo, podemos interpretar su significado fsico
Si a < 0 y b > 0, se trata de una masa que parte de un punto bajo la posicin de equilibrio y
lleva una velocidad hacia arriba.
Si a < 0 y b = 0, se trata de una masa en reposo, que se suelta desde un punto | | unidades
arriba de la posicin de equilibrio.
Luego la formulacin matemtica de este problema es
Solucin de la Ecuacin de Movimiento y su Interpretacin

La ecuacin caracterstica asociados es:
.: La solucin general es y = c1 coswt + c2 senwt

En este tipo de problemas se usa la siguiente terminologa Se llama
Perodo de las vibraciones libres a:
Pgina 4

Frecuencia de las vibraciones u oscilaciones es:
Recordemos que estos nmeros indican:

El perodo indica que la grfica se repite cada
grfica cada
unidades y la frecuencia que hay w ciclos de la
unidades es decir la masa realiza
adems el perodo
oscilaciones completas por unidad de tiempo,
es el intervalo de tiempo entre dos mximos sucesivos de x(t).
Unidades
Dos posibles graficas a y b
Pgina 5

Dos posibles graficas
El movimiento que describe la figura es oscilatorio pero, a causa del coeficiente

amplitudes de vibracin tienden a cero cuando
Pgina 6
, las

Posible grafica
SOLUCIN EN HOJA DE CLCULO
Datos de Entrada
h=
=
w=
Pgina 7
0.2
5
3

k1
l1
0.2
2.6562
-1.8
-0.36
0.4 2.23179948
-1.98744
0.6 1.84353926 1.77379906
0.8 1.51401639 1.49411254
1 1.24090261 1.23522205
1.2 1.01634703 -1.0147258
1.4 0.83222528 0.83176258
1.6 0.68140142 0.68126937
1.8 0.55789486 0.55785717
2 0.45676963 0.45675888
2.2 0.37397324 0.37397017
2.4 0.30618456 0.30618369
2.6 0.25068357 0.25068332
2.8 0.20524301 0.20524294
3 0.1680393 0.16803928
3.2 0.13757937
-
k2
k3
-0.54
-1.18116 -0.478116
-0.397488 0.04235906 0.40172391
0.35475981 0.22922746 0.33183707
0.29882251 0.26299557 0.27252295
0.24704441 0.2368194 0.22336247
0.20294516 0.20002694 0.18294247
0.16635252 0.16551966 0.14980055
0.13625387 0.13601618 0.12265226
0.11157143 0.1115036 0.10042107
0.09135178 0.09133241 0.08221853
0.07479403 0.07478851 0.06731518
0.06123674 0.06123516 0.05511322
0.05013666 0.05013621 0.04512304
0.04104859 0.04104846 0.03694374
0.03360786 0.03360782 0.03024707
- 0.02751586
-
0.324
Pgina 8
-5.4
l2
l3
0
k4
-4.914
0.3577392
-0.3276 -1.0748556
-0.36171408 0.03854675
0.31928383
-0.32283143 0.20859699
0.26894026
-0.27192848 0.23932597
0.22233997
-0.22481041 0.21550566
0.18265064
-0.1846801 0.18202452
0.14971726
-0.15138079 0.15062289
0.12262849
-0.12399102 0.12377472
0.10041429
-0.10153001 0.10146827
0.0822166
-0.08313012
0.0831125
0.06731463
-0.06806257 0.06805754
0.05511306
-0.05572543
0.045123
0.055724
-0.04562436 0.04562395
0.03694373
-0.03735421
0.0373541
0.03024707
0.02476429
-0.03058315 0.03058311
-0.02503945 0.02503944
-0.9828
0.57497112
0.40519735
0.31304041
0.25095731
0.20394328
0.16654026
0.13622794
0.11149893
0.09127778
0.07472928
0.06118253
0.05009194
0.04101187
0.03357777
0.02749123
-
l4
4.428
1.5582312
0.68581978
0.39313006
0.2738149
0.21046683
0.16840208
0.1367593
0.11165058
0.09132106
0.07474163
0.06118605
0.05009294
0.04101216
0.03357785
0.02749126
0.02250799

3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
0.13757937
0.11264082
0.09222279
0.07550588
0.06181918
0.05061342
0.02751587
0.11264082
0.02252816
0.09222279
0.01844456
0.07550588
0.01510118
0.06181918
0.01236384
0.05061342
0.01012268
0.0414389 -0.0414389 0.00828778
0.0339274 -0.0339274 0.00678548
0.0277775 -0.0277775
0.02274236 0.02274236
0.01861993 0.01861993
0.01524476 0.01524476
0.01248139 0.01248139
0.01021893 0.01021893
0.00836658 0.00836658
-0.0055555
0.00454847
0.00372399
0.00304895
0.00249628
0.00204379
0.00167332
0.00685
0.02476429
0.02252816 0.02027535 0.02027535
0.01842804
0.01844456 -0.0166001 0.0166001

0.01510117 0.01359106 0.01359106
0.01236384 0.01112745 0.01112745
0.01012268 0.00911042 0.00911042
-0.01678455 0.01678455
0.00828778
-0.00754188 0.00754188 -0.0067794 0.0067794

-0.00617479 0.00617479 0.00555052 0.00555052
0.00678548
0.0055555
0.00454847
0.00372399
0.00304895
0.00249628
0.00204379
0.00167332
-0.007459
0.00610693
0.00499995
0.00409363
0.00335159
0.00274406
0.00224665
0.00183941
0.00150598
-0.00685
-0.00137
0.00137 -0.001233
6.4 0.00560832 0.00560832 0.00112166 0.00112166 -0.0010095
Pgina 9
-0.02050063 0.02050063
0.02250799
0.01842804
0.01508765
0.01235276
0.01011362
0.00828036
0.007459
0.00610693
-0.01374207 0.01374207
-0.01125109 0.01125109
-0.00921164 0.00921164
0.00499995
-0.0050555
0.00409363
-0.00413911
0.00335159
-0.00338883
0.00274406
-0.00277455
0.00224665
-0.00227161
0.00183941
-0.00185985
0.00150598
-0.00152272
0.001233
-0.0012467
0.0010095
-0.00102071
0.0050555 -0.0045444
0.00413911 0.00372065
0.00338883 0.00304622
0.00277455 0.00249404
0.00227161 0.00204196
0.00185985 0.00167182
0.00152272 0.00136877
0.0012467 0.00112066
0.00102071 0.00091752
0.01508765
0.01235276
0.01011362
0.00828036
0.0045444
0.00372065
0.00304622
0.00249404
0.00204196
0.00167182
0.00136877
0.00112066
0.00091752

6.6 0.00459172 0.00459172

6.8 0.00375939 0.00375939
7 0.00307794 0.00307794
7.2 0.00252001 0.00252001
7.4 0.00206322 0.00206322
7.6 0.00168923 0.00168923
7.8 0.00138303 0.00138303
8 0.00113233 0.00113233
8.2 0.00092708 0.00092708
8.4 0.00075903 0.00075903
8.6 0.00062144 0.00062144
8.8 0.00050879 0.00050879
9 0.00041657 0.00041657
9.2 0.00034106 0.00034106
9.4 0.00027924 0.00027924
9.6 0.00022862 0.00022862
9.8 0.00018718
-
0.00091834 0.00091834 0.00082651 0.00082651

0.00075188 0.00075188 0.00067669 0.00067669
0.00061559 0.00061559 0.00055403 0.00055403
-0.000504
0.00041264
0.00033785
0.00027661
0.00022647
0.00018542
0.00015181
0.00012429
0.00010176
-8.3313E05
-6.8211E05
-5.5847E05
-4.5724E05
-3.7436E-
Pgina 10
9.1583E-05
-0.00083569 0.00083569 0.00075121

-0.00068421 0.00068421 0.00061504
-0.00056019 0.00056019 0.00050355
-0.00045864 0.00045864 0.00041227
-0.00037551 0.00037551 0.00033754
-0.00030744 0.00030744 0.00027636
-0.00025171 0.00025171 0.00022626
-0.00020608 0.00020608 0.00018525
-0.00016873 0.00016873 0.00015167
-0.00013814 0.00013814 0.00012418
-0.0001131 0.0001131 0.00010167
-8.3239E-9.2601E-05 9.2601E-05
05
7.4982E-05
-7.5815E-05 7.5815E-05
-6.139E-05 6.139E-05
-5.0262E05 5.0262E-05
-4.1151E05 4.1151E-05
-3.3692E- 3.3692E-05
-6.2072E-05 6.2072E-05
0.000504 -0.0004536
0.00041264 0.00037138
0.00033785 0.00030406
0.00027661 0.00024894
0.00022647 0.00020382
0.00018542 0.00016687
0.00015181 0.00013662
0.00012429 0.00011186
-9.1583E0.00010176
05
-7.4982E8.3313E-05
05
6.8211E-05
5.5847E-05
4.5724E-05
3.7436E-05
0.0004536
0.00037138
0.00030406
0.00024894
0.00020382
0.00016687
0.00013662
0.00011186
-5.0821E-05 5.0821E-05
-4.1609E-05 4.1609E-05
-3.4066E-05 3.4066E-05
-6.815E-05
-5.5797E05
-4.5683E05
-3.7402E05
-3.0622E-
0.00075121
0.00061504
0.00050355
0.00041227
0.00033754
0.00027636
0.00022626
0.00018525
0.00015167
0.00012418
0.00010167
8.3239E-05
6.815E-05
5.5797E-05
4.5683E-05
3.7402E-05
3.0622E-05

10 0.00015325
10.2 0.00012547
10.4 0.00010273
10.6 8.4106E-05
10.8
6.886E-05
11 5.6378E-05
11.2 4.6159E-05
11.4 3.7792E-05
11.6 3.0941E-05
11.8 2.5333E-05
12 2.0741E-05
12.2 1.6981E-05
12.4 1.3903E-05
12.6 1.1383E-05
12.8 9.3195E-06
13 7.6302E-06
0.00018718
0.00015325
0.00012547
0.00010273
-8.4106E05
-6.886E-05
-5.6378E05
-4.6159E05
-3.7792E05
-3.0941E05
-2.5333E05
-2.0741E05
-1.6981E05
-1.3903E05
-1.1383E05
-9.3195E06
-7.6302E06
05
-3.065E-05
-2.5094E05
-2.0545E05
-1.6821E05
-1.3772E05
-1.1276E05
-9.2317E06
-7.5583E06
-6.1882E06
-5.0665E06
-4.1481E06
-3.3962E06
-2.7806E06
-2.2766E06
-1.8639E06
-1.526E-06
Pgina 11
3.065E-05
2.5094E-05
2.0545E-05
1.6821E-05
1.3772E-05
1.1276E-05
9.2317E-06
7.5583E-06
6.1882E-06
5.0665E-06
4.1481E-06
3.3962E-06
2.7806E-06
2.2766E-06
1.8639E-06
1.526E-06
05
-2.7585E05
-2.2585E05
-1.8491E05
-1.5139E05
-1.2395E05
-1.0148E05
-8.3085E06
-6.8025E06
-5.5694E06
-4.5599E06
-3.7333E06
-3.0566E06
-2.5025E06
-2.0489E06
-1.6775E06
-1.3734E06
2.7585E-05
-2.7891E-05 2.7891E-05
2.2585E-05
-2.2836E-05 2.2836E-05
1.8491E-05
-1.8696E-05 1.8696E-05
1.5139E-05
-1.5307E-05 1.5307E-05
1.2395E-05
-1.2533E-05 1.2533E-05
1.0148E-05
-1.0261E-05 1.0261E-05
8.3085E-06
-8.4009E-06 8.4009E-06
6.8025E-06
-6.8781E-06 6.8781E-06
5.5694E-06
-5.6313E-06 5.6313E-06
4.5599E-06
-4.6105E-06 4.6105E-06
3.7333E-06
-3.7748E-06 3.7748E-06
3.0566E-06
-3.0906E-06 3.0906E-06
2.5025E-06
-2.5303E-06 2.5303E-06
2.0489E-06
-2.0717E-06 2.0717E-06
1.6775E-06
-1.6961E-06 1.6961E-06
1.3734E-06
-1.3887E-06 1.3887E-06
05
-2.5072E05 2.5072E-05
-2.0527E05 2.0527E-05
-1.6806E05 1.6806E-05
-1.376E-05
-1.1266E05
-9.2234E06
-7.5515E06
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-1.2483E06
1.376E-05
1.1266E-05
9.2234E-06
7.5515E-06
6.1827E-06
5.062E-06
4.1444E-06
3.3932E-06
2.7781E-06
2.2745E-06
1.8622E-06
1.5247E-06
1.2483E-06

13.2 6.2471E-06
13.4 5.1147E-06
13.6 4.1876E-06
13.8 3.4285E-06
14
2.807E-06
14.2 2.2982E-06
14.4 1.8816E-06
14.6 1.5405E-06
14.8 1.2613E-06
15 1.0327E-06
15.2 8.4548E-07
15.4 6.9222E-07
15.6 5.6674E-07
15.8 4.6401E-07
16
3.799E-07
16.2 3.1104E-07
16.4 2.5466E-07
-6.2471E06
-5.1147E06
-4.1876E06
-3.4285E06
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-8.3751E07 8.3751E-07
-6.857E-07
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-3.7632E07
-3.0811E07
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-2.0653E07
6.857E-07
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3.7632E-07
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2.5226E-07
2.0653E-07
-1.691E-07 1.691E-07
-1.3844E07 1.3844E-07
-1.1335E07 1.1335E-07
-9.2802E08 9.2802E-08
-7.598E-08 7.598E-08
-6.2208E08 6.2208E-08
-5.0932E- 5.0932E-08
Pgina 12
-1.1245E06
-9.2064E07
-7.5376E07
-6.1713E07
-5.0527E07
-4.1368E07
-3.3869E07
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7.5376E-07
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6.1713E-07
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5.0527E-07
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4.1368E-07
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8.3522E-08
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8.445E-08
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1.6894E-07
1.3832E-07
1.1325E-07
9.2719E-08
7.5912E-08
6.2152E-08
5.0886E-08
4.1662E-08

07
16.6
2.085E-07
-2.085E-07
16.8
1.707E-07
-1.707E-07
-1.3976E07
-1.1443E07
-9.3685E08
-7.6703E08
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-2.3103E08
-1.8915E08
-1.5486E08
-1.2679E08
-1.0381E08
17 1.3976E-07
17.2 1.1443E-07
17.4 9.3685E-08
17.6 7.6703E-08
17.8 6.2799E-08
18 5.1416E-08
18.2 4.2096E-08
18.4 3.4465E-08
18.6 2.8218E-08
18.8 2.3103E-08
19 1.8915E-08
19.2 1.5486E-08
19.4 1.2679E-08
19.6 1.0381E-08
08
-4.1699E08
-3.4141E08
-2.7952E08
-2.2885E08
-1.8737E08
-1.5341E08
4.1699E-08
3.4141E-08
2.7952E-08
2.2885E-08
1.8737E-08
1.5341E-08
-1.256E-08 1.256E-08
-1.0283E08 1.0283E-08
-8.4192E09 8.4192E-09
-6.893E-09 6.893E-09
-5.6436E09 5.6436E-09
-4.6206E09 4.6206E-09
-3.783E-09 3.783E-09
-3.0973E09 3.0973E-09
-2.5359E09 2.5359E-09
-2.0762E09 2.0762E-09
Pgina 13
08
-3.7529E08
-3.0727E08
-2.5157E08
-2.0597E08
-1.6863E08
-1.3807E08
-1.1304E08
-9.2548E09
-7.5772E09
-6.2037E09
-5.0792E09
-4.1585E09
-3.4047E09
-2.7876E09
-2.2823E09
-1.8686E09
08
3.7529E-08
-3.7946E-08 3.7946E-08
3.0727E-08
-3.1068E-08 3.1068E-08
2.5157E-08
-2.5436E-08 2.5436E-08
2.0597E-08
-2.0826E-08 2.0826E-08
1.6863E-08
-1.7051E-08 1.7051E-08
1.3807E-08
-1.396E-08
1.396E-08
1.1304E-08
-1.1429E-08 1.1429E-08
9.2548E-09
-9.3577E-09 9.3577E-09
7.5772E-09
-7.6614E-09 7.6614E-09
6.2037E-09
-6.2727E-09 6.2727E-09
5.0792E-09
-5.1356E-09 5.1356E-09
4.1585E-09
-4.2047E-09 4.2047E-09
3.4047E-09
-3.4425E-09 3.4425E-09
2.7876E-09
-2.8185E-09 2.8185E-09
2.2823E-09
-2.3076E-09 2.3076E-09
1.8686E-09
-1.8893E-09 1.8893E-09
-3.411E-08 3.411E-08
-2.7927E08 2.7927E-08
-2.2865E08 2.2865E-08
-1.872E-08
-1.5327E08
-1.2549E08
-1.0274E08
-8.4116E09
-6.8869E09
-5.6385E09
-4.6164E09
-3.7796E09
-3.0945E09
-2.5336E09
-2.0743E09
-1.6983E09
1.872E-08
1.5327E-08
1.2549E-08
1.0274E-08
8.4116E-09
6.8869E-09
5.6385E-09
4.6164E-09
3.7796E-09
3.0945E-09
2.5336E-09
2.0743E-09
1.6983E-09

19.8 8.4992E-09
20 6.9586E-09
20.2 5.6972E-09
20.4 4.6645E-09
20.6
3.819E-09
20.8 3.1267E-09
21
2.56E-09
21.2 2.0959E-09
21.4
1.716E-09
21.6 1.4049E-09
21.8 1.1503E-09
22 9.4177E-10
22.2 7.7106E-10
22.4 6.3129E-10
22.6 5.1686E-10
22.8 4.2317E-10
23 3.4646E-10
-8.4992E09
-6.9586E09
-5.6972E09
-4.6645E09
-3.819E-09
-3.1267E09
-2.56E-09
-2.0959E09
-1.716E-09
-1.4049E09
-1.1503E09
-9.4177E10
-7.7106E10
-6.3129E10
-5.1686E10
-4.2317E10
-3.4646E-
-1.6998E09 1.6998E-09
-1.3917E09 1.3917E-09
-1.1394E09 1.1394E-09
-9.329E-10
9.329E-10
-7.638E-10 7.638E-10
-6.2535E10 6.2535E-10
-5.1199E10 5.1199E-10
-4.1919E10 4.1919E-10
-3.432E-10
-2.8099E10
-2.3006E10
-1.8835E10
-1.5421E10
-1.2626E10
-1.0337E10
-8.4634E11
-6.9293E-
Pgina 14
3.432E-10
2.8099E-10
2.3006E-10
1.8835E-10
1.5421E-10
1.2626E-10
1.0337E-10
8.4634E-11
6.9293E-11
-1.5299E09
-1.2525E09
-1.0255E09
-8.3961E10
-6.8742E10
-5.6281E10
-4.6079E10
-3.7727E10
-3.0888E10
-2.5289E10
-2.0705E10
-1.6952E10
-1.3879E10
-1.1363E10
-9.3035E11
-7.6171E11
-6.2364E-
1.5299E-09
-1.5469E-09 1.5469E-09
1.2525E-09
-1.2665E-09 1.2665E-09
1.0255E-09
-1.0369E-09 1.0369E-09
8.3961E-10
-8.4894E-10 8.4894E-10
6.8742E-10
-6.9506E-10 6.9506E-10
5.6281E-10
-5.6907E-10 5.6907E-10
4.6079E-10
-4.6591E-10 4.6591E-10
3.7727E-10
-3.8146E-10 3.8146E-10
3.0888E-10
-3.1231E-10 3.1231E-10
2.5289E-10
2.0705E-10
1.6952E-10
1.3879E-10
1.1363E-10
-2.557E-10
2.557E-10
-2.0935E-10 2.0935E-10
-1.714E-10
1.714E-10
-1.4033E-10 1.4033E-10
-1.149E-10
1.149E-10
9.3035E-11
-9.4069E-11 9.4069E-11
7.6171E-11
6.2364E-11
-7.7017E-11 7.7017E-11
-6.3056E-11 6.3056E-11
-1.3905E09
-1.1384E09
-9.3207E10
-7.6311E10
-6.2479E10
-5.1153E10
-4.1881E10
-3.4289E10
-2.8074E10
-2.2985E10
-1.8819E10
-1.5407E10
-1.2615E10
-1.0328E10
-8.4558E11
-6.9231E11
-5.6682E-
1.3905E-09
1.1384E-09
9.3207E-10
7.6311E-10
6.2479E-10
5.1153E-10
4.1881E-10
3.4289E-10
2.8074E-10
2.2985E-10
1.8819E-10
1.5407E-10
1.2615E-10
1.0328E-10
8.4558E-11
6.9231E-11
5.6682E-11

23.2 2.8366E-10
23.4 2.3224E-10
23.6 1.9015E-10
23.8 1.5568E-10
24 1.2746E-10
24.2 1.0435E-10
24.4 8.5439E-11
24.6 6.9952E-11
24.8 5.7272E-11
25
4.689E-11
10
-2.8366E10
-2.3224E10
-1.9015E10
-1.5568E10
-1.2746E10
-1.0435E10
-8.5439E11
-6.9952E11
-5.7272E11
-4.689E-11
11
-5.6732E11
-4.6449E11
-3.8029E11
-3.1136E11
-2.5492E11
-2.0871E11
-1.7088E11
5.6732E-11
4.6449E-11
3.8029E-11
3.1136E-11
2.5492E-11
2.0871E-11
1.7088E-11
-1.399E-11 1.399E-11
-1.1454E11 1.1454E-11
-9.3781E12 9.3781E-12
Pgina 15
11
-5.1059E11
-4.1804E11
-3.4226E11
-2.8022E11
-2.2943E11
-1.8784E11
-1.5379E11
-1.2591E11
-1.0309E11
-8.4403E12
5.1059E-11
-5.1626E-11 5.1626E-11
4.1804E-11
-4.2268E-11 4.2268E-11
3.4226E-11
-3.4606E-11 3.4606E-11
2.8022E-11
-2.8333E-11 2.8333E-11
2.2943E-11
-2.3198E-11 2.3198E-11
1.8784E-11
-1.8993E-11 1.8993E-11
1.5379E-11
-1.555E-11
1.555E-11
1.2591E-11
-1.2731E-11 1.2731E-11
1.0309E-11
-1.0423E-11 1.0423E-11
8.4403E-12
-8.534E-12
8.534E-12
11
-4.6407E11
-3.7995E11
-3.1108E11
-2.5469E11
-2.0852E11
-1.7072E11
-1.3978E11
-1.1444E11
-9.3697E12
-7.6712E12
4.6407E-11
3.7995E-11
3.1108E-11
2.5469E-11
2.0852E-11
1.7072E-11
1.3978E-11
1.1444E-11
9.3697E-12
7.6712E-12

Grafica Caso I
3.5
Desplazamiento (S)
2.5
1.5
0.5
0
0
10
15
Tiempo (t)
Pgina 16
20
25
30

Datos de Entrada
h=
=
w=
x
0
1
2
3
4
5
6
7
1
0.33333333
0.33333333
0.18518519
0.08641975
0.03703704
0.01508916
0.00594422
k1
0
l1
k2
-4
1.33333333 1.33333333
-4
2.22222222
1.03703704
0.44444444
0.18106996
0.07133059
0.02743484
0.88888889 0.88888889
0.44444444 0.44444444
0.19753086 0.19753086
0.08230453 0.08230453
0.03292181 0.03292181
0.01280293 0.01280293
Pgina 17
1
2
2
-2
0.66666667
0.22222222
0.07407407
0.02469136
0.00823045
0.00274348
0.00091449
l2
k3
l3
1.33333333
0.44444444 1.11111111
0.14814815 0.51851852
0.04938272 0.22222222
0.01646091 0.09053498
0.00548697 0.03566529
0.00182899 0.01371742
-8
5.33333333
2.66666667
1.18518519
0.49382716
0.19753086
0.07681756
0.02926383
k4
l4
-8
20
-4 9.33333333
1.77777778
4
0.74074074 1.62962963
-0.2962963 0.64197531
0.11522634 0.24691358
0.04389575 0.09327846
0.01646091 0.0347508

8 0.00228624
9 0.00086369
10 0.00032177
11 0.00011855
-4.3279E12
05
-1.5681E13
05
14
22
-5.645E-06
-2.0211E06
-7.2015E07
-2.5554E07
-9.0341E08
-3.1834E08
-1.1185E08
-3.9196E09
-1.3703E09
23
24
-4.78E-10
-1.6641E-
15
16
17
18
19
20
21
0.00487731 0.00487731 0.01036427 0.00030483 0.00060966 0.00518214 0.01097394 0.00609663

0.00182899 0.00182899 -0.0038612 0.00010161 0.00020322 0.0019306 0.00406442 0.00223543
0.0006774 0.0006774 0.00142255 -3.387E-05 6.774E-05 0.00071127 0.00149029 0.00081288
0.00024838 0.00024838 0.00051934 -1.129E-05 2.258E-05 0.00025967 0.00054192 0.00029354
9.032E-05 9.032E-05 0.00018817 -3.7634E-06 7.5267E-06 9.4084E-05 0.00019569 0.00010537
-7.0249E-3.7634E3.2616E-05 3.2616E-05 -6.774E-05 -1.2545E-06 2.5089E-06 3.387E-05
05
05
-2.4253E-2.5089E-1.3381E1.1708E-05 1.1708E-05
05 -4.1815E-07 8.363E-07 1.2126E-05
05
05
-8.6418E-8.9205E4.1815E-06 4.1815E-06
06 -1.3938E-07 2.7877E-07 4.3209E-06
06 -4.739E-06
-3.0664E-3.1594E-1.6726E1.4868E-06 1.4868E-06
06 -4.6461E-08 9.2922E-08 1.5332E-06
06
06
-1.0841E-1.1151E-5.8851E5.2656E-07 5.2656E-07
06 -1.5487E-08 3.0974E-08 5.4205E-07
06
07
-3.8201E-3.9234E-2.0649E1.8584E-07 1.8584E-07
07 -5.1623E-09 1.0325E-08 1.9101E-07
07
07
-1.3422E-1.3766E-7.2273E6.539E-08 6.539E-08
07 -1.7208E-09 3.4416E-09 6.7111E-08
07
08
-4.7035E-4.8182E-2.5238E2.2944E-08 2.2944E-08
08 -5.7359E-10 1.1472E-09 2.3517E-08
08
08
-1.6443E-1.6825E-8.7951E8.0303E-09 8.0303E-09
08 -1.912E-10 3.824E-10 8.2215E-09
08
09
-5.7359E-5.8634E-3.0592E2.8042E-09 2.8042E-09
09 -6.3733E-11 1.2747E-10 2.868E-09
09
09
-2.0394E-1.0622E9.7723E-10 9.7723E-10 -1.997E-09 -2.1244E-11 4.2488E-11 9.9848E-10
09
09
3.3991E-10 3.3991E-10
-6.9398E- -7.0814E-12 1.4163E-11 3.4699E-10
-7.0814E-3.6823E-
Pgina 18
0.01280293
0.00467408
0.00169351
0.00060966
0.00021827
7.7776E-05
2.7598E-05
9.7568E-06
3.4381E-06
1.208E-06
4.2331E-07
1.4799E-07
5.1623E-08
1.7973E-08
6.2458E-09
2.1669E-09
7.5063E-10

25
10
-5.7832E11 1.1802E-10 1.1802E-10
10
-2.4077E10 -2.3605E-12 4.7209E-12 1.2038E-10
10
-2.4549E10
10
-1.2747E10 2.5965E-10
Grafica Caso II
1.2
1
0.8
Desplazamiento (S)
0.6
0.4
0.2
0
0
10
15
-0.2
-0.4
-0.6
Tiempo (t)
Pgina 19
20
25
30

Datos de Entrada
h=
=
w=
x
0
k1
l1
0.5
3
5
k2
0
0
-12.5
-3.125
0.5 0.28385417 4.94791667 2.47395833 11.2955729 0.3499349
1 0.89870199 3.06667752 1.53333876 2.03374227 2.04177433
1.5 0.35184629 1.67551217 0.83775609 9.42461513 -1.5183977
2 0.43148482 3.25497571 1.62748785 4.37136689 0.53464613
2.5 0.52173518 0.80639431 0.40319715 4.10250681 1.42882386
3 0.01150217 1.85280733 0.92640366 -5.7021991 0.49914611
3.5 0.36996639 -1.7311934 -0.8655967 0.56900033 0.72334662
4 0.23761553 0.26617716 0.13308858 3.76872556 0.80909281
Pgina 20
l2
k3
6.25
9.81445313
10.6002384
0.52366798
7.98611565
4.57123562
2.93892335
5.1254792
1.05255915
l3
k4
l4
1.5625
-2.34375 -1.171875
-25
0.02034505 5.61319987 5.28055827 28.3894857
1.11672084 5.1730103 4.11984391 0.47372182
0.70683909 0.85087243 1.2631923 20.8127211
0.36904106 4.26626825 3.76062198 12.5571584
0.73961175 2.02921113 1.41780272 7.26002028
0.19167283 1.82584912 1.83932822 13.5756568
0.4157731 2.59830212 2.16474776 3.16674292
0.39622837 0.29073423 0.01227853 7.84937748

4.5 0.12012905
1.4162317 0.70811585 2.74708194 0.02134537

5 0.24619673 0.68538333 0.34269167 1.02130908 0.59801893
5.5 0.06576482 0.59881709 0.29940855 2.61851149 0.35521933
6 -0.1371835 0.86651741 0.4332587 0.88475847 0.21206909
6.5 0.13255813 0.10425232 0.05212616 1.34421965 0.38818107
7 0.01699391 -0.5616794 -0.2808397
7.5 0.10634192 0.42347482 0.21173741
8 0.05362713 0.14349972 0.07174986
8.5 0.04362327 0.39501548 0.19750774
9 -0.0657975 0.14110979 0.07055489
9.5 0.00925108 0.19804727
10 0.04182282 0.22473819
10.5 0.03260785 0.00385208
11 0.01001827 0.16482185
11.5 0.02977727 0.09937104
12 0.01121478 -0.0575392
12.5
- 0.10748488
0.09902364
0.11236909
0.00192604
0.08241093
0.04968552
-0.0287696
0.05374244
Pgina 21
-3.0521831 0.05492992 1.69778417

2.65247745 0.3204277 1.26240691
0.56204766 0.15889663 -0.4446808
2.26548766 0.13311321 1.18804124
0.99789832 0.19734842 0.41493543
1.47261438 0.08731389 1.01894094 0.02610447 0.60150887
0.05884959 0.22644981 1.35278361 0.12645849 0.6727137
1.10083825 -0.2034597 0.1019825 0.09724548 0.01781114
0.63975556 0.03756885 0.91454559 0.03112866 0.49725752
0.39913938 0.17033974 0.64053778 0.08957955 0.29532268
0.70978038 0.07842146 0.26400754 0.03302175 0.17636505
0.15142938 0.07451175 0.62659215 0.04427894 0.32276041
0.41915439 0.10286256 0.19753946 -0.0513109 0.07257258
0.36923724 0.00989838 0.33044967 0.00020149 0.18830216
-0.0741027 0.06821119 0.34758585 0.03721094 -0.1691615
0.3128024
0.049431 0.0234088 -0.0229174 0.03125455
-0.140313 0.01866419
- -0.012691 0.14163643
1.55700794 -7.1538104
0.97389512 1.23943456
0.52174894 5.93876177
1.02727932 2.78496703
0.25959387 2.56626862
0.58159413 3.60344681
0.54809426 0.37856038
0.08065543 2.36984023
-0.4461365 1.74241989
0.21821623 0.63291574
0.18720616 1.65164741
-0.2737493 0.56622382
0.03436025 0.84282293
0.17656201 0.93162509
0.13426627 0.03175496
0.04439688 0.69303356
0.12456065
-

0.01457133
0.26573374
13 0.01713692 0.02503053 0.01251527 0.13911985

-8.9575E13.5
05 0.06217329 0.03108665 0.18763957
14 0.01233056 0.05662585 0.02831292 0.01574557
14.5 0.00770712 0.00984086 0.00492043 0.12586156
15 0.00413537 0.04702683 0.02351342 0.0893884
15.5 0.00813355 0.02203415 0.01101707 0.03556697
16 0.00205218 0.02033308 0.01016654 -0.0866515
16.5 0.00460678 0.02852792 0.01426396 0.02799909
17 -0.0043385 0.0029852
17.5 0.00064068 0.01876897
18 0.00353283 0.01379047
18.5 0.00172655 0.00501846
19 0.00148333 0.01307776
19.5 0.00216726 0.00447829
20 0.00027114 0.00667663
20.5 0.00139838 0.0073749
0.0014926 0.04527564
0.00938448 0.04829841
0.00689523 0.00278899
0.00250923 0.03663731
0.00653888 0.02069169
0.00223914 0.01365586
0.00333831 0.02341916
0.00368745 0.00464493
Pgina 22
0.04729523 0.14778034 0.02442982

0.01582324 0.10047176 0.00596871
0.02437653 0.16908298 0.01395782
0.02654496 0.0321781 0.01296495
0.00116632 0.10226465 0.00205275
0.01990882 0.08664019 0.01064297
0.01149634 0.02021511 0.00511276
0.00726419 0.07515022 0.00452359
0.01281151 0.03196657 0.00649904
0.00269012 0.03450383 0.00075853
0.00759248 0.04448971 0.00422719
0.0066501 0.00263597 0.00316822
0.00136596 0.03052215 0.00109166
0.00565311 0.02082258 0.0029665
0.00251648 0.00915488 0.00104959
0.00252622 0.02072408 0.00149357
0.06519518
0.06196335
0.08552559
-0.0082227
-0.0567191
0.40658484
0.04511286 0.24890703
0.06206832 0.44813846
0.07107572 0.09784956
0.00080908 0.26325538
0.05187297 0.23388637
0.04109716 0.03156565 0.04531267

0.01552327 0.01792818 0.19713083
0.03932505 0.03392649 0.08942933
0.01315356 0.00806938
0.087053
0.02027056 0.01951977 -0.1185917

0.02207054 0.01793051 0.01058272
0.00097185 0.00299515 0.07915562
0.0165543 0.01481603 0.05670888
-0.0095578 0.00701804 0.02206374
0.00604114 0.00635888 0.05466249
0.01065235 0.00901362 0.01793231

21 0.00106268 0.00025478 0.00012739 0.01404781 0.00338456

21.5 0.00035207 0.00548827 0.00274414 0.01206394 0.00027185
22 0.00098628 0.00321744 0.00160872 0.00267623 0.00227778
22.5 0.00035682 0.00197262 0.00098631 0.01037818 0.00160823
23 0.0004917 0.00354809 0.00177405
0.004498 0.00064954
23.5 0.00056265 0.00077332 0.00038666 0.00471323 0.00156497
24 -6.659E-06 0.00208516 0.00104258 0.00617225 0.00050048
24.5 0.0004108 -0.0018513 0.00092565 0.00041892 0.00082092
25 -0.0002498 0.00035968 0.00017984 0.00420148 0.00087053
Pgina 23
0.00622773 0.00168432 0.00223588

0.01111888 3.5584E-05 0.00631344
0.01139262 0.00123943 0.00552904
0.00097536 0.00074247 0.00113632
0.00883878 0.00043565 0.00470052
0.00477324 0.00080665 0.00209204
0.00343001 0.00018508 0.00210077
0.00557585 0.00046831 0.00281411
0.00097675 0.00042403 0.00022578
0.00099055 0.02839419
0.00590086 0.03055945
0.00437324 0.00158202
0.00155447 0.02306802
-0.0041243 0.01315393
0.00143268 0.00852022
0.00209297 0.01478804
-0.0023327 0.00300734
-6.6946E05 0.00882446

Grafica Caso III

1.5
Desplazamiento (S)
0.5
10
15
-0.5
-1
Tiempo (S)
Pgina 24
20
25
30

SOLUCIN UTILIZANDO MATLAB

Solucin en extensin (*.m)
Pgina 25

Solucin en extensin (*.fig)
Pgina 26

Pgina 27

Solucin utilizando herramientas Simulink
Pgina 28

Pgina 29

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UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

FACULTAD DE INGENIERA CIVIL, ARQUITECTURA Y GEOTECNIA

amortiguamiento es directamente proporcional a la velocidad. Por la segunda ley de Newton

Donde es la constante de friccin o constante de amortiguamiento. La misma que se puede

, estamos en caso de un movimiento sobre-amortiguado.

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

, estamos en caso de un movimiento crticamente amortiguado.

, estamos en caso de un movimiento sub-amortiguado.

FORMULACIN DEL PROBLEMA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

Donde m es la masa y g la aceleracin, es decir la condicin de equilibrio significa que

, y la ecuacin diferencial describe el Movimiento armnico simple o Movimiento

vibratorio libre no amortiguado.

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

Las condiciones iniciales asociadas, son obvias

Luego la formulacin matemtica de este problema es

Solucin de la Ecuacin de Movimiento y su Interpretacin

.: La solucin general es y = c1 coswt + c2 senwt

Perodo de las vibraciones libres a:

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

Frecuencia de las vibraciones u oscilaciones es:

Recordemos que estos nmeros indican:

unidades y la frecuencia que hay w ciclos de la

unidades es decir la masa realiza

oscilaciones completas por unidad de tiempo,

es el intervalo de tiempo entre dos mximos sucesivos de x(t).

Dos posibles graficas a y b

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

Dos posibles graficas

El movimiento que describe la figura es oscilatorio pero, a causa del coeficiente

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

SOLUCIN EN HOJA DE CLCULO

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

0.01844456 -0.0166001 0.0166001

-0.00754188 0.00754188 -0.0067794 0.0067794

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

6.6 0.00459172 0.00459172

0.00091834 0.00091834 0.00082651 0.00082651

-0.00083569 0.00083569 0.00075121

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

0.00487731 0.00487731 0.01036427 0.00030483 0.00060966 0.00518214 0.01097394 0.00609663

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

1.4162317 0.70811585 2.74708194 0.02134537

-3.0521831 0.05492992 1.69778417

UNIVERSIDAD NACIONAL JORGE BASADRE GROHMANN - TACNA

13 0.01713692 0.02503053 0.01251527 0.13911985

0.04729523 0.14778034 0.02442982

0.04109716 0.03156565 0.04531267