MTODOS NUMRICOS BSICOS PARA INGENIERACarlos Armando De Castro Payares - Ing. Mec.Este archivo contiene los ejemplos de las implementaciones en Excel de algunos de los mtodos numricos presentados en el libro.

Contacto con el autor:cadecastro@gmail.com

http://matematicas.ingenieria.googlepages.com/metodos_numericos_ingenieria

0 20 40 60 80 100 1200.000

0.200

0.400

0.600

0.800

1.000

1.200

Iteracin

af [

in]

0 20 40 60 80 100 1200.000

0.200

0.400

0.600

0.800

1.000

1.200

Iteracin

af [

in]

a0 [in] t [in] w [in] L [in] P max [kip] P min [kip]0.25 0.45 2.5 9 28 8

min [ksi] max [ksi] [ksi] af [in]7.11 24.89 17.78 0.620

SOLUCIN GRIETA CRTICA POR MTODO DEL PUNTO FIJOa(i) Y a (i+1) a (i+1) - a (i) Iteracin

0.250 2.103 0.987 0.737 10.987 3.684 0.322 -0.665 20.322 2.180 0.919 0.597 30.919 3.437 0.369 -0.549 40.369 2.239 0.871 0.501 50.871 3.282 0.405 -0.465 60.405 2.288 0.834 0.428 70.834 3.172 0.434 -0.400 80.434 2.330 0.804 0.370 90.804 3.089 0.458 -0.347 100.458 2.366 0.780 0.322 110.780 3.024 0.477 -0.302 120.477 2.397 0.759 0.282 130.759 2.971 0.495 -0.265 140.495 2.425 0.742 0.248 150.742 2.928 0.509 -0.233 160.509 2.450 0.727 0.218 170.727 2.892 0.522 -0.205 180.522 2.472 0.714 0.192 190.714 2.861 0.533 -0.181 200.533 2.491 0.703 0.170 210.703 2.835 0.543 -0.160 220.543 2.509 0.694 0.150 230.694 2.813 0.552 -0.142 240.552 2.524 0.685 0.133 250.685 2.793 0.559 -0.126 260.559 2.538 0.677 0.118 270.677 2.777 0.566 -0.111 280.566 2.551 0.671 0.105 290.671 2.762 0.572 -0.099 300.572 2.562 0.665 0.093 310.665 2.749 0.577 -0.088 320.577 2.572 0.660 0.082 330.660 2.738 0.582 -0.078 340.582 2.581 0.655 0.073 350.655 2.728 0.586 -0.069 360.586 2.589 0.651 0.065 370.651 2.720 0.590 -0.061 38

0.590 2.596 0.648 0.058 390.648 2.712 0.593 -0.054 400.593 2.602 0.644 0.051 410.644 2.706 0.596 -0.048 420.596 2.608 0.642 0.045 430.642 2.700 0.599 -0.043 440.599 2.613 0.639 0.040 450.639 2.695 0.601 -0.038 460.601 2.618 0.637 0.036 470.637 2.690 0.603 -0.034 480.603 2.622 0.635 0.032 490.635 2.686 0.605 -0.030 500.605 2.625 0.633 0.028 510.633 2.682 0.607 -0.027 520.607 2.629 0.632 0.025 530.632 2.679 0.608 -0.024 540.608 2.631 0.630 0.022 550.630 2.676 0.609 -0.021 560.609 2.634 0.629 0.020 570.629 2.674 0.611 -0.019 580.611 2.636 0.628 0.018 590.628 2.672 0.612 -0.017 600.612 2.638 0.627 0.016 610.627 2.670 0.612 -0.015 620.612 2.640 0.626 0.014 630.626 2.668 0.613 -0.013 640.613 2.642 0.626 0.012 650.626 2.666 0.614 -0.012 660.614 2.643 0.625 0.011 670.625 2.665 0.615 -0.010 680.615 2.644 0.624 0.010 690.624 2.664 0.615 -0.009 700.615 2.645 0.624 0.009 710.624 2.663 0.616 -0.008 720.616 2.646 0.623 0.008 730.623 2.662 0.616 -0.007 740.616 2.647 0.623 0.007 750.623 2.661 0.616 -0.006 760.616 2.648 0.623 0.006 770.623 2.660 0.617 -0.006 780.617 2.649 0.622 0.005 790.622 2.660 0.617 -0.005 800.617 2.649 0.622 0.005 810.622 2.659 0.617 -0.005 820.617 2.650 0.622 0.004 83

0.622 2.658 0.618 -0.004 840.618 2.650 0.621 0.004 850.621 2.658 0.618 -0.004 860.618 2.651 0.621 0.003 870.621 2.658 0.618 -0.003 880.618 2.651 0.621 0.003 890.621 2.657 0.618 -0.003 900.618 2.652 0.621 0.003 910.621 2.657 0.618 -0.002 920.618 2.652 0.621 0.002 930.621 2.657 0.618 -0.002 940.618 2.652 0.621 0.002 950.621 2.656 0.619 -0.002 960.619 2.652 0.620 0.002 970.620 2.656 0.619 -0.002 980.619 2.653 0.620 0.002 990.620 2.656 0.619 -0.002 1000.619 2.653 0.620 0.001 1010.620 2.656 0.619 -0.001 1020.619 2.653 0.620 0.001 1030.620 2.656 0.619 -0.001 1040.619 2.653 0.620 0.001 1050.620 2.655 0.619 -0.001 1060.619 2.653 0.620 0.001 1070.620 2.655 0.619 -0.001 1080.619 2.653 0.620 0.001 1090.620 2.655 0.619 -0.001 1100.619 2.653 0.620 0.001 1110.620 2.655 0.619 -0.001 1120.619 2.654 0.620 0.001 1130.620 2.655 0.619 -0.001 1140.619 2.654 0.620 0.001 1150.620 2.655 0.619 -0.001 1160.619 2.654 0.620 0.001 1170.620 2.655 0.619 -0.001 1180.619 2.654 0.620 0.001 1190.620 2.655 0.619 0.000 120

m A2.25 6.60E-09 52

INTEGRACIN NUMRICA DE LA EC. DE PARISa [in] Y f (a) N [ciclos]

0.25 2.10 208547.52 00.26 2.11 197484.08 20300.27 2.12 187263.46 39540.28 2.13 177795.17 57790.29 2.14 169001.39 7513

0.3 2.15 160814.80 91620.31 2.17 153176.89 107320.32 2.18 146036.50 122280.33 2.19 139348.74 136550.34 2.20 133074.00 150170.35 2.21 127177.24 163190.36 2.23 121627.29 175630.37 2.24 116396.34 187530.38 2.25 111459.51 198920.39 2.27 106794.44 20983

0.4 2.28 102380.99 220290.41 2.29 98200.96 230320.42 2.31 94237.87 239940.43 2.32 90476.73 249180.44 2.34 86903.90 258050.45 2.35 83506.93 266570.46 2.37 80274.43 274760.47 2.39 77195.95 282630.48 2.40 74261.90 290200.49 2.42 71463.45 29749

0.5 2.43 68792.44 304500.51 2.45 66241.38 311250.52 2.47 63803.28 317760.53 2.49 61471.72 324020.54 2.50 59240.71 330060.55 2.52 57104.68 335870.56 2.54 55058.48 341480.57 2.56 53097.27 346890.58 2.58 51216.58 352100.59 2.60 49412.20 35714

0.6 2.62 47680.21 361990.61 2.64 46016.95 366680.62 2.66 44418.97 37120

KIC [ksiin]

0 20 40 60 80 100 1200.000

0.200

0.400

0.600

0.800

1.000

1.200

Iteracin

af [

in]

0 5000 10000 15000 20000 25000 30000 35000 400000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N [ciclos]

a [in

]

0 20 40 60 80 100 1200.000

0.200

0.400

0.600

0.800

1.000

1.200

Iteracin

af [

in]

0 5000 10000 15000 20000 25000 30000 35000 400000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N [ciclos]

a [in

]

x(i-1) x(i) x(i+1) x(i+1)-x(i) Iteracin10.000 6.673 8.361 1.689 1

6.673 8.361 8.098 0.264 28.361 8.098 8.176 0.078 38.098 8.176 8.176 0.000 48.176 8.176 8.176 0.000 58.176 8.176 8.176 0.000 68.176 8.176 8.176 0.000 7

0 1 2 3 4 5 6 7 80.0000.2000.4000.6000.8001.0001.2001.4001.6001.800

Iteracin

0 2 4 6 8 10 12

-4

-3

-2

-1

0

1

2

3

4

x0

0.51

1.52

2.53

3.54

4.55

5.56

6.57

7.58

8.59

9.510

0 1 2 3 4 5 6 7 80.0000.2000.4000.6000.8001.0001.2001.4001.6001.800

Iteracin

0 2 4 6 8 10 12

-4

-3

-2

-1

0

1

2

3

4

y(x)3

3.112173222.4623779

1.20970659-0.33914308

-1.8049587-2.82885748-3.16015329-2.71773336-1.60991752-0.10793772

1.4204692.601095363.144882872.918693361.977905930.55285815-1.0075486-2.3212723

-3.06666759-3.0612357

t [s] x [m] v [m/s] a [m/s^2] j [m/s^3]0 0.00 0.84 0.07 1.24

0.1 0.08 0.85 0.19 0.940.2 0.17 0.87 0.29 0.670.3 0.26 0.90 0.36 0.410.4 0.35 0.93 0.40 0.130.5 0.45 0.97 0.41 -0.190.6 0.54 1.02 0.39 -0.570.7 0.65 1.05 0.33 -1.040.8 0.76 1.09 0.23 -1.610.9 0.87 1.11 0.07 -2.31

1 0.98 1.12 -0.16 -3.171.1 1.09 1.10 -0.48 -4.191.2 1.20 1.05 -0.90 -5.391.3 1.30 0.96 -1.44 -6.751.4 1.39 0.82 -2.11 -8.231.5 1.46 0.61 -2.94 -9.731.6 1.51 0.31 -3.91 -11.111.7 1.53 -0.08 -5.02 -12.081.8 1.50 -0.58 -6.23 -12.241.9 1.41 -1.20 -7.45 -11.02

2 1.26 -1.95 -8.55 -7.632.1 1.02 -2.80 -9.32 239.722.2 0.69 -3.73 14.66 -56.282.3 0.27 -2.27 9.03 -194.462.4 0.24 -1.37 -10.42 94.552.5 0.00 -2.41 -0.96 -0.39

0 0.5 1 1.5 2 2.5 30.000.200.400.600.801.001.201.401.601.80

t [s]

Posi

cin

[m

]

0 0.5 1 1.5 2 2.5 3

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

t [s]Ve

loci

dad

[m/s

]

0 0.5 1 1.5 2 2.5 30.000.200.400.600.801.001.201.401.601.80

t [s]

Posi

cin

[m

]

0 0.5 1 1.5 2 2.5 3

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

t [s]

Velo

cida

d [m

/s]

0 0.5 1 1.5 2 2.5 3

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

t [s]

Acel

erac

in

[m/s

^2]

0 0.5 1 1.5 2 2.5 3

-300.00

-200.00

-100.00

0.00

100.00

200.00

300.00

t [s]So

brea

cele

raci

n [

m/s

^3]

0 0.5 1 1.5 2 2.5 3

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

t [s]

Acel

erac

in

[m/s

^2]

0 0.5 1 1.5 2 2.5 3

-300.00

-200.00

-100.00

0.00

100.00

200.00

300.00

t [s]

Sobr

eace

lera

cin

[m

/s^

3]

t [in] w [in] L [in] ao [in] Pmax [kip] Pmin [kip]0.45 2.5 9 0.25 28 8

SOLUCIN POR EULER DE LA EC. DE PARISN a(n) [in] a(n+1) [in]

0 0.250 0.250100 0.250 0.251200 0.251 0.251300 0.251 0.252400 0.252 0.252500 0.252 0.253600 0.253 0.253700 0.253 0.254800 0.254 0.254900 0.254 0.255

1000 0.255 0.2551100 0.255 0.2561200 0.256 0.2561300 0.256 0.2571400 0.257 0.2571500 0.257 0.2581600 0.258 0.2581700 0.258 0.2591800 0.259 0.2591900 0.259 0.2602000 0.260 0.2602100 0.260 0.2612200 0.261 0.2612300 0.261 0.2622400 0.262 0.2622500 0.262 0.2632600 0.263 0.2632700 0.263 0.2642800 0.264 0.2642900 0.264 0.2653000 0.265 0.2653100 0.265 0.2663200 0.266 0.2673300 0.267 0.2673400 0.267 0.2683500 0.268 0.2683600 0.268 0.2693700 0.269 0.2693800 0.269 0.2703900 0.270 0.270

0 5000 10000 15000 20000 25000 30000 35000 400000.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

N [ciclos]

a [in

]

4000 0.270 0.2714100 0.271 0.2714200 0.271 0.2724300 0.272 0.2724400 0.272 0.2734500 0.273 0.2734600 0.273 0.2744700 0.274 0.2754800 0.275 0.2754900 0.275 0.2765000 0.276 0.2765100 0.276 0.2775200 0.277 0.2775300 0.277 0.2785400 0.278 0.2785500 0.278 0.2795600 0.279 0.2805700 0.280 0.2805800 0.280 0.2815900 0.281 0.2816000 0.281 0.2826100 0.282 0.2826200 0.282 0.2836300 0.283 0.2836400 0.283 0.2846500 0.284 0.2856600 0.285 0.2856700 0.285 0.2866800 0.286 0.2866900 0.286 0.2877000 0.287 0.2887100 0.288 0.2887200 0.288 0.2897300 0.289 0.2897400 0.289 0.2907500 0.290 0.2907600 0.290 0.2917700 0.291 0.2927800 0.292 0.2927900 0.292 0.2938000 0.293 0.2938100 0.293 0.2948200 0.294 0.2958300 0.295 0.2958400 0.295 0.296

8500 0.296 0.2968600 0.296 0.2978700 0.297 0.2988800 0.298 0.2988900 0.298 0.2999000 0.299 0.3009100 0.300 0.3009200 0.300 0.3019300 0.301 0.3019400 0.301 0.3029500 0.302 0.3039600 0.303 0.3039700 0.303 0.3049800 0.304 0.3059900 0.305 0.305

10000 0.305 0.30610100 0.306 0.30610200 0.306 0.30710300 0.307 0.30810400 0.308 0.30810500 0.308 0.30910600 0.309 0.31010700 0.310 0.31010800 0.310 0.31110900 0.311 0.31211000 0.312 0.31211100 0.312 0.31311200 0.313 0.31411300 0.314 0.31411400 0.314 0.31511500 0.315 0.31611600 0.316 0.31611700 0.316 0.31711800 0.317 0.31811900 0.318 0.31812000 0.318 0.31912100 0.319 0.32012200 0.320 0.32012300 0.320 0.32112400 0.321 0.32212500 0.322 0.32212600 0.322 0.32312700 0.323 0.32412800 0.324 0.32512900 0.325 0.325

13000 0.325 0.32613100 0.326 0.32713200 0.327 0.32713300 0.327 0.32813400 0.328 0.32913500 0.329 0.32913600 0.329 0.33013700 0.330 0.33113800 0.331 0.33213900 0.332 0.33214000 0.332 0.33314100 0.333 0.33414200 0.334 0.33514300 0.335 0.33514400 0.335 0.33614500 0.336 0.33714600 0.337 0.33714700 0.337 0.33814800 0.338 0.33914900 0.339 0.34015000 0.340 0.34015100 0.340 0.34115200 0.341 0.34215300 0.342 0.34315400 0.343 0.34415500 0.344 0.34415600 0.344 0.34515700 0.345 0.34615800 0.346 0.34715900 0.347 0.34716000 0.347 0.34816100 0.348 0.34916200 0.349 0.35016300 0.350 0.35016400 0.350 0.35116500 0.351 0.35216600 0.352 0.35316700 0.353 0.35416800 0.354 0.35416900 0.354 0.35517000 0.355 0.35617100 0.356 0.35717200 0.357 0.35817300 0.358 0.35817400 0.358 0.359

17500 0.359 0.36017600 0.360 0.36117700 0.361 0.36217800 0.362 0.36317900 0.363 0.36318000 0.363 0.36418100 0.364 0.36518200 0.365 0.36618300 0.366 0.36718400 0.367 0.36818500 0.368 0.36818600 0.368 0.36918700 0.369 0.37018800 0.370 0.37118900 0.371 0.37219000 0.372 0.37319100 0.373 0.37419200 0.374 0.37519300 0.375 0.37519400 0.375 0.37619500 0.376 0.37719600 0.377 0.37819700 0.378 0.37919800 0.379 0.38019900 0.380 0.38120000 0.381 0.38220100 0.382 0.38320200 0.383 0.38320300 0.383 0.38420400 0.384 0.38520500 0.385 0.38620600 0.386 0.38720700 0.387 0.38820800 0.388 0.38920900 0.389 0.39021000 0.390 0.39121100 0.391 0.39221200 0.392 0.39321300 0.393 0.39421400 0.394 0.39521500 0.395 0.39621600 0.396 0.39721700 0.397 0.39721800 0.397 0.39821900 0.398 0.399

22000 0.399 0.40022100 0.400 0.40122200 0.401 0.40222300 0.402 0.40322400 0.403 0.40422500 0.404 0.40522600 0.405 0.40622700 0.406 0.40722800 0.407 0.40822900 0.408 0.40923000 0.409 0.41023100 0.410 0.41123200 0.411 0.41223300 0.412 0.41323400 0.413 0.41423500 0.414 0.41623600 0.416 0.41723700 0.417 0.41823800 0.418 0.41923900 0.419 0.42024000 0.420 0.42124100 0.421 0.42224200 0.422 0.42324300 0.423 0.42424400 0.424 0.42524500 0.425 0.42624600 0.426 0.42724700 0.427 0.42824800 0.428 0.42924900 0.429 0.43125000 0.431 0.43225100 0.432 0.43325200 0.433 0.43425300 0.434 0.43525400 0.435 0.43625500 0.436 0.43725600 0.437 0.43825700 0.438 0.44025800 0.440 0.44125900 0.441 0.44226000 0.442 0.44326100 0.443 0.44426200 0.444 0.44526300 0.445 0.44626400 0.446 0.448

26500 0.448 0.44926600 0.449 0.45026700 0.450 0.45126800 0.451 0.45226900 0.452 0.45427000 0.454 0.45527100 0.455 0.45627200 0.456 0.45727300 0.457 0.45927400 0.459 0.46027500 0.460 0.46127600 0.461 0.46227700 0.462 0.46427800 0.464 0.46527900 0.465 0.46628000 0.466 0.46728100 0.467 0.46928200 0.469 0.47028300 0.470 0.47128400 0.471 0.47328500 0.473 0.47428600 0.474 0.47528700 0.475 0.47628800 0.476 0.47828900 0.478 0.47929000 0.479 0.48029100 0.480 0.48229200 0.482 0.48329300 0.483 0.48529400 0.485 0.48629500 0.486 0.48729600 0.487 0.48929700 0.489 0.49029800 0.490 0.49129900 0.491 0.49330000 0.493 0.49430100 0.494 0.49630200 0.496 0.49730300 0.497 0.49930400 0.499 0.50030500 0.500 0.50130600 0.501 0.50330700 0.503 0.50430800 0.504 0.50630900 0.506 0.507

31000 0.507 0.50931100 0.509 0.51031200 0.510 0.51231300 0.512 0.51331400 0.513 0.51531500 0.515 0.51631600 0.516 0.51831700 0.518 0.52031800 0.520 0.52131900 0.521 0.52332000 0.523 0.52432100 0.524 0.52632200 0.526 0.52732300 0.527 0.52932400 0.529 0.53132500 0.531 0.53232600 0.532 0.53432700 0.534 0.53632800 0.536 0.53732900 0.537 0.53933000 0.539 0.54133100 0.541 0.54233200 0.542 0.54433300 0.544 0.54633400 0.546 0.54733500 0.547 0.54933600 0.549 0.55133700 0.551 0.55333800 0.553 0.55533900 0.555 0.55634000 0.556 0.55834100 0.558 0.56034200 0.560 0.56234300 0.562 0.56434400 0.564 0.56534500 0.565 0.56734600 0.567 0.56934700 0.569 0.57134800 0.571 0.57334900 0.573 0.57535000 0.575 0.57735100 0.577 0.57935200 0.579 0.58135300 0.581 0.58235400 0.582 0.584

35500 0.584 0.58635600 0.586 0.58835700 0.588 0.59035800 0.590 0.59235900 0.592 0.59536000 0.595 0.59736100 0.597 0.59936200 0.599 0.60136300 0.601 0.60336400 0.603 0.60536500 0.605 0.60736600 0.607 0.60936700 0.609 0.61136800 0.611 0.61436900 0.614 0.61637000 0.616 0.61837100 0.618 0.620

n C max [ksi] min [ksi] [ksi] af [in]2.25 6.60E-09 24.89 7.11 17.78 0.62

0 5000 10000 15000 20000 25000 30000 35000 400000.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

N [ciclos]

a [in

]

N N100 37100

g [m/s^2] [kg/m^3] A [m^2] m [kg] Vt teorica [m/s] Vt teorica [kph]9.8 1 3.14E-04 1.00E-02 24.98 89.92

t [s] V0 [m/s]1 0

t [s] V [m/s] k1 k2 k3 k40 0.00 9.80 9.42 9.45 8.401 9.32 8.43 6.92 7.23 5.492 16.36 5.59 4.03 4.49 2.973 20.63 3.11 2.07 2.43 1.454 22.89 1.57 1.00 1.21 0.685 24.00 0.75 0.47 0.58 0.316 24.52 0.35 0.22 0.27 0.147 24.77 0.16 0.10 0.12 0.078 24.88 0.08 0.05 0.06 0.039 24.93 0.03 0.02 0.03 0.01

10 24.96 0.02 0.01 0.01 0.0111 24.97 0.01 0.00 0.01 0.0012 24.97 0.00 0.00 0.00 0.0013 24.98 0.00 0.00 0.00 0.0014 24.98 0.00 0.00 0.00 0.0015 24.98 0.00 0.00 0.00 0.0016 24.98 0.00 0.00 0.00 0.0017 24.98 0.00 0.00 0.00 0.0018 24.98 0.00 0.00 0.00 0.0019 24.98 0.00 0.00 0.00 0.0020 24.98 0.00 0.00 0.00 0.00

Vt num [m/s] Vt num [kph]24.98 89.92

0 5 10 15 20 250.00

5.00

10.00

15.00

20.00

25.00

30.00

t [s]

v [m

/s]

PresentacinE3.1&E6.1E3.3E5.1E7.1E7.2