aula maple 2a parte

7/23/2019 Aula Maple 2a Parte

http://slidepdf.com/reader/full/aula-maple-2a-parte 1/6



0.0025050134

0.4974949866

> dX:=-Ji^(-1).R[0];

:=dX

-0.0191136376969470723

0.576502430715991498

> X[1]:=X[0]+dX;

:= X 1

-1.01911363769694718

2.07650243071599138

> X[0]:=X[1]:

Equações diferenciais ordinarias lineares

Equação de movimento amortecido de um corpo com 1 GDL.

> :=eqd m

d

d

t

d

d

t ( )x t 20

d

d

t ( )x t k ( )x t 20 ( )cos 20 t

:=eqd 150

d

d 2

t 2

( )x t 20

d d

t ( )x t 50 ( )x t 20 ( )cos 20 t

Solução analítica.

> k:=50; m:=150; c:=0.3;

:=k 50

:=m 150

:=c 0.3

> eqd;

150

d

d 2

t 2 ( )x t 20

d

d

t ( )x t 50 ( )x t 20 ( )cos 20 t

> resultado:=dsolve(eqd,x(t));

resultado ( )x t e

t

15

sin

74 t

15_C2 e

t

15

cos

74 t

15_C1:=

2398

7188325( )cos 20 t

16

7188325( )sin 20 t

Determinação de C1 e C2. Para x(0) = 0.8 e D(x)(0) = 1.0

> eq_aux:=rhs(resultado);

eq_aux e

t

15

sin

74 t

15_C2 e

t

15

cos

74 t

15_C1

2398

7188325( )cos 20 t :=

16

7188325( )sin 20 t

> evalf(subs(t=0, eq_aux)=0.8);

0.0003335964915 1. _C1 0.8

> isolate(%,_C1);

_C1 0.8003335965

> assign(%);

> subs(t=0, diff(eq_aux,t))=1.0;

1

15e

0( )sin 0 _C2

1

15e

0( )cos 0 74 _C2 0.05335557310 e

0( )cos 0

0.05335557310 e0

( )sin 0 749592

1437665( )sin 0

64

1437665( )cos 0 1.0

> isolate(%,_C2); evalf(%);

_C2 1.836673847

_C2 1.836673847

> assign(%);

Finalmente a solução da equação diferencial.

> eq_aux;

1.836673847 e

t

15

sin

74 t

150.8003335965 e

t

15

cos

74 t

15

2398

7188325( )cos 20 t

16

7188325( )sin 20 t

Em qualquer instante de tempo pode-se determinar a posição e a velocidade do sistema.

> evalf(subs(t=0.5,eq_aux)); evalf(subs(t=0.5,diff(eq_aux,t)));

1.245205591

0.7649618551

Solução numérica.

> dsys := {eqd, x(0)= 0.8,D(x)(0)=0.0};

dsys :=

{ }, ,

150

d

d 2

t 2 ( )x t 20

d

d

t ( )x t 50 ( )x t 20 ( )cos 20 t ( )x 0 0.8

( )( )D x 0 0.

Solução pelo método de Runge-Kutta.

> dsol := dsolve(dsys, numeric, method=rkf45,

output=procedurelist,range=0..30):

> dsol(0.5);

, ,t 0.5 ( )x t 0.768197563745935952

d

d

t ( )x t -0.130962361867184379

> with(plots):

> odeplot(dsol,[t,x(t)]);



> odeplot(dsol,[t,D(x)(t)]); Plano de fase

> odeplot(dsol,[x(t),D(x)(t)]);



> for i from 0 by 0.5 to 30 do

dsol(i);

od;

, ,t 0. ( )x t 0.80000000000000

d

d

t ( )x t 0.

, ,t 0.5 ( )x t 0.768197563745935952

d

d

t ( )x t -0.130962361867184379

, ,t 1.0 ( )x t 0.675982512402666824

d

d

t ( )x t -0.230038011523358205

, ,t 1.5 ( )x t 0.536052906814768804

d

d

t ( )x t -0.325659869901727962

, ,t 2.0 ( )x t 0.362430624143847357

d

d

t ( )x t -0.366168679348157444

, ,t 2.5 ( )x t 0.170201756173711305

d

d

t ( )x t -0.391783712259251127

, ,t 3.0 ( )x t -0.0221309348718498125

d

d

t ( )x t -0.378671062576175720

, ,t 3.5 ( )x t -0.201387204502363665 d

d

t ( )x t -0.328638149145211168

, ,t 4.0 ( )x t -0.352239987217594197

d

d

t ( )x t -0.273734610869820816

, ,t 4.5 ( )x t -0.465273035475648744

d

d

t ( )x t -0.177343468464130876

, ,t 5.0 ( )x t -0.534304572525855282

d

d

t ( )x t -0.0935471311991692700

, ,

t 5.5

( )x t -0.555098037248683008

d

d

t ( )x t 0.00375143682108820300

, ,t 6.0 ( )x t -0.531269002740448637

d

d

t ( )x t 0.0958902932846419088

, ,t 6.5 ( )x t -0.465510744305213998

d

d

t ( )x t 0.160615788899196699

, ,t 7.0 ( )x t -0.367133657795731538

d

d

t ( )x t 0.230041566270258707

, ,t 7.5 ( )x t -0.245957893365722020

d

d

t

( )x t 0.253898456133835881

, ,t 8.0 ( )x t -0.112081117217420117

d

d

t ( )x t 0.272207638418829445

, ,t 8.5 ( )x t 0.0209981009622273764

d

d

t ( )x t 0.262832740153134403

, ,t 9.0 ( )x t 0.144895410738542646

d

d

t ( )x t 0.224607865387992462

, ,t 9.5 ( )x t 0.248523161441303514

d

d

t ( )x t 0.189697050143371320

, ,t 10.0 ( )x t 0.325628959586297674

d

d

t ( )x t 0.118663392834282552

, ,t 10.5 ( )x t 0.372275182702799301

d

d

t ( )x t 0.0627933432392509816

, ,t 11.0 ( )x t 0.385074491532969244

d

d

t ( )x t -0.00501406632308102392

, ,t 11.5 ( )x t 0.367374376110609302

d

d

t ( )x t -0.0704866677429301232



:=sys_ode ,d

d

t ( )y t ( )x t

d

d

t ( )x t ( )x t

> dsolve([sys_ode]);

{ },( )x t _C4 e( )t

( )y t _C4 e( )t

_C3

condições iniciais.

> ics := x(0)=1, y(1)=0;

:=ics ,( )x 0 1 ( )y 1 0

Solve the system of ODEs subject to the initial conditions ics.

> dsolve([sys_ode, ics]);

{ },( )x t e( )t

( )y t e( )t 1

e

>

>

>>

aula maple 2a parte

Documents