* B4


by CO.H . TRAN   -    University of Natural Sciences , HCMC Vietnam -   &       

                                                        Copyright 2006

                                                        Feb 06   2006 

 **  Abstract : The system of non-linear differential quations is solved by Runge-Kutta method .

 ** Subjects:  Vibration Mechanics , The Differential equations .


NOTE: This worksheet demonstrates Maple's capabilities in the design and finding the numerical solution of the non-linear vibration system .                                     

 All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .


                                                              MAU DAO DONG PHI TUYEN


                                                          TRAN HONG CO   - Dai hoc Khoa hoc tu nhien - tp HCM Vietnam


A . Xac dinh he thong . [ System Definition ]
B. Mo hinh dao dong . [ Vibration Model ]

Khao sat mau vat the dang giai tich co hinh mo phong nhu tren [ Consider an analytical model which has the simulation figure above ]

He phuong trinh vi phan chuyen dong : [ System of differential equations ]


Xac dinh cac dieu kien dau . [ Define initial conditions ] :                                                          

Thay cac gia tri cua tham so m1,m2 , b , c1 , c3 , l , g , J . [ Substitute the parameter values m , b , c1 , c3 , l ,g , J ] .

> interface(warnlevel=0):
C .  Chuong trinh tinh toan mo hinh dao dong phi tuyen   . [ Calculation procedures for the vibration model ]

Bang hoat trinh Maple nay gom 2 phan . Ví du ve cach su dung trong cac bai toan thuc te , hay xem phan tiep sau .

* (khoi luong m1 , m2 , hang so can nhot b, he so lo xo c1, he so lo xo c3, solan T,chdai l ,moment J , giatoc trong truong g)

This Maple worksheet contains 2 parts. For examples of applying them to real problems, see the following action .

* (mass m1 , m2 ,viscous damping constant b,spring constant c1, spring constant c3,number of points , moment J ,acceleration of gravity g )


> restart;T:=5;m1:=1; m2:=1; b:=5; c1:= 1;c3:=1 ; l:= 0.05 ; J:= 0.5 ; g:=9.8;

> with(DEtools):with(plots):alias(y=y(t), phi=phi(t), y0=y(0),p0=phi(0), yp0=D(y)(0),pp0=D(phi)(0));

eq1:=(m1+m2)*Diff(y,t$2)*l*cos(phi)+(m1*l^2+J)*Diff(phi,t$2)+m1*g*l*cos(phi)=0;eq2:=(m1+m2)*Diff(y,t$2)+m1*l*cos(phi)*Diff(phi,t$2)-m1*l*Diff(phi,t)^2*cos(phi)+b*Diff(y,t)+c1*y+c3*y^3 =0;


print(" Loi giai so bang phuong phap RUNGE - KUTTA ");for i from 0 to T do print(G(i)); od;yy:=t-> rhs(G(t)[2]):

pp:=t-> rhs(G(t)[4]):

yyp:=t->rhs(G(t)[3]):ppp:=t->rhs(G(t)[5]):plot(yy,0..T,0..0.05,color=red,thickness=3,title=`tung do y(t)`);

plot(pp,0..T,-3.5..0.5,color=blue,thickness=3,title=`goc phi phi(t)`);plot(yyp,0..T,color=green,title=`daohamtungdo y'(t)`);

plot(ppp,0..T,color=black,title=`daohamgocphi phi'(t)`);


> interface(warnlevel=0):

> ;for k from 1 to T do print(" Do thi ham ",y(t)," voi :",t=k,s);plot(yy,0..k,thickness=4,title='hamy');od;

> for k from 1 to T do print(" Do thi dao ham ",diff(y(t),t)," voi :",t=k,s);plot(yyp,0..k,title='daohamhamy',color=green,thickness=2);od;

> for k from 1 to T do print(" Do thi ham ",phi(t)," voi :",t=k,s);plot(pp,0..k,thickness=4,color=blue,title='hamphi');od;

> for k from 1 to T do print(" Do thi dao ham ",diff(phi(t),t)," voi :",t=k,s);plot(ppp,0..k,color=black,title='daohamhamphi',color=black,thickness=2,title='daohamhamphi');od;


Activate the following procedure twice to obtain the result completely . ( in Maple 9.5 & 10 )

Animation Code
> mohinh(3,5);

Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 

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