This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.
BÀI TOÁN DAO ĐỘNG PHI TUYẾN VỚI NHẬP LIỆU RỜI RẠC
NUMERICAL - GRAPHICAL SOLUTIONS OF THE NON-LINEAR VIBRATION MODEL with discrete data input .
by CO.H . TRAN - University of Natural Sciences , HCMC Vietnam -
coth123@math.com & coth123@yahoo.com
Copyright 2007
May 06 2007
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** Abstract : The system of non-linear differential quations with discrete input_function is solved by Runge-Kutta method .
** Subjects : Vibration Mechanics , The Differential equations .
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NOTE: This worksheet demonstrates Maple's capabilities in the design and finding the numerical solution of the non-linear vibration sys tem .
All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .
LOI GIAI SO VA DO THI CUA
MAU DAO DONG PHI TUYEN voi so lieu roi rac .
TRAN HONG CO - Dai hoc Khoa hoc tu nhien - tp HCM Vietnam
cohtran@mail.com & coth123@yahoo.com
Step 1 : System Definition .
restart: eq1:=(m1+m2)*Diff(y,t$2)*l*cos(phi)+(m1*l^2+J)*Diff(phi,t$2)+m1*g*l*cos(phi)=f(t);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 =h(t);
with(plots): readlib(spline): with(inttrans): Warning, the name changecoords has been redefined
Step 2 : Fitting the experimental data by Spline function .
eq1:=(m1+m2)*Diff(y,t$2)*l*cos(phi)+(m1*l^2+J)*Diff(phi,t$2)+m1*g*l*cos(phi)=f(t);eq2:=(m1+m2)*Diff(y,t$2)+m1*l*cos(phi)*Diff(phi,t$2)-m1*l*Diff(phi,t)^2*cos(phi)+b*Diff(y,t)+c1*y+c3*y^3 =h(t);
> datax1:=[0,0.5,1,1.5,2,2.5,3,3.5,4];datay1:=[0.2,0.5,0.7,0.4,0.65,1.2,2.4,0.9,1.1];pldataf:= zip((x,y)->[x,y], datax1, datay1):dataplot1 := pointplot(pldataf, symbol=diamond);
> Ft:=spline(datax1, datay1, w, cubic);
> dothif:=plot(Ft, w=0..5, color=red):display(dataplot1,dothif, axes=frame);
> fnum:=subs(w=t,Ft);eq1:=subs(f(t)=fnum,eq1);
> datax2:=[0,0.5,1,1.5,2,2.5,3,3.5,4];datay2:=[0.3,0.5,0.58,0.4,0.85,1.2,1.4,0.9,1.55];Ht := zip((x,y)->[x,y], datax2, datay2):dataplot2 := pointplot(Ht, symbol=cross);
> Ht:=spline(datax2, datay2, w, cubic);
> dothih:=plot(Ht, w=0..5, color=blue):display(dataplot2,dothih, axes=frame);
> h1:=subs(w=t,Ht);eq2:=subs(h(t)=h1,eq2);
Step 3 : The non-linear vibration system with discrete data input .
> T:=5;m1:=1; m2:=1; b:=5; c1:= 1;c3:=1 ; l:= 0.05 ; J:= 0.5 ; g:=9.8;n:=2;
> with(DEtools):with(plots):alias(y=y(t), phi=phi(t), y0=y(0),p0=phi(0), yp0=D(y)(0),pp0=D(phi)(0));;;;;eq1 := .10*Diff(y,`$`(t,2))*cos(phi)+.5025*Diff(phi,`$`(t,2))+.490*cos(phi) = PIECEWISE([.2000000000+.559628129599999968*t+.161487481600000010*t^3, t < .5],[.1596281296+.680743740799999997*t+.242231222385861644*(t-.5)^2-1.60743740800000001*(t-.5)^3, t < 3 =" PIECEWISE([.3000000000+.401389911599999982*t-.555964654000000014e-2*t^3," y0="0,p0=" yp0="0,pp0="> 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..n*T,color=red,thickness=3,title=`tung do y(t)`);plot(pp,0..n*T,color=blue,thickness=3,title=`goc phi phi(t)`);plot(yyp,0..n*T,color=green,title=`daohamtungdo y'(t)`);plot(ppp,0..n*T,color=black,title=`daohamgocphi phi'(t)`);;;;;;;;;;;;;;;;;;;;;;;
Activate the following procedure twice to obtain the result completely . ( use Maple 9.5 & 10 )*******Animation Code*******
> mohinh:=proc(M,lan)
> mohinh(0.75,3);
Disclaimer: While every effort has been made to validate the solutions in this worksheet, the authors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.Legal Notice: The copyright for this application is owned by the author . 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 and the author .
by CO.H . TRAN - University of Natural Sciences , HCMC Vietnam -
coth123@math.com & coth123@yahoo.com
Copyright 2007
May 06 2007
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
** Abstract : The system of non-linear differential quations with discrete input_function 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 sys tem .
All rights reserved. Copying or transmitting of this material without the permission of the authors is not allowed .
LOI GIAI SO VA DO THI CUA
MAU DAO DONG PHI TUYEN voi so lieu roi rac .
TRAN HONG CO - Dai hoc Khoa hoc tu nhien - tp HCM Vietnam
cohtran@mail.com & coth123@yahoo.com
Step 1 : System Definition .
restart: eq1:=(m1+m2)*Diff(y,t$2)*l*cos(phi)+(m1*l^2+J)*Diff(phi,t$2)+m1*g*l*cos(phi)=f(t);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 =h(t);
with(plots): readlib(spline): with(inttrans): Warning, the name changecoords has been redefined
Step 2 : Fitting the experimental data by Spline function .
eq1:=(m1+m2)*Diff(y,t$2)*l*cos(phi)+(m1*l^2+J)*Diff(phi,t$2)+m1*g*l*cos(phi)=f(t);eq2:=(m1+m2)*Diff(y,t$2)+m1*l*cos(phi)*Diff(phi,t$2)-m1*l*Diff(phi,t)^2*cos(phi)+b*Diff(y,t)+c1*y+c3*y^3 =h(t);
> datax1:=[0,0.5,1,1.5,2,2.5,3,3.5,4];datay1:=[0.2,0.5,0.7,0.4,0.65,1.2,2.4,0.9,1.1];pldataf:= zip((x,y)->[x,y], datax1, datay1):dataplot1 := pointplot(pldataf, symbol=diamond);
> Ft:=spline(datax1, datay1, w, cubic);
> dothif:=plot(Ft, w=0..5, color=red):display(dataplot1,dothif, axes=frame);
> fnum:=subs(w=t,Ft);eq1:=subs(f(t)=fnum,eq1);
> datax2:=[0,0.5,1,1.5,2,2.5,3,3.5,4];datay2:=[0.3,0.5,0.58,0.4,0.85,1.2,1.4,0.9,1.55];Ht := zip((x,y)->[x,y], datax2, datay2):dataplot2 := pointplot(Ht, symbol=cross);
> Ht:=spline(datax2, datay2, w, cubic);
> dothih:=plot(Ht, w=0..5, color=blue):display(dataplot2,dothih, axes=frame);
> h1:=subs(w=t,Ht);eq2:=subs(h(t)=h1,eq2);
Step 3 : The non-linear vibration system with discrete data input .
> T:=5;m1:=1; m2:=1; b:=5; c1:= 1;c3:=1 ; l:= 0.05 ; J:= 0.5 ; g:=9.8;n:=2;
> with(DEtools):with(plots):alias(y=y(t), phi=phi(t), y0=y(0),p0=phi(0), yp0=D(y)(0),pp0=D(phi)(0));;;;;eq1 := .10*Diff(y,`$`(t,2))*cos(phi)+.5025*Diff(phi,`$`(t,2))+.490*cos(phi) = PIECEWISE([.2000000000+.559628129599999968*t+.161487481600000010*t^3, t < .5],[.1596281296+.680743740799999997*t+.242231222385861644*(t-.5)^2-1.60743740800000001*(t-.5)^3, t < 3 =" PIECEWISE([.3000000000+.401389911599999982*t-.555964654000000014e-2*t^3," y0="0,p0=" yp0="0,pp0="> 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..n*T,color=red,thickness=3,title=`tung do y(t)`);plot(pp,0..n*T,color=blue,thickness=3,title=`goc phi phi(t)`);plot(yyp,0..n*T,color=green,title=`daohamtungdo y'(t)`);plot(ppp,0..n*T,color=black,title=`daohamgocphi phi'(t)`);;;;;;;;;;;;;;;;;;;;;;;
Activate the following procedure twice to obtain the result completely . ( use Maple 9.5 & 10 )*******Animation Code*******
> mohinh:=proc(M,lan)
> mohinh(0.75,3);
Disclaimer: While every effort has been made to validate the solutions in this worksheet, the authors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.Legal Notice: The copyright for this application is owned by the author . 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 and the author .
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 United States License.
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Toán học thuần túy, theo cách của riêng nó, là thi ca của tư duy logic.
Pure mathematics is, in its way, the poetry of logical ideas.
Albert Einstein .
