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   

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 system .                                     

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 

coth123@math.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);

 

(1.1)

 

(1.1)

 

>	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);

 

(2.1)

 

(2.1)

 

>	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);

 

(2.2)

 

(2.2)

 

(2.2)

 

>	Ft:=spline(datax1, datay1, w, cubic);

 

(2.3)

 

>	dothif:=plot(Ft, w=0..5, color=red): display(dataplot1,dothif, axes=frame);

 

 

>	fnum:=subs(w=t,Ft);eq1:=subs(f(t)=fnum,eq1);

 

(2.4)

 

(2.4)

 

>	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);

 

(2.5)

 

(2.5)

 

(2.5)

 

>	Ht:=spline(datax2, datay2, w, cubic);

 

(2.6)

 

>	dothih:=plot(Ht, w=0..5, color=blue): display(dataplot2,dothih, axes=frame);

 

 

>	h1:=subs(w=t,Ht);eq2:=subs(h(t)=h1,eq2);

 

(2.7)

 

(2.7)

 

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 < 1],[.9826030927-.282603092700000002*t-2.16892488954344652*(t-1)^2+3.06826215099999988*(t-1)^3, t < 1.5],[.6254970546-.150331369699999995*t+2.43346833578792321*(t-1.5)^2-2.26561119299999980*(t-1.5)^3, t < 2],[-.5178571430+.583928571299999977*t-.964948453608247214*(t-2)^2+3.99418262299999994*(t-2)^3, t < 2.5],[-5.336542708+2.61461708300000017*t+5.02632547864506574*(t-2.5)^2-10.9111192900000002*(t-2.5)^3, t < 3],[4.027190721-.542396906999999984*t-11.3403534609720165*(t-3)^2+12.8502945499999992*(t-3)^3, t < 3.5],[8.757603092-2.24502945500000006*t+7.93508836524300420*(t-3.5)^2-5.29005890999999994*(t-3.5)^3, otherwise]); eq2 := 2*Diff(y,`$`(t,2))+.5e-1*cos(phi)*Diff(phi,`$`(t,2))-.5e-1*Diff(phi,t)^2*cos(phi)+5*Diff(y,t)+y+y^3 = PIECEWISE([.3000000000+.401389911599999982*t-.555964654000000014e-2*t^3, t < .5],[.3013899116+.397220176799999990*t-.833946980854194387e-2*(t-.5)^2-.932201767300000039*(t-.5)^3, t < 1],[.8902706185-.310270618499999984*t-1.40664212076583217*(t-1)^2+2.61436671600000015*(t-1)^3, t < 1.5],[.342065540e-1+.243862297300000003*t+2.51490795287187030*(t-1.5)^2-2.40526509599999994*(t-1.5)^3, t < 2],[-1.059642857+.954821428699999974*t-1.09298969072164964*(t-2)^2+1.16669366700000010*(t-2)^3, t < 2.5],[-.642129970+.736851987999999958*t+.657050810014727760*(t-2.5)^2-2.66150957300000002*(t-2.5)^3, t < 3],[3.206688144-.602229381300000033*t-3.33521354933726099*(t-3)^2+5.07934462299999990*(t-3)^3, t < 3.5],[1.347770617-.127934461999999998*t+4.28380338733431554*(t-3.5)^2-2.85586892500000022*(t-3.5)^3, otherwise]); G:=dsolve({eq1,eq2,y0=0,p0=0,yp0=0,pp0=0.1},[y,phi],'numeric'): 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..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)`);

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Animation Code 

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

 

Mohinh procedure 

 

>	mohinh:=proc(M,lan)

 

>	mohinh(0.75,3);

 

 

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