paramath

* B3

 

NUMERICAL - GRAPHICAL SOLUTIONS OF  THE
NON-LINEAR VIBRATION MODEL
with discrete data input 

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

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 .                                     

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 

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);

 

 

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.
 

B3.COHONGTRAN-NUGRADIS