Visualizing Free and Forced Harmonic Oscillations using Maple V R5*

Transcription

1 Int. J. Engng Ed. Vol. 15, No. 6,. 437±455, X/91 $ Printed in Great Britain. # 1999 TEMPUS Publications. Visualizing Free and Forced Harmonic Oscillations using Male V R5* HARALD PLEYM Telemark College, Deartment of Technology, Kjolnes Ring 56, 3914 Porsgrunn, Norway. harald.leym@hit.no This aer focuses on how the symbolic, numerical and grahical ower of the comuter algebra system Male V Release 5 can be used to exlore and visualize with animation free harmonic motion with and without daming and vibration of a mass-sring system forced by an external shar blow. Dirac's delta function is used to model the external force that acts for a very short eriod of time. It is also demonstrated how dislacement resonance in a forced oscillation can be readily obtained from the transfer function of the system reresented by a second-order differential equation. The information contained in the system frequency resonse may also be conveniently dislayed in animated grahical form. INTRODUCTION AT TELEMARK College, Deartment of Technology in Porsgrunn, Norway we have incororated the use of the comuter algebra system Male V in the engineering mathematics curriculum from August With the owerful software rogram including grahical, symbolic and numerical techniques, Male V Release 5 is the ideal rogram to make mathematics more relevant and motivating for engineering students and an excellent demonstration tool for the classroom. The use of Male has changed the way I teach and hoefully the way the students learn. I now have the oortunity to emhasize the learning of concets, the visualization of concets and the solution of more realistic engineering alications. Many of the henomena we observe in everyday life have a eriodic motion. And in many cases the eriodic motion is maintained by a eriodic driving force. For these forced oscillations the amlitude deends on the frequency of the driving force. There is usually some frequency of the driving force at which the oscillations have their maximum amlitude. This occurs when the natural frequency of the vibrating system and the frequency of the driving force are aroximately equal. This henomenon of resonance lays an imortant role in almost every branch of hysics. And the avoidance of destructive resonance vibrations is an imortant factor in the design of mechanical systems of all tyes. Alication of differential equations lays an imortant role in science and engineering. And the most imortant ste in determining the natural frequency of vibration of a system is the formulation of its differential equation. The urose of this article is to demonstrate how Male can be used to investigate and visualize with animations:. simle harmonic motion of a mass-sring system;. the resonse of a mechanical system (mass-sring system) to an imulse function;. the dislacement resonance in forced oscillations. ANIMATION Coding in Male does not require exert rogramming skills, so writing a Male rogram deals with utting a roc( ) and an end around a sequence of commands. And it is retty easy to ut re-witten routines together from Male's owerful building blocks and lotting facilities. The lottools ackage rovides many useful commands for roducing lotting objects, which can be scaled, translated and rotated. The following Male codes will be used in the coming sections. The first rocedure animates a mass-sring system with and without a dashot, and the second animates the resonse to a rectangular ulse by a linear differential second-order equation. Because it is imossible to show animation on aer, the animation figures show four consecutive frames. The animations in this aer can be seen live on the Journal's Web site. * Acceted 28 June

2 438 H. Pleym Male code for a mass-sring system Given m ˆ mass; r ˆ daming constant; k ˆ sring constant; x1; x2 ˆ ositions of the sring x0 ˆ x 0, x0 ˆ D x 0 ˆ initial conditions; f ˆ driving force; n ˆ number of frames. >mass_sring:=roc(m,r,k,x1,x2,x0,x0,tk,f,n) local mass,deq,init,sol,xk,xu,lt1,lt2,lt,ltxk,rect,base,sring,sring1,rod,cylinder,iston,fluid,dashot; with(lots):with(lottools): #loads the lots and lottools ackages sring:=roc(x1,x2,n) #rocedure for the sring local 1,2,3,4,n_1,n,; 1:=[x1,0];2:=[x1+0.25,0];3:=[x2-0.25,0];4:=[x2,0]; :=i->[x (x2-x1-0.5)/n*i,(-1)^(i+1)*0.5]; lot([1,2,seq((i),i=1..n-1),3,4],thickness=2,color=aquamarine); end: sring1:=x2->sring(x1,x2,12); rect:=t->translate(rectangle([-0.2,-0.2],[0.2,0.2],color=red),t,xk(t)+x2+1); #rect: dislay the osition of the mass on the dislacement curve x(t) mass:=x2->rectangle([x2,-1],[x2+2,1],color+red): if x0=0 and xk(0.1)=0 then xu:=1.5 elif x0=0 and xk(0.1)<>0 then xu:=3.5; else xu:x0; fi; rod:=x2->lot([[x2+2,0],[x *xu,0]],color=grey,thickness=2); cylinder:=lot([[x2+2.5xu,-1],[x2+2.5+xu,1],[x *xu,1],[x *xu,-1],[x2+2.5+xu,-1]],color=tan,thickness=2); iston:=x2->rectangle([x *xu,-1],[x2+3+2*xu,1],color=grey); fluid:=rectangle([x2+2.5+xu,-1],[x *xu,1],color=blue); dashot:=x2->dislay(rod(x2),cylinder,iston(x2),fluid); deq:=m*diff(x(t),t$2)+r*diff(x(t),t)+k*x(t)=f(t); rint(deq); init:=x(0)=x0, D(x)(0)=x0; sol:=dsolve({deq,init},x(t)); rint(combine(simlify(sol))); xk:=unaly(rhs(sol),t); #xk=x(t) : the dislacement of mass ltxk:=lot([xk(t)+x2+1,x2+1],t=0..max(tk,x2+4+3*xu)+0.5, color=[blue,grey],numoints=400); base:=lot(-1.1,t=0..max(tk,x2+4+3*xu)+0.5,color=brown,thickness=4); if r=0 then #undamed case lt1:=x->translate(dislay(sring1(x2+x),mass(x2+x),base),0,-3); else #with daming lt1:=x->translate(dislay(sring1(x2+x),mass(x2+x),dashot(x2+x),base),0,-3); fi; lt2:=i->dislay(ltxk,rect(tk/n*i)); lt:=i->dislay(lt1(xk(tk/n*i)),lt2(i)); dislay(seq(lt(i),i=0..n),insequence=true); dislay(%,args[11..nargs]); end: Malecode for a rectangular ulse Given m ˆ mass; r ˆ daming constant; k ˆ sring constant; a ˆ constant; es ˆ constant; n ˆ number of frames. >imulse_func:=roc(m,r,k,a,es,n) local h,dlign1,dlign2,sol1,sol2,lt2,ltd,lt,txt,txtd,kloss,e; with(lots): alias(u=heaviside): h:=es->(u(t-a)-u(t-a-es))/es; dlign1:=es->m*diff(y(t),t,t)+r*diff(y(t),t)+k*y(t)=5*h(es); dlign2:=m*diff(y(t),t,t)+r*diff(y(t),t)+k*y(t)=5*dirac(t-a); rint(dlign1(es), esilon=es); rint(dlign2); sol1:=es->simlify(dsolve({dsolve({dlign1(es),y(0)=0,d(y)(0)=0},y(t))); sol2:=dsolve({dlign2,y(0)=0,d(y)(0)=0},y(t)); e:=textlot([6,1.3,convert([101],bytes)], font=[symbol,12]): txt:=es->textlot([6.9,1.3,cat(`=',convert(evalf(es,2),string))]); align={right,above}); ltd:=es->lot(h(es),t=0..5*pi,color=green,thickness=2):

3 Visualizing Free and Forced Harmonic Oscillations using Male V R5 439 lt1:=es->lot(rhs(sol1(es)),t=0..5*pi,color=red,thickness=2): lt2:=lot(rhs(sol2),t=0..5*pi,color=blue,thickness=2): lt:=es->dislay(ltd(es),txt(es),e,txtd,lt1(es),lt2); dislay(seq(lt(es-(es-0.1)/n*i),i=0..n),insequence=true); dislay(%,args[7..nargs]); end: FREE HARMONIC OSCILLATIONS Machines with rotating comonents commonly involve mass-sring systems or their equivalents in which the driving force is simle harmonic motion. The motion of a mass attached to a sring serves as a simle examle of vibrations that occur in more comlex mechanical systems. From a teaching oint of view it is suitable to consider a body of mass m attached to one end of a sring that resists comression as well as stretching. A rod attached to the mass carries a disk moving in an oilfilled cylinder (a dashot). The other end of the sring could be attached to a fixed wall and vibrating horizontally. The resultant force on the body is the sum of the restoring force kx t and the daming force x t where k is the force constant, x(t) is the distance of the body of mass from its equilibrium osition, t is time and r is the daming constant. We take x t > 0 when the sring is stretched. The differential equation of motion is therefore: >deq1:=m*diff(x(t),t$2)+r*diff(x(t),t)+k*x(t)=0; deq1 @t 2 x t kx t ˆ 0 If we set r ˆ 0 in deq1 the motion is undamed. Otherwise the solution of deq1 resents three distinct cases of daming according to whether d ˆ r 2 4mk is greater than, equal to, or less than zero. >eq:=d=r^2-4*m*k: >solm:=solve(eq,m): Let us relace m with a new variable mv in deq1. >mv:=unaly(solm,d,r,k); >deq2:=collect(4*k*subs(m=mv(d,r,k),deq1),diff(x(t),t$2)); mv :ˆ d; r; k! 1 d r 2 4 k deq2 :ˆ d r 2 x 4k x t kx t ˆ 0 With this differential equation, deq2, the system is overdamed if d > 0, critically damed if d ˆ 0 and underdamed if d < 0. If we request Male to solve deq1 for the undamed case and deq2 for each of the damed cases, subject to initial condition x 0 ˆ x 0 x t j tˆ0 ˆ 0. >init:=x(0)=x[0],d(x)(0)=0: #initial conditions we get the following solutions. Undamed conditions With r ˆ 0 Ns=m we get: >deq1a:subs(r=0,deq1); deq1a 2 x t kx t ˆ 0 >dsolve({deq1a,init},x(t)); x t ˆ x 0 cos r! k t m

4 440 H. Pleym A tyical grah of x t with m ˆ 1 kg and k ˆ 1 N=m is shown in Fig. 1. Figure 2 shows the animation of the undamed motion >xk:unaly(rhs(%),m,k,x[0],t): #defines x=x(m,k,x[0],t) Overdamed conditions >interface(showassumed=0): >assume(d>0): >dsol:=dsolve({deq2,init},x(t)); >subs(eq,dsol); dsol :ˆ x t ˆ 1 x 0 r 2 x t ˆ 1 x 0 r 2 d kt= d r d e 2 r d x 0 r d e 2 r d d kt= d r 2 r 2 1=2 r r 4mk e 2 4mk t=m 1 x 0 r r 2 1=2 r r 4mk e 2 4mk r 2 4mk 2 r 2 4mk >xo:=unaly(rhs(%),m,r,k,x[0],t): #defines x=x(m,r,k,x[0],t) The solution consists of two exonential terms when r > 4mk. In all subsequent figures we take m ˆ 1 kg and k ˆ 1 N=m. Figure 3 show some tyical grahs of the osition function for the overdamed case. Figure 4 shows the animation of the motion with the daming constant r ˆ 4 Ns=m. Critically damed condition >lhs(dsol)=limit(rhs(dsol),d=0); x t ˆ x0 2kt r e 2 kt=r r t=m >subs(r=sqrt(4*m*k),%); x t ˆ 1 x 0 2kt 2 mk e kt= mk 2 mk >xc:unaly(rhs(%),m,k,x[0],t): #defines x=x(m,k,x[0],t) The solution consists of one exonential term when r ˆ 4mk. The grah in Fig. 5 resemble those of the overdamed case in Fig. 3. The animation of a critically damed motion with r ˆ 2Ns=m is shown in Fig. 6. Under-damed condition >assume(d<0): dsolve({deq2,init},x(t));! x 0 re 2rkt=d r2 d kt sin 2 d r 2! x t ˆ x 0 e 2rkt=d r2 d kt cos 2 d d r 2 >subs(eq,%); x 0 re 1=2 rt=m sin 1! r 2 4mkt 2 m x t ˆ x 0 e 1=2 rt=m cos 1! r 2 4mkt r 2 4mk 2 m >xu:unaly(rhs(%),m,r,k,x[0],t): #defines x=x(m,r,k,x[0],t) The solution reresents exonentially damed oscillations of the mass-sring system about its equilibrium osition as shown in both Fig. 7 and Fig. 8. Undamed motion results >with(lots): #load the lots ackage >lot(xk(1,1,1,t),t=0..19,color=black,labels=[`t',`x(t)'],labelfont=[times,bold,12]);

5 Visualizing Free and Forced Harmonic Oscillations using Male V R5 441 Fig. 1. Undamed motion. m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m. 2 x t x t ˆ 0 x t ˆ 2 cos t The small rectangle shows the osition of the mass on the dislacement curve. Overdamed motion results If we select m ˆ 1 kg, k ˆ 1 N=m and three different d-values: 12, 45, 96 Ns=m 2, r named rk becomes: >rk:=(d,m,k)->simlify(sqrt(d+4*m*k)): r=[seq(rk(d,1,1),d=[12,45,96])]; Fig. 2. Animation of an undamed motion. m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m. r ˆ 4; 7; 10Š >lt1:=d->lot(xo(1,rk(d,1,1),1,2,t),t=0..20,color=black): >txt1:=d->textlot([4,xo(1,rk(d,1,1),1,2,4),convert(r=rk(d,1,1),string)]): >ltod:=dislay(seq({lt1(d),txt1(d)},d=[12,45,96]),labels=[`t',`x(t)'],labelfont=[times,bold,12]):%;

6 442 H. Pleym Fig. 3. Overdamed motion, m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r: daming constant. Figure 3 shows three tyical grahs of the osition function x(t) for the overdamed case. Figure 4 shows the animation of the motion with r ˆ 4 Ns=m. >mass_sring(1,4,1,0,5,2,0,14,0,3,tickmarks=[5,3],labels=[`t',`'],scaling=constrained); t x t ˆ 2 3 e 2 2 x x t ˆ 0 3 e 2 3 t 2 3 e 2 3 t e t Fig. 4. Animation of an overdamed motion of a mass-sring system with dashot: m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r ˆ 4 Ns=m.

7 Visualizing Free and Forced Harmonic Oscillations using Male V R5 443 Critically damed motion results >lt2:=lot(xc(1,1,2,t),t=0..20,color=black): >txt2:=textlot([3,xc(1,1,2,3),convert(r=rk(0,1,1),string)]): >ltcd:=dislay(lt2,txt2):%; Fig. 5. Critically damed motion, m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r: daming constant. 2 x t x t ˆ x t ˆ 2e t 2e t t Fig. 6. Animation of an critically damed motion. m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r ˆ 2 Ns=m.

8 444 H. Pleym In this critically damed case, then resistance of the dashot is just large enough to dam out any oscillations. Underdamed motion results We select m ˆ 1 kg, k ˆ 1 N=m and two r values Ns=m : >lt3:=lot([xu(1,1/2,1,2,t),xu(1,1/5,1,2,t)],t=0..20,color=black): >txt3:=textlot([[6.5,xu(1,1/2,1,2,6.5),`r=1/2'],[6.5,xu(1,1/5,1,2,6.5),`r=1/5']]): >ltod:=dislay(lt3,txt3,labels=[`t',`x(t)'],labelfont=[times,bold,12]):%; Fig. 7. Underdamed motion, m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r: daming constant. 2 x t x t x t ˆ 0 x t ˆ 2e 1=10t 3 cos 11 t 2 11 e 1=10t sin 3 11 t Fig. 8. Animation of an underdamed motion of a mass-sring system with dashot: m ˆ 1 kg, k ˆ 1 N=m, r ˆ 1=5 Ns=m. The action of the dashot exonentially dams the oscillations in accord with the time-varying amlitude. The dashot also decreases the frequency of the motion from 1 in the undamed to case 3 11 =10 in the underdamed motion with the same mass and force constant. Comarison of the grahs in Figs 3, 5 and 7 (Fig. 9) shows that when the motion is critically damed the mass reaches its equilibrium osition in a shorter time than when the daming is larger. And for any daming constant less than r ˆ 2 Nm=m in our articular examle the motion becomes oscillatory. >dislay(lt1(45),lt2,lt3,txt1(45),txt2,txt3,labels=[`t',`x(t)'],labelfont=[times,bold,12]);

9 Visualizing Free and Forced Harmonic Oscillations using Male V R5 445 Fig. 8. Fig. 9. Overdamed, critically damed and damed oscillatory motion of a mass-sring system: m ˆ 1 kg, k ˆ 1 N=m, x 0 ˆ 2 m, r: daming constant.

10 446 H. Pleym FORCED HARMONIC OSCILLATIONS Mechanical systems are often acted uon by an external force of large magnitude that acts for only a short eriod of time. For examle a sring-mass system could be given a shar blow at some secific time t. In engineering ractice it is convenient to use the Dirac delta function as a mathematical model for such a blow. But it is my exerience that it is often difficult for engineering students to get a real understanding of what this imulse function stands for. With the use of Male it is easy and instructive to simulate Dirac's delta function t a. The resonse of a mass-sring system to an imulse function As said above the Dirac delta function t a could serve as a mathematical model for an external force of large magnitude that acts for only a very short eriod of time. >delta(t-a)=iecewise(t<a,0,t=a,infinity,t>a,0); 8 < 0 t < a t a ˆ 1 t ˆ a : 0 a < t where: >Int(delta(t-a),t=-infinity..infinity)=int(Dirac(t-a),t=-infinity..infinity); 1 1 t a dt ˆ 1 Obviously no real function can satisfy being zero excet at a single oint and have an integral equal to one. In Male t a ˆ Dirac t a is exressed as the derivative of the Heaviside standard unit ste function u t a. >alias(u=heaviside): u t a ˆ Dirac a It is instructive to use Male to model such an instantaneous unit imulse by starting with the function: >delta[esilon](t-a)=iecewise(t<a,0,t<a+esilon,1/esilon,0); 8 0 t < a >< 1 t a ˆ t < a >: 0 otherwise >h:=unaly(convert(rhs(%),heaviside),t,a,esilon); h :ˆ t; a;! A lot of the rectangular ulse for a ˆ and ˆ 3 gives: u t a u a t >lot(h(t,pi,3),t=0..10,thickness=3); >assume(esilon>): >A:=Int(h(t,a,esilon),t=a..a+esilon): >A=value(A); a a u t a u a t dt ˆ 1

11 Visualizing Free and Forced Harmonic Oscillations using Male V R5 447 Fig. 10. Rectangular ulse t a with a ˆ and ˆ 3 (width and height 1=. The area of the rectangular ulse is equal to 1. where u t is Heavisides unit ste function. We can comare the resonse of a damed mass-sring system to a rectangular ulse t a as! 0 with the resonse of the Dirac's delta function used by Male. The outut of the Male code imulse_func in Fig. 11 animate the resonse to the rectangular ulse t a by a linear second-order differential equation with constant coefficients and with daming. 2 y y t y t ˆ 5 3 u t 5 u 3 t ; ˆ 3 2 y y t y t ˆ 5 Dirac t Fig. 11. The behavior of the resonse to a rectangular ulse by a linear second-order differential equation with daming as! 0 comared to the resonse to Dirac's delta function.

12 448 H. Pleym The last animated frame in Fig. 11 shows little difference between the two resonses when ˆ 0:1 Figure 12 shows an animated resonse of the damed mass-sring system initially at rest. At time t ˆ the system is suddenly given a shar ``hammerblow'' modelled by f t ˆ 5 Dirac t. >f:=t->5*dirac(t-pi): 2 x x t x t ˆ 5 Dirac t x t ˆ 5 u t e 1=2 t u t e 1=2 t 3 5 Fig. 12. The motion of a mass-sring system with dashot under the influence of a shar blow at t ˆ rovided by 5 t. The dislacement resonance in forced oscillations In every oscillating system there is dissiation of mechanical energy, which results when the motion of the mass-sring system described in the revious sections die out. If the oscillations are to be maintained, energy must be sulied to the system. In this section we shall assume that the system is acted on by a eriodic driving force. Suose that the mass-sring system is subjected to a eriodic force F 0 sin!t, where F 0 is the maximum value of the alied force and f ˆ!=2 is its frequency. The equation of motion is 2 x x t kx t ˆ F 0 sin!t where m is the mass of the system, x(t) is the distance of the body of mass from its equilibrium osition, r is the daming constant and k is the force constant. The frequency resonse of this system can be readily obtained from the system transfer function H(s) by relacing s by i!.

13 Visualizing Free and Forced Harmonic Oscillations using Male V R5 449 >f:=`f': >with(inttrans): L:=x->lalace(x,t,s):invL:=X->invlalace(X,s,t): >alias(x(s)=l(x(t)),f(s)=l(f(t))): >dlign:=m*diff(x(t),t$2)+r*diff(x(t),t)+k*x(t)=f(t); >simlify(l(dlign)); dlign 2 x x t kx t ˆ f t ms 2 X s msx 0 md x 0 rsx s rx 0 kx s ˆ F s >X(s)=solve(%,X(s)); X s ˆ msx 0 md x 0 rx 0 F s ms 2 rs k >subs(x(0)=0,d(x)(0)=0,%); X s ˆ F s ms 2 rs k The transfer function of the mass-sring system is given by: >H:=s->1/(m*s^2+r*s+k): H s ˆ 1 ms 2 rs k It is easy to show that the steady-state frequency resonse to an inut q t ˆ F 0 cos!t becomes: >x(t)=f[0]*abs(`h'(i*omega))*cos(omega*t+arg(`h'(i*omega))); x t ˆ F 0 jh I! j cos!t arg H I! >value(%); 1 F 0 cos!t arg m! x t ˆ 2 Ir! k j m! 2 Ir! kj The steady-state system resonse is also a cosine having the same frequency! as the inut. And the amlitude of this resonse is F 0 jh I! j. The variation in both the magnitude jh I! j and argument arg H I! as the frequency! of the inut cosine is varied constitute the frequency resonse of the system, as the following examle shows. Let us first solve the differential equation using Males dsolve. >f:=(omega,t)->2*cos(omega*t); f :ˆ!; t! 2 cos!t >deq:=(m,r,k,omega)->m*diff(x(t),t$2)+r*diff(x(t),t)+k*x(t)=f(omega,t); deq :ˆ m; r; 2 x x t kx t ˆ f!; t >sol:=dsolve({deq(1,0.25,4,2),x(0)=0,d(x)(0)=0},x(t),method=lalace); >lot(rhs(sol),t=0..40,color=black); sol :ˆ x t ˆ 4: sin 2:t 4: e : t sin 1: t

14 450 H. Pleym Fig. 13. The dislacement x t of a mass-sring system undergoing forced oscillations lotted against the time: m ˆ 1 kg, r ˆ 0:25 Ns=m, k ˆ 4 N=m,! ˆ 2=s. The grah shows that the transient solution dies out as t increases. The transfer function of the system is given by: >H:=(m,r,k,s)->1/(m*s^2+r*s+k); H :ˆ m; r; k; s! With m ˆ 1 kg, k ˆ 4 N=m and s ˆ I!, we get: 1 ms 2 rs k >`H'(1,r,4,I*omega)=H(1,r,4,I*omega); H 1; r; 4; I! ˆ >Habs:=unaly(evalc(abs(H(1,r,4,I*omega))),r,omega); 1! 2 Ir! 4 1 Habs :ˆ r;!!! 4 8! 2 16 r 2! 2 There is a frequency called the resonance frequency at which the amlitude A ˆ 2Habs becomes a maximum. This resonance frequency can be recognized in many vibrating systems unless the daming force r is too large. >solve(omega^4-8*omega^2+16+r^2*omega^2=0,{r}); r ˆ I! 2! 2! ; r ˆ I! 2! 2! With zero daming force, r ˆ 0 and the resonance frequency! ˆ 2 in our examle. See also Fig. 15. The hase angle is given by: >hi:=unaly(evalc(argument(h(1,r,4,i*omega))),r,omega); :ˆ r;!! arctan r!;! 2 4

15 Substituting the values r ˆ 0:25 Ns=m,! ˆ 2=s and F 0 ˆ 2N, gives the steady-state resonse, xs ˆ x t >xs:=unaly(2*habs(0.25,2)*cos(2*t+hi(0.25,2)),t); `xs'(t)=xs(t); #steady state resons >sol; #Male's solution As t! 1, x t! xs t. Visualizing Free and Forced Harmonic Oscillations using Male V R5 451 xs t ˆ 4: cos 2t 1: x t ˆ 4: sin 2:t 4: e 1: t sin 1: t >control:=`4*sin(2*t)-4*cos(2*t-pi/2)'=simlify(4*sin(2*t)-4*cos(2*t-pi/2)); control :ˆ 4 sin 2t 4 cos 2t 1 2 ˆ 0 The system is lagging because the hase shift between inut and outut arg H i! is ˆ 1:57... Figure 14 shows this hase shift. >with(lots): lt1:=lot([f(2,t),xs(t)],t=30..39,color=black,thickness=[1,2]: lttext1:=textlot([[39,xs(39),`steady-state resonse'],[39,f(2,39),`inut']],align=right): dislay({lt1,lttext1},labels=[`t',`x(t)'],labelfont=[times,bold,12]); Fig. 14. Phase shift between steady-state resonse and inut dislacement x(t). >A:=`A': Habsi:=i->lot(Habs(i,omega),omega=1..3,A=0..2.5,color=black): >lt2:=i->dislay(habsi(i),text2(i)): >text2:=i->textlot([2,habs(i,2),convert(r=i,string)],align=above,font=[times,bold,12]: >dislay(seq(lt2(i),i=[0.1,0.25,0.5,0.5,0.75,1,1.41]),labels=[`omega',`a'],labelfont=[times,bold,12]); 4 Fig. 15. Variation of the amlitude A ˆ 2=! 8! 2 16 r 2! 2 against the angular velocity! of the forced oscillationsdislacement resonance for different values of the daming constant r. Figure 15 shows that the smaller the value of r, the sharer the resonance of the vibrating system to the alied force. And the resonance frequency varies with the value of the daming constant r. The amlitude A tends towards infinity when the alied frequency becomes equal to the resonance frequency and the system has zero daming.

16 452 H. Pleym Fig. 15. Figure 16 animates the dislacement resonance of the forced oscillations and Fig. 17 shows the variation of the hase shift against the angular velocity, which is animated in Fig. 18. Figure 19 shows what haens with the mass-sring system when the daming constant r ˆ 0:25 N=s and Fig. 20 when the daming constant is aroximately equal to zero, r ˆ 0:01 N=s. >dislay(seq(lt2(0.05*i),i=1..28),insequence=true,labels=[`omega',`a'],labelfont=[times,bold,12]); Fig. 16. Animation of Figure 15.

17 Visualizing Free and Forced Harmonic Oscillations using Male V R5 453 >Harg:=i->lot(hi(i,omega),omega=1..3,color=black): >text3:=i->textlot([2.4,hi(i,2.4),convert(r=i,string)],align={right,above},font=[times,bold,12]): >lt3:=i->dislay(harg(i),text3(i)): >dislay(seq(lt3(i),i=[0.1,0.25,0.5,0.75,1,1.41]),labels=[`omega',`hi'],labelfont=[times,bold,12]); Fig. 17. Variation of the hase shift ˆ arctan! 2 4=r! against the angular velocity! of the forced oscillations-dislacement resonance for different values of the daming constant r. As the alied velocity is increased from zero to the natural angular velocity of the system, the hase angle increases from zero to =2. When r ˆ 0 the angle changes abrutly from 0 to. >dislay(seq(lt3(0.05*i),i=1..28),labelfont=[times,bold,12],insequence=true); Fig. 18. Animation of Fig. 17.

18 454 H. Pleym >f:=t->2*sin(2*t): 2 x x t 4x t ˆ 2 sin 2t x t ˆ 4: cos 2: t sin 2: t 4e 0: t cos 1: t 0: e 0: t sin 1: t Fig. 19. The dislacement x(t) of the mass-sring system undergoing forced oscillations with daming constant r ˆ 0: x x t 4x t ˆ 2 sin 2t x t ˆ 99: cos 2: t 0: sin 2: t 99: e 0: t cos 1: t 0: e 0: t sin 1: t Fig. 20. The dislacement x(t) of the mass-sring system undergoing forced oscillations with daming constant r ˆ 0:01 Ns=m.

19 Visualizing Free and Forced Harmonic Oscillations using Male V R5 455 Fig. 20. SUMMARY This aer discusses the use of the comuter algebra system (CAS) Male in calculating and animating the resonses of free and force vibrations systems. CAS has changed the fundamental way in which many math and engineering courses are taught. The user can enter a mathematical exression which in this aer Male subsequently executes, and one is freed from tedious comutations. This allows the student to exlore how the solution of an engineering roblem deends on various arameters of the roblem under study. The grahics caabilities are extremely helful for visualizing the behaviour of the system under investigation. Male's lots and lottools ackages enable the user to utilize lotting objects for animating uroses. This is highlighted and demonstrated in this aer. It is hoed that this feature can attract the attention of both lecturers and students to visualize dynamic resonses, tyical in engineering, more realistically. Harald Pleym is Associate Professor in Deartment of Technology at Telemark College in Porsgrunn, Norway. His rofessional background is geohysics and atmosheric sciences, but now he regularly teaches courses in engineering mathematics. He initiated the use of the Comuter Algebraic System, Male, as a tool in all math courses since August His current main interest is rearing lessons and develoing sulementary, interactive course material in Male. The Male system enables him to develo both simle and comlex mathematical models in the classroom and laboratory, making mathematics more dynamic and meaningful.