COULD A VARIABLE MASS OSCILLATOR EXHIBIT THE LATERAL INSTABILITY?

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1 Kragujevac J. Sci. 3 (8) UDC COULD A VARIABLE MASS OSCILLATOR EXHIBIT THE LATERAL INSTABILITY? Nebojša Danilović, Milan Kovačević and Vukota Babović Institute of Physics, Faculty of Science, University of Kragujevac, P.O. Box 6, 34 Kragujevac; Republic of Serbia (Received April, 8) ABSTRACT. The paper deals with the physical states of an elastic pendulu near its resonant point. As far as we know, in the rich literature on the subject there is a lack of experiental studies of variable ass haronic oscillations. We perfor a set of such easureents in order to investigate the behavior of non-stationary oscillators including autoparaetric resonance conditions. Our experients reveal that the lateral instability really starts when the bob ass equals the critical value predicted by the linear theory of the phenoenon. The subsequent evolution of the otion is generally very coplex and strongly depends on any subtle details of the physical syste at the early oents in the post-resonant period. We also undertook the detailed coputational job to copare the full theory of the elastic pendulu with the linear approxiation and found that the ain predictions of the linear theory, as stated in [], are correct. The characteristic coparisons of vertical and horizontal oscillations are presented.. INTRODUCTION The elastic pendulu (also known as the extensible pendulu or spring pendulu) is a siple pendulu with a spring incorporated in its string. This sort of haronic oscillator placed in a gravitational field soeties exhibits loss of lateral stability. The horizontal or pendulu oscillations of such kind are due to paraetric resonance. As we will describe below, if the pendulu weight is forced to start with soe vertical displaceent, but no horizontal displaceent, vertical oscillations are set up and the bob will siply bounce up and down. However, if the frequency of the vertical haronic oscillations is exactly twice the pendulu frequency, the vertical otion becoes paraetrically unstable. It eans that any sall initial horizontal deviation of the bob fro the vertical equilibriu position results in an exponentially increasing pendulu otion. Evidently, the energy of the spring oscillations goes into the pendulu oscillations. After the full aount of the energy is transferred, the centrifugal force initiates the reverse process. The initial state will be soon restored. Of course, then begins a novel cycle of this recurrence process, and so on. The elastic pendulu is seeingly a siple physical syste. Nevertheless, it can exhibit very coplex phenoena, which require rather sophisticated analytical ethods for the satisfactory explanations). Therefore, this ass-spring-sting cobination has been treated any ties in the past, as a challengeable scientific proble. ANIČIN, DAVIDOVIĆ

2 3 and BABOVIĆ [] give a literature survey of the subject, with short critical notes of soe extent. In addition, the elastic pendulu has been often used as an efficient odel when treating different probles in several fields of physics, e.g. in accelerators physics (theory of orbital oscillations), nonlinear optics and plasa physics (wave-wave interactions). In the cited paper [] the authors have presented the linear theory of the elastic pendulu. The standpoint there adopted is the Ince-Strutt stability chart of the Mathieu equation. They have deterined graphically the range of ass leading to instability for particular aplitude of the initial vertical oscillation and the corresponding growth coefficient. In the beginning, they have derived the relevant syste of two nonlinear coupled differential equations. Further, the authors have restricted their analyses to the case of otion liited to a relatively sall region around the origin in the ( x, y ) plane, i.e. for x and y. The linear syste of equations appeared fully capable to deonstrate the paraetric resonance at ω = ω and practically to reove the possibility of the paraetric resonance at ω = ω (here, ω and ω are, respectively, the angular frequencies of horizontal and vertical haronic oscillations). In this paper, we are going to perfor two tasks. First, we want to study nuerically the original, nonlinear syste of coupled differential equations for the given lateral instability proble without any linearization procedure. So far, a direct nuerical analysis has not been perfored and it is doubtful how uch the linear approxiation is reliable. Secondly, we have done an experient with variable ass elastic pendulu. Our ai was to study the resonant process while passing the critical cobinations of relevant paraeters during tie. As far as we know, no one has yet reported on such type of observations when dealing with a laterally unstable haronic oscillator. In the next section we will, in the sake of readers cofort, present the detailed theory of the elastic pendulu and outline the relevant equation published in several papers on the subject, aong the in above cited reference [].. THEORY In order to write down all the necessary expressions, let us take the advantage of the figure, where is sketched the elastic pendulu: a bob hanged on string with an incorporated spring. Fig. Elastic pendulu in the position when its string akes x y point defines the the angle ϕ with the vertical Y axis; (, ) instantaneous position of the pendulu weight; the point O is the origin of the Cartesian right-handed syste and represents the locus of weight in the static equilibriu.

3 33 The Cartesian axes XY, point of the attached weight. The coordinates start fro the origin that physically represents the equilibriu ( x, y ) give the instantaneous position of the weight. We will designate the spring stiffness constant with k. For the ass of the bob let us write. This ass depends on tie. The angular frequency of the vertical haronic oscillations is then k ω =. () Assue that g is the acceleration due to gravity. The natural length of the pendulu we will denote with l. When the bob is attached, the spring would becoe longer, according to the relation l= l + l. () Of course, the additional length in fact l is deterined with the above-defined paraeters, and g l = (3) k In the operating position depicted on the figure, the bob is for the distance apart fro the point of suspension (therefore, is the pendulu length at a given instant). The Cartesian coordinates of the bob in this position are x and y, whereas the polar ones are and ϕ. Using the geoetry (sketched on the figure ) of the two forces which are r Fig. Forces acting on the pendulu bob in the position sketched in Fig. ; T is the string tension, g r the acceleration due to gravity. acting on the oscillating bob, we write according to the Newton second law a r r = T + g r. (4) The projection of this vector equation on the abscissa yields = Tsinϕ. (5)

4 34 Here we ay put sin ϕ = x /. The force in string, according to Hook s law reads Hence, we obtain ( ) ( ) T = k l. (6) x = k( l+ l l ). (7) Recall that k l l = g. The above equation could be divided with the length l ; in what follows we shall consider that x and are the noralized quantities. Therefore, k x g x ( ) =. (8) l We can extent the both sides of this equation with the expression ω x, where ω = g / l. The equation which originate fro such a procedure could be put in the for Using the abbreviation we arrive at the expression + ωx = ( ω ω ) x. (9) β ω ω =, () + ω x= βx. () Now we shall seek what would be the y -coponent of the vector equation (4). First, we have d y = k ( l ) cosϕ+ g. () The extension used above in the equation (8) gives now once ore fro () d y = k ( cos ) ϕ+ g( cosϕ ). (3) Also, we shall incorporate here our failiar quantities ω, ω and β. Therefore, we arrive at d y + y + ω y = β + β. (4) We have obtained the two necessary equations. As an interediate résué, let us group here these equations, () and (4), calling the now the equations (I) and (II): d y + ω x= βx, (I) + y + ω y = β + β. (II)

5 35 The possible linearization process assues the validity of approxiations y. It leads to the approxiated equation pair (see the Appendix A) x and d y + ω x = β xy, (I- l ) + ω y = β x, (II- l ) which were the basis for the linear analysis conducted in []. As we have already pointed out, we shell shall here nuerically treat the original pair of coupled equations (I) and (II), in its full nonlinear content. Nevertheless, since it will be useful in our subsequent analysis of experiental data, we shall now in short repeat the basic conclusions of the linear theory. Suppose we start the vertical siple haronic oscillation ( ω ) of soe aplitude A. Therefore, we have fro (I- l ) Consequently, the differential equation y = Acos t (5) cos ( ) + ω x = βxa ω t. (6) + ω + βacos( ωt) x = has to be copared with the well-known Mathieu equation [6] in its standard for ( τ ) We conclude that the paraeters satisfy these relations: The condition acts when we have (7) + a+ 6qcos x dτ =. (8) τ = ωt, (9) l 4 4 g a = = l kl, () A a q =. () 4 4 a = corresponds to the paraetric resonance ω = ω. This condition 4g R = kl, or k 3g R = l. () Therefore, the forula () specifies the resonant ass. By this value of ass, the syste spring-ass quickly develops the lateral instability. Fro the very definition of the spring stiffness constant k, we ay write

6 36 g k =. (3) l After putting this value into the result (), we obtain iediately R = 3 ll. If M D is that ass which would double the natural length of spring, l, a siple and retentive forula arises: M D R =. (4) 3 Finally, we would like to reark what follows. In soe applications, using the latest version of the Matheatica coputational syste, we found that the appropriate solutions are of the for: l = ω v x, (5) p d y ( l )( l+ ) y p = ωv + g, (6) instead of (I) and (II). The diensionless paraeter p, when passing through unity, defines the nonlinear coupling resonance condition. The derivation of the above equations is straightforward and we oit here the proof. 3. COMPUTATIONS As it already has been announced, we shall now nuerically treat the set of two nonlinearly coupled equations (I) and (II). The figure 3 shows the tie evolution of the elastic pendulu otion when the conditions of the autoparaetric resonance are nearly or fully satisfied. Here our choice of paraeters was: the spring stiffness constant 5 N/ and the equilibriu length of string/spring cobination c. Fig. 3 Matheatica 6. plot of the full nonlinear syste of equations (I), (II); tie evolution of the vertical oscillations (continuous line) and horizontal oscillations (dashed line); k = 5 N/, l =, = R =.74 kg. In fact, we have presented two curves on the sae plot. The vertical oscillations start with a given aplitude. The initial horizontal displaceent is very sall, but cannot be put to zero. After soe periods of oscillations, a considerable aount of energy of this ode is

7 transferred to the horizontal, or pendulu ode of oscillations. The aplitude of the pendulu ode exponentially grows. At soe instant, designated on the figure 3 with the tick 8, alost all the energy is in this horizontal ode of otion; siultaneously, the aplitude of vertical oscillations is at its iniu. The process is now reversed. There are two cycles of vertical oscillations in each pendulu period. Physically, the centrifugal force acts as an excitation ter to the vertical oscillations. Soon the initial state is restored. The oscillating energy is flowing fro the vertical otion into the horizontal otion and vice versa without interruption. The linear theory, under the sae condition (conditions) valid for the figure 3, gives the result we plotted in the figure Fig. 4 Matheatica 6. plot of the linearized syste of equations ( I l ), ( II l ); tie evolution of the vertical oscillations (continuous line) and horizontal oscillations (dashed line); k = 5 N/, l =, = R =.74 kg. Let us now exaine the figure 5. There we have plotted the horizontal coponent of the above collective graphic and added the horizontal curve obtained according to the approxiated pair of equations (I- l ) and (II- l ). Fig. 5 The coparison of solutions for horizontal oscillations; exact theory (continuous line) and linear theory (dashed line); k = 5 N/, l =, = R =.74 kg. The dashed line corresponds to linear solution. The difference between the solutions sees to be hardly discernible. Evidently, the linearization has in this case its full justification, because it spreads analytical possibilities in the proble treated.

8 38 Fig. 6 The coparison of solutions for vertical oscillations; exact theory (continuous line) and linear theory (dashed line); k = 5 N/, l =, = R =.74 kg. Finally, let us see in the figure 6 the coparison of the two solutions concerning the vertical oscillations. The dashed curve is the linear solution. Again, we have the good agreeent between the exact and approxiated forulae. 4. APPARATUS We ust begin the experiental work by perforing two preliinary easureents: a) the deterination of the spring stiffness constant, and b) the deterination of the string length. The both tasks ust be done as carefully as possible, because the reliability of the results depends on it. law Deterination of the spring stiffness constant In fact, this procedure is standard and could be easily perfored applying Hook s g k =. Nevertheless, one ust be cautious. The spring will not stretch at all under l the action of a sall gravitational force g. The easureents are to be perfored well in the region of full linearity, sufficiently above the critical force. We did so, applying seven asses, one by one, in the range fro g to 8 g. Each tie we have readout the corresponding lengthening of the spring. The pairs of such easured values were indicated into a graphic. Through the set of points we have plotted a strait line by eans of the least squares ethod. The spring we finally choose as appropriate for our easureents was proved, according to this procedure, to have the stiffness constant k = 9. N/. Deterination of the natural pendulu length In our notation, l is the distance, with no weight attached, fro the hook where the spring is attached to its other end. We found that this length was nearly c. To this value ust be added, as we have estiated, additional 4 c ( c fro the hook on the vessel to the center of ass of the vessel containing water and c for two sall pieces of string on both sides of spring). The vessel is the plastic cylinder with diaeter 6 c and height 4.5 c. It could optially contain g of water.

9 The effective length of the spring with no ass attached, i.e. the natural length of the pendulu was so deterined to be l = 5 c. Deterination of the spring ass The ass of our spring is = spring g. The ass of the epty vessel is tank = 9.5g. The bob ass of the elastic pendulu in action is the su = tank + water. We see that the relation is nearly satisfied. spring In our theory, the ass of the spring is regarded as negligible. 5. EXPERIMENT We ade in the center on the botto of the vessel a sall cyclic opening in diaeter. On the top side of the vessel, there is a hole for the inlet of air while the water leaks out through the opening. We have proved, in non-oscillating state, that the initially vessel full of water would be epty in nearly 5 seconds. Making our variable ass oscillator, we were fully aware of experiental troubles, which could follow the usage of a fluid-weight changing in tie, even if in a controlled anner (FLORES, SOLOVEY & GIL, 3b). y = Acos ω t by pulling down the We excite a periodic oscillation of the type ( ) weight and setting it free with the zero velocity at t =. As the oscillations progress, the ass of the weight diinishes and the otion of vessel becoes quicker. At soe instant, the pendulu starts to develop the horizontal oscillations. The instability grows fast and soon we see only the pendulu ode of otion. We have taken a set of photographs of this phenoenon and one of these is shown on the picture. Photograph of the apparatus in action; the instant of axiu horizontal deviation; the flow of water is visible; in this pendulu ode of oscillations, the flow rate depends on the water height in vessel, but otherwise is constant. Obviously, the oscillator goes through the point of its paraetric resonance. We have easured the corresponding critical ass of weight (the ass of vessel with the ass of reaining water) in a series of observations. The ean easured critical ass was found to be R = 48.5 g. Now we shall copute the theoretical critical ass. Using the above-entioned values for the free pendulu length and the spring stiffness constant, we obtain t kl R = = g. 3g

10 4 The percent error is δ =.8 %. At this point, we have undertaken a coputational siulation of the phenoenon. We have odeled the process of water leaking down adopting the linear ass dependence of tie, = C ( C t ). In the literature, one can find the evidence that sand sifts very R close to a constant rate (FLORES, SOLOVEY & GIL (3a)). For water, this is only but an approxiation (certainly, there are laboratory techniques, which would help to overcoe the trouble [5]). Fig. 7 The tie evolution of the horizontal oscillations according to nonlinear theory; paraeters used in experients: k = 9. N/, l =.5 c, C =.3, C =. ; =.467 kg; = C ( C t). R R Fro a vertically oscillating vessel, the flow rate also oscillates. (We plan to study particularly this phenoenon in a separate paper.) Nevertheless, near the resonant point we shall try with the indicated linearization. Assuing the pair of coupled equations is sufficiently true in the case of a variable ass, we have looked for the solution in the Matheatica 6. progra. The results are shown on Figs Fig. 8 The tie evolution of the vertical oscillations according to nonlinear theory; paraeters used in experients: k = 9. N/, l =.5 c, C =.3, C =. ; =.467 kg; = C ( C t). R R

11 4 Fig. 9 The coparison of oscillations; vertical oscillations (continuous line), horizontal oscillations (dashed line); paraeters used in experients: k = 9. N/, l =.5 c, C =.3, C =. ; =.467 kg; = C ( C t). R R Again, we have the excellent global agreeent in the frae of our physical odel. The process of the lateral instability developent entirely coincides with the growth of the pendulu aplitude as well as with the vertical aplitude diinishent. However, let us see the Fig.. Fig. The pendulu weight is supposed to decrease linearly with tie; = C ( C t), =.467 kg; C =.3, C =. ; R R This figure gives the variable ass versus tie. The dependence is of the type t ( ) = R f( t), where f () t = C( Ct) and the resonant ass = kl / ( 3g) R. The constants are. 3 and., respectively. We already know that the resonant ass which corresponds to the paraeters adopted in easureents is approxiately 49 gras. Fro Fig. 7 we read that the axiu lateral weight deviation occurs 3.5 seconds after the otion started. Fro Fig. we ay conclude that for t = 3.5 s the ass is 55 g. The coputed ass value, obviously, does not coincide very well with the easured ass value. However, the error of about 3 % could not be regarded as quite unsatisfactory. Factors, which supposedly could cause the discrepancy, are nuerous; the ain, as we believe, is the rude and incoplete odel of the vessel with leaking water. In addition, the physicists in this area of experiental physics, well know it is extreely iportant to set up the syste as precisely as possible a sall error can have a large negative effect on the quality of the results. A carelessly started process of the vertical oscillations could initiate an undesired evolution of otion. There are two types of qualitative otion of the elastic pendulu. These are a) crescent and b) leon oscillations (Rusbridge (98);

12 4 Davidović, Aničin & Babović (996)). None of the both is of priary interest in the current paper; we are aidst the job of searching how the variable ass could affect the otion of an elastic pendulu in connection with the autoparaetric resonance: in other words, could the variable ass elastic pendulu deonstrate unstable otion. As we see, it can and, besides, would help deeper insights into this coplex physical phenoenon. 6. CONCLUSION This paper deals with the variable ass elastic pendulu. We proved it exhibits the lateral instability, under the conditions known to be valid in stationary experients. When passing through the resonant ass, the strong autoparaetric resonance in the syste is evident. While preparing the apparatus, we experiented with any and any strings seeking the optial cobination of string paraeters (stiffness constant, length, diaeter, ass and so on). Our ipression is that the strings of about k = N/ and l = c are preferable, at least in preliinary experients. Leaking fluids, presuable water, are attractive in designing the variable weights, but we do not exclude other possibilities (first of all, fine sorts of sands offer interesting possibilities). We ust be aware that the fluid jet streaing fro an oscillating vessel gives a flow rate far fro being a constant one. This is a proble deserving its own right. The non-linear coupled equations (I, II) explain the various details in experients with the elastic pendula very well. Our analyses and coputations show that the linearized equations retained the basic capability of the standard elastic pendulu theory; this is iportant because the siplified equations enable the rich area of analytical activity. In the paper [4], the authors investigated the proble of the variable ass oscillator rather thoroughly. However, they have not coented at all the possibility that the oscillator passes the region of the lateral instability connected with the autoparaetric instability, although their pendulu is evidently elastic (extensible). Our experience is that the variable ass haronic oscillator can exhibit the lateral instability and this aspect of the proble can not be disregarded. Acknowledgents Thanks are due to Professor D. Knežević for the attentive reading of the paper. She gave a nuber of suggestions which ade the text ore legible. References: [] ANIČIN, B.A., DAVIDOVIĆ D.M., and BABOVIĆ V. M.: On the linear theory of the elastic pendulu, Eur. J. Phys. 4 (993) The Physics Departent of The University of Maryland included this article into Deonstrations Reference File as good article on the elastic pendulu. [] DAVIDOVIĆ, D.M., ANIČIN, B.A., BABOVIĆ, V.M.: The libration liits of the elastic pendulu, Aerican Journal of Physics, 64 (3), March 996. [3] FLORES, J., SOLOVEY, S., GIL, S.: Flow of sands and a variable ass Atwood achine, A. J. Phys. 7 (7), July 3a, p. 75.

13 [4] FLORES, J., SOLOVEY, S., GIL, S.: Variable ass oscillator, A. J. Phys. 7 (7), July 3b, p. 7. The authors reark: For fluids, the tie rate of the ass discharged through an orifice depends on the height of the colun and thus is tie dependent. In contrast, we show that the flow rate of sand is constant in tie and does not depend on its height. However, we decided to carry out this series of experients with fluids (water), and postpone the use of sand for a future experiental activity. In addition, what is ost iportant for us, the authors proved that ω was well described by the relation ω = k/ t. [5] Mariotte s vessel keeps the escape velocity constant. The vessel has a hole in the wall of a Heron Ball (or siilar vessel). The hole is below the lower end of the tube. In addition, the discharge velocity could be kept constant by various siphon arrangeents. [6] MCLACHLAN, N.W.: Theory and Application of Mathieu Functions, Oxford: Clarendon, 947. [7] RUSBRIDGE, M.G.: Motion of the Spring Pendulu, A. J. Phys. 48, (98), p. 46. () 43 Appendix A The linearization of the coupled equations of the elastic pendulu Let us first see the equation (I). We need to approxiate the rhs expression: + ω x= βx. (I) βx βx ( y) ( o) + + o. (A-) + + o It eans β x β xy, (A-) in agreeent with the equation (I- l ) which reads Let us now focus our attention on the second equation (II) The rhs converges to d y + ω x = β xy. (I- l ) + y + ω y = β + β. (II)

14 44 + y + y β + β β + β ( + y) + y β + β + + y x o This is in agreeent with the equation (II- l ): ( ) x ( + y), (A-3), (A-4) β β + x (A-5) x. (A-6) d y + ω y = βx. (II- l ) The linear set of the two equations, which we have just derived applying the reasonable approxiations to the original equations of the elastic pendulu are easy adaptable to the fors rightly required by the Mathieu equation. Appendix B The listing of the Matheatica 5. prograe used in the Experient section