Pendulum with a square-wave modulated length

Transcription

1 Penduum with a square-wave moduated ength Eugene I. Butikov St. Petersburg State University, St. Petersburg, Russia Abstract Parametric excitation of a rigid panar penduum caused by a square-wave moduation of its ength is investigated both anayticay and with the hep of computer simuations. The threshod and other characteristics of parametric resonance are found and discussed in detai. The roe of noninear properties of the penduum in restricting the resonant swinging is emphasized. The boundaries of parametric instabiity ranges are determined as functions of the moduation depth and the quaity factor. Stationary osciations at these boundaries and at the threshod conditions are investigated. The feedback providing active optima contro of pumping and damping is anayzed. Phase ocking between the drive and the penduum at arge ampitudes and the phenomenon of parametric autoresonance are discussed. Keywords: parametric resonance, optima contro, phase ocking, autoresonance, bifurcations, instabiity ranges 1. Introduction: the investigated physica system Periodic excitation of a physica system is caed parametric forcing if it is reaized by variation of some parameter that characterizes the system. In particuar, a penduum can be excited parametricay by a given vertica motion of its suspension point. In the frame of reference associated with the pivot, such forcing of the penduum is equivaent to periodic moduation of the gravitationa fied. This apparenty simpe physica system exhibits a surprisingy vast variety of possibe reguar and chaotic motions. Hundreds of texts and papers are devoted to investigation of the penduum with verticay osciating pivot: see, for exampe, Refs. [1], [2] and references therein. A widey known curiosity in the behavior of an ordinary rigid panar penduum whose pivot is forced to osciate aong the vertica ine is the dynamic stabiization of its inverted position, occurring when the driving ampitude and frequency ie in certain intervas (see, for exampe, Refs. [2] [5]). Another famiiar method of parametric excitation which we expore in this paper consists of a periodic variation of the ength of the penduum. In many textbooks and papers (see, for exampe, Refs. [6] [1]) such a system is considered as a simpe mode of a payground swing. Indeed, the swing can be treated as a physica penduum whose effective ength changes periodicay as the chid squats at the extreme points, and straightens each time the swing passes through the equiibrium position. It is easy to iustrate this phenomenon of the swing pumping by the foowing simpe experiment. Let a thread with a bob hanging on its end pass through a itte ring fixed in a support. You can pu by some sma distance the other end of the thread that you are hoding in your hand each time the swinging bob passes through the midde position, and reease the thread to its previous ength each time the bob reaches its extreme positions. These periodic variations of the penduum s ength with the frequency twice the frequency of natura osciation cause the ampitude to increase progressivey. A fascinating description of an exotic exampe iustrating this mode of parametric excitation can be found in Ref. [11], p. 27. In Spain, in the cathedra of a northern town Santiago de Preprint submitted to Internationa Journa of Non-Linear Mechanics February 11, 216

2 Compostea, there is a famous O Botafumeiro, a very arge incense burner suspended by a ong rope, which can swing through a huge arc. The censer is pumped by periodicay shortening and engthening the rope as it is wound up and then down around the roers supported high above the foor of the nave. The pumping action is carried out by a squad of priests, caed tiraboeiros or ba swingers, each hoding a rope that is a strand of the main rope that goes from the penduum to the roers and back down to near the foor. The tiraboeiros periodicay pu on their respective ropes in response to orders from the chief verger of the cathedra. One of the more terrifying aspects of the penduum s motion is the fact that the ampitude of its swing is very arge, and it passes through the bottom of its arc with a high veocity, spewing smoke and fames. In this paper we consider a penduum with moduated ength that can swing in the vertica pane in the uniform gravitationa fied. To aow arbitrariy arge swinging and even fu revoutions, we assume that the penduum consists of a rigid massess rod (rather than a fexibe string) with a massive sma bob on its end. The effective ength of the penduum can be changed by shifting the bob aong this rod. Periodic moduation of the effective ength by such mass redistribution can cause, under certain conditions, a growth of initiay sma natura osciations. This phenomenon is caed parametric resonance. 2. The square-wave moduation of the penduum ength In this paper we are concerned with a periodic square-wave (piecewise constant) moduation of the penduum ength. The square-wave moduation provides an aternative and may be more straightforward way to understand the underying physics and to describe quantitativey the phenomenon of parametric resonance in comparison with a smooth (e.g., sinusoida) moduation of the penduum ength [6] [1]. A computer program [12] deveoped by the author simuates such a physica system and aids greaty in investigating the phenomenon. In the case of the square-wave moduation, abrupt, amost instantaneous increments and decrements in the ength of the penduum occur sequentiay, separated by equa time intervas. We denote these intervas by T/2, so that T equas the period of the ength variation (the period of moduation). It is easy to understand how the square-wave moduation can produce considerabe osciation of the penduum if the period and phase of moduation are chosen propery. For exampe, suppose that the bob is shifted up (toward the axis) at an instant at which the penduum passes through the ower equiibrium position, when its anguar veocity reaches a maximum vaue. Whie the weight is moved radiay, the anguar momentum of the penduum with respect to the pivot remains constant. Thus the resuting reduction in the moment of inertia is accompanied by an increment in the anguar veocity, and the penduum gets additiona energy. The greater the anguar veocity, the greater the increment in energy. This additiona energy is suppied by the source that moves the bob aong the rod. On the other hand, if the bob is instanty moved down aong the rod of the swinging penduum, the anguar veocity and the energy of the penduum diminish. The decrease in energy is transferred back to the source. In order that increments in energy occur reguary and exceed the amounts of energy returned, i.e., in order that, as a whoe, the moduation of the ength reguary feeds the penduum with energy, the period and phase of moduation must satisfy certain conditions. In particuar, the greatest growth of the ampitude occurs if effective ength of the penduum is reduced each time the penduum crosses the equiibrium position, and is increased back at greatest eongations, when the anguar veocity is amost zero. Therefore this radia dispacement of the bob into its former position causes neary no decrement in the kinetic energy. The resonant growth of the ampitude occurs if two cyces of moduation are executed during one 2

3 period of natura osciations. This is the principa parametric resonance. The time history of such osciations for the case of a very weak friction (Q = 15) is shown in Fig. 1 together with the square-wave variation of the penduum ength. 13 o -D j(t) o T Period of moduation T = T /2, depth of moduation m = 1%, quaity factor Q = 15 Figure 1: Initia exponentia growth of the ampitude of osciations at parametric resonance of the first order (n = 1) under the square-wave moduation, foowed by beats. In a rea system the growth of the ampitude at parametric resonance is restricted by noninear effects. In a noninear system ike the penduum, the natura period depends on the ampitude of osciations. As the ampitude grows, the natura period of the penduum becomes onger. However, in the accepted mode the drive period (period of moduation) remains constant. If conditions for parametric excitation are fufied at sma osciations and the ampitude is growing, the conditions of resonance become vioated at arge ampitudes the drive sips out of resonance. The drive wi then drift out of phase with the penduum. The phase reationships between the moduation and osciations of the penduum change graduay to those favorabe for the backward transfer of energy from the penduum to the source of moduation. This causes gradua reduction of the ampitude. The natura period becomes shorter, and conditions for the growth of the ampitude restore. Osciations of the penduum acquire the character of beats, as shown in Fig. 1. Due to friction these transient beats graduay fade, and the ampitude tends to a finite constant vaue. Detais of the process of resonant growth foowed by a noninear restriction of the ampitude for parametricay excited penduum (T = T /2) with considerabe vaues of the moduation depth and friction (m =15%, Q = 5.) are shown in Fig. 2. The vertica segments of the phase trajectory and of the φ(t) graph correspond to instantaneous increments and decrements of the anguar veocity φ at the instants at which the bob is shifted up and down respectivey. The curved portions of the phase trajectory that spira in toward the origin correspond to damped natura motions of the penduum between the jumps of the bob. The initiay fast growth of the ampitude (described by the expanding part of the phase trajectory) graduay sows down, because the natura period becomes onger. After reaching the maximum vaue of 78.3, the ampitude aternativey decreases and increases within a sma range approaching sowy its fina vaue of about 74. The initiay unwinding spira of the phase trajectory simutaneousy approaches the cosed imit cyce, whose characteristic shape can be seen in the eft part of Fig. 2. It is evident that the energy of the penduum is increased not ony when two fu cyces of variation in the parameter occur during one natura period of osciation, but aso when two cyces occur during three, five or any odd number of natura periods (resonances of odd orders). We sha see ater that the deivery of energy, though ess efficient, is aso possibe if two cyces of moduation occur during an even number of natura periods (resonances of even orders). 3

4 D T Period of moduation T = T /2, depth of moduation m = 15%, quaity Q = 5. Figure 2: The phase diagram (φ φ pane, eft) and time-dependent graphs (right) of anguar veocity φ(t) and ange φ(t) for the process of resonant growth foowed by noninear restriction of the ampitude. The square-wave moduation of the penduum ength is aso shown (up and down positions of the penduum bob). 3. The threshod of parametric excitation There are severa important differences that distinguish parametric resonance from the ordinary resonance caused by an externa force exerted directy on the system. Variations of the ength cannot take a resting penduum out of equiibrium: in contrast to the direct forcing, parametric excitation can occur ony if (even sma) natura osciations aready exist. Parametric resonance is possibe when one of the foowing conditions for the frequency ω (or for the period T ) of a parameter moduation is fufied: ω = ω n = 2ω n, T = T n = nt, n = 1, 2,... (1) 2 Parametric resonance is possibe not ony at the frequencies ω n given in Eq. (1), but aso in ranges of frequencies ω ying on either side of the vaues ω n (in the ranges of instabiity). These intervas become wider as the depth of moduation is increased. An important distinction between parametric excitation and forced osciations is reated to the dependence of the growth of energy on the energy aready stored in the system. Whie for a direct forced excitation the increment in energy during one period is proportiona to the ampitude of osciations, i.e., to the square root of the energy, at parametric resonance the increment in energy is proportiona to the energy itsef, stored in the system. Energy osses caused by friction are aso proportiona to the energy aready stored. In the case of direct forced excitation, energy osses restrict the growth of the ampitude because these osses grow with the energy faster than does the investment in energy arising from the work done by the externa force. In the case of parametric resonance, both the investment in energy caused by the moduation of a parameter and the frictiona osses are proportiona to the energy stored, and so their ratio does not depend on the ampitude. Therefore, parametric resonance is possibe ony when a threshod is exceeded, that is, when the increment in energy during a period (caused by the parameter variation) is arger than the amount of energy dissipated during the same time. The critica (threshod) vaue of the moduation depth depends on friction. However, if the threshod is exceeded, the frictiona osses of energy cannot restrict the growth of the ampitude. With friction, stationary osciations of a finite ampitude eventuay estabish due to noninear properties of the penduum. 4

5 We can use arguments empoying the conservation aws to evauate the moduation depth which corresponds to the threshod of the principa parametric resonance. Let the changes in the ength of the penduum occur between 1 = (1 + m ) and 2 = (1 m ), where m is the dimensioness depth of moduation (or moduation index). To cacuate the change in tota energy of the penduum during a period, we shoud not worry about the potentia energy. Indeed, after a period the penduum occurs again in the vertica position with the bob at the same height, hence after a period its potentia energy is the same. Thus we shoud cacuate ony the change in kinetic energy. Next we cacuate the fractiona increment in energy E/E during one cyce of moduation, namey, between two consecutive passages through the equiibrium position in opposite directions. At the first passage, the energy E 1 equas v1/2 2 per unit mass of the bob, where v 1 is the bob s veocity. At this time moment the bob is shifted up, so the ength of the penduum changes from (1 + m ) to (1 m ). During abrupt radia dispacements of the bob aong the penduum rod, the anguar momentum L = J φ = M 2 φ is conserved (M is mass of the bob, J = M 2 is the moment of inertia about the pivot). Therefore the anguar veocity changes at this moment from φ 1 to (1 + m ) 2 /(1 m ) 2 φ 1. This means that the inear veocity v of the bob changes from v 1 = (1 + m ) φ 1 to (1 + m )/(1 m ) v 1. Then the penduum moves from the vertica φ = up to the maximum defection φ m, whose vaue can be cacuated using the energy conservation: 1 2 v2 1 ( ) m = g (1 m )(1 cos φ m ). (2) 1 m When the frequency and phase of the moduation have those vaues which are favorabe for the most effective deivery of energy to the penduum, the abrupt backward dispacement of the bob toward the end of the rod occurs at the instant when the penduum attains its greatest defection (more precisey, when the penduum is very near it). At this instant the anguar veocity of the penduum is amost zero. Hence this action produces no change in the kinetic energy. At this time moment the bob is shifted down, and the ength of the penduum becomes (1 + m ). The penduum starts its backward motion with zero veocity. Veocity v 2 in the equiibrium position which is gained during this motion, again can be cacuated, ike in Eq. (2), on the basis of energy conservation: 1 2 v2 2 = g (1 + m )(1 cos φ m ). (3) From Eqs.(2) (3) we find: ( ) 3 ( ) v2 2 = v1 2 m 1 + m, E 2 = E 1, (4) 1 m 1 m where E 2 = v2/2 2 is the kinetic energy (per unit mass) after a period T of moduation. Hence E E = E ( ) m 1 = 1 6 m. (5) E 1 1 m The ast approximate expression in Eq. (5) is vaid for sma vaues of the moduation depth m 1. That is, the fractiona increment of tota energy E/E during one period T of moduation approximatey equas 6m. The sequence of energy vaues E n at consecutive passages through the equiibrium position forms a geometric progression. A process in which the increment in energy E during a period is proportiona to the energy E stored (de/dt 6m E/T ) is characterized on average by the exponentia growth of the energy with time: E(t) = E exp( 6m T t) = E exp(αt). (6) 5

6 In this case the index of growth α is proportiona to the depth of moduation m of the penduum ength: α = 6m /T. When the moduation is exacty tuned to the principa resonance (T = T /2), the decrease of energy is caused amost ony by friction. Dissipation of energy due to viscous friction during an integer number of natura haf-cyces (for t = nt = nt /2) is described by the foowing expression: E(t) = E exp( 2γt), (7) where γ is the damping constant for the ampitude. Comparing equations (6) and (7), we obtain the foowing estimate for the threshod (minima) vaue (m ) min of the depth of moduation corresponding to the excitation of the principa parametric resonance: (m ) min = 1 3 γt = 1 6 γt = π 6Q. (8) Here we introduced the dimensioness quaity factor Q = ω /(2γ) to characterize the strength of viscous friction in the system..12. ϕ( t ) -D -.12 o 1 T =.5 T m = 1% Q = 5.2 o ϕ( t) T Figure 3: The phase trajectory (eft) and the time-dependent graphs of stationary osciations (right) at the threshod condition m π/(6q) for T = T /2. The phase trajectory and the pots of time dependence of the ange and anguar veocity of parametric osciations of a sma ampitude occurring at the threshod conditions, Eq. (8), are shown in Fig. 3. We can see on the graphs and the phase trajectory ony abrupt increments in the magnitude of the anguar veocity occurring twice during the period of osciation (when the bob is shifted upward). The downward shifts of the bob occur at instants when the anguar veocity is amost zero. Therefore the corresponding decrements in veocity are too sma to be visibe on the graphs. This mode of steady osciations (which have a constant ampitude in spite of the dissipation of energy) is caed parametric regeneration. Computer simuations show that regime of parametric regeneration is stabe with respect to sma variations in initia conditions: at different initia conditions the phase trajectory and graphs acquire after a whie the same characteristic shape. However, this regime is unstabe with respect to variations of the penduum parameters. If the friction is sighty greater or the depth of moduation sighty smaer than Eq. (8) requires, osciations graduay damp in spite of the moduation. Otherwise, the ampitude grows. For the third resonance (T = 3T /2) the threshod vaue of the depth of moduation is three times greater than its vaue for the principa resonance: (m ) min = π/(2q). In this instance two cyces of the parametric variation occur during three fu periods of natura osciations. 6

7 Radia dispacements of the penduum bob again happen at the time moments most favorabe for pumping the penduum up at the equiibrium position, and down at the extreme positions. The same investment in energy occurs during an interva that is three times onger than the interva for the principa resonance. When the depth of moduation exceeds the threshod vaue, the energy of initiay sma osciations during the first stage increases exponentiay with time. For the principa parametric resonance this initia growth is shown in Fig. 4. The growth of the energy again is described by 64. o j(t) o T Period of moduation T = T /2, depth of moduation m = 5%, quaity factor Q = 5 Figure 4: Graduay fading beats of the ampitude of osciations at parametric resonance of the first order (n = 1). Eq. (6). However, now the index of growth α is determined by the amount by which the energy deivered through parametric moduation exceeds the simutaneous osses of energy caused by friction: α = 6m /T 2γ. If the swing is sma enough, the energy is proportiona to the square of the ampitude. Hence the ampitude of parametricay excited osciations initiay aso increases exponentiay with time: a(t) = a exp(βt). The index β in the growth of ampitude is one haf the index of the growth in energy. For the principa resonance, when the investment in energy occurs twice during one natura period of osciation, we have β = 3m /T γ = 6m /T γ = 3m ω /π γ. If the threshod is exceeded, the ampitude grows, conditions of resonance vioate, and this causes a gradua reduction of the ampitude. At sma ampitudes the natura period becomes shorter, conditions of resonance restore, so that osciations of the penduum acquire the character of beats. Due to friction, these transient beats graduay fade, and eventuay steady-state osciations of a finite ampitude estabish (Fig. 4). We note again that the growth of ampitude is restricted by noninear properties of the penduum, namey, by the dependence of the natura period T on the ampitude. For sma and moderate vaues of the ampitude φ m this dependence is approximatey given by T (φ m ) T sma (1 + φ 2 m/16), where T sma is the period of infinitey sma natura osciations. In contrast with the ordinary resonance caused by direct periodic forcing in a inear isochronous system, friction aone cannot restrict the growth of the ampitude at parametric resonance. In an ideaized inear system the ampitude of parametric osciations over the threshod grows indefinitey (Refs. [14] [15]). 4. Feedback, parametric autoresonance, bifurcations, mutistabiity In the above anaysis we assumed that the period T of moduation remains the same as the ampitude increases. At exact tuning to the principa resonance this period equas T /2, where T is the period of sma natura osciations. When we appy the mode of the penduum with moduated ength for expaining the pumping of a payground swing, we shoud take into 7

8 account that the chid on the swing may notice the engthening of the natura period as the ampitude increases, and can react correspondingy, increasing the period of pumping to stay in phase with the swing. This intuitive reaction may be considered as a kind of feedback oop: the chid determines the time instants to squat and to stand depending on the actua position of the swing. We can incude this feedback oop in our mode by requiring that instantaneous upward shifts of the bob of the penduum occur exacty at the time moments, at which the penduum crosses the equiibrium position, and that backward shifts of the bob to the end of the rod occur exacty at extreme positions of the penduum. Such manipuations provide the optima active contro for the most effective and rapid pumping. 18 o j(t) -D o T Period of moduation T = T /2 and feedback, depth of moduation m = 5%, quaity factor Q = 5 Figure 5: Parametric pumping of the penduum with the usage of a feedback oop that provides the most effective deivery of energy to the penduum. Figure 5 shows the graph of progressivey growing osciations occurring under this optima contro with a feedback. Initiay the period of moduation T satisfies conditions of the principa parametric resonance at sma swing (T = T /2). We note how the period T of the square-wave moduation increases with the ampitude due to the feedback. After the ampitude reaches 18, the penduum executes fu revoutions. Certainy, the priests that pump O Botafumeiro aso use the feedback (maybe intuitivey) for controing the penduum behavior. They increase graduay the period of moduation as the ampitude grows, and then probaby reduce the depth of moduation to the eve, sufficient to compensate for frictiona osses and to maintain the desirabe swing. The strategy of optima active contro for the most rapid damping of existing osciations on the basis of feedback consists in reversing the phase of moduation. Namey, the ength of the penduum must be increased at moments of crossing the equiibrium position, and the ength must be reduced at extreme positions. Is it possibe to excite arge osciations of the penduum at a sma excess of the drive over the threshod without the feedback, that is, without appropriatey adjusting the period and phase of moduation as the ampitude grows? It occurs that under certain conditions a spontaneous phase ocking between the drive and the penduum motion becomes possibe: the penduum can automaticay adjust its ampitude to stay matched with the drive. By sweeping the drive period appropriatey, we can contro the ampitude of the penduum. This phenomenon is caed parametric autoresonance. Autoresonance aows us to both excite and contro a arge resonant response in noninear systems by a sma forcing. We can start pumping the penduum by moduating its ength with period T = T /2, which corresponds to resonant condition at an infinitey sma swing. Then, in the process of osciations, we sowy increase the period of moduation. This can be done in sma steps. After 8

9 each increment of the period we wait for a whie so that transients amost fade away. During this time the ampitude increases just to the amount which provides matching of an increased natura period of the penduum with the new period of moduation. Thus in each step of this sweeping the penduum remains ocked in phase with the drive. To iustrate the phenomenon of parametric autoresonance in a computer simuation (Fig. 6), we choose the foowing vaues for the penduum parameters: depth of moduation 5%, quaity factor Q = 2. When the period of moduation is graduay increased from T =.5T up to T =.9T, the penduum swings with an ampitude of 153. At T =.9T a bifurcation of symmetry breaking occurs: the penduum swings to one side through an ange of 161, whie its excursion to the other side is ony 146. This asymmetry in the swing increases up to T =.913T, when a bifurcation of period doubing occurs: during two cyces of moduation the penduum executes one asymmetric osciation between the vaues and , whie during the next two cyces the penduum swings between and Then the process repeats. Thus one period of the penduum motion covers now four periods of excitation. These osciations are iustrated in Fig. 6. We note that the cosed phase trajectory is formed by two nearby amost merging oops. Such asymmetric regimes exist (for the same vaues of m and Q) in pairs, whose phase orbits are mirror images of one another ο ο -D ο T Period of moduation T =.913 T, depth of moduation m = 5%, quaity Q = 2, initia defection 142, initia anguar veocity.38 ο ω Figure 6: Bifurcation of period doubing in parametric autoresonance. Further increasing of the drive period by tiny steps causes a whoe condensing cascade of nearby period doubing bifurcations, which ends at T =.9148T by a crisis: osciations of the penduum become unstabe, finay it turns over the upper equiibrium, and then, after ong irreguar transient osciations with graduay diminishing ampitude, eventuay comes to rest in the downward vertica position. Stationary parametric osciations of the penduum with arge ampitude, ocked in phase with the drive and occurring at a sma or moderate moduation (ike those described above and shown in Fig. 6), can be excited not ony by sowy sweeping the drive period, but aso by appropriate initia conditions. The system eventuay comes to a certain periodic regime (imit cyce, or attractor), if initia conditions are chosen within the basin of attraction of this regime. In noninear systems different periodic regimes may coexist at the same vaues of parameters. This property is caed mutistabiity. An exampe of mutistabiity is shown in Fig. 7. Curve 1 (upper side of Fig. 7) describes stationary periodic osciations of the penduum with a finite ampitude corresponding to the principa parametric resonance. One period of these osciations covers two cyces of excitation. 9

10 D ο ο T Period of moduation T =.56 T, depth of moduation m = 15%, quaity Q = 2. Figure 7: Stationary periodic osciations and rotations, occurring at the same vaues of the system parameters. Curves 2 and 3 (ower side of Fig. 7) correspond to period-1 unidirectiona rotations of the penduum in cockwise and countercockwise directions respectivey. The penduum makes one revoution during each period of moduation. One more attractor is represented by a singe point at the origin of the phase pane, which describes the state of rest of the penduum in the downward vertica position. Each of these different stationary modes, coexisting at the same vaues of a parameters of the penduum and the drive, is characterized by a certain basin of attraction in the phase pane of initia states. 5. Governing equation for parametric osciations and the mean natura period at arge depth of moduation Next we consider a more rigorous mathematica treatment of parametric resonance under square-wave moduation of the parameter. During the time intervas (, T/2) and (T/2, T ), the ength of the penduum is constant, and its motion can be considered as a free osciation described by a corresponding differentia equation. However, the coefficients in this equation are different for the adjacent time intervas (, T/2) and (T/2, T ): φ + 2γ φ + ω 2 1 sin φ =, ω 1 = φ + 2γ φ + ω 2 2 sin φ =, ω 2 = ω 1 + m for < t < T/2, (9) ω 1 m for T/2 < t <. (1) Here ω = g/ is the natura frequency of sma osciations for the penduum with mean ength, and γ is the damping constant characterizing the strength of viscous friction. For a sow penduum traveing in air, the inear dependence of drag on veocity is a reasonabe approximation. When damping is caused by the drag force exerted on the penduum bob, and this force is proportiona to the inear veocity of the bob, the frictiona torque about the pivot is proportiona to 2. Since the moment of inertia is aso proportiona to 2, the damping constant γ in this mode remains the same when the ength of the penduum changes, that is, its vaues in Eqs. (9) and (1) are equa. At each instant t n = nt/2 (n = 1, 2,... ) of an abrupt change in the ength of the penduum, we must make a transition from one of these equations (9) (1) to the other. During each haf-period T/2 the motion of the penduum is a segment of some natura osciation. An anaytica investigation of parametric excitation can be carried out by fitting to one another known soutions to equations (9) (1) for consecutive adjacent time intervas. 1

11 The initia conditions for each subsequent time interva are chosen according to the physica mode in the foowing way. Each initia vaue of the anguar dispacement φ equas the vaue φ(t) reached by the osciator at the end of the preceding time interva. The initia vaue of the anguar veocity φ is reated to the anguar veocity at the end of the preceding time interva by the aw of conservation of the anguar momentum: (1 + m ) 2 φ 1 = (1 m ) 2 φ 2. (11) In Eq. (11) φ 1 is the anguar veocity at the end of the preceding time interva, when the moment of inertia of the penduum has the vaue J 1 = J (1 + m ) 2, and φ 2 is the initia vaue for the foowing time interva, during which the moment of inertia equas J 2 = J (1 m ) 2. The change in the anguar veocity at an abrupt variation of the inertia moment from the vaue J 2 to J 1 can be found in the same way. We may use here the conservation of anguar momentum, as expressed in Eq. (11), because at sufficienty rapid dispacement of the bob aong the rod of the penduum, the infuence of the torque produced by the force of gravity is negigibe. In other words, we can assume the penduum to be freey rotating about its axis. This assumption is vaid provided the duration of the dispacement of the bob constitutes a sma portion of the natura period. Considering conditions for which equations (9) (1) yied soutions with increasing ampitudes, we can determine the ranges of frequency ω near the vaues ω n = 2ω /n, within which the state of rest is unstabe for a given moduation depth m. In these ranges of parametric instabiity an arbitrariy sma defection from equiibrium is sufficient for the progressive growth of sma initia osciations. The threshod for the parametric excitation of the penduum is determined above for the resonant situations in which two cyces of the parametric moduation occur during one natura period or during three natura periods of osciation. The estimate obtained, Eq. (8), is vaid for sma vaues of the moduation depth m of the penduum ength. For arge vaues of the moduation depth m, the notion of a natura period needs a more precise definition. Let T = 2π/ω = 2π /g be the period of osciation of the penduum when its massive bob is fixed in the midde position, for which the effective ength equas. The period is somewhat onger when the weight is moved further from the axis: T 1 = T 1 + m T (1 + m /2). The period is shorter when the weight is moved coser to the axis: T 2 = T 1 m T (1 m /2). It is convenient to define the natura average period T av not as the arithmetic mean 1 2 (T 1+T 2 ), but rather as the period that corresponds to the arithmetic mean frequency ω av = 1 2 (ω 1 + ω 2 ), where ω 1 = 2π/T 1 and ω 2 = 2π/T 2. So we define T av by the reation: T av = 2π ω av = 2T 1T 2 (T 1 + T 2 ). (12) Indeed, the period T of the parametric moduation which is exacty tuned to any of the parametric resonances is determined not ony by the order n of the resonance, but aso by the depth of moduation m. In order to satisfy the resonant conditions, the increment in the phase of natura osciations during one cyce of moduation must be equa to π, 2π, 3π,..., nπ,.... During the first haf-cyce the phase of osciation increases by ω 1 T/2, and during the second haf-cyce by ω 2 T/2. Consequenty, instead of the approximate condition expressed by Eq. (1), we obtain: ω 1 + ω 2 T = nπ, or T = T n = n π = n T av 2 ω av 2. (13) Thus, for a parametric resonance of some definite order n, the condition for exact tuning can be expressed in terms of the two natura periods, T 1 and T 2. This condition is T = nt av /2, where 11

12 T av is defined by Eq. (12). For sma and moderate vaues of m it is possibe to use approximate expressions for the average natura frequency and period: ω av = ω ( ) ω (1 + 3 ) m 1 m 8 m2, T av T (1 3 ) 8 m2. (14) The difference between T av and T reveas itsef in terms proportiona to the square of the depth of moduation m. 6. Frequency ranges for parametric resonances of odd orders To find the boundaries of the frequency ranges of parametric instabiity surrounding the resonant vaues T = T av /2, T = T av, T = 3T av /2,..., we can consider stationary osciations of indefinitey sma ampitude that occur when the period of moduation T corresponds to one of the boundaries. These periodic stationary osciations can be represented as an aternation of natura osciations with the periods T 1 and T 2.. ϕ ( t ) m = 1% T =.457 T -1 -D ϕ ( t ) 1 2 3T Figure 8: Phase trajectory and time-dependent graphs of stationary parametric osciations at the ower boundary of the principa interva of instabiity (near T = T av /2) Main interva of parametric instabiity We examine first the vicinity of the principa resonance occurring at T = T av /2. Suppose that the period T of the parametric square-wave moduation is a itte shorter than the resonant vaue T = T av /2, so that T corresponds to the eft boundary of the interva of instabiity. In this case a itte ess than a quarter of the mean natura period T av eapses between consecutive abrupt increases and decreases of the penduum ength. Stationary regime with a constant swing in the absence of friction can be reaized ony if the abrupt increments and decrements of the anguar veocity are equa in magnitude. The graphs of the ange φ(t) and anguar veocity φ(t) for this periodic stationary process have the characteristic symmetric patterns shown in Fig. 8. The segments of the graphs of free osciations (which occur within time intervas during which the ength of the penduum is constant) are aternating parts of sine or cosine curves with the periods T 1 and T 2. These segments are symmetricay truncated on both sides. To find conditions at which such stationary osciations take pace, we can write the expressions for φ(t) and φ(t) during the adjacent intervas in which the osciator executes natura osciations, and then fit these expressions to one another at the boundaries. Such fitting must provide a periodic stationary process. 12

13 We et the origin of time, t =, be the instant when the bob is shifted downward. The anguar veocity is abrupty decreased in magnitude at this instant (see Fig. 8). Then during the interva (, T/2) the graph describes a natura osciation with the frequency ω 1 = ω / 1 + m. Since the graph is symmetric with respect to time moment T/4, we can write the corresponding time dependencies of φ(t) and φ(t) in the foowing form: φ 1 (t) = A 1 cos ω 1 (t T/4), φ 1 (t) = A 1 ω 1 sin ω 1 (t T/4), < t < T/2. (15) Simiary, during the interva ( T/2, ) the graph in Fig. 8 is a segment of natura osciation with the frequency ω 2 = ω / 1 m: φ 2 (t) = A 2 sin ω 2 (t + T/4), φ 2 (t) = A 2 ω 2 cos ω 2 (t + T/4), T/2 < t <. (16) To determine the vaues of constants A 1 and A 2, we use the conditions that must be satisfied when the segments of the graph are joined together, and take into account the periodicity of the stationary process. At t = the ange of defection is the same for both φ 1 and φ 2, that is, φ 1 () = φ 2 (). The anguar veocity at t = undergoes a sudden change, which foows from the conservation of anguar momentum: (1 + m ) 2 φ 1 () = (1 m ) 2 φ 2 (), see Eq. (11). From these conditions of fitting the graphs we find the foowing equations for A 1 and A 2 : A 1 cos(ω 1 T/4) = A 2 sin(ω 2 T/4). (17) A 1 (1 + m ) 2 ω 1 sin(ω 1 T/4) = A 2 (1 m ) 2 ω 2 cos(ω 2 T/4). (18) These homogeneous equations (17) (18) for A 1 and A 2 are compatibe ony if the foowing condition is fufied: (1 + m ) 2 ω 1 sin(ω 1 T/4) sin(ω 2 T/4) = (1 m ) 2 ω 2 cos(ω 1 T/4) cos(ω 2 T/4). (19) This is the equation that determines period T of moduation (for a given vaue m of the depth of moduation) which corresponds to the eft boundary of the interva of parametric instabiity. Next we rearrange Eq. (19) to the foowing form which is convenient for obtaining its numeric soution for the unknown variabe T : (q + 1) cos(ω av T/2) = (q 1) cos( ωt/4), (2) where ω av = (ω 1 + ω 2 )/2, and ω = ω 2 ω 1. In Eq. (2) we have introduced a dimensioness quantity q which depends on the depth of moduation m : ( ) 3/2 1 + m q =. (21) 1 m To find the eft boundary T of the instabiity interva which contains the principa parametric resonance, we search for a soution T to Eq. (2) in the vicinity of T = T /2. We repace T in the argument of the cosine on the eft-hand side of Eq. (2) by T av /2 + T. Since ω av T av = 2π, we can write the cosine as sin(ω av T/2). Then Eq. (2) becomes: sin(ω av T/2) = q 1 q + 1 cos ω(t av/2 + T ). (22) 4 This equation for the unknown quantity T can be soved numericay by iteration. We start with T = as an approximation of the zeroth order, substituting it into the right-hand side of Eq. (22). Then the eft-hand side of Eq. (22) gives us the vaue of T to the first order. We 13

14 m (%) T/T Figure 9: Intervas of parametric instabiity at square-wave moduation of the penduum ength in the absence of friction. substitute this first-order vaue into the right-hand side of Eq. (22), and on the eft-hand side we obtain T to the second order. This procedure is iterated unti a sef-consistent vaue of T for the eft boundary is obtained. Performing such cacuations for various vaues of the moduation depth m, we obtain the whoe eft boundary T (m ) for the first interva of parametric instabiity. Beow we expain how the right boundary of this interva can be cacuated, as we as the boundaries of other intervas. The intervas of instabiity in the pane T m for the first six parametric resonances, cacuated numericay with the hep of the above described procedure, are shown in Fig. 9. This is an anaog of the Incze-Strutt diagram of parametric instabiity for a system which is described by Mathieu equation, say, for a penduum with vertica osciations of the suspension point. To observe stationary osciations that correspond to the eft boundary of the instabiity interva (see Fig. 8) in the simuation, it is insufficient to choose for period T of moduation a sef-consistent soution to Eq. (22) for a given vaue of moduation depth m. After period T is cacuated, aso the initia conditions shoud be chosen propery. This can be done on the basis of Eq.(15), according to which for an arbitrary initia dispacement φ() the initia anguar veocity shoud have the vaue φ 1 () = ω 1 tan(ω 1 T/4)φ 1 (). For the right boundary of the main interva of instabiity, the period T of the parametric square-wave moduation is a itte onger than the resonant vaue T = T av /2. In this case a itte more than a quarter of the mean natura period T av eapses between consecutive abrupt increases and decreases of the penduum ength. The graphs of the ange φ(t) and anguar veocity φ(t) for this periodic stationary process are shown in Fig. 1. We can write the corresponding time dependencies of φ(t) and φ(t) for the time interva (, T/2) in the foowing form: φ 1 (t) = B 1 sin ω 1 (t T/4), φ 1 (t) = B 1 ω 1 cos ω 1 (t T/4), < t < T/2. (23) During the interva ( T/2, ) the graph in Fig. 1 is a segment of natura osciation with the frequency ω 2 = ω / 1 m: φ 2 (t) = B 2 cos ω 2 (t + T/4), φ 2 (t) = B 2 ω 2 sin ω 2 (t + T/4), T/2 < t <. (24) Further cacuations are simiar to those for the eft boundary described after Eqs. (15) (16). It occurs that T for the right boundary is determined as a soution to equation which differs from Eq. (22) by the opposite sign on its right-hand side. Soving it numericay by iterations 14

15 . ϕ ( t ) m = 1% T =.5455 T -D ϕ ( t ) T Figure 1: Stationary parametric osciations at the upper boundary of the principa interva of instabiity (near T = T av /2). for various vaues of m, we obtain the right boundary of the principa interva (n = 1) of parametric instabiity, Fig. 9. To obtain approximate anaytica soutions to Eq. (22) that are vaid for sma vaues of the moduation depth m, we can simpify the expression on its right-hand side by assuming that q 1 + 3m, q 1 3m. We may aso assume the vaue of the cosine to be approximatey 1. On the eft-hand side of Eq. (22), the sine can be repaced by its sma argument, in which ω av = 2π/T av. This yieds the foowing approximate expressions for both boundaries of the main interva that are vaid up to terms to the second order in m : T = 1 2 ( 1 3m π ) T av = 1 2 ( 1 3m π 3m2 8 ) T. (25) 6.2. Third-order interva of parametric instabiity The boundaries of the instabiity intervas that contain higher order parametric resonances can be determined in a simiar way. At the third order resonance (n = 3) two cyces of variation of the penduum ength occur during approximatey three natura periods of osciation (T 3T av /2). The phase trajectories and the time-dependent graphs of stationary osciations at the eft and right boundaries of the third interva are shown in Fig. 11. The phase orbit of the periodic osciation coses after two cyces of moduation. This orbit is formed by two concentric eipses which correspond to sma natura osciations of the penduum with frequencies ω 1 and ω 2. The representative point moves cockwise aong this orbit, jumping from one eipse to the other each time the bob is shifted aong the penduum rod. The numbers in Fig. 11 make easier foowing how the representative point describes this orbit: equivaent points of the phase orbit and the graph of anguar veocity are marked by equa numbers. Considering conditions at which the graphs of natura osciations with frequencies ω 1 and ω 2 on the eft boundary fit one another for adjacent time intervas and produce the periodic process shown in Fig. 11, we get the same Eqs. (17) (18) for A 1 and A 2, as we as Eq. (22) for the period of moduation. Actuay, this is true for a intervas of parametric instabiity of odd orders. Simiary, for the right boundary we get the same equations for B 1 and B 2 as in case n = 1, and aso Eq. (22) with the opposite sign for determination of the corresponding period of moduation T. However, if we are interested in the third interva, we shoud search for a soution to these equations in the vicinity of T = 3T av /2, as we as for any other interva of odd order n in the vicinity of T = nt av /2. The boundaries of intervas of the third and fifth orders, obtained by a numerica soution, are aso shown in Fig

16 Left boundary: m = 2% -D T = T T. 8 ϕ( t) m = 2% Right boundary: -D T = T T Figure 11: The phase trajectory and the graphs of the anguar veocity and the defection ange of stationary parametric osciations at the eft and right boundaries of the interva of instabiity near T = 3T av /2. For sma vaues of the depth of moduation m, we can find approximate anaytica expressions for the ower and the upper boundaries of the third interva that are vaid up to quadratic terms in m : T = 3 2 ( 1 m π ) T av = 3 ( 1 m ) 2 π 3m2 T, m 1. (26) 8 In this approximation, the third interva has the same width (3m /π)t as does the interva of instabiity in the vicinity of the principa resonance. However, this interva is distinguished by greater asymmetry: its centra point is dispaced to the eft of the vaue T = 3 2 T by 9 16 m2 T. 7. Parametric resonances of even orders For sma and moderate square-wave moduation of the penduum ength, parametric resonance of the order n = 2 (one cyce of the moduation during one natura period of osciation) is reativey weak compared to the above considered resonances n = 1 and n = 3. In the case in which n = 2 the abrupt shifts of the bob induce both an increase and a decrease of the energy ony once during each natura period. The growth of osciations occurs ony if the increase in energy at the instant when the bob is shifted up is greater than the decrease in energy when the bob is shifted down. This is possibe ony if the bob is shifted up when the anguar veocity of the penduum is greater in magnitude than it is when the bob is shifted down. For T T av, these conditions can fufi ony because there is a (sma) difference between the natura periods T 1 and T 2 of the penduum, where T 1 = T 1 + m is the period with the bob shifted down and 16

17 T 2 = T 1 m is the period with the bob shifted up. This difference in the natura periods is proportiona to m ϕ ( t ) 1.3 o T =.985 T m = 2% Q = 5 o -D ϕ ( t ) o T Figure 12: The phase trajectory and the graphs of anguar veocity φ(t) and ange φ(t) of osciations corresponding to parametric resonance of the second order n = 2 (T = T av ). The growth of osciations at parametric resonance of the second order is shown in Fig. 12. We note the asymmetric character of osciations at n = 2 resonance: the anguar excursion of the penduum to one side is greater than to the other. In this case, the investment in energy during a period is proportiona to the square of the depth of moduation m, whie in the cases of resonances with n = 1 and n = 3 the investment in energy is proportiona to the first power of m. Therefore, for the same vaue of the damping constant γ (the same quaity factor Q), a consideraby greater depth of moduation is required here to exceed the threshod of parametric excitation. The growth of the ampitude again is restricted by the noninear properties of the penduum. The interva of instabiity in the vicinity of n = 2 resonance (for sma vaues of m ) is consideraby narrower compared to the corresponding intervas of n = 1 and n = 3 resonances. Its width is aso proportiona ony to the square of m. To determine the boundaries of this interva of instabiity, we can consider, as is done above for other resonances, stationary osciations for T T formed by aternating segments of free osciations with the periods T 1 and T 2. The phase trajectory and the graphs of the anguar veocity φ(t) and the ange φ(t) of such stationary periodic osciations for one of the boundaries are shown in Fig. 13. During osciations occurring at the boundary of the instabiity interva, the abrupt increment and decrement in the anguar veocity exacty compensate each other. To describe these stationary osciations with sma ampitude, we can use the foowing expressions for φ(t) and φ(t) in the interva (, T/2) (see Fig. 13): φ 1 (t) = A 1 cos ω 1 (t T/4), φ 1 (t) = A 1 ω 1 sin ω 1 (t T/4), < t < T/2, (27) and during the interva ( T/2, ) φ 2 (t) = A 2 cos ω 2 (t + T/4), φ 2 (t) = A 2 ω 2 sin ω 2 (t + T/4), T/2 < t <. (28) The conditions for joining the graphs at t = are the same as for other resonances, namey, at t = we require φ 1 () = φ 2 (), and the anguar veocity undergoes a sudden change, which 17

18 . T =.9336 T m = 25% -D ϕ ( t ) T Figure 13: Stationary parametric osciations at the eft boundary of the interva of instabiity of the second order n = 2 (near T = T av T ). foows from the conservation of anguar momentum (see Eq. (11)). From these conditions we find the foowing equations for A 1 and A 2 : A 1 cos(ω 1 T/4) = A 2 cos(ω 2 T/4), (29) A 1 (1 + m ) 2 ω 1 sin(ω 1 T/4) = A 2 (1 m ) 2 ω 2 sin(ω 2 T/4). (3) These homogeneous equations (29) (3) for A 1 and A 2 are compatibe ony if the foowing condition is fufied: (1 + m ) 2 ω 1 sin(ω 1 T/4) cos(ω 2 T/4) = (1 m ) 2 ω 2 sin(ω 2 T/4) cos(ω 1 T/4). (31) This is the equation that determines period T of moduation (for a given vaue m of the depth of moduation) which corresponds to the eft boundary of the 2nd interva of parametric instabiity. We transform Eq. (31) to the foowing form which is convenient for a numeric soution by iteration: (q + 1) sin(ω av T/2) = (q 1) sin( ωt/4), (32) where q depends on the depth of moduation m according to Eq. (21). Next we repace T in the argument of the sine on the eft-hand side of Eq. (32) by T av + T. Since ω av T av = 2π, we can write this sine as sin(ω av T/2). Then Eq. (32) becomes: sin(ω av T/2) = q 1 q + 1 sin ω(t av + T ). (33) 4 This equation for T can be soved numericay by iteration with the hep of the above described procedure. Its sef-consistent soutions for various vaues of the moduation depth m give the eft boundary of the n = 2 instabiity interva. After period T for this boundary is cacuated, the initia conditions that provide stationary osciations can be chosen on the basis of Eq. (27), according to which, for an arbitrary initia dispacement φ(), the initia anguar veocity shoud have the vaue φ 1 () = ω 1 tan(ω 1 T/4)φ 1 (). The right boundary of the 2nd interva is given by equation which differs from Eq. (33) by the opposite sign on its right-hand side. Both boundaries are shown on the diagram in Fig. 9 together with intervas of higher even orders, which are obtained with the hep of simiar numeric cacuations. 18