LECTURE 10. The world of pendula

Transcription

1 LECTURE 0 The word of pendua For the next few ectures we are going to ook at the word of the pane penduum (Figure 0.). In a previous probem set we showed that we coud use the Euer- Lagrange method to derive the equations of motion for the pane penduum. Figure 0.: The pane penduum. In particuar, since the kinetic energy (T) and the potentia energy (U) are, respectivey, = T m θ U = mg( cos θ ) (0.) where m is the mass, is the ength, and θ is the ange, the equation of motion is where g θ θ θ ω θ + sin = ( t) + sin ( t) = 0 (0.) ω = g Equation (0.) is a noninear ordinary differentia equation. It is non-inear because of the term sin θ, i.e θ θ θ sinθ θ + + 3! 5! 7! The critica point of (0.) is the soution to the equation

2 sinθ = 0 and hence the soutions are θ θ = 0 = π First we consider the critica point at θ = 0. The inearized equation is () t + ω θ = 0 (0.3) θ The characteristic equation λ = ω and λ =± iω Thus know that the critica point is a center and that the soution is xt () = Asin( ωt+ δ) (0.4) Thus, as is we known, when there is a dispacement from θ = 0 the dispacement shows sinusoida osciations. When θ = π, we have the inverted penduum. The inearized equation becomes θ() t ω θ() t = 0 (0.5) for which we find λ =± ω Thus this critica point is unstabe. It is in fact a sadde point. Let us ook at the inearized version of the penduum a itte coser to make sure that we understand two things:

3 Phase shifts. What is δ?. How good is the estimate of the period? To understand the significance of the parameter, δ, we consider Figure 0.. The ine SS represents the x component of the motion of the penduum bob, and A is the imum vaue of x measured from the mid-point O of SS. Construct the circe of radius A, tangent to SS at x=0. At any time the position x of the partice can be represented by the projection, on SS, is the distance from the ine OO of a point traveing in a circuar path, i.e. A sin γ. Suppose we et γ = ω t 0, that is, the ange γ is increasing uniformy at a rate ω 0. Thus x = Asinω t 0 (0.6) is represented by the uniform motion of the point aong the circuar path, the anguar veocity of the point being. ω 0 Figure 0.: A spring approximation for the pane penduum. It is cear from this construction that A is the ampitude, or imum vaue, of the dispacement x. The anguar frequency, ω 0, is reated to the actua frequency, f, by the reation ω0 = πf 3

4 The quantity ω t in (0.6), or ω + δ 0 in (0.4), is caed the phase ange, or simpy 0 t the phase of the motion. The meaning of the phase constant, δ, in (0.4) is now cear: it is the ange at zero time in the circe of reference. Soutions with different initia phase anges are shifted by different amounts with respect to the soution obtained when δ = 0. The period The initia reason that it was important to understand the penduum was because of its reationship to time keeping devices, such as grandfather cocks. Interestingy we do not have to inearize the equation for the penduum in order to cacuate its period. Let s see how this is done. Define θ = θ to be the highest point of the motion. Then from (0.) we have T = 0 U = mg( cos θ ) Since for the inearized penduum is a conservative dynamica system (i.e. the critica point is a center), we have Looking at some math tabes we see ( ) E = m g cosθ cosθ = sin ( θ / ) and hence sin θ E = m g Now T = E U and hence m () t mg sin( /) sin( ()/) t θ = θ θ or 4

5 dθ () t g = sin ( θ / ) sin ( / ) dt θ (0.7) We can integrate (0.7) to determine the period, T, of the penduum, i.e. dt = sin ( θ / ) sin ( θ / ) d g θ (0.8) Now there is a itte trick invoved in the evauation of (0.8). Since the motion of the penduum is symmetric, integrating from 0 to θ = θ gives us one-quarter of the period. In other words θ T = sin ( / ) sin ( / ) g d θ (0.9) θ θ 0 Equation (0.9) can be re-written as 0 where sin( θ / ) z = sin( θ / ) k = sin( θ / ) T = 4 ( )( k z ) g z dz (0.0) dz = kz k This integra is caed an eiptic integra of the first kind. Tabes that give numerica vaues of the integra exist. If sin( θ / ) = k <<, then it can be shown that dθ T π θ g + + θ Phase pane portraits Another way to represent the soution of this equation is to draw the phase portrait. The phase portrait can be represented in a number of ways and the 5

6 choice of which way to do it depends on the probem. For mechanica systems, such as the penduum, we want to pot something reated to the dispacement versus something reated to the veocity. Sometimes we can be a itte cever. We notice that (0.7) gives us a reation between θ and θ. For sma θ and we have θ g θ + θ θ This is the equation for a circe. Thus if we choose the coordinates for the phase pane as θ and θ / g/, then the phase pane trajectories near the critica point θ = 0 are circes (as expected since we have a center). Figure 0.3 shows the phase pane for the pane penduum. Figure 0.3: Phase pane portrait for the pane penduum. The doube penduum One of the most interesting penduums for pay and toys is the doube penduum. Euer and Bernoui introduced this penduum to the scientific word in the 730s. The doube penduum (Figure 0.4) consists of one penduum suspended freey 6

7 Figure 0.4: A schematic representation of a doube penduum from another, but with both constrained to swing in the same vertica pane. In the word of pay, the doube penduum arises every time we try to hit a ba with a racket or a cub. My favorite exampe, of course, is the gof swing (Figures 0.5 and 0.6). Look at the reative positions of the eft arm and the cub during the swing: the hinge of this doube penduum is the wrist; one penduum the arm, the other the cub. Another exampe is the baseba swing of a good hitter (Figure 0.7). Figure 0.5: Gof swing of Se Ri Pak (LPGA). Figure 0.6: Gof swing of Nick Price. As usua we obtain the equations of motion using the Lagrangian method ( ) ( ) U = ( cos φ) mg + cosφ + cosφ mg 7

8 T = ( m + m) φ + m φ + m φφ cos( φ φ) where the meanings of the parameters are shown in Figure 0.4. Figure 0.7: Baseba swing of Tony Gwynn (San Diego Padres). For simpicity we take = = and m m = m m + we obtain (Acheson, 997) g φ+ m φcos( φ φ) m( φ) sin( φ φ) + sinφ = 0 g φ + φcos( φ φ) + ( φ) sin( φ φ) + sinφ = 0 (0.) These equations are obviousy quite compicated. However, we can ook at the sma ange case, i.e. sin φ φ and cos φ to get 8

9 g φ+ m φ + φ = 0 g φ + φ+ φ = 0 (0.) Now take (Acheson, 997) φ = Acosωt φ = Bcosωt and we see that the doube penduum has two natura frequencies of osciation, ω,, given by where the corresponding motions are g/ ω = ± m φ φ =± m In the ow frequency mode (i.e. the upper sign in the above two equations), the two penduums swing in the same direction at any given moment, whie in the high-frequency mode they swing in opposite directions. In the case when the two masses are equa ( m = 0.5 ) the frequency of the fast mode is.5 times that of the sow mode. If m/ mis arge then m is amost equa to. In this case the two natura frequencies are much more widey separated. In this imit, the sow mode has φ and φ amost equa, and the penduums swing ike a singe penduum of ength. On the other hand, the faster mode has φ and φ amost equa, but opposite. No the upper mass osciates to and fro whie the ower, much greater mass remains amost stationary. Once the ange become arger, it is necessary to anayze the behavior of the doube penduum equations using computer simuations. In the eary 990s investigations used motion anaysis methods to study the motion of the arge ange doube penduum (Levien and Tan, 993; Shinbrot, et a, 99). There interest was on the behavior of the penduum in the arge ange imit (ike the gof swings in Figures 0.5 and 0.6) for different choices of the initia conditions. The found that even in two initia conditions were chosen to be very cose the trajectories of the doube penduum rapidy diverged (Figure 0.8). 9

10 Figure 0.8: Divergence of the two anges as a function of time for two trias with different initia conditions. Figure from Levien and Tan, 993. To estimate how rapidy the two trajectories diverged they measured the separation between them as a function of time (Figure 0.9) and assumed that Separ ation( t) e λt Figure 0.0: Measurement of the separation between the trajectories for two different choices of the initia conditions (a tria). A tota of 4 trias are shown together with mean (soid ine). Figure reproduced from Shinbrot, et a, 99. Surprisingy they observed that λ > 0, i.e. Se paration() t e 7.5t 0

11 This means that the trajectories rapidy diverged from each other (What happens in a stabe inear dynamica system?). This type of observation demonstrates an important property of a dynamica system that exhibits chaos, i.e. sensitivity to initia conditions. Such dynamica systems are truy chaotic since it is not possibe to repeat an experiment and get the same resuts (or even cose to it). Thus we have reached the surprising concusion that two of America s favorite pastimes, namey hitting a baseba and hitting a gof ba, are inherenty uncontroabe task. How do they do it? References Acheson D (997). From Cacuus to Chaos: An Introduction to Dynamics. Oxford University Press, Toronto, Chapter. Boyce WE and DiPrima RC (969). Eementary Differentia Equations and Boundary Vaue Probems. John Wiey & Sons, Toronto, pp Gwynn T and Rosentha J (99). Tony Gwynn s Tota Baseba Payer. St. Martin s Griffin, New York. Levien RB and Tan SM (993). Doube penduum: An experiment in chaos. Amer. J. Physics 6: Marion JB (970). Cassica Dynamics of Partices and Systems, Second Edition. Academic Press, New York, Chapter 5. Shinbrot T, Grebogi C, Wisdom J and Yorke JA (99). Chaos in a doube penduum. Amer. J. Physics 60: