Synchronous Oscillations of Asymmetric Coupled Pendulums

Synchronous Oscillations of Asymmetric Coupled Pendulums John Haine1 Introduction Double-pendulum clocks seem to be attracting increasing interest, but their principles remain obscure. A recent article in Horological Journal2 by David Walter describing his superb clocks in homage to Breguet for example, contains several statements that show that the theory of the double pendulum is little understood. Coupled pendulums have been a particular interest of mine since I became fascinated by the possibilities of making a clock that was insensitive to outside disturbances, and in some articles in HJ3 and on the BHI website I described some researches into double pendulum clocks and how they can be much less sensitive to horizontal vibrations than ones with single pendulums; and why they give little or no benefit with respect to vertical disturbances. Reading the article mentioned above led me to think fu the a out dou le pe dulu s a d ho to add ess the mysteries that see to su ound them – this article presents the results. We are interested in clocks which have two identical pendulums swinging in anti-phase with the same amplitude and in exact synchronism. It has been known since the time of Huygens that two such pendulums will slowly come into and remain in synchronism. If they also swing in the same plane, then the forces they exert on the clock through their suspensions also cancel out, which means that the clock will not be rocked by the pendulums and should need a much less rigid mounting. The Project 150 clock at Upton Hall carries this to extremes by having 3 pendulums swinging at 120° (in space) and free-stands in the centre of a suspended floor 4. “o e DP lo ks B eguet’s a d Walte ’s ei g e a ples ha e thei pe dulums coaxial but in different pa allel pla es, so the ea tio fo es do ’t completely cancel out but twist the clock around a vertical axis – this seems to me to be slightly perverse as it throws away some of the main benefit of the system. On the other hand the clock can be narrower and perhaps more aesthetically pleasing. Earlier DP clocks seemed to be extremely rigid in their construction, but as Walter says this can make it harder to get the pendulums to synchronise. We shall find out why this is. An EM-maintained clock by Bigelow5 suspends the pendulums from a platform which itself is suspended on short tapes, giving much 1 [email protected] Horological Journal, October 2010 3 Horological Journal, July / August 2007. 4 However, as they are insensitive only to horizontal vibrations but the suspended floor will mainly move vertically as people alk a out o it, the ai e efit is that the lo k does ’t eed a igid ou ti g – indeed it is suspended on cables from its stand. 5 HSN 2002-2, May 2002, p2 – 6. 2 2 greater coupling, as well as ensuring the pendulums are co-planar. This is a much superior arrangement in my view. Mysteries The pe dulu s sy chro ise through reso a ce . This is not the case. Resonance is just a property of any resonator, whether it is a single pendulum, a balance, tuning fork, quartz crystal, Tacoma Narrows suspension bridge, or organ pipe. Wikipedia says: In physics, resonance is the tendency of a system to oscillate with larger amplitude at some frequencies than at others. These are known as the system's resonant frequencies. At these frequencies, even small periodic driving forces can produce large amplitude oscillations, because the system stores vibrational energy. Resonances occur when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a pendulum). Resonance is a property of a single resonator – it is ’t a o ept that i itself can explain why two pe dulu s ill s h o ise, o fo e a ple h the a e ha d to ake s h o ise if the oupli g is ’t very great. We need to explain what the limits are on synchronisation and how they depend on factors such as the degree of coupling. The pe dulu s correct each other. I’ ot su e hat this ea s. It see s plausi le that the but what is the mechanism and what are the limits on how much they can be corrected? ight, Placi g a weight to tri the period of o e pe dulu i ediately has the sa e a d i ediate effect on the period of the other. But if you change the rate of one, why do they synchronise at all? And surely changing the natural frequency of one but not the other must introduce some asymmetry into their motions? Why identical pendulums synchronise A pair of identical pendulums with coupling can be represented as shown below. Essentially we just connect their bobs together with a very light a d e flopp sp i g, so that he eithe pe dulu deviates from its rest position it causes a force to be exerted by the spring on the other (as well as itself experiencing an equal and opposite force). The spring force on the first weight acts in the same di e tio as g a it ; ut the fo e e e ted o the se o d eight as a esult of the fi st’s o e e t opposes gravity – so e a e pe t the oupli g to ha e so e affe t o the pe dulu s’ pe iod. 3 It will be objected that actually the coupling is through small amounts of flexing of the frame of the clock because of reaction forces through the suspension points. This is quite true, but it has been shown before that such a coupling5 is exactly equivalent to the floppy spring connecting the bobs – the stiffer the frame of the real clock, the floppier is the equivalent spring. This is the mechanical equivalent of the tee to pi t a sfo i ele t i al i uits. The key to understanding why identical double pendulums synchronise is to recognise that they can oscillate in two modes, one in anti-phase and one in-phase. In each mode, the two pendulums swing with identical amplitude. In the anti-phase mode though, the spring is alternately stretched and compressed, so that the spring force always acts with gravity, slightly increasing the frequency of oscillation. However, CG of the system does ’t o e so it loses u h less e e gy to the environment and has highe Q. This I ha e alled the odd ode in the past through analogy to electrical systems that are analysed in a similar way. In the in-phase o e e ode ho e e the sp i g is eithe st et hed o o p essed, a d the pe dulu s’ ates a e ot affe ted at all o pa ed to thei atu al rate. The CG however does move, and the reaction forces on the clock are doubled so the Q is lower. These t o odes a e the o al , atu al o eige odes of os illatio of the DP s ste . Reall they are more significant than the individual oscillations, as any steady-state oscillatory motion can be expressed as a linear combination of the two normal modes. As the passage from Wikipedia indicates, a resonator, having two modes of energy storage, is a second-order system; two coupled pendulums make a fourth order system and normally analysing it would be much more complicated and generally messy. But by recognising that any steady-state oscillation of coupled pendulums can be represented as the sum of two second-order modes makes life much easier. Essentially, the reason why a pair of identical coupled pendulums synchronise is this: any even mode oscillation will decay much faster than the odd mode because of increased support loss; only odd-mode energy persists, and so the pendulums apparently synchronise in anti-phase. I say apparently because this is what is observed but the underlying reality is that the other mode is damped out. Often, as in the Breguet and Walter clocks, each pendulum is impulsed by a separate movement. But as both Bigelow and Gagnier have shown, it is unnecessary to impulse both pendulums – if only one is impulsed and the coupling is high, then both will swing in anti-phase with virtually the same amplitude. This explanation of synchronisation in DPs is perfectly sufficient to analyse6, for example, why they are much less sensitive to horizontal vibrations; why it is extremely difficult to maintain two pendulums in quadrature; and why, even if you could, there is little reduction in sensitivity to vertical vibrations. It does not however help us to understand the limits to synchronisation. For example: for a given degree of coupling what degree of asymmetry either in length or bob mass is acceptable? or (other than through simulation) understand how changing the length of o e pe dulu to t i the lo k’s ate would affect the other. For that we need a more general analysis and have to go back to the differential equations. 6 Well actually, to simulate. 4 Asymmetric coupled pendulums The diagram below shows the system of two pendulums, of unequal bob weights and lengths coupled by a spring, to be analysed. l2 l1 k m1 m2 x1 x2 The differential equations of motion, for small swings, are: If there are any modes of oscillation for which both pendulums swing sinusoidally at the same frequency, then they could be represented as: and where A is a constant (which might be complex). Substituting these values in the original equations, and dividing right through by , we get: (1) (2) Rearranging both equations to give and equating we get: Now at this point it is worth noting that most of the terms here are effectively natural frequencies of the system. So and are just the squares of the radian resonant frequencies of the two pendulums, 5 assuming no coupling between them; and and are the radian resonant frequencies of the two o s o the oupli g sp i g. Let’s all these f e ue ies respectively. Then with a bit of manipulation the equation becomes: Now, in practice there seems to be no loss of generality if we assume that the two bobs have the same mass m, since the key point seems to be that the two lengths can be different and also the natural u oupled f e ue ies of the pe dulu s. “o e a ite ωc to replace both ωa and ωb and the second bracketed term in the square brackets in the middle of the equation vanishes. Multiplying right through by A/ωc2 we get: If we say that: then Note that, because the coupling spring is very floppy the value of k is very low, and the resonant f e ue of the o s o the sp i g is also e lo . This ea s that the le gths do ’t ha e to e e different for the value of Y, and A, to be rather large. There are two possible values of A that satisfy this equation, corresponding to the + and – signs in front of the square-root. The square-rooted bit is always slightly bigger than Y si e it has added to it, so one of the values is positive and the other negative. When A is positive the pendulums swing in-phase; when negative they swing in anti-phase. Interestingly the product of the two values of A is always -1. If the two pendulums have equal length, Y=0, and A is either +1 or -1. This corresponds to the normal DP ase. To find the frequencies of oscillation corresponding to the two values the expression for A is substituted in(1): 6 For equal lengths, the bracket multiplying evaluates to either 0 or -2; the first case corresponds to the e e ode he e the f e ue e uals its u -coupled value; the second corresponds to the odd mode where the oscillation frequency is , a known result. The analysis above demonstrates that a pair of coupled pendulums, even of different lengths, always have synchronised modes of oscillation, where they swing at the same frequency and either in-phase or in anti-phase; but the pendulums swing with different amplitudes if they have different lengths. So the e is othi g e spe ial a out the usual DP ode that is e ploited i lo ks, other than that the amplitudes are equal. It is easy to see why the modes can exist. Considering the anti-phase mode, the shorter higher-frequency pendulum has to swing in the opposite direction to the longer lower-frequency one and with higher amplitude so that the spring connecting them can supply extra restoring force to the bob of the longer one to make its frequency higher. At the same time the spring is also increasing the restoring force to the bob on the shorter pendulum, increasing its frequency too. To demonstrate synchronism of asymmetric pendulums, I set up a simulation of a system as shown in the VisSim network below. This is an extreme case where one pendulum has a length of 25 cm and the other of 50 cm, and the spring coupling constant is 0.5 N/m. The (lossless) pendulums start out with amplitude 1 and -0.02547 at time zero (the latter being calculated using the formula for A above), and are allowed to swing for 1000 seconds. The plot shows x1 and x2, the latte ei g a plified a fa to 9. the e ip o al of the amplitude), for the last 10 seconds of the run. The pendulums are swinging in exact antiphase even after 1000 cycles, showing that even coupled pendulums of grossly unequal lengths can oscillate in synchrony at the appropriate relative amplitudes. 7 The choice of coupling spring constant above o espo ds to the alue used i Bigelo ’s dou lependulum clock. The graph below shows how the amplitude ratio for synchronous antiphase oscillation, and the synchronous frequency, varies with length of one pendulum over a ±1mm range around equality of lengths. Amplitude ratio is plotted on a log scale, and varies from about 0.85 to 1.15. Pendulum length (m) 0.249 0.2495 0.25 0.2505 0.251 0.998 0.9975 0.997 1 0.9965 0.996 0.1 Antiphase mode frequency (linear scale) Amplitude ratio (log scale) 10 Ampl. Ratio Freq. 0.9955 If the coupling factor is much less, which seems to have been the case with the older very rigid DP clocks, the situation is very different. The next graph plots the ratio for a coupling factor 100 times less. Pendulum length (m) 0.2495 0.25 0.2505 0.251 0.9975 Amplitude ratio (log scale) 100 0.997 10 0.9965 1 0.996 0.9955 0.1 0.995 0.01 Antiphase mode frequency (linear scale) 0.249 Ampl. Ratio Freq. 0.9945 The amplitude ratio grows very quickly as the lengths deviate from equality. The implication is that, unless the pendulum frequencies are almost exactly equal, synchronous equal-amplitude oscillation is 8 not possible. If the two pendulums are impulsed by identical escapements (which seems usually to be the case), adjustment of the pendulums will be extremely critical as the escapements will be expecting equal amplitudes which the pendulums will only deliver if they are very close to being synchronously tuned. This explains the difficulty that Walter has described in making DP clocks work satisfactorily. The other implication is that that it will be difficult to rate the clock if the coupling is small. Whilst it can be done by adjusting just one pendulum, the shape of the f e ue u e a o e sho s that this is only a one-way street – both need to be trimmed to get the right rate and equal amplitudes. In the calculations above I skipped over the situation where the bob masses were unequal. It seems that if the lengths are unequal then the bob masses are of little importance. If however the lengths are equal it turns out that the pendulums will fairly happily swing synchronously, with relative amplitudes that adjust themselves so that the pendulums have equal energy. So if one bob has twice the mass of the other, it will swing with 0.7071 the amplitude. So precise equality of bob mass is not nearly so critical. De-mystification It should e lea that the e is o eed to appeal to so e st a ge o ept of eso a e to e plain why the pendulums synchronise. Two pendulums will always synchronise provided that the ratio of their amplitudes is right, whatever their lengths. Synchronised antiphase equal-length pendulums keep better time because the system has higher Q than a single pendulum, since the support motion and loss is minimised. In addition, as previous work has shown, they are nearly insensitive to horizontal vibrations, and would be completely insensitive were it ot fo i ula e o . I do ’t a t to deal i detail here with what happens if you slightly disturb one pendulum as it has been covered elsewhere3, but essentially a small nudge to one can be resolved as a su of i -phase a d a tiphase udges to oth pe dulu s, each of amplitude equal to half the origi al udge. The i phase udge does ’t affe t ti ekeepi g much (as it is equivalent to any other horizontal vibration); but the antiphase one has exactly the same effect as it would to a single pendulum, so the effect on timekeeping depends in the same way on where in the cycle it occurs, but is only half as great. If you slightly de-tune one pendulum, for example by placing a weight on the weight tray, the common oscillation frequency of the two will slowly change, and the relative amplitude of the two pendulums will also slowly change to a new value which depends on the difference in effective lengths. As long as the movements can accommodate the amplitude difference, synchronisation will persist. The change in amplitude can be minimised by having a high degree of coupling between the pendulums, which will also facilitate adjustment. Conclusions 1. The phenomenon of coupled pendulums synchronising is very easily explained through the existence of two normal modes of oscillation, in-phase and anti-phase. The in-phase mode has 9 lower Q because its support loss is much bigger, and dies away more quickly than the anti-phase mode. 2. Coupled pendulums of different length also have in-phase and anti-phase normal modes, but they are of different amplitude. The weaker the coupling, the more different the amplitudes are, and amplitude ratios of 10:1 or greater for minimal differences of length will be found for weakly coupled pendulums. 3. To minimise the amplitude difference and make synchronisation easy, the coupling should be large. An arrangement such as that adopted by Bigelow where the suspensions are mounted on a plate itself suspended from the clock frame seems ideal. 10