From Huygens' odd sympathy to the energyFrom Huygens odd sympathy to the energy extraction from the sea waves

Tomasz Kapitaniak

Technical University of Lodz

Co workersCo-workers

• Przemysław Perlikowski• Krzysztof Czołczynski• Andrzej Stefański• Andrzej Stefański• Marcin Kapitaniak

Our publicationsOur publications • K. C., P. P., A. S. and T. K.: "Clustering and synchronization of n Huygens'

clocks", Physica A 388 (2009)• K. C., P. P., A. S. and T. K.: "Clustering of Huygens' clocks", Progress of, , g yg , g

Theoretical Physics Vol. 122, No. 4 , (2009)• K. C., P. P., A. S. and T. K.: "Clustering of non-identical clocks", Progress

of Theoretical Physics Vol 125 No 3 (2011)of Theoretical Physics Vol.125, No. 3 (2011)• K.C., P.P., A.S. and T.K: ‘Why two clocks synchronize: Energy balance of

synchronized clocks”, Chaos, 21, 023129 (2011)• K. C., P. P., A. S. and T. K.: "Huygens' odd sympathy experiment

revisited", International Journal Bifurcation and Chaos (2011) – in press

Salisbury Cathedral 1386Salisbury Cathedral - 1386

John Constable ca 1825John Constable – ca. 1825

Pendulum clock conceived by Galileo Galilei around 1637Pendulum clock conceived by Galileo Galilei around 1637

Christiaan HuygensChristiaan Huygens

by Bernard Vaillant, Museum Hofwijck, Voorburg

The first pendulum clock, invented by Christiaan Huygens in 1656

The pendulum clock was invented in 1656 by DutchThe pendulum clock was invented in 1656 by Dutchscientist Christiaan Huygens, and patented the followingyear. Huygens was inspired by investigations of

d l b G lil G lil i b i i d 1602pendulums by Galileo Galilei beginning around 1602.Galileo discovered the key property that makespendulums useful timekeepers: isochronism, which

th t th i d f i f d l imeans that the period of swing of a pendulum isapproximately the same for different sized swings.Galileo had the idea for a pendulum clock in 1637, partly

d b hi i 1649 b i h li d fi i hconstructed by his son in 1649, but neither lived to finishit.The introduction of the pendulum, the first harmonicoscillator used in timekeeping, increased the accuracy ofp g yclocks enormously, from about 15 minutes per day to 15seconds per day leading to their rapid spread as existing'verge and foliot' clocks were retrofitted with pendulums.g p

Sketch of basic componentsSketch of basic components of a pendulum clock with anchor escapement. (From Martinek & Rehor 1996.) )

Parts:- pendulum; p- anchor escapement arms; - escape wheel; - gear train, - gravity-driven weight.

Edward East winged lantern clockEdward East winged lantern clock

Th l l k d t th i t h dThese early clocks, due to their verge escapements, hadwide pendulum swings of up to 100°. In his 1673analysis of pendulums, Horologium Oscillatorium,y p , g ,Huygens showed that wide swings made the penduluminaccurate, causing its period, and thus the rate of theclock to vary with unavoidable variations in the drivingclock, to vary with unavoidable variations in the drivingforce provided by the movement. Clockmakers'realization that only pendulums with small swings of afew degrees are isochronous motivated the invention ofthe anchor escapement around 1670, which reduced thependulum's swing to 4°-6°pendulum s swing to 4 6

Verge escapementVerge escapement

Verge escapement showing (c) crown wheel, g p g ( )(v) verge rod, (p,q) pallets.

It was used in the first mechanical clocks and was originally controlled byfoliot, a horizontal bar with weights at either end. The escapement consistsof an escape heel shaped some hat like a cro n ith pointed teethof an escape wheel shaped somewhat like a crown, with pointed teethsticking axially out of the side, oriented vertically. In front of the crownwheel is a vertical shaft, the verge, attached to the foliot at top, which

i l l ( ll ) i ki lik fl f fl lcarries two metal plates (pallets) sticking out like flags from a flag pole,oriented about ninety degrees apart, so only one engages the crown wheelteeth at a time. As the wheel turns, one tooth pushes against the upperpallet, rotating the verge and the attached foliot. As the tooth pushes pastthe upper pallet, the lower pallet swings into the path of the teeth on theother side of the wheel. A tooth catches on the lower pallet, rotating thep , gverge back the other way, and the cycle repeats. A disadvantage of theescapement was that each time a tooth lands on a pallet, the momentum ofthe foliot pushes the crown wheel backwards a short distance before thethe foliot pushes the crown wheel backwards a short distance before theforce of the wheel reverses the motion.

Anchor escapement1660 by Robert Hooke1660 by Robert Hooke

Reuleaux-Voight model X-3

Escapement mechanismEscapement mechanism

First step 0<ϕi<γN (i=1,2) then MDi=MNi and when ϕ1<0 then MDi=0. p ϕi γN ( ) Di Ni ϕ1 Di

Second stage -γN <ϕ1<0 MDi=-MNi and for ϕ1>0 MDi=0.

Works on clock dynamicWorks on clock dynamic• Huygens, C. Letters to Father (1665)

• Blekham, I.I., “Synchronization in Science andT h l ” (ASME N Y k 1988)Technology,” (ASME, New York,1988).

• Bennet, M., et al. “Huygens’s clocks,” Proc. Roy.S L d A 458 563 579 (2002)Soc. London, A 458, 563-579 (2002).

• Roup, A. et al.: “Limit cycle analysis of the vergeand foliot clock escapement using impulsiveand foliot clock escapement using impulsivedifferential equations and Poincare maps,” Int. J.Control, 76, 1685-1698 (2003).

• Moon, F. and Stiefel, P., “Coexisting chaotic andperiodic dynamics in clock escapements,” Phil.Trans. R. Soc. A, 364, 2539 (2006).Trans. R. Soc. A, 364, 2539 (2006).

Metronome clockMetronome - clock

Shortly after The Royal Society’s founding in 1660, Christiaan Huygens, in partnership with the Society, set out to solve the outstanding technological challenge of the day: the longitude problem, i.e. - finding a robust, accurate method of determining longitude for maritimep g g gnavigation (Yoder 1990). Huygens had invented the pendulum clock in 1657 (Burke 1978) and, subsequently, had demonstrated mathematically that a pendulum would follow an isochronous path, independent of amplitude, if cycloidal-shaped plates were used to confine the pendulum suspension (Yoder 1990). Huygens believed that cycloidal pendulum clocks, suitably modified to withstand the rigours of sea travel, could provide timing of sufficient accuracy to determinelongitude reliably. Maritime pendulum clocks were constructed by Huygens in collaborationwith one of the original fellows of The Royal Society, Alexander Bruce, 2nd Earl of Kincardine. Over the course of three years (1662-1665) Bruce and the Society supervised sea trials of the clocks. Meanwhile, Huygens, remaining in The Hague, continually corresponded with the S i t th h Si R b t M b th t i i b t th t f th t i l d tSociety through Sir Robert Moray, both to inquire about the outcome of the sea trials and to describe the ongoing efforts Huygens was making to perfect the design of maritime clocks. On 1 March 1665, Moray read to the Society a letter from Huygens, dated 27 February 1665, reportingof (Birch 1756):of (Birch 1756):an odd kind of sympathy perceived by him in these watches [two maritime clocks] suspended by the side of each other.] p y

Huygens’s study of two clocks operating simultaneously arose from the practical requirement of redundancy for maritime clocks: if one clock stopped (or had to be cleaned), then the other could be used to provide timekeeping (Huygens 1669). In a contemporaneous letter to his father, p p g ( yg ) pHuygens further described his observations made while confined to his rooms by a brief illness. Huygens found that the pendulum clocks swung in exactly the same frequency and 180o out ofphase (Huygens 1950a; b). When he disturbed one pendulum, the anti-phase state was restored within half an hour and remained indefinitely. Motivated by the belief that synchronization could be used to keep sea clocks in precise agreement (Yoder 1990), Huygens carried out a series of experiments in an efort to understand the phenomenon. He found that synchronization did not occur when the clocks were removed at a distance or oscillated in mutually perpendicular planes. Huygens deduced that the crucial interaction came from very small movements of the common frame supporting the two clocks. He also provided a physical explanation for how the frame

ti t th ti h ti b t th h hi t hi t l li it d himotion set up the anti-phase motion, but though his prowess was great his tools were limited: his discovery of synchronization occurred in the same year when young Isaac Newton removed to his country home to escape the Black Plague, and begin the work that eventually led to his Principia published some 20 years later The Royal Society viewed Huygens’s explanation ofPrincipia, published some 20 years later. The Royal Society viewed Huygens s explanation of synchronization as a setback for using pendulum clocks to determine longitude at sea (Birch 1756). Occasion was taken here by some of the members to doubt the exactness of the motion of these watches at sea since so slight and almost insensible motion was able to cause an alterationthese watches at sea, since so slight and almost insensible motion was able to cause an alteration in their going. Ultimately, the innovation of the pendulum clock did not solve the longitude problem (Britten 1973). However, Huygens’s synchronization observations have served to inspire study of sympathetic rhythms of interacting nonlinear oscillators in many areas of science.

Huygens experimentHuygens experiment

The pendulum in each clock measured ca. 9 in. in length, corresponding to an oscillation period of ca. 1 s. Each pendulum weighed 1/2 lb. and regulated the clock through a verge escapement, which required each pendulum to execute large angular displacement amplitudes of ca. 20o or more from vertical for the clock to function for a detailed description of the verge escapement). The amplitude dependence of the period in these clocks was typically corrected by use of cycloidal-shaped boundaries to

fi th i (H 1986) E h d l l kconfine the suspension (Huygens 1986). Each pendulum clock was enclosed in a 4 ft{ long case; a weight of ca. 100 lb was placed at the bottom of each case (to keep the clock oriented aboard a ship.)

An original drawing of Huygens illustrating his experimentswith pendulum clocks

M B l (2002)M. Bennett et al. (2002) Proc. R. Soc. Lond. A (2002) 458, 563-579

Van der Pol oscillator

( )2( ) 012 =+−+ ykyydym yy &&&

Two coupled clocks

,sincos2iDiiiiiiiiii mMglmmclxmlm =+++ ϕϕϕϕ ϕ &&&&&

( ) 0sincos2

22

+++⎟⎞

⎜⎛

+ ∑∑ lmxkxcxmM ϕϕϕϕ &&&&&& ( ) ,0sincos11

=−+++⎟⎠

⎜⎝

+ ∑∑== i

iiiiixxi

i lmxkxcxmM ϕϕϕϕ

Parameters

BeamPendulum

m1=1.0 [kg], M=10.0 [kg],

BeamPendulum

m2=controling parameter, l=g/4π2=0.2485 [m],

0 0083× [N ]

cx=1.53 [Ns/m], kx=3.94 [N/m],

cϕ1=0.0083×m1 [Ns], cϕ2=0.0083×m2 [Ns], Escapement mechanism

γN=5.0o

MN1=0.075×m1 [Nm] MN2=0.075×m2 [Nm]

Assuming the small amplitudes of the pendulums’ oscillations (typically for pendulum clocks Φ<2π/36 and for clocks with long pendulums Φ is even smaller one can describe h d l ’ i i h f ll i fthe pendulum’s motion in the following form:

( ),sin iii t βαϕ +Φ=( )( ).sin

,cos2

iii

iii

t

t

βααϕ

βααϕ

+Φ−=

+Φ=

&&

&

( )22

∑∑ ⎟⎞

⎜⎛

Substituting above eqs to equation of motion:

( ).)sin()(cos)sin(1

2322

1∑∑==

++Φ++Φ=++⎟⎠

⎞⎜⎝

⎛+

iiiiiiiixx

ii ttlmtlmxkxcxmM βαβααβαα&&&

,3sin25.0sin25.0sincos2 αααα +=

2

∑

Considering we get:

,25.0),25.0(, 323

321

1iiiiiii

ii lmFlmFmMU Φ=Φ+Φ=+= ∑

=

αα

( )∑2

)33i ()i ( FFkU ββ&&& ( )∑=

+++=++1

31 .)33sin()sin(i

iiiixx tFtFxkxcxU βαβα&&&

Assuming the small value of the damping coefficient cx previous equation can be rewritten in the following form

( )∑ +++=2

31 )33sin()sin( iiii tXtXx βαβα( )∑=

+++1

31 ,)33sin()sin(i

iiii tXtXx βαβα

)250( 32lF α Φ+Φ

where:

25.0

,)25.0(

323

2

3

21

1

lmFX

Uklm

UkFX

iii

x

iii

x

ii

α

αα

α

Φ

−Φ+Φ

=−

=

.99 22

33 UkUk

Xx

ii

x

ii αα −

=−

=

implies the following acceleration of the beam M

( )∑=

+++=2

131 ,)33sin()sin(

iiiii tAtAx βαβα&&

)250( 34lm α Φ+Φ

p g

25.0

,)25.0(

34

21

lmA

UklmA

ii

x

iiii

α

αα

Φ

−Φ+Φ

−=

.9 23 Uk

Ax

iii α−

−=

Energy balanceThe work done by the escapement mechanism during tone period of pendulum’s oscillations can be expressed as

.2200

NNiiNi

T

iDiDRIV

i MdMdtMWN

γϕϕγ

=== ∫∫ &

E di i t d i th d i i b

.)(cos 22222ii

T

iii

T

iiDAMP

i cdttcdtcW Φ=+Φ== ∫∫ ϕϕϕ παβααϕ&

Energy dissipated in the damper is given by

00∫∫

The energy transferred from the i-th pendulum to the beam M (pendulum looses t f it t f th b t ill t ) h

.cos0∫=T

iiiSYN

i dtlxmW ϕϕ &&&

part of its energy to force the beam to oscillate), so we have:

Energy balance for the i-th pendulum

.SYNi

DAMPi

DRIVi WWW +=

Energy balance during the anti-phase synchronization (identical pendulums)

In the case of the anti-phase synchronization of two identical pendulums the beam M is in resttwo identical pendulums the beam M is in rest (Czolczynski et al., 2009(a,b)). There is no energy transfer between pendulums

.DAMPi

DRIVi WW =

22 iiNNi cM Φ= ϕπαγ

.2

i

NNii c

M

ϕπαγ

=Φ

Energy balance - non-identical pendulums

Setting β1=0.0 (one of the phase angles can be arbitrarily chosen) and linearizing pendulum motion

,sin 22221

421

1SYNSYN Wm

UklmW =Φ

Φ−= β

απα

.sin 21122

422

2SYN

x

SYN

x

WmUk

lmW

Uk

−=Φ−

Φ=

−

βαπα

α

x

Both synchronization energies are equal and the energy balance of both d l h f ll i fpendulums have following form:

DAMPSYNDRIV

SYNDAMPDRIV

WWW

WWW

22

11

=+

+=

Energy balance - non-identical pendulums

Finally we get:

,sin2 22221

4212

111 βπαπαγ ϕ ΦΦ

−Φ= mUk

lmcM NN0.9

Φ

y g

,

422

2222111

πα

βα

γ ϕ

Φ

−

lm

UkxNN

0.6

0.7

0.8Φi

,sin2 2112222

222 βαπαπαγ ϕ Φ

−Φ

+Φ= mUk

lmcMx

NN

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 70

0.1

0.2

β2

We get 2 equations, as a parameter we take sin(β),we plot angles as a function of parameter sin(β)p g p (β)and then we numerically find the phase shift β .

Energy balance of pendulums; (a) anti-phase synchronization of identical pendulums – there is no transfer of energy between pendulums, (b) phase p gy p , ( ) psynchronization of the pendulums with different masses: m1=1.0 [kg] and

m2=0.289 [kg] and Φ1≈γN=5.0o – pendulum 1 transfer energy to the pendulum 2 via the beam M.pendulum 2 via the beam M.

Analytically we can find condition of in-phase andAnalytically we can find condition of in-phase andphase synchronization for both cases: identical andnon-identical masses of pendulumsnon-identical masses of pendulums.

But this not the end of the story…

Parameters

BeamPendulum

m1=1.0 [kg], M=10.0 [kg],

BeamPendulum

m2=controling parameter, l=g/4π2=0.2485 [m],

0 0083× [N ]

cx=1.53 [Ns/m], kx=3.94 [N/m],

cϕ1=0.0083×m1 [Ns], cϕ2=0.0083×m2 [Ns], Escapement mechanism

γN=5.0o

MN1=0.075×m1 [Nm] MN2=0.075×m2 [Nm]

Identical clocks

Nonidentical clocks

Decreasing mass of second pendulum

Increasing mass of second pendulum

Comparison with analytics results

How harmonic are clocks?

m =1 0 [kg];m1=1.0 [kg];m2=11.0 [kg]

Long period synchronizationLong period synchronization

ϕ

Long period synchronization

Escapement mechanism – are the parametersimportant ?

γNi − maximum angle below which escapement mechanism generate moment

MNi - constant moment of escapement mechanism

Assumption:

ConstMdM NiNiiDi ==∫ γϕ

Basin of attraction fordifferent sets of escapement, d e e t sets o escape e tmechanism parameters

I iti l diti

,0.0)0(,0)0( == xx &

cossin 0000 iiii βαϕβϕ Φ=Φ= &

Initial conditions:

.cos,sin 0000 iiii βαϕβϕ ΦΦ

Parameters: 4 8 o ( )γN =4.8 o (a);

γN =4.9o (b);γN =5.0o (c);γ =5 05o (d);γN =5.05o (d); γN =5.1o (e);γN =5.2o (f).

Poincare maps for chosen attractors

4 8o T 23 ( )γN =4.8o, T=23 (a),γN =4.9o, T=6 (b), γN =4.9o, T=11 (c), γ =5 0o T=11 (d);γN =5.0o, T=11 (d);

Poincare maps for chosen attractors

5 0o h ( )γN =5.0o, chaos (e),γN =5.1o, T=59 (f), γN =5.2o, T=13 (g), γ =5 2o T=35 (h);γN =5.2o, T=35 (h);

Rare attractorsRare attractors• Blekhman, I., and Kuznetsova, L. "Rare events - rare attractors; formalization andexamples", Vibromechanika, Journal of Vibroengineering, 10, 418-420 (2008)

Z k h k M S h ki I d Y i j V "R i d i li• Zakrzhevsky, M., Schukin, I. and Yevstignejev V. "Rare attractors in driven nonlinearsystems with several degree of freedom", Sci. Proc. Riga Tech. Uni. 6(24), 79-93, (2007)

• Chudzik, A., P. P., A. S. and T. K.: "Multistability and rare attractors in van der Pol -Duffing oscillator", International Journal Bifurcation and Chaos (2011), accepted forpublication

Definition – open problemOur proposal

As an example of the system whichpossesses multistability and rarepossesses multistability and rareattractors we consider an externallyexited van der Pol-Duffing oscillator

where: α=0.2, F=1.0, ω=0.955.For simplicity set of accessibleparameters is following

Sets of possible initial conditions:

• Long Period Synchronization• MultistabilityMultistability• Sensitivity on escapement mechanism

parameters• Rare attractorse c o s• Chaos

More Clocks ?More Clocks ?

Possible configurationsPossible configurations

• the complete synchronization in which all pendula behave identically, p y,

• pendula create three or five clusters of synchronized pendulasynchronized pendula,

• anti-phase synchronization in pairs (for even n and identical clocks),

• uncorrelated behavior of all pendula• uncorrelated behavior of all pendula

Energy extraction from the sea waves

Thank you !Thank you !