AACIMP 2011 Summer School. Science of Global Challenges Stream. Lecture by Tomasz Kapitaniak.

2. From Huygens odd sympathy to the energy
Huygens'
extraction from the sea waves
Tomasz Kapitaniak
Technical University of Lodz

3. Co-workers
Co workers
• Przemysław Perlikowski
• Krzysztof Czołczynski
• Andrzej Stefański
• Marcin Kapitaniak

4. Our 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)
Vol.125, No.
• 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

5. Salisbury Cathedral - 1386

6. John Constable – ca 1825
ca.

8. Pendulum clock conceived by Galileo Galilei around 1637

9. Christiaan Huygens
by Bernard Vaillant, Museum Hofwijck, Voorburg

10. The first pendulum clock, invented by Christiaan Huygens
in 1656

11. The pendulum clock was invented in 1656 by Dutch
scientist Christiaan Huygens, and patented the following
year. Huygens was inspired by investigations of
pendulums b G lil
d l by Galileo G lil i b i i
Galilei beginning around 1602
d 1602.
Galileo discovered the key property that makes
pendulums useful timekeepers: isochronism, which
means th t th period of swing of a pendulum i
that the i d f i f d l is
approximately the same for different sized swings.
Galileo had the idea for a pendulum clock in 1637, partly
constructed b hi son i 1649 b neither li d to fi i h
d by his in 1649, but i h lived finish
it.The introduction of the pendulum, the first harmonic
oscillator used in timekeeping, increased the accuracy of
p g y
clocks enormously, from about 15 minutes per day to 15
seconds per day leading to their rapid spread as existing
'verge and foliot' clocks were retrofitted with pendulums.
g p

12. Sketch 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.

13. Edward East winged lantern clock

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

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

16. It was used in the first mechanical clocks and was originally controlled by
foliot, a horizontal bar with weights at either end. The escapement consists
of an escape wheel shaped some hat like a cro n with pointed teeth
heel somewhat crown, ith
sticking axially out of the side, oriented vertically. In front of the crown
wheel is a vertical shaft, the verge, attached to the foliot at top, which
carries two metal plates ( ll ) sticking out lik fl
i l l (pallets) i ki like flags f
from a fl pole,
flag l
oriented about ninety degrees apart, so only one engages the crown wheel
teeth at a time. As the wheel turns, one tooth pushes against the upper
pallet, rotating the verge and the attached foliot. As the tooth pushes past
the upper pallet, the lower pallet swings into the path of the teeth on the
other side of the wheel. A tooth catches on the lower pallet, rotating the
p , g
verge back the other way, and the cycle repeats. A disadvantage of the
escapement was that each time a tooth lands on a pallet, the momentum of
the foliot pushes the crown wheel backwards a short distance before the
force of the wheel reverses the motion.

17. Anchor escapement
1660 by Robert Hooke

19. Reuleaux-Voight model X-3

20. Escapement mechanism

21. First step 0<ϕi<γN (i=1,2) then MDi=MNi and when ϕ1<0 then MDi=0.
p ( )
Second stage -γN <ϕ1<0 MDi=-MNi and for ϕ1>0 MDi=0.

22. Works on clock dynamic
• Huygens, C. Letters to Father (1665)
• Blekham, I.I., “Synchronization in Science and
Technology,” (ASME N Y k 1988)
T h l ” (ASME, New York,1988).
• Bennet, M., et al. “Huygens’s clocks,” Proc. Roy.
Soc. London, 458, 563-579 (2002).
S L d A 458 563 579 (2002)
• Roup, A. et al.: “Limit cycle analysis of the verge
and foliot clock escapement using impulsive
differential equations and Poincare maps,” Int. J.
Control, 76, 1685-1698 (2003).
• Moon, F. and Stiefel, P., “Coexisting chaotic and
periodic dynamics in clock escapements,” Phil.
Trans. R. Soc. A, 364, 2539 (2006).

23. Metronome - clock

24. 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
p
problem, i.e. - finding a robust, accurate method of determining longitude for maritime
g g g
navigation (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 determine
longitude reliably. Maritime pendulum clocks were constructed by Huygens in collaboration
with 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
Society th
S i t through Sir Robert Moray, both t inquire about the outcome of the sea trials and to
h Si R b t M b th to i i b t th t f th ti l dt
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, reporting
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

25. 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 ( yg
p p g (Huygens 1669). In a contemporaneous letter to his father,
) p
Huygens 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 of
phase (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
motion set up the anti-phase motion, but though his prowess was great his tools were limited: his
ti t th ti h ti b t th h hi t hi t l li it d hi
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 of
Principia, later. Huygens’s
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 alteration
sea
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.

26. Huygens 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
confine th suspension (Huygens 1986). E h pendulum clock was
fi the i (H 1986) Each d l l k
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.)

27. An original drawing of Huygens illustrating his experiments
with pendulum clocks

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

29. Van der Pol oscillator
( )
&& + d y y 2 − 1 y + k y y = 0
my &

34. Two coupled clocks
mi l 2ϕ i + mi &&l cos ϕ i + cϕi miϕ i + mi gl sin ϕ i = M Di mi ,
&& x &
⎛ ⎞
( )
2 2
⎜ M + ∑ mi ⎟ && + cx x + k x x + ∑ mi l ϕi cos ϕi − ϕi2 sin ϕi = 0,
x & && &
⎝ i =1 ⎠ i =1

35. Parameters
Pendulum Beam
m1=1.0 [kg], M=10.0 [kg],
m2=controling parameter, cx=1.53 [Ns/m],
l=g/4π2=0.2485 [m], kx=3.94 [N/m],
cϕ1=0.0083×m1 [N ]
0 0083× [Ns],
cϕ2=0.0083×m2 [Ns], Escapement mechanism
γN=5.0o
MN1=0.075×m1 [Nm]
MN2=0.075×m2 [Nm]

36. 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
the pendulum’s motion in the following form:
h d l ’ i i h f ll i f
ϕ i = Φ i sin (αt + β i ),
ϕ i = αΦ i cos(αt + β i ),
&
ϕ i = −α 2 Φ i sin (αt + β i ).
&&
Substituting above eqs to equation of motion:
⎛ ⎞
( )
2 2
⎜ M + ∑ mi ⎟ && + c x x + k x x = ∑ mi lα 2 Φ i sin(αt + β i ) + mi lα 2 Φ i3 cos 2 (αt + β i ) sin(αt + β i ) .
x &
⎝ i =1 ⎠ i =1
Considering cos 2 α sin α = 0.25 sin α + 0.25 sin 3α , we get:
2
U = M + ∑ mi , F1i = mi lα 2 (Φ i + 0.25Φ i3 ), F3i = 0.25mi lα 2 Φ i3 ,
i =1
2
U&& + c x x + k x x = ∑ (F1i sin(αt + β i ) + F3i sin(3αt + 3β i ) ).
x & i ( i (
i =1

38. Assuming the small value of the damping coefficient cx previous equation can be
rewritten in the following form
2
x = ∑ ( X 1i sin(αt + β i ) + X 3i sin(3αt + 3β i ) ),
i =1
where:
F1i mi lα 2 (Φ i + 0.25Φ 3 )
X 1i = = i
,
kx −α U2
kx −α U 2
F3i 0.25mi lα 2 Φ 3
X 3i = = i
.
k x − 9α U
2
k x − 9α U
2
implies the following acceleration of the beam M
p g
2
&& = ∑ ( A1i sin(αt + β i ) + A3i sin(3αt + 3β i ) ),
x
i =1
mi lα 4 (Φ i + 0.25Φ 3 )
A1i = − i
,
kx −α U 2
0.25mi lα 4 Φ 3
A3i = − i
.
k x − 9α U
2

39. Energy balance
The work done by the escapement mechanism during tone period of pendulum’s
oscillations can be expressed as
T γN
Wi DRIV
= ∫ M Diϕi dt = 2 ∫ M Ni dϕi = 2M Niγ N .
&
0 0
Energy dissipated in the damper is given by
E di i t d i th d i i b
T T
Wi DAMP
= ∫ cϕiϕ dt = ∫ cϕiα 2 Φ i2 cos 2 (αt + β i )dt = παcϕi Φ i2 .
& i
2
0 0
The energy transferred from the i-th pendulum to the beam M (pendulum looses
part of its energy to force the beam to oscillate), so we have:
t f it t f th b t ill t ) h
T
Wi SYN
= ∫ mi &&l cosϕ iϕ i dt.
x &
0
Energy balance for the i-th pendulum
Wi DRIV = Wi DAMP + Wi SYN .

40. 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 rest
(Czolczynski et al., 2009(a,b)). There is no
energy transfer between pendulums
Wi DRIV = Wi DAMP .
2 M Ni γ N = παcϕi Φ i2
2 M Niγ N
Φi = .
παcϕi

41. Energy balance - non-identical pendulums
Setting β1=0.0 (one of the phase angles can be arbitrarily chosen) and linearizing
pendulum motion
m1l 2α 4πΦ1
W SYN
=− m2 Φ 2 sin β 2 = W SYN ,
k x − α 2U
1
m2 l 2α 4πΦ 2
W SYN
= m1Φ1 sin β 2 = −W SYN .
k x − α 2U
2
Both synchronization energies are equal and the energy balance of both
pendulums have following form:
d l h f ll i f
W1DRIV = W1DAMP + W SYN
W2DRIV + W SYN = W2DAMP

42. Energy balance - non-identical pendulums
Finally we get:
y g
m1l 2α 4πΦ1
2 M N 1γ N = παcϕ1Φ −
2
m2 Φ 2 sin β 2 , Φi
0.9
kx − α U
1 2
0.8
0.7
0.6
m2l α πΦ 2
2 4
2 M N 2γ N = παcϕ 2 Φ 2 + m1Φ1 sin β 2 , 0.5
k x − α 2U
2
0.4
0.3
0.2
0.1
We get 2 equations, as a parameter we take sin(β), 0
β2
0 1 2 3 4 5 6 7
we plot angles as a function of parameter sin(β)
p g p (
and then we numerically find the phase shift β .

43. Energy balance of pendulums; (a) anti-phase synchronization of identical
p
pendulums – there is no transfer of energy between p
gy pendulums, ( ) p
, (b) phase
synchronization 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.

44. Analytically we can find condition of in-phase and
phase synchronization for both cases: identical and
non-identical masses of pendulums
pendulums.
But this not the end of the story…

45. Parameters
Pendulum Beam
m1=1.0 [kg], M=10.0 [kg],
m2=controling parameter, cx=1.53 [Ns/m],
l=g/4π2=0.2485 [m], kx=3.94 [N/m],
cϕ1=0.0083×m1 [N ]
0 0083× [Ns],
cϕ2=0.0083×m2 [Ns], Escapement mechanism
γN=5.0o
MN1=0.075×m1 [Nm]
MN2=0.075×m2 [Nm]

46. Identical clocks

47. Nonidentical
clocks

48. Decreasing mass of second pendulum

49. Increasing mass of second pendulum
Comparison with analytics results

50. How harmonic are clocks?
m1=1 0 [kg];
=1.0
m2=11.0 [kg]

51. Long period synchronization
ϕ

52. Long period synchronization

53. Escapement mechanism – are the parameters
important ?
γNi − maximum angle below which escapement mechanism generate moment
MNi - constant moment of escapement mechanism
Assumption:
∫M Di dϕ i = M Ni γ Ni = Const

54. Basin of attraction for
,
d ee t
different sets of escapement
o escape e t
mechanism parameters
Initial
I iti l conditions:
diti
x(0) = 0, x(0) = 0.0,
&
ϕ i 0 = Φ sin β i 0 ,ϕ i 0 = αΦ cos β i 0 .
&
Parameters:
γN =4.8 o ( )
4 8 (a);
γN =4.9o (b);
γN =5.0o (c);
γN =5 05o (d);
=5.05
γN =5.1o (e);
γN =5.2o (f).

55. Poincare maps for
chosen attractors
γN =4.8o, T 23 (a),
4 8 T=23 ( )
γN =4.9o, T=6 (b),
γN =4.9o, T=11 (c),
γN =5 0o, T=11 (d);
=5.0

56. Poincare maps for
chosen attractors
γN =5.0o, chaos ( )
50 h (e),
γN =5.1o, T=59 (f),
γN =5.2o, T=13 (g),
γN =5 2o, T=35 (h);
=5.2

57. Rare attractors
• Blekhman, I., and Kuznetsova, L. "Rare events - rare attractors; formalization and
examples", Vibromechanika, Journal of Vibroengineering, 10, 418-420 (2008)
• Z k h k M S h ki I and Y
Zakrzhevsky, M., Schukin, I. d Yevstignejev V "R
i j V. "Rare attractors i d i
in driven nonlinear
li
systems 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 for
publication

58. Definition – open problem
Our proposal

59. As an example of the system which
possesses multistability and rare
attractors we consider an externally
exited van der Pol-Duffing oscillator
where: α=0.2, F=1.0, ω=0.955.
For simplicity set of accessible
parameters is following
Sets of possible initial conditions:

60. • Long Period Synchronization
• Multistability
• Sensitivity on escapement mechanism
parameters
• Rare attractors
e cos
• Chaos

61. More Clocks ?

79. Possible configurations
• the complete synchronization in which all
p
pendula behave identically,
y,
• pendula create three or five clusters of
synchronized pendula
pendula,
• anti-phase synchronization in pairs (for
even n and identical clocks),
• uncorrelated behavior of all pendula

80. Energy extraction from the
sea waves

85. Thank you !