Selected topics in nonlinear dynamics and theoretical electrical engineering

Chapter 1
Synchronization of Two Nonidentical Clocks:
What Huygens was Able to Observe?

Krzysztof Czolczynski, Przemysaw Perlikowski,
Andrzej Stefanski, and Tomasz Kapitaniak

Abstract. We consider the synchronization of two clocks which are accurate (show
the same time) but have pendulums with different masses. We show that such clocks
hanging on the same beam can show the almost complete (in-phase) and almost
antiphase synchronizations. By almost complete and almost antiphase synchroniza-
tion we defined the periodic motion of the pendulums in which the phase shift be-
tween the displacements of the pendulums is respectively close (but not equal) to
0◦ or 180◦ . We give evidence that almost antiphase synchronization was the phe-
nomenon observed by Huygens in XVII century. We support our numerical studies
by considering the energy balance in the system and showing how the energy is
transferred between the pendulums via oscillating beam allowing the pendulums’
synchronization.

1.1 Introduction
In the 60-ties of XVII century the longitude problem, i.e., finding a robust, accurate
method of the longitude determination for marine navigation was the outstanding
challenge. Huygens believed that pendulum clocks, suitably modified to withstand
the rigors of the sea, could be sufficiently accurate to reliably determine the longi-
tude (The discrepancy in the clock rate equal to one oscillation of the second pendu-
lum (pendulum with a period of oscillations equal to 1 [s] in a day corresponds to the
error in the longitude determination, approximately equal to 500 m in a day (at the
equator).). In a letter to the Royal Society of London of 27 February 1665 Huygens
described his famous experiment which showed the tendency of two pendulums (of
the clocks) to synchronize, or anti-synchronize when mounted together on the same
beam (Huygens, 1665). Originally, he used the phrase “an odd kind of sympathy” to
Krzysztof Czolczynski · Przemysław Perlikowski · Andrzej Stefanski · Tomasz Kapitaniak
Division of Dynamics, Technical University of Lodz,
Stefanowskiego 1/15, 90-924 Lodz, Poland
e-mail: tomasz.kapitaniak@p.lodz.pl

K. Kyamakya et al. (Eds.): Selected Topics in Nonlinear Dynamics, SCI 459, pp. 3–18.
DOI: 10.1007/ 978-3-642-34560-9 1 c Springer-Verlag Berlin Heidelberg 2013

4 K. Czolczynski et al.

describe the observed behavior in two maritime clocks. The original drawing show-
ing this experiment is shown in Figure 1.1. Two pendulums, mounted together, will
always end up swinging in exactly opposite directions, regardless of their respective
individual motion. This was one of the first observations of the phenomenon of the
coupled harmonic oscillators, which have many applications in physics (Pikovsky
et al., 2001, Blekham, 1988; Strogatz & Steward, 1993, Golubitsky et al. 1999).
Huygens originally believed the synchronization occurs due to air currents shared
between two pendulums, but later after performing several simple tests he dismissed
this and attributed the sympathetic motion of pendulums to imperceptible movement
in the beam from which both pendulums are suspended.
Huygens’ study of two clocks operating simultaneously arose from the very prac-
tical requirement of the redundancy: if one clock stopped, had to be cleaned or
winded up, then the other one provided the proper timekeeping (Huygens 1669).
Ultimately, the innovation of the pendulum did not solve the longitude problem,
since slight and almost insensible motion was able to cause an alteration in their
going (Birch, 1756; Britten, 1973).
Recently, this idea has been validated by a few groups of researchers which tested
Huygens’ idea (Bennet et al., 2002, Pogromsky et al. (2003), Kanunnikov & Lam-
per, 2003, Senator, 2006, Dilao, 2009, Kumon et al., 2002, Fradkov, and B. An-
drievsky, 2007, J. Pantaleone, 2002, Ulrichs et al., 2009, Czolczynski et al., 2009a,b,
2011a,b). These studies do not give the definite answer to the question; what Huy-
gens was able to observe, e.g., Bennet et al. 2002 stated that to repeat Huygens’
results, the high precision (the precision that Huygens certainly could not achieve)
is necessary and Kanunnikov & Lamper, 2003 showed that the precise antiphase
motion of different pendulums noted by Huygens cannot occur. Different pendu-
lums (possibly with different masses) were definitely used by Huygens as can be
seen in his drawing shown in Figure 1.2. In this paper we consider the synchro-
nization of two clocks which have pendulums with the same length but different
masses. Such clocks are accurate, i.e., show the same time as both pendulums have
the same length. We show that two such clocks hanging on the same beam can show
the almost complete (in-phase) and almost antiphase synchronizations. By almost

Fig. 1.1 An original drawing of Huygens illustrating his expermiments with pendulum clocks

1 What Huygens Was Able to Observe? 5

complete and almost antiphase synchronization we deﬁned the periodic motion of
the pendulums in which the phase shift between the pendulums displacements is
respectively close (but not equal) to 0◦ or 180◦ . We give evidence that the almost
antiphase synchronization of the clocks was the phenomenon which Huygens ob-
served in XVII century. This paper is organized as follows. Sec. 1.2 describes the
model of the clocks which has been used. In Sec. 1.3 we derive the energy balance
of the synchronized pendulums. Section 1.4 presents the results of our numerical
simulations and describes the observed synchronizations states. Finally, we discuss
the energy balance of the synchronized pendulums and summarize our results in
Sec. 1.5.

1.2 Model
The analyzed system is shown in Figure 1.3. It consists of the rigid beam and two
pendulum clocks suspended on it. The beam of mass M can move in a horizontal
direction, its movement is described by coordinate x. The mass of the beam is con-
nected to the refuge of a linear spring and linear damper k x and c x . The clocks’
pendulum consists of the light beam of the length l and mass mounted at its end. We
consider the pendulums with the same length l but different masses m1 and m2 . The
same length of both pendulums guarantees that the clocks are accurate, i.e., both
show the same time. The motion of the pendulums is described by angles ϕ1 andϕ2
and is damped by the dampers(not shown in Figure 1.3) with the same damping
coefﬁcients c ϕ .
The pendulums are driven by the escapement mechanism described in details in
(Rawlings, 194, Lepschy et al., 1993, Roup et al., 2003, Moon and Stiefel, 2006,
Czolczynski et al. 2009b). Notice that when the swingng pendulums do not exceed

Fig. 1.2 Details of the Huygens’ experiment; it is clearly visible that the clocks used in the
experiment have not been identical, D denotes the weight used to stabilize the clock case


Fig. 1.3 The model of the system – two pendulum clocks are mounted to the beam which
can move horizontally

certain angle γ N , the escapement mechanisms generate the constant moments M N
(the same for both pendulums). This mechanism acts in two successive steps i.e.,
the first step is followed by the second on and the second one by the first one
(detailed description of the escapement mechanism has been given in our previ-
ous work (Czolczynski el al., 2009b)). In the first step if 0 < ϕi < γ N (i=1,2)
then MD = M N and when ϕi < 0 then MD = 0. For the second stage one has for
−γ N < ϕi < 0MD = − M N and for ϕi > 0MD = 0. The energy supplied by the
escapement mechanism balance the energy dissipated due to the damping. The pa-
rameters of this mechanics have been chosen in the way that for the beam M at the
rest both pendulums perform the oscillations with the same amplitude. Typically
pendulum clocks oscillate with amplitude smaller then 2π/36 and for the clocks
with long pendulums like marine clocks this amplitude is even smaller (Rawlings,
1994). Since the damping coefficients of the two pendulums are identical, in the case
of unmovable beam the pendulums oscillate with the same amplitude. the move-
ment of the beam may change both the period and the amplitude of pendulums’
oscillations.
The equations of motion of considered systems are as follow:

m1 l 2 ϕ + m1 xl cos ϕ1 + c ϕ ϕ1 + m1 gl sin ϕ1 = MD
¨ ¨ ˙
(1.1)
m2 l 2 ϕ + m2 xl cos ϕ2 + c ϕ ϕ2 + m2 gl sin ϕ2 = MD
¨ ¨ ˙

2 2
M + ∑ mi ¨ ˙ ¨ ˙
x + c x x + k x x + ∑ mi l ( ϕi cos ϕi − ϕ2 sin ϕi ) = 0 (1.2)
i
i =1 i =1

Note that the escapement mechanism acts only when the amplitude of the pendulum
oscillations is larger than γ N .


1.3 Energy Balance
1.3.1 Energy Balance of the Pendulum
Multiplication of both sides of eq.(1.1) by the angular velocity of the i-th pendulum
gives:

¨ ˙ ˙ ˙
mi l 2 ϕi ϕi + mi gl ϕi sin ϕi = MD ϕ2 − mi xl cos ϕi ϕi ,
¨ ˙ i = 1, 2 (1.3)
i

In the case of the periodic motion of the pendulums after integration eq.(1.3) gives
the energy balance of the i-th pendulum in one period of motion:
T T
mi l 2 ϕi ϕi dt +
¨ ˙ mi gl ϕi sin ϕi dt =
˙
0 0
T T T
(1.4)
MD ϕi dt −
˙ ˙
c ϕ ϕ2 dt − mi xl cos ϕi ϕi dt,
¨ ˙ i = 1, 2
i
0 0 0

Left hand side of eq.(1.4) represents the decrease of the total energy of the i-th
pendulum. In the case of the periodic behavior of the system(1,2) this decrease is
equal to zero, so
T T
mi l 2 ϕi ϕi dt +
¨ ˙ mi gl ϕi sin ϕi dt = 0,
˙ i = 1, 2 (1.5)
0 0

The work done by the escapement mechanism during one period of pendulum’s
oscillations can be expressed as
T γN
W DRIV E = MD ϕi dt = 2
˙ M N dϕi = 2M N γ N , i = 1, 2 (1.6)
0 0

Note that W DRIV E does not depend on the pendulum’s masses, displacement
ϕ1,2 ( ϕ1,2 > γ N ) and velocity. Energy dissipated in the damper is given by
T
WiDAMP = cφ φi2 dt,
˙ i = 1, 2 (1.7)
0

The last component of eq.(1.7) represents the energy transferred from the i-th pen-
dulum to the beam M (the pendulum loses part of its energy to force the beam to
oscillate), so we have:
T
WiSYN = mi xl cos φi φi dt,
¨ ˙ i = 1, 2 (1.8)
0

Note that in the case of pendulums in the state of complete synchronization (φ1 (t) =
φ2 (t)), energies dissipated by dampers are equal and energies transmitted to the
beam are proportional to pendulums’ masses. Substituting eqs. ((1.5)-(1.8)) into
eq.(1.4),


W DRIV E − W1
DAMP
− W1
SYN
= 0,W DRIV E − W2
DAMP
− W2
SYN
= 0, (1.9)

one gets energy balances for pendulums.

1.3.2 Energy Balance of the Beam and Whole System (1,2)
˙
Multiplying equation of the beam motion (2) by beam velocity x one gets:
2 2
M + ∑ mi x x + cx x2 + k x x x +
¨˙ ˙ ˙ ∑ mi l (φi cos φi − φ˙i2 sin φi
¨ x=0
˙
i =1 i =1
(1.10)
Integrating eq.(1.10)) over the period of oscillations we obtain the following energy
balance:
2 2 2
M + ∑ mi x xdt +
¨˙ k x x xdt+ =
˙
0 i =1 0
(1.11)
T 2 T
−
0
∑ mi l (φi cos φi − φ˙i2 sin φi
¨ xdt = −
˙
0
c x x˙2 dt
i =1

Left hand side of eq.(1.11) represents the increase of the total energy of the beam
which for the periodic oscillations is equal to zero:

2 2 2
M + ∑ mi x xdt +
¨˙ k x x xdt+ = 0
˙ (1.12)
0 i =1 0

The ﬁrst component on the right-hand side of eq.(1.11) represents the work per-
formed by the horizontal component of the force with which the pendulums act on
the beam causing its motion:

T
Wbeam E = −
DRIV
0
∑ 2mi l (φi cos φi − φ˙i2 sin φi )
¨ ˙
xdt. (1.13)
i =1

The second component on the right hand side of eq.(1.11) represents the energy
dissipated by the damper c x :
T
Wbeam =
DAMP
c x x2 dt.
˙ (1.14)
0

Substituting eqs.((1.12)-(1.14)) into eq.(1.11) one gets energy balance in the follow-
ing form
Wbeam E − Wbeam = 0.
DRIV DAMP
(1.15)
In the case of the periodic oscillations it is possible to prove that


SYN
W1 + W2
SYN
= Wbeam E .
DRIV
(1.16)

Adding eqs.(1.9) and (1.15)

2W DRIV E − W1
DAMP
− W2
DAMP
− W1
SYN
− W2
SYN
+ Wbeam E − Wbeam = 0,
DRIV DAMP

and considering eq.(1.16) one obtains

2W DRIV E − W1
DAMP
− W2
DAMP
− Wbeam = 0,
DAMP
(1.17)

Eq.(1.17) represents the energy balance of the whole system (1,2).

1.4 Numerical Results
In our numerical simulations eqs ((1.1),(1.2)) have been integrated by the Runge-
Kutta method. The stability of the obtained synchronous states has been investigated
using the variational equations as described in (Czolczynski et. al. 2009(a,b)).
We used the following system parameters: mass of pendulum 1 m1 = 1.0[kg];
the length of the pendulums l = g/4π 2 =0.2485 [m] (it has been selected in such
a way when the beam M is at rest, the period of pendulum oscillations is equal to
T=1.0 [s] and oscillations frequency to α = 2π [s−1 ]), g=9.81[m/s2] is an accelera-
tion due to the gravity, beam mass M=10.0[kg], damping coefficients cφ =0.01 [Ns]
and c x =1.53 [Ns/m] and stiffness coefficient k x =4.0 [N/m]. When the displacements
of the pendulums are smaller than γ N =5.0◦ , the escapement mechanisms generate
driving moments M N = 0.075[Nm] and allow the pendulums to oscillate with am-
plitude Φ1 = Φ2 = Φ = 0.2575 (≈ 14.75◦) when beam M is at rest. The mass m2 of
the pendulum 2 has been taken as a control parameter. It varied in the wide interval
to examine the influence of its changes on the type of the observed synchronization.

1.4.1 From Complete to (Almost) Antiphase Synchronization
The evolution of system (1,2) behavior starting from complete synchronization of
identical clocks (m1 =m2 =1.0 [kg]) and the increase of the values of control parame-
ter m2 is illustrated in Figure 1.4(a-d). In Figure 1.4(a) we present the bifurcation di-
agram of the system (1,2). The mass of pendulum 2 − m2 has been taken as a control
parameter and it increases in the interval [1.0, 41.0]. In the initial state (m1 = m2 )
the pendulums exhibit complete synchronization (φ1 = φ2 ) and the beam moves in
contrary phase to the pendulums. The increase of the bifurcation parameter leads
to the reduction of the oscillations amplitudes of both pendulums which are in the
state almost complete synchronization; their movements are not identical, but very
close to identical as shown in Figure 1.4(b). Figure 1.4(b) shows the displacements
of the pendulums φ1 , φ2 and the beam x (for better visibility enlarged 10 times) for
m1 =1.0 [kg] and m2 =3.0 [kg]. Time on the horizontal axis is given in the follow-
ing way t = NT, where N=1,2,3,. . . and T is a period of pendulum’s oscillations
when the beam is at rest. Notice that φ1 ≈ φ2 as the differences are hardy visible.


Further increase of the mass m2 results in the further reduction of pendulums’ am-
plitudes and the increase of the beam amplitude as can be seen in Figure 1.4(c) for
the m2 =20.0 [kg]. The period of pendulums’ oscillations decreases (in Figure 1.4(b)
we observe 11.5 periods while in Figure 1.4(c) 17.5 in the same time interval). This
reduction is due to the fact that while increasing mass of pendulum 2, the center
of the mass moves towards the ends of the pendulums, i.e., towards the material
points with masses m1 and m2 and moves away from the beam with constant mass
M. For m2 =30.3 [kg] the amplitude of pendulums’ oscillations decreases to limit
Φ ≈ γ N = 5◦ , below which the escapement mechanism is turned off. For larger
values of m2 , we observe oscillations of pendulum 1 with amplitude Φ ≈ 15◦ , and
small oscillations of pendulum 2 (whose escapement mechanism is turned off as can
be seen in Figure 1.4(d). The pendulum moves due to the energy supplied to it by
pendulum 1 via the beam. The bifurcation diagram of Figure 1.4(a) shows the ex-
istence of: (i) complete synchronization for m1 =m2 =1.0 [kg], (ii) almost complete
synchronization for 1.0 [kg] < m2 < 30.3 [kg], (iii) almost antiphase synchroniza-
tion, with one pendulum with turned off escapement mechanism (in Figure 1.4(d)
pendulum 2 has turned off mechanism) for m2 > 30.3 [kg].

Fig. 1.4 Evolution from complete to almost antiphase synchronization; (a) bifurcation dia-
gram for increasing values of m2 , (b) time series of almost complete synchronization m1 =1.0
[kg] and m2 =3.0 [kg]; (c) time series of almost complete synchronization m1 =1.0 [kg] and
m2 =20.0 [kg]; (d) time series of almost antiphase synchronization for m1 =1.0 [kg] and
m2 =35.0 [kg]– the escapement mechanism of pendulum 2 is turned off


1.4.2 From Complete Synchronization to Quasiperiodic
Oscillations
Evolution of system (1,2) behavior starting from complete synchronization of iden-
tical pendulums (m1 =m2 =1.0 [kg]) and the decrease of the values of the control
parameter m2 is illustrated in Figure 1.5(a-d). Figure 1.5(a) shows the bifurcation
diagram for decreasing values of mass m2 (m2 ∈[0.01,1.00]). In the interval 1.0 [kg]
> m2 >0.25 [kg], both pendulums are in a state of almost-complete synchroniza-
tion. Their oscillations are ”almost identical” as can be seen in Figure 1.5(b) for
m1 =1.0 [kg] and m2 =0.3 [kg], the differences between the amplitudes and phases of
φ1 and φ2 are close to zero, both pendulums remain in the (almost) antiphase to the
oscillations of the beam. Further reduction of mass m2 leads to the loss of synchro-
nization and the motion of the system becomes quasiperiodic as shown in Figure
1.5(c). Figure 1.5(d) presents the Poincare map (the displacements and velocities
of the pendulums has been taken at the moments of the greatest positive displace-
ment of the ﬁrst pendulum) for m2 =0.2 [kg]. In summary, the bifurcation diagram of
Figure 1.5(a) shows the existence of: (i) complete synchronization for m1 =m2 =1.0
[kg], (ii) almost complete synchronization for 1.0 [kg] > m2 >0.25 [kg], (iii) the
lack of synchronization and a quasiperiodic oscillations for m2 <0.25 [kg].

Fig. 1.5 Evolution from complete synchronization to quasiperiodic oscillations; (a) bifurca-
tion diagram for increasing values of m2 , (b) time series of almost complete synchronization
for m1 =1.0 [kg] and m2 =0.3 [kg], (c) time series of quasiperiod oscillations for m1 =1.0 [kg]
and m2 =0.2 [kg], (d) Poincare map showing quasiperiodic oscillations for m1 =1.0 [kg] and
m2 =0.2 [kg]


1.4.3 From Antiphase to Almost-Antiphase Synchronization
The evolution of system (1,2) behavior starting from antiphase synchronization of
identical pendulums (m1 =m2 =1.0) and the increase of the values of the control pa-
rameter m2 is illustrated in Figure 1.6(a-c). Figure 1.6(a) presents another bifurca-
tion diagram for the increasing values of m2 (m2 ∈[1.0,41.0]). This time we start
with a state of antiphase synchronization of pendulums with masses m1 =m2 =1.0
[kg], during which two pendulums are moving in the opposite way, such that
φ1 (t)=−φ2 (t) and the beam is at rest. The increase of bifurcation parameter m2
leads to the increase of the oscillations amplitude of pendulum 1 (with constant
mass (m1 =1.0 [kg]) and decrease of the amplitude of the pendulum 2 (with increas-
ing mass m2 ). The pendulums remain in the state of almost-antiphase synchroniza-
tion as phase shift between their displacements is close to 180◦ as shown in Figure
1.6(b) for m1 =1.0 [kg] and m2 =2.0 [kg]. In Figure 1.6(b) displacements φ1 and
φ2 are almost in antiphase (difference between antiphase and almost antiphase is
hardly visible). The beam is oscillating and its displacement x is shifted by approx-
imately 90deg (respectively forward and backward) in relation to the pendulums’
displacements. The beam oscillations are caused by the energy transmitted to the
beam by pendulum 2. Part of this energy is dissipated in the beam damper c x and

Fig. 1.6 Evolution from antiphase to almost antiphase synchronization; (a) bifurcation di-
agram for increasing values of m2 , (b) time series of almost antiphase synchronization for
m1 =1.0 [kg] and m2 =2.0 [kg], (c) Poincare map of quasiperiodic oscillations for m1 =1.0
[kg] and m2 =5.0 [kg]


part is transferred to pendulum 1. As the results of this transfer pendulum 1 os-
cillates with the amplitude larger than initial Φ ≈ 15◦ (see discussion in Sec.1.3).
When mass m2 reaches a value equal to 4.45 [kg] the amplitude of pendulum 2
oscillations decreases below the critical value Φ ≈ γ N = 5◦ . In the interval 4.45
[kg] < m2 <5.3 [kg] we observe a quasiperiodic oscillations of the system (1,2),
in this range the escapement mechanism of pendulum 2 is turned off. The exam-
ple of quasiperidic oscillations is shown in Figure 1.6(c). There are two irregular
phases of these oscillations: (i) after turning off the escapement mechanism of pen-
dulum 2, it does not provide energy to pendulum 1, the amplitude of pendulum
1 oscillations decreases and simultaneously the amplitude of pendulum 2 oscilla-
tions increases and its escapement mechanism is turned on, (ii) after turning on of
the escapement mechanism of pendulum 2, it provides the energy to pendulum 1,
which causes the reduction of pendulum 2 amplitude and leads to the turn off of
the escapement mechanism. For m2 >5.3 [kg] the escapement mechanism of pen-
dulum 2 is still turned off and the system tends to almost antiphase synchronization.
The bifurcation diagram of Figure 1.6(a) shows the existence of: (i) antiphase syn-
chronization for m1 =m2 =1.0 [kg], (ii) almost-antiphase synchronization for 1.0 [kg]
< m2 <4.45 [kg], (iii) the lack of synchronization and quasi-periodic oscillations
for 4.45 [kg]< m2 <5.3 [kg].(iv) almost-antiphase synchronization with the turn off
of the escapement mechanism of pendulum 2 for m2 >5.3 [kg].

1.4.4 From Antiphase Synchronization to Quasiperiodic
Oscillations
The evolution of system (1,2) behavior starting from antiphase synchronization of
identical pendulums (m1 =m2 =1.0 [kg]) and the decrease of the values of the control
parameter m2 is illustrated in Figure 1.7(a,b)). Figure 1.7(a) shows the bifurcation
diagram of the system (1,2) for decreasing values of m2 (m2 decreases from an
initial value 1.0 [kg] up to 0.01 [kg]). We start from the state of antiphase synchro-
nization observed for m1 =m2 =1.0 [kg]. In the interval 1.0 [kg]> m2 >0.38 [kg]

Fig. 1.7 Evolution from antiphase synchronization to quasiperiodic oscillations; (a) bifurca-
tion diagram of system (1,2) for decreasing m2, (b) time series of almost antiphase synchro-
nization for m1 =1.0 [kg], m2 =0.5 [kg]


both pendulums are in the state of almost antiphase synchronization. Its displace-
ments are out of phase by an angle close to 180◦ as shown in Figure 1.7(b) showing
the time series of φ1 , φ2 and x for m1 =1.0 [kg] and m2 =0.5 [kg]. When the mass m2
decreases from 1.0 [kg] to 0.56 [kg], the amplitude of oscillations of pendulum 1
decreases and the amplitude of oscillations of pendulum 2 increases. In the interval
0.56 [kg]> m2 >0.38 [kg] we observe fast decrease of pendulum 2 amplitude up
to the limit value Φ ≈ γ = 5◦ . For m2 =0.38 [kg] system (1,2) changes the type of
the synchronization to almost complete (previously observed in Figure 1.5(b) and
described in Sec.1.4.3). Further reduction of m2 leads to the quasi-periodic motion
of the system (as described in Sec.1.4.3). The bifurcation diagram of Figure 1.7(a)
shows the existence of: (i) antiphase synchronization for m1 = m2 =1.0 [kg], (ii) al-
most antiphase synchronization for 1.0 [kg]> m2 > 0.38 [kg], (iii) almost complete
synchronization for 0.38 [kg]> m2 >0.25 [kg], (iv) the lack of synchronization and
quasiperiodic oscillations for m2 <0.25 [kg].

1.5 From Complete to (Almost) Antiphase Synchronization
In the considered system of two clocks suspended on the horizontally movable beam
we identified the following types of synchronizations: (i) the complete synchroniza-
tion during which the pendulums’ displacements fulfill the relation φ1 (t) = φ2 (t)
and the phase shift between pendulums displacements φ1 (t) and φ2 (t) is equal to
zero, (ii) the almost complete synchronization during which the pendulums’ dis-
placements fulfill the relation φ1 (t) ≈ φ2 (t) and the phase shift between pendulums’
displacements φ1 (t) and φ2 (t) is close to zero, (iii) antiphase synchronization during
which the pendulums’ displacements fulfill the relation φ1 (t) = φ2 (t) and the phase
shift between the pendulum displacements is equal to 180◦ , (iv) almost antiphase
synchronization during which the phase shift between the pendulum displacements
is close to 180◦ . Additionally, in our previous work we find the possibility of long pe-
riod synchronization during which the difference of the pendulums’ displacements
φ1 − φ2 is a periodic function of time and chaotic behavior of the clocks’ pendu-
lums (Czolczynski et al., 2011b). Note that types (i) and (iii) are possible only for
non robust case of identical pendulum masses (m1 = m2 ). For cases (ii) and (iv)
pendulums’ energy balance looks differently as can be shown in Figure 1.8(a-c).
(i) Energy balance in the state of almost complete synchronization
In the state of complete synchronization pendulums’ displacements fulfill the
relation φ1 (t) = φ2 (t), both dampers dissipate the same amount of energy and both
pendulums transfer the same amount of energy to the beam:
T T
DAMP
W1 = cφ φ1 dt =
˙2 cφ φ2 dt = W2
˙2 DAMP
,
0 0
(1.18)
T T
SYN
m2 W1 = m2 m1 xl cos φ1 φ1 dt = m1
¨ ˙ m2 xl cos φ2 φ2 dt = m1 W2 .
¨ ˙ SYN
0 0

After substituting the energy values satisfying eqs.(1.18) into eqs.(1.9), eqs.(1.9)
are not contradictory equations only in two cases: (i) masses of both pendulums are


equal (m1 = m2 ) and both pendulums transmit the same amount of energy to the
SYN
beam (see Figure 1.8(a)), (ii) synchronization energies W1,2 are equal to zero, i.e.,
both pendulums dissipate the whole energy supplied by the escapement mechanism
(see Figure 1.8(b)). In general case when m1 = m2 and c x = 0.0, instead of com-
plete synchronization we observe almost complete synchronizations during which
the pendulums’ displacements are not identical (but close to each other) and one
gets the following expressions for considered energies
T T
DAMP
W1 = cφ φ1 dt ≈
˙2 cφ φ2 dt = W2
˙2 DAMP
,
0 0
T T
(1.19)
SYN
W1 = m1 xl cos φ1 φ1 dt ≈
¨ ˙ m2 xl cos φ2 φ2 dt = W2 .
¨ ˙ SYN
0 0

which fulﬁlls eqs.(1.9). The scheme of the energy balance is similar to the one in
Figure 1.8(a).
(ii). Energy balance in the state of almost antiphase synchronization
In the state of antiphase synchronization the pendulums’ displacements fulﬁll the
relation φ1 (t) = −φ2 (t) and both dampers dissipate the same amount of energy.
The energies transmitted to the beam have the same absolute values but opposite
signs, i.e.,
T T
DAMP
W1 = cφ φ1 dt =
˙2 cφ φ2 dt = W2
˙2 DAMP
,
0 0
T T
SYN
m2W1 = m2 m1 xl cos φ1 φ1 dt = −m1
¨ ˙ m2 xl cos φ2 φ2 dt = −m1 W2
¨ ˙ SYN
0 0
(1.20)
After substituting the energy values satisfying eqs.(1.20) into eqs.(1.9), eqs.(1.9) are
not contradictory equations only when the beam acceleration is zero, which implies
the zero value of its velocity and displacement (in the synchronization state of the
behavior of the system is periodic). This condition requires a balancing of the forces
which pendulums act on the beam, and this in turn requires that the pendulum have
the same mass. The scheme of the energy balance is similar to the one in Figure
1.8(b). If the pendulums’ masses are different, instead of antiphase synchronization
we observe an almost-antiphase synchronization, during which the displacement
pendulums have different amplitude and phase shift between these displacements is
close, but not equal to 180◦ . Hence
DAMP
W1 = W2
DAMP SYN
,W1 = W2
SYN
(1.21)

The energy balance for the case of almost antiphase synchronization is shown in
Figure 1.8(c). Part of the energy supplied by the escapement mechanism of pen-
dulum 1 (let us assume that it has smaller mass) W1 DRIV E is dissipated by the

damper of this pendulum (W1 DAMP ) and the rest (W SYN ) is transmitted to the beam.
1
The damper of pendulum 2 dissipates the energy W2 DRIV E supplied by the escape-

ments mechanism and energy (W2 SYN ) transmitted from the beam (mathematically


Fig. 1.8 Energy balance schemes: (a) almost complete synchronization – both pendulums
driver the beam, (b) antiphase synchronization – beam at rest, (c) almost antiphase synchro-
nization – pendulum 1 drives the beam and supplies energy to pendulum 2


this energy is negative). The damper of the beam dissipates the rest of the energy
W1 SYN : W SYN = W SYN − (−W SYN ). Finally we give answer to the initial question
beam 1 2
(posed in the title): our results give evidence that Huygens in his famous experiment
was unable to observe antiphase synchronization as stated in his letters (Huygens,
1665) but observed almost antiphase synchronization of two pendulum clocks. In
his times the distinction between antiphase and almost antiphase synchronization
for the clock with similar masses of the pendulums was impossible.

Acknowledgements. This work has been supported by the Foundation for Polish Science,
Team Programme – Project No TEAM/2010/5/5. PP. acknowledges the support from Foun-
dation for Polish Science (the START fellowship).

References
1. Bennet, M., Schatz, M.F., Rockwood, H., Wiesenfeld, K.: Huygens’s clocks. Proc. Roy.
Soc. London, A 458, 563–579 (2002)
2. Blekham, I.I.: Synchronization in Science and Technology. ASME, New York (1988);
Birch, T.: The history of The Royal Society of London for improving of natural knowl-
edge, in which the most considerable of those papers communicated to the Society, which
have hitherto not been published, are inserted in their proper order, as a supplement to the
Philosophical Transactions, vol. 2, pp. 19, 21, 23–24. Johnson, London (1756) (reprint
1968)
3. Britten, F.J.: Britten’s old clocks and watches and their makers; a historical and descrip-
tive account of the different styles of clocks and watches of the past in England and
abroad containing a list of nearly fourteen thousand makers. Methuen, London (1973)
4. Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Clustering of Huygens’
Clocks. Prog. Theor. Phys. 122, 1027–1033 (2009a)
5. Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Clustering and Synchro-
nization of Huygens’ Clocks. Physica A 388, 5013–5023 (2009b)
6. Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Huygens’ odd sympathy
experiment revisited. Int. J. Bifur. Chaos 21 (2011a)
7. Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Why two clocks synchro-
nize: Energy balance of the synchronized clocks. Chaos 21, 023129 (2011b)
8. Dilao, R.: Antiphase and in-phase synchronization of nonlinear oscillators: The Huy-
gens’s clocks system. Chaos 19, 023118 (2009)
9. Fradkov, A.L., Andrievsky, B.: Synchronization and phase relations in the motion of
twopendulum system. Int. J. Non-linear Mech. 42, 895 (2007)
10. Huygens, C.: Letter to de Sluse. In: Oeuveres Completes de Christian Huygens (letters;
no. 1333 of 24 February 1665, no. 1335 of 26 February 1665, no. 1345 of 6 March 1665).
Societe Hollandaise Des Sciences, Martinus Nijhoff, La Haye (1893)
11. Huygens, C.: Instructions concerning the use of pendulum-watches for ﬁnding the lon-
gitude at sea. Phil. Trans. R. Soc. Lond. 4, 937 (1669)
12. Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: Symmetry in locomotor central
pattern generators and animal gaits. Nature 401, 693–695 (1999)
13. Kanunnikov, A.Y., Lamper, R.E.: Synchronization of pendulum clocks suspended on an
elastic beam. J. Appl. Mech. & Theor. Phys. 44, 748–752 (2003)


14. Kumon, M., Washizaki, R., Sato, J., Mizumoto, R.K.I., Iwai, Z.: Controlled synchroniza-
tion of two 1-DOF coupled oscillators. In: Proceedings of the 15th IFAC World Congress,
Barcelona (2002)
15. Lepschy, A.M., Mian, G.A., Viaro, U.: Feedback control in ancient water and mechanical
clocks. IEEE Trans. Education 35, 3–10 (1993)
16. Moon, F.C., Stiefel, P.D.: Coexisting chaotic and periodic dynamics in clock escape-
ments. Phil. Trans. R. Soc. A 364, 2539 (2006)
17. Pantaleone, J.: Synchronization of metronomes. Am. J. Phys. 70, 992 (2002)
18. Pikovsky, A., Roesenblum, M., Kurths, J.: Synchronization: An Universal Concept in
Nonlinear Sciences. Cambridge University Press, Cambridge (2001); Pogromsky, A.Y.,
Belykh, V.N., Nijmeijer, H.: Controlled synchronization of pendula. In: Proceedings of
the 42nd IEEE Conference on Design and Control, Maui, Hawaii, pp. 4381–4385 (2003)
19. Roup, A.V., Bernstein, D.S., Nersesov, S.G., Haddad, W.S., Chellaboina, V.: Limit cycle
analysis of the verge and foliot clock escapement using impulsive differential equations
and Poincare maps. Int. J. Control 76, 1685–1698 (2003)
20. Rawlings, A.L.: The science of clocks and watches. Pitman, New York (1944)
21. Senator, M.: Synchronization of two coupled escapement-driven pendulum clocks. Jour-
nal Sound and Vibration 291, 566–603 (2006)
22. Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Scientiﬁc
American 269(6), 102–109 (1993)
23. Ulrichs, H., Mann, A., Parlitz, U.: Synchronization and chaotic dynamics of coupled
mechanical metronomes. Chaos 19, 043120 (2009)

Selected topics in nonlinear dynamics and theoretical electrical engineering

More Related Content

What's hot (19)

Similar to Selected topics in nonlinear dynamics and theoretical electrical engineering (20)

More from Springer (20)

Recently uploaded (20)

Selected topics in nonlinear dynamics and theoretical electrical engineering