Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 3, pp. 729-741, Warsaw 2012 50th Anniversary of JTAM ENERGY BALANCE OF TWO SYNCHRONIZED SELF-EXCITED PENDULUMS WITH DIFFERENT MASSES Tomasz Kapitaniak, Krzysztof Czołczyński, Przemysław Perlikowski, Andrzej Stefański Technical University of Lodz, Division of Dynamics, Łódź, Poland e-mail: tomasz.kapitaniak@p.lodz.pl; krzysztof.czołczyński@p.lodz.pl; przemyslaw.perlikowski@p.lodz.pl; andrzej.stefanski@p.lodz.pl We consider the synchronization of two self-excited pendulums with different masses. We show that such pendulums hanging on the same beam can show almost-complete (in-phase) and almost-antiphase synchronizations in which the difference of the pendulums displace- ments is small. Our approximate analytical analysis allows one to derive the synchronization conditions and explains the observed types of synchronizations as well as gives an approxi- mate formula for amplitudes of both the pendulums and the phase shift between them.We consider the energy balance in the system and show how the energy is transferred between the pendulums via the oscillating beam allowing synchronization of the pendulums. Key words: coupled oscillators, pendulum, synchronization 1. Introduction Currently, we observe growing interest in the studies of coupled oscillatory systems which are stimulated by various applications in physics, engineering, biology, medicine, etc. (Andronov et al., 1966; Blekhman, 1988; Pikovsky et al., 2001). Synchronization is commonly observed to occur between oscillators. It is a process where two or more systems interact with each other and come to oscillate together. Groups of oscillators are observed to synchronize in a diverse variety of systems, despite inevitable differences between the oscillators. The phenomenon of synchronization of clocks hanging on a common movable beam (Kapitaniak et al., 2019) has been recently the subject of research by a number of authors (Bennet et al., 2002; Czolczynski et al., 2009a,b, 2011; Dilao, 2009; Fradkov andAndrievsky, 2007; Huygens, 1893; Kanunnikov et al., 2003; Kumon et al., 2002; Pantaleone, 2002; Perlikowski et al., 2012; Senator, 2006; Ulrichs et al., 2009). These studies give the definite answer to the question; what Huygens was able to observe, e.g., Bennet et al. (2002) state that to repeat Huygens’ results, high precision (the precision that Huygens certainly could not achieve) is necessary, and Kanunnikov et al. (2003) show that the precise antiphasemotion of different pendulums noted byHuygens cannot occur. Our studies (Czolczynski et al., 2009a,b, 2011;Dilao, 2009) prove that in the case of nonidentical clocks, only almost-antiphase synchronization can be observed. In this paper, we consider the synchronization of two self-excited pendulumswhich have the the same length but different masses. Oscillations of each pendulum are self-excited by van der Pol’s type of damping. We show that two such pendulums hanging on the same beam, besides the complete (in-phase) and antiphase synchronizations already demonstrated for the case of pendulumswith the samemasses in Blekhman (1988), Czolczynski et al. (2009b), Fradkov and Andrievsky (2007), Perlikowski et al. (2012), Ulrichs et al. (2009), performalmost-complete and almost-antiphase synchronization in which the phase differences of oscillations are respectively close (but not equal) to 0 or π.Weperforman approximate analytical analysis which allows one 730 T. Kapitaniak et al. to derive the synchronization conditions and explains the observed types of synchronizations. The energy balance in the system allows one to show how the energy is transferred between the pendulums via the oscillating beam. This paper is organized as follows. Section 2 describes the considered model of the coupled pendulums. In Section 3we derive the energy balance of the synchronized pendulums. Section 4 presents the results of numerical simulations and describes the observed synchronization states. Finally, we summarize our results in Section 5. 2. Model Theanalyzed system is shown inFig. 1. It consists of a rigidbeamand twopendulumssuspended on it. The beam of mass M canmove in the horizontal direction, its movement is described by the coordinate x. Themass of the beam is connected to the refuge of a linear spring and linear damper kx and cx. Each pendulum consists of a light beam of length l and amass mounted at its end. We consider the pendulums with the same length l but different masses m1 and m2. The motion of the pendulums is described by angles ϕ1 and ϕ2 and is self-excited by van der Pol’s type of damping (not shown inFig. 1) given bymomentum (torgue) cϕvdpϕ̇l(1−ζ 2 l ), where cϕvdp and ζ are constant. Fig. 1. The model of the system – two self-excited pendulums mounted to the beamwhich canmove horizontally The object of studies, whose results are presented in this paper differ from the earlier ones (Czolczynski et al., 2009a,b, 2011; Kapitaniak et al., 2012; Perlikowski et al., 2012) – as instead of the clocks with pendulums driven by a discontinuous escapement mechanism, we consider two self-excited pendulums with van der Pol’s type of damping. The mathematical description of these pendulums contains the self-excited component cϕvdpϕ̇ and energy-dissipating compo- nent −cϕvdpζϕ̇ϕ 2. The balance of these components results in creation of a stable limit cycle (Andronov et al., 1966). The equations of motion of the considered system are as follows m1l 2ϕ̈1+m1ẍlcosϕ1+ cϕvdpϕ̇1(1− ζϕ 2 1)+m1glsinϕ1 =0 m2l 2ϕ̈2+m2ẍlcosϕ2+ cϕvdpϕ̇2(1− ζϕ 2 2)+m2glsinϕ2 =0 (2.1) and ( M+ 2 ∑ i=1 mi ) ẍ+ cxẋ+kxx+ 2 ∑ i=1 mil(ϕ̈i cosϕi− ϕ̇ 2 i sinϕi)= 0 (2.2) Equations (2.1) and (2.2), contrary to the equations considered in Czolczynski et al. (2009a,b, 2011), Kapitaniak et al. (2012), Perlikowski et al. (2012), are continuous. Energy balance of two synchronized self-excited pendulums... 731 3. Energy balance of the system Multiplying both sides of Eq. (2.1) by the angular velocity ϕi, one gets mil 2ϕ̈iϕ̇i+miglϕ̇i sinϕi =−cϕvdpϕ̇ 2 i + cϕvdpζϕ̇ 2 iϕ 2 i −miẍlcosϕiϕ̇i i=1,2 (3.1) In the case of periodic oscillations with period T , integration of Eq. (2.2) gives the following energy balance T ∫ 0 mil 2ϕ̈iϕ̇i dt+ T ∫ 0 miglϕ̇i sinϕi dt=− T ∫ 0 cϕvdpϕ̇ 2 i dt+ T ∫ 0 cϕvdpζϕ̇ 2 iϕ 2 i dt − T ∫ 0 miẍlcosϕiϕ̇i dt i=1,2 (3.2) The left hand side of Eq. (3.2) represents the increase of the total energy of i-th pendulum, which in the case of periodic oscillations is equal to zero T ∫ 0 mil 2ϕ̈iϕ̇i dt+ T ∫ 0 miglϕ̇i sinϕi dt=0 i=1,2 (3.3) The energy supplied to the system by van der Pol’s damper in one period of oscillations is given by WSELFi =− T ∫ 0 cϕvdpϕ̇ 2 i dt i=1,2 (3.4) The next component on the right hand side of Eq. (3.2) represents the energy dissipated by the van der Pol damper WVDPi =− T ∫ 0 cϕvdpζϕ 2ϕ̇2i dt i=1,2 (3.5) The last component of Eq. (3.2) represents the energy transfer from the pendulum to the beam or to the second pendulum (via the beam) WSYNi = T ∫ 0 miẍlcosϕiϕ̇i dt i=1,2 (3.6) Substituting Eqs. (3.3)-(3.6) into Eq. (3.2), one gets energy balances of the pendulums in the form WSELF1 −W VDP 1 −W SYN 1 =0 WSELF2 −W VDP 2 −W SYN 2 =0 (3.7) Multiplying equation of motion (2.2) by the beam velocity ẋ, one gets ( M+ 2 ∑ i=1 mi ) ẍẋ+ cxẋ 2+kxxẋ+ ( 2 ∑ i=1 mil(ϕ̈icosϕi− ϕ̇ 2 i sinϕi) ) ẋ=0 (3.8) 732 T. Kapitaniak et al. Integrating Eq. (3.8) over the period of oscillations, we obtain the following energy balance T ∫ 0 ( M+ 2 ∑ i=1 mi ) ẍẋ dt+ T ∫ 0 kxxẋ dt=− T ∫ 0 ( 2 ∑ i=1 mil(ϕ̈icosϕi−ϕ̇ 2 i sinϕi) ) ẋ dt− T ∫ 0 cxẋ 2 dt (3.9) The left hand side of Eq. (3.9) represents the increase of the total energy of the beam,which for the periodic oscillations is equal to zero T ∫ 0 ( M+ 2 ∑ i=1 mi ) ẍẋ dt+ T ∫ 0 kxxẋ dt+=0 (3.10) The first component on the right-hand side of Eq.(3.9) represents the work performed by the horizontal component of the forcewithwhich the pendulumsact on the beamcausing itsmotion WDRIVEbeam =− T ∫ 0 ( 2 ∑ i=1 mil(ϕ̈icosϕi− ϕ̇ 2 i sinϕi) ) ẋ dt (3.11) The second component on the right hand side of Eq.(3.9) represents the energy dissipated by the damper cx WDAMPbeam = T ∫ 0 cxẋ 2 dt (3.12) Substituting Eqs. (3.10)-(2.12) into Eq. (3.9), one gets the energy balance in the following form WDRIVEbeam −W DAMP beam =0 (3.13) In the case of periodic oscillations, it is possible to prove that WSYN1 +W SYN 2 =W DRIVE beam =W DAMP beam (3.14) so adding Eqs. (3.7) and (3.13) and considering Eq.(3.14) one obtains WDRIVE1 +W DRIVE 2 −W DAMP 1 −W DAMP 2 −W DAMP beam =0 (3.15) Equation (3.15) represents the energy balance of the whole system (1,2). 4. Numerical results Weperform a series of numerical simulations in which Eqs. (2.1) and (2.2) have been integrated using the Runge-Kuttamethod. The primary objective of these simulations is to investigate the influence of nonidentity of the pendulums on the observed types of synchronization. In our numerical studies, we consider the following parameters: mass of pendulum 1 m1 = 1.0kg; pendulums length l = g/4π 2 = 0.2485m (g = 9.81m/s2) (chosen so that the- ir period of free oscillations in the case of unmovable beam is T = 1.0s and the frequency of free oscillations α = 2π s−1), negative damping coefficient causing self-excited oscillations cϕvdp =−0.01Nms; van der Pol coefficient ζ = 60.0; beam mass M = 10.0kg, beam damping coefficient cx =1.53Ns/m, beam stiffness coefficient kx =4.0N/m.We assume themass of the second pendulum m2 as a control parameter. Note that because the coefficients of self-oscillations cϕvdp and damping ζ of the two pendu- lums are the same, in the case of an unmovable beamboth pendulums have the same amplitude Φ = 0.26 (≈ 15◦), regardless of their masses. The motion of the beam may change both the period and amplitude of oscillations. Energy balance of two synchronized self-excited pendulums... 733 4.1. From complete to almost-antiphase synchronization The evolution of system (1,2) behavior starting from the complete synchronization of iden- tical pendulums (m1 = m2 = 1.0kg) and increasing the value of control parameter m2 is illustrated in Figs. 2a-2f. Figure 2a presents the bifurcation diagram for the increasing values of m2 (m2 ∈ [1.0,6.0]). On the vertical axis, we show the maximum displacement ϕ1 of pen- dulum 1, and the displacements of pendulum 2 – ϕ2 as well as of the beam x recorded at moments when ϕ1 is maximum.Creating this diagram, we start with the state of complete syn- chronization of the pendulums with masses m1 = m2 = 1.0kg, during which they are moving in the same way (ϕ1 =ϕ2) in antiphase to the movement of the beam. Fig. 2. Evolution from the complete to almost-antiphase synchronization; (a) bifurcation diagram for increasing values of m2, (b) time histories of almost-complete synchronization m1 =1.0kg and m2 =2.0kg; (c) plots of system energy; (d) time histories of almost-complete synchronization for m1 =1.0kg and m2 =3.5kg; (e) time histories of almost-antiphase synchronization: m1 =1.0kg, m2 =5.0kg; (f) nonstationary complete synchronization: m1 =m2 =3.0kg 734 T. Kapitaniak et al. Increasing the value of m2, we observe that initially both pendulums are in the state of almost-complete synchronization. Figure 2b found for m1 = 1.0kg, m2 = 2.0kg shows the displacements ϕ1 ≈ ϕ2 and the displacement of the beam x (for better visibility enlarged 10 times) as a function of time (on the horizontal axis, the time is expressed as the number of periods of free oscillations of pendulums suspended on an unmovable beam – N). Notice that the differences ϕ1-ϕ2 are hardly visible. Further increase of themass m2 causes an increase of theamplitudeofpendulumsoscillations andan increase of theamplitudeof beamoscillations as canbeseen inFig. 2d (m2 =3.5kg).One also observes an increase of the period of pendulum oscillations (Fig. 2b presents 11.25 periods of oscillations while Fig. 2d – 12 periods in the same time). This is due to the fact that with the increasing mass of pendulum 2, the center of mass moves towards the ends of the pendulums, i.e., towards thematerial points withmasses m1 and m2, andmoves away from the beamwith the constant mass. Noteworthy is the fact that in the state of complete synchronization, when the displacements of both pendulums fulfill the relation ϕ1(t)=ϕ2(t), the energy transmitted to the beamby each pendulum is proportional to its mass. Therefore, these energies satisfy the following equations WSELF1 = T ∫ 0 cϕvdpϕ̇ 2 1 dt= T ∫ 0 cϕvdpϕ̇ 2 2 dt=W SELF 2 WVDP1 = T ∫ 0 cϕvdpζϕ̇ 2 1ϕ 2 1 dt= T ∫ 0 cϕvdpζϕ̇ 2 2ϕ 2 2 dt=W VDP 2 WSYN1 = T ∫ 0 m1ẍlcosϕ1ϕ̇1 dt= m1 m2 T ∫ 0 m2ẍlcosϕ2ϕ̇2 dt= m1 m2 WSYN2 (4.1) After substituting Eqs. (4.1) into Eqs. (3.7), Eqs. (3.7) become contradictory (except for spe- cial non-robust case of two identical pendulums when m1 = m2). In the general case when m1 6=m2, instead of the complete synchronization, an almost-complete synchronization occurs during which the displacements and velocities of the pendulums are almost-equal, and appro- priate energies satisfy the following equations WDAMP1 = T ∫ 0 cϕϕ̇ 2 1 dt≈ T ∫ 0 cϕϕ̇ 2 2 dt=W DAMP 2 WSYN1 = T ∫ 0 m1ẍlcosϕ1ϕ̇1 dt≈ T ∫ 0 m2ẍlcosϕ2ϕ̇2 dt=W SYN 2 WSELF1 = T ∫ 0 cϕvdpϕ̇ 2 1 dt≈ T ∫ 0 cϕvdpϕ̇ 2 2 dt=W SELF 2 WVDP1 = T ∫ 0 cϕvdpζϕ̇ 2 1ϕ 2 1 dt≈ T ∫ 0 cϕvdpζϕ̇ 2 2ϕ 2 2 dt=W VDP 2 WSYN1 = T ∫ 0 m1ẍlcosϕ1ϕ̇1 dt≈ T ∫ 0 m2ẍlcosϕ2ϕ̇2 dt=W SYN 2 (4.2) After substitutionofEqs. (4.2), the energy equations (3.7) are satisfied for pendulumsof different masses. Figure 2c shows the values of all energies as a function of themass m2. As one can see, Energy balance of two synchronized self-excited pendulums... 735 for m2 < 4.0kg all energies are positive. This means that both pendulums transfer a part of their energy to the beam, causing its motion (see Eq. (3.14)). For m2 = 4.0kg, the system undergoes bifurcation, an attractor of an almost-complete synchronized state loses its stability and we observe the jump to the co-existing attractor of almost-antiphase synchronization as shown in Fig. 2e (m2 =5.0kg). The amplitudes of oscilla- tions are different but the phase shift between the pendulums is close to π. The oscillations of the beam are so small that they are not visible in the scale of Fig. 2e. One can show that when one changes themass of pendulum 1 to m1 =2.0kg, m1 =3.0kg, m1 = 4.0kg, the bifurcation from almost-complete to almost-antiphase synchronization occurs respectively for m2 = 3.0kg, m2 = 2.0kg and m2 = 1.0kg. This bifurcation occurs when the total mass of both pendulums reaches the critical value mcr = 5.0kg, which depends on the system parameters, particularly on the beam ones M, cx and kx. Figure 2f shows the time histories of beam vibrations and oscillations of two pendulums in the case of identical masses m1 =m2 =3.0kg in the state of complete synchronization. These results have been obtained for identical initial conditions, so that they constitute de facto the pendulum of mass m = 6.0 > mcr. It is easy to see that this synchronized state is unstable: small disturbances lead to a stable coexisting attractor of antiphase synchronization. Notice that for the pendulums with slightly different masses (e.g., m1 =2.99kg and m2 =3.01kg) it is impossible to obtain a result similar to that shown in Fig. 2f, even for the identical initial conditions.Different pendulummasses cause that initially analmost-complete synchronization is observed, but small differences in ϕ1 and ϕ2 lead to the stable almost-antiphase synchronization. To summarize the bifurcation diagram in Fig. 2a, the existence of three different types of synchronization can be disinguished; (i) complete for m1 = m2 = 1.0kg, (ii) almost-complete for 1.0kg 4.0kg. 4.2. From complete synchronization to quasiperiodic oscillations Evolution of the behavior of system (1,2), starting from the complete synchronization of identical pendulums (m1 =m2 =1.0kg) anddecreasing the values of the control parameter m2, is illustrated in Figs. 3a-3d. Figure 3a shows the bifurcation diagram for decreasing values of mass m2 (m2 ∈ [0.01,1.00]). In the interval 1.0kg > m2 > 0.0975kg, both pendulums are in the state of almost-complete synchronization. Their oscillations are “almost-identical” as can be seen in Fig. 3b for m1 =1.0kg and m2 =0.01kg – the differences between the amplitudes and phases of ϕ1 and ϕ2 are close to zero, both pendulums remain in (almost) antiphase to the oscillations of the beam. Figure 3c shows values of different energies. Like in the interval 1.0kg < m2 < 4.0kg of Fig. 2c, all energies are positive and both pendulums drive the beam. Further reduction of the mass m2 leads to the loss of synchronization, and motion of the system becomes quasi- -periodic. Figure 3d presents the Poincaré map (the displacements and velocities of the pendu- lumshave been taken at themoments of greatest positive displacement of thefirstpendulum) for m2 = 0.07kg. The mechanism of the loss of stability is explained in Fig. 3c. In the interval 0.35kg>m2 > 0.07kg, the energy dissipated by the first pendulum W VDP 1 approaches the le- vel of the energy supplied by the self-exited component of this pendulum WSELF1 . Consequently, the energy supplied by the first pendulum to the beam WSYN1 decreases. The energy supplied to the system by the second pendulumalso decreases WSELF2 , which drives the pendulum from the beam. For m2 < 0.07kg, the energy balance is disrupted: pendulum2 has not enough ener- gy to cause its oscillations, the oscillations of the beam additionally support the oscillations of pendulum1. In this case, the almost-antiphase synchronization is not possible (see Section 3.4), and system (1,2) exhibits quasiperiodic oscillations. 736 T. Kapitaniak et al. Fig. 3. Evolution from the complete synchronization to quasiperiodic oscillations; (a) bifurcation diagram for increasing values of m2, (b) time histories of almost-complete synchronization for m1 =1.0kg and m2 =0.1kg, (c) energy plots, (d) Poincarémap showing quasiperiodic oscillations for m1 =1.0kg and m2 =0.07kg In summary, thebifurcationdiagram inFig. 3a showsthe existenceof: (i) complete synchroni- zation for m1 =m2 =1.0kg, (ii) almost-complete synchronization for 1.0kg>m2 > 0.0975kg, (iii) the lack of synchronization and quasi-periodic oscillations for m2 < 0.0975kg. 4.3. From antiphase to almost-antiphase synchronization The evolution of the system (1,2) behavior starting from antiphase synchronization of iden- tical pendulums (m1 =m2 =1.0kg) and the increase of the values of the control parameter m2 are illustrated in Figs. 4a-4d. Figure 4a presents another bifurcation diagram for the increasing values of m2 (m2 ∈ [1.0,6.0]). This time we start with a state of antiphase synchronization of the pendulumswithmasses m1 =m2 =1.0kg, during which two pendulums are moving in the same way (ϕ1 =−ϕ2) and the beam is at rest. The increase of the control parameter m2 leads to the reduction of pendulum 2 amplitude of oscillations but the amplitude of oscillations of pendulum 1 remains nearly constant. The pendulums remain in a state of almost-phase synchronization: the phase shift between the di- splacements is close to π, as shown in Fig. 4b (m1 =1.0kg, m2 =1.5kg). The displacement of the beam is practically equal to zero. In the state of antiphase synchronization when the pendulums’ oscillations satisfy the con- dition ϕ1(t) = −ϕ2(t), two van der Pol’s dampers dissipate the same amount of energy. The energies transmitted by both pendulums to the beam have absolute values proportional to pen- dulumsmasses and opposite signs Energy balance of two synchronized self-excited pendulums... 737 Fig. 4. Evolution from antiphase to almost-anitphase synchronization; (a) bifurcation diagram for increasing values of m2, (b) time series of almost-antiphase synchronization for m1 =1.0kg and m2 =1.5kg, (c) energy plots, (d) time series of almost-antiphase synchronization for m1 =1.0kg and m2 =20.0kg WSELF1 = T ∫ 0 cϕvdpϕ̇ 2 1 dt= T ∫ 0 cϕvdpϕ̇ 2 2 dt=W SELF 2 WVDP1 = T ∫ 0 cϕvdpζϕ̇ 2 1ϕ 2 1 dt= T ∫ 0 cϕvdpζϕ̇ 2 2ϕ 2 2 dt=W VDP 2 WSYN1 = T ∫ 0 m1ẍlcosϕ1ϕ̇1 dt=− m1 m2 T ∫ 0 m2ẍlcosϕ2ϕ̇2 dt=− m1 m2 WSYN2 (4.3) After substituting the energy values satisfying Eqs. (4.3) into Eqs. (3.7), Eqs.(3.7) are not contradictory equations only when the beam acceleration is zero, which implies the zero value of its velocity and acceleration (in the synchronization state of the behavior of the system is periodic). This condition requires the balancing of the forces which act on the pendulumbeam, and this in turn requires that the pendulums have the samemass. If the pendulums’masses are different, instead of antiphase synchronization we observe an almost-antiphase synchronization, during which the pendulums’ displacements have different amplitudes and phase shift between these displacements is close, but not equal to π. Hence WSELF1 6=W SELF 2 W VDP 1 6=W VDP 2 W SYN 1 6=W SYN 2 (4.4) The values of each considered energy is shown in Fig. 4c. In a state of almost-antiphase synchro- nization we have WSELF1 W VDP 2 . W SELF 2 part of the energy W SELF 2 738 T. Kapitaniak et al. supplied by van der Pol’s damper of pendulum 2 (with a greater mass) is transferred via the beam as WSYN2 to the pendulum1 (for this pendulum it is negative energy denoted by W SYN 1 ) and together with the energy WSELF1 dissipated as W VDP 1 by van der Pol damper. Van der Pol’s component of pendulum2 dissipates the rest of the energy WSELF2 , as W VDP 2 . The energy dissipated by the beam damper is negligibly small, because the beam virtually does not move. Figure 4d shows the time series of the system oscillations for the m2 =20.0kg.We observe that further increase m2 causes the reduction of the amplitude of pendulum 2 oscillations, the amplitude of oscillations of pendulum 1 remains unchanged. It can be observed that when the mass m2 increases, the equality of forces, with which the pendulums act on the beam occurs at decreasing amplitude of oscillations of pendulum 2. Pendulum 1 (with a smaller mass) has a virtually constant amplitude of oscillations and works here as a classical dynamical damper. The comparison of Fig. 2a and Fig. 4a indicates that in the interval 1.0kg < m2 < 4.0kg almost-complete and almost-antiphase synchronization coexist (which of them takes place the initial conditions decide). In summary, the diagram shown in Fig. 4a shows the existence of: (i) antiphase synchroniza- tion for m1 =m2 =1.0kg, (ii) almost-antiphase synchronization for 1.0kg m2 > 0.45kg, both pendulums are in the state of almost-antiphase synchronization, as shown in Fig. 5b for m2 =0.5kg. We observe a phenomenon similar to that of Fig. 4a, i.e., when decreasing mass m2, the amplitude of oscillations of pendulum 1 decreases (in this case pendulum1 has a larger mass), the amplitude of pendulum2 oscillations is practically constant and pendulum2 acts as a dynamical damper. In Fig. 5c one can see the negative energy WSYN2 – there is a transfer of energy from pendulum 1 to pendulum 2. For m2 =0.45kgwe observe the loss of synchronization due to the fact that energy W SELF 1 becomes equal to energy WSYN1 whichmeans that all the energy supplied to pendulum1by van derPol’s damper is transmitted to pendulum2. For smaller values of m2, pendulum2 is not able to supply the energy needed to maintain a state of almost-antiphase synchronization and the systemfirst obtains the state of almost-complete synchronization, andnextwhen m2 < 0.095kg exhibits unsynchronizedquasi-periodicoscillations. Thebehavior of the system for m2 < 0415kg has been described in Section 2.2. In thenarrow interval between the state of almost-antiphase and the state of almost-complete synchronization, i.e., for 0.45kg > m2 > 0415kg we observe quasiperiodic oscillations of the system, as shown on the Poincaré map of Fig. 5d (m2 =0.44kg). The bifurcation diagram of Fig. 5a shows the existence of: (i) antiphase synchronization for m1 = m2 = 1.0kg, (ii) almost-antiphase synchronization for 1.0kg > m2 > 0.45kg, (iii) the lack of synchronization and quasi-periodic oscillations for 0.45kg>m2 > 0.415kg, (iv) almost- -complete synchronization for 0.415kg > m2 > 0.095kg, (v) the lack of synchronization and quasi-periodic oscillations for m2 < 0.095kg. Energy balance of two synchronized self-excited pendulums... 739 Fig. 5. Evolution from antiphase synchronization to quasiperiodic oscillations; (a) bifurcation diagram of system (1,2) for decreasing m2, (b) time series of almost-antiphase synchronization for m1 =1.0kg, m2 =0.5kg, (c) energy plots, (d) Poincaré maps showing quasiperiodic oscillations for m1 =1.0kg and m2 =0.44kg 5. Conclusions Our studies show that the system consisting of a beam and two self-excited pendulums with van der Pol’s type of damping can perform four types of synchronization: (i) complete syn- chronization (possible only for nonrobust case of identical masses of both pendulums), i.e., the periodic motion of the system during which the displacements of both pendulums are identical (ϕ1(t) = ϕ2(t)), (ii) almost-complete synchronization of the pendulums with different masses, in which phase difference between the displacements ϕ1(t) and ϕ2(t) is small (not larger than a fewdegrees), (iii) antiphase synchronization (possible only for nonrobust case of identicalmasses of both pendulums), i.e., the periodic motion of the system, during which the phase difference between the displacements ϕ1(t) and ϕ2(t) is equal to 180 ◦, (iv) almost-antiphase synchroni- zation, during which the phase difference between the displacements ϕ1(t) and ϕ2(t) is close to 180◦ and the amplitude of oscillations of both pendulums are different. Theobservedbehavior of the system (1,2) canbe explainedby the energy expressionsderived in Section 3. The examples of the energy flow diagrams are shown in Figs. 6a,b. In the state (ii) both pendulums drive the beam (transferring to it part of the energy obtained from van der Pol’s dampers) as seen in Fig. 6a. In the case (iv) the pendulumwith larger mass and smaller amplitude of oscillations transmits part of its energy to the pendulum lower mass. The beam motion is negligibly small and the pendulumwith lowermass reduces the amplitude of vibration of the pendulumwith larger mass, acting on the classical model of the dynamic damper. We identified two reasons for the sudden change of the attractor in system (1,2); (i) loss of stability of one type of synchronization afterwhich the system trajectory jumps to the coexisting 740 T. Kapitaniak et al. Fig. 6. Energy balances of the system (1,2); (a) almost-complete synchronization – both pendulums driver the beam, (b) almost-antiphase synchronization – pendulum 1 drives pendulum 2 via the beam synchronization state, (ii) inability of van der Pol’s damper of one of the pendulums energy necessary to drive the second pendulum. Acknowledgement This work has been supported by the Foundation for Polish Science, TEAM Programme – Project No. TEAM/2010/5/5. References 1. Andronov A., Witt A., Khaikin S., 1966,Theory of Oscillations, Pergamon, Oxford 2. BennetM., SchatzM.F.,RockwoodH.,WiesenfeldK., 2002,Huygens’s clocks,Proc. Roy. Soc. London, A 458, 563-579 3. Blekhman I.I., 1988, Synchronization in Science and Technology, ASME, NewYork 4. CzolczynskiK., Perlikowski P., Stefanski A.,Kapitaniak T., 2009a,Clustering and Syn- chronization of Huygens’ Clocks,Physica A, 388, 5013-5023 5. Czolczynski K., Perlikowski P., Stefanski A., Kapitaniak T., 2009b, Clustering of Huy- gens’ Clocks,Prog. Theor. Phys., 122, 1027-1033 6. Czolczynski K., Perlikowski P., Stefanski A., Kapitaniak T., 2011, Why two clocks synchronize: Energy balance of the synchronized clocks,Chaos, 21, 023129 7. Dilao R., 2009, Antiphase and in-phase synchronization of nonlinear oscillators: The Huygens’s clocks system,Chaos, 19, 023118 8. Fradkov A.L., Andrievsky B., 2007, Synchronization and phase relations in the motion of twopendulum system, Int. J. Non-linear Mech., 42, 895 9. Huygens C., 1893, 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 10. Kapitaniak M., Czolczynski K., Perlikowski P., Stefanski A., Kapitaniak T., 2012, Synchronization of clocks, Physics Report, published on-line, http://dx.doi.org/10.1016/ j.physrep.2012.03.002 11. Kanunnikov A.Yu., Lamper R.E., 2003, Synchronization of pendulum clocks suspended on an elastic beam, J. Appl. Mech. and Theor. Phys., 44, 748-752 12. Kumon M., Washizaki R., Sato J., Mizumoto R.K.I., Iwai Z., 2002, Controlled synchroni- zation of two 1-DOF coupled oscillators,Proceedings of the 15th IFACWorld Congress, Barcelona 13. Pantaleone J., 2002, Synchronization of metronomes,Am. J. Phys., 70, 992 Energy balance of two synchronized self-excited pendulums... 741 14. Perlikowski P., Kapitaniak M., Czolczynski K., Stefanski A., Kapitaniak T., 2012, Chaos on coupled clocks, Int. J. Bif. Chaos, in press 15. Pikovsky A., Roesenblum M., Kurths J., 2001, Synchronization: An Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge 16. PogromskyA.Yu., Belykh V.N., Nijmeijer H., 2003, Controlled synchronization of pendula, Proceedings of the 42nd IEEE Conference on Design and Control, Maui, Hawaii, 4381-4385 17. Senator M., 2006, Synchronization of two coupled escapement-driven pendulum clocks, Journal Sound and Vibration, 291, 566-603 18. Strogatz S.H., 2004, Sync: The Emerging Science of Spontaneous Order, Penguin Science, London 19. Ulrichs H., Mann A., Parlitz U., 2009, Synchronization and chaotic dynamics of coupled mechanical metronomes,Chaos, 19, 043120 Bilans energii dwóch zsynchronizowanych wahadeł samowzbudnych o różnych masach Streszczenie Artykuł prezentuje analizę zjawiska synchronizacji dwóch wahadeł samowzbudnych o różnych ma- sach. Pokazano, że jeśli takie wahadła zostaną zawieszone na wspólnej, ruchomej podstawie, zachodzi zjawisko ich (prawie) zupełnej lub (prawie) antyfazowej synchronizacji. Analiza bilansu energetycznego układupozwalana określenie parametrówukładuwstanie synchronizacji (amplitudydrgań i przesunięcia fazowe). Analiza bilansu energetycznegowyjaśnia takżemechanizm synchronizowania się ruchuwahadeł: stały przepływ strumienia energii od jednego wahadła, viawspólna ruchoma podstawa, do drugiego wa- hadła powoduje, że ruch układu jest okresowy, a przesunięcia fazowe pomiędzy wahadłami przyjmują stałe, charakterystyczne wartości. Manuscript received February 15, 2012; accepted for print March 9, 2012