AB24 3UE, Aberdeen, Scotland. PACS: f, a Keywords: coupled systems, synchronization, rotating pendula.

Transcription

1 Synchronous states of slowly rotatng pendula Marcn Kaptanak 1,, Krzysztof Czolczynsk 1, Przemysław Perlkowsk 1, Andrzej Stefansk 1, and Tomasz Kaptanak 1* 1 Dvson of Dynamcs, Techncal Unversty of Lodz, Stefanowskego 1/15, Lodz, Poland Centre for Appled Dynamcs Research, School of Engneerng, Unversty of Aberdeen AB4 3UE, Aberdeen, Scotland PACS: f, a Keywords: coupled systems, synchronzaton, rotatng pendula. Abstract: Coupled systems that contan rotatng elements are typcal n physcal, bologcal and engneerng applcatons and for years have been the subject of ntensve studes. One problem of scentfc nterest, whch among others occurs n such systems s the phenomenon of synchronzaton of dfferent rotatng parts. Despte dfferent ntal condtons, after a suffcently long transent, the rotatng parts move n the same way complete synchronzaton, or a permanent constant shft s establshed between ther dsplacements,.e., the angles of rotaton - phase synchronzaton. Synchronzaton occurs due to dependence of the perods of rotatng elements moton and the dsplacement of the base on whch these elements are mounted. We revew the studes on the synchronzaton of rotatng pendula and compare them wth the results obtaned for oscllatng pendula. As an example we consder the dynamcs of the system consstng of n pendula mounted on the movable beam. The pendula are excted by the external torques whch are nversely proportonal to the angular veloctes of the pendula. As the result of such exctaton each pendulum rotates around ts axs of rotaton. It has been assumed that all pendula rotate n the same drecton or n the opposte drectons. We consder the case of slowly rotatng pendulums and estmate the nfluence of the gravty on ther moton. We classfy the synchronous states of the dentcal pendula and observe how the parameters msmatch can nfluence them. We gve evdence that synchronous states are robust as they exst n the wde range of system parameters and can be observed n a smple experments. *Correspondng author. E-mal address: tomaszka@p.lodz.pl (T. Kaptanak). 1

2 Contents 1. Introducton. Synchronous states of rotatng pendula 3. Synchronzaton mechansm 3.1 Equatons of moton 3. Synchronzaton condtons 3..1 Energy balance of pendula 3.. Energy balance of the beam 3..3 Synchronzaton condtons - lnearzed model 4. Numercal examples 4.1 Two pendula rotatng n the same drecton 4. Three pendula rotatng n the same drecton 4.3 Large system of pendula rotatng n the same drecton 4.4 Two pendula rotatng n the opposte drectons 4.5 Three pendula rotatng n varous drectons 4.6 Large system of pendula rotatng n dfferent drectons 4.7 Two pendula wth dfferent drvng torques, rotatng n the same drecton 5. Synchronzaton extends the lfe tme of rotaton 6. Dscusson and conclusons References Nomenclature α [rad/s] -angular frequency of pendulum rotaton; α x [rad/s] -angular frequency of the beam-pendula system oscllatons-rotatons; β [rad], -phase shft between pendula; Δ - logarthmc decrement of dampng Φ,Φ [rad] - ampltudes of oscllatons of the pendulum; ϕ, & ϕ, & ϕ - dsplacement [rad], velocty [rad/s] and acceleraton [rad/s ] of the -th pendulum; ϕ 0,ϕ& 0 - ntal condtons of the -th pendulum moton; ξ - scale factor of pendulums; [rad/s] - velocty of -th pendulum; [rad/s] - nomnal velocty of -th pendulum; A [m] - ampltude of parametrc exctaton; A b [m/s ] - ampltude of beam acceleraton; b [rad] - ampltude of harmonc component of rotaton angle; c ϕ [Nsm] - dampng coeffcent of the -th pendulum damper; c x [Ns/m] - dampng coeffcent of the damper between the beam and the bass; T, U [Nm] - knetc and potental energy; F [N] - resultng force wth whch pendulums act on the beam; g [m/s ] - gravtatonal acceleraton; k x [N/m] - stffness coeffcent of the sprng between the beam and the bass; l, l [m] - length of the pendulum; m b [kg] - mass of the beam; m, m [kg] - mass of the pendulum; n - number of pendulums n the system; N - number of perods T of pendula oscllatons (NT - unt of tme);

3 p = p0 ϕ& p1 [Nm] - drvng torque; s[m] - length of the arc of the crcle; t [s] - tme; U [kg] - global mass of the system (beam plus pendula); W beam [Nm] - energy dsspated by the beam durng one perod of moton; DAMP W [Nm] - energy dsspated by the -th pendulum durng one perod of moton; DRIVE W beam [Nm] - energy delvered to the beam durng one perod of moton; DRIVE W [Nm] - energy delvered to the -th pendulum durng one perod of moton; SYN W [Nm] - energy delvered from the -th pendulum to the beam durng one perod of moton; X [m] - ampltude of the beam oscllatons; X 1, X 3 [m] - ampltudes of frst and thrd harmonc component of the beam oscllatons; x, x&, & x - dsplacement [m], velocty [m/s] and acceleraton [m/s ] of the beam; x, x & ntal values of dsplacement and velocty of the beam; * In ths paper all values of the parameters and state varables are gven n the above unts. For smplcty of the presentaton the unts are omtted n the text. 3

4 1. Introducton A pendulum s an archetype for strongly nonlnear dynamcal systems, whch naturally has been gven a great deal of attenton n lterature [1,50 and ref. wthn]. In the last few decades, partcularly snce the expermental verfcaton of chaotc moton of pendulum [46], there has been an exploson of work n ths area. The plane pendulum s a constraned system: a mass pont m moves on a crcle of constant radus l, as sketched n Fgure 1. We denote by φ(t) the angle that measures the devaton of the pendulum from the vertcal lne and by s(t)=lφ(t) the length of the correspondng arc on the crcle. We then have knematc T and potental U energes gven respectvely as and 1, where g s the acceleraton due to the gravty. Let us ntroduce the constant 1 1 1, where [3]. For ε<1 the pendulum performs oscllatons around the stable equlbrum pont φ=0, whle for ε>1 the pendulum always swngs n one drecton,.e., t rotates ether clockwse or counterclockwse. The boundary ε=1 between these qualtatvely dfferent domans s a sngular value and corresponds to the moton where the pendulum reaches the uppermost poston but cannot swng beyond t. Ths sngular trajectory s called the separatrx, t separates the doman of oscllatory and rotatonal behavor. Fgure 1. The planar pendulum. Fgure. Physcal model of the pendulum: (a-c) pendulum excted at the pvot pont (a) vertcally, (b) ellptcally, (c) along a tlted axs, (d) pendulum excted by the external torque p(t): m s the mass of the pendulum bob, l s the length of the pendulum arm. 4

5 In the dsspatve case, e.g. when the energy dsspates due to the frcton n the pvot, one has to excte the pendulum to preserve oscllatory or rotatonal moton. There are two possbltes of exctaton: () by the moton of the pvot,.e., the parametrcal exctaton as shown n Fgure (a-c), () by the torque appled drectly to the mass m Fgure (d). The parametrcal exctaton can be mplemented by the vertcal (Fgure (a)) [43,44], ellptcal (Fgure (b)) [3,73] or along a tlted lne (Fgure (c)) moton of the pvot [56]). Despte the fact that both oscllatory and rotatonal motons are robust only a few percent of the publshed papers on pendula dynamcs refer to the rotatonal moton. Rotatng solutons of a pendulum have been studed manly n the case of pendulum parametrc exctaton. Koch and Leven [39] used Melnkov s method to detect bfurcatons where harmonc and subharmonc rotatng solutons are born. The closed form expresson for the lower boundary n frequency-ampltude exctaton parameters space has been derved for the regon of exstence of the stable rotatng soluton. Capecch and Bshop [1,13] studed rotatng solutons analytcally usng harmonc balance method. Approxmate analytcal solutons have been compared wth numercal results. They also constructed basns of attracton for dfferent types of moton. Later Clfford and Bshop [15] and Garra and Bshop [9] nvestgated rotatng solutons numercally and ntroduced the classfcaton of such trajectores dstngushng purely rotatng, oscllatng rotatng, straddlng rotatng and large ampltude rotatng orbts. Szemplnska-Stupncka et al. [71,7] conducted numercal studes of the parametrcally excted pendulum, whch have been llustrated usng basns of attracton, bfurcaton dagrams and attractor manfolds phase portrats. They have dentfed global bfurcatons responsble for the onset of complex transent and/or steady state dynamcs and varous other aspects ncludng fractal basn boundares and coexstence of rotatng solutons wth other (ncludng nonrotatng) attractors. Extensve numercal smulatons have been performed by Xu et al. [78-81]. Varous parameter space plots for dfferent sets of ntal condtons and dampng whch allow to follow the development of attractors n the exctaton ampltude-frequency parameters space have been calculated. The problem of dynamcal ntegrty of both rotatng and oscllatng competng trajectores has been studed by Lenc and Rega [43] usng a systematc constructon of basns of attractons for varyng parameters. The cross-eroson and the effects of secondary attractors n reducng the attractors safety, and thus ts practcal relablty, has been ponted out. Some analytcal and expermental studes of the rotatng solutons were also conducted by Xu et al. n [80,81]. In 008, Lenc et al. [44] consdered perod-1 rotatons analytcally and obtaned the approxmaton for the lower stablty boundary assocated wth these perod-1 rotatons n the forcng parameter planes. De Paula et al. [55] have appled chaos control methods to avod bfurcatons that destablze the rotatng moton keepng the desred rotaton over the extended parameter range. Pendula excted by a combnaton of the vertcal and horzontal forcng at the pvot have been consdered n the lterature but they are far less researched [6,30,31,3,45,48,73]. Ge and Ln [30] numercally studed the response of a pendulum, whose pvot has been vertcally excted and has been free to move horzontally. Mann and Koplow [48] used a combnaton of expermental measurements and analytcal predctons (based on the method of multple scales) to understand the results of the ptchfork bfurcaton. Thompson et al. [73] have nvestgated the dynamcs of the ellptcally excted pendulum numercally concentratng on the change of stablty boundares of rotatonal moton due to the ntroducton of horzontal component. A pendulum excted by the combnaton of vertcal and horzontal forcng at the pvot pont has been consdered by Pavlovskaa et al. [56]. Analytcal approxmatons of perod-1 rotatons and ther stablty boundary on the exctaton parameters plane have been derved usng asymptotc analyss for the pendulum excted ellptcally and along a tlted axs. One should menton the studes whch have been concentrated on the oscllatons of the nverted pendulum excted along a tlted axs (see e.g. [70,8]). Lythoof [47] have explaned 5

6 why when a smple harmonc pendulum s vewed bnocularly wth a neutral-tnt flter n front of one eye, the pendulum, nstead of swngng to and fro n a plane, appears to swng n an ellpse, frst advancng towards and them recedng from the observer. A smple mechancal system consstng of the rotatng pendulum has been used n the expermental studes of Lorenz chaos [14]. A model whch comprses a rotatng pendulum lnked by an oblque sprng pnned to ts rgd support s nvestgated n [65]. Ths model provdes a cylndrcal dynamcal system wth both smooth and dscontnuous regmes dependng on the value of a system parameter and also the dynamcs transent relyng on the couplng strength between the rotatng pendulum and the lnked sprng. Fnally, one should menton the studes n whch the pendulum s excted parametrcally by the random sgnal [8,83,84]. To summarze these studes t should be ponted out that the rotatonal solutons exst over lmted parameters range and there are numerous bfurcatons of the system that destablze a rotatonal moton. The study of synchronzaton of oscllatng pendula can be traced back to the works of the Dutch researcher Chrstan Huygens n XVIIth century [33,34]. He showed that a couple of mechancal clocks hangng from a common support were synchronzed. Huygens had found that the pendulum clocks swung n exactly the same frequency and π out-of-phase,.e., n antphase synchronzaton. After the external perturbaton, the antphase state was restored wthn half an hour and remaned ndefntely. Recently, several research groups revsted the Huygens experment [7,17-19,-4,36-38,59,66,74,76,77]. Pogromsky et al. [61] desgned a controller for synchronzaton problem for two pendula suspended on an elastcally supported rgd beam. To explan Huygens observatons, Bennett et al. [4] bult an expermental devce consstng of two nteractng pendulum clocks hanged on a heavy support whch was mounted on a low-frcton wheeled cart. The devce moves by the acton of the reacton forces generated by the swng of two pendula and the nteracton of the clocks occurs due to the moton of the clocks base. It has been shown that to repeat the results of Huygens, hgh precson (the precson that Huygens certanly could not acheve) s necessary. Another devce mmckng Huygens clock experment, the so-called coupled pendula of the Kumamoto Unversty [41], conssts of two pendula whose suspenson rods are connected by a weak sprng, and one of the pendula s excted by an external rotor. The numercal results of Fradkov and Andrevsky [8] show smultaneous approxmate n-phase and antphase synchronzaton. Both types of synchronzaton can be obtaned for dfferent ntal condtons. Addtonally, t has been shown that for the small dfference n the pendula frequences they may not synchronze. A very smple demonstraton devce was bult by Pantaleone [54]. It conssts of two metronomes located on a freely movng lght wooden base. The base les on two empty soda cans whch smoothly rolls on the table. Both n-phase and antphase synchronzatons of the metronomes have been observed. Synchronous confguratons of a par of double pendula has been dentfed n [40]. Fnally, one should mentoned the frst expermental observaton of chmera states n mechancal system of a number of coupled metronomes [49]. Mechancal systems that contan rotatng parts (for example vbro-excters, unbalance rotors) are typcal n engneerng applcatons and for years have been the subject of ntensve studes [16,4,75]. One problem of scentfc nterest, whch among others occurs n such systems, s the phenomenon of synchronzaton of dfferent rotatng parts [,5,51] and references wthn]. Despte dfferent ntal condtons, after a suffcently long transent, the rotatng parts move n the same way - complete synchronzaton or a permanent constant shft s establshed between ther dsplacements,.e., the angles of rotaton - phase synchronzaton [,5,1,,51]. Synchronzaton occurs due to dependence of the perods of rotatng elements moton and the dsplacement of the base on whch these elements are mounted [7]. Prasad [6] consders the system of coupled counter-rotatng oscllators and observes mxed synchronzatons,.e., some systems varables are synchronzed n-phase, whle others are 6

7 out-of-phase. The dynamcs of the system consstng of n rotatng (n the same drecton) pendula mounted on the movable beam have been consdered n [5,0]. The pendula have been excted by the external torques whch are lnearly dependent on the angular veloctes of the pendulums. As the result of such exctaton, each pendulum rotates around ts axs of rotaton. It has been shown that both complete and phase synchronzatons of the rotatng pendula are possble. The approxmate analytcal condtons for both types of synchronzatons and equatons whch allow the estmaton of the phase dfferences between the pendula have been derved. Contrary to the case of the oscllatory pendula[17-19,,3,37,38], phase synchronzaton s not lmted to three and fve clusters confguratons. The case of slowly rotatng pendula and the nfluence of the gravty on ther moton have been consdered. The obtaned results have been compared to those of Blekhman [5]. The dynamcs of the smlar system n whch one pendulum rotates counter-clockwse,.e., has a postve angular velocty, whle the remanng pendula rotate clockwse wth negatve angular velocty has been studed n [1]. Two cases have been consdered: () pendula rotate n the horzontal plane,.e., the gravty has no nfluence on ther moton, () pendula rotate n the vertcal plane and ther weght causes the unevenness of ther rotaton,.e., each pendulum slows down when the center of ts mass goes up and accelerates when the center of ts mass goes down. It has been shown that n such systems, despte opposte drectons of rotaton dfferent types of synchronzaton occur. The dynamcs of the pendula suspended on the nonlnear oscllators has been studed n [9,10,35]. The regons of stable synchronous rotatonal moton have been dentfed. In [69] the dynamcs of the set of two pars of double pendula mounted on the platform whch oscllates vertcally has been studed. Usng a custom desgned expermental rg dfferent types of synchronous moton of rotatng pendula have been dentfed. The extreme sensblty of the synchronzed state on the system parameters and ntal condtons has been ponted out. In ths revew we consder the dynamcs of the system consstng of n pendula mounted on the movable beam. The pendula are excted by the external torques whch are lnearly dependent on the angular veloctes of the pendula. As the result of such exctaton each pendulum rotates around ts axs of rotaton. We consder two cases: () all pendula rotate n the same drecton, () one pendulum rotates n the opposte drecton to the other pendula. It has been shown that both complete and varous types of phase synchronzatons of the rotatng pendula are possble. The synchronzaton mechansm base on the energy transfer between pendula va the oscllatng beam has been dentfed. We derve the approxmate analytcal condtons for each type of synchronzatons and equatons whch allow the estmaton of the phase dfferences between the pendula. We consder the case of slowly rotatng pendula and consder the nfluence of the gravty on ther moton. Our results have been compared to those of [5]. Dfferences of both analyses have been ponted out and explaned. The case when the exctaton of one pendulum s weakenng or even stops operatng s also consdered. We gve evdence that the ntal synchronzaton and the energy transfer between pendula can extend the rotatonal moton of ths pendulum. We gve evdence that our results are robust as they exst n the wde range of system parameters. Ths revew paper s organzed as follows. Secton explans how the synchronzaton can be acheved n the system of the coupled pendula. We consder two cases: () externally excted pendula are mounted to the movable beam, () unexcted pendula are mounted on oscllatng platform. Both cases are llustrated by the examples of expermentally observed synchronous states. In Sec. 3 we descrbe the consdered model and dentfy the synchronzaton mechansm. The approxmate analytcal condtons for each type of synchronzatons are derved. The man Secton 4 gves several numercal examples of the synchronous behavor. The cases of both dentcal and nondentcal pendula as well as of 7

8 dentcally and dfferently excted pendula are consdered. Sec. 5 descrbes the behavor of the system when the exctaton of one of the pendula weakens or vanshes. Fnally, we summarze our results n Sec. 6.. Synchronous states of rotatng pendula Let us consder the systems shown n Fgure 3(a-c). Each system conssts of a rgd beam of mass m B on whch n rotatng pendula are mounted. In Fgure 3(a,b) the beam s connected to a statonary base by the sprng (or sprngs) wth stffness coeffcent k x and a damper (or dampers) wth a dampng coeffcent c x. Due to the exstence of the forces of nerta, whch act on each pendulum pvot, the beam can move n horzontal (Fgure 3(a)) or vertcal (Fgure 3(b)) drectons (ths moton s descrbed by coordnate x). The masses of the pendula are ndcated as m ; l are the lengths of the pendula. The rotaton of the -th pendula s descrbed by ϕ. The rotatons of the pendula are damped by lnear dampers (not shown n Fgure 3) wth dampng coeffcent c ϕ. Each pendulum s drven by the drve torque nversely proportonal to ts velocty: p0 ϕ& p1. If any other external forces do not act on the pendulum, then under the acton of such a moment t rotates wth constant angular velocty. As the system s n a gravtatonal feld (g= acceleraton of gravty), the weght of the pendulum causes the unevenness of ts rotaton: the pendulum slows down, when the center of mass rses up and accelerates when the center of mass falls down. The effect of gravty s mportant n the case of slow rotatons of the pendula. For hgh rotatonal speed t can be neglected as n the studes of rotor dynamcs [16,4,75 and references wthn]. It s assumed that p 1 >0.0. If p 0 torque s postve, the pendulum rotates to the left havng a postve value of the nstantaneous angular velocty, f p 0 <0.0, the pendulum rotates to the rght wth a negatve angular velocty. In the system shown n Fgure 3(c) pendula are forced to rotate (and oscllate) by the parametrcal exctaton,.e., the perodc moton of the base to whch they are mounted. To explan how the synchronzaton can be acheved n the systems of Fgure 3(a,b) frst consder the case of dentcal pendula and nonmovable beam. In ths case all pendula have the same perod of rotatons (the pendula have the same masses and lengths). The rotatons of the pendula are ntated by non-zero ntal condtons and the pendula s evolutons tend to the lmt cycles. The pendula are not coupled and the phase angles between ther dsplacements have fxed values, dependng on ntal condtons. Any perturbaton of the pendula results n the changes of these angles. (a) 8

9 (b) Fgure 3. (a,b) externally forced pendula mounted to the beam whch can move; (a) horzontally (b) vertcally, parametrcally excted pendula mounted to the beam whch oscllates perodcally. When the beam can move (horzontally or vertcally), the oscllatons of the beam excted by the forces wth whch pendula act on t, cause the changes of the phase shfts between the pendula s dsplacements and dfferentate the angular velocty of ther rotatons. When after the transent tme, all pendula have the same angular velocty of rotaton and there are constant phase shfts between the pendula s dsplacements, we can say that the pendula acheve synchronzaton [6,57,60,63,64]. The state of synchronzaton s acheved when the moton of the system s perodc and there are constant phase shfts between the pendula dsplacements [5,0,1]. The values of the phase shfts characterze the synchronous confguraton and are ndependent of the ntal condtons (unless the ntal condtons belong to the basn of attracton of the partcular confguraton). For systems lke Fgure 3(a,b) the synchronzaton of the rotatng pendula has been observed expermentally n [85]. A smple rg consstng of three drect-current electrcal motors mounted on the wooden plate whch can oscllate vertcally (shown n Fgure 4) has been consdered. The pendula are mounted at the end of the motor's rods. The control system (for detals see [85] has been used to vary the pendula's angular velocty. The sponges are used as sprngs and vcous dampers. 9

10 Fgure 4. The expermental rg: three drect-current electrcal motors mounted on the wooden plate whch can oscllate vertcally. Fgure 5. Expermentally observed synchronous confguratons, (a) complete synchronzaton, (b) synchronzaton wth the phase shfts between pendula equal to π/3. The examples of expermentally observed synchronous confguratons are shown n Fgure 5(a,b). Fgure 5(a) presents the complete synchronzaton of three pendula. The confguraton n whch the phase shfts between pendula's dsplacement are equal to π/3 s llustrated n Fgure 5(b). In the case of Fgure 3(c) we cannot speak about the synchronzaton between the pendula but the pendula can synchronze wth the perodc parametrcal forcng (phase lockng) [5]. Ths can lead to the occurrence of the varous synchronous states of rotatng pendula [69]. Fgure 6. Expermental mplementaton of the parametrcally excted pendula of Fgure 3(c). In [69] the system lke one n Fgure 3(c),.e., the system of four pendula arranged nto a cross structure as shown n Fgures 6 has been consdered. The base, mounted on the shaker, s excted n the vertcal drecton by a parametrc exctaton. In the experment, the rg has been mounted on the shaker LDS V780 Low Force Shaker. The shaker ntroduces 10

11 practcally knematc perodc exctaton cos, where A and ω are the ampltude and the frequency of the exctaton, respectvely. At ntal moments the pendula have been assumed to be n the upper poston,.e., /36. We fx the value of the exctaton ampltude A=0.01±0.005 [m] and consder exctaton frequency ω as a control parameter. Rotatng pendula can be 1:1 and 1: synchronzed wth the oscllatons of the platform. In the consdered system one can observe the synchronous states of both clockwse and counter-clockwse rotatng pendula. In the experment usng a smple mechancal rg, the exstence of dfferent types of synchronous confguratons of rotatng nondentcal pendula, has been confrmed. Typcal examples of dfferent types of synchronous states are shown n Fgures 7(a-d), where yellow arrows ndcate the drecton of rotaton. For a qualtatve classfcaton of the pendula behavor, we use the followng nomenclature; the pendula whch rotate clockwse or counter-clockwse are marked by + and -, respectvely, the pendula whch are at rest are marked by 0. The angular velocty of the pendulum s gven as follows: sn, where =1,,4, for the case of clockwse rotaton and sn, where the harmonc component descrbes the nfluence of the gravty on the moton of pendula (b s constant for all pendula as ther masses are the same) [0,1]. Fgure 7(a) presents the case when all pendula rotate n the same drecton,.e., (+,+,+,+). The pendula s dsplacements fulfll the relaton 0, where,j=1,,3,4, j. In Fgure 7(b) one observes the synchronous moton when 3 pendula (1,, and 3) rotate n the same drecton, whle the fourth n the opposte one (+,+,+,-). In ths case, 0 and pendulum 4 s n the state of mrror synchronzaton [1] wth the cluster of synchronzed pendula 1, and 3. In Fgure 7(c), we present the varaton of the case (+,+,+,-) when three pendula rotate n the same rotaton velocty whle the fourth one rotates twce slower. Pendula 1,, and 3 are synchronzed. The case when two pendula rotate clockwse and two counterclockwse s presented n Fgure 7(d). The pars of the pendula whch rotate n the same drectons are synchronzed and are n the state of cluster antphase synchronzaton [1],.e.,,, sn. All the observed synchronous states are stable but ther basn of attracton are very small and the small perturbatons (smaller than the accuracy of the shaker) can lead the system to the other confguraton. The lkelhood that the system wll reman n a gven confguraton s very small and n the experments practcally equal to zero. In such systems one has to use the concept of basn stablty [87]. Ths has been confrmed n the numercal smulatons summarzed n Fgure 8 (for detals see [67]). 11

12 Fgure 7. Dfferent types of expermentally observed synchronous states; (a) pendula rotate clockwse (+,+,+,+), ω=0.00 [rad/s], (b) 3 pendula rotate clockwse whle the fourth one counterclockwse (+,+,+,-), ω=4.00 [rad/s], (c) 3 pendula rotate clockwse (ω=9.00 [rad/s]) whle the fourth one counterclockwse (+,+,+,-) wth twce slower angular velocty, (d) pendula rotate clockwse and counterclockwse (+,+,-,-), ω=35.00 [rad/s]. The extreme senstvty of the synchronzed state on the system parameters and the ntal condtons whch ntroduces pseudo-randomness to the predctablty of the synchronous state has been observed. The basns of attracton of dfferent synchronous states are presented n Fgure 8. It can be seen that the type of pendula synchronzaton very strongly depends on the exctaton parameters. Generally, synchronous rotaton of pendula s robust as t exsts for the wde range of exctaton parameters, but partcular synchronous states are very senstve to the changes of system parameters as shown n Fgure 8. 1

13 Fgure 8. Basns of attracton of the combned synchronous states of 4 parametrcally excted pendula. In practcal applcaton of the systems based on the parametrcally excted pendula one has to apply a feedback control mechansm. Ths mechansm should be capable to keep the pendula rotatng permanently n the desred synchronous confguraton. 13

14 3. Synchronzaton mechansm 3.1. Equatons of moton Let consder the system shown n Fgure 3(a). The equatons of moton descrbed above are as follows: ml & ϕ + ml && xcosϕ + cϕ & ϕ + ml g snϕ = p0 p1 & ϕ, (1) n n mb + m & x + cxx& + kxx = ml ( && ϕ cosϕ + & ϕ snϕ ), () = 1 = 1 where =1,, n. In our numercal smulatons eqs.(1,) have been ntegrated by the 4 th order Runge-Kutta method. The obtaned results confrmed the exstence of the phenomenon of phase synchronzaton n the consdered system and allowed the determnaton of phase angles between the synchronzed pendula. Addtonally the numercal ntegraton of eqs.(1,) allows the determnaton of the basns of attracton of dfferent coexstng confguratons of the synchronzed pendula. 3.. Synchronzaton condtons Energy balance of pendula Multplyng eq. (1) by the angular velocty of the pendula we obtan the equaton of the energy balance: m l & ϕ & ϕ + m l g & ϕ sn ϕ = c & ϕ ϕ m l && x cosϕ & ϕ + p & ϕ p & ϕ, (3) 14 where =1,,n. Assume that the moton of both pendula s perodc wth perod T and ntegratng eq.(3) over the perod T we obtan the equaton of the energy balance: T 0 m l && ϕ & ϕ dt + = T 0 ϕ c & ϕ dt T 0 T 0 m l g & ϕ snϕ dt = m l && xcosϕ & ϕ dt + T 0 ( p 0 & ϕ p & ϕ ) dt. Left hand sde of eq.(4) represents the ncrease of the total energy of the -th pendulum. For the perodc oscllatons of the pendula (and the beam) the angular veloctes of the rotatng pendula fluctuate around the constant mean value so ths ncrease has to be equal zero T 0 T ml & ϕ & ϕdt + ml g & ϕ snϕdt = 0. (5) 0 The frst component of the rght hand sde of eq. (4) gves the energy dsspated by the vscous dampers c φ : W DAMP = T 0 1 c ϕ & ϕ dt. (6) The next component of the rght hand sde of eq. (4) descrbes the energy transferred by the - th pendulum to the beam (when t s postve) or the energy transferred from the beam to the - th pendulum: W SYN = T 0 m l & x cos ϕ & ϕ dt. (7) The last component of the rght hand sde of eq.(4) gves the energy suppled to the -th pendulum by the drvng torque: 0 1 (4)

15 W DRIVE = T 0 ( p 0 & ϕ p & ϕ ) dt. (8) 1 Substtutng eqs. (5-8) nto eq. (4).e., DRIVE DAMP W SYN = W + W. (9) one obtans pendula energy balances Energy balance of the beam Multplyng eq.() by beam s velocty one gets: n n mb + m & xx && + cx x& + k x xx& = ml x& ( && ϕ cosϕ + & ϕ snϕ ). (10) = 1 = 1 and assumng that beam s oscllatons are perodc and ntegratng eq.(10) over perod T we obtan the energy balance of the beam: T n T T T n mb + m & xxdt && + k x xxdt & = cx x& dt + ml ( && ϕ cosϕ + & ϕ snϕ ) x& dt. (11) 0 = = 1 Left hand sde of eq.(11) represents the ncrease of the total energy of the beam. As the oscllatons are perodc ths ncrease should be equal zero: T n T m B + m & xxdt && + k x xx& dt = 0 (1) 0 = 1 0 Frst component on the left hand sde of eq.(11) descrbes the energy dsspated by the vscous damper c x durng one perod of oscllatons: T W beam = cx x& dt. (13) 0 The next component gves the energy whch s suppled to the beam by the pendula (the sum of the works performed durng the perod T by the forces wth whch pendula act on the beam): = n T n SYN ( W ) = ml ( & ϕ cosϕ + & ϕ snϕ ) x& dt. (14) = 1 0 = 1 Substtutng eqs.(1-14) nto eq. (11) we get the energy balance of the beam n the followng form: = = n SYN W beam ( W ). (15) =1 Substtutng eq.(9) nto eq. (15),.e., DRIVE DRIVE DAMP DAMP W 1 + W = W1 + W1 + Wbeam. (16) we get the energy balance of system (1,) Synchronzaton condtons - lnearzed model In ths secton we derve the approxmate analytcal condtons for synchronzaton of rotatng pendula. Followng the dea of Blekhman [5] to explan the phenomena of synchronzaton we determne and analyze the work done by the momentum wth whch the -th pendulum acts on beam - W SYN. Let us assume that the dampng n the system s small,.e., 15

16 W beam 0.0, DAMP W 0.0. When the oscllatons are perodc and there s no energy dsspaton there s no need for the energy supply, so DRIVE W = 0. (18) From eqs. (9) we have W = 0. (19) SYN () Pendula rotatng n the same drectons As n [5], we assume that the pendula's angular veloctes are constant,.e., the fluctuatons of the pendula's angular veloctes caused by the moton n the gravtatonal feld are so small that can be neglected. Hence, the pendula's acceleratons are equal to zero and n the case, when the pendula rotate n the same drecton, one gets lnear functons descrbng the pendula's angles of rotaton: & ϕ = ω, ϕ = ωt + β, && ϕ = 0. Rght hand sde of eq.() descrbes the force wth whch n pendula are actng on beam: F = n = 1 m l ( & ϕ cosϕ + & ϕ snϕ ). (1) Substtutng eq. (0) nto eq.(1) one gets: n F = ( m l ω sn( ωt + β )). () = 1 Substtutng eq. () nto eq. () and denotng U = m B + m, n = 1 one gets: U& x + c x& + k x = n ( m l ω sn( ωt + β )). (3) x x = 1 Assumng that the dampng coeffcent c x s small the oscllatons of the beam can be descrbed n the followng way: n ω ( m l sn( ωt + β )) x =, kx ω U = 1 (4) 4 n ω && x = ( sn( + )). ml ωt β kx ω U = 1 In the equaton of moton of each pendula (1) we have the component whch has been dentfed as a synchronzaton momentum. Substtutng eqs. (0,4) nto eq. (7) and denotng 4 ω A = kx ω U one gets: (17) (0) 16

17 T n SYN Wk = mklk A ( ml sn( ω t + β) ) ω cos( ωt + βk ) dt = 0. (5) 0 = 1 After some calculatons one gets n n = cos sn sn cos = n SYN Wk mklkπ A βk ml β βk ml β mklkπa ml sn( β βk ) = 0. (6) = 1 = 1 = 1 Eq. (6) allows the calculaton of the phase angles β k for whch the synchronzaton takes place and the pendula rotate perodcally. Eq. (6) s fulflled for β 1 = β =... = β n (7) and the pendula reach complete synchronzaton or when n n ml cos β = 0, ml sn β = 0, (8) = 1 = 1 and we observe phase synchronzaton. Note that the synchronzaton condtons gven by eqs. (7-8) are dentcal to those obtaned by Blekhman usng small parameter methods [5]. In the case of dentcal pendula condton (8) s smplfed to the followng form cosβ + cosβ cosβn = 0, (9) sn β + sn β sn β n = 0. It can be shown that eqs. (9) are fulflled for the followng phase angles ( 1) β = π, = 1... n. (30) n For n= (two pendula) eq.(30) gves β 1 =0, β =0 and β 1 =0, β =π. For n=3 (three pendula) eq.(30) gves β 1 =0, β =0 (as observed n Fgure 5(a)) and β 1 =0, β =π/3 and β 3 =4π/3 (Fgure 5(b). In the consdered case the synchronzaton state of the pendula moton s the perodc moton of the system (1,) n whch phase angles β fluctuate around constant mean values (characterstc for a gven confguraton). The mean values of β are ndependent of ntal condtons (n the basn of attracton of the partcular confguratons) and not senstve to the external perturbatons. In ths state, when the pendula are dentcal there s no energy transfer between the pendula va the beam. () Pendula rotatng n dfferent drectons In the case when m of the total n pendula rotates n the opposte drecton to the rest (.e., n-m) of the pendula the phase dfferences between pendula n the synchronous confguratons have to be calculated from dfferent equatons. Let assume the pendulum 1 rotates clockwse and the other pendula counterclockwse. In ths case the lnearzed equatons descrbng pendula's dsplacements and angular veloctes are as follows: & ϕ = ω, 1 & ϕ = & ϕ =... = & ϕ = ω, ϕ = ωt + β, 1 ϕ = ωt + β,... ϕ = ωt + β, n && ϕ = && ϕ =... = && ϕ = n n n (31) 17

18 Repeatng the calculatons presented n Sec one gets the followng equatons (equvalent to eqs. (6)): W W W... W SYN 1 SYN SYN 3 SYN n = Aπ = Aπ = Aπ = Aπ ( m l m l sn( β β ) + m l m l sn( β + β ) m m sn( β + β )) 1 1 = 0 ( m l m l sn( β + β ) + m l m l sn( β β ) m m sn( β β )) ( m l m l sn( β + β ) + m l m l sn( β β ) m l m l sn( β β )) = 0 = 0 ( m l m l sn( β + β ) + m l m l sn( β β ) m l m l sn( β β )) = 0 n n n n n 1 Eqs.(3) allow the calculaton of the value of phase angles β at whch the moton of pendulums synchronzaton occurs, and thus the moton of the system s perodc. In the case of n= dentcal pendula, and assumng that β 1 =0, eqs.(3) get the form of two dentcal equatons: sn( β ) = 0 (33) sn( β ) = 0 whch are fulflled n two cases: () β =0 - the mrror-synchronzaton (M), () β =π - the antphase synchronzaton (A) as gven n Table 1. type of synchronzaton phase dfference between pendula mrror-synchronzaton (M) β 1 =0, β =0. 3 n 1 n 3 3 n n n 1 n n n n n 3 n n pendula's confguraton n n (3) antphase synchronzaton (A) β 1 =0, β =π Table 1. Types of synchronzaton observed for n= pendula rotatng n opposte drectons. type of synchronzaton tree-synchronzaton (T) phase dfference between pendula β 1 =0, β =-π/3, β 3 =-5π/3 pendula's confguraton cluster-antphase synchronzaton (CA) β 1 =0, β =β 3 = π Table. Types of synchronzaton observed for n=3 pendula (pendulum 1 rotates n opposte drecton to pendula and 3). For three dentcal pendula, assumng that β 1 =0, eqs.(3) get the followng form: 18

19 sn( β ) + sn( β3) = 0 sn( β ) sn( β β3) = 0 (34) sn( β3) sn( β3 β ) = 0 whch are fulflled for: () β =-π/3 and β 3 =-5π/3 - the tree-synchronzaton (T), () β =β 3 = π the cluster-antphase synchronzaton (CA) as gven n Table. 19

20 4. Numercal examples 4.1. Two pendula rotatng n the same drecton In ths example we consder the system (1-) wth the followng parameter values: m 1 =m =1.00, l 1 =l =0.5, c ϕ1 =c ϕ =0.01, p 01 =p 0 =5.00, p 11 =p 1 =0.50, m B =6.00. One can calculate that 10.0 and U=8.0. We consder dfferent values of stffness coeffcent k x of the sprng connectng the beam m B wth a fxed foundaton so the beam can oscllate above or below the resonance,.e., the frequency k x α = x U (35) s smaller or larger than the pendulum's 1 angular velocty. The dampng coeffcent c x has been selected n such a way as to be equvalent to the arbtrarly selected logarthmc decrement of dampng Δ=ln(1.5). As such a dampng does not sgnfcantly change the perod of the beam's free oscllatons c x can be calculated from the formula Δ k xu cx =. (36) π Typcal tme seres of pendula veloctes and dsplacements are shown n Fgure 9(a,b). The unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty. Fgure 9(a) shows the angular veloctes of pendula ϕ& 1and ϕ& for a system wth low stffness coeffcent k x =100.0, so 10.0/ The followng ntal condtons have been consdered: ϕ 10 =0, ϕ 0 =π/4, & ϕ 10 = & ϕ0 = As one can see, after the transent the phase dfference between the pendula veloctes tends to π. Fgure 9(b) shows the angular dsplacement of pendulum, ϕ ϕ1, related to the dsplacement of the frst pendulum. One can notce that after the decay of the transent, ths angle oscllates around a constant average value π and the system reaches the state of antphase synchronzaton. Ths state for small values of k x s reachable for any ntal condtons. Shown n both fgures the fluctuatons of angular veloctes and dsplacements, are caused by the weght of pendula,.e., pendulum. speeds durng the moton down and slows when ts mass rse up. 0

21 Fgure 9. Tme seres of pendula veloctes and dsplacements calculated from eqs (1,): m 1 =m =1.00, l 1 =l =0.5, c ϕ1 = c ϕ =0.01, p 01 =p 0 =5.00, p 11 =p 1 =0.50, m B =6.00, 10.0 and U=8.0 (the unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty ); (a) angular veloctes ϕ& 1and ϕ& for a system (1,) wth low stffness coeffcent k x =100.0, ϕ 10 =0, ϕ 0 =π/4, & ϕ 10 = & ϕ0 = 0 ; (b) angular dsplacement of pendulum : ϕ ϕ1 related to the dsplacement of the frst pendulum. The pendula confguratons characterstc for the system (1-) wth n= pendula and ts basns of attracton are shown n Fgure 10(a-d). Fgure 10(a) presents the confguraton of antphase synchronzaton wth β 1 =0 and β =π. Notce that the same values of β 1 and β can be calculated analytcally from eq.(30) and condton (9) s fulflled. The confguraton complete synchronzaton s presented n Fgure 10(b). Ths confguraton s observed for larger values of coeffcent k x when condton (7) s fulflled. Fgure 10(c) shows the basns of attracton of the complete (whte color) and ant-phase (gray color) synchronzaton states n the system wth a stffness coeffcent of k x = The basns are shown n the ϕ 10 -ϕ 0 plane wth fxed ntal veloctes & ϕ 10 = & ϕ0 = 0. These basns for systems wth dfferent values of the stffness coeffcent k x, shown on the plane k x -ϕ 0 (ϕ 10 =0, & ϕ 10 = & ϕ0 = 0 ) are presented n Fgure 10(d). The results of Fgure 10(d) and predctons of [5] are sgnfcantly dfferent. Blekhman [5] predcts the exstence of complete (for 800.0) and antphase (for 800.0) ndependently of ntal condtons. Meanwhle, Fgure 10(d) shows, that the boundary between the basns of attracton of complete and antphase synchronzatons s located sgnfcantly below the value of and s not horzontal. For k x <k x , ndependently of ntal condtons one observes the antphase synchronzaton whle for k x1 <k x <k x there exsts the coexstence of complete and antphase synchronzatons. In the nterval k x <k x <k x ndependently of ntal condtons the system (1,) reaches the state of complete synchronzaton and for larger values of k x >k x3 we have the coexstence of both synchronzaton states agan. 1

22 Fgure 10. The pendula confguratons characterstc for the system (1,) wth n= pendula and ts basns of attracton; (a) confguraton of antphase synchronzaton wth β1=0 and β=π, (b) complete synchronzaton, (c) basns of attracton of the complete (whte color) and ant-phase (gray color) synchronzaton states, k x = (the basns are shown n the plane wth fxed ntal veloctes 0, (d) basns attracton for dfferent values of the stffness coeffcent k x, shown on the plane, ( 0) 4.. Three pendula rotatng n the same drecton Let us consder the system (1-) wth the followng parameter values: m 1 =m = m 3 =1.00, l 1 =l =l 3 =0.5, c ϕ1 =c ϕ =c ϕ3 =0.01, p 01 =p 0 = p 03 =5.00, p 11 =p 1 = p 13 =0.50, m B =6.00. One can calculate that 10.0 and U=9.0 (due to n=3). The values of stffness and dampng coeffcents k x and c x have been taken as n prevous secton. Typcal tme seres of pendula veloctes and dsplacements n the case of phase synchronzaton are shown n Fgure 11(a,b). Fgure 11(a) shows the angular veloctes of pendula, and for a system wth low stffness coeffcent k x=100.0 and 10.0/ The followng ntal condtons have been consdered: 0,,, 0. As n the prevous plots (Fgure 9(a,b)) the unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty. As one can see, after the decay of transents the phase dfference between the pendula veloctes tends to

23 the constant value of π/3. Fgure 11(b) shows the angular dsplacement of pendula, and, related to the dsplacement of the frst pendulum. One can notce that these angles n what follows referred as the relatve dsplacements oscllate around a constant average values π/3 and 4π/3. Such a state of phase synchronzaton s obtaned for k x =100.0 and arbtrary ntal condtons. Shown n both fgures the angular velocty fluctuatons and movements n relatve terms, are caused by the moton n gravtatonal feld. Numercally estmated phase shfts β=π/3 and β3=4π/3 are n good agreement wth the values calculated analytcally from eqs.(30). Fgure 11. Tme seres of pendula veloctes and dsplacements n the case of phase synchronzaton,. (a) angular veloctes of pendula, and for a system (1,) wth low stffness coeffcent k x =100.0 and 3.33, 0,,, 0 (the unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty ), (b) angular dsplacement of pendula and related to the dsplacement of the frst pendulum, β=π/3, β3=4π/3. In another example, t s assumed that k x =3600.0, so / and the system of the beam and three pendula eqs. (1,) s below the resonance. We consder the followng ntal condtons: 0,,, Fgure 1(a,b) shows that after a transtonal perod the angular veloctes of all three pendula are the same and the relatve dsplacements and tend to zero, so one observes the state of complete synchronzaton. Due to the exstence of gravtatonal feld we observe the fluctuatons of the pendula moton caused by ther weghts. 3

24 Fgure 1. Tme seres of pendula veloctes and dsplacements n the case of the complete synchronzaton,. (a) angular veloctes of pendula, and for a system (1,) wth low stffness coeffcent k x = and 0.0, 0,,, 0 (the unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty ), (b) angular dsplacement of pendula and related to the dsplacement of the frst pendulum, β=β3=0. In the system (1,) wth a stffness coeffcent of k x = and dfferent ntal condtons (for example 0,,, 0 one observes a dfferent type of synchronzaton as shown n Fgure 13(a,b). After a transtonal perod angular veloctes ϕ& 1 andϕ& 3 tend to each other and are dfferent than ϕ& ; relatve dsplacement ϕ3 ϕ1 reaches a constant value π, so ϕ 3 = ϕ1, and ϕ ϕ1 =π. Two pendula 1 and 3 create a cluster whch s n ant-phase wth pendulum. Fgure 13. Tme seres of pendula veloctes and dsplacements n the case of the antphase synchronzaton of pendulum wth a cluster consstng of pendula 1 and 3; (a) angular veloctes of pendula, and for a system (1-) wth low stffness coeffcent k x = and 0.0, 0,,, 0 (the unt of tme on the horzontal axs s the number /,.e., the number of complete revolutons of the pendulum rotatng wth constant angular velocty ), (b) angular dsplacement of pendula and related to the dsplacement of the frst pendulum, β=π, β3=π.. 4

25 Fgure 14. Synchronzaton confguratons n the system (1,) wth n=3 pendula; (a) phase synchronzaton wth phase shfts between pendula: β1=0, β=π/3 and β3=4π/3, (b) complete synchronzaton (β1=β=β3=0), (c) antphase synchronzaton of a sngle pendulum wth the cluster of two other pendula, β1=0, β=π and β3=0. Fgure 15. Basns of attracton of the dfferent states of pendula synchronzaton, (a) basns of attracton of complete (whte color) and ant-phase (gray color wth dfferent shades for dfferent pars of pendula n the cluster) synchronzaton states for a system (1,), k x = (the basns are shown n the plane, 0, 0, (b) basns of attracton of complete (whte color), ant-phase (gray color n dfferent shades for dfferent pars of pendula n the cluster) and phase (dark gray color at the bottom) for dfferent values of stffness coeffcent k x, 0,, 0. 5

26 Our numercal results show that n the system (1-) wth n=3 pendula three dfferent confguratons of synchronzed pendula are possble, as shown n Fgure 14(a-c). Fgure 14(a) presents the phase synchronzaton wth phase shfts between pendula: β 1 =0, β =π/3 and β 3 =4π/3 (condton (1) s fulflled) whch exsts for suffcently small values of k x <370 (regardless of ntal condtons). Complete synchronzaton (β 1 =β =β 3 and condton (0) s fulflled) whch exsts for the approprate values of k x (370<k x <1880) regardless of ntal condtons and whch for suffcently large values of k x (k x >1880) coexsts wth antphase synchronzaton s descrbed n Fgure 14(b). In contrast to the prevously studed systems wth oscllatng pendula [17-19] one can observe the phenomenon of antphase synchronzaton of a sngle pendulum wth the cluster of two other pendula. Fgure 14(c) presents the ant-phase synchronzaton β 1 =0 (or β =0 or β 3 =0) and two other phase shft angles equal to π (condton (1) s fulflled). Ths confguraton co-exsts wth a complete synchronzaton for suffcently large values of k x (k x >1880). Dependng on ntal condtons the cluster s created of pendula 1-, 1-3 or -3. The basns of attracton of dfferent states of pendula synchronzaton are shown n Fgure 15(a,b). Fgure 15(a) shows the basns of attracton of complete (whte color) and antphase (gray color wth dfferent shades for dfferent pars of pendula n the cluster) synchronzaton states for a system wth stffness coeffcent k x = The basns are shown n the plane ( 0, 0). Fgure 15(b) shows the basns of attracton of complete (whte color), ant-phase (gray color n dfferent shades for dfferent pars of pendula n the cluster) and phase (dark gray color at the bottom) for dfferent values of stffness coeffcent k x. The followng ntal condtons have been consdered: 0,, 0. The results obtaned by the numercal ntegraton of equatons of moton (1,) are sgnfcantly dfferent from the results obtaned by the method of small parameter [5]. For example Blekhman [5] predcts the exstence of complete and phase synchronzaton (the second one wth the same phase shfts as n our studes,.e., π/3 and 4π/3). It has been stated that for the systems wth stffness coeffcent ndependently of ntal condtons the phase synchronzaton occurs whle for larger values of the complete synchronzaton takes place. Contrary to ths statement Fgure 15(a,b) shows that the boundary between the basns of attracton phase and complete synchronzaton takes place at the level k x =370.0 (almost three tmes lower). Another sgnfcant dfference between our results and these of [5] s the exstence of ant-face synchronzaton of a sngle pendulum and a cluster consstng of two pendula ([5] does not prescrbe such confguraton). For k x > ths confguraton co-exsts wth a complete synchronzaton of all pendula. Notce that the method of small parameter used n [5] does not allow the dentfcaton of the coexstng confguratons Large system of pendula rotatng n the same drecton We studed the systems wth up to 100 rotatng pendula. It has been found that for larger n same types of synchronzaton are observed. Ther examples are shown n Fgures Fgure 16(a,b) presents the phase synchronzaton of n=0 pendula n the system (1-) wth π k x =1000.0, M=0.0, and the followng ntal condtons: ϕ 0 =, ϕ& =. Fgure 16(a) shows that pendula veloctes & ϕ,...,ϕ& oscllate around the average value close to 1 0. Angular dsplacements ϕ ϕ1 tend to the constant values whch dffer by π/10 as can be seen n Fgure 16(b). The complete synchronzaton of 0 pendula s descrbed n Fgure 17(a,b). 6

27 7π We consder the system (1-) wth k x =0000.0, M=0.0 and ntal condtons : ϕ 0 =, 180 ϕ& = The veloctes of all pendula & ϕ,...,ϕ& 1 0 oscllate around the constant average value (Fgure 17(a)) and angular dsplacements ϕ ϕ1 tend to zero,.e., the dsplacements of all pendula are the same (Fgure 17(b)). The example of the synchronzaton n clusters s presented n Fgure 18(a,b). We consder the same system as n the prevous example wth the π followng ntal condtons: ϕ 0 =, ϕ& = 0.. Fgure 18(a) shows that pendula veloctes & ϕ,...,ϕ& oscllate around the average value close to 1 0. The angular dsplacements ϕ ϕ1 tend to two constant values 0 or π as can be seen n Fgure 18(b). Two clusters of synchronzed pendula have been created (they consst of 7 and 13 pendula). The clusters are synchronzed n antphase. Contrary to the case of oscllatng pendula [17,18] rotatng pendula are not grouped n three or fve clusters only. The lack of ths restrcton causes that n the system (1,) dependng on ntal condton one can observe a great varety of dfferent clusters confguratons. The number of confguratons grows wth a number of pendula n. Fgure 16. Phase synchronzaton of n=0 pendula n the system (1-): m 1 =m = =m 0 = 1.00, l 1 =l = =l 0 =0.5, c ϕ1 =c ϕ = =c ϕ0 =0.01, p 01 =p 0 = =p 00 =5.00, p 11 =p 1 = = p 10 =0.50, m b π =0.00, k x =1000.0, ϕ 0 =, ϕ& = 0. ; (a) pendula veloctes ϕ&, (b) angular dsplacements ϕ ϕ 1. 7

28 Fgure 17. Complete synchronzaton of 0 pendula n system (1-): ): m 1 =m = =m 0 = 1.00, l 1 =l = =l 0 =0.5, c ϕ1 =c ϕ = =c ϕ0 =0.01, p 01 =p 0 = =p 00 =5.00, p 11 =p 1 = = p 10 =0.50, m B 7π =0.00, k x =0000.0, ϕ 0 =, ϕ& = 0. ; (a) pendula veloctes ϕ&, (b) angular dsplacements ϕ ϕ1. Fgure 18. Cluster synchronzaton of 0 pendula n system (1-): m 1 =m = =m 0 = 1.00, l 1 =l = =l 0 =0.5, c ϕ1 =c ϕ = =c ϕ0 =0.01, p 01 =p 0 = =p 00 =5.00, p 11 =p 1 = = p 10 =0.50, m b π =0.00, k x =0000.0, ϕ 0 =, ϕ& = 0. ; (a) pendula veloctes ϕ&, (b) angular dsplacements ϕ ϕ 1. Clusters of 7 and 13 pendula are synchronzed n antphase. 4.4 Two pendula rotatng n the opposte drectons Now, let us consder the system (1-) wth the followng parameter values: m 1 =m =1.00, l 1 =l =0.5, c ϕ 1=c ϕ =0.01, p 01 =5.00, p 0 = -5.00, p 11 =p 1 =0., m B =6.00. One can calculate that 10.0, 10.0 and U=8.0. The values of stffness and dampng coeffcents k x and c x have been taken as n prevous secton. Fgure 19(a) presents tme seres of the pendula's angular veloctes ϕ& 1and ϕ& for the small value of the stffness coeffcent k x =500.0, so 500.0/ The pendula rotate n opposte drectons wth constant veloctes & ϕ 10 = & ϕ1 N = 10. 0[s -1 ], & ϕ & ϕ 10.0 [s -1 ] startng from the ntal postons: ϕ 10 =0, ϕ 0 =π/4. One can see that after 8 0 = N

29 the ntal transent (several rotatons) caused by the oscllatons of the beam the pendula's veloctes fluctuate (due to the gravty) around the ntal values and. Fgure 19(b) shows tme seres of the sum of pendula's dsplacements ϕ + ϕ1. One can see that ths sum (after the ntal transent) s constant and equal to zero so ϕ = ϕ1. Ths type of synchronzaton s the mrror-synchronzaton (M) as the rotatonal moton of pendulum s the mrror mage of the rotatons of pendulum 1 as can be seen at the dagram shown n Fgure 19(b). For the small value of the stffness coeffcent k x =500.0 and dfferent ntal condtons ϕ 10 =0, ϕ 0 =-43π/36 one can observe dfferent type of synchronzaton as shown n Fgure 1(c,d). After the ntal transent the pendula's veloctes (as n the prevous case) fluctuate around the ntal values and (Fgure 19(c)). The sum of pendula's dsplacements ϕ + ϕ 1 fluctuates around the constant averaged value < ϕ + ϕ1 >= π as shown n Fgure 19(d). We call ths type of synchronzaton the antphase-synchronzaton (A). For the larger values of k x the next type of synchronzaton can be observed. Fgure 0(a,b) llustrates pendula's synchronzaton for large values of the stffness coeffcent k x = and / We consder the followng ntal condtons: ϕ 10 =0, ϕ 0 =-3π/. After the ntal transent the sum of pendula's dsplacements ϕ + ϕ1 fluctuates around constant averaged value < ϕ + ϕ1 > close to -3π/ as shown n Fgure 0(a). We call ths type of synchronzaton the thrd-quarter-synchronzaton (3Q). When pendulum 1 passes through the statc equlbrum poston pendulum approaches the horzontal plane of symmetry (ϕ 1 = 0 ϕ -3π/). Fgure 19. Mrror-synchronzaton (M) and antphase-synchronzaton (A) of pendula; (a) pendula's veloctes durng mrror-synchronzaton, k x =500, ϕ 10 =0, ϕ 0 =-π/4; (b) pendula's dsplacements durng mrror-synchronzaton, k x =500, ϕ 10 =0, ϕ 0 =-π/4; (c) pendula's veloctes durng antphase-synchronzaton, k x =500, ϕ 10 =0, ϕ 0 =-1.19; (d) pendula's dsplacements durng antphase-synchronzaton, k x =500, ϕ 10 =0, ϕ 0 =

30 Fgure 0. Thrd-quarter-synchronzaton (3Q) and frst-quarter-synchronzaton (1Q) of pendula; (a) pendula's dsplacements durng thrd-quarter-synchronzaton: k x =3000.0, ϕ 10 =0, ϕ 0 =-3π/; (b) pendula's dsplacements durng frst-quarter-synchronzaton: k x =3000.0, ϕ 10 =0, ϕ 0 =-π. Fgure 1. The nfluence of the stffness coeffcent k x on the type of synchronzaton of pendula: (a) average value of the sum of the pendula's dsplacements <ϕ +ϕ 1 > versus k x : ϕ 10 =0, ϕ 0 =-3π/; (b) average value of the sum of the pendula's dsplacements <ϕ +ϕ 1 > versus k x : ϕ 10 =0, ϕ 0 =-π; (c) mrror- and antphase- synchronzaton for dfferent ntal condtons: k x =500; (d) thrd- and frst- quarter-synchronzaton for dfferent ntal condtons: k x =

31 Fgure. Dfferent types of synchronzaton of pendula versus stffness coeffcent k x and ntal poston of pendulum ϕ 0 : ntal poston of pendulum 1 ϕ 10 =0. For dfferent ntal condtons: ϕ 10 =0, ϕ 0 =-π/, after the ntal transent the sum of pendula's dsplacements ϕ + ϕ1 fluctuates around constant averaged value < ϕ + ϕ1 > closed to -π/ as shown n Fgure 0(b). Ths type of synchronzaton has been called the frstquarter-synchronzaton (1Q). When pendulum 1 passes through the statc equlbrum poston pendulum leaves the horzontal plane of symmetry (ϕ 1 =0 ϕ -π/). The pendula's confguratons durng (1Q) and (3Q) synchronzatons are shown at dagrams n Fgure 13(a,b). (3Q) and (1Q) synchronzatons are not observed when one neglects the effect of gravty or when the pendula rotate n the horzontal plane. In both cases the pendula's veloctes oscllate around the ntal values and as n the examples shown n Fgure 1(a,c). The nfluence of the stffness coeffcent k x and ntal condtons on the type of synchronzaton s dscussed n Fgure 1(a-d). Fgure 1(a) presents the averaged value of the sum of pendula's dsplacements <ϕ +ϕ 1 > versus stffness coeffcent k x. For all values of k x the moton of the system s ntated from the same ntal condtons. In Fgure 1(a) we show the averaged value of the sum of pendula's dsplacements <ϕ +ϕ 1 > versus the stffness coeffcent k x for the followng ntal condtons: ϕ 10 =0, ϕ 0 =-3π/. One can see that for the small values of the stffness coeffcent k x <360.0, the value of <ϕ +ϕ 1 >=0 and the system s n the state of mrror-synchronzaton (M). For k x =360.0 the value of <ϕ +ϕ 1 > jumps to -π and n the nterval 360.0<k x < we observe antphase-synchronzaton (A). For k x = the next jump of <ϕ +ϕ 1 > (to the value of -4π/3) occurs and the type of synchronzaton s changed to the thrd-quarter-synchronzaton (3Q). In the nterval <k x < [N/m] n the state of the thrd-quarter-synchronzaton (3Q) the value of <ϕ +ϕ 1 > ntally decreases down to the value -1.41π and later ncreases up to the value -π, so we observe the return to the state of antphase-synchronzaton (A). Fgure 1(b) shows the value of <ϕ +ϕ 1 > versus k x for dfferent ntal condtons (we change the value of ϕ 0 from ϕ 0 =-3π/ to ϕ 0 =3π/). As n Fgure 1(a) for small values of k x frst we observe the state of the mrror-synchronzaton (M) and next the antphase-synchronzaton (A). The jump of the value of <ϕ +ϕ 1 > to -0.65, observed for k x =1960.0, ndcates the change of the type of synchronzaton to the frstquarter (1Q). In the nterval <k x < n the state of the frst-quartersynchronzaton, the value of <ϕ +ϕ 1 > ntally ncreases up to and next decreases down to -π, so we observe the return to the state of antphase-synchronzaton (A). Fgure 1(c) 31

32 presents the nfluence of the ntal condtons ϕ 10 and ϕ 0 on the type of synchronzaton for the small value k x =500.0 (types (M) and (A) are observed) whle Fgure 1(d) shows basns of (3Q) and (1Q) for large value of k x = The nfluence of the stffness coeffcent k x and ntal poston of pendulum - ϕ 0 on the type of synchronzaton s dscussed n Fgure. We assume the ntal poston of the pendulum 1 - ϕ 10 =0. One can see that for small values of the stffness coeffcent k x <45.0 and any value of ϕ 0 the mrror-synchronzaton (M) occurs. In the nterval 45.0<k x <760.0, dependng on ntal condton ϕ 0 one observes mrror (M) or antphase (A) synchronzaton. (A) and (M) types of synchronzaton observed for k x =500.0 are shown n Fgure 1(c). In the nterval 760.0<k x < for any value of ϕ 0 antphase-synchronzaton (A) occurs. For k x > dependng on ntal condton ϕ 0 we observe the thrd-quarter (3Q) or the frstquarter (1Q) synchronzaton. (1Q) and (3Q) types of synchronzaton observed for k x = are shown n Fgure 1(d) Three pendula rotatng n varous drectons In the smulatons of the system of three pendula, we use the same parameter values as n prevous example, and addtonally consder p 03 =-5.00,.e., pendulum 1 rotates counterclockwse and pendula and 3 clockwse. Fgure 3(a-d) shows tme seres of the sum of pendula's dsplacements ϕ + ϕ1 and ϕ 3 + ϕ 1 durng four dfferent synchronous states. In Fgure 3(a) we present tme seres for the case of small stffness coeffcent k x =00.0, so 00.0/ and the followng ntal condtons: ϕ 10 =0, ϕ 0 =-π/3, ϕ 30 =- 4π/3. After the ntal transent the sum of pendula's dsplacements ϕ + ϕ1 fluctuates around constant averaged value approxmately equal to - π/3, and the sum ϕ 3 + ϕ1, fluctuates around the averaged value close to Ths type of synchronzaton we call the tree synchronzaton (T). The pendula's confguraton for ϕ 1 =0 s shown at the dagram n Fgure 3(a). Increasng the value of the stffness coeffcent to k x = (so / and the beam oscllatons are above the resonance) and changng the ntal postons of the pendula to ϕ 10 =0, ϕ 0 =-π, ϕ 30 =-π/ (other ntal condtons are the same as n Fgure 3(a) one observes the synchronous state n whch pendula and 3 (rotatng to the left) create the cluster (ther dsplacements are dentcal) as shown n Fgure 3(b).The sum of the dsplacements of any pendulum n cluster and pendulum 1 ϕ + ϕ1 = ϕ 3 + ϕ1 fluctuates around constant average value < ϕ + ϕ1 >=< ϕ 3 + ϕ1 > approxmately equal to -π and we observe the cluster-antphasesynchronzaton (CA). The pendula's confguraton durng ths type of synchronzaton for ϕ 1 =0 s shown at the dagram n Fgure 3(b). Addtonally n the system wth three pendula one can observe four new types of synchronzaton whch occur due to the exstence of gravty and the change of the ampltude and phase of the beam's oscllatons (as the result of the ncreased value of the stffness coeffcent k x ). For k x = and ntal condtons ϕ 10 =0, ϕ 0 =-1.38, ϕ 30 =-1.44, we observe the type of synchronzaton smlar to (T) synchronzaton but wth obtuse angles between pendulum 1 and pendula,3 as shown at the dagram n Fgure 3(c). We call ths synchronzaton the yankee 3 (Y3) -synchronzaton. For dfferent ntal condtons one can observe the pendula's confguraton whch s the mrror mage of (Y3),.e., pendulum 3 s on the rght sde and pendulum on the left sde of the dagram. Ths confguraton s called the yankee 3 (Y3) synchronzaton. For the same value of the stffness coeffcent k x and ntal condtons: ϕ 10 =0, ϕ 0 =-1.38, ϕ 30 =-π we observe the type of 3

33 synchronzaton shown n Fgure 3(d). Ths synchronzaton s smlar to (CA) synchronzaton, but the angle between the cluster (of pendula and 3) and pendulum 1 s approxmately equal to -3π/ (the cluster-rght-synchronzaton (CR)) or -π/ (the clusterleft-synchronzaton (CL)). Fgure 3: Dfferent types of synchronzaton of 3 rotatng pendula: (a) tree synchronzaton (T), k x =00.0[N/m], ϕ 10 =0, ϕ 0 =-π/3, ϕ 30 =-4 π/3; (b) cluster-antphase-synchronzaton (CA), k x =1000.0[N/m], ϕ 10 =0, ϕ 0 =-π, ϕ 30 =-π/; (c) yankee-3-synchronzaton (Y3), k x =3000.0[N/m], ϕ 10 =0, ϕ 0 =-1.38, ϕ 30 =-1.44; (d) cluster-rght-synchronzaton (CR), k x =3000.0[N/m], ϕ 10 =0, ϕ 0 =-1.38, ϕ 30 =-π. 33

34 Fgure 4: The nfluence of the stffness coeffcent k x on the type of synchronzaton of 3 pendula shown as the average values of the sums of pendula's dsplacements <ϕ +ϕ 1 > and <ϕ 3 +ϕ 1 > versus k x : (a) ϕ 10 =0, ϕ 0 =-0.75, ϕ 30 =-1.87; (b) ϕ 10 =0, ϕ 0 =-0.75, ϕ 30 =-π; (c) ϕ 10 =0, ϕ 0 =-0.75, ϕ 30 =-0.44; (d) ϕ 10 =0, ϕ 0 =-0.75, ϕ 30 =

35 Fgure 5: Dependence of the type of synchronzaton on the stffness rato k x and ntal condtons: (a) type of synchronzaton versus k x and ϕ 30 : ϕ 10 =0, ϕ 0 =-0.75; A-A cross-secton of map (c); (b) the enlargement of map (a) for small values of k x, B-B cross-secton map (d); (c) the type of synchronzaton for dfferent ntal condtons ϕ 0 and ϕ 30 ; ϕ 10 =0, k x =3000, C- C cross-secton map (a); (d) the type of synchronzaton as functon ϕ 0 and ϕ 30 ; ϕ 10 =0, k x = 35, D-D cross-secton map (b). Fgure 4(a-d) shows the nfluence of stffness coeffcent k x on the type of the synchronous state. The averaged values of the sums of pendula's dsplacements < ϕ + ϕ1 > and < ϕ 3 + ϕ1 > versus k x are shown. For all values of k x the moton of the system s ntated from the same ntal condtons. In the system wth small stffness coeffcent k x one observes (T) type synchronzaton as can be seen n Fgure 4(a-d). For larger values of k x we observe (CA) synchronzaton and fnally for k x > (exact value depends on φ 30 ) two other types of synchronzaton (CR) and (Y3) and ther mrror mages (CL) and (Y3) are possble. Fgure 4(a-d) shows that the values < ϕ + ϕ1 > and < ϕ 3 + ϕ1 > are changng wth the change of k x, so the descrptons of the pendula's confguratons n dfferent types of synchronzatons wth the statements about the angles close to π n the case of (CA) or 3π/ and π/ n the case of (CR) 35

36 and (CL) present only the qualtatve dfferences. Partcularly n Fgure 4(c) the dstncton between (CA) and (CL) synchronzaton s arbtrary, due to the contnuous change of the angle between cluster (pendula and 3) and pendulum 1. In the other cases the dstncton between dfferent types of synchronzaton s justfed by the jump changes of angles < ϕ + ϕ1 > and (or) < ϕ 3 + ϕ1 >. The cross sectons shown n Fgure 4(a-d) are ndcated n Fgure 5(a). Fgure 5(a) shows the basns of exstence of dfferent types of synchronzaton on the plane k x -ϕ 30. We consder the followng ntal condtons ϕ 10 =0, ϕ 0 = Fgure 5(b) shows the enlargement of Fgure 5(a) for 00<k x <400. From Fgure 5(a,b) one can conclude that for stffness coeffcent k x < one can observe ether (T) (k x < ) or (CA) synchronzaton ( <k x <00 400). The type of synchronzaton depends on the value of k x only n the neghborhood of the boundares between basns (T) and (CA) as shown n 5(b) and 5(d). Fgure 5(d) presents the cross secton of Fgure 5(b) on level k x =35 whch shows the coexstence of (T) and (CA) synchronzatons for dfferent ntal condtons ϕ 0 and ϕ 30. For larger values of k x we observe the coexstence of four types of synchronzaton as can be seen n Fgure 5(a,c). Fgure 5(c) s the cross secton of Fgure 18(a) at level k x = Large system of pendula rotatng n dfferent drectons In an attempt to generalze the results of prevous sectons to the system wth larger number of pendula we perform smulatons of such systems. In Fgure 6(a) we show the sums of pendula's dsplacements ϕ +ϕ1(= 6) for the system wth small stffness k x =00.0. The followng ntal condtons have been used: ϕ 10 =0, ϕ 0 =-0.16, ϕ 30 =-0.3, ϕ 40 =-0.48, ϕ 50 =- 0.64, ϕ 60 = One can see that after the ntal transent (several rotatons) the sums of pendula's dsplacements fluctuate around constant values close to 0, ±π and ±π/3. Ths type of synchronzaton s equvalent to the tree-synchronzaton (T). Fgure 6(b) presents that the ncrease of the stffness coeffcent to k x =000.0 leads to the change of the type of synchronzaton to the cluster-antphase (CA). Fgure 6(c,d) shows that wth further ncrease of the stffness coeffcent, e.g. to the value k x = the type of synchronzaton depends on the ntal condtons as one can observe ether (CL) (Fgure 6(c)) or Yankee (Fgure 6(d)) synchronzaton. 36

37 Fgure 6: Dfferent types of synchronzaton for the system wth sx pendula; (a) pendula's dsplacements durng the tree-synchronzaton, k x =00.0, ϕ 10 =0, ϕ 0 =-0.16, ϕ 30 =-0.3, ϕ 40 =- 0.48, ϕ 50 =-0.64, ϕ 60 =-0.80, (b) pendula's dsplacements durng the cluster-antphasesynchronzaton, k x =000.0, ϕ 10 =0, ϕ 0 =-0.7, ϕ 30 =-0.55, ϕ 40 =-0.83, ϕ 50 =-1.11, ϕ 60 =-1.38, (c) pendula's dsplacements durng the cluster-left-synchronzaton, k x =3000.0, ϕ 10 =0, ϕ 0 =-0.7, ϕ 30 =-0.55, ϕ 40 =-0.83, ϕ 50 =-1.11, ϕ 60 =-1.38,, (d) pendula's dsplacements durng the yankeesynchronzaton, k x =3000.0, ϕ 10 =0, ϕ 0 =-0.44, ϕ 30 =-0.5, ϕ 40 =-π, ϕ 50 =-3π/, ϕ 60 =-1.5. Generally n the systems wth large number of pendula we observe the same types of synchronzatons as descrbed for three pendula n prevous sectons. Probablty of the appearance of the Yankee type of synchronzaton decreases wth the ncrease of the number of pendula Two pendula wth dfferent drvng torques, rotatng n the same drecton In the last example, let us consder the case of two pendula wth the same lengths and masses but wth dfferent drvng torques. In our numercal smulatons eqs.(1,) have been ntegrated by the 4 th order Runge-Kutta method. The obtaned results confrmed the exstence of the phenomenon of phase synchronzaton n the consdered system and allowed the determnaton of phase angles between the synchronzed pendula. Addtonally, the numercal 37

38 ntegraton of eqs.(1,) allows the determnaton of the basns of attracton of dfferent coexstng confguratons of the synchronzed pendula. We use the followng parameters' values: l 1 =l =5.0, c ϕ1 =c ϕ = 0.01, m 1 =m =1.0, m b =10.0, c x =1.5. In dfferent examples we consder two values of k x =400.0 or Pendulum 1 s drven by the torque gven by p 01 =5.0, p 11 =0.4. The parameters of the drvng torque of pendulum have been taken as control parameters. Fgure 7. Two types of synchronous confguratons of dentcally drven pendula: l 1 =l =0.5, c ϕ1 =c ϕ = 0.01, m 1 =m =1.0, m b =10.0, c x =1.5, p 01 =5.0, p 11 =0.4, p 0 =5.0, p 1 =0.4; (a) tme seres of the pendula angular veloctes, durng the state of complete synchronzaton 0, k x =400.0, (b) tme seres of the pendula angular veloctes, durng the state of antphase synchronzaton 0, k x =400.0, (c) pendula confguraton durng the complete synchronzaton, (d) pendula confguraton durng the antphase synchronzaton, (e) basns of attracton of complete and antphase synchronzatons; p 01 =5.0, p 11 =0.4, p 0 =5.0, p 1 =0.4, ϕ 10 =0, 0, 0, x 0 =0, 0. 38

39 Fgure 7(a-e) presents two types of the synchronous confguratons whch have been predcted n Sec. 3. In Fgure 7(a) we show the tme seres of the pendula angular veloctes, durng the state of complete synchronzaton 0 obtaned for p 01 =p 0 =5.0, p 11 =p 1 =0.4, k x = Notce that these veloctes are equal ( ) and fluctuate (due to the gravty and the beam s oscllatons) around the nomnal value 1.5. The tme seres x s also shown (for better vsblty t has been enlarged 5 tmes). The pendula confguraton durng ths type of synchronzaton s shown n Fgure 7(c). Fgure 7(b) shows the tme seres of the pendula angular veloctes, durng the state of antphase synchronzaton 0 obtaned for p 01 =p 0 =5.0, p 11 =p 1 =0.4, k x = These veloctes fluctuate around nomnal value 1.5 and the fluctuatons are n the antphase. The dfference between the pendula dsplacements fluctuates around mean value equal to π,.e.,. The ampltudes of the beam s oscllatons are much smaller than n the prevous case. The pendula confguraton durng ths type of synchronzaton s shown n Fgure 7(d). In Fgure 7(e) we show the basns of attracton of dfferent types of synchronzaton. The basns of complete and antphase synchronzatons are shown respectvely n whte and red colors. The basns are presented n the plane showng the stffness coeffcent k x versus ntal value of dsplacement ϕ 0. The rest of the ntal condtons are as follows ϕ 10 =0,, 0, x 0 =0, 0. For small values of k x, (smaller than 19) the system reaches the state of antphase synchronzaton for all values of ϕ 0. Smlarly n the nterval 1600<k x <4680 ndependently of ϕ 0 we observe complete synchronzaton. In the ntervals 760<k x <1600 and 4680<k x <8000 the type of synchronous confguraton depends on ϕ 0. These results can be compared to that of [5]. On the base of analytcal analyss he has shown that the basn boundary between complete and antphase synchronzaton s determned by the constant value of k x (ndependent of ntal condtons), for whch angular velocty, 1.5 s the resonant frequency of the lnear oscllator consstng of the sprng wth stffness coeffcent k x and mass m b =10.0. For the consdered parameters values t s equal to k x = = Ths crteron s ndcated n Fgure 7(e) by the horzontal lne. In further studes we consder the pendula whch are drven by dfferent torques. In Fgure 8(a) we show the tme seres of the pendula angular veloctes, durng the state of almost complete synchronzaton 0 obtaned for: p 01 =5.0, p 11 =0.4, p 0 =3.75, p 1 =0.3. Notce that the values of the parameters are dfferent but the ratos are the same, 1.5. In comparson to the case of Fgure 7(a) we observe only small dfferences n angular veloctes of both pendula. Wth the ncrease of p 1 (the decrease of the nomnal angular velocty of pendula ) these dfferences ncrease and the phase shft between pendula becomes clearly vsble. Fnally, for p 1 =0.355 synchronzaton mechansm based on the energy transfer between the pendula (va the beam) fals and the synchronzaton s no longer observed and we observe the quasperodc moton. The example of quasperodc moton s shown n Fgure 8(b) for p 1 =0.36. The acton of the synchronzaton mechansm s stll vsble. The energy transfer from pendulum 1 to pendulum tres to ncrease the angular velocty of pendulum and allows both pendula to rotate wth the same velocty. Ths acton s manfested by long ntervals n whch the dfference of pendula dsplacements ϕ -ϕ 1 fluctuates between constant values. However, the energy transfer s not suffcent and we observe the rapd ncrease (nearly a jump) of the dfference ϕ -ϕ 1 by π. One can say that at some nstances pendulum loses one rotaton n comparson to pendulum 1. Fgure 8(c) presents the pendula and the beam energy balances versus parameter p 1. Pendulum 1 ncreases the common angular velocty to the value larger than the nomnal velocty of pendulum so the value of energy W DRIVE decreases up to W DAMP (for p 1 =0.319), energy 39

40 W SYN becomes zero and takes no part n the exctaton of the beam s oscllatons. Next energy W SYN becomes negatve,.e., the drver torque of pendulum supples less energy than the energy dsspated by the damper of pendulum. To keep both pendula rotatng wth the same velocty pendulum 1 has to supply energy to pendulum. The energy W 1 SYN s dvded between the beam and pendulum,.e., pendulum 1 drves both the beam and pendulum. Fgure 8(d) presents the example of the energy balance for the case wth negatve W SYN for p 1 =0.33. Fgure 8. Almost complete synchronzaton n the system wth pendula wth dfferent drvng torques: 1.0, p 01 =5.0, p 0 =3.75, p 11 =0.4; (a) tme seres of the pendula angular veloctes, durng the state of almost complete synchronzaton 0, p 1 =0.3, (b) tme seres of the pendula angular veloctes, at the threshold between almost complete synchronzaton and quasperodc moton, p 1 =0.356; (c) energy balance of the system versus p 1, (d) the example of the energy balance n the case when W SYN <0. 40

41 Fgure 9. Bfurcaton dagrams:, (at the tme nstances when pendulum 1 passes through the equlbrum poston,.e., when ϕ 1 = πj, j=1,, ) versus p 1 : k x =400.0, p 01 =5.0, p 11 =0.4, p 0 =3.75; ϕ 10 =0, 0, 0, x 0 =0, 0; (a) the value of p 1 s ncreased from 0.3 to 0.8 and next decreased from 0.8 to 0.3, the ntervals of 1/1 almost complete synchronzaton and ntervals of 1/, 1/3, /3, 3/4 synchronzatons are ndcated, (b) the value of p 1 s decreased from 0.3 to 0.05, nterval of 1/1 almost complete synchronzaton s ndcated, (c) the value of p 1 s ncreased from 0.05 to 0.3, ntervals of 3/1, /1 and 1/1 synchronzatons are ndcated. 41

42 Fgure 30. The examples of dfferent types of system behavors shown n Fgure 9(a-c): 1.0, k x =400.0, p 01 =5.0, p 11 =0.4, p 0 =3.75; (a) tme seres of the pendula angular veloctes, durng the state of 3/1 synchronzaton, p 1 =0.088, ϕ 10 =0, 1.0, ϕ 0 =1.57,.0, x 0 =0, 0, (b) tme seres of the pendula angular veloctes, durng the state of almost complete synchronzaton 0) 1/1 synchronzaton (coexstng wth 3/1 synchronzaton), p 1 =0.088,, ϕ 10 =0, 1.0, 14.8, ϕ 0 =1.80, 13., x 0 =0.05, 0.95, (c) tme seres of the pendula angular veloctes, durng the state 1/3 synchronzaton, p 1 =0.76, ϕ 10 =0, 0, ϕ 0 =1.57, 0, x 0 =0.0, 0, (d) Poncare map of quasperdc moton, p 1 =0.36, ϕ 10 =0.0,,.0, ϕ 0 =0.0,.0, x 0 =0, 0. 4

43 Fgure 31. Bfurcaton dagrams:, (at the tme nstances when pendulum 1 passes through the equlbrum poston,.e., when ϕ 1 = πn) versus p 1, k x =400.0, p 01 =5.0, p 11 =0.4, p 0 =0.36; (a) the value of p 1 s ncreased from 0.15 to 0.8, ϕ 10 =0, 1.0, ϕ 0 =0, 0, x 0 =0, 0, (b) the value of p 1 s decreased from 0.15 to 0.05, ϕ 10 =0, 1.0, ϕ 0 =0,.0, x 0 =0, 0, (c) the value of p 1 s ncreased from 0.05 to 0.15, ϕ 10 =0, 36.0, ϕ 0 =0, 36.0, x 0 =0, 0. 43

44 Fgure 3: The examples of dfferent types of coexstng attractors shown n Fgure 31(a-c): k x =400.0, p 01 =5.0, p 11 =0.4, p 0 =0.36, p 1 =0.068; (a) tme seres of the pendula angular veloctes, durng the state of 1/1 almost synchronzaton, p 1 =0.088,, ϕ 10 =0, 0, ϕ 0 =3.14, 18.0, x 0 =0, 0, (b) tme seres of the pendula angular veloctes, durng the state of /1 synchronzaton, ϕ 10 =0, 0, ϕ 0 =3.14, 4.0, x 0 =0, 0, (c) tme seres of the pendula angular veloctes, durng 0/1 state, ϕ 10 =0, 0, ϕ 0 =1.57, 0, x 0 =0, 0, (d) basns of attracton of the attractors of (a-c), ϕ 10 =0.0,, 1.0, x 0 =0, 0. 44

45 Fgure 33. Type of synchronzaton for dfferent values of drvng torques: map p 0 versus p 1, ntally the drvng torques of both pendula are dentcal p 01 =p 0 =5.0, p 11 =p 1 =0.4, ϕ 10 =0, 1.0, ϕ 0 =0, 1.0, x 0 =0, 0 (n the ntal moment both pendula are passng through the equlbrum wth the same angular velocty equal to and after the transent the system reaches 1/1 complete synchronzaton). 45

46 Fgure 34. Bfurcaton dagrams:, (at the tme nstances when pendulum 1 passes through the equlbrum poston,.e., when ϕ 1 = πj, j=1,, ) versus p 1 : k x =400.0, p 01 =5.0, p 11 =0.4, p 0 =3.75; ϕ 10 =0, 0, 0, x 0 =0, 0; (a) the value of p 1 s ncreased from 0.3 to 0.8 and next decreased from 0.8 to 0.3, the ntervals of 1/1 almost complete synchronzaton and ntervals of 1/, 1/3, synchronzatons are ndcated, (b) the value of p 1 s decreased from 0.3 to 0.05 and next ncreased from 0.05 to 0.3, ntervals of 3/1, 3/1, /1 and 3/ synchronzatons are ndcated. The bfurcaton dagrams shown n Fgure 9(a-c) descrbe the behavor of the system wth drve torques (p 01 =5.0, p 11 =0.4, p 0 =3.75) n the wde range of the values of p 1. We show the values of, (at the tme nstances when pendulum 1 passes through the equlbrum poston,.e., when ϕ 1 =πj, j=1,, ) versus p 1. The calculatons have been started wth the state of complete synchronzaton. The dagram of Fgure 9(a) has been calculated for ncreasng values of p 1 n the nterval [0.3, 0.8]. The sngle lnes determne the regons of synchronzaton whle the blurred columns of markers ndcate the lack of synchronzaton. One can observe the followng types of synchronzaton: () 1/1 almost complete synchronzaton for 0.3<p 1 <0.355,() 1/ synchronzaton for 0.5<p 1 <0.58, () 1/3 synchronzaton for 0.75<p 1 <0.78, () /3 synchronzaton for 0.41<p 1 <0.44, (v) 3/4 synchronzaton for 0.78<p 1 < Symbol 1/1 ndcates that pendulum rotate wth the same mean angular velocty as pendulum 1,.e., n the tme nterval of one rotaton of pendulum pendulum 1 performs one rotaton, and the symbol r/q, where r,q=,3 ndcates that n the tme nterval of r rotaton of pendulum pendulum 1 performs q rotatons. r/q 46

47 synchronous regons exst n the wndows n the nterval of exstence of quasperodc regme. Our numercal calculatons ndcate that such wndows (most of them very small) are dense n the consdered nterval. In the range 0.3<p 1 <0.8 we have not observed other attractors coexstng wth those shown n Fgure 9(a). The decrease of the values p 1 from 0.8 to 0.3 gves an dentcal bfurcaton dagram as n the case of the ncrease. The bfurcaton dagram of Fgure 9(b) has been calculated for the decreasng values of p 1 (from 0.3 to 0.05). One observes 1/1 almost complete synchronzaton n the whole nterval 0.3> p 1 >0.05. The ncrease of p 1 from 0.05 to 0.5 allows the calculaton of Fgure 9(c). In the nterval 0.05<p 1 <0.16 we observe dfferent behavor to that shown n Fgure 9(b), whch ndcates the coexstence of dfferent attractors, e.g.; n the nterval 0.085<p 1 <0.09 we have the coexstence of 1/1 and 3/1 synchronzatons and n the nterval 0.114<p 1 <0.155 synchronous confguratons 1/1 and /1 coexst. One can also observe the coexstence of 1/1 synchronzaton (shown n Fgure 9(b)) and quas-perodc rotatons. Fgure 30(a-c) shows the examples of tme seres of the pendula angular veloctes, showng dfferent types of synchronzatons. Fgure 30(a) presents tme seres durng 3/1 synchronzaton for p 1 =0.088 (as shown n Fgure 9(c)). In Fgure 30(b) 1/1 synchronzaton (whch coexsts wth quasperodc behavor for p 1 =0.088) s descrbed. 1/3 synchronzaton for p 1 =0.76 s llustrated n Fgure 30(c) (compare wth Fgure 9(a)). Poncare map of the quasperodc behavor s shown n Fgure 30(d) (p 1 =0.36). Notce that the ponts of the map forms closed curve wth the complcated structure. The bfurcaton dagrams shown n Fgure 31(a-c) for the pendula whch are drven by dfferent torques. We take p 01 =5.0, p 11 =0.4 (as n calculatons shown n Fgure 9(a-c)) but consder dfferent value of p 0 =0.36. Note that n ths case the value of maxmum dmensonless torque whch can be generated by the drvng devce p 0 s smaller than the maxmum value of the torque generated by the weght of pendulum (equal to m gl ). Ths ndcates that there exst ntal condtons for whch pendulum stops to rotate at ϕ crt gven by relaton m gl snϕ crt =p 0. Calculatng the dagram shown n Fgure 31(a) we ncrease the value of p 1 n the nterval [0.15, 0.8] and observe three basc types of system (3,4) behavor: () 1/1 almost complete synchronzaton for 0.15<p 1 <0., () quasperodc moton 0.8 <p 1 <0.7, () the state ndcated as 0/1,.e, pendulum does not rotate (s at rest) n the nterval 0.7<p 1 <0.8. In further calculatons we decrease the value of p 1 from 0.8 to 0.05 and observe that n the whole nterval pendulum s at rest. Ths ndcates that n the whole nterval 0.15<p 1 <0.7 there s the coexstence of the state 0/1 wth the states shown n 31(a).When calculatng the dagram shown n Fgure 31(b) we decrease the value of p 1 from 0.15 to In the whole nterval one observes 1/1 almost complete synchronzaton. The dagram presented n Fgure 31(c) has been calculated for p 1 ncreasng from 0.05 to Note that n the nterval 0.05<p 1 <0.084 we observe dfferent behavor to that presented n Fgure 31(b), e.g., very narrow ntervals of 4/1 synchronzaton and 3/1 synchronzaton and for 0.058<p 1 <0.08 /1 synchronzaton. Fgure 31(a-c) shows that the system (3-4) has the followng coexstng attractors:() for 0.05<p 0 <0.08 we observe three attractors: 0/1, 1/1 and one of the attractors shown n Fgure 31(c), () for 0.08<p 0 <0.7 we have two attractors: 0/1 and one of the attractors shown n Fgure 31(a,b), () for 0.7<p 0 <0.8 there s only one attractor 0/1,.e., pendulum stops ndependently of ntal condtons. In Fgure 3(a-c) we show the tme seres of the pendula angular veloctes, for dfferent type of coexstng attractors whch exst for p 1 = Fgure 3(a,b) presents respectvely 1/1 and /1 synchronzatons. The state 0/1 (wth non-rotatng pendulum ) s shown n Fgure 3(c). The basns of attracton of these attractors are presented n Fgure 3(d). The basns are shown on the plane ϕ 0 - (other ntal condtons ϕ 10,, x 0, 0 are equal to zero). 47

48 Fgure 33 presents map p 0 versus p 1 whch descrbes how the type of synchronzaton s changng wth the change of the parameters of drvng torques. We assume that ntally the drvng torques of both pendula are dentcal p 01 =p 0 =5.0, p 11 =p 1 =0.4. We consder the followng ntal condtons: ϕ 10 =0, 1.0, ϕ 0 =0, 1.0, x 0 =0.0, 0,.e., n the ntal moment both pendula are passng through the equlbrum wth the same angular velocty close to and after the transent the system reach 1/1 almost complete synchronzaton. Next, we change (the jump change) the values of parameters p 0 and p 1. We observed the followng cases. In the regon ndcated n whte the synchronzaton 1/1 has been preserved. The type of synchronzaton has been changed (regons ndcated by r/q). Pendulum stops (black regon ndcated) as 0/1 or the pendula start to rotate quasperodcally (red regon). For example, along lne p o =0.9 the system shows 1/1 almost complete synchronzaton or pendulum stops (0/1 confguraton) and along the lne p o =3.75 one observes the behavor descrbed n Fgure 9(a,c). Fnally, Fgure 34(a,b) presents the bfurcaton dagrams calculated n the same way as the dagram shown n Fgure 9(a-c) but n ths case we start wth the antphase synchronzaton. The dagram of Fgure 34(a) has been calculated for ncreasng values of p 1 n the nterval [0.3, 0.8]. One can observe the followng types of synchronzaton: () 1/1 complete synchronzaton for 0.3<p 1 <0.364, () 1/ synchronzaton for 0.464<p 1 <0.488, () 1/3 synchronzaton for 0.716<p 1 < r/q synchronous regons exst n the wndows n the nterval of exstence of quasperodc regme. The decrease of values p 1 from 0.8 to 0.3 gves the dentcal bfurcaton dagram as n the case of the ncrease. The bfurcaton dagram of Fgure 34(b) has been calculated for decreasng values of p 1 (from 0.3 to 0.05). One observes 1/1 antphase synchronzaton n the nterval 0.3>p 1 >0.8. The ncrease of the values p 1 from 0.05 to 0.3 gves dentcal bfurcaton dagram as n the case of the decrease. Contrary to the case when the calculatons started from the complete synchronzaton we have not observed the coexstence of attractors. In Fgure 9(a-c) 31(b,c) and 34(a-b) all presented perodc solutons represents resonances on the torus. They orgnates and termnates as the result of the saddle-node bfurcatons (as n the classcal Arnold's tongues). In Fgure 31(a) for 0<p 1 <0.1 we observe 1/1 resonance. Next the perodc soluton s destroyed n saddle-node bfurcaton and we observe the small nterval of behavor. Another saddle-node bfurcaton leads to /1 resonance and fnally after the sequence of perod-doublng bfurcatons (we observe 4/, 8/4 and 16/8 resonances) the system show chaotc behavor. Bfurcatons have been dentfed usng path followng software AUTO-07P and confrmed by the calculatons of Lyapunov exponents. 48

49 5. Synchronzaton extends the lfe tme of rotaton Synchronzaton occurs wdely n natural and technologcal world, from the rhythm of applause [53], rhythm of the crowd of walkers on the brdge [5,68], flashng of frefles [11] to the nanomechancal or chemcal oscllators [67,86], but t has not been shown that the synchronzaton can extend the lfe tme of the desrable behavor of the coupled systems. In ths secton we gve evdence that the ntal synchronous state extends the lfetme of rotatonal behavor of the coupled pendula n the case when the exctaton of one or few pendula s suddenly (breakdown of energy supply) or gradually (as the effect of agng and fatgue) swtched off. We show that for the properly chosen couplng (n our system parameters k x and m b ) the energy transfer from the excted pendula allow unexcted pendula to rotate. The ntal synchronous confguraton s replaced by phase synchronzaton wth dfferent phase shfts between pendula and the rotatonal velocty of the synchronzed pendula s decreased. These two factors can be consdered as the ndcator of the breakdown of exctaton n one or few pendula. As a proof of concept we examne the followng examples. The presented results have been obtaned from the numercal ntegraton of equatons of moton (1,). Example 1. Two pendula rotatng n the same drecton In ths example we reconsder the system (1,) consstng of two dentcal pendula rotatng n the same drecton. We consder system (1-) wth the followng parameter values: m 1 =m =1.00, l 1 =l =0.5, c ϕ1 =c ϕ =0.03 and dfferent values of the mass of the beam m b and the stffness coeffcent k x. The dampng coeffcent c x has been selected n such a way as to be equvalent to the arbtrarly selected logarthmc decrement of dampng Δ=ln(1.5) (the decrement characterstc for the lnear oscllator wth mass equal to the total mass of the system shown n Fgure 3(a) mounted to the sprng wth the stffness coeffcent k x. The exctaton parameters of pendulum 1 are equal to p 01 =5.0, p 11 =0.. These parameters of pendulum,.e., p 0 and p 1 are taken as control parameters. Fgure 35(a-d) presents three dfferent types of synchronous confguratons for the exctaton torques characterzed by: p 01 = p 0 =5.0, p 11 = p 1 =0.. In Fgure 35(a) we show the tme seres of pendula's angular veloctes, for k x =7000 and m b =1 n the state of the complete synchronzaton (C). Angular veloctes of both pendula are dentcal, ther oscllatons are caused by the effect of gravty and oscllatons of the beam. The pendula's dsplacements fulfll the relaton ϕ ϕ1 = 0 and the beam performs small perodc oscllatons (dsplacement x(t) s magnfed 10 tmes). Addtonally, the synchronous confguraton at the tme when the pendula are movng through the lower stable equlbrum s shown. Fgure 35(b) shows the same tme seres n the case of antphase synchronzaton (A) for k x =1000 and m b =16. Pendula's angular veloctes, oscllate around the same mean value. The phase shft between these oscllatons s equal to π. The phase shft between pendula's dsplacements ϕ ϕ1 oscllate around π (.e., 0). The ampltudes of the beam's oscllatons are smaller than n the case of complete synchronzaton. In Fgure 35(c) both pendula perform quasperodc rotatons for k x =6500 and m b =17. The pendula's average angular veloctes (calculated for the large number of rotatons) are equal, so the pendula perform the same number of rotatons n the gven tme. As both pendula perform the same number of rotatons n the gven tme we can call ths case the quasperodc synchronzaton (QS). Quasperodc oscllatons of the pendula's angular veloctes are clearly vsble at the Poncare map shown n Fgure 35(d). Ths map shows the pendula's angular veloctes, versus the phase shfts between pendula's dsplacements 0 49

50 calculated at the tme when pendulum 1 s movng through the lower equlbrum poston (.e., for ϕ 1 =jπ, j+1,, ). Fgure 35. Tme seres of pendula's dsplacements,, angular veloctes, and beam's dsplacement x (magnfed 10 tmes): p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (a) complete synchronzaton k x =7000.0, m b =1.0, (b) antphase synchronzaton k x =1000.0, m b =16.0, (c,d) synchronous quasperodc rotaton k x =6500.0, m b =17.0, (d) Poncare map for the case of (c). Fgure 36(a,b) shows the dependence of the synchronous confguraton on the parameters k x and m b. The calculatons have been performed for p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0. and ntal condtons ϕ 10 =0, ϕ 0 =0.5π, 5, 0 (Fgure 36(a)) and ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0,.e., the pendula ntally rotate n dfferent drectons (Fgure 36(b)). The regons of complete (C), antphase (A), quasperodc (QS) synchronzatons are ndcated respectvely n green, navy blue and volet colors. Notce the navy blue volet band n the vcnty of the dagonal. In ths regon the complete synchronzaton (see Fgure 36(a)) coexsts wth ether antphase or quasperodc synchronzaton. Determnaton of the doman n parameters space (k x, m b ), n whch dfferent synchronous confguratons coexst, s not straghtforward and requre precse calculatons. Blekhman [5] suggested that the boundary of the complete and antphase s gven by the condton p 01 /p 11 = a x, where a x s the resonant frequency of the lnear oscllator consstng of mass m b +nm suspended on the sprng wth stffness coeffcent k x. Ths boundary s shown as a black lne n Fgure 36(a,b). One can see that t s located away from the real boundary as n the consdered system dampers dsspate part of the energy suppled by the exctaton torques. Ths dsspaton causes the reducton of the actual angular velocty of the pendula below ther nomnal velocty /. Generally, desgnng a system supportng the pendula, the weght of the beam m b and the stffness coeffcent k x should be chosen n such a way that the resonance frequency of the system s smaller than the angular velocty of the pendula to obtan antphase synchronzaton 50

51 or sgnfcantly hgher than the angular velocty of pendula to acheve complete synchronzaton. Unfortunately, the angular velocty of the pendula can be determned only by numercal ntegraton of equatons of moton of a partcular system (1,). Fgure 36. The regons of the parameters k x - m b space of complete (C) (green), antphase (A) (navy blue), quasperodc (QS) (volet) synchronzatons for the system of two dentcal and dentcally drven pendula rotatng n the same drecton: p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., (a) ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (b) ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0. Now let us nvestgate the effect of sudden exctaton swtch off n one of the synchronzed pendula,.e., system (1,) of two dentcal and dentcally drven pendula s n the synchronzed state and the exctaton of pendulum s suddenly swtched off. The results of our calculatons are shown n Fgure 37(a,b) and Fgure 38(a-c). Then for gven values of k x and m b the pendula are n the synchronzed state descrbed n Fgure 36(a,b) when the exctaton of pendulum s swtched off. After the ntal transent pendula reach the state descrbed n Fgure 37(a,b). In the green and navy blue regons respectvely ntal complete and antphase synchronzaton s replaced by the phase synchronzaton wth varous phase shfts between pendula. In the red regon there s no synchronzaton and pendulum stops. In small volet and blue regons we observed respectvely quasperodc and multperodc synchronzaton. In Fgure 38(a) we show the tme seres of pendula's angular veloctes, for k x =7000.0, m b =1.0 (pont I n Fgure 36(a) and 37(a)). Intally complete synchronzaton (C) s replaced by the phase synchronzaton wth the phase shft between pendula equal to 0.4π. The tme seres for k x = and m b =16.0 (pont II) s shown n Fgure 38(b). In ths case ntally antphase synchronzaton (A) s replaced by the phase synchronzaton wth the phase shft between pendula equal to 0.7π. In both cases pendulum 1 transfers suffcent amount of energy to keep pendulum rotatng. The phase shfts between pendula dfferent from orgnal 0 and π can be taken as ndcators of the exctaton swtch off. In regon (N) ndcated n red color the synchronzaton s lost. Pendulum slows down and fnally stops. The tme seres of pendula's angular veloctes, are shown n Fgure 38(c) (k x =1000.0, m b =1.0 - pont III). Pendulum performs oscllatons caused by the oscllatons of the beam x and angular velocty of pendulum oscllates around zero. In Fgure 37(a,b) one can see small regons (volet) when quasperodc synchronzaton and the 51

52 synchronzaton durng whch pendulum rotates n dfferent drecton than the excted pendulum 1 occur. The tme seres showng the transent behavor of pendula's angular veloctes,, the dfference of pendula's dsplacement and beam's dsplacement x (magnfed 100 tmes) are shown n Fgure 39(a,b). At N=50 the exctaton of pendulum s swtched off. In the case of Fgure 39(a) (k x =7000, m b =1) after the short nterval of the lack of synchronzaton the pendula became synchronzed agan but the phase shft between the pendula s larger than zero (phase synchronzaton wth the phase shft equal to 0.4π). In the second case (k x =1000, m b =1), when the exctaton of pendulum s swtched off t stops to rotate. Pendulum starts to oscllate as the result of the beam's oscllatons x (see the fluctuatons of the angular velocty ). The dfference of the angular dsplacements of the pendula grows to nfnty. Fgure 37. Synchronous states of the system of two dentcal pendula (the exctaton of pendulum s swtched off after tme 60N): p 01 =5.0, p 11 =0., p 0 =0, p 1 =0 (the regons n parameters k x - m b space of synchronous pendula's rotaton are shown n green, navy blue and volet and the regon n whch pendulum stops n red colors). (a) ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (b) ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0. 5

53 Fgure 38. Tme seres of pendula's dsplacements,, angular veloctes, and beam's dsplacement x (magnfed 100 tmes): p 01 =5.0, p 11 =0., p 0 =0, p 1 =0, ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (the exctaton of pendulum s swtched off after the tme 60N), (a) phase synchronzaton wth phase shfts between pendula equal to π/4 (ntally complete synchronzaton) k x =7000.0, m b =1.0, (b) phase synchronzaton wth phase shfts between pendula equal to π/ (ntally antphase synchronzaton) k x =1000.0, m b =16.0, (c) pendulum stops k x =1000.0, m b =

54 Fgure 39. Tme seres showng the transent behavor of pendula's angular veloctes,, the dfference of pendula's dsplacement and beam's dsplacement x (magnfed 100 tmes) n the case when the exctaton of pendulum s swtched off at N=50: p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., p 1 =p =0, 0,, 0, 5, 0, (a) k x =7000.0, m b =1.0, (b) k x =1000.0, m b =

55 Fgure 40. Bfurcaton dagrams of pendula's angular veloctes, and the dfference of pendula's dsplacement versus parameter ξ, exctaton of pendulum gradually decays to zero,.e., (1-ξ)(p 0 p 1 ), (a) k x =7000.0, m b =1.0, 0,, 0, 5, 0, (b) k x =1000.0, m b =1.0, 0,, 0, 5, 0, (c) k x =3700.0, m b =18.0, 0,, 0, 5, 0. Now let us consder the case when the exctaton of pendulum gradually decays to zero. The exctaton decay can be descrbed as (1-ξ)(p 0 p 1 ), where ξ ( 0,1 ) s a control parameter. The bfurcaton dagrams shown n Fgure 40(a,b) present the values of the pendula's angular veloctes, (at the moments when pendulum 1 moves through the lower equlbrum poston) and the dfference of pendula's dsplacement versus parameter ξ. In the case of Fgure 40(a) (k x =7000.0, m b =1.0, both pendula are n the state of complete synchronzaton) synchronzaton s preserved up to the value ξ=0.85. The phase shft between pendula s vsble for larger values of ξ (complete synchronzaton s replaced by phase synchronzaton. In the whole nterval of ξ pendulum 1 transfers enough energy to pendulum to ensure the pendula's synchronzaton. The dfference of the pendula's dsplacements s so small that t s not vsble (n the scale of Fgure 40(a)). Fgure 40(b) presents bfurcaton dagram for k x = and m b =1.0. In ths case the phase synchronzaton s observed for ξ<0.95. At ths pont pendulum stops to rotate (pendulum 1 s unable to transfer enough energy to keep t rotatng) and one observes the ncrease of the dfference ϕ ϕ1. The case of k x =3700 and m b =18 (pont IV n Fgure 36(b) and Fgure 37(b) located close to the area of the coexstence of dfferent synchronous confguratons) s llustrated n Fgure 40(c). In the nterval 0.0<ξ<0.49 one observes perodc rotatons of the 55

56 synchronzed pendula (wth the phase shft larger than zero). For larger values of ξ (0.49<ξ<0.70) pendula perform perodc synchronous rotatons wth hgher perods or quasperodc synchronous rotatons. Further ncrease of ξ<0.70 stops pendulum. Example. Two pendula rotatng n dfferent drectons Let us consder the same system as n Example 1 but assume that pendula rotate n dfferent drectons. We consder system (1-) wth the followng parameter values: m 1 =m =1.00, l 1 =l =0.5, c ϕ1 =c ϕ =0.03. The exctaton parameters of pendulum 1 are equal to p 01 =5.0, p 11 =0.. In the ntal state t has been assumed that p 0 =-5.0, p 1 =0.,.e., pendulum 1 rotates counterclockwse and pendulum rotates clockwse. Later the parameters of pendulum : p 0 and p 1 are taken as control parameters. Fgure 41. The regons the parameters space k x - m b of mrror M (lght red), antphase A (blue), quasperodc Q (volet) synchronzatons for the system of two dentcal and dentcally drven pendula rotatng n dfferent drectons: p 01 =5.0, p 11 =0., p 0 = -5.0, p 1 =0., (a) ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (b) ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0. Fgure 4. Synchronous states of the system of two dentcal pendula rotatng n dfferent drectons (the exctaton of pendulum s swtched off after the tme 60N): p 01 =5.0, p 11 =0., p 0 =0, p 1 =0 (the regons n the parameters k x - m b space of synchronous pendula's rotaton are shown n lght red, blue and volet and the regon n whch pendulum stops n red colors). (a) ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (b) ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0. 56

57 Fgure 43. Tme seres of pendula's dsplacements,, angular veloctes, and beam's dsplacement x (magnfed 100 tmes) for the system of two pendula rotatng n dfferent drectons; ϕ 10 =0, ϕ 0 =0.5π, 5, 0, (a) mrror synchronzaton, both pendula excted: k x =000.0, m b =0.0, p 01 =5.0, (b) antphase synchronzaton, both pendula excted: k x =000.0, m b =0.0, p 01 =5.0, p 11 =0., p 0 =-5.0, p 1 =0., (c) ntally mrror synchronzaton, at N=60 the exctaton of pendulum swtched off: k x =000.0, m b =0.0, p 01 =5.0, p 11 =0., p 0 =0.0, p 1 =0.0, (d) ntally antphase synchronzaton, at n=60 the exctaton of pendulum s swtched off : k x =000.0, m b =0.0, p 01 =5.0, p 11 =0., p 0 =0.0, p 1 =0.0. To nvestgate the effect of sudden exctaton swtch off n one of the synchronzed pendula we assume that system (1,) of two dentcal and dentcally drven pendula s n the synchronzed state and the exctaton of pendulum s suddenly swtched off. The results of our calculatons are shown n Fgure 41(a,b), 4(a,b) and 43(a-d). For gven values of k x and m b the pendula are n the synchronzed state descrbed n Fgure 41(a,b) when the exctaton of pendulum s swtched off. The calculatons have been performed for p 01 =5.0, p 11 =0., p 0 =-5.0, p 1 =0. and ntal condtons ϕ 10 =0, ϕ 0 =0.5π, 5, 0 (Fgure 41(a)) and ϕ 10 =0, ϕ 0 =0.5π, 5, 5, 0,.e., pendulum ntally rotate n dfferent drectons (Fgure 41(b)). The regons of mrror M, antphase A, quasperodc (QS) synchronzatons are ndcated respectvely n lght red, blue and volet colors. Notce the navy blue volet band n the vcnty of the dagonal n Fgure 41(b). In ths regon the antphase synchronzaton (see Fgure 41(a)) coexsts wth ether mrror or quasperodc synchronzaton. After the sudden breakdown of pendulum exctaton 57

58 synchronzaton s not lost n the regons ndcated n lght red, blue (phase synchronzaton) and volet (quasperodc synchronzaton) colors shown n Fgure 4(a,b). In the lght red and blue regons respectvely mrror and antphase synchronzaton s replaced by the phase synchronzaton wth varous phase shfts between pendula. In regon (N) ndcated n red color the synchronzaton s lost. Pendulum 3 slows down and fnally stops. In Fgure 43(a,c) we show the tme seres of pendula's angular veloctes, for k x =000.0, m b =0.0 (pont I n Fgure 41(a)). In the case of Fgure 43(a) both pendula are excted and reach the state of mrror synchronzaton. After the swtchng off of pendula exctaton ntally mrror synchronzaton (M) s replaced by the phase synchronzaton wth the phase shft between pendula equal to π/4 as shown n Fgure 43(c). The tme seres for k x = m b =0.0 (pont II) s shown n Fgure 43(b,d). In ths case ntally antphase synchronzaton (A) (Fgure 43(b)) s replaced by the phase synchronzaton wth the phase shft between pendula equal to π/ as shown n Fgure 43(d). In both cases pendulum 1 transfers suffcent amount of energy to keep pendulum rotatng. The phase shfts between pendula dfferent from orgnal 0 and π can be taken as ndcators of the exctaton swtch off. In regon (N) ndcated n red color the synchronzaton s lost. Pendulum slows down and fnally stops. Example 3. Three pendula rotatng n the same drecton In ths secton we reconsder system (1,) consstng of three dentcal pendula rotatng n the same drecton. The presented results have been obtaned from the numercal ntegraton of equatons of moton (1,). In ths example we consder the system (1-) wth the followng parameter values: m 1 =m =m 3 =1.00, l 1 =l =l 3 =0.5, c ϕ1 =c ϕ = c ϕ3 =0.03. We consder dfferent values of the mass of the beam m b and the stffness coeffcent k x. The dampng coeffcent c x has been selected n such a way as to be equvalent to the arbtrarly selected logarthmc decrement of dampng Δ=ln(1.5) (the decrement characterstc for the lnear oscllator wth mass equal to the total mass of the system (m b +nm) mounted to the sprng wth the stffness coeffcent k x ). The exctaton parameters of pendulum 1 are equal to p 01 = p 0 =5.0, p 11 = p 1 =0.. The exctaton parameters of pendulum 3 are ntally the same but after N=50 the exctaton s swtched off,.e., p 03 =0, p 13 =0.. To nvestgate the effect of sudden exctaton swtch off n one of the synchronzed pendula we assume that system (1,) of three dentcal and dentcally drven pendula s n the synchronzed state and the exctaton of pendulum 3 s suddenly swtched off. The results of our calculatons are shown n Fgure 44(a,b), and 45(a-b). Fgure 44(a) shows the dependence of the synchronous confguraton on the parameters k x and m b. One observes complete (green regon), Yankee (navy blue regon) and quasperodc (volet regon) synchronous states (all pendula are excted). Addtonally the synchronzaton s not observed for k x and m b from the red regon. Then for gven values of k x and m b the pendula are n the synchronzed state descrbed n Fgure 44(a) when the exctaton of pendulum 3 s swtched off. After the sudden swtch off of pendulum 3 exctaton synchronzaton s not lost n the regons ndcated f Fgure 44(b) n green, navy blue (phase synchronzaton), blue (synchronous state n whch pendula 1 and create cluster whch s n antphase to nonexcted pendulum 3) and volet (quasperodc synchronzaton) colors. In the green and navy blue regons respectvely complete and Yankee synchronzaton s replaced by the phase synchronzaton wth varous phase shfts between pendula. In regon (N) ndcated n red color the synchronzaton s lost. Pendulum 3 slows down and fnally stops. The tme seres showng the transent behavor of pendula's angular veloctes,,, the dfferences of pendula's dsplacements, and beam's dsplacement x (magnfed 100 tmes) are shown n Fgure 45(a,b). At N=50 the exctaton of 58

59 pendulum 3 s swtched off. In the case of Fgure 45(a) (k x =1000, m b =6, pont I n Fgure 44(a)) after the short nterval of the lack of synchronzaton the pendula became synchronzed agan but the phase shft between the pendula s larger than zero (phase synchronzaton wth the phase shft equal to 0.3π). In the second case (k x =000, m b =6, pont II) shown n Fgure 45(b), when the exctaton of pendulum 3 s swtched off t stops to rotate. Pendulum 3 starts to oscllate as the result of the beam's oscllatons x (see the fluctuatons of the angular velocty ). The dfference of angular dsplacements of the pendula grows to nfnty. Fgure 44. Synchronous states n k x m b parameter space of three pendula rotatng n the same drecton: p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., p 03 =p 13 =0, 0,,, 0, 5, 0,(a) all pendula excted, (b) pendulum 3 swtched off at N=50. Fgure 45. Tme seres showng the transent behavor of pendula's angular veloctes,,, the dfference of pendula's dsplacement, and beam's dsplacement x (magnfed 100 tmes) n the case when the exctaton of pendulum 3 s swtched off at N=50: p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., p 03 =p 13 =0, 0,,, 0, 5, 0, (a) k x =1000.0, m b =6.0 (I), (b) k x =000.0, m b =6.0 (II). 59

60 Example 4. Two pendula wth dfferent masses rotatng n the same drecton In ths example we consder the case of two pendula wth dfferent masses rotatng n the same drecton. Pendulum 1 has mass (1+η)m and pendulum mass (1-η)m, where η s constant. Dampng coeffcents c φ and exctaton torques are proportonal to the pendula's masses. In ths case equatons of moton can be rewrtten n the followng form: ( 1+ η)( ml && ϕ1 + mlx && cosϕ1 + cϕ & ϕ1 + mgl snϕ1) = ( 1+ η)( p01 p11 & ϕ1), ( 1 η)( ml ϕ + mlxcosϕ + c ϕ + mgl snϕ ) = ( 1 η)( p p ϕ ) ( m + ) b = ml && nm && x + c x && x& + k x x = ϕ & ( + η )( && ϕ cosϕ + & ϕ snϕ ) + ml( 1 η)( && ϕ cosϕ + & ϕ sn ) ϕ In our numercal calculatons we consder the followng parameter's values: l=0.5, m=1.0, c ϕ = 0.03, p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0.. Fgure 46(a,b) shows the dependence of the synchronous confguraton on the parameters k x and m b. The calculatons have been performed for ntal condtons 0, 0, 5, 5, 0, (compare wth Fgure 36(a) and 37(a) calculated for η=0). The regons of complete (C), antphase (A), quasperodc (QS) synchronzatons are ndcated respectvely n green, navy blue and volet colors. Fgure 46(a) llustrates the case of η=0.9 (m 1 =1.9, m =0.1). After the ntal tme 50N the exctaton of pendulum s swtched off. For k x and m b n the green regon the synchronzaton s preserved but the complete synchronzaton (phase shft between pendula equal to zero) s replaced by phase synchronzaton (phase shft larger than zero). For the wde range k x and m b parameters the rotaton of both pendula s preserved. Contrary to ths for η=-0.1,.e., m 1 =0.9 and m =1.1 the set parameters for whch both pendula rotate s very small as shown n Fgure 46(b) (green regon). Notce that n ths case the exctaton of the heaver pendulum has been swtched off. Wth the further ncrease of pendulum mass we observe the loss of synchronzaton and the unexcted pendulum stops. Fgure 47(a,b) shows tme seres of the transent behavor of pendula's angular veloctes,, the dfference of pendula's dsplacement and beam's dsplacement x (magnfed 100 tmes) n the case when the exctaton of pendulum s swtched off. The case of η=0.9 (m 1 =1.9, m =0.1) and m b =1.0, k x =7000 (pont I n Fgure 46(a) s llustrated n Fgure 47(a). Both pendula are n the state of complete synchronzaton when at tme 0N the exctaton of pendulum s swtched off. After the transent tme the system reaches the state of phase synchronzaton wth nonzero phase shft between pendula. Notce that ths phase shft s smaller than n the case of dentcal pendula (Fgure 39(a)). Fgure 47(b) shows the case for η=-0.15,.e., m 1 =0.85 and m =1.15. At tme 0N the exctaton of pendulum s swtched off. Synchronzaton and rotaton of both pendula are preserved but the phase shft ϕ ϕ1 ncreases to the value larger than π/. Further decrease of η, (down to η=-0.17,.e., m 1 =0.83 and m =1.17) leads to the loss of synchronzaton and pendulum stops. 0 1 &, (35) (36) Example 5. Larger number of pendula In the state of complete synchronzaton the forces wth whch pendula act on the beam are algebracally added so ths example can be generalzed to the case of any number of pendula of total mass equal to m. Consder the case of n pendula n the state of complete synchronzaton. The effect of the swtch off of the exctaton of p pendula s the same as the effect of swtch off of the exctaton of pendulum wth mass mp/n n the system of two 60

61 pendula (the second one wth mass m(n-p)/n). As an example consder the system of 0 dentcal pendula rotatng n the same drecton wth masses (m 1-0 =0.1, m b =1.0, k x =7000) shown n Fgure 48(a-d). All pendula are n the state of complete synchronzaton when at tme 50N the exctaton of one (Fgure 48(a)), ten (Fgure 48(b)), eleven (Fgure 48(c)) and twelve (Fgure 48(d)) pendula s swtched off. Up to the case of 11 pendula the ntal complete synchronzaton s replaced by the phase synchronzaton and all pendula rotate. The phase shft between the clusters of excted and unexcted pendula ncreases wth the ncrease of the number of unexcted pendula. When the exctaton of the 1-th pendulum s swtched off the synchronzaton s lost and all unexcted pendula stop to rotate. Fgure 46. The regons of the parameters k x - m b space of complete (C) (green), antphase (A) (navy blue), quasperodc (QS) (volet) synchronzatons for the system of two pendula wth dfferent masses rotatng n the same drecton; l=0.5, m=1.0, c ϕ = 0.03, p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., 0, 0, 5, 5, 0, (a) η=0.9,.e., m 1 =1.9 and m =0.1, after the ntal tme equal to 50N, exctaton of pendulum s swtched off, (b) η=-0.1,.e., m 1 =0.9 and m =1.1, after the ntal tme equal to 50N, exctaton of pendulum s swtched off. Fgure 47. Tme seres showng the transent behavor of pendula's angular veloctes,, the dfference of pendula's dsplacement and beam's dsplacement x (magnfed 100 tmes) n the case when the exctaton of pendulum s swtched off: p 01 =5.0, p 11 =0., p 0 =5.0, p 1 =0., p 1 =p =0, 0, 0, 5, 5, 0, k x =7000.0, m b =1.0, (a) η=0.9 (m 1 =1.9, m =0.1) the exctaton of pendulum swtched off at 0N, (b) η= (m 1 =0.85, m =1.15) the exctaton of pendulum swtched off at 50N. 61

62 Fgure 48. Tme seres showng the transent behavor of pendula's angular veloctes,,, the dfference of pendula's dsplacement,, and beam's dsplacement x (magnfed 100 tmes) n the case when the exctaton of a number of pendula s swtched off for the system of 0 dentcal pendula wth mass m=0.1 rotatng n the same drecton, m b =1.0, k x =7000,,, 0,,, 5, 0. After tme equal to 50N, exctaton of some pendula s smultaneously swtched off, (a) exctaton of pendulum 1 s swtched off, (b) exctaton of ten pendula 1,,10 s swtched off, (c) exctaton of eleven pendula 1,,11 s swtched off, (d) exctaton of twelve pendula 1,,1 s swtched off. 6

63 Fgure 49. Tme seres showng the transent behavor of pendula's angular veloctes,,, the dfference of pendula's dsplacement,, n the case when the exctaton of a number of pendula s swtched off for the system of 0 dentcal pendula wth mass m=0.1 rotatng n the same drecton, m b =1.0, k x =7000,,, 0,,, 5, 0. At the moments ndcated by the arrows the exctaton of a number of pendula s swtched off, (a) exctaton of 11 pendula s swtched off at N=50, 60, 150, exctaton of the 1-th pendulum s swtched off at N= 300, (b) exctaton of 1 pendula are swtched off for N= 50, 60, 150,160, exctaton of the 13-th pendulum s swtched off for N=0. In the consdered examples a number of pendula losses exctaton smultaneously, f the pendula's exctaton s swtched off one by the scenaro can be dfferent. In the case descrbed n Fgure 49(a,b) (m 1-0 =0.1, m b =1.0, k x =7000) the exctaton s swtched off at the moments ndcated by arrows. In Fgure 49(a) eleven pendula are losng exctatons n the tme ntervals of 10N startng at 50N. The ncrease of the phase shft between the clusters of excted and unexcted pendula s vsble. For N=300 the exctaton of the 1-th pendulum s swtched off leadng to the loss of synchronzaton (1 pendula stop to rotate). Dfferent scenaro s descrbed n Fgure 49(b). The 1-th pendulum loses exctaton just after 11-th at N=160. Shortly after t 11 pendula (whch lost exctaton before) stop to rotate but the 1-th pendulum stll rotates and s phase synchronzed wth the cluster of 8 excted pendula (the phase shft s close to π/4). Later at N=0 the 13-th pendulum loses exctaton and two clusters of 7 excted and unexcted are created. The consdered example shows that t s possble to estmate the crtcal number of pendula whch exctaton can be swtched off and the rotaton of all of them s preserved. In the case when the pendula's exctatons are swtched off non-smultaneously t s possble to observe the case n whch unexcted pendula form two groups one of them stops to rotate and the second one rotates and s phase synchronzed wth the excted pendula. 63