Analysis of Synchronization of n Metronomes on a Hanging Plate via Describing Function Method without Assumption on Amplitudes of Metronomes
详细信息 查看官网全文
- 英文论文题名：Analysis of Synchronization of n Metronomes on a Hanging Plate via Describing Function Method without Assumption on Amplitudes of Metronomes
- 论文作者：Xin Xin ; Yoshinori Muraoka ; Shinsaku Izumi ; Taiga Yamasaki
- 英文论文作者：Xin Xin ; Yoshinori Muraoka ; Shinsaku Izumi ; Taiga Yamasaki ; Faculty of Computer Science and Systems Engineering Okayama Prefectural University
- 年：2017
- 作者机构：Faculty of Computer Science and Systems Engineering Okayama Prefectural University;
- 英文论文关键词：Synchronization ; metronomes ; pendulum clocks ; describing function ; phasor representation
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：TN752
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：589k
- 原文格式：D
- 会议级别：全国
摘要
This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes on a hanging plate by using the describing function(DF) method, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. Different from a previous study on the synchronization of the system with the first harmonic of the angle of each metronome having the same amplitude, this paper analyzes the synchronization without any assumption on the amplitudes of n metronomes. By investigating the relation between the motion of the hanging plate(common base) and n metronomes, this paper characterizes the synchronization phenomena by the following two types of synchronization:One is the inherent synchronization, whose frequency and amplitude of all synchronized metronomes are the same as those of a single metronome on a fixed base. The other is the non-inherent synchronization, which includes the in-phase synchronization with the same amplitude, and the synchronization with different amplitudes. The analytical results are validated by numerical simulation for three metronomes about the synchronization with the same or different amplitudes.
This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes on a hanging plate by using the describing function(DF) method, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. Different from a previous study on the synchronization of the system with the first harmonic of the angle of each metronome having the same amplitude, this paper analyzes the synchronization without any assumption on the amplitudes of n metronomes. By investigating the relation between the motion of the hanging plate(common base) and n metronomes, this paper characterizes the synchronization phenomena by the following two types of synchronization:One is the inherent synchronization, whose frequency and amplitude of all synchronized metronomes are the same as those of a single metronome on a fixed base. The other is the non-inherent synchronization, which includes the in-phase synchronization with the same amplitude, and the synchronization with different amplitudes. The analytical results are validated by numerical simulation for three metronomes about the synchronization with the same or different amplitudes.
This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes on a hanging plate by using the describing function(DF) method, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. Different from a previous study on the synchronization of the system with the first harmonic of the angle of each metronome having the same amplitude, this paper analyzes the synchronization without any assumption on the amplitudes of n metronomes. By investigating the relation between the motion of the hanging plate(common base) and n metronomes, this paper characterizes the synchronization phenomena by the following two types of synchronization:One is the inherent synchronization, whose frequency and amplitude of all synchronized metronomes are the same as those of a single metronome on a fixed base. The other is the non-inherent synchronization, which includes the in-phase synchronization with the same amplitude, and the synchronization with different amplitudes. The analytical results are validated by numerical simulation for three metronomes about the synchronization with the same or different amplitudes.
引文
[1]A.Pikovsky,M.Rosenblum,and J.Kurths,Synchronization:A Universal Concept in Nonlinear Sciences.Cambridge:Cambridge University Press,2003.
[2]M.Bennett,M.F.Schatz,H.Rockwood,and K.Wiesenfeld,“Huygens’s clocks,”Proceedings:Mathematics,Physical and Engineering Sciences,vol.458,no.2019,pp.563–579,2002.
[3]W.Oud,H.Nijmeijer,and A.Y.Pogromsky,“A study of Huijgens’synchronization:Experimental results,”in Group Coordination and Cooperative Control.Springer,2006,pp.191–203.
[4]N.V.Kuznetsov,G.A.Leonov,H.Nijmeijer,and A.Y.Pogromsky,“Synchronization of two metronomes,”in Proceedings of the 3rd IFAC Workshop on Periodic Control Systems,2007,pp.49–52.
[5]K.Czolczynski,P.Perlikowski,A.Stefanski,and T.Kapitaniak,“Clustering and synchronization of n Huygens’clocks,”Physica A:Statistical Mechanics and Its Applications,vol.388,no.24,pp.5013–5023,2009.
[6]J.P.Ramirez,L.A.Olvera,H.Nijmeijer,and J.Alvarez,“The sympathy of two pendulum clocks:beyond Huygens’observations,”Scientific Reports,vol.6,2016.
[7]H.Ulrichs,A.Mann,and U.Parlitz,“Synchronization and chaotic dynamics of coupled mechanical metronomes,”Chaos:An Interdisciplinary Journal of Nonlinear Science,vol.19,no.4,p.043120,2009.
[8]R.Dil?ao,“Antiphase and in-phase synchronization of nonlinear oscillators:The Huygens’s clocks system,”Chaos:An Interdisciplinary Journal of Nonlinear Science,vol.19,no.2,p.023118,2009.
[9]A.Gelb and W.E.Vander Velde,Multiple-Input Describing Functions and Nonlinear System Design.New York:Mc Graw-Hill,1968.
[10]D.P.Atherton,Nonlinear Control Engineering:Describing Function Analysis and Design.London:Van Nostrand Reinhold,1975.
[11]X.Xin and Y.Liu,“Analysis of synchronization phenomena of two metronomes on a cart using describing function approach,”in Proceedings of the 2015 American Control Conference,2015,pp.1345–1350.
[12]X.Xin,Y.Muraoka,and S.Hara,“Analysis of synchronization of n metronomes on a cart via describing function method:New results beyond two metronomes,”in Proceedings of the 2016 American Control Conference,2016,pp.6604–6609.
[13]Y.Muraoka,X.Xin,S.Izumi,and T.Yamasaki,“Analysis of synchronization of two metronomes hanging from a plate via describing function approach,”in Proceedings of the 35th Chinese Control Conference,2016,pp.1110–1115.
[14]Y.Sato,K.Nakano,T.Nagamine,and M.Fuse,“Synchronized phenomena of oscillators:Experimental and analytical investigation for two metronomes,”Transactions of the Japan Society of Mechanical Engineers(in Japanese),vol.66,no.642,pp.363–369,2000.
[15]T.Kandou,Y.Bonkobara,H.Mori,and S.Ishikawa,“Selfsynchronized phenomena generated in pendulum-type oscillators:1st report,analysis for self-synchronized phenomena between two metronomes by using improved shooting method,”Transactions of the Japan Society of Mechanical Engineers(in Japanese),vol.68,no.676,pp.3499–3506,2002.
[16]S.Skogestad and I.Postlethwaite,Multivariable Feedback Control:Analysis and Design.New York:Wiley,2007.
[2]M.Bennett,M.F.Schatz,H.Rockwood,and K.Wiesenfeld,“Huygens’s clocks,”Proceedings:Mathematics,Physical and Engineering Sciences,vol.458,no.2019,pp.563–579,2002.
[3]W.Oud,H.Nijmeijer,and A.Y.Pogromsky,“A study of Huijgens’synchronization:Experimental results,”in Group Coordination and Cooperative Control.Springer,2006,pp.191–203.
[4]N.V.Kuznetsov,G.A.Leonov,H.Nijmeijer,and A.Y.Pogromsky,“Synchronization of two metronomes,”in Proceedings of the 3rd IFAC Workshop on Periodic Control Systems,2007,pp.49–52.
[5]K.Czolczynski,P.Perlikowski,A.Stefanski,and T.Kapitaniak,“Clustering and synchronization of n Huygens’clocks,”Physica A:Statistical Mechanics and Its Applications,vol.388,no.24,pp.5013–5023,2009.
[6]J.P.Ramirez,L.A.Olvera,H.Nijmeijer,and J.Alvarez,“The sympathy of two pendulum clocks:beyond Huygens’observations,”Scientific Reports,vol.6,2016.
[7]H.Ulrichs,A.Mann,and U.Parlitz,“Synchronization and chaotic dynamics of coupled mechanical metronomes,”Chaos:An Interdisciplinary Journal of Nonlinear Science,vol.19,no.4,p.043120,2009.
[8]R.Dil?ao,“Antiphase and in-phase synchronization of nonlinear oscillators:The Huygens’s clocks system,”Chaos:An Interdisciplinary Journal of Nonlinear Science,vol.19,no.2,p.023118,2009.
[9]A.Gelb and W.E.Vander Velde,Multiple-Input Describing Functions and Nonlinear System Design.New York:Mc Graw-Hill,1968.
[10]D.P.Atherton,Nonlinear Control Engineering:Describing Function Analysis and Design.London:Van Nostrand Reinhold,1975.
[11]X.Xin and Y.Liu,“Analysis of synchronization phenomena of two metronomes on a cart using describing function approach,”in Proceedings of the 2015 American Control Conference,2015,pp.1345–1350.
[12]X.Xin,Y.Muraoka,and S.Hara,“Analysis of synchronization of n metronomes on a cart via describing function method:New results beyond two metronomes,”in Proceedings of the 2016 American Control Conference,2016,pp.6604–6609.
[13]Y.Muraoka,X.Xin,S.Izumi,and T.Yamasaki,“Analysis of synchronization of two metronomes hanging from a plate via describing function approach,”in Proceedings of the 35th Chinese Control Conference,2016,pp.1110–1115.
[14]Y.Sato,K.Nakano,T.Nagamine,and M.Fuse,“Synchronized phenomena of oscillators:Experimental and analytical investigation for two metronomes,”Transactions of the Japan Society of Mechanical Engineers(in Japanese),vol.66,no.642,pp.363–369,2000.
[15]T.Kandou,Y.Bonkobara,H.Mori,and S.Ishikawa,“Selfsynchronized phenomena generated in pendulum-type oscillators:1st report,analysis for self-synchronized phenomena between two metronomes by using improved shooting method,”Transactions of the Japan Society of Mechanical Engineers(in Japanese),vol.68,no.676,pp.3499–3506,2002.
[16]S.Skogestad and I.Postlethwaite,Multivariable Feedback Control:Analysis and Design.New York:Wiley,2007.