Nonlinear dynamics of a classical rotating pendulum system with multiple excitations

doi:10.1088/1674-1056/ab9df2

中国物理B ›› 2020, Vol. 29 ›› Issue (11): 110502-.doi: 10.1088/1674-1056/ab9df2

Ning Han(韩宁)¹^,†(), Pei-Pei Lu(鲁佩佩)¹

收稿日期:2020-04-06 修回日期:2020-06-09 接受日期:2020-06-18 出版日期:2020-11-05 发布日期:2020-11-03

Nonlinear dynamics of a classical rotating pendulum system with multiple excitations

Ning Han(韩宁)^† and Pei-Pei Lu(鲁佩佩)

Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Received:2020-04-06 Revised:2020-06-09 Accepted:2020-06-18 Online:2020-11-05 Published:2020-11-03
Contact: ^†Corresponding author. E-mail: hanning.bing@163.com; ninghan@hbu.edu.cn
Supported by:
the National Natural Science Foundation of China (Grant Nos. 11702078 and 11771115), the Natural Science Foundation of Hebei Province, China (Grant No. A2018201227) and the High-Level Talent Introduction Project of Hebei University, China (Grant No. 801260201111).

摘要/Abstract

Abstract:

We report an attempt to reveal the nonlinear dynamic behavior of a classical rotating pendulum system subjected to combined excitations of constant force and periodic excitation. The unperturbed system characterized by strong irrational nonlinearity bears significant similarities to the coupling of a simple pendulum and a smooth and discontinuous (SD) oscillator, especially the phase trajectory with coexistence of Duffing-type and pendulum-type homoclinic orbits. In order to learn the effect of constant force on this pendulum system, all types of phase portraits are displayed by means of the Hamiltonian function with large constant excitation especially the transitions of complex singular closed orbits. Under sufficiently small perturbations of the viscous damping and constant excitation, the Melnikov method is used to analyze the global structure of the phase space and the feature of trajectories. It is shown, both theoretically and numerically, that this system undergoes a homoclinic bifurcation and then bifurcates a unique attracting rotating limit cycle. Finally, the estimation of the chaotic threshold of the rotating pendulum system with multiple excitations is calculated and the predicted periodic and chaotic motions can be shown by applying numerical simulations.

Key words: rotating pendulum, Melnikov method, rotating limit cycle, chaotic dynamics

Ning Han(韩宁), Pei-Pei Lu(鲁佩佩). [J]. 中国物理B, 2020, 29(11): 110502-.

Ning Han(韩宁) and Pei-Pei Lu(鲁佩佩). Nonlinear dynamics of a classical rotating pendulum system with multiple excitations[J]. Chin. Phys. B, 2020, 29(11): 110502-.

图/表 16

参考文献 49

[1]	Peters R D 2004 Sci. Educ. 13 279 DOI: 10.1023/B:SCED.0000041838.62582.ce
[2]	Matthews M R, Gauld C F, Stinner A 2005 The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives Dordrecht Springer 139
[3]	Parks H V, Faller J E 2010 Phys. Rev. Lett. 105 110801 DOI: 10.1103/PhysRevLett.105.110801
[4]	Carroll S P, Owen J S, Hussein M F M 2014 J. Sound Vib. 333 5865 DOI: 10.1016/j.jsv.2014.06.022
[5]	Stachowiak T, Okada T 2006 Chaos Solitons Fract. 29 417 DOI: 10.1016/j.chaos.2005.08.032
[6]	Roy J, Mallik A K, Bhattacharjee J K 2013 Nonlinear Dyn. 73 993 DOI: 10.1007/s11071-013-0848-1
[7]	Eissa M, Sayed M 2008 Commun. Nonlinear Sci. Numer. Simul. 13 465 DOI: 10.1016/j.cnsns.2006.04.001
[8]	Gitterman M 2010 Physica A 389 3101 DOI: 10.1016/j.physa.2010.03.008
[9]	Paula A S D, Savi M A, Wiercigroch M, Pavlovskaia E 2012 Int. J. Bifur. Chaos 22 1250111 DOI: 10.1142/S0218127412501118
[10]	Horton B, Wiercigroch M, Xu X 2008 Philos. Trans. R. Soc. A 366 767 DOI: 10.1098/rsta.2007.2126
[11]	Bartuccelli M V, Gentile G, Georgiou K V 2001 Proc. R. Soc. London A 457 3007 DOI: 10.1098/rspa.2001.0841
[12]	Han N, Cao Q J 2016 Int. J. Non-Linear Mech. 88 122 DOI: 10.1016/j.ijnonlinmec.2016.10.018
[13]	Xu X, Wiercigroch M. 2007 Nonlinear Dyn. 47 311 DOI: 10.1007/s11071-006-9074-4
[14]	Das S, Wahi P 2016 Proc. R. Soc. A 472 20160719 DOI: 10.1098/rspa.2016.0719
[15]	Zhang H, Ma T W 2012 Nonlinear Dyn. 70 2433 DOI: 10.1007/s11071-012-0631-8
[16]	Koch B P, Leven R W 1985 Physica D 16 1 DOI: 10.1016/0167-2789(85)90082-X
[17]	Ji J C, Chen Y S 1999 Appl. Math. Mech. 20 350 DOI: 10.1007/BF02458560
[18]	Albert Luo C J, Min F H 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 4704 DOI: 10.1016/j.cnsns.2011.01.028
[19]	Leven R W, Koch B P 1981 Phys. Lett. A 86 71 DOI: 10.1016/0375-9601(81)90167-5
[20]	Kapitaniak M, Czolczynski K, Perlikowski P, Kapitaniak T 2012 Phys. Rep. 517 1 DOI: 10.1016/j.physrep.2012.03.002
[21]	Najdecka A, Kapitaniak T, Wiercigroch M. 2015 Int. J. Non-Linear Mech. 70 84 DOI: 10.1016/j.ijnonlinmec.2014.10.008
[22]	Wojna M, Wijata A, Wasilewski G, Awrejcewicz J. 2018 J. Sound Vib. 430 214 DOI: 10.1016/j.jsv.2018.05.032
[23]	Alevras P, Brown I, Yurchenko D. 2015 Nonlinear Dyn. 81 201 DOI: 10.1007/s11071-015-1982-8
[24]	Wu S T, Siao P S 2012 J. Sound Vib. 331 3020 DOI: 10.1016/j.jsv.2012.02.021
[25]	Mahmoudkhani S 2018 J. Sound Vib. 425 102 DOI: 10.1016/j.jsv.2018.03.025
[26]	Najdecka A, Narayanan S, Wiercigroch M 2015 Int. J. Non-Linear Mech. 71 30 DOI: 10.1016/j.ijnonlinmec.2014.12.008
[27]	Jia M M, Jiang H G, Li W J 2019 Acta Phys. Sin. 68 130503 in Chinese
[28]	Cao Q J, Han N, Tian R L 2011 Chin. Phys. Lett. 28 0605021
[29]	Han N, Cao Q J, Wiercigroch M 2013 Int. J. Bifur. Chaos 23 1350074 DOI: 10.1142/S0218127413500740
[30]	Han N, Cao Q J 2016 Commun. Nonlinear Sci. Numer. Simulat. 36 431
[31]	Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2006 Phys. Rev. E 74 046218 DOI: 10.1103/PhysRevE.74.046218
[32]	Li Z X, Cao Q J, Alain L 2016 Chin. Phys. B 25 010502 DOI: 10.1088/1674-1056/25/1/010502
[33]	Hong L, Xu J 2004 Commun. Nonlinear Sci. Numer. Simulat. 9 313
[34]	Ueda Y 2006 Ann. New York Acad. Sci. 357 422 DOI: 10.1111/j.1749-6632.1980.tb29708.x
[35]	Zhang X H, Zhou L Q 2018 Appl. Math. Model. 61 744 DOI: 10.1016/j.apm.2018.05.003
[36]	Hou L, Chen H Z, Chen Y S, Lu K, Liu Z S 2019 Mech. Syst. Signal Process. 125 65 DOI: 10.1016/j.ymssp.2018.07.019
[37]	Hou L, Su X C, Chen Y S 2019 Int. J. Bifur. Chaos 29 1950173 DOI: 10.1142/S0218127419501736
[38]	Cao Q J, Wiercigroch M, Pavlovskaia E E, Grebogi C, Thompson J M T 2008 Philos. Trans. R. Soc. A 366 635 DOI: 10.1098/rsta.2007.2115
[39]	Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields New York Springer 184
[40]	Melnikov V K 1963 Trans. Moscow. Math. Soc. 12 1
[41]	Hu H Y 2000 Applied Nonlinear Dynamics Beijing Aviation Industry Press of China 38 in Chinese
[42]	Strogatz S H 2015 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Beijing China Machine Press 272
[43]	Kwekt K H, Li J B 1996 Int. J. Non-Linear Mech. 31 277 DOI: 10.1016/0020-7462(95)00068-2
[44]	Sun J C 2016 Chin. Phys. Lett. 33 100503 DOI: 10.1088/0256-307X/33/10/100503
[45]	Gao F, Li W Q, Tong H Q et al. 2019 Chin. Phys. B 28 090501 DOI: 10.1088/1674-1056/ab38a4
[46]	Seydel R 1985 Physica D 17 308 DOI: 10.1016/0167-2789(85)90213-1
[47]	Albert Luo C J 2017 Resonance and Bifurcation to Chaos in Pendulum Beijing Higher Education Press of China 220
[48]	Liu Y Z, Chen L Q 2001 Nonlinear Vibrations Beijing Aviation Industry Press of China 286 in Chinese
[49]	Feng J Q 2018 Nonlinear Dynamics of a Typical Vibro-Impact System Beijing Science Press of China 95 in Chinese

Nonlinear dynamics of a classical rotating pendulum system with multiple excitations

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