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Rotor-pendulum systems are widely applied to aero-power plants, mining screening machineries, parallel robots, and other high-speed rotating equipment. However, the investigation for synchronous behavior (the computation for stable phase difference between the rotors) of a rotor-pendulum system has been reported very little. The synchronous behavior usually affects the performance precision and quality of a mechanical system. Based on the special background, a simplified physical model for a rotor-pendulum system is introduced. The system consists of a rigid vibrating body, a rigid pendulum rod, a horizontal spring, a torsion spring, and two unbalanced rotors. The vibrating body is elastically supported via the horizontal spring. One of unbalanced rotors in the system is directly mounted in the vibrating body, and the other is fixed at the end of the pendulum rod connected with the vibrating body by the torsion spring. In addition, the rotors are actuated with the identical induction motors. In this paper, we investigate the synchronous state of the system based on Poincar method, which further reveals the essential mechanism of synchronization phenomenon of this system. To determine the synchronous state of the system, the following computation technologies are implemented. Firstly, the dynamic equation of the system is derived based on the Lagrange equation with considering the homonymous and reversed rotation of the two rotors, then the equation is converted into a dimensionless equation. Further, the dimensionless equation is decoupled by the Laplace method, and the approximated steady solution and coupling coefficient of each degree of freedom are deduced. Afterwards, the balanced equation and the stability criterion of the system are acquired. Only should the values of physical parameters of the system satisfy the balanced equation and the stability criterion, the rotor-pendulum system can implement the synchronous operation. According to the theoretical computation, we can find that the spring stiffness, the installation title angle of the pendulum rod, and the rotation direction of the rotors have large influences on the existence and stability of the synchronous state in the coupling system. Meanwhile, the critical point of synchronization of the system can lead to no solution of the phase difference between the two rotors, which results in the dynamic characteristics of the system being chaotic. Finally, computer simulations are preformed to verify the correctness of the theoretical computations, and the results of theoretical computation are in accordance with the computer simulations.
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Keywords:
- rotors /
- pendulum rod /
- stability /
- synchronous behavior
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[12] Marcheggiani L, Chacn R, Lenci S 2014 Eur. Phys. J.: Spec. Top. 223 729
[13] Wen B C, Fan J, Zhao C Y, Xiong W L 2009 Synchronization and Controled Sychronization in Engineering (Beijing: Science Press)
[14] Zhao C Y, Zhang Y M, Zhang X L 2010 Chin. Phys. B 19 030301
[15] Zhang X L, Wen B C, Zhao C Y 2012 Acta Mech. Sin. 28 1424
[16] Sperling L, Ryzhik B, Linz C, Duckstein H 2000 Math. Comput. Simulat. 58 351
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[18] Balthazar J M, Felix J L P, Reyolando M L R F 2005 Appl. Math. Comput. 164 615
[19] Djanan A A N, Nbendjo B R N, Woafo P 2014 Eur. Phys. J.: Spec. Top. 223 813
[20] Lacarbonara W, Arvin H, Bakhtiari-Nejad F 2012 Nonlinear Dyn. 70 659
[21] Stoykov S, Ribeiro P 2013 Finite Elem. Anal. Design. 65 76
[22] Warminski J, Szmit Z, Latalski J 2014 Eur. Phys. J.: Spec. Top. 223 827
[23] Andreas M, Peter M 2007 Multibody Syst. Dyn. 18 259
[24] Hou Y J, Zhang Z L China Patent 201110115274 [2012-12-26]
[25] Fang P, Hou Y J, Yang Q M, Chen Y 2014 J. Vibroeng. 16 2188
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[1] Blekhman I I 1988 Synchronization in Science and Technology (New York: ASME Press)
[2] Pikovsky A, Rosenblum M, Kurths J 2001 Synchronization, an Universal Concept in Nonlinear Sciences (Cambridge: Cambridge University Press)
[3] Arenas A, Albert D G, Kurths J, Moreno Y, Zhou C S 2008 Phys. Rep. 469 93
[4] Li Y S, L L, Liu Y, Liu S, Yan B B, Chang H, Zhou J N 2013 Acta Phys. Sin. 62 020513 (in Chinese) [李雨珊, 吕翎, 刘烨, 刘硕, 闫兵兵, 常欢, 周佳楠 2013 62 020513]
[5] Yuan W J, Zhou C S 2011 Phys. Rev.E 84 016116
[6] Yu H J, Liu Y Z 2005 Acta Phys. Sin. 54 3029 (in Chinese) [于洪洁, 刘延柱 2005 54 3029]
[7] Qin W Y, Yang Y F, Wang H J, Ren X M 2008 Acta Phys. Sin. 57 2068 (in Chinese) [秦卫阳, 杨永锋, 王红瑾, 任兴民 2008 57 2068]
[8] Pea R J, Aihara K, Fey R H B, Nijmeijer H 2014 Physica D 270 11
[9] Jovanovic V, Koshkin S 2012 J. Soun. Vib. 331 2887
[10] Kapitaniak M, Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T 2014 Phy. Rep. 541 1
[11] Dilo R 2014 Eur. Phys. J.: Spec. Top. 223 665
[12] Marcheggiani L, Chacn R, Lenci S 2014 Eur. Phys. J.: Spec. Top. 223 729
[13] Wen B C, Fan J, Zhao C Y, Xiong W L 2009 Synchronization and Controled Sychronization in Engineering (Beijing: Science Press)
[14] Zhao C Y, Zhang Y M, Zhang X L 2010 Chin. Phys. B 19 030301
[15] Zhang X L, Wen B C, Zhao C Y 2012 Acta Mech. Sin. 28 1424
[16] Sperling L, Ryzhik B, Linz C, Duckstein H 2000 Math. Comput. Simulat. 58 351
[17] Balthazar J M, Felix J L P, Reyolando M L R F 2004 J. Vib. Control. 10 1739
[18] Balthazar J M, Felix J L P, Reyolando M L R F 2005 Appl. Math. Comput. 164 615
[19] Djanan A A N, Nbendjo B R N, Woafo P 2014 Eur. Phys. J.: Spec. Top. 223 813
[20] Lacarbonara W, Arvin H, Bakhtiari-Nejad F 2012 Nonlinear Dyn. 70 659
[21] Stoykov S, Ribeiro P 2013 Finite Elem. Anal. Design. 65 76
[22] Warminski J, Szmit Z, Latalski J 2014 Eur. Phys. J.: Spec. Top. 223 827
[23] Andreas M, Peter M 2007 Multibody Syst. Dyn. 18 259
[24] Hou Y J, Zhang Z L China Patent 201110115274 [2012-12-26]
[25] Fang P, Hou Y J, Yang Q M, Chen Y 2014 J. Vibroeng. 16 2188
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