On nonlinear dynamics of a parametrically excited pendulum using both active control and passive rotational (MR) damper

Abstract

This paper presents two control strategies for a parametrically excited pendulum with chaotic behavior. One of them considers active control obtained by nonlinear saturation control (NSC) and the other a passive rotational magnetorheological (MR) damper. Firstly, the active control problem was formulated in order to design the external torque for the pendulum, considering the NSC. Numerical simulations were carried out in order to show the effectiveness of this method for the active control of the pendulum oscillation. The ability of the control of the proposed NSC in suppression of the chaotic behavior, considering the proposed parameters, was tested by a sensitivity analysis to parametric uncertainties. In the case of the passive rotational MR damper, firstly the influence of the introduction of the MR in a pendulum was performed considering the 0-1 test. Different electric currents are applied to suppress the chaotic behavior of the system. The numerical results showed that the simple introduction of a passive rotational MR damper without electric current did not change the chaotic behavior of the system. However, it is possible to keep the pendulum oscillating with periodic behavior using the rotational MR damper with energizing discontinuity.

Keywords

Electromechanical system nonlinear saturation control Dahl model rotational magnetorheological damper nonlinear control chaos 0-1 test

Get full access to this article

View all access options for this article.

References

Aguirre Carvajal N, Ikhouane F, Rodellar Benedé J, et al. (2010) Viscous+ Dahl model for MR damper characterization: a real time hybrid test (RTHT) validation. In: Proceedings of the 14th European conference on earthquake engineering, Ohrid, Republic of Macedonia, 30 August–3 September 2010, pp.1–8.

Alevras

Brown

Yurchenko

(2015) Experimental investigation of a rotating parametric pendulum. Nonlinear Dynamics 81(1–2): 201–213.

Altenborn KA, Klausen A, Tordal SS, et al. (2014) Firefly optimization used to identify hysteresis parameter on rotational MR-damper. In: Proceedings of the international conference on mechatronics and control (ICMC), Jinzhou, China, 3–5 July 2014, pp.2302–2307. IEEE.

Avraam

Horodinca

Romanescu

et al. (2010) Computer controlled rotational MR-brake for wrist rehabilitation device. Journal of Intelligent Material Systems and Structures 21(15): 1543–1557.

Balthazar

Bassinello

Tusset

et al. (2014a) Nonlinear control in an electromechanical transducer with chaotic behaviour. Meccanica 49(8): 1859–1867.

Balthazar

Tusset

Bueno

(2014b) Nonlinear TM-AFM control considering parametric errors in the control signal evaluation. Journal of Theoretical and Applied Mechanics 52(6): 93–106.

Balthazar

Tusset

Souza

SLTD

et al. (2012) Microcantilever chaotic motion suppression in tapping mode atomic force microscope. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 227(8): 1730–1741.

Bernardini

Litak

(2015) An overview of 0-1 test for chaos. Journal of the Brazilian Society of Mechanical Sciences and Engineering 38(5): 1433–1450.

Bernardini

Rega

Litak

et al. (2013) Identification of regular and chaotic isothermal trajectories of a shape memory oscillator using the 0-1 test. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 227(1): 17–22.

10.

Bishop

Clifford

(1994) Approximating the escape zone for the parametrically excited pendulum. Journal of Sound and Vibration 172(4): 572–576.

11.

Clifford

Bishop

(1995) Rotating periodic orbits of the parametrically excited pendulum. Physics Letters A 201(2–3): 191–196.

12.

Clifford

Bishop

(1996) Locating oscillatory orbits of the parametrically-excited pendulum. Journal of the Australian Mathematical Society Series B: Applied Mathematics 37(3): 309–319.

13.

Felix

JLP

Balthazar

Brasil

RMLRF

(2005) On saturation control of a non-ideal vibrating portal frame foundation type shear-building. Journal of Vibration and Control 11(1): 121–136.

14.

Gopal

Venkatesan

Lakshmanan

(2013) Applicability of 0-1 test for strange nonchaotic attractors. Chaos 23(2): 023123.

15.

Golnaraghi

(1991) Vibration suppression of flexible structure using internal resonance. Mechanics Research Communications 18(2–3): 135–143.

16.

Gottwald G and Melbourne I (2004) A new test for chaos in deterministic systems. Proceedings of the Royal Society of London A 460(2042): 603–611.

17.

Gottwald G and Melbourne I (2005) Testing for chaos in deterministic systems with noise. Physica D: Nonlinear Phenomena 212(1–2): 100–110.

18.

Huang

Zhang

Yang

et al. (2002) Analysis and design of a cylindrical magneto-rheological fluid brake. Journal of Materials Processing Technology 129(1): 559–562.

19.

Iliuk

Balthazar

Tusset

et al. (2013) A non-ideal portal frame energy harvester controlled using a pendulum. The European Physical Journal Special Topics 222(7): 1575–1586.

20.

Kecik

Warminski

(2012) Chaos in mechanical pendulum-like system near main parametric resonance. Procedia IUTAM 5: 249–258.

21.

Lenci

Rega

(2011) Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: a dynamical integrity perspective. Physica D: Nonlinear Phenomena 240(9): 814–824.

22.

Lenci

Pavlovskaia

Rega

Wiercigroch

(2008) Rotating solutions and stability of parametric pendulum by perturbation method. Journal of Sound and Vibration 310(1): 243–259.

23.

Litak

Bernardini

Syta

et al. (2013) Analysis of chaotic non-isothermal solutions of thermomechanical shape memory oscillators. The European Physical Journal Special Topics 222(7): 1637–1647.

24.

Litak

Borowiec

Kasperek

(2008a) Response of a magneto-reological fluid damper subjected to periodic forcing in a high frequency limit. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 88(12): 1000–1004.

25.

Litak

Borowiec

Wiercigroch

(2008b) Phase locking and rotational motion of a parametric pendulum in noisy and chaotic conditions. Dynamical Systems 23(3): 259–265.

26.

Litak

Syta

Wiercigroch

(2009) Identification of chaos in a cutting process by the 0–1 test. Chaos, Solitons and Fractals 40(5): 2095–2101.

27.

Litak

Wiercigroch

Horton

et al. (2010) Transient chaotic behaviour versus periodic motion of a parametric pendulum by recurrence plots. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 90(1): 33–41.

28.

(2006) Chaotic motions of a parametrically excited pendulum. Communications in Nonlinear Science and Numerical Simulation 11(7): 861–884.

29.

Nandakumar

Wiercigroch

Chatterjee

(2012) Optimum energy extraction from rotational motion in a parametrically excited pendulum. Mechanics Research Communications 43: 7–14.

30.

Nayfeh

(2000) Nonlinear Interactions, New York: Wiley.

31.

Nozaki

Balthazar

Tusset

et al. (2013) Nonlinear control system applied to atomic force microscope including parametric errors. Journal of Control, Automation and Electrical Systems 24(3): 223–231.

32.

Oueini SS (1999) Techniques for controlling structural vibrations. PhD Thesis, Virginia Tech, USA.

33.

Oueini

Nayfeh

Golnaraghi

(1997) A theoretical and experimental implementation of a control method based on saturation. Nonlinear Dynamics 13(2): 189–202.

34.

Pai

Schulz

(2000) A refined nonlinear vibration absorber. International Journal of Mechanical Sciences 42(3): 537–560.

35.

Pai

Wen

Naser

et al. (1998) Structural vibration control using PZT patches and non-linear phenomena. Journal of Sound and Vibration 215(2): 273–296.

36.

Piccirillo

Balthazar

Tusset

et al. (2015) Characterizing the nonlinear behavior of a pseudoelastic oscillator via the wavelet transform. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230(1): 120–132.

37.

Senkal

Gurocak

(2010) Serpentine flux path for high torque MRF brakes in haptics applications. Mechatronics 20(3): 377–383.

38.

Sieber

Gonzales-Buelga

Neild

et al. (2008) Experimental continuation of periodic orbits through a fold. Physical Review Letters 100(24): 244101.

39.

Stilling DSD and Szyszkowski W (2002) Controlling angular oscillations through mass reconfiguration: a variable length pendulum case. International Journal of Non-Linear Mechanics 37(1): 89–99.

40.

Tusset

Bueno

Nascimento

et al. (2013) Nonlinear state estimation and control for chaos suppression in MEMS resonator. Shock and Vibration 20(4): 749–761.

41.

Tusset

Piccirillo

Balthazar

et al. (2015) On suppression of chaotic motions of a portal frame structure under non-ideal loading using a magneto-rheological damper. Journal of Theoretical and Applied Mechanics 53(3): 653–664.

42.

Tusset

Piccirillo

Bueno

et al. (2016) Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker. Journal of Vibration and Control 22(17): 3621–3637.

43.

Wang

Jing

(2004) Chaos control of chaotic pendulum system. Chaos, Solitons and Fractals 21(1): 201–207.

44.

Wiercigroch M (2005) A new concept of energy extraction from waves via parametric pendulor. Patent application, UK.

45.

Wolf

Swift

Swinney

et al. (1985) Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena 16(3): 285–317.

46.

Wiercigroch

Cartmell

(2005) Rotating orbits of a parametrically-excited pendulum. Chaos, Solitons and Fractals 23(5): 1537–1548.

47.

Yokoi

Hikihara

(2011) Tolerance of start-up control of rotation in parametric pendulum by delayed feedback. Physics Letters A 375(17): 1779–1783.