Sage Journals: Discover world-class research

Abstract

The stationary dynamics of an unbalanced rotor (vibrator) on a movable base (linear oscillator) under excitation by a driving torque is studied with focusing on the stability of 1:1 stationary regimes of rotation and oscillation. This problem was well studied previously in the first approximation, dealing, in fact, with averaged regimes, mainly in the framework of asymptotic procedures. We use an efficient analytical procedure, proposed in our previous works for another problem, which sequentially separates the averaged regimes and deviations from them. Describing in the first approximation the known features of the synchronous stationary regimes under consideration, this approach in the second approximation results in the analytical solution for nonuniform rotation whose exactness is confirmed in the numerical simulation. The solution enables us to reveal possibility of parametric instability for oscillations of the rotor angular velocity and to describe two possible mechanisms of this instability. It is shown that the known condition of stability of stationary synchronous rotation-oscillation regimes is only necessary but not sufficient criterion, and two additional necessary conditions of stability are obtained and confirmed by the numerical simulation.

Keywords

Oscillator-vibrator systems unbalanced rotor on elastic base stationary synchronous regimes stability

Get full access to this article

View all access options for this article.

References

Rocard

. Dinamique Generale des vibrations, Paris: Masson, 1949, pp. 439–439.

Mazet

. Mecanique vibratoire, Paris, Liege: Libr. politechn. ch. Beranger, 1955, pp. 280–280. (in French).

Kononenko

. Vibrating systems with limited power supply, New York: Scripta Technica, Inc., 1969. (Russian ed.: Moscow, Nauka, 1964, p. 254).

Blekhman

. Synchronization of dynamic systems, Moscow: “Nauka”, 1971, pp. 894–894. (in Russian).

Blekhman

. What vibration can do. Vibrational mechanics and vibrational technology, Moscow: Nauka, 1988, pp. 208–208. (in Russian).

Blekhman

I . Vibrational mechanics. Nonlinear dynamic effects, general approach, applications, Singapore: World Scientific, 2000, pp. 509–509.

Alifov

Frolov

. Interaction of nonlinear oscillatory systems with power source, Moscow: Nauka, 1985, pp. 328–328. (in Russian).

Petchenev

. On the motion of the oscillating system with limited excitation in the vicinity of resonance. Doklady Mathematics 1986; 290: 1–1. (Translated from: Doklady of Acad, Sciences of the USSR 1986; 290: 27-31).

Rand

Kinsey

Mingori

. Dynamics of spinup through resonance. Int. J. of Non-Linear Mechanics 1992; 27(3): 489–502.

10.

Dimentberg

McGovern

Norton

, et al. Dynamics of an unbalanced shaft interacting with a limited power supply. Nonlinear Dyn 1997; 13: 171–187.

11.

Balthazar

Mook

Weber

, et al. An overview on non-ideal vibrations. Meccanica 2003; 38: 613–621.

12.

Wiercigroch

Cartmell

. Rotating orbits of a parametrically-excited pendulum. Chaos Soliton Fract 2005; 23: 1537–1548.

13.

Wiercigroch

. Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dyn 2007; 47: 311–320.

14.

Bolla

Balthazar

Felix

JLP

, et al. On an approximate analytical solution to a nonlinear vibrating problem, excited by a nonideal motor. Nonlinear Dyn 2007; 50: 841–847.

15.

Blekhman

Indejcev

Fradkov

. Slow motions in systems with inertial excitation of oscillations. J Mach Manuf Reliab 2008; 37: 21–27.

16.

Lenci

Pavlovskaia

Rega

, et al. Rotating solutions and stability of parametric pendulum by perturbation method. J Sound Vib 2008; 310: 243–259.

17.

Wiercigroch

Najdecka

Vaziri

. Nonlinear dynamics of pendulum system for energy harvesting. In: The 10th international conference on vibration problems, Springer Proceedings in Physics 139, 2011, pp. 35–42.

18.

Manevich

. Dynamics of an oscillator-rotator system under harmonic excitation. In: The 4th international conference on localization, energy transfer and nonlinear normal modes in mechanics and physics, (NNM2012), Haifa, Israel, 1–5 July 2012, pp. 44–47.

19.

Piccirillo

Tusset

Balthazar

. Dynamical jump attenuation in a non-ideal system through a magnetorheological damper. J Theor Appl Mech 2014; 52: 595–604.

20.

Cveticanin

Zukovic

. Motion of a motor-structure non-ideal system. Eur J Mech A/Solids 2015; 53: 229–240.

21.

Awrejcewicz

Starosta

Sypniewska-Kaminska

. Decomposition of governing equations in the analysis of resonant response of a nonlinear and non-ideal vibrating system. Nonlinear Dyn 2015; 82: 299–309.

22.

Balthazar

. Nonlinear dynamic interactions and phenomena: vibrating systems with limited power supply: an emergent topic after Prof. Kononenko. In: Proceedings of the 5th international conference on nonlinear dynamics, (ND-KhPI2016), Kharkov, Ukraine, 27–30 September 2016, pp. 16–22.

23.

Manevich

. An oscillator-rotator system: vibrational maintenance of rotation, stationary synchronous regimes, stability, vibration mitigation. J Sound Vib 2018; 437: 223–241.

24.

Manevich

. Nonlinear interaction of oscillation and rotation in oscillator-vibrator systems. In: Proceedings of the 5th international confererence on nonlinear dynamics, (ND-KhPI2016), Kharkov, Ukraine, 27–30 September 2016, pp. 149–156.

Stability of synchronous regimes in unbalanced rotors on elastic base

Abstract

Keywords

Get full access to this article

References