Nonlinear oscillations of rotor active magnetic bearings system

Abstract

In this paper, an analytical approximate solution is constructed for a rotor-AMB system that is subjected to primary resonance excitations at the presence of 1:1 internal resonance. We obtain an approximate solution applying the method of multiple scales, and then we conducted the system bifurcation analyses. The stability of the system is investigated applying Lyapunov’s first method. The effects of the different parameters on the system behavior are investigated. The analytical results showed that the rotor-AMB system exhibits a variety of nonlinear phenomena such as bifurcations, coexistence of multiple solutions, jump phenomenon, and sensitivity to initial conditions. Finally, the numerical simulations are performed to demonstrate and validate the accuracy of the approximate solutions. We found that all predictions from analytical solutions are in excellent agreement with the numerical integrations.

Appendices

Appendix A

Introducing nondimensional parameters $u = c_{0}{\stackrel{\frown}{u}}, v = c_{0}{\stackrel{\frown}{v}}, i_{u} = I_{0}{\stackrel{\frown}{i}} _{u}, i_{v} = I_{0}{\stackrel{\frown}{i}} _{v}$, and $t = \zeta{\stackrel{\frown}{t}}, \varOmega= \zeta^{ - 1}{\stackrel{\frown}{\varOmega}}$, omitting the hat for brevity, Eqs. (5) and (7) can be rearranged as

(45)

(46)

$$ c_{0}\zeta^{2}\ddot{u} - \frac{1}{m}f_{u} + \frac{cc_{0}\zeta}{m}\dot{u} = e\cos(\varOmega t) $$

(47)

$$ c_{0}\zeta^{2}\ddot{v} - \frac{1}{m}f_{v} + \frac{cc_{0}\zeta}{m}\dot{v} = e\sin(\varOmega t) $$

(48)

Substituting Eq. (45) into (47), and (46) into (48), respectively, we get

(49)

(50)

Putting

in Eqs. (49) and (50), we get

(51)

(52)

where

Appendix B