Active Control of Internal Damping Instabilities in a Cracked Rotor with Magnetic Bearing

Abstract

In high-speed rotating machinery, the internal damping becomes predominant that further aggravates the appearance of cracks in the rotor due to severe unstable vibrations. The internal damping comes into existence in thick shafts when the fibers of the material are alternately compressed and stretched due to the asynchronous whirling motion. The presence of internal damping in rotors is also influenced by the appearance of the crack due to rubbing of fatigue crack fronts during its opening and closing, which has an additional effect of reducing the stiffness of the shaft. The aim of the present paper is to actively control through magnetic bearings (MB), the unstable vibrations induced by the internal damping in the presence of switching crack and unbalances in a rotor system. Equations of motion of a simple Jeffcott rotor are derived considering both external and internal damping, switching crack model, unbalance force, and active MB force. The chosen crack model gives multiple harmonics not only in the forward whirl but also in the backward whirl. The active MB system, which is used here as a controller and not for supporting the rotor static load, utilizes PID control law, which requires tuning of control law parameters for stable control of the rotor system. Rotor responses are obtained through a numerical simulation to study interplay between instability of rotor due to internal damping and its active control through the MB. Full (or directional) spectrum plots are utilized to demonstrate both the forward and backward harmonics of the rotor whirling at different rotor speeds due to the presence of switching crack with and without active MB. Nyquist plots are provided to check the stability of the rotor system at different operating speeds.

Appendix 1 Transfer Function of the Rotor-AMB Model

The closed-loop transfer function of the rotor-bearing system,

$$G = \frac{{G_{CMA} }}{{1 + G_{CMA} G_{sn} }}$$

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with

$$G_{CMA} = G_{C} G_{M} G_{A} = \left( {\frac{{K_{P} s + K_{I} + K_{D} s^{2} }}{s}} \right)\left( {\frac{{k_{i} }}{{\left\{ {ms^{2} + (c_{V} + c_{H} )s + (k - j\omega c_{H} ) + k_{s} } \right\}}}} \right)k_{a}$$

and

$$G_{sn} = k_{sn}$$

where $G_{CMA}$ be the transfer function of the system containing controller, sensor and magnetic bearing in series as shown in Fig. 8, and $G_{sn}$ is the overall correction sensor gain. The overall closed-loop transfer function of the system becomes

$$G = \frac{{s^{2} \left( {K_{P} + \frac{{K_{I} }}{s} + K_{D} s} \right)k_{i} k_{s} }}{{\left( {\left\{ {ms^{3} + (c_{V} + c_{H} + K_{D} k_{i} k_{s} k_{sn} )s^{2} + (k - j\omega c_{H} + k_{s} + K_{P} k_{i} k_{x} k_{sn} )s} \right\}} \right) + K_{I} k_{i} k_{s} k_{sn} }}$$

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where k_x is the amplifier gain of the power amplifier. The characteristics equation of motion of the system becomes

$$ms^{3} + (c_{V} + c_{H} + K_{D} k_{i} k_{s} k_{sn} )s^{2} + (k - j\omega c_{H} + k_{s} + K_{P} k_{i} k_{s} k_{sn} )s + K_{I} k_{i} k_{x} k_{sn} = 0$$

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