Attenuation of Sommerfeld effect in an internally damped eccentric shaft-disk system via active magnetic bearings

Abstract

Eccentric shaft-disk system with internal damping driven by a non-ideal power source exhibits Sommerfeld effect characterized by nonlinear jump phenomena of amplitude and rotor speed upon exceeding a critical power input around the critical speed. This effect causes instability in high speed rotors. So the diminution of such effect is extremely important in order to smooth running of the rotors at high speeds. The aim of this paper is to attenuate the Sommerfeld effect of an internally damped unbalanced flexible shaft-disk system via linearized active magnetic bearings. The shaft-disk system is excited through a brushed DC motor which acts as a non-ideal energy source. The characteristic equation of fifth order polynomial in rotor speed is obtained through energy balance of supplied power and the mechanical power dissipated at steady-state condition. Using MATLAB simulations, amplitude frequency responses are obtained close to system resonance for several values of bias current of active magnetic bearings. Thus the Sommerfeld effect is found to be attenuated as bias current increases gradually. The complete disappearance of Sommerfeld effect is also reported when the bias current reaches a specific value under certain conditions. A few numerical results are validated with established results when bias current is made to zero.

Appendix

$$\dot{x}_{1} = x_{2}$$

(21)

$$\dot{x}_{2} = - \frac{{\left( {R_{e} + R_{i} + C_{AMB} } \right)}}{m}x_{2} - \frac{{R_{i} \theta_{2} }}{m}y_{1} - \frac{{\left( {K + K_{AMB} } \right)}}{m}x_{1} + e\theta_{2}^{2} \cos \left( {\theta_{1} + \varphi } \right)$$

(22)

$$\dot{y}_{1} = y_{2}$$

(23)

$$\dot{y}_{2} = - \frac{{\left( {R_{e} + R_{i} + C_{AMB} } \right)}}{m}y_{2} + \frac{{R_{i} \theta_{2} }}{m}x_{1} - \frac{{\left( {K + K_{AMB} } \right)}}{m}y_{1} + e\theta_{2}^{2} \sin \left( {\theta_{1} + \varphi } \right)$$

(24)

$$\dot{\theta }_{1} = \theta_{2}$$

(25)

$$\dot{\theta }_{2} = \frac{{\left( {T_{m} - T_{L} } \right)}}{I} - \frac{{R_{r} \theta_{2} }}{I}$$

(26)

where $x = x_{1} , \, y = y_{1} , \, \theta = \theta_{1} , \, T_{m} ,$ the torque supplied by the DC motor (shown in Eq. 14) and source loading torque is

$$T_{L} = R_{i} \left( {x_{1} y_{2} - x_{2} y_{1} } \right) + \left[ {\left( {R_{e} + R_{i} + C_{AMB} } \right)x_{2} + \left( {K + K_{AMB} } \right)x_{1} } \right]e\sin \theta - \left[ {\left( {R_{e} + R_{i} + C_{AMB} } \right)y_{2} + \left( {K + K_{AMB} } \right)y_{1} } \right]e\cos \theta .$$