Transient Excitation Suppression Capabilities of Electromagnetic Actuators in Rotor-Shaft Systems

Abstract

Impulsive forces cause sudden variations in dynamic systems and may result in fatal damage to the system especially in turbines where the whole system is enclosed inside a casing with very small clearance between the blades and the casing. Electromagnetic actuators have been known to apply non-contact electromagnetic force on a rotor- shaft section preventing rotor-shaft vibrations. This work attempts to investigate the comparison of two different control laws in controlling the vibrations caused by noises such as sudden flow rate change, blade loss, or excitations occurring due to seismic vibrations. A rotor-shaft system is developed within a simulation framework which includes an actuator placed away from the bearings. Literature shows the use of conventional PD (Proportional Derivative) control law which is equivalent to a 2-element support model. This work novels the 3-element viscoelastic support model, which is found to offer better vibration mitigation abilities in terms of controlling transient excitations. Preliminary theoretical simulation using linearized expression of electromagnetic force and the accompanying example show good reduction in transverse response amplitude, postponement of instability caused by viscous form of rotor internal damping.

Appendix

System of Eq. (24.7) has been derived using conservation of linear and angular momentum in each direction [11]. Of course, there is no change in linear velocity along the axis of the rotor. That can be assumed to be constant and equal to zero. All the factors are defined as follows:

$$ \left[\begin{array}{ccccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}& {\mathrm{a}}_{14}& {\mathrm{a}}_{15}\\ {}{\mathrm{a}}_{21}& {\mathrm{a}}_{22}& 0& 0& {\mathrm{a}}_{25}\\ {}{\mathrm{a}}_{31}& 0& {\mathrm{a}}_{31}& {\mathrm{a}}_{34}& 0\\ {}0& 0& 0& 1& 0\\ {}0& 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}{\Omega}_{\mathrm{f}}\\ {}{\dot{\mathrm{u}}}_{\mathrm{f}}\\ {}{\dot{\mathrm{v}}}_{\mathrm{f}}\\ {}{\dot{\Phi}}_{\mathrm{f}}\\ {}{\dot{\upvarphi}}_{\mathrm{f}}\end{array}\right\}=\left\{\begin{array}{c}{\mathrm{b}}_1\\ {}{\mathrm{b}}_2\\ {}{\mathrm{b}}_3\\ {}{\mathrm{b}}_4\\ {}{\mathrm{b}}_5\end{array}\right\} $$

where,

$$ {a}_{11}={J}_{pf}+\left(m-{m}_b\right){e}^{\prime}\left\{r-e\cos \left({\gamma}^{\prime }-\gamma \right)\right\} $$

$$ {a}_{12}=\left(m-{m}_b\right)\left\{{e}^{\prime}\sin \left(\varOmega t+{\gamma}^{\prime}\right)-e\sin \left(\varOmega t+\gamma \right)\right\} $$

$$ {a}_{13}=\left(m-{m}_b\right)\left\{{e}^{\prime}\cos \left(\varOmega t+{\gamma}^{\prime}\right)-e\cos \left(\varOmega t+\gamma \right)\right\} $$

$$ {a}_{14}={e}^{\prime}\sin \phi $$

$$ {a}_{15}={e}^{\prime}\sin \psi $$

$$ {a}_{21}=-{e}^{\prime}\sin \left(\varOmega t+{\gamma}^{\prime}\right) $$

$$ {a}_{31}={e}_f\cos \left(\varOmega t+{\gamma}_f\right) $$

$$ {\displaystyle \begin{array}{c}{b}_1=\varOmega \left[{J}_p+m\left(r-e\right)r+{m}_b{r}_b\left\{{r}_b-e\cos \left({\gamma}_b-\gamma \right)\right\}\right]\\[8pt] {}+{m}_b\left\{\left({r}_b\sin \left(\varOmega t+{\gamma}_b\right)-e\sin \left(\varOmega t+{\gamma}_b\right)\right)\right\}\dot{y}\\[8pt] {}-{m}_b\left\{\left({r}_b\cos \left(\varOmega t+{\gamma}_b\right)-e\cos \left(\varOmega t+{\gamma}_b\right)\right)\right\}\dot{z}\end{array}} $$

$$ {b}_2=\dot{u}+\frac{\varOmega }{m-{m}_b}\left[{m}_b{r}_b\sin \left(\varOmega t+{\gamma}_b\right)-{m}_e\sin \left(\varOmega t+\gamma \right)\right] $$

$$ {b}_3=\dot{v}-\frac{\varOmega }{m-{m}_b}\left[{m}_b{r}_b\cos \left(\varOmega t+{\gamma}_b\right)-{m}_e\cos \left(\varOmega t+\gamma \right)\right] $$

$$ {b}_4=\dot{\phi} $$

$$ {b}_5=\dot{\psi} $$