Stability of Parametrically Excited Active Magnetic Bearing Rotor System Due to Moving Base

Abstract

Active magnetic bearings (AMBs) offer contact-less functioning and active vibration control capability while supporting and levitating a rotor. This is the reason that the AMBs are being progressively researched for novel and challenging applications in the industry. In application areas, such as ships, airplanes and space crafts, the rotor is mounted on a moving base, which causes parametric excitation to the system. This, in turn, is generally known to cause stability issues in a rotor shaft system. The present work thus attempts to conduct stability analysis of a rotor shaft system supported by an AMB and is parametrically excited due to the presence of periodically varying base motion. The finite element model for a generic rotor shaft system mounted on a moving base is first presented, and the time-periodic state matrix for the system is found. The Floquet–Liapunov method of analyzing stability of a periodically varying system is used to find the stability boundaries for the system with two widely used control laws for the AMB. The analysis reveals that it is important to consider the parametric excitation caused to the system when the AMBs are being designed for applications involving large base motions.

Appendix

Details of the matrices used for finding the global matrices given in Eqs. (1)–(3).

Shaft Inertia matrix: $\left[ M \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} m\left[ {\psi \left( x \right)} \right]^{T} \left[ {\psi \left( x \right)} \right] {\text{d}}x + \mathop \smallint \limits_{0}^{l} i_{d} \left[ {\psi^{\prime}\left( x \right)} \right]^{T} \left[ {\psi^{\prime}\left( x \right)} \right] {\text{d}}x$where $\left[ \psi \right]$ is the shape function matrix, gyroscopic matrix: $\left[ G \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} i_{p} \left[ {\psi^{\prime}} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime}} \right]{\text{d}}x$; $\left[ H \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} i_{p} \left[ {\psi^{\prime}} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime}} \right]{\text{d}}x$ bending stiffness matrix: $\left[ {K_{B} } \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} EI\left[ {\psi^{\prime\prime}} \right]^{T} \left[ {\psi^{\prime\prime}} \right]{\text{d}}x$; $\left[ {\psi^{\prime\prime}} \right] = \frac{{d^{2} \left[ {\psi \left( x \right)} \right]}}{{{\text{d}}x^{2} }}$.

Circulatory matrix: $\left[ {K_{C} } \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} EI\left[ {\psi^{\prime\prime}} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime\prime}} \right]{\text{d}}x$

Coriolis matrix: $\left[ C \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} m\left[ {\psi \left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi \left( x \right)} \right]{\text{d}}x + \mathop \smallint \limits_{0}^{l} i_{d} \left[ {\psi^{\prime}\left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime}\left( x \right)} \right]{\text{d}}x$

Parametric stiffness matrix due to base motion:

$$\left[ {K_{p11} } \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} m\left[ {\psi \left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]\left[ {\psi \left( x \right)} \right]{\text{d}}x + \mathop \smallint \limits_{0}^{l} i_{p} \left[ {\psi^{\prime}\left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 1 \\ \end{array} } \right]\left[ {\psi^{\prime}\left( x \right)} \right]{\text{d}}x$$

$$\left[ {K_{p22} } \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} m\left[ {\psi \left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 1 \\ \end{array} } \right]\left[ {\psi \left( x \right)} \right]{\text{d}}x + \mathop \smallint \limits_{0}^{l} i_{p} \left[ {\psi^{\prime}\left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime}\left( x \right)} \right]{\text{d}}x$$

$$\left[ {K_{p12} } \right]_{S}^{e} = \mathop \smallint \limits_{0}^{l} m\left[ {\psi \left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi \left( x \right)} \right]{\text{d}}x + \mathop \smallint \limits_{0}^{l} \left( {i_{p} - i_{d} } \right)\left[ {\psi^{\prime}\left( x \right)} \right]^{T} \left[ {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right]\left[ {\psi^{\prime}\left( x \right)} \right]{\text{d}}x$$

Rotor disk finite element matrices

Inertia matrix: $\left[ M \right]_{D} = \left[ {\begin{array}{*{20}c} {m_{D} } & 0 & 0 & 0 \\ 0 & {m_{D} } & 0 & 0 \\ 0 & 0 & {I_{d} } & 0 \\ 0 & 0 & 0 & {I_{d} } \\ \end{array} } \right]$;

Gyroscopic matrix: $\left[ G \right]_{D} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - I_{p} } \\ 0 & 0 & {I_{p} } & 0 \\ \end{array} } \right]$;

Coriolis effect matrix: $\left[ C \right]_{D} = \left[ {\begin{array}{*{20}c} 0 & { - m_{D} } & 0 & 0 \\ {m_{D} } & 0 & 0 & 0 \\ 0 & 0 & 0 & { - I_{d} } \\ 0 & 0 & {I_{d} } & 0 \\ \end{array} } \right]$; Parametric stiffness matrix: $\left[ {K_{p11} } \right]_{D} = \left[ {\begin{array}{*{20}c} {m_{D} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & {I_{p} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right];\left[ {K_{p22} } \right]_{D} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & {m_{D} } & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {I_{p} } \\ \end{array} } \right]$; $\left[ {K_{p12} } \right]_{D} = \left[ {\begin{array}{*{20}c} 0 & {m_{D} } & 0 & 0 \\ {m_{D} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {I_{p} - I_{d} } \\ 0 & 0 & {I_{p} - I_{d} } & 0 \\ \end{array} } \right]$