On the robust stability of segmented driveshafts with active magnetic bearing control

Many researchers and engineers have employed active control techniques, such as active magnetic bearings (AMBs), to suppress imbalance vibration in various subcritical and supercritical speed rotor-dynamic applications. One issue that has not yet been addressed in previous AMB driveline control studies is the effect of non-constant velocity (NCV) flexible couplings, such as U-joint or disk-type couplings, present in many segmented drivelines. The NCV effects introduce periodic parametric and forcing terms into the equations of motion that are functions of shaft speed, driveline misalignment, and load-torque, resulting in a linear periodically time-varying system. Previous research has found that both internal damping and NCV terms greatly impact stability; thus, they must be accounted for in the control law design in order to ensure closed-loop stability of any AMB-NCV-driveline system. In this paper, numerical Roquet theory is used to explore the closed-loop stability of a flexible segmented NCV-driveline supported by AMBs with a proportional-derivative (PD) type controller. To ensure robust stability with respect to internal damping and NCV effects, the robust P and D gains and AMB locations are selected based on maximizing a stability index over a range of shaft speeds, driveline misalignments, and load-torques. It is found that maximum robustness occurs within a finite range of P and D gains for several different AMB locations. Finally, the range of robustly stabilizing P gains versus the shaft speed is examined for several misalignment and load-torque bounds.