High performance open loop control of scanning with a small cylindrical cantilever beam

The steady-state response motion of a base excited cantilever beam with circular cross-section excited by a unidirectional displacement will fall along a straight line. However, achieving straight-line motion with a real cantilever beam of circular cross-section is difficult to accomplish. This is due to the fact that nonlinear effects, small deviations from circularity, asymmetric boundary conditions, and actuator cross coupling can induce whirling. The vast majority of previous work on cantilever beam whirling has focused on the effects of system nonlinearities. We show that whirling is a much broader problem in the design of resonant beam scanners in that the onset of whirling does not depend on large amplitude of motion. Rather, whirling is the norm in real systems due to small system asymmetries and actuator cross coupling. It is therefore necessary to control the growth of the whirling motion when a unidirectional beam motion is desired. We have developed a novel technique to identify the two eigen directions of the beam. Base excitation generated by virtual electrodes along these orthogonal eigen axes of the cantilever beam system generates tip vibration without whirl. This leads to accurate open loop control of the motion of the beam through the combined actuation of two pairs of orthogonally placed actuator electrodes.