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.

中文翻译：

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使用小型圆柱悬臂梁扫描的高性能开环控制

由单向位移激发的基部激发的圆形截面悬臂梁的稳态响应运动将沿直线下降。然而，用真正的圆形截​​面悬臂梁实现直线运动是很难实现的。这是因为非线性效应、与圆度的小偏差、不对称边界条件和致动器交叉耦合会导致旋转。以前关于悬臂梁旋转的绝大多数工作都集中在系统非线性的影响上。我们表明，在共振光束扫描仪的设计中，旋转是一个更广泛的问题，因为旋转的开始不依赖于大振幅的运动。相反，由于小的系统不对称和执行器交叉耦合，旋转是实际系统中的常态。因此，当需要单向光束运动时，有必要控制旋转运动的增长。我们开发了一种新技术来识别光束的两个本征方向。虚拟电极沿悬臂梁系统的这些正交本征轴产生的基础激励产生无旋转的尖端振动。这导致通过两对正交放置的致动器电极的组合致动对光束的运动进行精确的开环控制。