Vibration suppression of a piezoelectrically driven cantilever beam by input shaping

Input shaping is an effective method for suppressing the vibration of structures where passive damping cannot be incorporated. In this paper, the input shaping method is used to modify the command signal of a continuous structure in order to minimize the system settling time. The governing equation of the cantilever beam is derived and the solution is expanded in terms of its normal modes. The result shows that without damping, each mode is decoupled from the rest and thus can be controlled individually. Mathematical expressions characterizing the performances of existing discrete type zero-vibration (ZV) and zero-vibration-and-derivative (ZVD) shapers are also derived from this continuous formulation. Meanwhile, a system consisting a steel beam, a piezoelectric actuator, a capacitive displacement sensor, and a computer for shaping command generation serves as the experimental platform for examining the effectiveness of both shapers. In parallel, a finite element dynamics model is also constructed to simulate the behavior of the beam subjected to the shaped input. The experimental and computer simulation results agree with each other to a large extent, and indicate that when properly designed, both ZV and ZVD shaper could effectively reduce the residual vibration of flexural structures. However, the results also show that the ZVD shaper exhibits a more robust performance and hence is recommended for future applications.