Design and modeling of a motion amplifier using an axially-driven buckling beam

by Jiang, Jie.

iii For active materials such as piezoelectric stacks, which produce large forces and small displacements, motion amplification mechanisms are often necessary -- not simply to trade force for displacement, but to increase the output work transferred through a compliant structure. Here, a motion amplifier for obtaining large rotations from small linear displacements produced by a piezoelectric stack is built and tested. The concept for this motion amplifier uses elastic (buckling) and dynamic instabilities of an axially driven buckling beam. The optimal design of the buckling beam end conditions was determined from a static analysis of the system using Euler’s elastica theory. This analysis was verified experimentally. A stack-driven, buckling beam prototype actuator consisting of a pre-compressed PZT stack (140 mm long, 10 mm diameter) and a thin steel beam (60 mm x 12 mm x 0.508 mm) was constructed. The buckling beam served as the motion amplifier, while the PZT stack provided the input actuation. The experimental setup, measuring instrumentation and method, the beam preloading condition, and the excitation are fully described. The frequency response of the system for three preloading levels and three stack driving amplitudes was obtained. A maximum 16° peak-to-peak rotation was measured when the stack was driven at amplitude of 325 V and frequency of 39 Hz. The experiments on the details of period-n motions and the effects of beam preload were also conducted. Since the amplifier is driving a large mass at the pinned end, for simplicity, the mass of the buckling beam is neglected and the system is modeled as a single-degree-offreedom, non-linear system. The beam simply behaves a non-linear rotational spring iv having a prescribed displacement on the input end and a moment produced by the inertial mass acting on the output end. The moment applied to the mass is then a function of the beam end displacement and the mass rotation. The system is then modeled simply as a base-excited spring-mass oscillator. Results of the response for an ideal beam using the SDOF model agree with the experimental data to a high degree. Loading and geometric imperfections are also studied to determine the sensitivity of the actuator. The behavior with slight imperfection is similar to the response for the ideal beam and the experimental results; the response is not particularly sensitive to imperfection. Parameter studies for the ideal buckling beam amplifier were conducted using the validated spring-beam model; these can be used as guidance for improving the design of the motion amplifier and finding the optimal operational conditions for different applications.