Modelling and experimental validation of resonant frequency shift in piezoelectric micro-cantilever beams

Cantilever microbeams play a critical role as key components in scanning probe microscopies (SPMs), bio and chemical sensors. Adding active layers such as piezoelectric thin film to form smart cantilever microbeams with sensing and actuation capabilities is required to facilitate miniaturization, enhance performance and functionalities of SPMs, and sensors. But it results in a complex multi-layered prestressed piezoelectric cantilever microbeam for which there is no reported mathematical model that predicts their dynamic behaviour. Here, we present a mathematical model with experimental validation that captures the dynamic behaviour and estimates the shift in the resonance frequencies of such cantilever microbreams. A modified Euler-Bernoulli beam equation considering axial load and bending effect due to residual stress and driving voltage has been developed to model the multi-layered microcantilever beam to obtain a closed-form solution. A very good agreement between the model predictions and experimental measurements has been obtained. The shift in resonance frequencies has displayed a linear characteristic for a wide range of driving voltages. Analytical model that describes the linear behaviour has been formulated and applied to investigate the shift in the fundamental resonant frequency of the cantilever microbeam as it is scaled down. The results show that the shift in a fundamental resonance frequency becomes significant as the cantilever microbeam is miniaturized from micro to nano scale. • A dynamic model of a multi-layered prestressed piezoelectric microcantilever beam has been developed and validated experimentally. • PZT based piezoelectric microcantilever has been fabricated and characterized for experimental verification of the dynamic mode. • A linearized formulation has been obtained to predict the shift in resonance frequencies of the PZT cantilever due to driving voltage. • Accurate dynamic model of PZT microcantilever beam is necessary for a range applications including scanning probe microscopies, smart sensing, and precision actuation.