Vibration Analysis of Piezoelectric Microcantilever Sensors

The main objective of this dissertation is to comprehensively analyze vibration characteristics of microcantilever-based sensors with application to ultra small mass detection and low dimensional materials characterization. The first part of this work focuses on theoretical developments and experimental verification of piezoelectric microcantilevers, commercially named Active Probes, which are extensively used in most today\u27s advanced Atomic Force Microscopy (AFM) systems. Due to special geometry and configuration of Active Probes, especially multiple jump discontinuities in their cross-section, a general and comprehensive framework is introduced for forced vibration and modal analysis of discontinuous flexible beams. More specifically, a general formulation is obtained for the characteristics matrix using both boundary and continuity conditions. The formulation is then reduced to the special case of Active Probes with intentional geometrical discontinuities. Results obtained from experiment are compared with the commonly used uniform beam model as well as the proposed discontinuous beam model. It is demonstrated that a significant enhancement on sensing accuracy of Active Probes can be achieved using the proposed discontinuous beam model compared to a uniform model when a multiple-mode operation is desired. In the second part of this dissertation, a comprehensive dynamic model is proposed for vector Piezoforce Microscopy (PFM) system under applied electrical loading. In general, PFM is considered as a suspended microcantilever beam with a tip mass in contact with a piezoelectric material. The material properties are expressed in two forms; Kelvin-Voigt model for viscoelstic representation of the material and piezoelectric force acting on the tip as a result of response of material to applied electric field. Since the application of bias voltage to the tip results in the surface displacement in both normal and in-plane directions, the microcantilever is considered to vibrate in all three directions with coupled transversal/longitudinal and lateral/torsional motions. In this respect, it is demonstrated that the PFM system can be governed by a set of partial differential equations along with non-homogeneous and coupled boundary conditions. Using the method of assumed modes, the governing ordinary differential equations of the system and its state-space representation are derived under applied external voltage. The formulation is then reduced to vertical PFM, in which low dimensional viscoelestic and piezoelectric properties of periodically poled lithium niobate (PPLN) material can be detected. For this purpose, the experimental and theoretical frequency responses along with a minimization strategy for the percentage of modeling error are utilized to obtain optimal spring constant of PPLN. Finally, the step input responses of experiment and theory are used to estimate the piezoelectric and damping coefficients of PPLN. Overall in this dissertation, a precise dynamic model is developed for piezoelectric microcantilever for ultra small mass detection purpose. This model can also be utilized in AFM systems to replace laser-based detection mechanism with other alternative transductions. Moreover, a comprehensive model is proposed for PFM system to simultaneously detect low dimensional viscoelastic and piezoelectric properties of materials. This model can also be utilized for data storage purpose in ferroelectric materials