This paper presents a mathematical model of a self-sensing microcantilever beam for mass sensing applications. Equations of motion are derived for a microcantilever beam with a tip mass and a piezoelectric patch actuator deposited on the cantilever surface. In the self-sensing mode, the same piezoelectric patch is used for actuation and sensing. Self-induced voltage signals, which are extracted using a capacitive bridge mechanism, reveal frequency information of the vibrating beam, which in turn, reveals the particle mass. Equations of motion are obtained using the extended Hamilton's principle by considering the microcantilever as a distributed- parameters system. Two methods to estimate the unknown tip mass are presented. The first one is based on an inverse solution to the characteristic equation problem, while the second method uses a constraint-based optimization approach to estimate the tip mass. To improve the self-sensing performance, the need for adaptive estimation of the piezoelectric capacitance is stressed and an online estimation mechanism is presented. Simulations are presented to demonstrate the ability of the model to detect tip mass up to 0.1 femtogram (1 femtogram= 10~(-15) gm). Further simulationresults demonstrate the working of constraint optimization method and adaptive self-sensing mechanism.
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