A theoretical study for the vibration of a cantilever microbeam as a free boundary problem

Vibration of a beam has extensive applications in engineering and industry. The objective of this article is to provide a novel analytical model by free boundary approach to derive a more exact model for the small oscillations of a cantilever microbeam in contact with an incompressible bounded fluid in a cavity. First, a system of integral equations is proposed to solve and interpret the problem. Then, the existence of the solution is proved by the use of the Banach fixed point theorem. At the end, an example is presented for verifying the proposed method and comparing the results with the fixed boundary approach.