Review of Euler–Bernoulli Rectangular Beam Theory

Beams are among the most fundamental structural elements that are used in MEMS devices. Many complex MEMS structures can be modeled as beams connected in series or parallel, with or without a proofmass. Given that the boundary conditions at either end of a beam that occurs in MEMS devices are either free or clamped, the beams that commonly occur in a MEMS energy harvester are either a clamped-free cantilever beam or a doubly clamped beam. The modal analysis of the beam is performed as a differential eigenanalysis of the governing equation of the beam. The Euler–Bernoulli beam equation describes the spatial relationship between the beam’s deflection and the applied load. It can be derived using Newtonian (force/moment balancing) or Hamiltonian (Lagrange) approaches. In this chapter, we provide a review of the Euler–Bernoulli rectangular beam theory and use it to derive the modal response of the rectangular cantilever beam and the lumped model of its dynamic system.