Solution of Euler-Bernoulli equation using a semi-analytical technique for steady-state deflection of microelectromechanical cantilever under electrostatic actuation

A technique to solve Euler-Bernoulli equation for microcantilever beams under electrostatic actuation to find its static deflection is presented. Euler-Bernoulli equation contains fourth-order derivative of deflection with respect to position. Owing to the nonlinearities imposed by the electrostatic force, this cannot be solved analytically. The authors double integrate it using Leibniz integration rule to reduce it to a second-order integro-differential equation, which has then been numerically solved to find the deflection of the beam. This reduces the computational complexities of finite-difference analysis. The results have been compared with finite difference analysis and also verified with the experimental results.