Recovering external forces on vibrating Euler-Bernoulli beams using boundary shape function methods

Abstract

In this paper, we recover unknown space–time dependent forces imposed on the vibrating Euler–Bernoulli beams under different boundary conditions, including simply supported, clamped-hinged, cantilevered, and two-end fixed conditions. The data overspecified to recover the external force are the final time displacement and the right-side strain or right-side moment of the beam. We develop a family of boundary shape functions, which automatically satisfy the initial conditions, final time condition, and homogeneous boundary conditions for each type of beam. When the solution is obtained using the method of superimposing boundary shape functions and solving a small-scale linear system to satisfy an extra right-side boundary condition, the unknown force can be recovered through back-substitution of the solution into the Euler–Bernoulli beam equation. The accuracy and robustness of the proposed methods are confirmed by comparing the recovered results of seven examples to the exact forces, even though considerable noise is present in the overspecified data.