Extending Timoshenko Beam Theory for Large Deflections in Compliant Mechanisms

Abstract

Compliant mechanisms (CMs) have presented its inherently advantageous properties due to the fact that CMs utilize elastic deformation of the elementary flexible members to transfer motion, force, and energy. Previously, the classic Euler–Bernoulli beam theory is the most used theory in terms of modeling large beam deflections in CMs. However, it has some assumptions that may decrease the modeling accuracy, such as ignoring the shear strain and the axial strain of cross sections. In this article, to take into account the shear and axial strains, we adopt the Timoshenko beam theory along with some modifications to consider the axial elongation. To simplify the complexity of the proposed governing boundary value problem (BVP), we transform the BVP into an explicit formulation and use weighted residual methods to numerically approximate the solution. We first focus on the single-beam deflection of a straight beam and an initially curved beam (ICB) using Euler–Bernoulli beam theory, Timoshenko beam theory, and solid mechanics to analyze the contributions of the influences of shear and axial strains in beam deflections. Then, we prove the feasibility of the proposed modeling strategy via mechanism synthesis for a bi-stable mechanism and an ICB-based parallelogram mechanism. Finally, the deduction of the mathematical model and the numerical results are provided along with brief analysis on the mechanical performances of the studied CMs.