Extending Timoshenko Beam Theory for Large Deflections in Compliant Mechanisms


 Compliant Mechanisms (CMs) have presented its inherently advantageous properties due to the fact that the nature of CMs is about utilizing elastic deformation of elementary flexible members to transfer motion, force and energy. In the previous work of CM community, the classic Euler-Bernoulli beam theory is the most used theory in terms of modeling large beam deflections. However, it has some assumptions that may decrease the modeling accuracy, such as ignoring the shear deformation of cross sections and the axial elongation. In this paper, to take into account the shear influences and the axial elongation, we adopt Timoshenko beam theory along with some modifications to consider the axial elongation. To simplify the complexity of the proposed governing 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 shear influences and axial elongation in beam deflections. Then, we have proved the feasibility of mechanism synthesis for a bi-stable mechanism and an ICB-based parallelogram mechanism using the proposed modeling strategy. 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.