A Pseudorigid-Body 3R Model for Determining Large Deflection of Cantilever Beams Subject to Tip Loads

In this paper, a pseudorigid-body (PRB) 3R model, which consists of four rigid links joined by three revolute joints and three torsion springs, is proposed for approximating the deflection of a cantilever beam subject to a general tip load. The large deflection beam equations are solved through numerical integration. A comprehensive atlas of the tip deflection for various load modes is obtained. A three-dimensional search routine has been developed to find the optimal set of characteristic radius factors and spring stiffness of the PRB 3R model. Detailed error analysis has been done by comparing with the precomputed tip deflection atlas. Our results show that the approximation error is much less than that of the conventional PBR 1R model. To demonstrate the use of the PRB 3R model, a compliant four-bar linkage is studied and verified by a numerical example. The result shows a maximum tip deflection error of 1.2% compared with the finite element analysis model. The benefits of the PRB 3R model include that (a) the model parameters are independent of external loads, (b) the approximation error is relatively small for even large deflection beams, and (c) the derived kinematic and static constraint equations are simpler to solve compared with the finite element model.