Chained Spatial Beam Adomian Decomposition Model: A Novel Model of Flexible Slender Beams for Large Spatial Deflections

Abstract

The main element of compliant mechanisms and continuum robots is flexible slender beams. However, the modeling of beams can be complicated due to the geometric nonlinearity becoming significant at large elastic deflections. This paper presents an explicit nonlinear model called the spatial beam Adomian decomposition model (SBADM) for intermediate spatial deflections of a slender beam with uniform, bisymmetric sections subjected to general end-loading. Specifically, the elongation, bending, torsion, and shear deformations of the beams are modeled based on Timoshenko’s assumptions and Cosserat rod theory. Then, the quaternion transformation and Adomian decomposition are used to solve the nonlinear governing differential equations for the beam by truncating the higher-order terms, yielding an explicit expression for spatially deflected beams within intermediate deflection ranges. Simulations demonstrate the accuracy and time-wise efficiency of the SBADM, as well as its advantages over the state-of-the-art. In addition, this paper also introduces a discretization-based scheme called the chained SBADM (CSBADM) for large spatial deflections of flexible beams. Real-world experiments with two different configurations have also been performed to validate the effectiveness of the CSBADM. The results indicate that the CSBADM can accurately calculate the load–displacement relations for large deformed beams.