A low-dimension, and numerically stable transfer matrix method to predict the dynamics of thin-walled beams with rigid bodies under intricate topologies

Abstract

Structures comprising mono-symmetric thin-walled beams (TBs) and rigid bodies (RBs) with complex topologies are commonly used in various surgical applications, including continuum robots and sensors. However, research about the dynamic modelling of these structures is rarely performed. In this paper, we develop a three-dimensional (3D) dynamic model for the system consisting of mono-symmetric TBs and RBs in a general manner utilizing a transfer matrix method (TMM). Firstly, the governing equations of the TB in the 3D space is derived by fully incorporating warping and shear effects, and the stability during the process of computing the transfer matrix (TM) is discussed. It is discovered that while the TM for the axial vibration is stable, the computations for bending and bending-torsion vibrations are numerically unstable due to the presence of the exponent terms in the TM, when the analysis frequency and TB length increase. Interestingly, the computation of the axial vibration response in a complex structure becomes unstable due to the coupled vibration among the TBs and RBs, compared to a single-span TB. To address this issue of numerical instability, a low-dimension TMM (14 × 14) is proposed by reducing the length of transfer path by using the substructure synthesis method. Several typical topologies, such as “chain-like”, “branch”, and cascaded “closed-loop” structures are addressed by the proposed method. Case studies have been conducted to illustrate the accuracy of the developed 3D dynamic model and the effectiveness of the proposed method in addressing the numerical difficulties of the conventional TMM at high frequencies. The proposed approach exhibits several advantages, including easy programming, accurate solution, and small degrees-of-freedom for intricate topologies, enabling the analysts to design structures more efficiently.