The Use of Gaussian Elimination to Deduce Recurrence Relation of Reduced Transformation for an Open-Loop Multibody System

The reduced multibody system transfer matrix method (Reduced MSTMM) improves both numerical stability and efficiency to study a long-chain multibody system. It further circumvents the limitations of the new version multibody system transfer matrix method (NV-MSTMM) where hinge transfer equation depends on the type of the outboard body’s outboard hinge resulting in a combinatorial problem of hinge modeling. The key of the Reduced MSTMM is to find the recurrence relation of the reduced transformation (RT) at connecting points of adjacent elements along the transfer path of a system. In the existing literatures, however, the derivation of the recurrence relation is rather laborious, error-prone, and highly dependent on one’s mathematical skill. In this paper, by using Gaussian elimination, we propose a structured and systematic interpretation to deduce the RT at an element’s output point based on that at the element’s input point and basic element equations combining spatial vector algebra for an open-loop system. Elements considered include rigid body undergoing spatial motion with multiple-input and single-output (MISO) ends, general elastic body undergoing large overall spatial motion but small deformation with MISO ends, and general hinge with single-input and single-output ends. First of all, the fundamental kinematic and kinetic equations in a spatial vector notation for the three types of elements are established based on first principles, respectively. Then, together with the RT at the element’s input point, these element equations are assembled into a composite equation in a block-matrix sense with some simple rules such that the composite matrix is as close to upper-triangular as possible. Finally, a Gaussian elimination process is performed, which leads to the desired RT at the element’s output point thus develops the recurrence relation. The proposed method greatly releases the reliance on the experience and mathematical skills of the formulation developers, thus reliably establishing the recurrence relation of the RT at the connection points in an open-loop rigid-flexible coupled multibody system. With the developed recurrence relation, a recursive algorithm can further be used to solve the dynamics of an open-loop multibody system.