Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.

On the use of linear graph theory in multibody system dynamics

Sensitivity analysis is the process of apportioning changes in the response of the system into the perturbations of the system parameters. In this chapter, we will present an overview of various issues regarding sensitivity analysis of multibody systems using symbolic formulations. Symbolic formulation for multibody system simulation has been demonstrated to have a number of benefits (Samin and Fisette in Symbolic Modeling of Multibody Systems, 2004), and has proven itself to be a very important tool for efficient sensitivity analysis. We present a detailed literature review of the subject, highlighting the challenges and the diverse applications of sensitivity analysis. We identify direct differentiation as a key approach towards symbolic sensitivity for its simplicity and accuracy. We will discuss software implementation and issues regarding the efficiency of the process of sensitivity analysis. We will present an overview of generation of sensitivity equations and using numerical examples, we will outline different approaches towards the evaluation of sensitivity information.

Symbolic Sensitivity Analysis of Multibody Systems

In this paper, theorems are proved that relate the degree of staticaI indeterminacy of space structures to their planar topological properties. An algorithm is developed for suboptimal layout of an arbitrary space structure in a plane, and a lower bound is established to the crossing number of its graph model.

Space structures and the crossing number of their graphs

A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented, Both problems are shown to be NP-hard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NP-complete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems. bence itself an NP-hard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed.

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On the Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis

The main focus of this research is to use graph-theoretic formulations to develop an automated algorithm for the generation of sensitivity equations. The idea is to combine the benefits of direct differentiation with that of graph-theoretic formulation. The primary deliverable of this work is the developed software module which can derive the system equations and the sensitivity equations directly from the linear graph of the system. Sensitivity analysis refers to the study of changes in system behaviour brought about by the changes in model parameters. Due to the rapid increase in the sizes and complexities of the models being analyzed, it is important to extend the capabilities of the current tools of sensitivity analysis, and an automated, efficient, and accurate method for the generation of sensitivity equations is highly desirable. In this work, a graph-theoretic algorithm is developed to generate the sensitivity equations. In the current implementation, the proposed algorithm uses direct differentiation to generate sensitivity equations at the component level and graph-theoretic methods to assemble the equation fragments to form the sensitivity equations. This way certain amount of control can be established over the size and complexity of the generated sensitivity equations. The implementation of the algorithm is based on a commercial software package MapleSim[Multibody] and can generate governing and sensitivity equations for multibody models created in MapleSim. In this thesis, the algorithm is tested on various mechanical, hydraulic, electro-chemical, multibody, and multi-domain systems. The generated sensitivity information are used to perform design optimization and parametric importance studies. The sensitivity results are validated using finite difference formulations. The results demonstrate that graph-theoretic sensitivity analysis is an automated, accurate, algorithmic method of generation for sensitivity equations, which enables the user to have some control over the form and complexity of the generated equations. The results show that the graph-theoretic method is more efficient than the finite difference approach. It is also demonstrated that the efficiency of the generated equations are at par or better than the equation obtained by direct differentiation. The developed software module can be used to generate sensitivity equations for multibody systems which can subsequently be used for design optimization, parameter identification, model simplification, and other related applications.

Graph-theoretic Sensitivity Analysis of Dynamic Systems

Graph theory is used to characterize the topology of multibody systems for high speed parallel computer dynamic simulation. Decoupled loops are defined for use in the recursive dynamics formulations, wherein Langrange multipliers can be eliminated concurrently within each decoupled loop. A row-scanning process for the nondirected adjacency matrix of a graph is described, to generate an identification array that facilitates compact storage and defines system connectivities. A decision tree search algorithm is employed to generate all spanning trees associated with a closed loop graph. For each spanning tree generated from the decision tree search algorithm, a precedence array is defined. A path matrix for the spanning tree and an outdegree of junction nodes are defined to assist in parallel implementation of recursive dynamics algorithms and are then constructed from the precedence array. Finally, an optimal tree search algorithm that identifies a spanning tree that leads to best solution sequence...

R. S. Hwang

Papers

Topological analysis of multibody systems for recursive dynamics formulations