Flexible multibody dynamics, a new approach using virtual work and graph theory

Abstract

A new approach to flexible multibody dynamics is presented. Its most prominent feature is that it extends the existing graph-theoretic (GT) method for multibody dynamics to include flexible bodies. This is accomplished by extending the traditional form of system graph and by using the novel idea of adopting virtual work as a through variable. The validity of virtual work (VW) as a through variable is demonstrated philosophically, mathematically, and with general examples, for the graph-theoretic models of elements presented in the thesis. An additional advantage of the new approach is that it can reduce the number of system equations as compared with conventional absolute or joint coordinate formulations for multibody systems with closed loops. The new VW graph-theoretic approach encompasses most existing graph-theoretic approaches to multibody dynamics.

New GT elements are created. They include the flexible body element, the flexible arm element, and the dependent VW element. Terminal equations for conventional multibody elements (rigid bodies, joints, forces) are derived in terms of VW. Construction of a system graph is explained and demonstrated with examples. In addition to the VW through variable, the conventional across and through variables for each element in the system all satisfy the topological cutset and circuit equations from the system graph.

A systematic procedure for formulating system equations, including kinetic and kinematic constraint equations, is put forward that preserves the methodical nature of the traditional GT method. A symbolic-numeric computer package (DynaFlex) is developed for a formulation in which joints are selected into the tree of the system graph.

The three-dimensional kinematics of a Bernoulli-Euler beam is revisited so that a suitable model is developed for it to be included in the new graph-theoretic approach. It is found that using a commonly-used first-order deformation field causes some first-order inertial force terms to be missed from system equations. A new remedy of using a complete second-order deformation field is proposed, and a methodical approach to generating a deformation field that is complete up to any order is given. The use of the proposed complete second-order deformation field is validated in Chapter 7.

Various other issues relating to symbolic implementation of the new approach, Rayleigh-Ritz discretization of the deformation variables of a Bernoulli-Euler beam, numerical solution of differential-and-algebraic equations, and future research directions are discussed.