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The application of linear graph theory to the modelling of flexible multibody systems is described. When combined with symbolic computing methods, linear graph theory leads to efficient dynamic models that facilitate real-time simulation of systems of rigid bodies and flexible beams. The natural extension of linear graphs to the modelling of mechatronic multibody systems is presented, along with a recently-developed theory for building complex system models from models of individual subsystems.

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Linear graph theory, invented in 1736 by Leonhard Euler, has been combined with principles of physics to develop algorithms for formulating the dynamic equations for multibody multi-domain systems. This graph-theoretic formulation allows electrical, mechanical, and hydraulic systems to be modelled within a common framework. The formulation has been implemented in a symbolic computer program, DynaFlexPro, that automatically generates compact and efficient sets of system equations that lead to reduced simulation times compared with most commercial multi-body dynamics software.In this article, models of pneumatic tyres are incorporated into the symbolic computer implementation, which is used to create real-time simulations of vehicle dynamics. The tyre component forms a list of symbolic expressions for important tyre variables, such as inclination and slip angle, that are used to calculate tyre forces and moments during simulation. If the transient behaviour of the tyre is important, the user can request that additional relaxation length equations be included in the model. The tyre component allows the user to choose from several tyre model functions that describe the generation of forces and moments at the tyre contact patch and can also accommodate user-developed tyre model functions.A brief introduction to the linear graph formulation procedure used by DynaFlexPro is given, as well as an explanation of how the tyre component works within the linear graph framework. As an example, optimized simulation code is generated for a three-dimensional vehicle model, and results are validated using an equivalent model in the MSC.ADAMS ® commercial software package.