Introduction
DynaFlexPro is a Maple software package for modeling and simulating the dynamics of multibody multi-domain systems. In this paper, DynaFlexPro is used to develop dynamic models of the John Deere Timberjack Grapple Skidder articulated-steer forestry vehicle and the Chevy Equinox sports utility vehicle.
ModelBuilder, a graphical user interface, is first used to define the topology of a system in block diagram form. The system and tire model equations are then generated from the resulting model definition file with DynaFlexPro and DynaFlexPro/Tire, which use a graph-theoretic formulation procedure to automatically generate the governing dynamic equations for systems with arbitrary topology.
By developing these equations symbolically in Maple, DynaFlexPro is able to produce computationally efficient simulation code that is ideal for real-time applications. The system equations for each of the vehicles are used to perform numerical simulations of their dynamic responses, and the results are validated using equivalent models developed in the MSC Adams software package.
Background
A fundamental objective in the field of multibody dynamics is the automatic generation of the governing equations of motion for a system, given a description of its components and the interconnections between them, or its topology. These equations can be formulated numerically or symbolically.
Numerical formulation techniques produce matrices that are only valid for a given instant of time, so the equations must be reformulated at every time step of a simulation. While numerical formulations are popular, the slow process of constantly reformulating the system equations may prohibit their use in real-time applications.
Multibody simulation packages that use numerical formulations, such as MSC Adams, generally use absolute coordinates to model a system, which results in large sets of differential-algebraic equations (DAEs) that are computationally "expensive" to solve. In contrast, joint coordinate formulations generally lead to smaller, more efficient sets of DAEs or, for open-loop mechanisms, ordinary differential equations (ODEs), which can be solved much faster than an equivalent set of DAEs.
Symbolic formulation techniques combine the system parameters and modeling variables to create sets of differential and algebraic equations that describe the system for all time. To be suitable for real-time applications, such as hardware-in-the-loop testing of automotive components, a model must be capable of simulating an event faster than it would occur in reality. Because the governing dynamic equations must only be generated once if formulated symbolically, such formulations are ideal for real-time applications.
Symbolic equations can also be greatly simplified in many ways, such as identifying and removing repeated calculations, and can easily be exported to any desired simulation language. Finally, in contrast to numerically formulated models, symbolic models allow the user to examine the governing equations in a meaningful form, which can facilitate the design and analysis of multibody systems.
Efficient procedures
Despite their many advantages, symbolic formulation techniques produce sets of equations that can become quite large and unwieldy for complex systems; very large systems may require more memory than is available to the symbolic computation package. An efficient symbolic package and an intelligent formulation procedure, however, can alleviate this difficulty [Ref. 1].
DynaFlexPro uses a formulation procedure based on linear graph theory, which provides a suitable framework for modeling the interactions between different energy domains; previously, many existing products have been restricted to the mechanical domain. A complete description of the graph-theoretic formulation procedure is given in Ref. 2. Existing products have also been limited in the operations that can be used to develop system models and post-process the resulting symbolic equations. The implementation of DynaFlexPro in Maple provides users with the flexibility to develop and analyze systems with arbitrary topologies and dynamic equations.
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