Bombardier, W., McPhee, J., and Schmitke, C., "Symbolic Formulation of Multibody Dynamic Equations for Wheeled Vehicle Systems on Three-Dimensional Roads," SAE Int. J. Mater. Manuf. 3(1):454-467, 2010, doi:10.4271/2010-01-0719.
A method to improve the computational efficiency of analyzing wheeled vehicle systems on three-dimensional (3-D) roads has been developed. This was accomplished by creating a technique to incorporate the tire on a 3-D road in a multibody dynamics model of the vehicle with an approach that formulates the governing equations using symbolic formulation. For general handling analysis performed on the vehicle, the tire forces and moments are determined using a tire model that represents the tire as a set of mathematical expressions. Since these expressions need numerical values to determine the forces and moments, a symbolic solution does not exist. Therefore, the evaluation of the tire forces and moments needs to be done during simulation. However, symbolic operations can be used when the governing equations are formulated to develop an efficient method to evaluate these forces.A method to automatically construct the procedures necessary to evaluate the tire forces and moments has been developed. This approach includes a technique to automatically generate an optimized road model procedure that calculates the contact point between the tire and the ground. The method is based on the thin disk tire model with variable radius, resulting in two non-linear equations that define the point of contact. The road model procedure is developed and optimized during automated formulation of the equations of motion.The road model procedure was implemented in the DynaFlexPro software package by creating a simulation code structure to evaluate the tire forces and moments during simulation. The structure was included in a tire/road component that is a linear graph representation of the tire and a three-dimensional (3-D) road. A vehicle system was analyzed on two different road profiles performing a cornering maneuver using DynaFlexPro and repeated in MSC.ADAMS to test the accuracy of the approach. Good agreement was achieved between the symbolic computing method presented here, and the purely numerical algorithm along with significant improvements in the simulation time.