In the ongoing search for mathematically efficient methods of predicting the motion of vehicle and other multibody systems, and presenting the associated results, one of the avenues of continued interest is the linearization of the equations of motion. While linearization can potentially result in reduced fidelity in the model, the benefits in computational speed often make it the pragmatic choice. When representing the governing equations of any linear system, one of the relevant problems is the determination of the mathematically equivalent formulation of the smallest size, and solving this system is the most mathematically efficient manner. This paper discusses and explores the potential of an approach based on modal identification, and the Jordan form of the system matrix.

To explore the applicability of the Jordan form to vehicle dynamics, a simple vehicle model is considered. Also widely  
known as the bicycle model, the yaw plane model has been used extensively for vehicle handling studies since its introduction, and has appeared in many variations in the literature. The degrees of freedom of the yaw plane model are the lateral velocity v, and the yaw velocity r of the vehicle. The forward speed u is assumed to be under driver control, and held constant, and as a result is treated as a parameter of the model rather than a variable. The motion is assumed to take place on a flat and level road, so all other motions, e.g., roll or heave, are ignored. Despite using only two degrees of freedom, the model provides a great deal of insight into vehicle handling.

It is shown by example that augmenting the equations of motion with the kinematic differential equations, as would be common in a vehicle dynamics context, can result in a system with a Jordan block larger than the 2 x 2 size that would normally be expected in a multibody vibration problem. This result is somewhat unfortunate, as the algorithm for computing the Jordan form in the general case could be simplified significantly with foreknowledge that the largest block expected was 2 x 2. Also, when computing minimal realizations (the smallest equivalent system matrix), one potential savings is the removal of repeated roots. The calculation of the equivalent system is quite straightforward if the blocks are only 2 x 2. Accounting for the possibility of blocks of size three (or potentially larger) imposes an extra layer of complexity. The result suggests that the Jordan form could be useful in simulation of vehicle dynamics, but does not show the most effective way to compute the Jordan form, as clearly the algorithm must be capable of accommodating systems with triply repeating roots. Nevertheless, the sparse nature of the Jordan form suggests that it remains a promising candidate for efficient numerical solution of the equations of motion, and further research into the area is warranted.  
 


Jordan Forms in Vehicle Dynamics

Division / Sponsors: 
Design Engineering Division and Computers & Information In Engineering Division


Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2020/volume/83907

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