Daily, J., "Analysis of Critical Speed Yaw Scuffs Using Spiral Curves," SAE Technical Paper 2012-01-0606, 2012, doi:10.4271/2012-01-0606.
Total station equipment, triangulation, or some other mapping technique can generate x-y coordinates describing curved tire marks on the pavement. These marks may result from a critical speed maneuver. Traditionally, these marks are assumed to follow a circular arc and a radius can be determined for use in the critical speed yaw formula. However, critical speed yaw marks typically have a decreasing radius in the direction of travel and a spiral is a more precise fit to the data. In this paper, a total least squares fitting approach is presented to fit the parameters of three types of spiral curves to coordinate data. These are a clothoid spiral, a logarithmic spiral, and an Archimedean spiral which are evaluated and compared for usability in a critical speed yaw analysis. A spreadsheet implementation is presented that makes use of the Microsoft Excel Solver Add-in to perform the minimization of the total least squares fit for the spirals. The goodness of fit was evaluated and showed an average deviation of close to 2 cm over a 44 meter arc length. An example is presented to demonstrate feasibility and application. Once the best fit spiral geometry was determined, the radius of curvature was evaluated and it was found the clothoid spiral may overestimate the radius of curvature at the beginning of the mark. A particle dynamics solution including normal and tangential components of friction is developed to show how speeds evolve while the vehicle travels in a spiral path. It was found that the logarithmic spiral matched the speed and acceleration data from the test example the closest. From the fitted parameters and a friction model, the velocity and acceleration of the vehicle can be determined. It can be concluded that spiral geometry better represents the actual path of travel and analysis based on spiral geometry has the potential to render accurate results over the length of the arc when using coordinate data for critical speed yaw analysis.