Analysis of Critical Speed Yaw Scuffs Using Spiral Curves

Transcription

1 Analysis of Critical Speed Yaw Scuffs Using Spiral Curves Jeremy Daily Department of Mechanical Engineering University of Tulsa PAPER #

2 Presentation Overview Review of Critical Speed Yaw Analysis Motivational Example Fitting Spiral Geometry Clothoid Spirals Logarithmic Spirals Archimedean Spirals Application to a Critical Speed Yaw Radius of Curvature Particle Dynamics 2

3 Critical Speed Yaw Evidence A vehicle performs a near steady-state maximum performance lateral maneuver Evidence shows striated curved tire marks. Front Outside Tire Rear Outside Tire 3

4 Radius Measurement Assume a circular arc 2 Measure 2 chords and 2 middle ordinates Show a decreasing radius Subtract half the track width 8 2 4

5 Vehicle Free Body Diagram Only tire forces and gravity act on the vehicle mph feet 5

6 Typical CSY Test Data 6

7 Changing Radius Data shows vehicle slows during a turn. Radius must decrease Measured tire marks show a decreasing radius. Most research shows the circular assumptions works for estimating speeds. Friction factor is the most influential input into the CSY equation. Try to find a curve that smoothly changes radius of curvature. A spiral does this. 7

8 Example Evidence and Data 8

9 Fitting Spiral Geometry to Data Cubic splines do not work Sawtooth shaped radius of curvature. Three spiral families to consider: Clothoid Spiral Logarithmic Spiral Archimedean Spiral Fit spiral to raw total station coordinate data X(m) Y(m)

10 Curve Fitting in General Function fitting Least squares assuming ordinate data is good. Total Least Squares Minimize perpendicular distance between data and curve. min ~ denotes data points and are functions of the parameters to be fit. The origin,, Scale parameter, Initial angle, Curve parameter, s. Non-linear fitting requires an iterative solution * * * * * * * * 10

11 Fitting Clothoid Spirals Clothoid Curve Equations: s cos ssin scos s sin where 1 cos 2 1 sin 2 are the Fresnel Integrals. Raw Points Fitted Curve Fitted Points X Y 11

12 Execute the Solver Curve parameters radians, 17.72, , Curve attributes m Arc Length: m m Note: radius from chord and middle ordinate was 33 m 12

13 Fitting Logarithmic Spirals 13

14 Fitting Archimedean Spirals coscos sin sin sin cos cossin / 14

15 Goodness of Fit Determine the distance from the fitted point to the data point Determine the slope Which side of the tangent line does the data lie? Use this to give + or Plot residuals 15

16 Clothoid Spiral Residuals The average residual magnitude is 4.3 cm 16

17 Logarithmic Spiral Residuals Average residual magnitude is 2.4 cm 17

18 Archimedean Residuals Average residual distance of 1.9 cm 18

19 Radius of Curvature Comparison 19

20 Test case 20

21 Modeling Assumptions The vehicle is considered a point mass with forces acting at its center of mass. The radius of curvature for the center of mass comes from the spiral less ½ the track width. The spiral (Not a Circle) will be considered the path of travel for the center of mass of the vehicle. The forces acting on the vehicle particle are a result from friction and, to a lesser degree, gravity. The longitudinal force is acting to accelerate or decelerate the vehicle. The vehicle maintains a yaw rate consistent with the change in direction of the spiral. 21

22 Dynamic Equations Use Tangent and Normal Coordinates Velocity at any point is Initial Velocity 2 Velocity along curve 2 Tangential Acceleration estimated as 2 22

23 Clothoid Spiral Speed Results 2 23

24 Logarithmic Spiral Speed Results 2 24

25 Archimedean Spiral Speed Results 2 25

26 Acceleration Data and Predictions Constant longitudinal acceleration ~ -0.2g Vehicle steering wheel sensor was monitored 26

27 Constant Lateral Acceleration 27

28 Observations Clothoid spirals are prone to over predict initial speeds. Once tires are saturated, more steering does not change acceleration. Implementation with a spreadsheet is possible. Speed determination through maneuver is possible. Logarithmic spirals are preferred. Least complicated math Uniform longitudinal and lateral accelerations Scaling also rotates spiral Only 4 parameters need fit 28

29 Conclusions Technique enables reliable extraction of radius of curvature from x-y coordinates Can be used with raw CAD data Possible to use with photogrammetry results Should still gather chord and m.o. for comparison Further testing with different vehicles, tires and speed regimes is needed. Friction is still important. 29