Modeling Edging Forces in Skiing using Merchant's Theory for Metal Cutting

1. Modeling Edging Forces in Skiing using Merchant's Theory for Metal Cutting Christopher A. Brown Mechanical Engineering Department Worcester Polytechnic Institute Worcester, Massachusetts, USA

2. outline • Lean and edge angle • speed, radius, side cut and angulation • Ski-snow forces • Merchant theory • friction, edge angle and penetration

3. Lean and edge angle • Lean angle and balancing centrifugal forces • changes with speed and slope • Edge angle and geometric turning • considering side cut radius • Angulation • difference between edge and lean angles

4. lean angle mv²/r lean angle mg cos 

5. edge angle edge angle

6. lean angle vs. turn radius for 5 slopes V= const 20m/s 90 75 lean angle (deg) 60 50° 45 10° 30 0 10 20 30 40 50 60 turn radius (m)

7. lean angle vs. turn radius for 5 speeds Slope= const 15 deg. 90 75 35m/s 60 30m/s lean angle (deg) 15m/s 20m/s 25m/s 45 30 15 0 10 20 30 40 50 60 turn radius (m)

8. r Length (L) Cd

9. waist ski edge angle  sidecut snow Cd

10. Type Model Length (m) Sidecut (m) max. radius (m) Rossignol SL 95 Pro 1.631 0.00921 36 GS 1.641 0.00978 34 Volkl SL P 40 1.576 0.01238 24 GS P 40 1.746 0.01122 32 SG P 30 1.906 0.00938 48 DH P 20 1.936 0.00702 66 K2 GS Biaxial 1.670 0.00850 40

11. edge angle vs. turn radius for different skis 90 80 70 60 Volkl DH 50 edge angle (deg) 40 Volkl SG Volkl SL 30 20 Volkl GS K2 GS 10 Rossignol GS Rossignol SL 0 0 10 20 30 40 50 60 turn radius (m)

12. angulation = edge - lean angulation angle lean angle edge angle

13. angulation vs. radius speed=20m/s slope=15° 5 -5 angulation (deg) -15 Volkl DH VolklSL -25 Volkl SG Volkl GS -35 K2 GS Rossignol SL Rossignol GS -45 40 10 30 20 50 60 70 0 turn radius (m)

14. Ski snow forces -Machining analogy • Tool = Ski • Workpiece = Snow • Cutting = Skidding • limiting condition on carving • Cutting force = Turning force • Rake angle = Edge angle (+90 deg)

15. (negative rake)  EDGE ANGLE (90+rake) Ft SKI (tool) M Fr SIDE WALL (relief face) SPRAY (chip) Shear Angle ø Fc p SHEAR PLANE

16. Critical Angle F from Brown and Outwater 1989

17. from Brown and Outwater 1989 On the skiability of snow,

18. Objectives of machining calculations- minimum conditions for carving • Turning force from mass, speed and radius • Edge penetration • as a function of edge angle and friction • Thrust force (normal to the snow) • can be influenced by body movements

19. Force relationships   Ski Snow p Fs Fc Fn F  R -- Ft N  - edge angle shear angle Forces Fc = centrifugal (cutting) Ft = thrust Fs = shear Fn = normal to shear plane F = friction on ski N = normal to ski

20.   ski snow p Fs Fc Fn F  R -- Ft N  - Merchant solution modified for edge angle Fc = Fs cos  + Fn sin  Fn = Fs / tan(--) Fc = Fs(cos  + sin  / tan(--))  = (-)/2 Merchant’s solution predicts where the snow will fail when skidding starts - essential for the solution

21. Fc tan(--) p >  Ls (cos  tan(--) + sin ) Conditions for carving Fs =  As As = Ls p / sin  As: area of the shear plane p: edge penetration Ls: length of the edge in the snow : shear strength of the snow Fc < p  Ls / (cos  + (sin  / tan(--)))

22. discussion • Negative now angulation predominates • Edge roundness, penetration and length • shorter skis should hold better • Penetration can be a function of snow strength • Leg strength should put a lower limit on edge angle

23. acknowledgements Thanks to Chris Hamel and Mike Malchiodi of WPI for help in preparation and equation checking. Thanks to Dan Mote for explaining that skiing is machining. Thanks to Branny von Turkovich for teaching me machining.