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

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

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

lean angle mv²/r mg cos  lean angle

edge angle edge angle

lean angle vs. turn radius for 5 slopes V= const 20m/s turn radius (m) lean angle (deg) 50° 10°

lean angle vs. turn radius for 5 speeds Slope= const 15 deg turn radius (m) lean angle (deg) 15m/s 20m/s 30m/s 25m/s 35m/s

Length (L) r Cd

waist edge angle  Cd sidecut snow ski

Rossignol Volkl K2 SL Type SG GS Model DH P Pro Length (m) Biaxial GS SL GS P 40 P 20 P Sidecut (m) max. radius (m)

edge angle vs. turn radius for different skis turn radius (m) edge angle (deg) Volkl SG Volkl GS Volkl DH Volkl SL Rossignol SL Rossignol GS K2 GS

angulation angle edge angle lean angle angulation = edge - lean

angulation vs. radius turn radius (m) angulation (deg) speed=20m/s slope=15° Volkl SL Rossignol GS Volkl GS Rossignol SL K2 GS Volkl SG Volkl DH

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)

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

F Critical Angle from Brown and Outwater 1989

On the skiability of snow,

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

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

Fc Fn FtR N F  ---- --    p Fs snow ski 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 Merchant solution modified for edge angle

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(  -  -  ))) p > Fc tan(  -  -  )  Ls (cos  tan(  -  -  ) + sin  )

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

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.