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 Ski-snow forces
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 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 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 36 GS
1.641 34
Volkl SL
P 40
1.576 24 GS
P 40
1.746 32 SG
P 30
1.906 48 DH
P 20
1.936 66 K2 GS
Biaxial
1.670 40

11. edge angle vs. turn radius for different skis90 80 70 60
Volkl DH
edge angle (deg) 50 40
Volkl SL
Volkl SG 30 20
Volkl GS
K2 GS 10
Rossignol GS
Rossignol SL 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
Volkl SL
-25
Volkl SG
Volkl GS
-35
K2 GS
Rossignol SL
Rossignol GS
-45 10 20 30 40 50 60 70
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. Ft Fc ø (negative rake)  EDGE ANGLE (90+rake) 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   Fc Fn Ft R N F  -- - p Fs Snow Ski
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. Merchant solution modified for edge angleFc Fn Ft R N F 
--
-   p Fs
snow
ski
Fc = Fs cos  + Fn sin 
Fn = Fs / tan(--)
Fc = Fs(cos  + sin  / tan(--))
 for min Fc:  = (-)/2
- predicts where the snow will fail when
skidding starts - essential for the solution

21. 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 sin  tan(--)
 Ls (cos  tan(--) + sin )

22. 

23. 

24. 

25. 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

26. 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