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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
)cos(**)tan(
2
gr
vLeanAngle
edgeangle
edge angle
lean angle vs. turn radius for 5 slopesV= const 20m/s
30
45
60
75
90
0 10 20 30 40 50 60
turn radius (m)
lea
n a
ng
le (
de
g)
50°
10°
lean angle vs. turn radius for 5 speedsSlope= const 15 deg.
15
30
45
60
75
90
0 10 20 30 40 50 60turn radius (m)
lea
n a
ng
le (
de
g)
15m/s 20m/s30m/s
25m/s
35m/s
Length (L)
r
Cd
rCd
LCd
*24
22
waist
edge angle
Cd
sidecut
snow
ski
cos
sidecutCd
Rossignol
Volkl
K2
SL
Type
SG
GS
Model
DH
P 40
95 Pro
Length (m)
Biaxial
GS
SL
GS
P 40
P 20
P 30
0.00921
Sidecut (m)
0.01238
0.00978
0.00938
0.01122
0.00702
0.00850
1.631
1.641
1.670
1.906
1.746
1.576
1.936
max. radius (m)
36
24
32
48
66
40
34
edge angle vs. turn radius for different skis
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50 60
turn radius (m)
ed
ge
an
gle
(d
eg
)
Volkl SG
Volkl GS
Volkl DH
Volkl SL
Rossignol SLRossignol GSK2 GS
angulation angle
edge angle
lean angle
angulation = edge - lean
angulation vs. radius
-45
-35
-25
-15
-5
5
0 10 20 30 40 50 60 70turn radius (m)
an
gu
lati
on
(d
eg)
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 Angleø
SPRAY
(chip)
F
Critical Angle
from Brown and Outwater 1989
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
FcFn
FtR
N
F
--
-
pFs
SnowSki
Force relationships
ForcesFc = centrifugal
(cutting)Ft = thrustFs = shearFn = normal to
shear planeF = friction on skiN = normal to ski
shear angleedge angle
FcFn
FtR
N
F
--
-
pFs
snowski
Fc = Fs cos + Fn sin
Fn = Fs / tan(--)
Fc = Fs(cos + sin / tan(--))
for min Fc: = (-)/2
- predicts where the snow will fail whenskidding 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 planep: 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 )
Edge Angle vs. Snow Penetration
0
2
4
6
8
10
12
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Edge Angle, Theta (deg)
Sn
ow
Pe
ne
tra
tio
n (
mm
)
velocity = 20 m/sradius = 20 mmass = 90 kgslope = 15 deg.snow strength = 0.06 mPa
Increasing the friction coefficient
= 0.10
= 0.02
Edge Angle vs. Thrust Force
0
1000
2000
3000
4000
5000
6000
7000
8000
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Edge Angle, Theta (deg)
Th
rus
t F
orc
e (
N)
Increasing the friction coefficient
= 0.10
= 0.02
velocity = 20 m/sradius = 20 mmass = 90 kgslope = 15 deg.snow strength = 0.06 MPa
Edge Angle vs. Axial Force
0
1000
2000
3000
4000
5000
6000
7000
8000
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Edge Angle, Theta (deg)
Ax
ial F
orc
e (
N)
velocity = 20 m/sradius = 20 mmass = 90 kgslope = 15 deg.snow strength = 0.06 MPa
Increasing the friction coefficient
= 0.10
= 0.02
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 ofWPI for help in preparation and equation checking.
Thanks to Dan Mote for explaining that skiing is machining.
Thanks to Branny von Turkovich for teaching memachining.