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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

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

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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

lean angle

mg cos 


edge angle

edge

angle


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)


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)


r

Length (L)

Cd


waist

ski

edge angle 

sidecut

snow

Cd


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


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)


angulation = edge - lean

angulation angle

lean angle

edge angle


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)


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)


  • (negative rake)

     EDGE ANGLE

    (90+rake)

    Ft

    SKI

    (tool)

    M

    Fr

    SIDE WALL

    (relief face)

    SPRAY

    (chip)

    Shear Angle

    ø

    Fc

    p

    SHEAR PLANE


    Critical Angle

    F

    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


    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


    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


    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(--)))


    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.


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