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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 l.jpg

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 l.jpg
outline Metal Cutting

  • Lean and edge angle

    • speed, radius, side cut and angulation

  • Ski-snow forces

    • Merchant theory

    • friction, edge angle and penetration


Lean and edge angle l.jpg
Lean and edge angle Metal Cutting

  • 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


Slide4 l.jpg

lean angle Metal Cutting

mv²/r

lean angle

mg cos 


Slide5 l.jpg

edge angle Metal Cutting

edge

angle


Slide6 l.jpg

lean angle vs. turn radius for 5 slopes Metal Cutting

V= const 20m/s

90

75

lean angle (deg)

60

50°

45

10°

30

0

10

20

30

40

50

60

turn radius (m)


Slide7 l.jpg

lean angle vs. turn radius for 5 speeds Metal Cutting

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)


Slide8 l.jpg

r Metal Cutting

Length (L)

Cd


Slide9 l.jpg

waist Metal Cutting

ski

edge angle 

sidecut

snow

Cd


Slide10 l.jpg

Type Metal Cutting

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


Slide11 l.jpg

edge angle vs. turn radius for different skis Metal Cutting

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)


Slide12 l.jpg

angulation = edge - lean Metal Cutting

angulation angle

lean angle

edge angle


Slide13 l.jpg

angulation vs. radius Metal Cutting

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 l.jpg
Ski snow forces -Machining analogy Metal Cutting

  • Tool = Ski

  • Workpiece = Snow

  • Cutting = Skidding

    • limiting condition on carving

  • Cutting force = Turning force

  • Rake angle = Edge angle (+90 deg)


  • Slide15 l.jpg

    (negative rake) Metal Cutting

     EDGE ANGLE

    (90+rake)

    Ft

    SKI

    (tool)

    M

    Fr

    SIDE WALL

    (relief face)

    SPRAY

    (chip)

    Shear Angle

    ø

    Fc

    p

    SHEAR PLANE


    Slide16 l.jpg

    Critical Angle Metal Cutting

    F

    from Brown and Outwater 1989


    Slide17 l.jpg

    from Brown and Outwater 1989 Metal Cutting

    On the skiability of snow,


    Objectives of machining calculations minimum conditions for carving l.jpg
    Objectives of machining calculations Metal Cutting- 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


    Slide19 l.jpg

    Force relationships Metal Cutting

    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


    Slide20 l.jpg

    Metal Cutting

    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


    Slide21 l.jpg

    Fc Metal Cuttingtan(--)

    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 l.jpg
    discussion Metal Cutting

    • 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 l.jpg
    acknowledgements Metal Cutting

    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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