Appendix F - Graphs of the difference between the spin required for rolling and Ihe actual spin after impact versus spin before impact.

(1)

Appendix

F - Graphs

of the difference

between

the spin

required

for rolling and Ihe actual

spin after

impact versus spin

before impact

t; ~

100 AU;'TF.RFlF.LDPARK

::!

:~

:\o~"'~j:

~ t.1 ·100

~~

z:1

·200 ~ ~ .300 '-_ ••.•• _-'---:'-_-'---:" ·800 -600 ·400 -200 0 200

SPIN BEFORE IMPACT (ndls)

!o-o •••• 200

u:;,

~.; 0

!

a5

-200

~e;

$~

-400

z:i

-600 ;: 0 C.C-..•..•.--'---'__ L-_'---> (IJ ~ -80$00 .600 .400 ·200 0 200 SPIN BEFORE IMPACT (nd/s)

BELFRY

.'

.

...

_.

....

I: ,

..

:

-

...

,.

so _BINGLEY

:

~

~.

-so

"to

.

t~

:.;

!z

:~

$~

-l Z-l

e; ~

.ioo

L'__ '"-_--' __

-:'-_---J

-JOO -200 ·100 0 100

SPIN BEFORE IMPACT (rad/s) 100

t~

< •

•.

_..,

~z

1:1' ~ -'00

t:~

o<!:

·200

z:j

••••

~ ~ '~800 ·600 ·400 ·200 0 200

SPIN BEFORE IMPACT (ndls)

..

CRF.WE

..

_....

(2)

FORMBY

•••

j&

.

,

:

·100 ·200 .300 '-'-'--'--'--'---' ·800 ·600 -400 ·200 0 200

SPIN BEFORE IMPACT (radls)

'00 _CANTON

}

_{i ~}

·100

,

·200

"

·300 -400 ·~L.OO""':-:.6.LO-:O-.4":O-:O-.2:'OC:Oc--"0'----::'200 SPIN BEfORE IMPACT (rad/s)

HALLOWF.S

..

'

"

·'00 ·150 '---'----'---''----' -300 -200 ·100 0 100

SPIN BEFORE IMPACT (rad/s)

t~

200 < •

•.

~

:liz

",Ii:

e~

·200

<'5

z~

-400

e; ~

.600 L-_ .••..•'--'-_ ...•..._-'-_--' -800 ·600 ·400 -200 0 200

SPIN BEfORE IMPACT (rud/s)

.

..

HILL VALLEY

• i-.

(3)

t~

100

~S

0

!~

-100

~e;

S ~

·200

z:i

-300

S. ~

.400

'-_...1.__ ...•...

__ '--_

.

-600 .400 -200 0 200

SPIN OF-FORE IMPACT (rad/s) KFJGHLRY •• '" 0

L

.LINDRJCK U~ ~.; -20 • ~.. ~ :

:; 2';

·40 • .- .•

: e;

-60

!::~

.8()

: 3

-100 : ~ ~ ·'~600 _5 00· "L~0-.-3 •.0-0 .-2"0-0 .-'''0-0 -0:'- .••.•' 00 SPIN BEFORE IMPACT (rud/s)

MOOR ALLERTON

·20 •

,

.

~.L.o O::-.""5LOO::-.-.l:'00'-.-:-3':-00"'.':2"00::-.-'''00~0~~'00 SPIN BEFORE IMPACT (rad/s)

t~

40 •• _20

..

~

!

z

0

~s

-20

$~'O

z

~-60 ~ ~ .8~LO::-O---'•• :':0-:-0-.':':0:':0-.-:'20::-0:O--:0!---=200 SPIN BEFORE IMPACT (rad/s)

'",

..

... .

;.

,

MOORIIAU • 181

(4)

20 MOORTOWN

....

'

.

·20

_:i

•

·40

:

·60

'.

·80 .'00 L...;;...-'-_-'--_ ....•.._-'-_-' ·800 ·600 -400 ·200 0 200

SPIN BEFORE IMPACT (radfs)

'00 _NEWCASTLE

I.

',' ·100 .200 I.- •...•••

_'---'-_'-....l.._'-....I

·600·500·400·300·200·100 0 100

SPIN BEFORE IMPACT (rad/s)

SANDMOOR ·'00

" .I.

t

I

·200 ·300 .40.08LOO::--: •• .L0"'0-_4:':0C:0-_2:'0:-:0::--:0~~200

SPIN BEFORf. IMPACT (rad/s)

'00

o

·'00 ·200 ·300 ._ •• "00 .•.•• -50. -600 '---'----'---''---''---' ·800 ·600 -400 ·200 0 200

SPIN BEFORE IMPACT (radh.)

SUTTON PARK

::.,

:~.

~.~

~.

(5)

Appendix

G . Playing

quality

test results

G.. Clegg Impact Hardness

Tester-This section contains the results from the Clegg Impact Hardness Tester using thefOUT

different indenters. i.e. the O.5kg and 1.0kg cylindrical indenters. the indenter with a real golf ball attached to its end and the indenter with a metal end shaped like a golf ball. The readings are given in tens of graveties (98.1 ms·2),

(i) Clegg 0.5kg Austerfield Park Belfry Bingley Dirkdale Crewe Formby Ganton Hallowes Hill Valley Keighley Lindrick

(I)

Lindrick (2) Moor Allerton Moorl·lall Moortown Newcastle-u-Lyme Sandmoce STRIGreen Sutton Park (ii) Clegg 1.0kg Austerfield Park Belfry Bingley BirkdaJe Crewe Formby Ganton Hallowes Hill Valley Keighley Lindrick (I) Lindrick (2) MoorAlienon Moor Hall Moortown Newcastle-u-Lyme Sandmoor STRIGrecn Sutton Park. (iii) Clegg

RB

Austerfield Park Belfry Bingley Birkdale Crewe

~~~~~~~~~~~~~~~~~~4

~~~~~~~~~~~~~~~~~~~6

No readings No readings

~~LL~LLL~Ll~LL~7

~Ll~~~l9

LLL~~~~lllLL~llll~~8

6.8. II. 12, 11,9,8,8.9,8,7,7,8,7.8.8,6.7,9.9

5,5,5,5,4,4, S. S. S.

6,

5, 5,

7, 7, 7,S.

6, 7, 8, 8

L~~~~~3

7,6,8,6,6,6,6,7,7,7,8,9,

10.10,8,8,8,8,8,7

LLLLLLL~~~~~~~~LLL~6

L~LlLLLL~~L7

~~~~~~~~~~~~~~~~~~~S

~L~~L~L~L~~4

~~~~LL~LL~~~~~~LLL~8

~LLLL~~6~LL6

L~~L~L~~lllllllLlLl6

~~~~~LL~l~~LL~7

3,3,2,2,2.

1,2,2.3.3. 1,2,2,7.2,

1,2,

I,

2 ~~~~~~~~~~~~~~~~~~~4

No readings No readings

~~~~~~~~~~~~~~~~~~~S

~~~~~~~~~~~~~~~~~~~~3

~~~~~~~~~~~~~~~~~~L7

6.6,5, S. 5, 5, 5, 4, 5, 7, 6, 7, 6, 6. 6, 5, 5, 4, 6, 7

~~~~~~~~~~~~~~~~~~~3

No readings

~LLLLL6AL~~~~L~6

4,4,5,5,5,5,5,5,5,6,4,4,4,4.4.4,4,5,3,3

LL~~~~~3

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,3,3, 3, 4, 4, 4, 4, 4

3,4,3.3,3,3,2,2,2,

1,2, 1,2,2,

i,

2, 2, 3, 3, 3

~~~~~~~~~~~~~~~~LLL7

~~~~~~~~~~~~~~~~~~~3

L~LLL~L~LLLLL~L6

~~~~~~~~~~~~~~~~~~~3

0,0,0,0,0,1,0,1,0,0,1,0.0,0,0,0,0,0,0,1

1,1,1,1,1.1,1,1,1,1,0,0,0,0.0,0,0,0.0.0

Norcadings No readings I, I, I, I, I, I, I, I, I, I, 1,1, I, I, I, I. I, 1,0,2

183

(6)

Fonnby Gantcn Hallowes Hill Valley Keighley Lindrick (1) Lindrick(2) Moor Allerton Moor Hall Moonown Newcastle-u-Lyrnc Sandmoor STRl Green Sulton Park (Iv) Clegg MB Austerfield Fark Belfry Bingley BirkdaJc Crewe Formby Ganton Hallcwes Hill Valley Keighley Lindrick(I) Undrick(2) Moor Allerton Moor Hall Moortown Newcastlc-u-Lyme Sandmoor STRI Green Sutton Park G.2 Penetrometer I, I, I, I, I. 1,0,0,0,0, I, I, I, I, I, I, 1,0,0,2 2,2,2,2,2,2,2,2,2,2,3,3,3,3, I, 1,1,2,2,4 3,1,2,1,1,2,2,2, 1,2,2,2,3,3,2,3,2.2,2,2 0,0,0,0,0,0,0,0,0,0,0, O. 0, 0, 0, 0, 0, I, I, 1 No readings 2,2,2,2,3,2,3,2,2,2,2,2, I,1,2, 1,2,2,2,2 I,

r,

I. I, I, I, I, I, I, I, I, I. I, I. I, I, 1,2,2,2 I, I, I, I, O.O.I, I, I, 1,0,0, I, I, I, 1,0,0, 1,0 0,0,0,0,0,0 I, I, 1. I, I. I, I, 1. I, I, I, I. I, 1 1,2, I, I, I. 0, 0, 1,0, 1,0,0,0,0,0,0, O. 0, 0, 1 0,0,0,0,0,0,0,0,0,0, I, I, I, I, I, I, I, I, I, 1 0,0,0,0,0, I, I, 1,0, I, I, 1,0,0, I, I, I, I, I, 1 1,2,3,2,2,2.2, 1.3,3,3,3,3,3,1,2,2,2,2,2 1,1.1,1.0,0,0,0,0,0,0.0,0,0,0,0,0,0,0,0 1.1.1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0 0,0,0,0,0,0,1,1,1,1,1,1.1,1,1,1,1,1,1,1 No readings No readings 1.1,1,1,1,1,1,1,1,1.1.1.1,1,1,1,1.1,1,1 0,0,0,0,0,1.1,1,1,1,1,1,1,1,1,1,1,2,2,2 1,1,1,1,1.1.1.2.2,2,2,2,2,2,2,2,2.2,2,2 2,2.2, 1,2.2,2,3,2, 1,2,1. I, I, 1,2, 1,3,1.1

~~~~~~~~~~~~~~~~~~~I

No readings 2,2,3,2,3.2,3,2,3,2, 3,2, 1.2,2,2,2,2,2,2 I, I, I, I, 1,2,2,2,2,2, I, I, I, I, 1.2,2,2,2,2 1. 1.0, I, 1,0, I, I, I, 1.0, I,I, I, I,1,0, 1,2,I 0,0,0,0,0,0,0,1. I, I,1,1,I,1,1,1,1,1, 1.1 I, 1,0, 1,0,0, 1,0, 1,0, I, I, I, 1,2, I, I, 1,0,1 0,0,0,0,0,0,0,0,1,1,1.1,1,1,1,1,1,1,2,3 1,0,0,0,0,0,0, I, 1,0, I. I, I, I, I, I, 1,0,0,1

~~~~~""~'~~~~3

_{1,1,1.1,1.1,1,1,1,1,0.0,0,0,2,1.1,1,1,1} The following are readings for each green using the penetrometer described in Chapter 2. TIIC units arc dimensionless.

Austerfield Park Belfry Bingley Birkdale Crewe Formby Garuon Hallowes Hill Valley Keighley Undrick(l) Lindrick (2) Moor Allerton Moor Hall Moortown Newcastle-u-Lyme 9,7,8,8,9,7,6.7, 10.7 6, 7, 7, 8, 8. 7, 7, 7, 7, 7 ~~~~~~~t~~~~~~~~~~'4 No readings 1,6.7,

t.

7. 7, 6, 7. 7.8 6,8,6,6,7,5,5,8,5,6 ~~~t~~~ttt~~~t~~~~~7 10,6,9, 10,5,6,8,8,7.8 7, 7, 7, 9, 10, 8, 8, 9, 9, 9 4.6,5,7,8,9,6,7,7,7,6, 12. 12,4, 10,4.6.9, 10,6,6,6. 8, 7. 6, 6, 6, 8, 10 3. 5, 4, 6, 5. 6, 6. 2. 3. 3 8.5,6,5,6.6.5.6,5,5 11,11,10,10,9,9.9,8.8,10.10,10,10,11,11.11.12,6, 10, 10,9,8,8, 12, 10,9, 12, II, 10,9 5.6,7,7, 11,4,5,5.6,7 6,6,7, 8. 10,9. 10.9, 4.6,7,4,7. 7,7,6. II, 8, 10. 10 5,5.6.4, S, S. S. 6, 7, 6 184

(7)

Sandmoor STRJ Green Sutton Park G.4 Traction 5. IS. 16.7. 9,8. 14,7.9,6,6, IS, 10,8,6, 10,5.6,9.6 12, 13, 11. 10, 11 5,5,5,6,7,7,6,7.7.8

The following surface traction values arc given in Newton metres. Austerfield Park Belfry Bingley Biri:.dale Crewe Formby Ganron Hallowes Hill Valley Keighley Lindrick (I) Lindrick (2) Moor Allerton MoorH.lI M00l10WO Newcastle-u-Lyrne Sandmoor STRIG",en Sutton Park G.4 Ball Bounce 17.18,20, 18, 16, 16. 16. 16, IS. 15 11,12, II. 13, 12 12. 14, IS. 16, IS, No readings 13. 13, IS. IS, 15 13. 14. 14, 14, 13 10. 10, IS, IS, 14 14. 16, 14, 16, 16, 14, 14. 16, 14. 13 10, 12, 13, 13, 14 12, II. 10, 10, 13. 13, 12. 10, 12. 13, 13, 12. 12, 13. 13, II, 10.12,10, II 1~1~1~1~1~1~1~1~1~1~14 14, 14. IS, 11, IS 12, 10, 12, 14, 12, 16, 14. 13. 14, 14, 13. 13, 14, 12, 13, 14, 13.12, 14. 14 15,16, 17.18, 18 18, 19, 17,18,18,18,18.17, 16, 18 10, 13, 14. IS, 16 I~I~ 1~1~1~1~ll1LI~15 No readings IS. IS. 14, 14. 14 Austerfield Park

This section contains the ball bounce values for each green. The numbers given arc percentages of the drop height (5m).

Belfry Bingley Birkdale Crewe Formby Ganton Hallowes Hill Valley Keighley Lindriek (I) Lindrick (2) Moor Allerton Moor Hall Moonown Newcastle-u-Lyme Sandmoor STRI Green Sutton Park 5.5. 5.5. 4.8, 6.4, 5.2. 5.4. 5.8. 5.6, 5.2, 6.0 (Wound) 6.0,4.8,5.8.5.4.4.2,5.6,5.7,5.0,4.2,5.2, 5.0 (Two-piece) 2.5,2.0.2.8,2.5,2.1.2.3,2.6,2.0,2.2,2.1.2.3 4.2, 4.0, 4.2, 4.0, 4.2, 4.0, 3.2. 2.5, 2.8, 3.4 3.8.4.2,3.8.4.2,4.4,4.1. 3.7, 3.8, 3.8, 3.2. 3.8. 3.7 4.9,5.1,4.9.5.5,5.8,5.3,5.2.6.2.5.6.5.5,5.6 4.7,5.0,6.6,4.6,7.0,5.4,5.4.5.2,6.2,5.8 4.0, 4.0, 4.5, 4.3, 3.8, 4.0, 4.0, 4.0, 4.5. 4.2, 4.2 6.8, 5.5, 5.3, 5.5, 5.6. 5.6, 5.2. 4.5, 5.3, 4.8. 5.8 6.3.6.8.5.7.5.2,6.0,7.0.6.7.6.8,6.7.6.3.6.8 9.0.8.4.7.2, 10.2,6.8.7.2,8.6. 12.0.9.0,5.4,5.8 5.0, 4.8, 4.2. 4.5. 4.2. 4.5, 5.3. 4.2, 4.3 4.0, 4.0, 3.7,4.2. 4.2, 4.2, 4.3, 4.3, 4.0. 4.2 No readings 4.2, 5.1, 4.5, 4.6, 4.0. 3.8, 3.8. 4.4, 4.0. 4.0 No readings 5.4.5.4,5.5,4.6,4.6,4.4,4.7.4.0,4.8,4.2,4.0,4.1 No readings No readings 5.3,4.8.4.7,4.7.5.0,4.3,4.1,4.3,4.3.3.9,4.6 Austerfield P'.U'k Belfry

G.S Botanical analysis (oplical potnt quadrat}

Poa onnuo ·79%; Mosa- 7%: Dead - 14%

Poaonn.ua·56%; Agroslis-41%: Dead - 3%

(8)

Bingley Birkda.le Crewe Formby Ganton Ballowe!' Hill Valley Keighley Lindrick (1) Undlick(2) Moor Allerton Moor Hall Moortown Newcastle-u-Lyme Sandmoor STRIG...,n Sutton Park G.6 Moisture content

Poa annua - 79%;Agrostis -18%: Dead - I%: Bare - 2%

No readings

Poa annua - 34%; Agrostis - 50%: Yorkshire Fog - 16%

POD.annua - 43%: Agrosns - 57%

Poa annua» 17%; Agrostis - 50%; Ryegrass - 3%; Clover - 1% PoaQnnUQ -20%:

Fessuca -

60%; Ryegrass - 15%: Dead - 5%

Poa annua - 5%; Agrosris - 95%

Poaannua - 65%; AgTOSlis - 28%; Dead - 7% No readings

Poa annua- 20%; Agrostis - 49%; Fescue - 31 %

Poa annua -61%: Agrostis -31%; Dead - 2%: Bare - 6%

Poa annua - 67%: Agrostls - 31 %; Dead - I%: Bare - 1%

POGannua - 100%

Poa

annua - 37%; Agrostls - 56%; Dead - 2%: Ryegrass - 2%:

Yorkshire fog· 3%

Poa

aMUQ.-97%: Poa pretensis - 3% No readings

Poa annua - 60%; Agrosns . 34%: Fescue- 2%: Dead - 2%

The following numbers are the moisture content of five sets of two cores taken on a single green at each course. The moisture contents are given as a percentage of the weight of the original mass of soil.

Austerfield Park Belfry Bingley Birkdale Crewe Fcrmby Gaaton Hallowes HiU Valley Keighley Lindrick If ) Lindlick (2) Moor Allerton Moor Hall Moortown Newcastle-u-Lyme Sandmoor STRIG...,n Sulton Park 26.97, 18.21,20.42,24.19,31.89 22.18,19.23,22.01,18.41,19.33 24.09,27.67,24.11,29.57.25.86 26.11,22.96,22.72,26.17,25.85 24.81.22.49,23.90,25.19.23.17 26.41,29.80,28.69,24 ..11,26.40 22.59,25.12,31.41,24.09 25.13,25.91,27.19,25.60,34.86,32.21 12.69, 15.00, 18.27, 15.77,21.32 29.75.30.57,22.12,21.30,31.40 33.06,33.27,33.26,31.42.32.35 36.90, 35.22, 41.10, 38.62 12.96,13.85,14.10,13.70,13.35 21.49, 24.33, 29.05, 26.58, 25.50 44.97,37.02,42.53,45.96,42.80 33.84, 35.93, 36.64, 33.13 26.41, 25.35, 24.66, 24.90, 23.03 23.05,24.66,21.89,21.33.24.23 30.84,29.16,31.55,31.81.22.59 G.7 Particle size Distribution

The following information details the particle size distribution of the ten cores taken from each green. The data is collated in the following order; stone!' (>8mm), coarse gravel (8 -4mm), fine gravel (4. 2mm), very coarse sand (2 - Imm), coarse sand (I - 0.5mm), medium sand (0.50 - O.25mm) fine sand (0.150 - O.125mm), silt (0.050 - 0.OO2mm), clay (<O.OO2mm), loss on ignition (%of oven-dry fine-earth), calcium carbonate (%of air-dry fine earth), Willer dispersibility of the clay(%). The soil from each green is also classed in terms of texture. Fines are taken as fine sand. silt and clay and are written in italics for clarity.

Austerfield t'ark loam Belfry Bingley

0,2, 1,2,6,23,21, IS, 12.2/. 12.2,0.1, 19. Sandy clay 0,5,7,4.14,38,33,8. J, 0, 4.1, 0, O. Sand

No readings .

(9)

Birkdale Crewe Formby Ganton Hallowes lIiU Valley Keighley Lindrick(I) Lindrick (2) Moor Allerton Moor Hall Moortown Newcastle-u-Lyme Sandmoor STRIGreen Sutton

Park

0,0,0,0,1,20,61,4,7,7,6.1,0.3,0. Loamy sand 0, 1,2,2,3,31,32,9,11,9,3.7,0.1,0. Loamy sand 0,0,0,2,5,26,54,3,5,5,6.4,0.2,0. Sand 0,0,0,2,9,29,36, 11,5,3,6.1,0,0. Loamy sand 0,2, I, 3, 9, 15,9,9.31,24,5.6,0.1. 30. Loam 0, I, I, 1,7,48,33,5,3,3,2.8,0,0. Sand No readings

0,0,0,2,10,20,15,10,22, 2/, 8.4, 9.8, 18. Sandy clay loam 0,0,0, I, 11,30, 19,9, 17, 13,9.8,3.0, 7. Sandy loam 0,4,11,13,38.28,9,3,4,5,2.1,0.0. Sand 0,0, 1,4, 11,29,25,11,12,8,6.5,0.2,0. Loamy sand 0, 1,2,7, 18,20,20,5,13,17,3.0,0.3,21. Sandy loam 0.0, I, 3. 8, 29, 27. 6,14,13,7.3.0.1,30. Sandy loam 1,0, 1,4, 10,20,24,16,12,14,6.3,0.1,46. Sandy loam No readings 0,0, 1,2,7,34,31,8, 12, 6, 7.9, 0.1, 0. Loamy sand G.8 Surface evenness (levels apparatus)

The following information details the ten readings of the vertical displacements of thelen

rods located on the profile gauge described in Chapter 2. The readings are given in millimetres. Austcrfield Park: Belfry Bingley BirkdaJe Crewe I. 2,2,1,2,0,0,1,0,0,0 2. 0, 2, 3, 3, 4, 3, 4. 4, 2, 2 3. 0,2, 1,0,1,2,3,3,2, I 4. 3,3,3.2,2.0.2.2. 1,-1 5. I, 1,2.3,4,3.4. 3, 1. 1 6. 1, 1. 2, 2. 2. 3, 4, 3, 3,

°

7.0,1,2, I, 1.0,3,2,2,2 8. 1, 2, 3, 2. 3. 2. 4, 3, 0.

°

9. 2, 0, 2, 1, 4. 2, 3, 4, 2, 2 10. 2, 3, 3, 1,4, 3,5,3,5,2 I. 1,1,1,-1,·1,0,0,0,2,0 2.0,1,1,0,0.0,1.0,1,0 3. -1,0,0, 1,0, 1. 1,0, 1, I 4.0,0,1,0,0,0,1,1,1,0 5. -1,0. 1,0,0, 1.3.1, 1.

°

6. 0.0, 1,0.0. 1,3. 1,0,

°

7.0,0,0,0,0,0,1,1,3,1 8.0,0.1,1,1,0, \,0,1,0 9. 2.0,1, 1,0,0, 1,2,2, 1 10.0,2,2,1,2, I, I, -I, 0,

°

No readings No readings I. 1.0, 1,2, 1,2, 1,0, 1, \ 2.0,1,0,-1,0.0,-1,0,-1.-' 3. 0,0, I, ·1, ·1, ·1, ·1, 0. 0,

°

4. 1. O. 1,0, -I, -I. 0, 0, -I.

°

5. 0.0.2,0,0, -I. 0, -I. 0. -I 6. 2.0,0.0,0. -I, 0, -I, 0,

°

7. 0,1.2. 1. 1, I, 1,0,0,

°

8. ·1,2,2,1,2,2, I, 1,0,0 9. 0, I. I, 1,0,0, 1,0,0,

°

10. -I, -I, 0, 0, -I, -I, 1,0,0,

°

(10)

Formby Garucn Hallowes Hill Valley Keighley Undock (1) 1. -2. O. O. -1. -1. O. O. O. O. 0 2. -1.0.0.1.1.0.0.0,1.2 3.0.0,0.0,0.1.1.0.1.1 4. O. O. O. -1. -I. 0.·1. -I. ·2.·1 5. -I. O. O. 0, O. O. ·1. O. 0.1 6.0.0,1,1,0.0.0.0.2.1 7. O. O. O. O. O. O. O. O. O. I 8. I. O. O. '1.0, O. 0.0. O. 0 9. O. O. 0, ·1. -I. O. O. O. O. 0 10. -I. O. ·1. O. -1. -I. O. ·1. 0.1 1. 0.0.·1,0,0.0.0.1,0.0 2. I, ·1. 0, -I. -I. O. ·1. -I. 0, 0 3. O. 1.0,0, -I.·1, -I, ·1.0,-1 4. 0, O. 0, O. 0, 0, O. -I, 1.0 5.0,0,-1.-1,-1,-1,-1.0.0,1 6. O. O. O. O. -I. O. -I. 0, 0, 0 7. -1,0, -I, -I. -I. ·1, ·1, O. O. 0

8.

0, O. 0, 0, 0, -1,-1, O.

O.0

9. 0, I, O. O. 0, 0,0.0,0. O. 0 10.2.2. I, 1.0.0, I, I, -1. 0 1. 2,3.4.2.2.2, I, O. -I, 3 2. O. I, 2. 2, 2. 0, I, 2, 2, 3 3. 3.3. I, O. 3, 3.3, 1,0,0 4. 1.3. S. 3. 3.2,2.2.2.1 5. 2.2,3,3.2.2.2, 1. O. 0 6. I, I. 1. 0.1.0. -1, O. 0.-1 7. I. I.I. 1.1.0. O. 1. 1. I 8. -1.0, O. -I. O. O. O. O. 0, 0 9.2.2.2.0,1.0.0.0,-1.0 10. 1.0, O. O. 0, O. 1,3.2,0 1. O. ,I. O. -1. -I. -I. 1,0,0,0 2. -1.-2.0,1.-1.-1.0.0.0.1 3. O. O. O. I. 1. I. O. -I, 1.0 4. O. O. O. O. O. 1.2. 1,0,0 5. O. I, 1. I. O. 1. O. O. 1.0 6.0.-1.0.0.1.0.0.1.1.1

7. O. -1. -I, -I. -I, ·2. ·1. 0, O. O. 0 8. -I, -1. O. ·1. -1. -I, O. -I, 0,1 9. -I. -I, -I. -I, -I. 0, O. O. O. I 10.0, -I, -I, O. O.O. O.0, 1,0 I. 2.2. O. O. -I. -I, ·1, -I. 0,-2 2. -I. -I, 0, -I. -2. -I, O. -I, O. -I 3.0,0.1,0.-1,1.1,0.0,0

4. 1,2. 1.0, -I, -2,I,-3. -I, 0 S.I,I.I.O,I,2.2,4,2,2 6.0.0.0.-1.-1,0,1.1,2.1 7. ·1, -I. O. ·2. -I. -I, ·1. ·2. 0.-1 8. 0,0, ·1, -2. -2, -2. -I, O. -2. 0 9.0.0,0.0.0,-1,0,-1.1.0. 10.1.1.1,0,0,0,1.1,1.0 I. 0.0,1.0,0.0.-1,-1.-1,0 2. 2.2. O. 0, 0, 1,0, -I, 0, I 3.0.0,1.0,0.0,0.1,1,1,1 4.0.1,1.0.0,0.0.1.0.0 S.I,2.2.0.I,I.I.2,I,2 188

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6.0,0,0,-1,0,-1,0.0.0, I 7. 0, 0, 0, 0, 0, 0, -2, 0, -I,

°

8.0,1,1,0,0,0,0,-1,-2,0 9.0,0,1,0,1,0,0,0,0,1 10.0,0,0,0,0,0,0,0,

-I,

°

Lindrick (2) 1. 0, 1,0,0,0,0,0,0,0,

°

2.0,-1,0,-1,0,-1,0,·1,-1,-1 3.1,1,0,0,0,0,0,0,-1,0 4.0,0,0,0,0,0,0,0,0.0 5. 0,0, I, -I, 0, 0, 0, 0, 0,

°

6. I, 1,0,0,0,0,0,0, I,

°

7. 0,0,0, 1,0,0,0,0, -1,-2 8. 1,0,0,0,0, -I, 0, 0, 0,0 9. -1,0,0,0,0,0, -1,0, -I,

°

10.0,0.0,0.0,0,0, -1, 0,

°

MoorAllerton I. 1,4.3. I, 1,2,3.2,2, I 2. 4,3, 2, 1,2,3,4,4,3,2 3. 0,2, 1,0, 1,2,0, I, 1,3 4.3,1,1,0,1,0,2,1,3,3 5. -1,0, -I, -1, -I, -I, 0, I. 1,2 6. 0,2,3,0,3,4,2, I,1,

°

7. 0, 1,0,0. -I, -I, -I, -2, -I,

°

8. 2, 3. 3, 3, 2, 2, 2, I,I,0, 0, 9. 1, 1,2,0, -I, -I, 3, -I, -2,

°

10.1,1,2,2,1,0,0,2,1,0 Moor Hall 1.I,1,0,0, 1,0,0, 1,0,

°

2. 0,0,0,0,0,0, 1,0,0, - 1 3. I. I, 0, 0, 0, 0, 0, 0, 0,

°

4. 0,0, -I, -I, 0, 0, -2, -I, 0,

°

5. 1,0,0, -1,0,0, -I, 0, 0,

°

6. 0,0,0,0,0, -I, -I, -I, 0,

°

7. 0,0,0, 1,0,0,0, 1,0,

°

8. 2,0, 1.2, I, 1,0, I, I,

°

9.0,-1,0,0,·1,0,0, I, 1,0 10. -I, -I, 0, 0, -I, 0. 0, I, I,

°

Moonown 1. 0. 0, 0, 0, I, 0, 0, 0, 0, 1 2. -1,0,0,0,0,0,0,0,1, I 3. 1,2,1,0,1,0,1,0,-1,0 4. 1,1,1,0,0,0,0,0,-1,0 5. 0, 1,0,I,I, -I, 0, -2, -I,

°

6. -1,-1,0,2,-1,0,0,0,0,0 7. 1,2, 1,0,2,2,0,0, -I,

°

8. 1,0, I, 1,0, I, I, 1,0, I 9. 1,3,3,1,1,1,3,1,0,1 10. -I, 2, 1,0,0,0,0,0,0, I Ncwcastle-u-Lymc 1. -1, -1,0, I,O.l,l,O,O,l 2.0,0,0,1,1,2,1,0,1.0 3. 1,2, I, I, I, I, I, I, I, I 4. 0,0, I, I, 1,2, 1,0, I, I 5. 0,0,0,0,0,0,0, -I, 0,

°

6. 0,0, 0, 0, 0, 0, 0, 0, I, I 7. 0, -I, 0, 0, 0, -I, -I, 0, I, I 8. I, 1.0, -I, -I. 0, -I, -I, 0,1 9.2,2,1,1,2,1,2,1,1,0 10.1. 0, 0, 0, 0, 0, 0, 0, 0, I

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Sandmoor STRI Green Sutton Park I. I,O,i,0,3,3,3,2, 1,2 2. 1,1, 1,0,0,0,2,2, I,1 3.1,1,1,0,0,0,2,2,0,0 4.1,1,2,1,2,2,1,2,1,2 5. 4, 3, 3, 1, 2, 1, 1. 1, 0, 0 6.2,2,2,2,1,0,0,-1,0,0 7. 1,1,2,2,3,2,2, 1,0,0 8. 3,2,2, 1,3, 1,2, 1,0,0 9. I, J, 1, 1,0, -I, 1,3,1, I 10. 2, 4, 2, 0, 3, I, i, 2, 1, 1 I. 0,1, C, 1,2, I, I, 1,0,1 2. 0, -I, 2, -I, 0, 0, 0, I, O. 1 3. 0,2,3,2, 1,2, 1. O. 0, 1 4.2,-1,0,1,2,1,2,0,0,1 5. -1,-1,0,-1,-1,0,0,0,1,0 6.2,3,1,0,2,0,1,0,3,0 7. 1,2, 1. 0, 1, 2, 0, O. 1, I 8. 1, I, 1, 1,2, I, 1,0, I, J 9. i, 0, 0, -I. -i, i, 0,1,0, i 10. ·1. -2,0, -I, 0, -I, 0,

-r,

0, J I. 0,0,1,1,1,0,0,0,0,0 2. 1,1, I, 1, 1,0,0, I, I, I 3. 0,0, 1,0,0,0, J, i, 1, J 4. 0, I, 1, I, I, 1,1, 1,0,0 5. 0, -1,0,0,0, -I, -I, -I, 0,-1 6. I, I, I, J, 2, I, 1, 1,2, I 7. -1,0,2, I,1, 1,2,2, I, I 8. 1,0,0, O. 0, 0,1, 0, 0, 1 9. 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 10.0,0, 1, 1,0,0,0, 1, I, 1 G.9 Stimpmeter

This section contains the distances rolled by golf balls across the green and set in motion using the stimpmeter. There are six readings on each line. The first two are the distances rolled by two balls travelling along exactly the same line while the third number indicates the distance between the resting positions of the two balls. The second set of three numbers on the same line is

me

replicate of the readings in the opposite direction. Austcrfield Park Belfry Bingley 1.98, 1.92,0.10 1.73, 1.79.0.09 1.42, 1.57,0.:7 1.74, 1.67. 0.08 1.70, 1.70,0 1.80, 2.09, 0.29 2.00, 1.99. 0.08 1.88, 1.98, 0.10 2.15,2.30,0.15 2.07,2.15,0.15 2.80, 2.98, 0.23 2.82, 2.68, 0.35 2.17,2.87,0.12 2.73, 2.93, 0.23 2.57, 2.68, 0.35 1.90, 1.90,0 1.97, 1.80, 0.17 2.22,2.14,0.10 1.25, 1.36, 0.13 1.91, 1.91,0 2.02, 1.99, 0.08 2.06, 2.01, 0.10 2.10, 1.92,0.10 2.00, 2.05, 0.12 2.07, 2.06, 0.05 1.71, 1.74,0.10 1.58, 1.63, 0.05 1.70, 1.77, 0.Q7 1.66, 1.17,0.17 1.62, 1.71,0.10 i90

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Birkdale No readings Crewe 2.21.2.21.0.11 2.43. 2.51. 0.08 2.15,2.23,0.15 2.25.2.07,0.18 2.38, 2.42, 0.04 2.33, 2.24, 0.09 2.35, 2.34. 0.20 2.50,2.41,0.24 2.39, 2.49, 0.12 2.20, 2.42, 0.34 Fomby 2.39. 2.54, 0.25 2.42. 2.44, 0.D7 2.67, 2.73,0.08 2.55, 2.62, 0.D7 2.41, 2.55, 0.08 2.41.2.45.0.04 2.40. 2.40. 0 2.37. 2.39, 0.05 2.43. 2.48. 0.18 2.13,2.19,0.14 Ganton 2.12. 2.15, 0.03 3.07,3.14,0.1 I 2.19, 1.96, 0.28 3.24,3.21, .021 2.02, 2.05, 0.15 3.45, 3.46, 0.30 1.98, 1.91,0.19 2.99, 3.09. 0.36 2.19,2.21,0.28 3.00, 3.26. 0.26 Hallowes No readings Hill Valley 2.40, 2.62, 0.31 1.83. 2.08, 0.27 2.50, 2.55, 0.22 1.87,2.12.0.25 2.55, 2.42, 0.21 2.13, 2.13 2.63, 2.66, 0.18 2.05, 2.18. 0.26 2.76,2.93,0.17 2.00, 1.91. 0.09 Keighley 2.16.2.17,0.14 2.63, 2.74, 0.29 2.19,2.23,0.10 2.37, 2.47, 0.10 2.26. 2,36, 0.10 2.30. 2.40. 0.11 2.11,2.28,0.17 2.29, 2.54, 0.29 2.04, 2.06, 0.06 2.64, 2.68, 0.08 Lindrick(I) 2,90, 3.00, 0.10 1.91, 1.83,0.09 2.54,2.91 1.75. 1.83 2.54, 2.46, 0.20 1.21,2.27 2.87,3.35,0.10 1.99, 2.29. 0.26 2.85, 3.04, 0.21 1.97, 2.00. 0.04 I.indrick (2) 2.05, 2.26, 0.25 1.89. 2.06. 0.18 2.63. 2.63, 0.15 2.03. 2.05. 0.05 3.00,3.16,0.16 2.12,2.12,0.12 2.77,2.74,0.03 2.11,2.11,0.20 3.06. 2.86, 0.26 2.11,2.24,0.13 Moor Allerton 2.43, 2.53, 0.1 1.87,2.01,0.14 2.35, 2.50, 0.31 1.82, 1.87. 0.09 2.91,3.02,0.11 1.72, 1.81,0.10 2.80, 2.94, 0.1 1.97, 1.99, 0.20 2.96.3.10,0.14 1.85, 1.86, 0.06 2.54, 2.83, 0.29 1.89, 1.96, 0.D7 2.54, 2.63, O.I 8 1.59,1.69,0.11 2.70, 2.93, 0.2 1.64, 1.69, 0.09 2.88.2.89,0.15 1.67,1.75.0.16 2.79, 3.63, 0.29 1.51, 1.51,0.12 Moor Hall 2.06, 2.11, 0.D7 2.72, 3.06, 0.38 2.08, 2.07. 0.04 2.63, 2.83, 0.20 2. J 3, 2.20. 0.07 2.70. 2.85, 0.15 191

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2.08, 2.04, 0.28 2.57, 2.45, O.I~ 2.01, 1.97,0.09 2.52, 2.52,0.13 Moortown 2.28, 2.34. 0.Q7 1.89, 1.95.0.10 2.55, 2.69, 0.15 2.09, 2.05, 0.10 2.75, 2.67,0.23 1.99, 1.83,0.16 2.11,2.18,0.18 2.07, 1.85, 0.35 2.90, 2.90,0 2.23,2.19,0.11 Newcastlc-u-Lyme 2.30, 2.39, 0.09 2.47, 2.42, 0.14 2.15,2.24,0.09 2.12,2.20,0.01 2.15, 2.15, 0 2.17,2.13.0.04 1.84, 2.07, 0.24 2.12,2.12,0 1.98, 1.84, 0.14 1.95,2.15,0.20 Sandmoor 2.05,2.01,0.13 2.39, 2.39. 0 2.11,2.13,0.11 2.62,2.75,0.13 2.17,2.17.0.05 2.25, 0.15, 2.39 2.01,2.15,0.17 2.02,2.01,0.08 STRIGreen 2.00, 1.96, 0.20 2.06, 1.96,0.18 2.14,2.15,0.33 2.00, 1.87,0.17 2.54, 2.36, 0.36 1.98. 1.88,0.10 2.22,2.31,0.10 2.21,2.21,0.05 2.25, 2.21, 0.32 1.88, 1.79, 0.20 Sutton Park: 2.01,2.00,0.10 2.23, 2.13, 0.10 1.95, 1.88,0.11 2.22, 2.23, 0.08 1.97, 2.08, 0.11 2.34.2.31,0.12 1.99. 2.06, 0.Q7 2.18,2.24,0.14 1.84, 1.98,0.14 2.26,2.35,0.12 G.I0 Friction

This section contains the results from the sliding friction apparatus, measured in Newtons. Austcrfield Park Belfry Bingley Birkdale Crewe Fonnby Gamon Hallowes Keighley Hill Valley Lindrick(I) Lindrick (2) Moor Allerton Moor Hall Moortown Newcastle-u-Lyme Sandmoor STRIGrecn Sunon Park 75, 70, 80, 80, 75, 75, 75, 80. 55. 55 80, 80, 85, 89, 79 80, 80, 85, 80, 78 No readings 95, 90, 95, 90, 95 100, 70, 90. 85, 90 75.72,71,70,78 70, 60, 50, 50, 65, 70, 70. 55. 50, 60

.~~~~~~~~~~~n~.~

81,76,71,85 80, 82, 85. 75, 75 85, 70, 75, 70,75, 75, 70, 60, 70, 60 82,85,81,90,85 90, 83, 75, 89. 78, 82, 85. 80. 74, 88, 85. 75. 70. 84, 82, 71. 68, 82, 68, 75 86, 90, 88, 85, 85 70, 65, 75, 75. 80. 80. 80. 75, 75. 85 82. 85. 85. 85. 90 70. 65, 60. 70, 70, 75, 75, 75, 75, 85 50. 55, 45, 50, 45, 80, 80, 90, 90. 92 192

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Appendix

H • The regression

eoeffielents

for the

relationships

between

velocity,

angle

and

spin

before

and

after

impact

for rolling

impacts,

The following tables contain data complimentary to the information in Tables 3.6, 3.7 and 3.8. Using the graphs in Appendix H. impacts in which slip occurred throughout impact were found and removed from the sets of data in Appendix E. Stepwise regressions were then carried out on the resultant sets of data, the

resultsof which are

shown here. nt .1 A e ocu dt..:t.) r curse II'IXI A I

r

ou,

.7

t-1

T-1 I I 1 4 I •.•.ter rm act IBelfrv Bm Ie

rewe

1 .5 •. 1

Austetticld

Formby Janron -lallcwes .1 Hill· euev 1 .1 - '.10 14 1 ei.c.hley IfIXEC A 1 I Lindrick .14I0.04IFIXEI; A 06.7 I M. At erton Moor a 5. 5 Moortown 0.06 IFIXEC AN FIX. AN L I I 04 . 4 I

Newcastle

ancmoor I 1.

o.

5

r om

.70 .0.1 10m IXEC

-'.I.

ANliLE

o.Ot~

.

- .1.

O.OJ •

I I

.uu

.se

o.oon

unon ark 1

.sz

.0.1 10.OJ

TABLE H.l. The coefficients for the regressions between the velocity of lhe rebounding bail and the incoming velocity, angle and spin for rolling Impacts only. The coefficients in italics are significantly different from the corresponding coefficients in Table 3.6 where all impactsweft included in the regression.

n eater tm act ourse nt v v

e

d

e

.,

d

.,

r Auste ield

..

,

1.1 A I [Be I .7 O. I. I

Binelc'

.9\ 0.6 IXEC A L I

rewe

I. 1.1 .1 I Pormb .7 .1 I 7 I anion 5. .7 FI A 5 1 Ha lowes .1 'alley I. _10. I. It I

xerenrev

I 7 1.5 I

u.

I A . O. I 1 Lindrick A

.

. U5 I M. Allerton -1 7 .1 I I I Moor Hall I.1 I. _10. A I Moonown I I I. _10. FI A I Newcastle -5.. 7 I. _10. I. - ).0]. I0.005 I 193

(16)

cur-se nt

.1

n e a ter rm ect -ecnttnue

andmoor

uuon ark -5. 1. I.

TABLE H.2. The coefficients for the regressions between the angle of rebound and the incoming velocity, angle and spin for rolling impacts only. The coefficients in italics are significantly different from the corresponding coefficients in Table 3.7 where aJl impacts were included in the regression.

nurse

nt vdLv IX ED ANGL .1 M.A lerton 1.5

-,.u

.

'<

Austcrfield Hclfrv In lc I 5. 5.0 I. I II. I. I. 11 1 7. I. rewe Formby anton Hallcwes In 8 ter Impact d r .!6 HII al e Keighley .5 1 L

FIxE ANtiL I U. U.04 .7

FIXED ANtiL I 0.18 Moor Hal '.0 I. 10. I A .1 r'XE[ ANGL I O. .5 5. Moortown ewcastle andmoor 681.6 1

Sutrcn P, - •• J

<.<

i

u .e

u.ue

.•

5

wool

TABLE H.3. Tables containing the regession coefficients of the equations relating the spin afterimpact to the initial velocity and spin for rolling impacts only.

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Appendix

J -

The equations

of motion

in a model

of oblique

impact

in which

the forces are proportion

to the area

of contact.

This model was first used by Rickerby and Macmillan to calculate the volume of the crater caused by the impact of a steel sphere on a rigid-plastic surface. Near the beginning of the impact the sphere is in contact with the whole of the surface of the crater formed by the impact and the equations of motion are.

d'.

"'(jiT

= -I'!' cqn. J.I

m~

=

P cqn. J.2

where P = 1ta2Pd' P;t is the "dynamic hardness" and is related to the properties of the surface. At a later time in the impact the equations of motion become more complex and arc,

ct'.

"'(jiT=

-psin(e, + {ll -I'!'cas(e, + {ll m~= pcas(e, + {ll - /lPSin(ei + {ll P=P,(A,sin("y - eil + A,cos"Y

eqn. J.3

where

eqn.1.4 eqn. J.5 The areas AI and A2. (3 and

y

are found in terms of the depth of the crater I, the radius of the ball r and the angle of incidence. These are,

. 1I-r (I- COSej)

"Y =

SIn-

I

2"in{l J

(3 :': sin-I

tr(l·

r (1-cos8j))2 + (a .rsinej)2}l;\f eqn. J.6

A,= ~~, -sin~l) cqn. J.7 ~l=2cos·'{~J eqn. J.B ccse,

.'

A2 =

¥2. -~,

+sin~,) eqn. J.9 ~,=2cos·l{(I-r)laO(~)) eqn.I.IO

.=

{1(2r·I)J~ cqn. J.ll

TIle equations above were calculated iteratively using a program written in BASIC for the Apple Macintosh. The program is shown below.

PRINT TAB(1};"This program uses the Hutchings equations (using m and r for" PRINT T.AB(I);"a golf ball to calculate the velocities and angles of a sphere" PRINT TAR{I );"on a rigid- plastic surface after impact There is also a" PRINT TAB(l);"modification (or the rebound of the ground which incorporates" PRINT TAIl(ll:". spring constant, k."

REM .••• ::.:::::::::::::::::::,., ... ,.... , :::::Read in the initial values READ Vi.I,mu,L,TSUM,H

READ PD,K

READ FSUM.XSUM,ZSUM. YSUM,XDDO,YDDO.WSUM.WDDO.WDSUM

~~~:;~::::::::::::::::::.:

•••

:·:·vi~·;;;~;/;~~

;,;k::::~~~::';~h;:;;::::;:;::.;:.;.:::::::::;::::::

.•· .

REM···The following equations calculate the accelerations in the X and Y direc-REMu"rions. "Theforces essentially arise from the a force normal to the area REM···of contact The areas of contact are approximated. The program jumps

ReM···

to a subroutine when t'te impact isnOIdetached.

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~~~~;;;~Vl~cos(;;:::

YDSUM.Vi·SIN(I) 101=ATN(yDSUM/XDSUM) A,SQR(L ·(,0426·L)) Kl=I·L·46,948 IF KI .1 THEN 400 DELTA.·ATN(Kl/SQR(-Kl·K 1+ 1))+ 1.5708 GOT0405 4OODELTA:O

40S IF ABS(I»DELTA THEN GOSUB 1050 ELSE 410 GOT0660 410 N",N+t IF A.O THEN 470 Cl.(L-.0213)·TAN(I)/A IF C1>1 OR C1<-1 THEN 470 PIIl=2'(-A TN(C I/SQR(-CI'C 1+1))+ 1.5708) GOT0480 470 PHI=O 480 A2.(6,283195-PHI+SlN(PHI))' A' A '.5 C2=(]·46.948'L)/COS(I) IF ChI OR C2<-1 THEN 530

THET A.2·( ·ATN(C2tSQR( -C2'C2+ 1))+ 1.5 708) GOT054O 530THETA=0 540 A :.(SIN(THETA)-THETA)'.OOO2268 AI=ABS(Al) C3.L-.0213·(I-COS(I)) C3=ABS(C3) C4=SQR«A-.0213'SIN(I))'2+C3'2)·23.474 IF ·C4'C4+ 1<0 THEN GOTO 620 B=ATN(C4/SQR(-C4'C4+1)) C5=C3·23.474/SIN(B) IF ·C5·C5+ 1<0 THEN GOTO 630 G.ATN(C5ISQR(-CS'C5+1)) GOT0640 620 B.O 630 G=O 640 Q=K'L P=PD'«A I'SIN(G·I»)+(A2'COS(G))) F.P+Q m=(WDSUM·V/.0213)'H'IOOO XDD=(·PSIN(I.B.m)-mu·F'COS(I+B.m»·22.222 YDD=(PCOS(I+B.m)-mu·PSIN(I.B+m»·22.222 WDD=mu' ,0213'FI.OOOO082 PRINT "Calculations are being made"

660 REM •• ·::::::: ::::::SIMPLE INTEGRATION ROUTINES :::::::::::::::::::::::::: WD=(WDDO+WDD)'H' .5 W=(WDSUM • .5·WD)·H WDDO=WDD WDSUM.WDSUM.WD WSUM=WSUM.W XD=(XDOO.XDD)·H·.5 X.(XDSUM+.S'XD)'H XDDO=XDD XOSUM=XDSUM.XD XSUM.XSUM.X YD=(YDDO+ YDD)'H·.5 Y.(YDSUM • .5·YD)·H YDDO=YDD

:::::The main program:::::::::::::: .

(19)

YDSUM.YDSUM.YD Lo<YSUM·Y L=ABS(L) YSUM,YSUM.Y TSUM,TSUM+H V,SQR(XDUM*XDSUM+ YDSUM*YDSUMj

IF WDSUM·V/.0213<O THEN m,(WDSUM·V/.0213)*H*loo ELSE moO If 1>0 AND ABS(I),DELTA THEN 1000 ELSE 10

IF YSUM ,0 THEN 1000 ELSE 10

REM· •• :::: , ::::::::::::0UTPlJf OF DATA:::::::::... . '., .. 1000 PRINT "ANGLE IN IS ",57.2957*·.7854:"DEGREES"

PRINT "VELOCITY L'IIS",Vi:'·MlS"

PRINT "ANGLE OUT IS ",S'.2957*ATN(YDSUM/XDSUM):"DEGREES" PRINT "VEL OUT IS",SQR«XDSUM'XOSUM)+(YDSUM*YDSUMll:"M/S" PRINT "CONTACT TiME IS",TSUMI.ool:"MILLISECS"

PRINT "PINAL SPIN IS",WDSUM PRINT:PRINT

DATA 20,·.7854,0.5,0,0,IE·5 DATA 3E6,0 DATA 0,0,0,0,0,0,0,0,0 END

REM· ••. : ::::::::Thi5 subroutine is used when the impact is not detacbedxc:.: 1050000 P::::3.141S93"PO·A*A B,O Q=K'L P=P+Q XDD"..-rr.u·282.(XHS*F YDDo282.oo78*P WDDcmu*.0213'P/.OOOOO82

PRINT

"The impact is not detached" RETURN

(20)

Appendix

K - BASIC

routines

for calculating

the rebound

of a

golf ball

from

a surface

using

visco-elastic

models

K.l A routine for solving differential equations using the Runga-Kutte fourth order set of equations

The folowing routine uses the Runga-Kutte fourth order set of equations to solve

numerically" set of differential equations. The routine indicated bythe command

GOSUB 200 contains the equations to be solved. FOR R=OTO N LET y(R)=yo(R) NEXT R GOSUB 200 FORR=OTON LET K l(R).H*F(R) LET y(R)=yo(R)+KI (R)/2 NEXTR GOSUB 200 FORR=OTON LET K2(R).H*F(R) LET y(R).yo(R)+K2(R)/2 NEXTR GOSUB 200 FOR R.OTON LET K3(R)=H*F(R) LET y(R)-yo(R)+K3(R) NEXT R GOSUB 200 FOR R.()TO N LET yo(R).yo(R)+(Kl(R)+2*(K2(R)+K3(R»+H*F(R))/6 NEXTR

K.2 Kelvin-Voigt model of vertical Impact

The equations of motion are shown in Chapter 5 (equation 5.15) and were translated into the formal required in order to be solved using the routine above. These equations were located in a subroutine at the end of the program and were

F(O).I

(*.

I)

F(I ).y(2) (V •

*")

F(2).-Cl*y(l)-C2*y(2) (acceleration -

%tl

where Ct and C2 are the spring and damper constants,

yell

is the vertical displacement and y(2) is the vertical velocity. These equations were modified to incorporate an area of contact and required a separate set of equations for when the centre of the ball was below the leve1 of the ground. 'These were,

area 1=(.0426-y(l »*3.l4159*y(l) F(O)·l

F(I)=y(2)

F(2)=(-C l*y( 1 )-C2*y(2))*a= I

for the centre of the ball above the 1e.•.el of the ground and, arealo.OOl4253

F(O).l F(I).y(2)

F(2).( -C 1*.0213-C2*y(2))*area 1

for the centre of me ball below the level of the ground 198

(21)

K.3 A two- layered viscoelastic model of vertical impact

The equations in the previous secdoe were modified so that the surface was mack:up of two layers. TIle whole subroutine was written as follows,

200

**REM··· .•···*···SUBROUT1NE···**

...•••••••• •..••••.••••••

REM··· .•···This section defines the equations used in the model··· •• •

IF

y(I)<OTHEN 1100

IF

y(I)<L1 THEN 300

IF

y(I»L1 AND y(I)<L2 THEN 400

IF

y(l»L2 THEN 500

300 REM·· •••• ··THE BALL IS IN CONTACT WITH THE FIRST LAyER··· ••u ••

IF

y(l)<.0213 THEN 310 ELSE 320

310 REM···The centre of the ball is above the level of the ground··· •.• area 1=(.0426·y(1 ))·3.14159·y(l)

area2=O 3f'C,(13••0 F(O)•.I F(I)=y(2) F(2)=( ·Cl·y(1)·C2*y(2))·areal GOTO 600

320

REM ••••.•• • •.•••

The centre of the ball is below the level of the ground··· areal:.OOI4253 areahO area3=O F(O)=I F(I)=y(2) F(2H·Cl·.0213·C2·y(2))·areal GOT0600

400 REM· •• ••• ••. ···THE BALL IS IN CONTACf

Wlrn

lWO LAyERS·· •.••.• • •.•

IF

y(l)<.0213 THEN 410

IF

y(I».0213 AND y(I)·L1<.0213 THEN 420

IF

y(I».0213 AND y(I)·Lb.0213 THEN 4}0

410 REM.H""*The centre of the ball is above Ll and above the level of the ground" •••• d=y(I)·l.1 area3=O area2=(.0426·d)·3.14159·d areal.(.0426-y(I»'3.14l59·y(l j-creaz F(O).l F(I).y(2) F(2)=(·CI·L1·C2·y(2»·areal· C3'y(l)'area2 GOT0600

420 REM· .•••• The centre of the ball is above L1 but below the level of the ground"'··· d.y(l)·L1 area~",O area2.(.0426·d)·3.14159·d area 1=.00 14253 F(O)=l F(I).y(2)

F(2).(·C J O(LI·y( 1)·.02l3)·C2·y(2))·area J. O'y(2)'an:a2 GOT0600

(22)

430 REM**"·1l1ecentre of the ball is below L1 and below the level of the ground" .•.•• d~y(l)·Ll area 3=0 area2=.0014253 areal=O F(OH F(I)=y(2) F(2).·CJ·y(2)'aJ<32 GOTO 600

600 REM .•."Next two lines set the vertical resistance to zero when it starts to become negative.

IF F(2)<=0 THEN 700 700 RETURN

K.4 A two-layered visco-elastic model

or

oblique impact

The model of vertical impact above was modified to take into account horizontal forces and was used to simulate oblique impacts. F( ..;' and F(4) are the differential equuations for the horizontal velocity and acceleration wuile F(5) and F(6) represent the rotational velocity and acceleration. The whole program, including all printing and formatting statements is shown below.

REi\f··· ..···IMPACT MODEL··· ..·••••··•· ..•···•

RI::M··· .••• ...••••• .•···"'····SJ.Haake· .•••• •.•••••••• •.•• •••••••••••• N=6: DIM y(N),yo(N),F(N) CI.4E+08 C2.900000! C3.0 C4.1000001 C5.0 C6.0 Vi.18 ANGi.1.308997 Wi •• IOO XF •. 5: H=.oool: Ll •. 008:

REM"'·The number of dependent variables.

REM··Maximum impact lime of 500msccs REM"Step length ofO.lmsecs REM"First discontinuity (mm) PRINT "INITIAL CONDITIONS"

PRINT'Velocit)' """vi" mlscc: Angle'" "ANGi*S7.29578" degrees" PRINT"EQUATION PARAMETERS"

PR INT "spring constant :%"C I"; dashpct constant :="C2 PRINT "Layer! ••."Ll·I()(X)" mm"

PRINT" TIME VEL ANGLE DEM'f1 Spin"

PRINT" (msecs) (mI,) deg mm radrs"

PRINT"

10Wi=·950---mue.S: REM" Coefficient of friction 1=0 yo(O).O yo(l)=O yo(2)=Vi'SIN(ANGi) yo(3)=0 yo(4)= Vi'COS(ANGi) yo(5)=0 yo(6)=Wi 200

(23)

mm~O m=O

REM••••

If •••••••••••• "' •••••••••••••••

STEp••••••••••••••

If ••••••.••••••••

HJOx=yo(3) mmcm

IF yo(2»0 THEN m=O ELSE m~ 1 IF mm=O ANI> m=\ THEN YMAX=yo(I) V=SQR(yo(2)'yo(2)+yo(4)*yo(4» ANG=ATN(yo(2)/yo(4»

PRINT USING "#######.###"; yo(O)*looo, V, ANO*S7.29S8, yo(l)*looo, yo(6), F(6)/IE6

IF yo(O»XF-.OOOOOOI THEN 1000 FORR=OTON

[,ET y(R) .•yo(R) NEXTR GOSUB 200 FOR R=OTO N LET KI(R)=B*F(R) LET y(R)=yo(R)+KI(R)/2 NEXT R GOSUB 200 FORR=OTON LET K2(R)=H*F(R) LET y(K)=yo(R)+K2(R)12 NEXTR GOSUB 200 FORR=OTON LET K3(R)=H'F(R) LET y(R)=yo(R)+K3(R) NEXT R GOSUB 200 FOR R=OTO N ~~~tOr)=yo(R)+ (K 1(R)+2·(K2(R). K3(R»+H*F(R»16 GOTO 100

1000 PRINT "XF rcached":PRINT ··••• •.•.XF reached···"

1100

PRINT:PRINT

"The ballleft the surface and the calculation stopped" PRINT

...

" PRINT USINO "#######.###"; Wi, V, ANO·S7.29S8, yo(6), YMAX'IOOO, Piteh·lOOO, yo(O)'"1000 PRINT

...

" IF wi--rocc THEN 1200 GOTO 10 1200 END 200

REM···SUBROUTINE···

•••••••

REM•• ••••

···This section defines the equations used in

the

model .••••••••

RE1\.f

••••••••••••••••••••••••••••••••••••••••••••••••••

~••

ot ••••••••

IFy(I)<OTHEN 1100 IF y(I)<l.1 THEN 300 IF y(l »LI THEN 400

300 REM •••• ····THE BALL IS IN

CONTACf

WITH

1llE

FIRST LAYER·· •• ••• ~N~~~~~\~:r2~;~(x~4).yO(4»

IF yo(6»=VI.0213 THEN mu~O:yo(6)=VI.02\3 IF yo(2»O THEN A.·I ELSE A= I

(24)

IF y(I)<.0213 THEN 310 ELSE 320

310 REM···· ...•.•• .•The centreof theball is above the level of theground ••

*···...···~

statee Ll

area 1.(.0426-y(1 »'3.14159'y(l) areaz-o area3.0 wl.l-y(I)1.0213 wl·SQR(l-wl'wl) area4=.0004537*(ATN(w2lwl)-wt*w2) ma5.0 area6=O dx.yo(3)-x F(O).! F(I).y(2) F(2H-CI'y(I)-C2'y(2»'areal+A'.00852'ABS(F(6»'SIN(ABS(ANG» F(3).y(4)

F( 4 ).( -C I "dx -C2 'y(4» "aread- .00852' ABS(F(6»'COS(ABS(ANG» F(5)=y(6)

F(6). 117.37

"mu'(ABS(F(2»'COS(ABS(ANG».A'

ABS(F( 4 »'SIN(ABS(ANG))) GOT06OO

320 REM··· •• The centre of the ball is below the level of the ground" ••.•.••• •.••.• state""I.2 are.koo14253 area2=O area3=O area4"".OOO7127 ma5=O area6,,{j dx.yo(3)-x F(O).I F(I).y(2)

F(2)o(·C I' .0213·C2 'y(2»'.re. 1+A' .00852' ABS(F(6»'SIN(ABS(ANG» F(3)=y(4)

F(4)=( -C l'dx·C2 'y(4»'.re.4-.oo852' ABS(F(6»'COS(ABS(ANG» F(5)=y(6)

F(6).1 17.37' mu'(A BS(F(2»' COStA BS(ANG »+A' ABS(F(4 »'SIN(ABS(ANG))) GOTO 600

400 REM .•• •• •• •••• •.rnE BAU..JS IN CONTAcr WITH lWOLAYERS· •.• •• •.• •• ~N~~~~~2):{2t;~~~~4).yo(4»

IF yo(2)>O¥HEN A.·I ELSE A=I IF yo(6».VI.0213 THEN muoO;yo(6).VI.0213 IF y(I)<.0213 THEN 410

IF y(l».0213 AND y(l)·LI<.0213 THEN 420 IF y(I».0213 AND y(I)·Lb.0213 THEN 430

410 REM"'·"'*The centre of the 0011 is above LI and above the level of the ground"'··· state=2.J

d.y(I)·Ll area3=O

area2.(.0426-d)'3.14159'd

area 1.(.0426-y( I »·3.14159·y( I )-area2 wl.l-y(l)I.0213 w2=SQR(I-wl"'wl) w3.1-<II.0213 w4.SQR(l-w3'w3) area600 are35 •. 0004537'(A TN(w41w3 )-w3'w4) area4=.0004537*(ATN(w2/w I)ow1 *w2)·areaS dx.yo(3)·x

(25)

1'(0)=1 F(1):y(2)

F(2)=(-Cl'Ll-C2'y(2»'areal-C3'y(2)'area2. A'.0085 'ABS(F(6»' SIN(ABS(ANG)) F(3):y(4) F(4).(-Cl'dx-C2'y(4))'area4-C3'y(4)'areaS-.OO852'ABS(F(6))' COS(ABS(ANG» F(5):y(6) 1'(6)= 117.37'mu·(ABS(F(2»'COS(ABS(ANG»+A 'ABS(F(4»'SIN(ABS(ANG)) GOT0600

420 REM*··"'''''''The centre of the ball is above L1 but below the level of the ground·"'''' sretc-z.z d.y(I)-LI area3=O area2=(.0426-d)·3.14159'd area 1=.0014253 w3.I-dl.0213 w4=SQR(I-w3'w3) area6=<l areaS=.0004537'(ATN(w4Iw3)-w3'w4) area4"".()(X)7J 27-areaS 1'(0).1 F(I)_.y(2)

1'(2).( -C j'(LI-y( I )-.0213)-C2·y(2))*area I-C4'y(2)'a",a2.A' .00852' ABS(F(6»)' SIN(ABS(ANG)) F(3).y(4) F(4).(-Cl'dx-C2'y(4))'area4-C3'y(4)'areaS-.00852·ABS(F(6))'COS(ABS(ANG») F(5)=y(6) 1'(6)= 117.37'mu'(ABS(F(2))'COS(ABS(ANG))+A'ABS(F(4))*SIN(ABS(ANG))) GOT0600

430 REM··· •.• The centre of the ball is below Ll and below the level of the grounds •.• • state=2.3 d.y(l)-Ll area3=0 area2=.0014253 areal.O area6=<l area5 •. 0007127 area4.--o 1'(0).1 F(1)=y(2)

F(2).-C3·y(2)' ,,,,ahA' .00852' A BS(F(6)'SIN(ABS(ANG)) F(3)~y(4)

F(4)=-C4'y(4)'areaS-.008528'ABS(F(6))'SIN(ABS(ANG)) F(5).y(6)

1'(6).117.37 'mu '(AB5(F(2))'COS(ABS(ANG))+A' A B5(F(4 »)'SIN(AB5(ANG))) 600 REM"'·Nexl two lines set the vertical resistance to zero when it starts(0become negative IF 1'(2)<=0 THEN 700 1'(2)=0 1=1+1 IF 1=1 THEN Piteh=yo(l) 700 RETURN 203