TP B.3 Throw Calibration and Contour Plots for Various Cut Angles, Speeds, English, and Roll

Transcription

1 technical proof T B.3 Throw Calibration and Contour lots for Various Cut Angles, Speeds, English, and Roll supporting: The Illustrated rinciples of ool and Billiards by David G. Alciatore, hd, E ("Dr. Dave") technical proof originally posted: 7/8/008 last revision: 8/7/008 Created at the request of Colin Colenso Equations from throw analysis (T A.14): R 1.15in m vsin( ϕ) R ω z Rω x cos ( ϕ ) v rel vω x ω z ϕ vω x ω z ϕ a b c min cv rel vω x ω z ϕ a be v v rel vω x ω z ϕ vcos( ϕ) cos( ϕ) 1 7 vsin( ϕ) Rω z 36 ω( ve) 5 v E 4 R Definitions: x, r : vertical plane roll rate (follow or draw) z, e : English spin rate E r : % roll (follow or draw) E e : % English (side spin) outside English (OE): e, E e > 0 inside English (IE): e, E e < 0 Speed scale: soft: v 1 4 km hr v m s v 1.485mph medium-soft: v 8 km hr v. m s v 4.971mph medium: v 3 1 km hr v m s v mph firm: v 4 0 km hr v m s v mph power: v 5 36 km hr v 5 10 m s v 5.369mph

2 Convert speeds to unitless m/s numbers: v 1 v v 3 v 4 v 5 v 1 v m v m 3 v m 4 v m 5 m s s s s s Calibration data (from experiment by Colin Colenso): measured throw values (at speeds: 1,, 3): 30deg cut angles: φ m 45deg in inches per yard: θ 1 θ 5.6 θ in degrees: tan θ deg tan θ θ deg tan deg sum of squares of errors in calibration data: SSE( ab c ) v φ m0 a b c v 0 0 φ m0 a b c v φ m0 a b c v φ m1 a b c v 0 0 φ m1 a b c v φ m1 a b c θ 10 θ 0 θ 30 θ 11 θ 1 θ 31 initial guesses for friction equation parameters: a 0.01 b 0.1 c 1 find best fit of friction model to calibration data: a b c Minimize( SSEa b c ) Throw equation in units of inches/yard: vω x ω z ϕ min cv rel vω x ω z ϕ a be v v rel vω x ω z ϕ vcos( ϕ) cos( ϕ) 1 7 vsin( ϕ) Rω z 36

3 lot of calibration data along with fit model: E r 0% E e 0% θ 1 θ 5.6 θ φ m 30deg 45deg ϕ 0deg 1deg 75deg 6 4 θ 1 θ θ 3 v 1 ω v 1 E r v ω v E r ω v 1 E e ϕ ω v E e ϕ v 3 ω v 3 E r ω v 3 E e ϕ φ m φ m φ m ϕ deg deg deg deg

4 lot other test data along with fit model: 50% E r 0% E e 100% 99% 100 % E m 50% ϕ 30deg θ 1 θ.5 3 θ θ 1 5 θ 3 θ 5 v 1 ω v 1 E r v 3 ω v 3 E r v 5 ω v 5 E r ω v 1 E e ϕ ω v 3 E e ϕ ω v 5 E e ϕ E m E m E m E e % % % % ϕ 45deg θ 1 θ.3 3 θ θ 1 5 θ 3 θ 5 v 1 ω v 1 E r v 3 ω v 3 E r v 5 ω v 5 E r ω v 1 E e ϕ ω v 3 E e ϕ ω v 5 E e ϕ E m E m E m E e % % % %

5 Generate contour plots of how throw varies with %English and %spin (top or bottom): rectangular grid of %English and %spin: i 1 41 E ei 100% ( i 1) 5% E e1 100% E e 95% E e40 95 % E e % j 1 1 E rj 0% ( j 1) 5% E r1 0% E r 5% E r0 95 % E r1 100 % NOTE - The figure below shows how all of the following contour plots should be interpreted: cue ball top/bottom spin max possible topspin stun contour plot max inside English micue limit English max outside English max possible bottom-spin contour plot color legend (ROYGBIV): red (max throw to right of line-of-centers), orange, yellow, green (very little or no throw), blue, indigo, violet (max throw to left of line-of-centers) NOTE - The diagram above illustrates tip contact positions on the ball, but be aware that the throw values correspond to the spin on the CB at OB contact. Spin changes due to drag on the cloth, as the CB approaches the OB, must be taken into account. For example, the "maximum possible bottom-spin" point at the bottom is possible only for a firm shot over a short distance; otherwise, drag action reduces the bottom spin some, in which case the effective point on the diagram would be a little higher. program to convert grid of calculated throw values to contour plot with fixed color range: M_to_( Mmax_throw) max_throw 5.7 for for i 1 41 j 1 1 i1j19 i1 1j max_throw M ij M ij max_throw

6 full-ball hit (no cut): φ deg v v 1 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) 4.5 min( M) 4.5 v v M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) 3.308

7 v v 3 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M).73 min( M).73 v v 4 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M).036 min( M).036 v v 5 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) 1.476

8 1/-ball hit (medium cut): φ 30deg v v 1 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) v v M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) 3.584

9 v v 3 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M).969 min( M).969 v v 4 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M).7 min( M).7 v v 5 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) 1.485

10 1/8-ball hit (thin cut): φ deg v v 1 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) 5.68 min( M) v v M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) 3.984

11 v v 3 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) min( M) v v 4 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M).94 min( M).94 v v 5 M ij vω ve rj ω ve ei φ M_to_( Mmax_throw) max( M) 1.99 min( M) 1.99

12 Write a dataset to a file for plotting outside of MathCAD: φ deg v v 5 M ij vω ve rj ω ve ei φ M i0 E ei % M 0j E rj % row and column headings WRITERN ("data/eigth-5.txt" ) M