**DIVING GEOMETRY
HANDBOOK
**Leif Zars

7/31/99

This Handbook is based upon the various studies conducted by Dr. Richard
Stone on behalf of the National Swimming Pool Foundation. The author has only attempted to
place the material in sequential order and provide some clarification so as to produce a
working handbook for those desirous of evaluating relationships between a diver and a
water envelope.

My personal thanks to the late Dr. Richard Stone for his sincere efforts and very learned
input on behalf of safety in aquatics.

**DIVING BOARD TESTING
SECTION 1**

DIVING BOARD RATING

SHOP FLOOR METHOD

1. Utilize a standard video Camera.

2. Firmly mount the board to be tested in accordance with manufacturer's

recommendations.

3. On the tip end of the board attach a small index so that the oscillations may be

easily captured on the video tape.

4. Oscillate the board first with no weight on the board, then with 100# added 6"

from the tip, then 200#, then 300#.

5. Capture the oscillations on video.

6. Using slow motion playback, count the number of video frames required for one

complete oscillation under each of the above four conditions. Average them as in Col J.

7. Divide the number of frames required for one complete oscillation by 30 seconds

per frame which is typical of all current VCRs. This gives seconds per oscillation. (Tp)

in Col. K.

8. On a graph similar to the one on the following pages, plot the time period in

seconds squared on the X axis against the load on the Y axis. This should yield more or

less a straight line.

9. From this graph determine the sprung weight of the board (Ws) by extending the

straight line of the graph towards the Y axis until in intersects the X
axis at some

negative value. This negative value (which is used as a positive in the calculation) is
the

sprung weight of the board (Ws) and ranged from 7.5 to 73.7 in earlier
tests on various

boards.

For example a 12' Board using a 59"' fulcrum is calculated as follows: (see also
this same example on the following calculation page):

If the period at 300 pounds of added weight was 0.832556
Seconds/Oscillation

Then the time period squared would be 0.693149 Seconds Squared

And if the time period at 0 pounds was 0. 185733 Seconds/Oscillation

This would be (squared) 0.034497 Seconds Squared

Sprung Weight (Ws) could then be calculated as follows:

__300(max weight) __= x = __15.71241__

0.693149 - 0.034497
0.034497

Finally divide the change in weight added (300 - 0) plus the Sprung
Weight by the change in the time period squared, and by g/4(Pi)squared. This yields the
spring constant (Kb)of the tested board mounting combination.

As a formula it is stated as:

Kb= __Added Weight + Sprung Weight __

Change in (Tp) squared x(g/4(Pi)squared)

The actual calculation for this would be:

Kb=__(300 + 15.71241) __= 588 Pounds/Foot

__(.658653)( 32.2 )__

(4)(3.1416)(3.1416)

Which is the Spring Constant for this Board at the 59" Fulcrum
Setting

RAW DATA DEVELOPED

FROM BOARD TESTING

CURVES DEVELOPED FROM

BOARD TESTING

DEVELOPMENT OF "SPRUNG WEIGHT"

FROM CURVES

WITH

RESULTANT SPRING CONSTANT

CALCULATION

CONVERTED SPRING CONSTANT

TO

EQUIVALENT FALL HEIGHT

USING

DR. STONE'S CURVE

CONVERTING

FALL HEIGHT

TO

VELOCITY

COMBINING BOARD "FALL HEIGHT"

WITH

BOARD HEIGHT

FOR

"COMBINED FALL HEIGHT"

AND

EQUIVALENT SPEED

CALCULATING AIR TRAJECTORIES

FOR VARIOUS BOARD, FULCRUM,

AND MOUNTING HEIGHTS

USING A

"WORST CASE"

6' DIVER

FORMULAS USED

TO DEVELOP

AIR DISTANCES, VELOCITIES

AND ENTRY ANGLES

RESULTANT TYPICAL

BASIC AIR DATA

WITH RESULTS IN COLUMNS L,M, &N

18 AIR CURVES

FROM 3 METER

TO 1' "ROCK" DIVES

SOURCE OF AIR ANGLE

TO WATER ANGLE

CONVERSION

CONVERTING AIR ENTRY ANGLES

TO WATER ENTRY ANGLES

USING DR. STONE'S WORK

IN FIRST LEXINGTON #2 REPORT

ON

UNDERWATER STEERING

TABLE OF RESULTANT

AIR ENTRY

TO

WATER ENTRY

ANGLES

FORMULA FOR

2G UNDERWATER

STEERING EFFORT

TABLES OF

1, 2, & 3G

STEERING CURVES

TYPICAL FORMULAS

FOR CALCULATING

UNDERWATER

VELOCITY REDUCTIONS

FROM DR. STONE'S WORK

IN ADL #2 & #4

CONVERTING

ENTRY VELOCITIES

TO WATER SPEED

REDUCTION WITH DISTANCE

FORMULAS FOR

CONVERTING WATER ENTRY ANGLE,

DISTANCE TO WATER ENTRY

TO UNDERWATER CURVES

AT SPECIFIED G EFFORTS

TYPICAL

RESULTANT

"PLOT" SHEET

PLOTTING

Up to this point the procedures are more or less easily defined. Plotting
the actual underwater curves is somewhat more difficult to outline.

You begin with the "Air Data" information on the particular board/height
combination. (In this worksheet you can enter the earlier calculated "Board Fall
Height", the "Board Height", and the "Diver's Height Ft." - all
of which are active entries in that they influence the outcome of the calculations).

For a specific "Jump Angle" (which is also an active number) you use the
"Combined Distance Out" Col. 1, the "Combined Entry Speed" Col. M, and
the "Entry Angle" Col. N and enter them into the worksheet called PLOTCALC.XLS.

In PLOTCALC.XLS enter "Distance Out" into G3, "Combined Entry Speed"
into C4, and "Entry Angle" into G4.

Also C5 is an active entry into which different steering efforts may be entered. Quoting
from First Lexington UNDERWATER STEERING STUDY dated June 1991 Dr. Stone stated "It
is shown that all of the divers studied steered with equal minimum steering radii of
approximately 2.1 feet, independent of entry speed!'- Page9. And then from ADL #5 'The
diver then -steering effort -- which generates an initial 3G steering force. I have
established that the assumed initial steering force is well within the comfortable
capability of recreational divers from the research that I've carried out for NSPF on
underwater steering." Further, plotting a 1, 2, and 3g underwater steering curve
shows the 3g curve to almost exactly fit the same curve as a 2.1 foot radii - which again
seems to fit the comfort level of the recreational diver.

Curves based on a 2g steering effort would appear to represent a conservative approach,
whereas curves based on a lg steering effort seem to indicate a very reduced steering
effort by the diver.

One other plotting factor is from ADL #5 wherein Dr. Stone states "--the diver
continues in a straight line for a distance equal to 40% of the height without slowing
down."

To keep track of the identity of the PLOTCALC sheet I usually enter the "Board",
"Height," "Fulcrum" (setting,) and "Dive Angle," although
these do not interplay with the calculations.

Start with the tip of the board and on a plot sheet in Auto Cad, move horizontally from
the tip of the board the distance in Col. G "Dist to Water Entry' to begin your plot.

Draw a line from the intersection of this distance with the water surface
36" long at the "Air Plot angle -of say 98.79 degrees (14) if we use the
following PLOTCALC sheet as an example. This represents the diver's air path to entry.

Next draw a line from the water's intersection with the air path, for a distance of
30" in the direction of 281.0 degrees (15). This represents 40% the diver travels
without rotation of steering effort.

Next draw a fine from the end point of the 30" line a distance of 5.17 feet (C 10) at
an angle of 11.0 degrees (F 10). This represents the radius point of the first rotational
curve for the steering effort expended.

From here draw a line back the 5.17 feet (C 10) at a return angle of 196.5 degrees (Ell).

Next draw a like from the end or the prior line 4.70 feet (C 11) at an angle of 16.5
degrees (F 11).

Then draw a line from that end point 4.70 feet (C 11) at an angle of 202.6 (E12).

Continue the above sequence until it is obvious the lines are approaching parallel to the
water surface - indicating the diver's trajectory is towards the surface.

Finally connect each line ending, starting with the end of the 30" line so as to form
visually an understandable curve. This then represents fairly well the underwater
trajectory of the diver's cg under the stated circumstances.

Each angle of dive take-off can then by sequentially plotted as above. In as much as each
underwater trajectory has the same shape and size of curve - only rotated so as to
accommodate the diver's altered entry angle - a little creativity will allow you to copy
your first curve, rotate it to accurately represent the new entry angle, Then position it
at the new water entry distance - thus saving considerable plotting time.

SERIES OF PLOT DATA

AND CURVES

FOR TESTED BOARDS

AT VARIOUS TAKE OFF

ANGLES AT 2G

TYPICAL

SWIMMING POOL

CROSS SECTIONS

TYPICAL

"WORST CASE" DIVER'S

UNDERWATER CURVES

OF CG AT 2G

FOR

VARIOUS POOL SECTIONS

CALCULATED TRAJECTORIES

AT 1G STEERING EFFORT

ADDITIONAL

UNDERWATER CURVES

FOR

GENERAL INFORMATION