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

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RAW DATA DEVELOPED
FROM BOARD TESTING

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CURVES DEVELOPED FROM
BOARD TESTING

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DEVELOPMENT OF "SPRUNG WEIGHT"
FROM CURVES
WITH
RESULTANT SPRING CONSTANT
CALCULATION

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CONVERTED SPRING CONSTANT
TO
EQUIVALENT FALL HEIGHT
USING
DR. STONE'S CURVE

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CONVERTING
FALL HEIGHT
TO
VELOCITY

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COMBINING BOARD "FALL HEIGHT"
WITH
BOARD HEIGHT
FOR
"COMBINED FALL HEIGHT"
AND
EQUIVALENT SPEED

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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

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RESULTANT TYPICAL
BASIC AIR DATA
WITH RESULTS IN COLUMNS L,M, &N

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18 AIR CURVES
FROM 3 METER
TO 1' "ROCK" DIVES

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SOURCE OF AIR ANGLE
TO WATER ANGLE
CONVERSION

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CONVERTING AIR ENTRY ANGLES
TO WATER ENTRY ANGLES
USING DR. STONE'S WORK
IN FIRST LEXINGTON #2 REPORT
ON
UNDERWATER STEERING

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TABLE OF RESULTANT
AIR ENTRY
TO
WATER ENTRY
ANGLES

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FORMULA FOR
2G UNDERWATER
STEERING EFFORT

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TABLES OF
1, 2, & 3G
STEERING CURVES

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TYPICAL FORMULAS
FOR CALCULATING
UNDERWATER
VELOCITY REDUCTIONS
FROM DR. STONE'S WORK
IN ADL #2 & #4

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CONVERTING
ENTRY VELOCITIES
TO WATER SPEED
REDUCTION WITH DISTANCE

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FORMULAS FOR
CONVERTING WATER ENTRY ANGLE,
DISTANCE TO WATER ENTRY
TO UNDERWATER CURVES
AT SPECIFIED G EFFORTS

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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.

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SERIES OF PLOT DATA
AND CURVES
FOR TESTED BOARDS
AT VARIOUS TAKE OFF
ANGLES AT 2G

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TYPICAL
SWIMMING POOL
CROSS SECTIONS

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TYPICAL
"WORST CASE" DIVER'S
UNDERWATER CURVES
OF CG AT 2G
FOR
VARIOUS POOL SECTIONS

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CALCULATED TRAJECTORIES
AT 1G STEERING EFFORT

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ADDITIONAL
UNDERWATER CURVES
FOR
GENERAL INFORMATION

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