**SIMPLIFIED SUB DESIGN MATH**

**February
20, 2013: SIMPLIFIED SUB DESIGN MATH is the program I devised in 1984 to design
and build my first submarine. At the
time, there were no “how-to” references for the backyard submarine builder, so I
had to figure it out as I went along. **

**I studied
the elements of submarine design, and reduced them to a few basic steps using a
combination of math I’d learned in school, industry, as a SCUBA diver, as a
Flight Instructor, and some that I went to the books and taught myself
specifically for this process. SSDM was
the result.**

**In the
1970’s and 80’s, information about the process of sub design was extremely
esoteric: limited to engineering tomes thick as a dictionary. And the mathematical equations in those books
are gibberish to the average craftsman.
I needed to reduce the whole process to the minimum number of steps; and
then describe that in plain language, rather than equations. So that’s what I did.**

**I used
SSDM to design the Nautilus Minisub, and all
that ornamental ironwork made her unusually complex in terms of Weight and
Balance. Nonetheless, that boat came out
right on the money. First time out of
the barn, she was properly balanced, and all we needed to submerge was the 170#
of additional weight I’d allotted for a passenger (not present during proving
tests). **

**And so,
in its first practical test, SSDM proved accurate within one pound of predicted
gross operating weight, for an extremely elaborate handmade steel submarine
weighing about 1.25 tons. **

**I first
brought SSDM online at two websites in Europe during 1997 and 1998, and on
websites in the United States in ’98 and ’99.
And, in 2001 The Subcommittee published SSDM in their magazine, The
Subcommittee Report. Since then,
SSDM has also been “adapted, repackaged, and renamed” (by persons other than
myself) for use with computers. As a
result, guys who once told me they “wouldn’t know where to start” when it came
to designing a submarine and didn’t even know what a square foot of steel plate
weighed; are now spouting numbers like engineers specialized in manned
submersibles. (I’m not grumbling about
it; that’s just an aspect of human nature I find funny.) Well, now you can, too. **

**DISCLAIMER: Below is the original Simplified Sub Design
Math program, as published back in 1998.
Notice it says: “After determining the general characteristics of the
sub I want to build, I...” **

**The key word
there, ladies and gentlemen, boys and girls, is “I”. As in “myself, me, not you or anyone else, etc.” **

**I am not
suggesting you use this to design a submarine; in fact, I am telling you not
to. IF you do so anyway, then you do it
of your own free will and at your own peril.
Submarines are inherently dangerous.
If you don’t know, don’t go. **

**I’m publishing
SSDM to show the World how I did it back in the 1980’s, for whatever
educational value that might have; and that’s all. **

**DO NOT USE SSDM TO BUILD A SUBMARINE; AND IF YOU TRY AND
FAIL, DON’T BLAME ME.**

**NOTE: SIMPLIFIED
SUB DESIGN MATH is a plain-language method of determining a submarine
design's Weight & Balance, Longitudinal Hydrostatic Center of Gravity
(location of the pitch axis when submerged immobile in a state of
equilibrium at neutral buoyancy), desired freeboard (surfaced waterline
location), and freeboard-required ballast tank volume (capacity required to
submerge and surface on soft ballast alone). SSDM can also be adapted to
determine Vertical Hydrostatic Center of Gravity (roll axis). This
process does not address hull crush-depth or hydrodynamic stability
considerations. **

** **

**SIMPLIFIED
SUB DESIGN MATH
**

Pat Regan

** **

*After
determining the general characteristics of the sub I want to build, I...** *

1. **DRAW THE SUB ON GRAPH
PAPER. **Drawing all six ends and sides enables me to accurately
determine the size, shape, and relationship of all major component parts used
in the design (valuable when making measurements for volume, area, and weight
& balance computations to come)

**COMPUTE PRESSURE HULL VOLUME.**Let's say it's a cylinder with hemispherical end-caps.*Vol**C = Pi times the radius squared, multiplied by the length.*And since two hemispherical ends comprise a sphere when considered together:*Vol**Sphere = 4/3 Pi times the radius cubed.*Adding both together gives us the pressure hull VOLUME expressed in cubic feet. (Use appropriate equations for other geometric shapes.)

**COMPUTE PRESSURE HULL DISPLACEMENT.**I multiply VOLUME (in cubic feet) times the weight of one cubic foot of water: (62.4# Fresh; 64# per cubic foot Seawater) to determine DISPLACEMENT (could be thought of as "buoyancy potential") expressed in pounds.

**COMPUTE PRESSURE HULL WEIGHT.**I calculate the surface area of the pressure hull in square feet, and multiply that times the precise weight of one square foot of the material being used. My rule-of-thumb for steel is 20# per square foot of 1/2-inch plate; 10# for 1/4-inch; 7.5# for 3/16; 5# for 1/8; and so on.*If DISPLACEMENT substantially exceeds WEIGHT, continue.*(If not, increase PHV.)

**COMPUTE DESIRED FREEBOARD PERCENTAGE AND REQUIRED BALLAST TANK VOLUME.**I decide how much of the hull I want above the waterline when surfaced; calculate the displacement-equivalent weight percentage of that section (expressed as the weight in pounds of that many cubic feet of water); and make External Ballast Tank Volume equal to that amount and usually a bit more: at least, 2 or 3 percent to enhance descent and enable bottoming capabilities. For example: if I want 10% of a 2,000-pound boat out of the water, ballast volume must equal 200 pounds.*200# divided by 64# CFSW = 3.125 CF EBTV.*(NOTE: In this example, the External Ballast Tanks would be situated below the surfaced waterline. Also, Internal ballast tanks can enhance system capabilities*without*adding their volume to the overall displacement of the hull.)

**DETERMINE GROSS VESSEL WEIGHT.**Generality and estimation are acceptable in preliminary design studies, but the accuracy of this process is relative to the precision of the calculations made. That means either computing or weighing each part, piece, and occupant. The precise weight of, say, angle steel, pipe, sheet metal, fiberglass, plastics, rubber, and other materials can be determined by actually cutting a foot-long (or square foot) piece and weighing it, and then multiplying that amount times the length (or area) of the part. Do this for every major part used to make up the submarine (or try your best to estimate it in preliminary design studies). Whenever you don't estimate, actually weigh components like air tanks, sonar sets, and the like; and don't forget to include the weight of the occupants when you sum it all up.*In the example of the 2,000-pound boat mentioned above: If Gross Vessel Weight equals Pressure Hull Displacement, the boat will float with a 10% waterline when surfaced with the ballast tanks dry; and submerge at neutral buoyancy with the external tanks flooded with an amount of water equaling 200#.*Generally, gross weight must be within the ballast-envelope (and preferably inside the low end of it) for the submarine to submerge and surface on air-water ballast alone.

**COMPUTE VESSEL LONGITUDINAL (DRY) CENTER OF WEIGHT.***(NOTE: In general vehicular design, the point at which structural weight "balances" is often called "Center of Gravity". However, in a submarine, the effective CG is actually determined by the relationship of structural mass to both gravity AND buoyancy. Therefore, to avoid confusion, I find it helps to refer to the dry CG as "Center of Weight"; the wet CB as "Center of Buoyancy"; and the average thereof as "Hydrostatic Center of Gravity".)*This is like an*aircraft weight and-balance*calculation. I assign an arbitrary "reference datum" line, usually somewhere near amidships, based on my visual estimation of how she'll balance around that point. Next, I compute a "Moment value" for each part, based on its weight multiplied by its distance from the reference datum, forward or aft, to produce a Moment for each part expressed in foot-pounds of force stationed at that "arm" from the reference datum, and acting*vertically downwards*. I assign a negative pitch value to forward Moments; a positive pitch value to aft Moments; and then average the two to determine the subs "out-of-water balancing point"*or Longitudinal Center of Weight*(LC/W).

**COMPUTE VESSEL LONGITUDINAL CENTER OF BUOYANCY.**If the pressure hull is a symmetrical cylinder, LC/B is at the longitudinal center of the tube. The same is true if a single "conning tower" cabin adaptation is added topside / amidships, or if two symmetrical towers are situated equidistant forward and aft of the longitudinal center. More complex shapes can be "broken up" into sections, and each section calculated as a cylinder of average dimensions. (For example, the Volume of a*truncated cone*measuring two feet long, and 12-inches or 8-inches on either end, is equal to a*cylinder*10-inches in diameter and two feet long.) Multiply section volume times the weight of water to get displacement; (and displacement equals buoyancy.) Again, I assign an arbitrary reference datum based on what my eye perceives as the approximate Longitudinal Center of Buoyancy; and then I compute the Moment of each buoyancy section (like the cone described above) in foot pounds exerted*vertically upwards*at that "arm" from the reference datum. However, in contrast to the procedure used in computing LC/W, this time I assign a*positive*pitch value to*forward*Moments; and a*negative*pitch value to*aft*Moments. Averaging the Moments determines LC/B.

**AVERAGE LC/W AND LC/B TO DETERMINE LONGITUDINAL HYDROSTATIC CENTER OF GRAVITY.**LHSC/G can be thought of as the location of the lateral pitch axis along the longitudinal plane, around which we will (in most cases) want the sub to balance in a level pitch attitude when submerged immobile in a state of equilibrium at neutral buoyancy. Generally, when LC/W, LC/B, and subsequent LHSC/G are all situated together (or within very close proximity of each other) the sub will adopt a level pitch attitude underwater. Averaging LC/W & LC/B determines LHSC/G. However, when these variables are NOT situated together, the sub will adopt a pitch attitude that is either bow high or low, depending on whether LC/B is forward or aft of LC/W respectively; and the axis for this pitch rotation will reside at the resultant median station of LHSC/G. In this case, an adjustment of either weight and/or buoyancy will achieve a level pitch attitude. To do this, we must modify the design to compensate. For example, if averaging LC/W and LC/B produces a result where LHSC/G is aft of LC/B by a moment equal to, say, 50 foot pounds; the boat will sit somewhat stern low. We can correct this by shifting weight forward enough to compensate. In this particular example, moving 50 pounds one foot forward; or 25 pounds two feet forward; or 5 pounds ten feet forward (a change which results in a forward shift equal to 50 foot pounds) will achieve the desired result. The change can also be achieved by modifying buoyancy (and this option may be within the capabilities of the pilot during actual operation simply by varying air-water ballast levels, shifting maneuverable hard-ballast devices, and so forth), but in most cases during the design phase it's probably desirable to simply shift the location of heavy objects (batteries, tanks, etc.) to achieve a boat which is inherently level to begin with. In cases of extreme design instability, it may be necessary to completely relocate certain components (say, shift a battery pack from the stern to the bow), or reposition / reconfigure pressure hull geometry and / or ballast tank location, to achieve a level pitch attitude.

**KEEP IN MIND,**even negatively buoyant external features (like the "ramming spur" on my NAUTILUS MINISUB, or the mechanical claw on a KITTEREDGE sub) will possess at least*some*buoyancy, even if it is generally nullified by the much greater weight of the object itself. Usually, I've found this small buoyant effect to be so minor that, for all practical intents and purposes inherent to the basic design of homebuilt subs, it could be ignored without noticeable effect. However, it is conceivable that such might not always be the case; and if the designer wants or needs greater accuracy when considering the performance of his sub in the water, he may want to include the small buoyancy of generally negatively buoyant external structures in his weight & balance computations.

**ALSO,**If the designer does not wish to go the "top buoyant / bottom heavy" route, the formula shown above can be adapted to calculate vessel attitude and stability about the roll axis, by relating variables of weight and buoyancy on, along, or adjacent to the vertical plane.

**AS A GENERAL RULE OF THUMB**, design the sub with obvious symmetry and positive dynamic stability in mind. Where symmetry is not possible or desirable, then seek to counterbalance components and forces to achieve the desired attitude and stability.

Copyright 2013, Pat Regan,
"All rights reserved."