How to Balance your Boat¹
with a Center of Gravity Database
¹ (or your airplane,
helicopter, blimp, submarine, ROV... or anything else that needs
to stay balanced despite having a bunch of junk on-board)
© 2004 by Steven K. Roberts
Nomadic Research Labs
I heard it from them all... every multihull marine architect who
had
ideas to contribute to the Microship project urged me, at one time or
another, to do a weight study. Robb Walker... Gino
Morrelli... John Marples... Jim Antrim... luminaries
in the field, every one. Each informed me that a critical first
step in designing a ship (or an airplane) is a thorough inventory of
gear and fixtures with at least a summary
of their weights and centers of gravity.
I really didn’t want to do this.
My first attempts were therefore pretty crude: one-page listings
of broad categories, with wild guesses about aggregate weights.
As I pointed out to my advisors, I could hardly come up with a detailed
inventory when I still didn’t know the size of the boat, now, could
I? “Tellya what,” I told ‘em cockily, “you tell me the constraints, and then I’ll
massage the inventory to fit!”
No dice. “Do a weight study,” they responded. “It affects
the entire design.”
Obviously, what we have here is a circular problem, and an iterative
one at that. You make a list, discover that the weight budget
won’t support a 100 gallon water tank (that’s 800 pounds of water),
change the capacity to 50 and nudge it forward in the hull a little
bit, then try again. There’s a lot of head-scratching and
guesswork involved, and whether you’re outfitting a voyaging yacht or a
racing skiff, the sheer volume of data can be overwhelming. If
said yacht happens to be a multihull, the weight and center-of-gravity
issue is even more critical... this is not a place for shortcuts.
And, I should point out, the same problem arises in aircraft design but
with even more critical implications.
So after a few false starts with a paper binder (tedious) and a
spreadsheet (ugly and a mess to edit), I decided to develop a
weight-study database that would maintain not only a running total but
also calculate, on-the-fly, the ship’s aggregate center of
gravity. This is derived from the weight of each object (starting
with the vessel itself), and its vertical, longitudinal, and transverse
centers of gravity (VCG, LCG, and TCG) – entered in any convenient
notation (such as height above or below waterline, distance to port or
starboard from centerline, and station,
or distance from the bow).
This works beautifully, and we now have a tool that objectively (not
optimistically) shows the effect of every added object – not only in
terms of bottom-line poundage, but effect on trim. This makes it
easy early in the design stages to modify decisions about battery bays,
tankage, spares inventory, and more – all with solid feedback on how
product choices and placement will affect performance.
This article will show you how to accomplish the same thing, using any
robust database package, on any computing platform. The example
here was created in FileMaker Pro for the Macintosh and is freely
available if you want it, but the theory and field definitions are
fully explained so you can implement it in other packages (or in a
spreadsheet such as Excel, if you prefer that over a database).
This tool will not only give you much greater control over the trim of
your boat, but also provide a convenient place to keep track of spares,
costs, insurable totals, vendors, serial numbers, service intervals,
and things to buy.
Let’s start with an introduction to the whole “center of gravity”
concept, see how to calculate the effect of an object’s weight and
location, then dive into the database design...
The Principle of Moments... A Few CG Basics
Every boat has a center of buoyancy
(CB), which is the center of the underwater volume of the vessel.
She also has a center of gravity
(CG) which is where all the mass would be concentrated if it had to be
compressed to a single point. If the boat is to float properly on
her design waterline, then the CG must be in line vertically with the
CB.... if it’s not, then the boat will correct for it by changing trim
(and thus underwater shape) until the new CB is in a vertical line with
the CG.
This pretty well sums up the problem. There’s not much you can do
about the CB for a given boat (which wanders around with heel angle,
but we won’t worry about that since it’s out of our control), and the
CG of the raw hull is pretty much a given as well. But the moment
you start installing equipment and people it all changes... often
dramatically. Just take a stroll from bow to stern of a canoe for
a quick demonstration of how much your weight can affect the trim of
something with 1,000 pounds or more of buoyancy.
I’m not going to go into the extensive mathematical analysis of all the
factors affecting this – it’s well covered in nautical textbooks and on
the Web (there are some useful links at the end of this file to get you
started if you want to explore the physics). What we do want to
discuss instead is how, exactly, you compute the resultant CG from the
locations and weights of random objects so that you control the process
from the beginning... instead of shoving stuff around later trying to
fix it.
If you have one object sitting alone, then determining net center of
gravity is obvious – it’s just the object’s own CG (fairly easy to
guess for most smallish things, not hard to measure for large
ones). But as soon as you have two or more attached the
same substrate, such as a boat, you have to do a mathematical trick to
find out their collective center of gravity.
An easy way to start visualizing this is with the seesaw in Figure 1. A heavy kid
and a light kid, to achieve balance, must arrange themselves in such a
way that the former is closer to the pivot point than the latter.
It happens that the math is simple: all they’re doing is
equalizing their moments, which are defined as their respective weights
times their distances from the pivot point. If we have an 80
pound kid and his 50-pound little brother sitting on a 12-foot seesaw,
they will automatically position themselves such that their moments are
equal. Assuming that the light kid is sitting on the very end,
his moment is 6 feet times 50 pounds, or 300 foot-pounds. To
balance, the 80-pounder just divides 300 by his weight, yielding
3.75... then moves to that spot on the board. Of course, he
doesn’t actually calculate this; he just scoots forward until the
seesaw balances... and that’s where he ends up: 3.75 feet from
the fulcrum.
Figure 1: The 80-pound kid finds the
correct spot on the seesaw to balance his little brother by achieving the same
moment -- the product of mass and distance from the fulcrum.
What we have here is a simple demonstration that a small object far
from the pivot point (CG) of a vessel (er, seesaw) has the same effect
on trim as a large object that’s closer. If the kids on the
seesaw happened to have instead been paddling a very tender canoe and
had the nautical sensibilities to keep her properly balanced on her
waterline, they would have positioned themselves similarly...
Now let’s see how moments can be combined to yield the aggregate center
of gravity of any arbitrary number of objects.
It’s a process much like averaging. Looking at part (a) of Figure 2 below, let’s pretend that
the horizontal line is a small 20-foot boat (we’ll ignore its own mass
for the moment, so to speak). It turns out that any point can be
used as a reference, so to keep measurement simple and avoid the added
confusion of negative numbers, let’s call the bow our “reference
datum,” which is a fancy name for zero.
I’ve arbitrarily placed two objects on board this imaginary
vessel: a 55-pound battery 6 feet back from the bow, and a
85-pound outboard motor whose center of mass is 1 foot forward of the
stern. Let’s calculate the moments:
Battery 55 lbs X 6’ = 330
Motor 85 lbs X 19’ = 1615
Where’s the resultant CG? Just add the moments and divide by the
total weight of the objects:
CG 330 + 1615 = 1945 = 13.89
55 + 85 140
The center of gravity of the battery and the motor is 13.89 feet back
from the bow... not very good trim!
Figure 2: These three images
illustrate the effect of moving a heavy object around on a boat.
In (a), the 55-pound battery and 85-pound motor combine to yield a
longitudinal center of gravity of 13.89 feet. In (b), the battery
weight is doubled, moving the CG forward by over 2 feet. In (c),
relocating the original battery to the bow has almost the same effect
on CG. Every object on a boat contributes to the collective
center of gravity through the combination of its weight and location.
In Figure 2(b), let’s add a
second battery to see how the CG is affected. When you run the
numbers with 110 pounds instead of 55, how much does the CG move?
It scoots forward by over 2 feet! Our new CG is 11.67 feet (try
this yourself to make sure you get the same answer).
But what if we instead kept the 55 pound battery but relocated it to a
point 1 foot back from the bow, as in Figure
2(c)? Let’s ignore the fact that this sort of behavior is
ludicrous from a nautical perspective (not only is that a poor place
for a battery for purely electrical reasons, but you generally want big
massy things near the boat’s CG to prevent hobby-horsing and other
pathological behavior... this is what “moment of inertia” is all
about). If you run the numbers, you see that the battery’s moment
is now only 55 (55 lbs X 1’). Adding that to the motor’s moment
and dividing by the total weight of the two yields a net CG of 11.93.
Now you can really see the interacting effects of an object’s weight
and its location... the result of doubling the weight of a battery
located 6 feet from the bow is almost identical to that of moving the
original battery to a point only 1 foot from the bow. This is a
key observation... for every object on your boat, you have two knobs to
twiddle when determining its effect on trim: weight and
location. Since the former is a little hard to adjust for most
things, your most powerful balancing tool is moving things around with
an understanding of moments.
This simple calculation scales to any number of objects, of course – we
can add a thousand moments, divide by the total weight of those 1,000
objects, and the number will be their net center of gravity.
Now, take everything I’ve told you and expand it into three
dimensions. CG comes in a trio of flavors, which add up to a
point in space, located somewhere inside your boat. What we’ve
been discussing so far is known as longitudinal
center of gravity, or LCG, and lays along a line from bow to
stern.
If you walk around to the stern and look at the boat from that
perspective, however, you can see that the same kind of issues are
present with side-to-side balance; you don’t want to be sitting all
wonky in the water (here’s one situation where it’s good to be
listless). The magic number here is called transverse center of gravity, or
TCG, and is affected by tankage and other heavy things that people like
to tuck out of the way to port or starboard. When
calculating TCG, it’s traditional to use the centerline of the boat as
the reference datum, with objects’ distances measured to port (-) or
starboard (+).
And finally, if you tilt your head sideways and think in terms of
stability, you have exactly the same set of phenomena operating
vertically, naturally called vertical
center of gravity, or VCG. In general, you want this to be
down as low as possible; if it’s way up in the air, you could have
serious problems! The reference datum for VCG calculations is
arbitrary, but is often the DWL (design waterline), as that’s just
about the only well-defined horizontal plane sliced through a
hull. I mean, how many flat surfaces and straight lines are there
in a boat? In all cases, the choice of reference datum is
ultimately irrelevant, so if you have a more convenient way to measure
from, by all means do so... just be absolutely consistent.
The foregoing quick introduction is all the background you need to
perform the magic math that yields the center of gravity of your boat,
including everything from the hull itself to that stainless steel
crescent wrench you just bought. All this adds up to much more
information than you’d ever want to keep in your head, so let’s
automate it.
Figure 3: The collective
center of gravity (whether longitudinal, transverse, or vertical) is
calculated by totaling the weights of all the objects, totaling all the
moments by multiplying the weight of each object by its own center of
gravity, then dividing the sum of those moments by the total
weight. Doing this calculation in all three axes for all objects
(including the substrate itself, such as a boat hull) will yield the
net CG of any system.
The Weight Study Database
First, I need to make a quick comment on the
implementation. It happens that I developed this database
on a Macintosh under FileMaker Pro version 3.0 (Nov 23, 2005
note: I'm now using version 7.0, which imported and converted
this just fine). Fortunately, this excellent
software is designed to be cross-platform, and with a few minor
cautions a database written on the Mac should run fine under Windows.
NOTE: If you are using a
PC, please be aware of a few caveats. First and foremost, I have
never tested this under Windows, so I don’t guarantee anything...
though theoretically it should work fine. (In theory, there is no
difference between theory and practice. In practice, there
is.) The differences between the platforms are minor, as far as
FileMaker is concerned: The Mac version supports AppleScript and
Windows has OLE, but we don’t use either here, so that’s not a
problem. You might have to adjust fonts, field and label sizes,
and possibly make other aesthetic tweaks to make it come out pretty on
a PC. And I hear rumors that the Windows version, at least in the
old days, required a specific filename extension for the file to be
compatible (even though that won’t show up in the title bar), and under
older versions, you were limited to the ancient DOS-style 8-character
filename. For all I know, these limitations are no longer the
case, so if you’re using a Windows box, just try it and see what
happens. Then tell me about it, so I can update this paragraph
and put your version on the site for others to download!
None of this should matter much anyway, since you’ll doubtless want to
configure the database to fit your own needs, using your own tools; by
the time you’ve done all that you’ve basically created your own
database (it’s not hard). But if you still want to play with my
version after all those warnings, you can download the 32K template
file here.
OK, now that the administrivia is out of the way, let’s talk about how
this actually works. Please take a look at Figure 4, which is a screen shot of
the test version of my Microship inventory database.
Figure 4: Screen shot of
database record. The 55-pound battery is located 6.5 feet back
from the bow, on the boat’s centerline. At the moment, the net
LCG (Longitudinal Center of Gravity) is a 9.12 feet from the bow, and
there is a very slight list to port.
The whole idea here is to have an inventory database for the ship, at
any level of detail you like. To keep it from becoming a
full-time job during initial planning, I just cluster whole piles of
things into single units – like “tool kit” instead of the 200 or so
entries that make up its contents. But the design does
accommodate grouping, as we shall see in a moment.
The database does a lot of traditional things in addition to the magic
center-of-gravity calculation – it lets you track your spares
inventory, record serial numbers and vendor information, make notes
about each item, and so on. And in my case, I also have a few
dollar-value fields that most people don’t need, as much of our
equipment is sponsored and it’s fun to keep track of that. I even
have a $/pound calculation, which is utterly useless but amusing.
(“Why do we do it? Because we CAN!”) I should also note
that the values you see here are from an ancient test phase of this
database, not reality... so please don’t draw any conclusions about the
Microship project from this bogus data.
Let’s take it from the top, field by field.
Title:
At the very top is a title, which appears on each record. This is
completely superfluous, as it’s also shown in the title bar... but it’s
pretty.
Item Name: This
identifies the widget under consideration, in this case a marine
deep-cycle battery.
Source: I find it useful
to associate a vendor or sponsor with each item, where
applicable... I bought this at West Marine.
Category: This field is a
pop-up menu of all possible categories of goods on the boat, which
makes it easy to look at just the contents of one locker or pack... or
see how much the electronics weighs by doing a find on a single
category. Here’s the full list from the test version of the
database; yours will look different:
ESTIMATES (temporary)
GEAR: Audiovisual
GEAR: Bedding/comfort
GEAR: Books
GEAR: Camping/shore
GEAR: Diving/fishing/etc
GEAR: Galley
GEAR: Goo
GEAR: Maintenance/cleaning
GEAR: Marine
GEAR: Miscellaneous
GEAR: Nav/mapping
GEAR: Office/business
GEAR: Parts/spares
GEAR: Safety/survival
GEAR: Clothing
GEAR: Manpack
GEAR: Personal
GEAR: Tools
INTEGRAL: Cockpit furnishings
INTEGRAL: Electrical/Solar
INTEGRAL: Electronics/AV-Comm
INTEGRAL: Electronics/Comp
INTEGRAL: Electronics/Packaging
INTEGRAL: Marine Misc
INTEGRAL: Landing Gear
INTEGRAL: Water System
RIGGING: Deck/misc
RIGGING: Ground tackle
RIGGING: Hydraulics
RIGGING: Sails & Running
STRUCTURE: Primary
STRUCTURE: Secondary
VARIABLE: Food-Water
ZERO-WEIGHT: Software
FileMaker allows you to edit a “value
list” for a field and present it as a pop-up list, pop-up menu, check
boxes, or radio buttons... it’s best to use this kind of approach for
categories so your searches won’t be thrown off by alternate spellings.
Contact,
Phone, and Serial #: These fields let you keep track of
additional details about the item in question, for insurance or support
purposes. This is the kind of data that can end up scattered to
the winds (usually in old notebooks or receipts) if you don’t make some
effort to put it in one place... and since you’re going to all the
trouble to inventory the boat for a weight study anyway, you might as
well put it here.
Cost and Value: I have
two fields where only one is needed, just to let me keep approximate
track of sponsorship, good deals, freebies, hand-me-downs, and so
on. Cost is actual out-of-pocket; Value is what it’s worth.
Item Weight: Here you
enter the weight of the item in pounds... and click a radio button to
indicate whether it’s estimated or real. This turns out to be
useful when you’re in the planning stages, and a quick find operation
can tell you what’s still ambiguous.
Item LCG: Longitudinal
Center of Gravity... this is where you record where the item is on your
boat, measured in feet from the bow.
Item TCG: Transverse
Center of Gravity... as above, here you record the location on either
side of centerline (use negative numbers for port and positive for
starboard).
(Item VCG: Vertical Center of Gravity. My boat, a
micro-trimaran, is so low that this turns out to be irrelevant for me,
so I left it out. It works exactly like the others, and adding it
is an exercise for the reader.)
Status: This line of
check boxes is a quick-and-dirty way to flag items that need to be
acquired, or otherwise create useful abstract subcategories for
searching.
PM Months: I haven’t
really done anything with this yet, but the idea is to periodically
print out a Preventive Maintenance schedule, with items sorted into
intervals (like checking the batteries in your strobe every
year). As with any other field in this database, if it’s not
useful to you, just delete it.
Batts/Spares: Here we
have a place to record what batteries or spare parts associated with
this particular item need to be on hand... and the two radio buttons to
the right indicate whether they are local or back at home base.
Notes: Anything you like
goes here. I tend to expand the description a bit and record any
other details I might like to know someday.
OK, here’s where it starts to get fun. Everything from this point
on is updated automatically by the database every time you add or
modify a record. Nothing below this line is ever manually entered
(and the database will stop you if you try).
Total
Value: This is a “summary” field that totalizes all the
“Value” fields in the database. A nice thing to know.
Total Cost: Likewise for
the real out-of-pocket...
% Spons: This is a
calculation based on the past two fields, and is defined as ((Total
Value - Total Cost) / Total Value). In practice, I find that the
separate sponsor database is a much more useful place to keep this
information, as it includes things that don’t appear in the ship
inventory. I’m leaving it in just to give another field
calculation example leading up to the CG stuff, but I don’t
particularly recommend that you use it.
$/lb: Be honest, now...
don’t you also wonder about completely useless things like this?
Item LM: Item
Longitudinal Moment, or the result of the item’s longitudinal (Weight X
LCG) calculation (55 X 6.5 = 357.5). This number by itself isn’t
particularly useful, though it does show you the “torque” applied to
the boat by the object in question. It exists as an intermediate
calculation step in the development of the value we’re really after.
Item TM: Item Transverse
Moment. Exactly as above, but reflecting instead the item’s
effect on side-to-side trim. Note that the battery is centered in
the boat, so even though it weighs 55 pounds its effect on the
transverse center of gravity is zero (55 X 0 = 0).
Total LM: This is the sum
of all the longitudinal moments in the database, at this instant
equaling 7,142.08 foot-pounds. If you tried to pick the boat up
by its very bow, this is the torque you would have to overcome.
Total TM: Likewise, the
sum of all the transverse moments. Since the center hull of my
boat is a canoe and things tend to more or less hover around the
centerline, this shows just a minor negative value. In a
millpond, this would translate into an almost imperceptible list to
port.
Total weight: This is the
beginning of the bottom line. Recall from our explanation that at
some point we’re going to need the total weight of everything on
board. This is a summary field that automatically increases –
frighteningly quickly, I might add – with each item you lug aboard.
Net LCG: Here we have the
result of the final longitudinal center of gravity calculation:
the database has totaled all the longitudinal moments and divided by
the total weight to yield a point just aft of center.
Net TCG: Similarly, this
is the result of the final transverse center of gravity calculation.
Using (and Enhancing) the CG Database
Putting all this to use is easy... just start throwing stuff at it and
watch the numbers converge! But it’s best to begin with the boat
itself.
This isn’t as hard as it sounds. You need a scale that can handle
about half the weight of your boat. Follow these steps:
- Weigh each end of the level boat, wherever and
however convenient.
- Note the exact distance from the bow of each
measurement point.
- Multiply each weight by the distance from the
bow.
- Add those two numbers.
- Divide the total by the sum of the two weights.
Sound familiar? You just performed the same old moment
calculation to determine the boat’s LCG! If she’s on a trailer,
you’ll have to measure the weight of the wheels and tongue, then make
the same measurements on the unloaded trailer and subtract.
Either way, it’s important to start with the substrate, the single
largest contributor to the CG numbers you’re after.
Once you have this, create a database record for the boat itself,
ideally with no equipment installed (I know, that’s not always
practical with an existing boat, but you need to draw a distinction
between things that are part of the boat and those that are not).
This is your starting point, and every subsequent component in the
growing calculation will nudge the CG figures back and forth, left and
right, up and down... giving you an evolving look at your static trim.
If the empty boat rested perfectly on her lines, you already know the
target values... you want the final net CG figures to be exactly the
same. If she didn’t float level, well, at least you know which
way you have to tweak things to achieve a proper balance. And
don’t forget the ephemeral variables: tankage, people, and
stores. It might be useful to include a yes-no field called
“empty” and use it to flag separate records for the contents of tanks,
expected personal gear brought on board by crew, and even the crew
themselves in estimated locations. When you do a FIND on the
database with empty=yes, you have the CG of the empty boat; if you do a
find-all (hitting command-J
in FileMaker), the additional weights of all the added cargo are
considered in the calculation.
Incidentally, there’s a lot more to this than just the CG. That
lets you determine static trim when sitting quietly in the water and
has a lot to do with stability, but there are dynamic issues as
well. In particular, the phenomenon known as moment of inertia
will affect the rolling and pitching action in a seaway: if all
the mass is concentrated at the center, the resulting motion is
quicker; if it’s at the ends, slower and deeper. There are all
sorts of trade-offs here that are further affected by your boat’s hull
design and purpose in life; see Link#2 below for a useful discussion on
the subject. Adding moments of inertia to the database is no big
deal (the basic difference is that you square the distance figures to
exaggerate their effect on the total).
I noted earlier that we happened to have done this in FileMaker Pro on
the Mac, and that you are encouraged to implement it using other
programs or computing platforms. One of the most interesting
database environments is the range of SQL-based tools, such as the
popular MySQL available free for Linux. The fundamental
difference is that this kind of relational database appears more as a
linked collection of tables, and there is no mechanism for internally
maintaining calculations as we do here. However, it’s quite easy
at the query level: you would write a query that computes the
various totals and net CG on demand and let the machine run the whole
process anytime you ask for it... such as
select sum(weight) as totalWeight from
myInventory
to compute the total weight of all items on the boat.
If you do generate an SQL variation, or implement similar methods in
another flavor of software (such as a spreadsheet or web-resident
tool), please consider passing it along to other readers in the spirit
of Open Source. Email me
to arrange adding your work and comments to this page. We are
putting the whole Microship project under GPL to do our bit to buck the
tide of increasingly oppressive corporate end user license agreements.
Finally, I should note that all this applies equally well to airplanes,
helicopters, submarines... anything for which balance is
important. You can even apply this tool to backpacking or
bicycle touring, for a nice low CG makes life much more pleasant when
you’re hauling something around with your body. Feel free to
experiment... the real point here is the basic calculation method;
there are countless ways to put it to work and this database is just
one convenient way to package it.
Good luck, and may your vessel always rest gracefully on her lines!
This little tool is "Tin Cup Ware,"
so if you find it useful and feel inclined to cast nickels in my direction,
this little button will facilitate the process:
Links
