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Author Topic: calculating displacement  (Read 7482 times)

Offline Roderick Smith

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calculating displacement
« on: November 19, 2006, 11:20:24 AM »
I just googled on 'formula for displacement of a boat': heaps of references.
One quicky, relevant to the size of boat being discussed in this forum:
waterline length x beam x draft x 0.6
If the measurements are in metres, the answer is in cubic metres; multiply by 1000 and you have an answer in litres (ie kilograms).
This is not an exact formula, it is an approximation which assumes that most different curved forms are comparable.
If your boat was built as a rectangular box, the factor of 0.6 would have to be eliminated (V = lwh is standard junior high mathematics).  Hence there will be different curvatures which require factors higher than 0.6, and possibly some requiring lower factors.

For James, we can assume that SJ (ie the boat) has typical curves, and that 0.6 is a good approximation.
After floating in your shallow dam, you could come up with an estimate of empty displacement (hopefully with a weighbridge result for a cross check).
Next you could take the measurements at minimum safe freeboard, and so measure the maximum safe displacement.
The difference becomes the amount of equipment and passengers which you can place in the hull.
Fuelling your boiler with hydrogen won't help the boat float higher out of the water (do ask your science teacher).

As an interesting excercise, I can steer you through two ways of estimating displacement with a spreadsheet which could help you win a mathematics talent quest prize at the same time as supplying useful data for SJ.  The nice outcome would be if all three methods correlated.

The easy method:  Keep filling the hull with water from a bucket, keeping a tally of the litres inserted, and the depth filled.  Keep the data, and bale or siphon all of the water back into the dam.

Roderick B Smith
Rail News Victoria Editor


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calculating displacement
« Reply #1 on: November 19, 2006, 11:47:14 AM »
Ah! Now that's just the kind of article I was looking for! Mind you I really can't be bothered loading Grebe up with water so I say I'll have to work it out mathematically.

So now I know roughly where the water line was...and the freeboard line so I should to be able to work out a truly rough estimate yes...?

How else do you get your boat weighed...? What's a weighbridge?

Offline anth

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calculating displacement
« Reply #2 on: November 19, 2006, 04:36:43 PM »
I am a little confused with this formula :oops:
it's making the assumption the beam is uniform through out the hull shape and a you know the draft of your hull..... :?

Offline Roderick Smith

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« Reply #3 on: November 19, 2006, 04:57:37 PM »
No, it isn't assuming uniform beam.
If it did, the factor of 0.6 would be 1.0 instead.
The 0.6 allows for the fact that it is slimmer at the ends than at the centre; the average is 0.6 rather than 0.5 because the shape is a curve bulging outwards, and not a straight line.
Perhaps it is assuming uniform draft rather than maximum draft, but again the 0.6 is an attempt to convert the maximum figures to average figures, allowing for curves rather than straight lines.

My own Jessie II is 9 m long (at waterline and elsewhere; vertical sides); 1.8 m beam and 0.1 m draft.
Hence, displacement by the formula is 9 x 1.8 x 0.1 x 0.6 = 0.97 cubic metres.
This converts to 970 litres or 970 kg.
I would believe this figure, as it is quoted as 1400 kg on the trailer.
Cruising with 7 passengers and crates of beer would add no more than 500 kg, and hence increase the draft only to 15 cm: still good to get through the waterways where I cruise.  My problem is far more likely to be weed ingestion or sand ingestion into the cooling system.  I did chicken out trying to reach the old jetty and launching ramp at Lake Tyers: 30 cm depth above sand, and possibly shallower ahead.  Your paddleboats would have gone through without fear.  I also chickened out on Barmah Lake in 2005: 30 cm depth, but lots of weeds.  Not even the bold PS Billy Tea ventured into the lake.

Readers could mock up the formula by carving a boat out of margarine, and measuring the displacement by pushing water out of a 5 litre icecream tub into a kitchen measuring jug.

A weighbridge is a large set of scales for measuring the weight of heavy equipment (eg trains or overloaded B-double highway trucks).  Traditionally, they were built mechanically (like a large set of kitchen scales); now they use calibrated load cells, and can record weights on the run.  It does become a bit tricky on mechanical ones if the whole vehicle can't fit in one go, as there is some error in measuring axle by axle.  I did attend a very interesting lecture once on measuring the mass of a horse when a full weighbridge was unavailable.  I think that the boat formula is easier than the horse one.

Roderick B Smith
Rail News Victoria Editor


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calculating displacement
« Reply #4 on: November 19, 2006, 06:11:29 PM »
We might have to wait a little bit, because the SJ wont be floating for a bit, but I can certainly weigh it and let you know!!
The displacement has been on my mind ever since Eddy mentioned it!!

Offline Roderick Smith

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calculating displacement
« Reply #5 on: November 20, 2006, 09:28:13 AM »
In junior maths, you cover the standard formulas:
Area of rectangle = length x width
Area of triangle = half base x height
Area of circle = pi r squared
You then move into three dimensions
volume of a prism = area of base x height
and volume of a pyramid = one third area of base x height.

Think of your boat made from cheese or margarine.
Slice it up with a knife, measure the slices and add them up: you now have the displacement volume.

Version 1: slice vertically from stem to stern: for a 4 m boat, either 20 slices of 20 cm (0.2 m), or 40 slices of 10 cm (O.1 m).  Each slice has a straight top, and two S-curved sides.  For practical purposes, think of it as a triangle: the bulging bit of the S curve balances approximately the bit which curves inwards.  So, measure the top and put it in column A in your spreadsheet; measure the height; put in in column B.  Put the formula into column C:
C1 = 0.5*A1*B1*0.2 (or*0.1 if you did thinner slices)
fill down
At the bottom, SUM column C, and you have the volume in cubic metres.
As before, multiply by 1000 and you have it litres (ie kilograms).

This method is limited, as you have to go back and do the job all over again to work out displacement at different drafts.

Version 2: slice horizontally from the bottom up.  Your boat has a total hull depth of about 75 cm: try 15 slices of 5 cm (0.05 m).
Each slice now looks like a chocolate box, with the base being approximately an ellipse.
Junior maths didn't give you a formula, but it quite easy.
Think of it as a sqashed circle with a long radius (a = half the length) and a short radius (b = half the width)
The area is pi x a x b
If you had bought a coracle, a and b would be the same, and the formula comes out to the familiar pi r squared again!

Once more, put a into column A, and b into column B and the formula into column C (C1 = 3.14 * A1 * B1 * 0.05)
This time, keep a progressive subtotal in column D
D1 = C1
D2 = D1 + C2
fill down
You now have the displacement at every draft (and hence the draft at every displacement), and can work out how much equipment and how many passengers can be added while still leaving a safe freeboard.

I can draw and scan diagrams to illustrate the two methods, but not today.

I have run a similar exercise to calculate the volume of a human skull, an idea which I got when eating at the original Island Trader restaurant chain, when it had a cannibal menu [I arrived late, and the head waiter said:'Sorry sir, everyone's eaten].  I asked my students to estimate the amount of icecream which could be held in a human skull.
I had something similar when I had a magnetic-inference scan for a research project: the computer sampled my brain, front to back in about 1 mm slices.

Roderick B Smith
Rail News Victoria Editor


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calculating displacement
« Reply #6 on: November 20, 2006, 09:34:00 AM »
I will go home and weigh the hull, then see if I can slice it up!! (Not literally).
If I can work out the displacement then I can work out how much weight it will be able take on board, I am hoping that it will be enough to support the rest of the equipment I intend of having onboard.


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