Lofting - 5.Calculation

Calculation.

Calculation for what?

It is to calculate Displacement of a vessel and diplacement is mass of the vessel(In this case, the dinghy we are building). Then, to calculate Displacement, we need to find out displaced volume of the vessel because...

* Displacement = Displaced Volume X Specific Gravity(of Fresh Water or Salt Water, it depends).

Then, Displaced Volume, how do we get it? As we all know,,
* Volume = Area X Height(Simply,,^^;;)

So if we know the area and height of the object, we could get the volume.
Then how do we get the area? Let me start with area calculation.


1.Area Calculation(Approximation)
Unlike rectangular, triangle, the surface we deal with is a fair line(Curve) and thus we do approximation for the calculation.

①Trapezoidal Rule

Using Trapezoidal Rule, we divide the object with same length(AE, EG, GI and IB are the same length) and regard the curve as a straight line, so simply think the object as a trapezoid.

Then, total area will be the sum of T(AEDF)+T(EGHF)+T(GIJH)+T(IBCJ)

T(AEDF) = 1/2*(y0+y1)*h
T(EGHF) = 1/2*(y1+y2)*h
T(GIJH) = 1/2*(y2+y3)*h
T(IBCJ)= 1/2*(y3+y4)*h

Area of ABCD = T(AEDF)+T(EGHF)+T(GIJH)+T(IBCJ) = 1/2*(y0+2y1+2y2+2y3+y4)*h = (1/2*y0+y1+y2+y3+1/2*y4)*h
=(10+29.24+35.51+39.06+20)*25= 3345.22
And I checked the area of ABCD in AutoCAD and it 3345. Voila!!

At first, I thought there should be differencde between the values I got from approximation and AutoCAD, but there's just slight difference and I think the reason is maybe AutoCAD is using the same logic to get the area.(not sure about this.)


②Simpson's First Rule

In Simpson's First Rule, to get the area of ABCD, you regard the curve DFC as a quadratic curve and multiply vertical length(y0, y1, y2) by a certain coefficient called Simpson's multiplier.

Area(ABCD) = 1/3*(y0+ 4y1+y2)*h

To apply Simpson's First Rule, you must divide the object into even number and Simpson's multiplier for Simpson's First Rule starts with 1 / 4 / 1.

Simpson's Multiplier for object with 2 equal segments : 1 4 1
Simpson's Multiplier for object with 4 equal segments : 1 4 2 4 1
Simpson's Multiplier for object with 6 equal segments : 1 4 2 4 2 4 1

Simpson's Multiplier

Then, let me calculate the area of object with 4 segments.

Area(ABCD) = Area(AGHD) + Area(GBCH) = 1/3*(yo+4y1+y2)h + 1/2(y2+4y3+y4)*h = 1/3*(y0+4y1+2y2+4y3+y4)*h


③Simpson's Second Rule
Simpson's Second Rule is similar to the First Rule.
Differences are like this.
- Regard the curve as a cubic curve.
- Simpson's multiplier starts with 1 3 1
- This is for more complicated objects.

But even for complicated objects, you could get area of it using the First Rule by dividing many more segements.


2.Volume Calculation(Approximation)

Here's Simpson's First Rule.
Area(ABCD) = 1/3*(y0+ 4y1+y2)*h
Here, y0, y1 and y2 are length of vertical line. We put lenght in there, it comes with area. If we put area in there, then, it comes with Volume.

So the Volum is
V=1/3*(AR0+ 4AR1+AR2)*h

For example, check the below.

Let's just say that it's a hull. AR0, AR1 and AR2 are the calculated value.(We can get it area calculation using Simpson's rule). With those value of AREAs, h, we can make new curve called "Area Curve" as below.

Here, ARo, AR1 and AR2 are the value of Area, not the lenth. If it is lenght, we could get the area from it but since it is the area, if we calculated the area of the object above, it becomes Volume.

3.Application of approximation.

Above is a body plan of the dinghy. Since the value we are fiding is a displaced volume, so we need to find the volume of the dinghy which is really in the water. So, that is, below LWL in Body Plan(as below)

* First, calculate area of each station in Body Plan.
* Get the full area value by multiplying 2
* Get the Function of each station by multiplying with Simpson's Multiplier and get the sum of Functions.
Then,, Since,
V=1/3*(AR0+ 4AR1+AR2)*h

= 1/3*(Sum of Functions)*C.I(Common Interval)

Note. the unit for area is "squrea mm" and unit for Volume is square M. So you have to convert "square mm" to "sauare M" by dividing it by 1,000,0000.

Now we have "Displaced Volume" and to get Displacement, we need to multiply the volume with S.G of saltwater which is 1.025t/m3(1,025kgf)

Thsus,,

Displacement =1/3*(AR0+ 4AR1+AR2)*h*1025

= 1/3*(Sum of Functions)*C.I(Common Interval)*1025


Cheers.