# Area under GZ curve

Discussion in 'Stability' started by claudiu01, Jul 28, 2010.

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### claudiu01New Member

Help me to calculate area under GZ curve for a 8 m sail boat.

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### claudiu01New Member

please some one 3. Joined: Jul 2010
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### latestarterSenior Member

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### claudiu01New Member

Thank`s

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### daiquiriEngineering and Design

A graph paper method is ok, but if you have the GZ curve in a form of a spreadsheet, you can let Excel preform the calculation for you.
As I understand that you already have the curve, the required area can be obtained through a simple integration. A general info on integrals can be found here: http://en.wikipedia.org/wiki/Integral
The simplest (and the least precize, but sufficient for your goal) way of calculating the integral of a function is the Rectangle Method: http://en.wikipedia.org/wiki/Rectangle_method

You basically have to subdivide the X-axis (heel-angle axis) in a sufficiently big number N of small intervals dH:
dH = ( H_max - H_min ) / N
where H is the heel angle.

In this way, you have discretized the heel range in N+1 values:
H,1 = H_min
H,2 = H,1+dH
H,3 = H,2+dH = H_min+2dH
H,4 = H,3+dH = H_min+3dH
etc.​

up to
H,N = H_min + (N-1)dH​

Then evaluate the GZ at each H,i point:
GZ,1 = GZ(H,1)
GZ,2 = GZ(H,2)
GZ,3 = GZ(H,3)
etc.​

up to
GZ,N = GZ(H,N)​

All you have to do now is to sum up the elemental rectangular areas, products of GZ(H) and dH, to obtain the total area under the curve:
Area,tot = GZ,1*dH + GZ,2*dH + GZ,3*dH + ... + GZ,N*dH​

Which can be simplified to:
Area,tot = (GZ,1 + GZ,2 + GZ,3 + GZ,4 + ... + GZ,N) * dH.​

And that's it. The numerical error will decrease as you increase N (say, 100, 500, 1000 or more - depending on the accuracy of your GZ curve and on the required precision).

Cheers

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### landord01New Member

If you have the Gz curve, it would be easier to draw it in Autocad, then you can measure the area under the GZ curve.

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### sandeepSandeep

hiii,
I think it is calculated by simple integration.you can use simpson's rules to find the are under gz curve.I suggest simpson's 1 4 1 rule.Divide the gz curve equally with a number of lines in vertical directions, if the number increases the result will be more accurate.The total no of lines should be odd no. including first and last.You can calculate the area by the following way

Suppose the lengths are a,b,c,d,e,f,g
then use the following method
a*1=a
b*4=4b
c*2=2c
d*4=4d
e*2=2e
f*4=4f
g*1=g

sum=a+4b+2c+4d+2e+4f+g
then area=(h/3)*sum

h=spacing between lines

k..good luck

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### MikeJohnsSenior Member

If your data points are equally spaced its very simple:The Trapezoid rule rules !
For any part of the curve you want the area under simply sum all the y data values and multiply by the x increment, ( the space between each x value). That will be close enough.

If your data points are not equally spaced you can still use the Trapeziod rule. Just look it up it's simple arithmetic..

If you want to play You can also run a curve fit program and derive and solve the definate integral between the ranges you are interested in, but that does need a knowledge of calculus. Although it is easy.

9. Planimeter

Daniel

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### DCockeySenior Member

For a smooth curve with an odd number of points (even number of intervals) Simpson's rule is more accurate than the Trapezoid rule. Simpson's rule is equivalent to fitting a second order curve through each set of three points.

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### MikeJohnsSenior Member

That sounds like a mathematician talking Trapezoidal will be perfectly accurate for this application, GZ curves are quite 'steady' as far as functions go and easily approximated by a constant slope over the sample interval. We don't want lots of decimal place anyway, that's a mistake often made in applied sciences; calculating something a little nebulous to far to many decimal places and imagining it's useful.

Even the Riemann Sums I described above ( add the Y values and multiply by the increment) are accurate enough for design work and quite valid if the data point are not too far apart. They do have the advantage of being very simple and can be performed manually quite quickly.
If you calculate the Reimann sum for the mid point of the value within the range then you have implemented a trapezoid anyway.

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### TeddyDiverGollywobbler

Just last week was rereading Skene (well pendling to work and few evenings away from home..)
anyway.. Trapezoidal rule neglects the curved part apart of the line btw the points. So depends a bit on the characteristics of the curve (ie the boat in question) and the amount of data points which is the "right" method..

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### MikeJohnsSenior Member

Just look up the error formula for the different methods and it becomes abundantly clear.

If you have a table of data from the PC with every 5 degrees then even Riemann will be accurate. Just sum and divide.

If you plotted (like skene) by hand than you have a coarse data set and it would be more prudent to use simpsons for sure, but the accuracy is poor to start with and the real GZ curve will be a different shape to the Simpsons parabola approximation.

If you really wanted to get carried away you could curve fit between 0 and 180 degrees with a 6th order polynomial and integrate it between any limits you wanted.

In reality stability curves are not that precise to start with, boat design for example half tanks and no free surface allowance! To worry about a few percent error of the area under an interpolated curve is losing sight of reality. IMHO

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### DCockeySenior Member

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### MikeJohnsSenior Member

DC
The 'formula' is not the Trapezoidal. I didn't make it very clear, sorry. If you read my post 11 above it should be a little more illuminating than my first post.

cheers

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