GZ curve calculation and fitting

Angle of Max GZ criterion is often more sensitive to the heel steps than it is to VCG so it is very important to have the same heel steps when comparing results from a Large Angle Stability analysis and a Limiting KG analysis.

It’s also important to understand the curve fitting that is done to smooth the GZ curve and how the angle of Max GZ is found. Piecewise cubic fitting is used which goes through exactly the computed GZ values at the specified heel angles. But because the curve fitting can get messed up by rapid changes of heel interval or if the heel interval is too high, there are certain conditions which must be met before the curve fitting is used:

a) All heel steps <= 10deg; and

b) No more than a factor of 3 or 1/3 between steps in the different heel angle step ranges

The curve fitting adds another 10 points between each pair of computed heel angles. And the Angle of Max GZ (and the Max GZ) are determined by looking at these interpolated points and the computed ones. So if you have a heel angle step of 1 deg; then the highest resolution for the angle of max GZ is 1/11 deg. (Roughly because it is a parametric curve and the interpolated points are equally-spaced parametrically which doesn’t necessarily map to exactly equal spacing in heel angle) You can double click on the graph and see the computed data and interpolated points:

This means that if the GZ curve is quite flat in the area of max GZ, then you can have some variation in VCG that still gives the same angle of max GZ

You can see this here:

(to get more decimal places in Maxsurf - it is a bit obscure but click Help|About menu with the shift key down and click [Yes] in the dialog)

Heel range intervals

So due to curve fitting to the individual heel angles at which the GZ curve is calculated some variation in the reported Angle of max GZ can be expected, especially if the GZ curve is quite flat over a broad range of heel angles near it maximum value. For example, below, the GZ curve is evaluated at 0.1 deg intervals to get an “exact” answer for my vessel which is 25.3deg. (This is accurate to approximately 0.01 deg due to the curve fitting and interpolation discussed in the previous section.) However it is not really practical to evaluate so many point exactly. Below can be seen the effect on the reported Angle of max GZ for different ranges of heel angles used. A variation of around 1.0 deg can be observed; this can have quite a significant impact on the maximum VCG computed by the Limiting KG analysis.