Prismatic Coeff?

[SeaDrive]

I've always been bothered by the rather ad hoc nature of the definition of Prismatic Coeffecient. It seems odd to me to use a volume ratio for what is effectively a measure of sailing length. Have the theorists out there done research on any alternates?

One possibility that seem logical to me are what a mathematician might call the moments of the distribution. The best known example would be the statisical concepts of mean, variance, skewness, etc.

Given an area distribution curve a and a total volume D (D = intregal of a), then the center of buoyancy (c) is analogous to the mean: integral of ax/D.

The second moment (e.g variance) is a measure of how spread out the distribution is . v = intregal of (x-c)**2. The square root (e.g. std deviation = s) is linear and should relate to sailing length. The ratio of s/lwl would replace Cp. It would take empirical evidence to determine if this value is more consistent from hull to hull than Cp.

The third moment, the integral of (x-c)**3 is a measure of symmetry. Currently the matter of whether (e.g.) the bow is too long for the stern is mostly a matter of judgement, as far as I know.

Any comments?

SeaDrive

p.s. sorry for the odd formulas, but it's not easy to do math in a forum like this.

[spoedvraat]

Originally Posted by SeaDrive

I've always been bothered by the rather ad hoc nature of the definition of Prismatic Coeffecient. It seems odd to me to use a volume ratio for what is effectively a measure of sailing length. Have the theorists out there done research on any alternates?

One possibility that seem logical to me are what a mathematician might call the moments of the distribution. The best known example would be the statisical concepts of mean, variance, skewness, etc.

Given an area distribution curve a and a total volume D (D = intregal of a), then the center of buoyancy (c) is analogous to the mean: integral of ax/D.

The second moment (e.g variance) is a measure of how spread out the distribution is . v = intregal of (x-c)**2. The square root (e.g. std deviation = s) is linear and should relate to sailing length. The ratio of s/lwl would replace Cp. It would take empirical evidence to determine if this value is more consistent from hull to hull than Cp.

The third moment, the integral of (x-c)**3 is a measure of symmetry. Currently the matter of whether (e.g.) the bow is too long for the stern is mostly a matter of judgement, as far as I know.

Any comments?

SeaDrive

p.s. sorry for the odd formulas, but it's not easy to do math in a forum like this.

Cp defines how "fine the ends of the boat are." This does not define how fine the bows are compared to the stern, or fineness of bow and stern absolutely.
If this is what you are referring to, then I agree, it is time for someone to explain to me how this works, 'cause I cannot figure out the sense in using a parameter which defines fineness of bows and stern, while they are obviously different and have a profound influence on the way the hull interacts with the water.
I know the curve of areas defines the volume of the extremities somewhat.

[sorenfdk]

Perhaps that is why someone invented the CPforward and CPaft i.e. the prismatic coefficient of the forward and aft part of the hull.
You have a point regarding using one parameter covering the whole hull, but in the real world it works OK. Take a look at the papers of Gerritsma et al., which formed the basis of most of the VPPs used today.
Regarding your statement that "...the curve of areas defines the volume of the extremities somewhat...", I'm a bit confused. As far as I know, the CoA defines the distribution of the volume quite precisely.

[SeaDrive]

You say the current definition "works OK", and in the context of comparing designs in one office, I'm sure it does. But, to me, it seems subject to a certain amount of "we do it this way."

One of the technical problems is that it depends so precisely on the max sectional area. A 2% error in the max section will result in a one point difference in the Cp, i.e. .54 instead of .55. I think errors of that magnitude must be common for designs drawn on paper. Indeed, errors of a similar magnitude must also arise from calculating displacement using Simpson's Rule.

I would not generalize about results from CAD systems, but having done a fair amount of computer programming, I am well aware of many possible sources of error.

SeaDrive

[sorenfdk]

You are quite right. That's why I said "...works OK." and not "...works to perfection."
But is it really that important to know the exact value of the Cp? A 2% difference can hardly be seen on the resistance curves.
Cp (and LCB and LCF etc.) does have some - not much - influence on the attainable speed, but other factors, which are beyond control and cannot - at least to my knowledge - be entered into a VPP, have much more influence. I'm of course thinking of things like wave spectra etc.

Søren Flening

[gonzo]

Prismatic coeficient is a very fast and fairly accurate way of calculating the fineness of a hull. However, it take experience to interpret the results.