Hull design

It's not a simple matter to choose the right conceptual procedure to obtain a hull shape that work well. 
A relevant point to consider is linked to the particular conformation of catamaran hulls, this type of hulls are very thin so the most part of drag that they creates in the water is given by the total wetted surface of the portion under the water of the  hulls and not, how instead is in monohulls, given by induction of hydrodinamic forces like creation of primary and secondary waves. However the major dvantage of a multihull is its intrinsic stability and thanks to this its light weight( no lead in the keel!), a multihull displacing the same weight of a same length monohull is not very much faster. The best shape for the section of this type of hulls is that link the best volume with the less wetted surface, by this way we willl obtain a U shape in wich the under water portion of the hull is a emi-circle.
However a good starting point is quite obiouvsly the length value of the boat: 7.62m.
We have to consider how many weigth the hull will carry, but we won't know exactly this weigth until we draw and then calculate the weigth upon a mathematical basis. Let's assume the whole boat weigth all up 300 Kg plus the crew weigth 180 Kg, it makes 480 Kg. We have to consider that we are talking of a catamaran, a type of multihull that spends much time on the leeward hull, so even only one hull has to float at least for this total weigth.
Now we have the whole weigth, hull length and the transverse shape of the ideal hull, is this sufficient to start the hull shaping? No, We have to know how distribute the volume longitudinally to the hull.
In a static world we should only find the center of gravity (CG) of the boat plus the crew, then distributes the volume of the hull around it. In the real world there are many forces that acts together un the boat: we are talking of  moments of inertia, sail (aerodynamic) forces, sea waves ecc....
I have found two ways to evaluate how and where the volume has to be positioned along the two hulls (note that i am talking about two hulls).
First approach: i could start considering only the pitching momet generated by the sail plus the anti-pitching moment component of the crew around a variable point of displacement on the hull. I shouldn't have to consider the lateral forces on the boat as the heeling moment of the sail plus that deriving from the daggerboard is costrained to be, at its maximum point, equal to the crew plus an emiboat weight, and so it is fixed even becouse when the heeling moment became to large on sail boat the helmsman is costrained to ease the tension on the sail.
It is easy to play a bit in order to find the point on the boat that puts the pitching and anti-pitching moments equal to zero, always remeber that the crew moment and the sail forward moment are fixed to certain values the only thing that change is the position of the center of effort of the hull volume. Remember also that we are talking of static forces, we will consider only in an empirical way dinamic forces later.

We are considering the sail's center of effort as it was positioned above the center of flotation, this is a semplification becouse the more it is far from F the more is hard to pitch the boat, it's a conservative semplification becouse we are considering by this way a greater than real pitching force, becouse it is above 130cm behind F. Limitations of this approach are due to the fact that we are not considering the hull volume and the hull center of flotation variation during pitching. This is consistent only for small changes of pitch angle ( a sort of metacentric approach), however with the rising of pitch angle the volume decreases but the lever rises, it would be very important the the product D*F remain equal in order to fulfil the equation. This is not always true, as is true that the bows can go down ito the water. A modern point of view to this problem is that is better to let the prow goes under the water at full speed (with a little resistance thanks to narrow prows) than make a thicker prow that doesn't allow pitch at expense of the drag ( this would increase resistance and by lowering abruptly speed will raise the pitchpoling moment).
But above all the biggest approximation is given from the fact that we are considering only the sail drive force as pitching force as if the boat was steady, stopped in a fixed point above the water. This for sure is not true as the drag of the hulls and appendages are not so great. The rotating couple (moment) in the real world is given by two opposites forces the first and the bigger is the sail drive, the second and the minor is given by the drag of the boat running in the water. The less is this resistance the less will be the pitching moment, it could be possible to say that if the drag tends to zero then the pitching moment tends to zero, while the more is the drag the more will be the pitcing moment, agin here it should be possible to say that if the boat is costrained to fixity then the pitching moment will be at its maximum (we are here not considering the inertia that the rig acquires during movement, if the boat is istantaneously stopped there will not be any thing that can hold the boat upright as the pitching moment becomes infinite).
Here the question is: are we able to determine the total drag of hulls plus appendages under various leeway angles and various speed and various sea condition?
Huuuummmmm........ the thing seems to become more and more difficoult.
First tip: take a look at this JAVA calculator. It calculates the drag of a hull given the length, width, prismatic coefficient for the fore and aft and given the displacement. It gives to me something like 75 Kg drag force for one hull displacing 480Kg going at 18Knots. It seems to me that it takes no count of the drag created by the appendages.

It should be quite easy to find the drag of a wing going at certain speed in the water with a certain angle.Please tell me if i go wrong: let's suppose we have a lift coefficient to drag table for a certain section ( i have one from Tom Speer webpages, this one is a asimmetrical section used for hydrofoils), we know from another table that an angle of 3 gives us a lift coefficient of 0.61 so we will choose the lift coefficient value of 0.61 (this means a lift for our foil that acts counter the sail's side force or sail's heeling force, the maximum side force the crew can handle should be  about 300Kg(link to a sail module of a VPP program)) and then watch what drag we obtain from the curve. Then we make an equation, we put 0.6 as our heeling force and make all equal to the drag coefficient  we obtain.
By this way we obtain a resistance value of 6.4 Kg for an "ideal" appendage. It seem very few to me, maybe (for sure i am gone wrong somewhere) the correct value is bigger, but not too much to me. Now we "have" the hull total drag and the appendage drag at 3 leeway, if we sum them we have the total drag of one hull displacing 450Kg and going at about 25Kn with a leeway of 3, all this calculations are well aproximated, but the point is that they shouldn't be very far from reality. However the total drag should be about 140-150Kg.
A more accurate method uses a formula that recently i have red. Di=(L^2/(3.14*e*AR*q)).Di stands for induced drag, L for lift, e is a factor that take count of taper of the planform usually is <=1, AR is the effective aspect ratio that is 2*Height^2/Area, q is dynamic pressure in water is 104.61 Kg m^-1 S^2.
Now L is always 300Kg, AR is 7,11, e is 0,9. These values give me 42.81Kg of induced drag for an appendage going at 18 Kn.
I have a table that compares two hull's total drag resistance (one with normal daggerboards the other with hydrofoils) counter the froude nummber of the two hulls (speed). I am not so sure that the table refers to  C Class cats. Well, the Froude number is equal to V/g*SQR Lwl, where g is 9,81 V is the speed (m/s). It is used this number to consider an adimensional value that can be pplied to boat different in length and by this way in speed.
Values upper than 1 are  easily in the speed range of our catamaran, as you can see
Values are not far away from those one we have "calculated".
What this hull drag value add to our calculations?
I believe that what makes the boat pitching is not the sail drive at its height, as it throw the boat forward if the hull would have zero resistance, i believe that we have to use only the hull drag value to evaluate the pitching moment as the value that came from drive force minus drag force really (to me ) throw the boat. Think at a pencil that you hold between two fingertips, make it moves in the air, yes it can pitch, but only for inertia that is neglegible, now make the pencil's tip touch a table, now it pitches, and the more it touchs the table(more drag) the more it pitches. By this way if we change SD with the total drag (hull drag+daggerboard drag= 75+42.8=117.8Kg) in our formula , we obtain a  D equal to 1.2m (rember that D stands for the Distance beetween the crew weight and the hull center of flotation(displacement)). This value to me is quite probable for a cat going upwind in 30 Kn apparent wind.
Should really our center of flotation be put sligtly forward the main crossbeams? 
However consider that if the hull could have zero drag, the hull center of displacement should be exactly under the center of gravity of the boat plus the crew. If the drag would be infinite the hull center of displacement should be where we have calculated with the first cheap approach.
Finally this hull center of displacement should be kept between this two extremes, maybe a little bit forward from the median point in order to compensate the inertia of the mast when the prow strikes a big wave
But Are we very sure that this is all we need to know to draw a good hull?
Ever red about Prismatic Coefficient? (Cp)
Cp is a value easily obtained that tells us how all that volume that we are estimated to be at 1.4m ahead the boat center of gravity is distribuited along the hull.

Practically a value of 1 rapresents a half cylinder, while a value of 0.5 represents the better penetrating shape for a certain volume. Ok , let's use 0.5! it seems easy said. It is not so simple. We have to evaluate the type of wave that an hull shape like this creates, becouse given a generated type of wave, as our cat is faster than this wave, and as it implies a graet amount of energy to overcome, we have to choose for sure a good penetrating shape, but also a shape that creates the longer wave that is possible to obtain, as the longer is the wave , the faster the hull can go without the problem of overcome  this wave.
However is very common to read that for fast and slender hulls like catamaran ones are very good Cp values of  0.7-0.8. The problem is that if you are looking for high Cp, soon will find fat prow and bottom sections, that are not so penetrating like we are looking for. Practically i have found that 0.68 is the best i can obtain in terms of penetrating sections and high Cp values. Consider that high Cp implies a lot of volume in the prow and in the bottom of our hull. If in the prow a lot of volume is wellcome (but it gives a very fat section) to avoid the prow becomes a submarine, is true that volume in the bottom is not wanted, as it tends to create static and dynamic forces that rises the bottom (lowering the prow)  when going fast. (very dangerous)
An interesting consideration is that in this type of little catamaran the crew position is a very determinant factor of how our hull stay in the water. So in light air the crew moves forward making a lower Cp, while in strong winds the crew can move aft (this make the  aft big sections goes underwater) rising the Cp.
However i have already said that 0.68 is a good compromise, not for hard types of calculation but simple becouse it gives  good sighgt of the hull's shape.
What does it rest for completing the hull design page?
Let me think a bit......
We have to talk a bit of the  properties of our hull when steering. It's easy to understand that when steering the more extreme sections of our hull tends to oppose to the movement. We have to avoid to put unnecessary skin areas especially very forward and after. It should be a compromise to have a beautifoul knife like forward section with a well rounded profile.
We will talk about this later.

Here you are some screenshots of the hull we are talking of, you will find also the hydrostatic properties of this hull. They are obtained for the two waterlines showed, more exactly for one hull displacing 470Kg(maybe a bit more than necesssary) and for one hull displacing 470/2Kg (practically two hulls on the water) The most interesting values are: sink, Water Line Beam, Draft (total immersion), Displ't ( Kg carried by the hull), Ctr. Buoy. X center of displacement, Wetd. Area wetted area, Prismatic coefficient, and finally easily obtained Length to Beam ratio See it at the end of the page

Future proposal:
I am going to apply this method for an A Class cat, when i will have the results, i will compare it to my A class Bim, this is a very concrete comparison to see if the thoughts are correct.
When i will see things going right, i intend to build 2 1/5 scale models with 4 different hulls. i would like to see them move on the water linked to my motor boat under a camera recorder  on the boat  and try to estrapolates something.
I have ,looked for and founded the address of the most important italian c class cat designer Giorgio Bergamini. I am preparing a large!?! document with all i have done here to see if him is disposed to give me a little bit of his experience of early '80 Little America's Cup campaign designs.

Click on the image below to see a larger rendering


Tell me what you are thinking of all this.

below the two hydrostatic output that multisurf gives

One hull displpacing the whole weight
MultiSurf Hydrostatics   Model file: marconuovo  27-Feb-2000  11:53:54
 Sta.     X        Area      Y-ctr.    Z-ctr.   Girth     Width
         m         m^2       m         m         m         m 
    1    0.050      0.00     0.000     0.000    0.378    0.000
    2    0.428      0.01     1.879    -0.094    0.451    0.084
    3    0.806      0.03     1.879    -0.102    0.524    0.148
    4    1.184      0.04     1.879    -0.109    0.587    0.202
    5    1.562      0.05     1.879    -0.114    0.638    0.248
    6    1.940      0.06     1.879    -0.118    0.678    0.285
    7    2.318      0.07     1.879    -0.120    0.708    0.318
    8    2.696      0.08     1.879    -0.121    0.732    0.348
    9    3.074      0.08     1.879    -0.122    0.750    0.372
   10    3.452      0.09     1.879    -0.121    0.762    0.394
   11    3.830      0.09     1.879    -0.119    0.768    0.412
   12    4.208      0.09     1.879    -0.116    0.766    0.426
   13    4.586      0.09     1.879    -0.112    0.758    0.434
   14    4.964      0.09     1.879    -0.107    0.742    0.437
   15    5.342      0.08     1.879    -0.101    0.717    0.434
   16    5.720      0.07     1.879    -0.094    0.684    0.426
   17    6.098      0.06     1.879    -0.085    0.641    0.409
   18    6.476      0.05     1.879    -0.076    0.587    0.385
   19    6.854      0.04     1.879    -0.066    0.523    0.353
   20    7.232      0.03     1.879    -0.055    0.448    0.314
   21    7.610      0.02     1.879    -0.043    0.362    0.264

MultiSurf Hydrostatics   Model file: marconuovo  27-Feb-2000  11:53:54
 21 stations,  1692 points
Sink            0.110 m             Spec. Wt.        1026.0 kg/m^3 
Trim, deg.       0.00               Z c.g.            0.000 m 
Heel, deg.       0.00
W.L. Length     7.560 m             W.L. Beam         0.440 m 
W.L. Fwd. X     0.050 m             Draft             0.283 m 
W.L. Aft  X     7.610 m 
Volume         0.4681 m^3           Ctr.Buoy. X       4.029 m 
Displ't.        480.3 kg            Ctr.Buoy. Y       1.879 m 
LCB (% w.l.)     52.6               Ctr.Buoy. Z      -0.106 m 
W.P. Area        2.48 m^2           Ctr.Flotn. X      4.338 m 
LCF (% w.l.)     56.7
      Wetted Surface
Wetd.Area        4.85 m^2           Ctr. W.S. X       3.831 m 
Ctr. W.S. Z    -0.146 m 
Waterplane      0.746
Prismatic       0.680
Block           0.497
Midsection      0.731
Disp/length      31.0

one hull displacing half weight

MultiSurf Hydrostatics   Model file: marconuovo  27-Feb-2000  11:53:02
 Sta.     X        Area      Y-ctr.    Z-ctr.   Girth     Width
         m         m^2       m         m         m         m 
    1    0.050      0.00     0.000     0.000    0.178    0.000
    2    0.428      0.01     1.879    -0.048    0.251    0.079
    3    0.806      0.01     1.879    -0.056    0.324    0.142
    4    1.184      0.02     1.879    -0.063    0.387    0.192
    5    1.562      0.03     1.879    -0.068    0.437    0.234
    6    1.940      0.04     1.879    -0.072    0.477    0.270
    7    2.318      0.04     1.879    -0.075    0.507    0.297
    8    2.696      0.04     1.879    -0.076    0.530    0.323
    9    3.074      0.05     1.879    -0.077    0.548    0.346
   10    3.452      0.05     1.879    -0.076    0.560    0.366
   11    3.830      0.05     1.879    -0.074    0.566    0.382
   12    4.208      0.05     1.879    -0.071    0.563    0.393
   13    4.586      0.05     1.879    -0.067    0.555    0.399
   14    4.964      0.04     1.879    -0.062    0.539    0.401
   15    5.342      0.04     1.879    -0.056    0.513    0.394
   16    5.720      0.03     1.879    -0.048    0.478    0.380
   17    6.098      0.03     1.879    -0.040    0.434    0.359
   18    6.476      0.02     1.879    -0.031    0.377    0.326
   19    6.854      0.01     1.879    -0.021    0.307    0.277
   20    7.232      0.00     1.879    -0.011    0.215    0.205
   21    7.610      0.00     0.000     0.000    0.000    0.000

MultiSurf Hydrostatics   Model file: marconuovo  27-Feb-2000  11:53:02
 21 stations,  1692 points
Sink            0.010 m             Spec. Wt.        1026.0 kg/m^3 
Trim, deg.       0.00               Z c.g.            0.000 m 
Heel, deg.       0.00
W.L. Length     7.560 m             W.L. Beam         0.401 m 
W.L. Fwd. X     0.050 m             Draft             0.183 m 
W.L. Aft  X     7.610 m 
Volume        0.23129 m^3           Ctr.Buoy. X       3.760 m
Displ't.       237.30 kg            Ctr.Buoy. Y       1.879 m 
LCB (% w.l.)     49.1               Ctr.Buoy. Z      -0.065 m 
W.P. Area        2.18 m^2           Ctr.Flotn. X      4.172 m 
LCF (% w.l.)     54.5
      Wetted Surface
Wetd.Area        3.27 m^2           Ctr. W.S. X       3.771 m 
Ctr. W.S. Z    -0.091 m 
Waterplane      0.719
Prismatic       0.605
Block           0.417
Midsection      0.689
Disp/length      15.3