# geometry of sailing

Discussion in 'Sailboats' started by Guest, Oct 8, 2002.

1. ### GuestGuest

Dear All,

I've been doing some general VPP study and come up with several questions regarding 'basic' geometric relations between various forces in steady state sailing.
Does anyone know exact functional relationship between these variables:

Vaw-component of relative wind speed in plane PERPENDICULAR to mast

Baw- relative wind angle in plane perpendicular to mast

Btw-true wind angle - horizontal plane

Vtw-true wind speed - horizontal plane

L - leeway angle

H - heel angle

in the form: Vaw = f( Btw, Vtw, L, H )
Baw = F( Btw, Vtw, L, H ) ?

Kind regards,

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### Steve HollisterJunior Member

Perhaps I don't understand, but there isn't any "exact" fuctional relationship between those variables. If you want to be exact, then you have to account for the variation in wind speed (Vtw) as a function of distance up from the water's surface (among other things). The VPP model takes this into account - sort of, but not for these variables.

That doesn't mean that these variables don't relate to each other. I'm just not sure what you are trying to calculate and I don't know what you mean by "perpendicular to the mast". Do you mean perpendicular to the mast-boom plane or the centerline plane of the boat? This is a complicated relationship, but you can set up something called a vector diagram to show the relationships.

This is what the VPP does with the forces on the boat. However, the VPP assumes that all of the forces pass through a single point so that there are no moments that have to be balanced. This means that the VPP model can't tell you where to put the mast to balance the helm.

Also, the relationship between L and H is not exactly known because it is a complicated relationship between the geometry of the sails and the shape of the hull and keel. In fact, the VPP model doesn't even try to calculate L.

Steve

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### tspeerSenior Member

You can find the equations for a bare-bones VPP at
http://www.tspeer.com/landyachts/performance/Single.html.

With only ASCII text, this is going to be a bit rough (and long), but here goes. You may have to copy this into a text editor to reformat it, putting back the spaces that the BBS strips out. If you're allergic to math, turn back now!

First, let's tackle the problem of determining the apparent wind vector in the horizontal plane. Define the true wind vector as the direction the wind is FROM, so that the wind vector points into the wind like a windvane. Let gamma be the angle between the true wind and the boat's velocity, positive when the boat is on port tack. gamma is the boat's point of sail; gamma=+-90 means the boat is on a beam reach, etc. Let Vb represent the speed of the boat.

Vectorially, the apparent wind is the true wind vector plus the boat velocity vector. Let beta be the angle between the apparent wind and the boat velocity (NOT the boat's centerline - that comes later), positive on the port tack. The true wind, boat velocity, and apparent wind form a triangle and we can use trigonometry's Law of Sines to form the following relationships:

Vaw/sin(gamma) = Vb/sin(gamma-beta) = Vtw/sin(beta)

In a VPP, you typically set Vtw and gamma. You can then iterate on Vb and calculate the other quantities until you get everything balanced. But I strongly recommend you iterate on beta instead. So assume Vtw, gamma and beta are known, and you want to know Vb and Va.

Vb = Vtw*sin(gamma-beta)/sin(beta)
Vaw = Vtw*sin(gamma)/sin(beta)
Vmg = Vb*cos(gamma)

I threw in the upwind velocity-made-good, Vmg, because you'll want it eventually. Note that the smaller beta is, the faster the boat goes. Closewindedness is close to Godliness!

Let L be the leeway angle, measured between the boat's centerline and the velocity vector, positive when drifting to starboard. Leeway points the bow more toward the wind. Let betaX be the angle between the true wind and the boat's centerline, and gammaX the angle between the true wind and the boat's centerline.

betaX = beta - L
gammaX = gamma - L

Ta da! First question answered - you now know the apparent wind and its orientation relative to the boat in the horizontal plane, given the true wind, point of sail, leeway, and beta. On to the second quesion: resolving the apparent wind relative to the mast.

To resolve the apparent wind velocity vectors and calculate the angle of attack of the sail, you have to define as many coordinate systems and rotation angles as you need. The trick is to be precise with your definitions and consistent in the transformations. Start with a local level earth coordinate system, with the X axis pointing North, the Y axis pointing East, and the Z axis pointing down toward the center of the (spherical) earth. This is the NED coordinate system. Let Psiw be the angle of the true wind with respect to North and Psia be the angle of the apparent wind with respect to North.

The components of the apparent wind in the NED coordinate system are:

[ VaxNED ] [ Va*cos(Psia) ]
[ VayNED ] = [ Va*sin(Psia) ]
[ VazNED ] [ 0 ]

Next define an axis system fixed to the boat. Naval architects typically use an axis system oriented forward-port-up, centered at the center fo gravity, for motion studies and aft-starboard-up, centered at the bow, for design. I come from an aeronautical background, so I'm going to pick the axis system I'm most comfortable with, which has the X axis forward, Y axis to starboard, and Z axis down, centered at the center of gravity (or amidships for a VPP). This will irritate the NA's no end, but it's a "bows right, noses left" thing.

Now imagine you have a toy boat that you're going to put through a series of rotations until it matches the real boat's orientation. Start with the toy boat level and pointing North, so the NED and boat axis systems are aligned together.

Rotate the toy boat about the Z axis through the angle Psi until it matches the heading of the real boat. This is axis system 1, and components in the NED system can be transformed into system 1 using the following direction cosine matrix:

[ X1 ] [ cos(Psi) sin(Psi) 0 ] [ XNED ]
[ Y1 ] = [-sin(Psi) cos(Psi) 0 ]*[ YNED ]
[ Z1 ] [ 0 0 1 ] [ ZNED ]

[ Vax1 ] [ cos(Psi) sin(Psi) 0 ] [ VaxNED ]
[ Vay1 ] = [-sin(Psi) cos(Psi) 0 ]*[ VayNED ]
[ Vaz1 ] [ 0 0 1 ] [ VazNED ]

[ Vax1 ] [ Va*cos(Psia)*cos(Psi) + Va*sin(Psia)*sin(Psi) ]
[ Vay1 ] = [-Va*cos(Psia)*sin(Psi) + Va*sin(Psia)*cos(Psi) ]
[ Vaz1 ] [ 0 ]

[ Vax1 ] [ Va*cos(Psia-Psi) ]
[ Vay1 ] = [ Va*sin(Psia-Psi) ]
[ Vaz1 ] [ 0 ]

[ Vax1 ] [ Va*cos(-betaX) ]
[ Vay1 ] = [ Va*sin(-betaX) ]
[ Vaz1 ] [ 0 ]

Next, rotate the toy boat about the Y1 axis, through the angle Theta (positive bow up), until it matches the pitch (trim) angle of the real boat, and call this axis system 2. The apparent wind can be transformed into system 2 by:

[ Vax2 ] [ cos(Theta) 0 -sin(Theta) ] [ Vax1 ]
[ Vay2 ] = [ 0 1 0 ]*[ Vay1 ]
[ Vaz2 ] [ sin(Theta) 0 cos(Theta) ] [ Vaz1 ]

[ Vax2 ] [ Vax1*cos(Theta) ]
[ Vay2 ] = [ Vay1 ]
[ Vaz2 ] [ Vax1*sin(Theta) ]

Then rotate the toy boat about the X2 axis through the roll angle, Phi (positive to starboard), until it matches the orientation of the real boat. Phi may or may not be the same as your heel angle, H, depending on how you define H. If H rotates about a horizontal axis, Phi and H will only be the same if Theta is zero. Which may be the case, since most VPP's don't calculate the pitch trim. The apparent wind transforms into the boat axis system by:

[ Vaxb ] [ 1 0 0 ] [ Vax2 ]
[ Vayb ] = [ 0 cos(Phi) sin(Phi) ]*[ Vay2 ]
[ Vazb ] [ 0 -sin(Phi) cos(Phi) ] [ Vaz2 ]

[ Vaxb ] [ Vax2 ]
[ Vayb ] = [ Vay2*cos(Phi) + Vaz2*sin(Phi) ]
[ Vazb ] [-Vay2*sin(Phi) + Vaz2*cos(Phi) ]

Now we have the apparent wind over the deck, but what we'd really like to know is the apparent wind relative to the plane of the sail (defined by tack, clew, and head). First, allow for canting of the mast to the side, with positive cant allowing the tip to move to starboard:

[ Vaxc ] [ 1 0 0 ] [ Vaxb ]
[ Vayc ] = [ 0 cos(cant) sin(cant) ]*[ Vayb ]
[ Vazc ] [ 0 -sin(cant) cos(cant) ] [ Vazb ]

[ Vaxc ] [ Vaxb ]
[ Vayc ] = [ Vay2*cos(cant) + Vaz2*sin(cant) ]
[ Vazc ] [-Vay2*sin(cant) + Vaz2*cos(cant) ]

Next rake the mast, with positive rake meaning the tip moves aft, completing the transformation to the mast axis system (for a non-rotating mast):

[ Vaxm ] [ cos(rake) 0 -sin(rake) ] [ Vaxc ]
[ Vaym ] = [ 0 1 0 ]*[ Vayc ]
[ Vazm ] [ sin(rake) 0 cos(rake) ] [ Vazc ]

[ Vaxm ] [ Vaxc*cos(rake) - Vazc*sin(rake) ]
[ Vaym ] = [ Vayc ]
[ Vazm ] [ Vaxc*sin(rake) + Vazc*cos(rake) ]

You now have the apparent wind components in the mast axis system. Vaxr is the component perpendicular to the mast, pointed forward. Vayr is the component perpendicular to the mast pointed to starboard. Vazr is the component that runs down the mast toward the step. But what you really need to know are the components in the plane of the sail.

Let delta be the angle through which the clew rotates about the mast, with positive delta meaning the clew moves to starboard. Rotating the apparent wind into the sail axis system:

[ Vaxs ] [ cos(delta) -sin(delta) 0 ] [ Vaxm ]
[ Vays ] = [ sin(delta) cos(delta) 0 ]*[ Vaym ]
[ Vazs ] [ 0 0 1 ] [ Vazm ]

[ Vaxs ] [ Vaxm*cos(delta) - Vaym*sin(delta) ]
[ Vays ] = [ Vaxm*sin(delta) + Vazm*cos(delta) ]
[ Vazs ] [ Vazm ]

Now that you know the apparent wind components in the sail coordinate system, the angle of attack, alpha, becomes:

alpha = - arctan(Vays/Vaxs)

The minus sign is because I've defined the angle of attack to be positive when the boat is on the port tack, to be consistent with the other definitions. The dynamic pressure, qbar, on the sail is effectively diminished by the vertical component of the apparent wind (rho is the air density):

qbar = 1/2 * rho * (Vaxs^2 + Vays^2)

Now that you have the angle of attack, you can get the aerodynamic coefficients of the sail, and these are multiplied by qbar and the sail area to get the force on the sail.

Be careful about the definition of the aerodynamic coefficients, however. Drag is always defined as being parallel to the apparent wind vector, and lift is perpendicular to the apparent wind vector - not perpendicular to the sail. The normal force coefficient IS perpendicular to the sail, meaning along the sail's Ys axis, as we've defined things here. The axial force coefficient will be along the sail's Xs axis.

You don't have to keep track of the apparent wind in all these intermediate coordinate systems. You can premultiply each matrix by the next transformation matrix to form one 3x3 matrix that transforms all the way from start to finish at once. So letting T21 be the transformation matrix that goes from axis system 1 to axis system 2,etc:

[ cos(Theta) 0 -sin(Theta) ]
T21 = [ 0 1 0 ]
[ sin(Theta) 0 cos(Theta) ]

[ Vax2 ] [ Vax1 ]
[ Vay2 ] = T21*[ Vay1 ]
[ Vaz2 ] [ Vaz1 ]

The transformation matrix that goes all the way from the horizontal plane coordinates of system 1 to the sail coordinate system is:

Ts1 = Tsm*Tmc*Tcb*Tb2*T21
Tsb = Tsm*Tmc*Tcb

Once you have the aerodynamic forces in the sail coordinate system, you transform them back to the horizontal coordinate system by simply transposing the elements of the transformation matrix:

T1s = transpose( Ts1 )
[ Fx1 ] [ Fxs ]
[ Fy1 ] = T1s*[ Fys ]
[ Fz1 ] [ Fzs ]

or, if you want to sum forces in the boat axis system:
Tbs = transpose( Tsb )

[ Fxb ] [ Fxs ]
[ Fyb ] = Tbs*[ Fys ]
[ Fzb ] [ Fzs ]

There you have it. The exact relationship between true wind and the apparent wind at the mast & sail.

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### tspeerSenior Member

Well, that depends on the VPP. The simplest VPP doesn't include moments or heel, just the lift/drag relationships of sail and hull - call this a 2 degree-of-freedom VPP because it has horizontal forces only. The next level of complication would be to add the rolling moment, making it a 3DOF VPP. After that, you might include pitch trim and sinkage for a 5DOF VPP, and finally yawing moments and rudder trim for a full 6DOF VPP. It all depends on how detailed your force and moment models are and the objectives for the analysis.

You get leeway angle by working the keel lift curve backward. The side force on the hull is set to match the side force applied by the rig. Then you go into the sideforce vs leeway relationship to get the leeway angle.

If your force and moment model works in the boat's body axis system, then calculating leeway angle is essential. If the forces are in the boat's velocity coordinate system, then you don't need leeway. Likewise, you don't need angle of attack of the sail unless you are working with the boat axis system. You can do basic performance work with just lift vs drag.

Cheers,

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Thank You guys, You've made my little grey cells very busy these days
I'll be back with some coments and further questions...

Best regards,

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### Steve HollisterJunior Member

When most people talk about the Velocity Prediction Program (VPP) in the marine field, they are referring to the performance model as described by US Sailing's implementation applied to the IMS rule. This performance model was originally developed at MIT in 1978 (Report 78-11 A Velocity Prediction Program for Ocean Racing Yachts Revised to June, 1978) and has been continually updated (see various Chesapeake Sailing Yacht Symposia papers). Larsson and Eliasson's book gives the closest published explanation of the model that I know of. Of course, this does not include the important Lines Processing Program (LPP) portion of the model which does appendage stripping and calculation of "LSM" (effective sailing length), keel span, and other basic hull dimensions. L&E also mention the use of "Reef"(R) and "Flat"(F) in the VPP model to change the sail forces for upwind sailing, but don't really explain the nonlinear optimization that must take place to maximize the velocity of the boat.

Since the original published work, many have attempted their own versions of a VPP-type program (some even for catamarans), trying to improve on the original assumptions. Many of these improvements (?) have been included in the official version of the VPP. Unfortunately, the only way that I know of to determine the current equations of the VPP is to purchase the code from US Sailing and try to decipher them by studying the code. (The LPP code is not available.) This is what I did when I wrote my own version of the VPP and tied it into my own version of the LPP. If there is now an official (current) document listing all of the equations used, I would like to know about it.

As I have mentioned in the past, there seems to be too high an expectation of the results from the standard VPP. Some even have put the polar speed values into the on-board instruments to use as target speeds. There are a lot of assumptions made in the standard VPP model and it is just not that accurate. You would be better off having instruments on board that keep track of previous best performances to compare with current performance.

TSpeer wrote:

"You get leeway angle by working the keel lift curve backward. The side force on the hull is set to match the side force applied by the rig. Then you go into the sideforce vs leeway relationship to get the leeway angle. "

This is obviously not an exact calculation. Hull leeway is dependent on the shape of the keel, the shape of the hull, the keel-hull interaction, the heel angle, the free surface waves that are generated. and more. Even the CFD RANS codes with free surface modeling struggle to get these answers, if at all.

You also have to keep in mind that a performance model is only as strong as its weakest link. Improving one part of the model isn't necessarily going to give you better answers. I usually tell people that the VPP is much better at evaluating changes in shape between boat A versus boat B, etc., than it is in giving exact performance numbers. However, systematic hull variation and analysis can be very enlightening.

Steve Hollister
New Wave Systems

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### tspeerSenior Member

The IMS VPP is a VPP but hardly the only VPP. I would say that the term "VPP" refers to a prediction of the steady-state performance of a sailing vehicle. The Pratt Project may have laid the groundwork for one of the first VPP's, but surely each design house has their own performance model that has been tweaked based on their own experience, analysis tools, and test data.

Fradrigo is clearly working on his own model, and he's looking for the constituent equations. These are a fairly straight-forward application of the basic principles of sailing theory.

One of my pet peeves about US Sailing is the technical data collected using OUR DUES is not disseminated, but closely held by US Sailing. When asked why, their reply was, "But then the designers would use it to make faster boats and the handicap ratings would not be as good. The handicappers have to stay ahead of the designers."

Sure - there are innacuracies in any VPP. They are only as good as the model data. But, no matter where you get the model data - whether it's CFD, handbook estimates, tow tank, or double-model wind tunnel test - you still end up with a curve of lift vs sideslip as a function of heel, Froude number, etc. And the way to reach closure in the VPP is to calculate the hull sideforce first, and then calculate the leeway. In the tank you do exactly the opposite - you set the leeway and measure the force.

This kind of modeling is routine in the aerospace industry. Nobody would think of trying to design an aircraft without having at least a good performance estimate, and any project bigger than a homebuilt will have a full 6 degree-of-freedom dynamic simulation as well.

I've been quite amazed at how little of this is done by yacht designers. As Lord Brabazon (British aviation pioneer) said, "Compared to designing a yacht, designing airplanes is child's play." I can't see how anyone makes rational design tradeoffs without some way of estimating their effects on performance, at least, if not seakeeping, too.

If the data aren't available for a VPP, then just what are the other design calculations based upon? It seems to me that if you've done your homework on engineering the design you'll have the makings for a VPP. If you haven't, then the "design" may be an interesting exercise in stylistics, but I'd wonder if it was an adequate basis for risking someone's investment or the lives of the crew.

Cheers,

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### Steve HollisterJunior Member

TSpeer wrote

"One of my pet peeves about US Sailing is the technical data collected using OUR DUES is not disseminated, but closely held by US Sailing. When asked why, their reply was, "But then the designers would use it to make faster boats and the handicap ratings would not be as good. The handicappers have to stay ahead of the designers.""

This is my biggest complaint. Since certain designers are involved with improving the rule, this information is not hidden from all designers. You just have to have the right connections. It is my contention, however, that all rules are development rules rather than handicapping rules. Given enough time, interest, and money, any rule can be beaten.

TSpeer wrote

"I've been quite amazed at how little of this is done by yacht designers. As Lord Brabazon (British aviation pioneer) said, "Compared to designing a yacht, designing airplanes is child's play." I can't see how anyone makes rational design tradeoffs without some way of estimating their effects on performance, at least, if not seakeeping, too. "

You don't make your living designing sailboats, do you? Not many can and I can assure you that for most, there is little in the budget for R&D. Sure, it's easy to buy and use a VPP program, but there is more to design than a VPP performance analysis.

TSpeer wrote

"If the data aren't available for a VPP, then just what are the other design calculations based upon? It seems to me that if you've done your homework on engineering the design you'll have the makings for a VPP. If you haven't, then the "design" may be an interesting exercise in stylistics, but I'd wonder if it was an adequate basis for risking someone's investment or the lives of the crew.'

Makes you wonder how they designed boats and ships in the old days, doesn't it? Airplanes too! You seem to dismiss history, common knowledge, and experience. I think you need to get off your soap box and chill.

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### giramontiJunior Member

Summary of VPP equations

There is a technial paper called "Predicting the Speed of Sailing Yachts", written by Peter van Oossanen and Associates. It's published in SNAME Transactions, Vol. 101, pp.337-397. It was published in 1993, but it is the most thorough coverage of VPP formulas I've seen since I began developing my own in-house VPP. Other sources I've referenced included L & E's Princicples of Yacht Design, which is a highly simplified method of predicting speed, but well organized. I used Claughton, Wellicome, & Shenoi's Sailing Yacht Design: Theory, which is somewhere in between L & E's and van Oossanen's perspectives. Marchaj also added some help with his study of aerodynamics of sails, which is highly lacking in the former three references.

Since I design very light displacement boats, usually around 80DLR, I added a planing resistance module to my VPP. I adapted planing powerboat speed prediction methods to my VPP, but I have no idea whether the results are even close to valid because I have no empirical data to support my calculations.

With all this said, knowing intimately the inner workings of a sophisticated VPP, I'm skeptical about the validity of VPP output. My VPP takes about 80 inputs. But only about a dozen of them are involved directly in the equations that determine the resistance of the yacht through the water. I can't believe that these parameters, the major ones being Lwl, Bwl, Disp, Cp, and Aws, can realistically define the drag on a yacht.

In other words, you can design a yacht with the perfect combination of these parameters, and it can still be slow because the shape of the hull just doesn't glide through the water like it should. I'm very wary of VPP's and though I tend to make sure I hit the numbers (sort of like a catholic getting baptised to make sure he goes to heaven), I'm more subjective when designing a hull. I always try to imagine the boat running through the water. I tried to visualize the flow of the water over the hull, to imagine the boat under sail, pounding in a seaway. I ask myself, what are the subjective qualities of the boat I'm designing? How will it feel to sail this boat? Can I make it go fast?

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