Larsson & Similitude

[Stephen Ditmore]

The similitude table in Principles of Yacht Design by Larsson & Eliasson, first edition, Fig 2.1 (p. 12)(attributed to Barkla) caught my attention a while back. I thought this was a wonderful thing that I'd use at my first opportunity - but now that the time has come I think Larsson/Barkla have it wrong.

I'm scaling down a (historic -> replica) sailing ship, and it surprises me that Larsson has fairbody draft scaling with beam. It seems to me b/d is typically higher on small craft than on large, so while it makes sense to me that freeboard scales with beam, I'm thinking fairbody draft should scale (more or less) with length.

In the stability work I've done it's struck me that GM(req'd) varies very little, if at all, with size. A rather extensive discussion of stability formulae was posted earlier at:
http://www.boatdesign.net/forums/showthread.php?threadid=272

Since GM = BM + GB, the fact that GB will change as the vessel is scaled may mess this up some, but for the moment I'm going to assume that the center of gravity is close to the center of bouyancy, so that the change in GB can be ignored.

Since BM is proportional to I/Displ and "I" is proportional to LB^3, BM is proportional to (B^2)/d. I'm thinking that if d scales with L, then B ought to scale with the square root of the scale factor.

Let n=the scale factor. Since displacement = Cp*L*B*d, displacement would scale with Ln*dn*bn^0.5, therefore with n^2.5
(which is close to Larsson/Barkla's 2.4 exponent for scaling displacement).

Does this seem right? Are there other sources on similitude that would scale beam and fairbody draft separately, in a manner resembling what I'm suggesting?

So that this is not completely dry, here's an image of the historic H.M.S. Surprise, and a link to more information.

http://www.modelships.co.uk/Models/HMS_Surprise/hms_surprise.html

 

[Stephen Ditmore]

Well, I need an answer to proceed, so here's my next stab at answering the question for myself. Check me on this, folks.

The answer I gave above would be right if I wanted to maintain the same stability to displacement ratio, but more likely I'm going to have sail area scale with L^2. This means:

Heeling moment scales with L^3
Righting Moment scales with L^3
Displ scales with L*d*B^y (I'll solve for y)
BM scales with [(B^y)^2]/d = (B^2y)/d
BM*Displ scales with [(B^2y)/d]*L*d*B^y
" " " " B^2y * L * B^y

3y therefore = 2 and y = .67

That's pretty close to the .70 in Larsson, and after taking the GB issue into account .70 might be right. But it's only right if d scales with L, which makes the exponent for scaling displacement 2.7, not 2.4

Agree?

 

[HOWdie]

Stephen:

I've not been through all the math, however, The only way that the various weights could scale down would be if I you could magically make wood/steel/fiberglass weigh differently per cb. ft. in your revision as it does in the original. Therefore I'm quite sure that there will be some if not major differences in the various centers. I think you would be better off just doing the math on your model rather than relying on the given displacement of the original craft.
IMHO
Terry

 

[Stephen Ditmore]

Thanks for your response, but I must say I couldn't disagree more. Until I develop a preliminary design, there's no model to do calculations on, and when a client comes to me and says they want a boat like x, only smaller, I feel I need to understand the implications of scaling a vessel down and scale the proportions in a way that makes sense technically before I encourage to client pay thousands of dollars for me to go forward with a weight study. In this case, the client is going to be raising money to have this boat built before anything resembling a final weight study is complete. I'm not expecting to be locked into the preliminary design if I find, down the road, that adjustments need to be made, but I want the preliminary to be in the right ballpark, and an understanding of similitude (and Coast Guard passenger vessel regulations) is important to that enterprise.

It occurs to me to compare some of the Bruce Farr designs by Carrol Marine to see how various proportions seem to scale among that line, and I'll post my observations here when I have some.

 

[Stephen Ditmore]

So far this has been a monologue, except for HOWdie, who I disputed. So much for social graces...

Anyway, I compared the Corel 45 with the CM 60, which the Carroll Marine web site says has "similar general characteristics."
The exponent for beam was 0.604, and for displacement was 2.62. This tells me that fairbody draft scales pretty much with length, which is my basic contention, and the beam exponent is close enough to my 0.667 that I regard this information as supporting my conclusion.

One minor correction: when I wrote displacement = Cp*L*B*d it should have been displacement = Cb*L*B*d with Cb = block coefficient.

 

[Stephen Ditmore]

In what follows I've changed the variable "n" to "s" for "scale factor."

Here's my latest algebra:

Ls(Bs^x)^3 = C(Ls)^3

L = vessel length
B = vessel beam
s = scale factor
solve for x

I've omitted the constant, C, as being irrelevant to what I'm interested in, and assuming C=1 set
Ls(Bs^x)^3 = (Ls)^3
(Bs^x)^3 = (Ls)^2
(B^3)(s^3x) = (L^2)(s^2)
(s^3x)/(s^2) = (L^2)/(B^3)
s^(3x-2) = (L^2)(B^-3)

I'm going to omit some algebra involving natural logs (Ln) and skip to my mathmatician cousin's answer:

x = [Ln(L^2/B^3)]/3Ln(s) + 2/3

So what does this mean?
To be continued...

 

[Stephen Ditmore]

Request a favor

I'd like to request a favor of someone with a decent hydrostatics module and x,y,z scaling. Take any monohull boat. Note it's beam, its whetted surface, and its transverse moment of inertia: "I(t)". Now double it's size, then scale the beam until the ratio of "I(t)" to whetted surface is DOUBLE what it was for the smaller version. What was the original beam, and what is the new one? For example, if I(t)/whetted surface was 10 on the original boat, I'd like to know at what beam I(t)/whetted surface = 20 after the boat has been doubled in size.

It'd be nice to get this result for several different boats. This'll really help me out.

Thanks!

 

[Stephen Ditmore]

Oops, I made a logical mistake in my post of 3/14. Now that I've corrected it I again come up with 2/3 as my exponent for scaling beam. Here's the algebra.

Basis:
RM1 = b(Awp^2)/(10*L) + (GB*Disp/57.3)
(approximately correct for most common hullforms)
or
RM1 = CL(b^3) + (GB*Disp/57.3)
(exactly correct if form constant, C, is correct)

Variables:
RM1 = righting moment at 1 deg of heel
C is a constant incorporating a form coefficient, the density of water and unit equivalencies
L = waterline length in feet
b = waterline beam in feet
Awp = waterplane area in feet^2
Disp = weight of vessel, ½ load condition, in pounds
GB = vertical distance from the center of gravity to the center of buoyancy, in feet, ½ load condition. (NOTE: IF THE CENTER OF GRAVITY IS ABOVE THE CENTER OF BUOYANCY, THIS VALUE IS NEGATIVE)

Notice that the first term is based on the transverse moment of inertia of the waterplane. When under sail equilibrium is reached when heeling moment = righting moment (HM=RM). When a design is scaled a design to a different size, heeling moment scales with the cube of the scale factor while stability scales with the scale factor to the fourth power. To compensate, beam should be scaled with the scale factor to some exponent.

For the moment I'm making the following assumptions, which I may want to question later:

A. L and d will scale with the cube root of the heeling moment.

B. GB is zero both before and after scaling, so the second term (GB*Disp/57.3) can be disregarded.

Let S = the scale factor, defined as the cube root of the heeling moment.

RM1= CLB^3 = heeling moment @ 1 deg
Let Q be some constant.
CQLb^3 = L^3
CQ = L^2/b^3
CQ remains constant as the dimensions scale.
(L^2/b^3)LS(bS^x)^3 = (LS)^3
(L^2/b^3)(bS^x)^3 = (LS)^2
(L^2/b^3)(b^3)(S^3x) = (L^2)(S^2)
(L^2/b^3)(S^3x)/(S^2) = (L^2)/(b^3)
S^(3x-2) = 1
3x-2 = 0
3x = 2
x = 2/3

 

[Guest]

Hi, Stephen. Your monologue seems very interesting. I, too, am a professional designer who has engaged in the "battle of similitude". I have been buried in design work for two classic yacht reproductions recently and have not visited here in a while, so have missed out on all the great debate you have been having with yourself. I'll look & think a bit on what you've posted and comment when I'm up to speed on it. Nice to have a real meaty thread to chew on...

Michael Mason
Mason Marine Designs Inc.

 

[Stephen Ditmore]

Letter to Lars

Dear Prof. Larsson:

I've just submitted the attached file as an abstract to the Chesapeake Sailing Yacht Symposium <http://wseweb.ew.usna.edu/nahl/csys/default.htm>. Since it opens by taking issue with the first edition of Principles of Yacht Design, I'd like to invite you to respond (note: I have not yet seen the second edition). Beyond that, I would welcome a co-author, particularly one more facile with logarithms than I. If you know anyone interested in working with me on this, or any sources I should consult, please let me know.

Thank you. Fair winds.

Stephen Ditmore
New York

 

[tspeer]

Unfortunately, I didn't see the Barkla paper listed in the references. Barkla contributed a lot to AYRS publications, so maybe it's there somewhere.

Without having seen the source, my speculation is that the relationships in the table aren't derived from first principles, but are regression analyses to statistical data covering a large number of boats of all sizes. This is a very typical practice in conceptual design, not just of boats but of aircraft as well (see for example, Raymer, Daniel P., "Aircraft Design: A Conceptual Appproach", AIAA or http://www.steamradio.com/JSYD/Articles.html for similarity and statistics of sailing multihulls). The idea is get a starting point for iterating the design and to have some reasonable guesses for numbers you need in one calculation before you get them from other interrelated calculations.

If they are regression fits, then they might not be perfectly consistent when you pick some numbers and then try to fit the others into a given formula. And you'd be able to find any number of counter examples that could lead you to pick different values. Each category of boats would have a somewhat different set of relationships.

When it comes to similitude, you have to look at the objectives and then choose the parameters that best meet those objectives. For example, I was looking at how to scale hydrofoils to make a subscale test boat. Assuming that the span scaled with the length, the chord scaled with length and the speed scaled with Froude number if the objective was to evaluate performance. But if one were interested in cavitation, the chord scaled as length*scale factor and the subscale speed was the same as for the full scale boat.

I think a more general exploration of possible similitude rules, taking into account different design objectives and types of boats, would be a valuable CSYS paper. Ideally, these would include both physics-based and statistical relationships.

 

[Stephen Ditmore]

Thanks for the thoughtful resposnse, Tom. I'll have to think on it a little. Would you be interested in co-authoring the paper?

I wish to continue to pursue my own line of reasoning as it relates to monohull sailing yachts, and so far my results seem to be consistant with the comparison of the two Bruce Farr designs I cite, though it'd be useful to get feedback from someone who has been tracking the IMS fleet using IMS certificate information, or some comparable class of sailing vessels that span a range of sizes.

Having said that, I'm very open to working with someone who would like to look at ways to further generalize the study and bring in other types of data and analysis (and, as I've stated, can help me solve equations using logarithms and generally bring a strong math/physics background). Also, once I get to a certain point I might want to open the door a crack to looking at how sail force coefficients, wind gradient, etc. might effect the equations, though this is outside my area of expertise, which is stability.

So, whadayathink?
S

 

[Stephen Ditmore]

Here's the file that used to be attached to my post of 05-16-2002, with some minor edits. (Jeff, if you can move the attached to appear with that previous post, please do so and delete this post. Thanks.)

 

[tspeer]

Sorry, I'm going to be maxed out writing my own paper if accepted.

 

[Stephen Ditmore]

Tom: Perhaps you've said in another thread, but I missed it. Is your paper for the CSYS? May I ask on what topic?

Thanks for telling me Barkla's connected with AYRS. I've e-mailed them asking to be put in touch with him.

S