Sail design study

I am interested in learning more about the Mini landyachts discussed in this forum.

I usually approach design problems by putting numbers onto a spreadsheet. Each line drawn requires answers to why and where things go. The decisions become mathematically linked together and what results is a parametric model that begins to look like something, in this case a sail. I started with the design presented by BenBoulder in the topic, 4.6 Mini Sail.

Here are the inputs and outputs for my first attempt. Any input(s) may be changed and the rest will automatically recalculate. The mast bend is a circular arc and there is no allowance for draft. I hope to get a little more information before finalizing these.

Please have a look, could be worthwhile.

General

Luff panel

LE Circular arc. This shape also defines the Mast curve.

Battens

5 Battens on sail extending from the Leech TE to the Luff pocket TE
4 Battens oriented horizontally and distributed evenly on the vertical span limited by the Head and Foot TE on the Leech.
Batten 5 is oriented below horizontal joining the Head TE to the Batten 4 LE.

Aft panels

Area behind Luff panel divided into three triangles and one trapezoid (upper).
The peaks of the three radial panels are located along the Z dimension of the Leech (quarters).
The peaks of the three radial panels are located along the X dimension at the same fraction of their local chord behind the Sail LE.

Luff panel overlap

When laid flat the Luff panel TE overlaps the Radial panel LE allowing for sail draft.
There is no overlap at the intersections with the Head and Foot.
Overlap determined as a percentage of the average sail chord between the Head and Foot TEs, applied to the local chords at the three radial panel peaks.

Input

Mast

Origin (X) 2500.0 mm
Origin (Z) 0.0 mm
Diameter 50.0 mm
Rake 10.0 deg

Luff pocket

Origin (Z) 500.0 mm
Luff pocket flattened 100.0 mm
Ratio of full to flattened 0.67
TE end to end (chord of arc) 4650.0 mm
TE peak at chord center 185.0 mm

Aft panels LE

Location on local sail chord line 0.30

Luff panel TE overlap

Overlap at average sail chord between the Head and Foot TEs 0.0 mm
Note: Sail is currently flat

Exterior Layout

Foot length 1445.0 mm
Foot angle -10.0 deg
Leech 3945.0 mm
Head 440.0 mm

Output

Layout

Sail area 4.595 m^2
Centroid (X) 1315.5 mm
Centroid (Z) 2416.0 mm
Aspect Ratio 4.25
Leech angle 96.9 deg
Head angle 31.2 deg
Luff panel LE angle (end to end) 109.0 deg

Radial panel 1 (bottom)

Length along Foot 1062.9 mm
Length along LE 1185.7 mm
Length along top 1277.0 mm

Radial panel 2

Length along Radial panel 1 1277.0 mm
Length along LE 1008.6 mm
Length along top 2041.9 mm

Radial panel 3

Length along Radial panel 2 2041.9 mm
Length along LE 1023.0 mm
Length along top 2951.1 mm

Trapezoid panel 1 (top)

Length along Radial panel 3 2951.1 mm
Length along LE 1264.2 mm
Head 440.0 mm
Length along Leech 3945.0 mm
Diagonal-Head TE to Radial 3 Peak 1235.0 mm

Luff panel

Length on Foot 382.1 mm
Length Foot LE to Head LE (straight) 4650.0 mm
Length on Head 57.8 mm
Length on TE (straight) 4461.0 mm

Hi Your mast will be long but doable with an extension top and or bottom on a 460 windsurf mast
Your sail has a quite high aspect ratio
The centre of effort on your sail will be forward of the centroid A rough rule of thumb is one third up from the foot and one quarter back from the mast Accurate calculations for centre of effort are complicated
The trend with minis is to move the mast more forward than previously thought to be the optimum
An oval shaped mast step allows mast rake and centre of effort to be adjustable and helps to get the balance right

I think you are trying to make this just come down to numbers, keep in mind the mast does NOT have an even curve as it is a tapered tube and not of constant diameter , 2 piece masts do flex slightly differently than a one piece mast.
The Centre Of Effort is constantly changing (within a region of the sail) with wind direction, yacht speed, and sheeting tension.

Sail leading edge curve

I'm not sure how the sail leading edge is determined. I have read that there is a relationship between the
mast bend characteristics and the shape so here is what I found out about masts.

Mast Bend characteristics

Indexed mast check system (IMCS) measures mast deflection (cm) with 30 kg suspended from mid mast

IMCS Stiffness = Length^3 / (Mid Point Deflection x 216225)

Bend Curve % = (deflection at .75 / deflection at .50) - (deflection at .25 / deflection at .50)

The higher the number the more flexible the mast is in the upper half relative to the bottom.

One mast manufacturer claims that a 76 - 64 (12%) bend characteristic suits 80% of the sails on the market.

A survey of available masts from two other manufacturers shows that the Bottom ratio varies from 62.0 to
64.5%. On a 460 cm mast the difference amounts to 4.5 mm... not really noticeable on a mast that long.?

Deflections for a sample mast

Length 460 cm
IMCS 25
Bend curve 12 %
Bottom ratio 64 %
Top Ratio 76 %

@ .25 Length 11.5 cm
@ .50 Length 18.0 cm
@ .75 Length 13.7 cm

Graphs

The top graph shows the sample mast plotted with the leading edge curve from the sail presented by BenBoulder
in the topic, 4.6 Mini Sail. The curves don't match, perhaps the Sail LE needs to be cut flatter at the top to increase tension there.

The bottom graph shows the same sail plotted with a mast that has the curvatures flipped, a better fit.

Looking for some qualitative wisdom on how the mast and Sail LE curves should relate.

The mast calculation for Imcs I believe where they use the 30 kg weight at the centre and measure deflection
is a means of comparing one mast with another and enable a buyer of a sail to match sail to mast
It still doesn't give us the correct curve for the luff In my opinion as we don't know how much the mast bends
under actual sailing conditions This is where the skill and magic of the sailmaker comes in to play and how he curves the luff and draft of the sail to match the shape he wants.
The work you are doing here is great and maybe you can demystify some of this for us

do not forget that booms that dont attach to the mast and booms that do bend the mast in very different ways. with no boom attachment and sheeting aft of midway on the boom the luff tension is reduced and the battens are in compression when unsheeted and go into tension to flatten the sail. very different from the boom on mast way of sheeting. im rather pleased that my By eye an a cane stick that feels right method is such a nice fit

Sail LE Shape

Here is my attempt to come up with a repeatable process for defining the Sail LE shape based on a given mast. The Sail LE curvature is based on increments to the mast curvature and are arbitrary. Examples that I have seen are bent further than the IMCS rating. It seems to make sense to increase the curvature at the quarter points; the top more than the bottom.

Mast length and rake angle.

Mast length = 4900 mm
Rake angle = 10 degrees

I drew the straight mast at its rake angle.

IMCS curvature profile.

Measured Length (ML = Mast length - 100 mm) = 4800 mm
IMCS stiffness (Mast length ^3 / (Mid Point Deflection x 4600^2) = 29
Mid Point deflection / Measured Length = 0.040
Bottom bend ratio (deflection at .25 ML / deflection at .50 ML) = 0.64
Top bend ratio (deflection at .75 ML / deflection at .50 ML) = 0.76

I drew the curved mast matching its LE angle with straight raked mast at base.

Curvature profile for the Sail LE based on increments to the Mast curvature profile

Delta Mid Point deflection (curve) / Length = 0.010
Delta Bottom bend (curve) ratio = 0.03
Delta Top bend (curve) ratio = 0.05

Desired exposed mast length = 500 mm
LE curve length (Mast length - exposed part) = 4400 mm
Measured Length (design: ML = LE curve length - 100 mm) = 4300.0 mm

Mid Point curve / Measured Length = 0.050
Measured Length / Mid Point curve = 20.0
Bottom curve ratio (curve at .25 ML / curve at .50 ML) = 0.67
Top curve ratio (curve at .75 ML / curve at .50 ML) = 0.81

I drew the Sail LE with its base LE angle matching the straight raked mast at the top of the exposed portion of the mast.

Thoughts on the Sail fabric

The sail fabric adjacent to and including the luff pocket is cut to match the desired LE shape.
The area is bounded by the luff panel leading edge and the fibers in the fabric connecting the head and the tack.
Under tension the fibers connecting the head and the tack form a chord line joining the LE curve ends.

The mast is fed into the luff pocket until it reaches the top.
A downhaul connects the sail tack at the bottom of the luff pocket to the base of the mast.
Downhaul tension draws the bottom of the luff pocket / tack towards the mast base forcing the mast into the luff pocket.
This causes the mast to bow outward toward the leading edge of the luff pocket.
As the mast is more flexible at the tip, it will meet the leading edge of the luff pocket before the base does.

Sufficient downhaul will cause the mast to be seated along the LE over its entire length.
Insufficient downhaul will cause looseness in the lower parts of the luff pocket.
Excess downhaul will not move the luff pocket along the mast any further, fabric tearing becomes an issue.

Any force that moves the luff pocket top towards it's bottom may require the downhaul to be adjusted.

I have a few of Marchai's books but not the one you mentioned. He made quite a contribution in his time.

I am a little closer to the experimental stage of this endeavor. My goal is to mathematically define the shape of a vehicle based on the experiences from an active group of users. Having that, the design becomes a parametric model that can be readily customized to meet specific requirements.

I am getting closer with the sail model. One set of characteristics for four sizes, change one and all four update. Just need a little more feedback on the areas I have been presenting. What looks good or otherwise?

Luff pocket sections

Was thinking a bit about the luff pocket. The mast tapers so why not the lull pocket too?

Shortening the chord increases the angle at the trailing edge of the pocket may have a negative impact aerodynamically. On the other hand a longer chord moves the batten end further away from the mast.

The spreadsheet does the hard work so I included the feature.


Tip

Mast diameter = 30.0 mm
Luff pocket chord with mast inside = 80.0 mm
Luff pocket chord / mast diameter = 2.7
Trailing edge angle = 26.7 degrees
Allowance for sewn TE each side = 20.0 mm
Material width required = 220.6 mm
Luff pocket chord flattened not including flange for sewing = 90.3 mm

Base

Mast diameter = 50.0 mm
Luff pocket chord with mast inside = 100.0 mm
Luff pocket chord / mast diameter = 2.0
Trailing edge angle = 38.9 degrees
Allowance for sewn TE each side = 20.0 mm
Material width required = 277.0 mm
Luff pocket chord flattened not including flange for sewing = 118.5 mm

Sail LE Shape 3

Here is a model based on what I found out about Luff shaping.

Mast length and rake angle.

Mast length = 4900 mm
Rake angle = 10 degrees

I drew the straight mast at its rake angle.

IMCS curvature profile.

Measured Length (ML = Mast length - 100 mm) = 4800 mm
IMCS stiffness (Mast length ^3 / (Mid Point Deflection x 4600^2) = 29
Mid Point deflection / Measured Length = 0.040
Bottom bend ratio (deflection at .25 ML / deflection at .50 ML) = 0.64
Top bend ratio (deflection at .75 ML / deflection at .50 ML) = 0.76

I drew the curved mast matching its LE angle with straight raked mast at base.

Curvature profile for Paul Day's sail.

This is the Luff curve Paul Day proposed in the "How I recut sailboard sails to suit landyachts" topic.

I drew the Luff curve with the Head matching that of the curved mast and the foot resting on the mast's LE.
It is hard to see at the scale drawn but the Luff curve leads the curved mast in the lower part of the sail.





To get a clearer picture I plotted the distance between the Sail LE and the curved mast. The S characteristic of the
Luff shaping is clearly evident.

Positive increments allow for increased fullness on the lower portion of the sail.

Negative increments cause looseness in the Leech allowing twist.





Curvature profile for a generic sail with the desired Luff curve shaping.

I drew a curve from the Mast head crossing the mast curve and joining the Mast curve some distance above the Mast Step.

Vertical location of Sail foot on mast 400 mm
Location of crossover in percent of distance between Head and Foot 75%
Initial angle relative to Mast curve at Foot 3 deg.

The method works well and in the case of the parameters above, it matches Paul Day's Sail curve.
It can be seen in the second plot where it nicely overlaps Paul's curve. You couldn't see the difference in the first plot if it were there.

Select to expand quote

Hiko said..
You are making me think about all this in ways I never have before !

That is the point of the exercise, its all part of the fun!

Select to expand quote

landyacht said..
no idea waht you are doing ! I do recall a sailmaker measuring one of my sails and asking why the top of the luff was curving the wrong way I said that was how you stopped the head from collapsing. he didnt get it. . gotta love a big slab of concrete to draw sails on

It does seem a little tricky. I was trying to reverse engineer your sail and didn't have the background to understand why the LE was curved differently from the mast. I think that I have a better idea now though.

The answer came from one of Gizmo's older posts (2009) referencing a link to "How to make a sailboard sail" www.sailrepair.co.uk/makingsail The article is gone from the net but I got a look at an archive copy and John from Sailrepair has given me permission to re-post or quote from it.

On the subject of Luff shaping...

Clipped from web.archive.org/web/20120421140714/http://www.sailrepair.co.uk/makingsail.htm

It is how the curve at the front of the sail relates to your mast when it is bent with down / out haul that will determine the whole nature of the sail.

In the diagram below we have 2 masts; Mast A with a even bend curve and Mast B with a flexy tip. THE LUFF CURVE ON THE SAIL DOES NOT MATCH THE CURVE OF THE MAST. If anything the luff curve is almost a upside down curve the mast takes. The curve in the top third of the sail should be very slight and yet down near the boom the luff curve may be greater than the bend in the mast and that's what produces the rotation or battens sitting to the side of the mast.

The greater the difference at the head of the sail between the bend of the mast and the sails straighter luff curve the greater the pre twist will be in the leech (the more floppy the head will be).

Sail designers tend to sit in specific camps on what the perfect luff curve should be like, in one camp you have Ezzy and North wave sails with very little rotation and then there is Neil Pryde ,Tushingham, Naish etc. with their well rotated sails.

How much should the luff curve away from the straight head to tack line? 25 cm may be a reasonable staring point on a 430 luff, more for bigger sails , less for smaller.

Hiko's masts

Hiko measured two masts at increments along a line 4600 mm tying the ends. I assume that the masts are somewhat longer to account for the curve length and that the 4600 is the measuring length to define the bend characteristics. The Serfiac is not as stiff overall and particularly at the top.

Kilwell 35%

Measured Length (ML) = 4600 mm
MCS (ML / deflection at .50 ML) = 22.7
Bottom bend ratio (deflection at .25 ML / deflection at .50 ML) = 0.634
Top bend ratio (deflection at .75 ML / deflection at .50 ML) = 0.767

Serfiac

Measured Length (ML) = 4600 mm
MCS (ML / deflection at .50 ML) = 21.6
Bottom bend ratio (deflection at .25 ML / deflection at .50 ML) = 0.631
Top bend ratio (deflection at .75 ML / deflection at .50 ML) = 0.818

Luff curve

To illustrate the Luff Shaping model and how different masts will change the sail shape, I created a Luff curve for the Killwell mast. The curvature profile plot shows the Kilwell as well as the Surfiac masts. Both will have fullness on the bottom part but the fit of the sail with the Serfiac mast will be very loose at the tip.

Example:

Location of Sail Head is common with that of the Mast.
The lower portion of the mast is exposed. The Sail foot is located on the Mast LE = 400 mm
LE curve crosses Mast a percentage of the distance between Head and Foot = 65 %
Initial LE slope at foot relative to Mast = 5 deg









Measuring the mast before designing the sail is the way to go. Selecting the Luff curve parameters to produce the right combination of draft down low and TE looseness above is going to take experience.

The masts measured above were both 4600 long but did have a 1000mm stiffener inserted in the base with 250 mm sticking out the bottom to go into the mast step
The zero measurement points were at each end of the 4600 mast however where they were supported also for the bend test

I am happy with what I have been able to put together on the sail layout. The spreadsheet will now easily produce the lines of a sail based on a small set of inputs. Time to build one and test out how it all comes together.

If anyone would like to have a go, here is the sample input. Let me know what you would change.

Typical Input

Mast

Total length = 4600.0 mm
Tip diameter = 30.0 mm
Base diameter = 50.0 mm

Center deflection / IMCS measured length = 0.040
IMCS Bottom bend ratio = 0.64
IMCS Top bend ratio = 0.76

Mast pivot X = 2400.0 mm
Mast pivot Z = 280 mm
Mast base above pivot = 50 mm
Rake = 5.0 deg

Sail LE

The Sail foot is located on the Mast LE at Z = 700.0 mm
LE curve crosses Mast; % distance between Head and Foot = 75 %
Initial LE slope at foot relative to Mast = 3.00 deg

Luff pocket tip chord / mast diameter = 2.67
Luff pocket base chord / mast diameter = 2.00

Sail TE

Leech roach / leech length (+, 0 or -) = 0.015
Bottom bend ratio = 75 %
Top bend ratio = 75 %

Aft panels LE

Location on local sail chord line = 0.30

Luff panel TE overlap

Overlap at average sail chord between the Head and Foot TEs = 25.0 mm

Exterior layout parameters

Foot length / Sail length on LE curve = 0.320
Foot angle WRT ground (independent of rake) = -15 deg

Head / Sail length on LE curve = 0.070
Head angle WRT ground (independent of rake) = 25 deg




Output

Layout

Sail area = 3.99 m^2
Geometric Centroid (X) = 1537.6 mm
Geometric Centroid (Z) = 2480.2 mm

Aspect Ratio = 4.24

Mean aerodynamic chord = 1070.4 mm
MAC / 4 (X) = 1810.3 mm
MAC / 4 (Z) = 2487.6 mm

Head length = 297.8 mm
Head angle = 25.0 deg
Leech (end to end length) = 3606.4 mm
Leech angle (end to end) = 268.9 deg
Foot length = 1361.5 mm
Foot Angle = -15.0 deg
Luff pocket length (end to end) = 4198.7 mm
Luff pocket TE angle (end to end) = 103.4 deg

Luff pocket

Tip chord = 80.0 mm
Base chord = 100.0 mm

Radial panel 1 (bottom)

Length along Foot = 1034.0 mm
Length along LE = 1175.2 mm
Length along top = 1258.9 mm

Radial panel 2

Length along Radial panel 1 = 1258.9 mm
Length along LE = 912.0 mm
Length along top = 1948.8 mm

Radial panel 3

Length along Radial panel 2 = 1948.8 mm
Length along LE = 918.7 mm
Length along top = 2762.2 mm

Trapezoid panel 1 (top)

Length along Radial panel 3 = 2762.2 mm
Length along LE = 1051.1 mm
Head = 297.8 mm
Length along Leech = 3606.4 mm
Head TE to Radial panel 3 Peak = 1027.0 mm

Luff panel

Length on Foot = 327.6 mm
Length Foot LE to Head LE (straight) = 4198.7 mm
Length on Head = 14.5 mm
Length on TE (straight) = 4049.1 mm