To calculate the dynamic jib force, the spreadsheet takes jib sail area, some
assumptions about lift and drag coefficients, the wind speed, the heading of the
boat (expressed as the angle of the apparent wind), and comes up with values for
side force and drive force. These two are combined together to yield a
total jib force, the force that is going to make the jibstay sag and the leech
bag out. Then the spreadsheet makes some assumptions about the bulb
weight, the length of the fin lever arm, the height of the centre of effort of
the sails (jib and main), and the contribution of the jib to the total sail
force, to estimate the heel. (As always, if you don't like any of the
assumptions, put in your own!)
There is a button to activate a "Heel" macro which populates some cells to show a graph of heel, seen on the "Heel graphs" worksheet. A heel of 30 degrees is shown as a horizontal line.
Then, the spreadsheet moves on to the "JS sag" worksheet. Given the total jib force, it estimates how much of this force operates to bring about sag in the jib luff and jibstay. The value of 40% shown, called the "Jib sag force distribution", is calculated from the pivot offset. Some of the total jib force goes into the head of the jib, some into the clew, and some into the tack. While 50% goes into the head, the remaining 50% is distributed between tack and clew in proportion to the pivot offset. Given that the offset is around 20% for most "reasonable" offsets, this means about 40% of the total jib force goes into the tack and 10% into the clew. The centrepiece of the "JS sag" worksheet is a large table of jibstay sags, where given a specific sag the force needed to produce it is calculated. So when the force acting on the jibstay is known, the spreadsheet looks up this table to yield an estimated sag. (It actually calculates a linear interpolation.) In the centre table, the sag causes the jibstay to stretch, and the amount of this stretch determines the increased dynamic jibstay tension using a modulus of elasticity. I've estimated this modulus at 20000, and this is one of the "black art" numbers in the spreadsheet. If you want to seriously pursue this, you'll have to calibrate the spreadsheet to reflect your own modulus value, because you'll be using a different mast with different materials for shrouds, backstay, and so on.
The top portion of the table illustrates what's going on. Imagine the jibstay laid horizontally, and imagine that the jib force is acting like a weight in the middle. For each amount of sag, the angle made by the jibstay to the horizontal is calculated -- it is this angle which allows the jibstay tension to support the jib force. The elongation of the jibstay is estimated, and this is turned into an additional tension in the jibstay, added to the static tension brought about by the backstay and shrouds. Finally, given the jibstay deflection (its angle) and its tension, the spreadsheet calculates the jib force that can be supported. So if we know that the force acting on the jib so as to cause sag is, say, 0.027 lb, we can then say that the jibstay is probably sagging by 1.5 mm according to the table.
While doing this, the spreadsheet keeps an eye on the topping lift tension. When the topping lift releases, the spreadsheet adds extra tension into the jibstay because the leech is now flying "freely". The extra tension is added as a function of the leverage of the boom. Assuming that the jib force acts at the centre of effort (CE), a point 40% along the chord, and given the pivot position, the spreadsheet can calculate the extra tension being put into the jibstay.
Finally, the spreadsheet attempts to estimate what happens to jib draft and jib twist as a result of jibstay sag. When the jibstay sags, it has the effect of pushing draft into the jib. The amount "pushed in" is regulated by the "sag angle". I have used a value of 30 degrees; you might prefer 45 degrees. The idea here is that the "full" amount of sag does not go into the increased draft, but only a portion of it. Some of the sag goes into reducing the jib twist (third Web page).
The graphs show estimated jibstay sag as the wind increases for a pivot offset of 72.5, and then the sag versus pivot offset for a wind speed of 14.5 ft/s. As before, we can see that the sag doesn't really vary according to pivot offset, because the jibstay tension doesn't really vary as a function of pivot offset.
(1) It is instructive to see that jibstay sag increases from the moment there is any wind, and that it increases systematically thereafter. That is, jibstay sag is unavoidable, no matter what tensions you put into your shrouds or backstay. The graph shows a sag of about 5mm with a (fairly modest!) wind speed of around 4 ft/sec, rising to about 13 mm when wind speed hits 14 or 15 ft/sec.
(2) The "Rig Parameters" table on the "Tensions" worksheet asked you to specify the luff allowance cut into your jib by your sailmaker. If the wind speed is not high enough to cause significant jibstay sag, a "tense" jibstay will in fact pull draft out of the sail. The graph shows this effect, where a nominal jib draft of 8% has been reduced to around 6% with a wind speed of 1 ft/sec, because there is not enough jibstay sag. The 3 mm luff allowance for this jib hits its stride when the wind is about 6 ft/sec, seen where the jib draft line intersects the nominal draft of 8%.
(3) Thereafter, as wind speed increases, draft is pushed into the jib as sag increases. The graph shows that for a wind speed of about 14 or 15 ft/sec, draft is around 10%. These calculations are carried out in the "Sag draft" worksheet.
This worksheet attempts to model the jib as two arcs. The forward or front arc is that part of the jib from the luff to the position of maximum draft, set in the Rig Parameters table. I've used a value of 40% here. The aft or rear arc is that part of the jib from the position of maximum draft to the leech. Each arc is treated as an arc of a circle, and this is obviously approximate. In the example shown here, a nominal draft of 8% in the middle of the jib (ie half-way up the jib) is caused by the chord of the jib at this point being shortened (called, somewhat misleadingly, "sag" as well, sorry) by about 2.09 mm in the fore part and 1.38 mm in the aft part. Then, the jibstay sag adds about 4.7 mm of extra sag, so the total sag of 8.19 mm now yields a new draft of approximately 9.95%. On the side, entry and exit angles are calculated as well. Entry angle of the 8% draft jib is estimated at 23.2 degrees, and of the 9.95% draft jib as about 29.1 degrees. (More details of the formulas on the Leech page.)
It is interesting to vary the position of maximum draft in the spreadsheet. It does make quite a difference to the changes in draft, twist, and entry angle as the wind picks up and the jibstay sags. In particular, the spreadsheet seems to show that the changes in draft are smaller, the more draft-forward the jib is cut. But this is only an indicative result, not a confirmed finding.
(4) Whatever jib luff allowance was cut into your jib by your sailmaker needs to be carefully maintained across almost the whole of the wind range for that jib. This means that we need to have some decent aft chainplate offset, so that we can crank up the shroud tension, and hence the jibstay tension, as wind speed rises. The graph shows, of course, that sag increases with wind speed, and that this increase looks to be basically linear. There are some skippers who would disagree with this, but that's what pops out of the spreadsheet.
Suppose that you and your sailmaker have agreed that your jib, with a luff allowance of 3 mm, sets best with a jibstay sag of 7 mm -- in other words, that the jib works best when the jibstay pushes an extra 4 mm of material into the body of the sail. According to the graph above, this occurs at a wind speed of around 6 ft/s. Now if the wind picks up to, say, 10 ft/s, and you don't change anything, your jibstay will sag about 9 mm and you'll have about 9% draft in the middle of your sail. Darn. The spreadsheet will allow you to calculate what increase in shroud or backstay tension or shroud offset you need in order to bring jibstay sag back down to 7 mm. But now the topping lift is excessively tight, so the pivot offset needs to reduce. The good news is that this change of pivot offset doesn't really affect jibstay tension or sag. The bad news is that mast compression has risen with the changes to the shrouds or backstay tensions.
(5) The approach using shroud changes allows us to either leave the backstay more or less alone, or to tweak it gently to give subtle mast bend changes to take draft out of or push draft into the main and to twist the main head off by the amount we want. The point here is that we are not using the backstay tension to manage the jibstay tension.
It might be worth mentioning that an alternative (to jacking up the tensions so high to avoid excessive material being pushed into the jib due to jibstay sag) is to flatten the jib. If you had left everything alone and allowed 9 mm of material to be pushed into the body of the jib instead of the optimal 7 mm, this would have the effect of increasing draft from, say, 8% to 9% in the middle of the jib. Not too serious, perhaps, except that the entry angle would have increased about 4 degrees, so you would not be able to point so well. Instead, just flatten the jib by moving the outhaul back 2 mm or so. More about the entry angle is given on the third Web page.
The dynamic tension calculations are approximations. On the "Tensions" worksheet, the jib lift and drag values are calculated from the "usual" formula, Area * Density * Coef * Wind Speed ^2, and are then converted into side force and driving forces by considering the apparent wind angle. The jib's contribution to the total side force is taken to be 10% more than what might be expected when considering the simple ratio of the area of the jib versus the area of the main. This 10% is called the "Jib contrib factor" on the spreadsheet. The resulting estimated heel takes no account of the fact that the side force is aimed increasingly downward, but that doesn't matter, since the heel values play no real part in any of the calculations.
The dynamic tensions are calculated on the "JS Sag" worksheet. The central table of this worksheet calculates, for a range of possible sags, what kind of jib force that sag can support. It is a "backwards" calculation, because what we really want to know is what is the sag as a result of a certain jib force, but this can't be calculated directly. So for each sag, the central table calculates the elongation in the jibstay, and multiplies this small elongation by the elastic "modulus" value to yield the new tension in the jibstay. The elongation is estimated using Pythagoras' theorem, and starts by assuming that the jibstay consists of two straight lengths with a kink in the middle, rather than the gentle curve it really is. This assumption rather under-estimates the elongation, so the spreadsheet just multiplies the result by 2, the "Elongation adjust factor" to get a better estimate! OK, you want something more accurate, use your own factor!
Now the fun begins. The spreadsheet estimates what the topping lift is currently doing to the jibstay because of its leverage, and there is a "fudge factor" called the "Jib lever wrt boom". In theory, only a fraction of the total drive and side force on the jib should apply to the topping lift and make it react against the jibstay, but the spreadsheet uses a value of 100%, which seems more consistent with what actually happens. (I'm not sure why, but an unformed idea says that the topping lift and the jib leech is the "easy" way out for the increasing jib forces when the wind rises.) First, the spreadsheet figures if the topping lift has "released" by comparing its static leverage with the leverage from the current jib forces due to the wind. If it hasn't released, nothing changes. If it has released, then the leverage it is exerting is added into the jibstay tension when calculating the jibstay elongation in the central sag table. Also, if it has released, the amount of its lift is calculated, called "TL release due to lever", to be used later when estimating the new twist in the jib. So, given the jib force acting to cause jibstay sag, the central table is looked up to see what sag results. An interpolation is made to give a better estimate. The central table is then looked up "backwards" to determine what kind of tension there is in the jibstay, and this is noted as the "Revised" jibstay tension. It is these revised jibstay and topping lift tensions that the tension graphs show.
©2011 Lester Gilbert