In case a new brace winch has to be manufactured, or if one wants to know if an existing winch will fit on a certain ship, one has to know the length of the braces and the proportional difference between the windward and leeward brace of each yard.
Of the winch one has to know the total circumference of the drum and if the range of the coning can cover the proportional difference in brace length. So there are two sets of calculations to be made which, in the end, has to match with each other.

        On the ships side the matters of interest are:

• the length of the concerning yard
• the distance between the yard and brace winch (horizontal)
• the maximum brace angle
• the dimension of the truss / parral
• the length of the brace pendant

Note: at first I believed that I took to many factors into account, but meanwhile I have learned that I even forgot some dimensions! Things like the bent in the hanging braces, the rake of the masts, etc. Two naval engineers checked my paper. Marijke de Jongh, rig-designer of the Stad Amsterdam made her own calculations with more dimensions and came to the same results as I did.
Jan Huerkamp, rig-designer of Navcon, checked my paper and found some imperfections, but they were to small to really worry about. Therefore I haven't changed my calculation nor spreadsheet.

        On the side of the winch, one has to know:

• the smallest and largest diameter of the drum
• the proportion between the small and the large diameter, which determines the coning
• the number of windings on the drum
• the diameter of the steel wire used for brace
• the length of the brace pendant

The length of the braces

   The drawing of the brace arrangement of Middendorf serves as a starting point for the calculation of the brace lengths. The standing parts of the braces are not touched during bracing. So the tackles on the ships rail do not take part in the calculations.
Of the lower yards, the running parts first go to the ships side, then up to the guiding block under the top. So the standing and running part of the lower braces are running almost parallel of each other. The equations for the lower yard therefore differ a little from the topsail yards equations.


a. Square braced

Click the image for a full size drawing

The drawing might look like a plate of coloured, uncooked spaghetti, but if you follow the text, it is not that difficult. The drawing represents a deck with two masts and one yard in square position. For the sake of clarity only the quantities on starboard side are drawn.
S_ur and S_lr are fictive braces, running straight to the yardarm. First S'_u and S'_l are calculated, which represent the projections of S_ur and S_lr.
Because the yard is in square position, calculation on one side are sufficient, the other side will give the same answers.

According to Pythagoras:

For S'_u counts:              (S'_u)² = (M + x )² + R²
For S_ur :                         S_ur = Ö (M + x)² + R² + (h2 - h1)²
For S'_l :                          (S'_l)² = (M + d + x)² + (R - ½Br - b)²
For S_lr :                          S_lr = Ö (M + d + x)² + (R - ½Br - b)² + (h1 - h3)²

The quantity d will have a negative value if the landing points of the standing parts are in front of the mast. In the equation for the S_ur of the lower yard brace, ½ R has to be reduced with half the ships width, because the running part stands also in the ships side.
For the lower yard accordingly counts:


                          S_ur = Ö (M + x)² + (R - ½Br)² + (h1 - h4)²
                          S^lr = Ö (M + d + x)² + (R - ½Br - b)² + (h1 - h3)²


    For the record: A brace pendant is a piece of rope or parcelled wire between the yardarm and the brace block. It prevents the brace block getting foul by lose ends like gaskets or reef tackle lines. Besides that, it diminishes the actual length of wire needed for the brace.
The pendant will always incline a division of the angle between the upper- and lower brace into two equal angles. This is because both ends of the brace draw with equal strength on the brace block. The block has to take in a position where the sum of the lengths u and l is the smallest. This position is the point where the tangent of the arc of the circle that the block can describe, is in right angles with the bisector of the angle between u and l.

See the square braced image.
In this diagram at starboards yardarm, several vertical angles are drawn, namely r, j and l.

cos j = h₁ / S_lr        Þ      j = arc cos (h₁ / S_lr)
sin r = S'_u / S_ur      Þ      r = arc sin (S'_u / S_ur)                           if h2 > h1.
cos r = S'_u / S_ur    Þ      r = arc cos (S'_u / S_ur) + 90°                if h2 < h1.

Hence it follows:            l = 180° - j - r
Now l is divided by 2     Þ    l' = l/ 2

This means:      cos l' = t / s     Þ     t = s . cos l'
Pythagoras again:      v² + t² = s²     Û     v = Ö s² - t²
S_l now can be determined by:   S_l = Ö(S_lr -t)² + v²      Û     S_l = Ö(S_lr - t)² + s² - t²
The same counts for S_u:    S_u = Ö (S_ur - t)² + v²     Û      S_u = Ö (S_ur - t)² + s² - t²

Here too the lower yard makes an exception. Because both parts of the brace run to the ships side, the angle between S_l and S_u is so small that it can be neglected. That means that the length of the brace pendant can be subtracted directly from S'_l and S'_u.
The total length of the brace , from brace block to and with the tackle on starboard is equal to S_l and S_u. Because the yard is squared, the brace on port side has an equal length.

Are you still with me?


b. Braced up sharp on starboards tack

Click the image for a full size drawing

While bracing, the length of the wire turned in on port side is increasing more then the length paid out on starboard side as explained previously. This effect is enlarged by the fact that the yard is fixed to the mast with a truss (or parrel). The centre of the yard is turning away from the line of the keel. This means that the yard length on starboard side diminishes with the line segment y. On the port side it is lengthened with the this amount.

Normally the brace angle is the sharp angle between the yard and the line of keel. In the calculations, however, the angle between the square yard and the braced-up yard is used. The yard remains her horizontal position, so h1 is constant.

For y counts:     y = x . tan a
For q counts:     q = x / cos a

Half the yard length on starboard is:     R_s = R - y     Þ     R_s = R - x . tan a
Half the yardlength on port side is:     R_b = R + y     Þ     R_b = R + x . tan a

See the braced up image.
     Here the yard is drawn braced up. The projection of the yard is seen on deck again. In this situation the length of the braces has to be worked out on both sides. On starboard side the distance between the masts is lengthened with factor N, which is the fore-and-aft component of the braced up yard. On port side M is diminished with N. In both cases the quantity q has to be added. L is the athwartships component of the braced up yard.

On starboard:    L_s = R_s . cos a     On portside:     L_b = R_b . cos a
                           N_s = R_s . sin a                                N_b = R_b . sin a

Starboard side:
For S'_u now counts:          (S'_u)² = (M + q + N_s)² + L_s²
For S_ur:                           S_ur = Ö (M + q + N_s)² + L_s² + (h2 - h1)²
For S'_l:                           (S'_l)² = (M + d + q + N_s)² + (L_s - ½Br - b)²
For S_lr:                           S_lr = Ö(M + d + q + N_s)² + (L_s - ½Br - b)² +(h1 - h3)²

The respective angles r, j and l can be determined in the same way as above.
For the lower yard counts:       S_ur = Ö (M + q + N_s)² + (Ls - ½Br)² + (h1 - h3)²
and:                                              B_lr = Ö (M + d + q - N_b)² + (L_b - ½Br - b)² + (h1 - h3)²

For the eventual 'upper'- and 'lower' brace counts:
                                             B_l = Ö(B_lr - t)² + v²      Û       B_l = Ö(B_lr - t)² + s² - t²
                                     B_u = Ö(B_ur - t)² + v²     Û       B_u = Ö(B_ur - t)² + s² - t²

Port side:
For the port side brace the same rules applies.
For the B'_u:                               (B'_u)² = (M + q - N_b)² + L_b²
For B_ur:                                      B_ur = Ö(M + q - N_b)² + L_b² + (h2 - h1)²
For B'_l:                                      (B'_l)² = (M + d + q - N_b)² + (L_b - ½Br - b)²
For B_lr:                                      B_lr = Ö(M + d + q - N_b)² + (L_b - ½Br - b)² + (h1 - h3)²

For the lower yard again:          B_ur = Ö(M + q - N_b)² + (L_b - ½Br)² + (h1 - h3)²
                                          B_lr = Ö(M + d + q - N_b)² + (L_b - ½Br - b)² + (h1 - h3)²

And for the eventual upper- and lower brace this counts:
                                          Bl = Ö(B_lr - t)² + v² Û Bl = Ö(B_lr - t)² + s² - t²
                                          Bu = Ö(B_ur - t)² + v² Û Bu = Ö(B_ur - t)² + s² - t²

Continue

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