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about the end of the standing ways, and as long as the buoyancy moment remains in excess of the weight moment about this point, there is no fear of the vessel tipping ; but if in any position the former moment falls short of the latter, it is evident that in order to restore equilibrium, the stern will drop, and thus increase the displacement until both moments are equal. Tipping, if occurring at all, must take place after the C.G. of the ship has passed the end of the standing ways, and before the commencement of the second period. In the example, the C.G. of the ship has passed

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the way ends when she has moved 174 feet. From about that point to a little beyond the end of the first period, the buoyancy and weight moments about the end of the standing ways are calculated at several intervals, and at each interval the latter moment, being deducted from the former, gives the moments against tipping. These moments are shown by curve E. If this curve at any part were to run below the base line, it would show that the vessel will tilt. The point where this curve is nearest to the base line gives the position of the vessel when she has least longitudinal stability, which in this case is when the vessel has travelled down the ways 189 feet, the minimum margin against tipping being 9,700 foot-tons.

It is desirable that the margin be not too small for uncertain vessels ; where this was the case they actually did tilt slightly, which shows that a moderate margin is required in calculation to allow for the error introduced by treating, as it is convenient to do in practice, those moments statically instead of dynamically. In calculating the buoyancy moments no account is taken of the cradle, which would only alter the results slightly ; the variations being on the right side, may be safely ignored. Besides, the after

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Outer 1804 1} 19"

.

Declivity of keel per foot . Declivity of standing ways per foot

if to 18 to 12 to 11 to 11 Camber of standing ways 2' 3'' 1'0" 1/2" 1'0" Length of standing s Inner 345'

367 395 370 ways

Outer 288 Length of sliding sliding s Inner 240

284' 330 305 .

1 Outer 165 Breadth of sliding Inner 1 10%

1'9'' 1'9" ways Area of sliding ways in square feet

1,430 994

1,155 1,067 Total fall in length of standing ways

23/0" 18 9' 19'7" 18° 6' Water on way ends .

8'7" 6'0" 4'4" 2 6" Draught of ship forward 11/2" 11'6" 7'0"

801" Draught of ship aft.

16'6" 14'0" 10' 104 105 Draught of ship inean

13' 10'' 12' 9" 90 925" Displacement in tons

2,850 2,500

2,157 2,240 Mean pressure per square foot

on sliding ways in tons . 2.00 2.51 1.9 2.09 Length of first period

278.0 283 250.5 279.5 Length of second period

67 84 144.5 90.5 Ratio of length of 2d period to

length of sliding ways 28% 30% 44% 30% Weight on sliding ways at com

mencement of 2d period 520 550 640 630 Weight on sliding ways at end

of second period (in tons) · 250 290 300 380 Margin against tipping : 10,500 33,250 80,000 | 35,300

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8''

to 16 i to 16 ito il 14 to 111 to 1d 1% to it is to il 1 to 12 1'11"11107 1' 10" 1 1'0 10%

1'0

6" 348 302 300 267' 250 195 259

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end of the sliding ways often rises to the surface shortly after the vessel has entered the water. In the diagram a complete set of curves has been given to fully illustrate the matter, but for practical purposes only that part of the diagram where the vessel is represented to be moving from the position where the C.G. is at the way ends, to the end of the second period, is required.

As the minimum moment against tipping is a very important thing, it will be useful to know what variation will be made in its amount by any alteration to the length and form of the standing ways of this vessel :

Lengthening the standing ways 10 feet increases the moment from 9,700 to 13,700 foot-tons.

Shortening the ways 10 feet decreases the moment to 5,300 foottons.

Increasing the camber from 12 inches to 18 inches increases the moment to 14,500 foot-tons.

Decreasing the camber to 6 inches decreases the moment to 4,000 foot-tons.

If with a certain declivity of ways for the launching of a vessel, it is found, by calculation, she will tilt, the standing ways must be extended further out into the water, or, if this cannot be done conveniently, their outer ends must be lowered, or ballast put into the fore end of the vessel. The first two increase the buoyancy moment about the end of the standing ways, and the third decreases the weight moment about the same point.

W sin 8-fW cos 8
Pressure on dog shores =

cos B
W = weight of vessel.
8 = mean angle of declivity of ways under vessel.
B = angle between ways and dog shores.
f = coefficient of friction (between 1.0 and .7).

The ratio of second period to length of sliding ways cannot be got lower than about 25 per cent without danger of tipping.

RUDDERS. In determining the most suitable area of rudder it is usual to take the same as a percentage of the immersed longitudinal plane of the ship, which percentage will vary with the degree of fineness of the vessel.

Percentage for Rudder Area in Various Types.

TYPE OF VESSEL.

PER CENT OF
IMMERSED LONGI-
TUDINAL PLANE.

Fast ocean liners
Freighters
Yachts
Paddle steamers

1.25 1.50 1.10 2.0

Having fixed upon the area, the diameter of stock may be calculated by various formulæ, some of them, unfortunately, of a very approximate character, and on this account, where high speed will be attained, it is advisable to carefully calculate the required diameter irrespective of the result obtained by the classification societies' formulæ. For this purpose it is necessary to know, (1) the hard over angle of rudder, (2) centre of pressure on rudder blade, (3) maximum pressure exerted at hard over with ship at full speed. The angle of helm being usually 35°, the pressure on blade at this angle at full speed may be found from the formula, — P representing the pressure in lbs.

P= AV2 x sin a X p. It should be stated that V = speed of vessel in knots per hour plus 20 per cent to allow for the slip ; A = area of rudder in square feet, including emerged surface; and p= pressure in lbs. per sq. foot at 1 knot, = 3.19 lbs. per sq. foot.

Before, however, the twisting moment on the stock can be solved, the centre of pressure must be located. This centre being } the breadth from the leading edge with the helm amidships, does not arrive at the centre of gravity of rudder until 90° is reached, and as 35o is the usual angle, it will be sufficiently close to take .37 of the breadth of the rectangle equalling the rudder area :

А
Centre of pressure from centre of stock=1= .37.

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