Remember that the weight and balance documentation is part of paperwork that must be on board at all times. It should show the empty weight, empty-weight CG, most forward and most rearward limits, and sample loadings. (The kit manufacturer or designer will provide the fore and aft CG limits and the maximum gross weight.)
Weight Times Arm Equals Moment Online Free
The first will show how many moments an item produces at each station. So each station of the airplane (front seats, rear seats, fuel, baggage, etc.) will have its own line on the graph. On the bottom will be weight, and on the side will be moments.
If you need to solve for CG, take your total moment (unindexed) and divide by the total weight. A common mistake is to add up all of the arms, but this will not provide a helpful number. Instead, you need to calculate it based on the weights and forces in the airplane, which are represented by the moments and weight.
Each value of n corresponds to a different moment: the 1st moment corresponds to n = 1; the 2nd moment to n = 2, etc. The 0th moment (n = 0) is sometimes called the monopole moment; the 1st moment (n = 1) is sometimes called the dipole moment, and the 2nd moment (n = 2) is sometimes called the quadrupole moment, especially in the context of electric charge distributions.
In addition to body, weight, and moment, there is a certain fourth power, which can be called impetus or force.[e] Aristotle investigates it in On Mechanical Questions, and it is completely different from [the] three aforesaid [powers or magnitudes]. [...]"
STABILITY CHARACTERISTICS 4-1. Foreword. As noted in Chapter III, vessels tend to remain upright and resist capsizing when inclined because they develop a righting moment. At any angle of heel, the value of the righting moment is displacement in tons times the distance between buoyant force and weight (called righting arm) in feet. In other words:Righting moment = W X GZ.Where: W = displacement in tons (2,240 pounds).GZ = righting arm in feet.There are certain factors which may cause either W or GZ to vary. It is the purpose of this discussion to consider some of these variations, and to develop a method of describing a ship's total stability characteristics.4-2. Righting arm and angle of heel. If the angle of heel is small, the metacenter may be regarded as being fixed. In this case a triangle is formed, whose sides are GM, GZ, and the line of action of the buoyant force (triangle GZM in fig. 4-1), in which the apex angle (GMZ) is the angle of heel. The angle between GZ and the buoyant force (GZM in fig. 4-1) is a right angle. Considering this right-angle triangle, it is evident that if GM remains constant, GZ will become larger as the angle of heel is increased.For convenience, designate the angle of heel (GMZ) by the Greek letter θ. Then the relation between metacentric height and righting arm for small angles may be expressed as:GZ = GM X sin θ.But, righting moment = W X GZ.Therefore, righting moment = W X GM X sin θ.In other words, righting moment equals displacement times metacentric height times the sine of the angle of heel. As the ship rolls, W remains constant. If it does not roll more than about 10 from the upright, GM remains practically constant. Initial stability consists of these two constants (GM and W) times a ratio called the sine of an angle. Consulting a table of sines will reveal that as the angle gets larger, the sine increases. Consequently, as the ship heels farther the righting moment gets larger.Figure 4-1. Diagram to illustrate the relationships existing among metacentric height, righting arm, and angle of heel. 38 Figure 4-2. A curve of stability.
4-3. Curve of stability. In view of the fact that stability varies with the angle of inclination, it is necessary to express stability at greater angles of heel in terms of a series of righting moments for successive angles of heel. With displacement (W) remaining constant as the ship rolls, the righting arms may be used instead of moments, bearing in mind that the moment can always be readily obtained if one multiplies the arm by displacement.Using graphic means, it is possible for a design activity to determine the distance GZ for any given angle of heel. A series of values of GZ for successive angles of heel is thus obtained, and it would be entirely feasible to supply them to the ship in tabular form. But it is more convenient and much more expressive to represent them as a curve of righting arms plotted against angles of heel. Such a diagram is called a curve of static stability. The word "static" means that it is not necessary for the ship to be in motion for this curve to apply; if the ship were momentarily arrested at any angle during its roll, the value of GZ (and righting moment) given by the curve would still remain in effect, assuming that the water is calm.Figure 4-2 is a typical curve of stability in which a series of righting arms is plotted vertically against corresponding angles of heel which are plotted horizontally. This curve indicates that initial stability increases with angle of heel at almost a constant rate, but at larger angles the increase in GZ begins to level off. Then GZ gradually diminishes, finally becoming zero when the ship is almost on her beam ends. 39 Figure 4-4. Diagram to show the effect of increased draft upon righting arm at a given angle of heel. 4-4. Righting arms and freeboard. A change of displacement will result in a change of draft and freeboard, causing the center of buoyancy to shift to a different location for any one given angle of inclination. At any angle except 0, a variation in draft causes B to shift both horizontally and vertically with respect to the water level. The horizontal shift in B changes the righting arm, GZ. As shown in figure 4-4, the change in GZ at a specified angle will be a reduction if the change of draft is an increase. In other words, as freeboard is lost, righting arms throughout the range of stability are reduced. The ship in figure 4-4 is inclined at a given angle (20). For the 18-foot draft the center of buoyancy is at B18 and the righting arm is GZ18. When the draft is increased to 26 feet, the center of buoyancy is at B20, which reduces the righting arm to GZ26.When weight is added to a ship, displacement and draft are increased at the expense of reserve buoyancy and freeboard. One result of this is a reduction in righting arms as shown in figure 4-4. Another result is a change in righting moments due to increasing W in the expression: righting moment = W X GZ. Although both of these variations always occur simultaneously, they are due to independent causes, and must be considered separately. The gain in W does not necessarily compensate for the loss in GZ when weight is increased. In order to avoid confusion in ensuing discussions these two separate results of a weight increase will be referred to as:1. Increased moments due to gain in displacement.2. Decreased righting arms due to loss of freeboard.There is a third factor to be considered later, which usually follows a change in displacement; namely, a shift in the center of gravity.4-5. The cross-curves of stability. The design activity inclines the line drawing of the ship at a given angle, and then draws on it a series of waterlines. These waterlines are chosen at evenly spaced drafts throughout the probable range of displacements. With each waterline inclined at the given angle, the center of buoyancy for this waterline can be located by graphic means, as well as the line of action of the buoyant force. The center of gravity of the ship is assumed at a fixed point called the axis, and the righting arm is now the perpendicular distance from the axis to the buoyant force.Figure 4-5 depicts the midship section of a ship inclined at a fixed angle, with the axis assumed at point A. The lines of buoyant force are sketched in for successive drafts. It is evident that the righting arm successively decreases as freeboard is lost, ranging from AZ10 at the 10-foot waterline down to AZ24 at the 24-foot waterline. The actual determination of AZ in the drafting room involves complicated graphic 40 Figure 4-5. Diagram to show a series of righting arms corresponding to successive drafts. or mathematical methods which take into account the changing shape of the hull throughout the ship's length.In this manner, a series of righting arms for successive drafts are obtained. For each inclination chosen, a curve is plotted of righting arms measured vertically against tons displacement measured horizontally. Values of displacement in the horizontal scale are chosen to correspond to the drafts used in obtaining the righting arms. The curve of righting arms versus displacement is known as a cross-curve of stability. It is customary to plot a cross-curve for 10, another for 20, another for 30, and so on by 10 intervals to 90, as shown in figure 4-6.Plotting all the cross-curves to the same scale on one sheet facilitates their use in obtaining stability curves, as described in Article 4-6. The entire set of cross-curves is based upon the assumption that the ship's center of gravity lies in the centerline plane and is located at point A. The axis A is chosen at a round number of feet above the keel, and for reasons to be demonstrated later, is usually chosen below any actual location of G for the conditions of loading in which the ship is to operate.It is noteworthy that the Z end of AZ is accurately and correctly determined in the cross-curves, but that the A end is an assumed, rather than an actual point. Correction for this assumption will be discussed later.4-6. Making use of the cross-curves. The cross curves are used for the purpose of finding a curve of stability for the ship at any required displacement. This is done by drawing a vertical line on the cross-curve sheet at the given displacement. At the intersection of this line with the 10 cross-curve, read the value of the righting arm (on the scale at the left in fig. 4-6). Plot this righting arm at 10 on the grid of the proposed stability curve. Repeat the process for 20, 30, etc., until a series of 9 points is plotted on the grid. Now fair a smooth line through these points from 0 to 90, forming the required curve of stability.For example, suppose that a stability curve for 11,500 tons displacement is to be taken from the cross-curves of U.S.S. MIDDLETOWN, and plotted on the grid in figure 4-7. First, draw a vertical line, MN in figure 4-6, at 11,500 tons. At point (a) on this line the righting arm is found to be 1.4 feet, as read on the scale at the left. Plot 1.4 at 10, establishing point (a) on figure 4-7. Now find the righting arm at point (b) in figure 4-6, as 2.8, and plot this value at 20 to establish point (b) in figure 4-7. In this manner points (a) through (i) are obtained in figure 4-7, and they correspond to points (a) through (i) 41 Figure 4-6. Cross-curves of stability. 42 Figure 4-7. An uncorrected stability curve taken from the cross-curves. respectively, in figure 4-6. The curve of stability is completed in figure 4-7 by drawing a smooth line through the points plotted.Any stability curve that is taken from the cross-curves in the above manner is said to be "uncorrected;" that is, not corrected for the actual height of the ship's center of gravity above the keel. It does, however, embody the effect on righting arm of the freeboard for a given position of the center of gravity. The method of correction for KG, the actual height of G above the base line, is presented in Article 6-4.A comparison of cross-curves and stability curves is effected in the three-dimensional representation of a surface obtained by plotting displacements in one dimension, angles of heel in a second, and righting arms in the third dimension as shown in figure 4-8.The two types of curves may be compared as follows:1. On cross-curves:Vertical values are righting arms.Horizontal values are displacements.Successive curves are for different angles of heel.2. On stability curves:Vertical values are righting arms.Horizontal values are angles of heel.Successive curves are for different displacements. 4-7. Righting moment curves. In Article 4-3 the curve of stability was introduced as a graph depicting righting arms at various angles of heel. This stability curve can be obtained from the cross-curves if the vessel's draft is known, and then must be corrected for the proper location of G. The latter correction is to be discussed later; assume for the present that it has been made. The result is a diagram on which any vertical distance from the base (or horizontal scale) to the curve is a righting arm. If this righting arm is multiplied by W (displacement) the product is the righting moment at that angle of heel. It is possible, therefore, to multiply all the verticals of the GZ curve by W, and thus obtain a series of righting moments. These can, in turn, be plotted on a grid to form a curve of righting moments, in which case the vertical scale at the left is graduated in foot-tons instead of feet. Figure 4-9 is a curve of righting moments obtained by multiplying the verticals of figure 4-2 by displacement and then plotting the results to a scale of foot-tons.A more simple method than re-plotting can be used to convert any curve of righting arms to a curve of righting moments. Merely multiply each figure in the scale at the left by the proper value of displacement. 43 Figure 4-8. Isometric projection of cross-curves and curves of stability showing relationship (20, 40, 60 and 80 cross-curves have been omitted). Note that the vertical scale represents righting arm in feet for the example used. Height of axis above keel is 20 feet.Figure 4-9. Curve of righting moments. Figure 4-10 is a righting-moment curve obtained by changing the scale of figure 4-2.WARNING: if two or more GZ curves for different displacements are plotted on the same grid, they can NOT be converted to a group of righting-moment curves by changing the scale at the left. In such a case, it is necessary to choose a new scale of foot tons and replot all the curves, multiplying each of them by its own proper displacement.A curve of static stability is either a curve of righting arms or righting moments, depending upon the scale units to which it is plotted or which are used in reading it. This is true only of individual curves, however, and not of groups of curves for different displacements.4-8. Features of a stability curve. To find the righting moment (or arm) at a given angle of heel on the stability curve, enter the horizontal scale at the required angle, proceed vertically upward to the curve, and then proceed horizontally and read the righting moment 44 Figure 4-10. Curve of righting moments obtained by changing the scale of figure 4-2.Figure 4-11. Properties of a stability curve. (or arm) on the vertical scale at the left. If it is a scale of arms in feet, and the moment is desired, multiply the arm by tons displacement.Thus, in figure 4-11, the righting arm at 23 is found to be 1.4 feet. Since displacement for that particular curve is 11,500 tons, the righting moment at 23 heel is 11,500 X 1.4 or 16,100 foot tons. Inspection of the figure will show that the maximum value of the righting arm is (in this case) about 2.2 feet, and that it occurs at an angle of approximately 38. Multiplying by displacement, the maximum righting moment is found to be 11,500 X 2.2, or approximately 25,300 foot tons. Further, the ship has a positive righting moment for all angles up to 71. This angle at which the righting moment becomes zero is known as the angle of vanishing stability, and we say that the ship has a range of stability of from 0 to 71. If she rolls beyond 71 in calm water the ship will capsize.Three important properties of any curve of stability are now evident (see fig. 4-11): 1. The maximum righting moment (or arm), AB.2. The angle at which the maximum righting moment (or arm) occurs, A.3. The range of stability, 0C.The curves of stability thus far presented apply only to the intact ship; their properties affect the ship's rolling characteristics, but do not apply to a vessel which has taken on a permanent list. The latter case is to be dealt with later.4-9. Dynamic stability. Dynamic stability is defined as the amount of work done in inclining a ship to a given angle. This work is done against the righting moment, and is done by some type of inclining moment. From physics we know that work is equal to moment times the angle through which the moment acts. Inspection of a righting-moment curve shows that any area on it represents work, since an area on the graph is a moment (measured vertically) times an angle (measured horizontally). Furthermore, the area under the curve up to any angle of heel represents the work done in inclining the ship to that angle of 45 Figure 4-12. The shaded area represents work done in inclining the ship 25; it also represents dynamic stability when inclined at this angle. 2ff7e9595c
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