SIMPSON’S First Rule
Most mono-hull forms for water craft are symmetric about the vessel’s centerplane (this is not always so, as there are some craft that are not symmetric because of specific needs for propulsion or other operational requirements). Because of the form symmetry, it has been traditional to draw vessel lines for only half of the body (vessel half-breadths or offsets are then taken from the lines). For this reason, the properties of a hull form are typically computed for the half-body and then doubled. This is included in the computation process as we will see below. Simpson’s First Rule is frequently used because it is quite accurate for typical fair hull form shapes and it can be readily applied to shapes that have significant variations in the shape such as when bulbs or other appendages are added or subtracted from the base form.
The development of Simpon’s First Rule is as follows:
If a curve, Y = F(X) is defined by its ordinates Y0, Y1, and Y2 at locations (abscissa) X0, X1, and X2 that are equally spaced, S, and we want to determine the area under the curve, we can make an assumption that the function (curve) may be reasonably represented by a constant plus a linearly varying amount, plus an amount that varies as the second degree of the abscissa. That is, it is assumed that the function (curve) can be represented by the equation Y = A + BX + CX2, a second degree parabola. We can then integrate this assumed function over the distance 2S, (X2-X0), and determine the area under that curve. Evaluating the equation coefficients in terms of the ordinates, we find that Y0 = A, Y1 = A + BS + CS2, and Y2 = A + 2BS + 4CS2. We can also assume that the area can be defined as a function of the ordinates as Area = k0Y0 + k1Y1 + k2Y2 . This is what was done to derive the Simpson’s First Rule coefficients. This derivation is given on page 24 of Volume I, Principles of Naval Architecture, Edited by E.V. Lewis and published by SNAME in 1988. Most textbooks will have a similar derivation. It is shown in the text that when the assumed function is integrated and evaluated at the three coordinate locations, X0, X1, X2, Area = 2AS +2BS2 + 8/3CS3 and you end up with an expression for the area under the curve.
Setting the two expressions for the area equal and setting the coefficients equal to each other we obtain k0 = S/3, k1 = 4S/3 and k2 = S/3.
We can factor out an S/3 and obtain the area under the curve as:
Area = (S/3) (Y0 + 4Y1 + Y2). We then sum the areas for each of the segments of the distribution of interest. Typically we break-up a ship’s waterplane into ten or twenty equal sections along its length and take the offsets (half-breadths) of the hull form as the basis for the calculations. When the area under the half-breadth curve is computed, it is necessary to double the result to obtain the full form value. We would then have, for the full form: Area = (2S/3)(Y0 + 4Y1 + Y3). We can then add elements for each segment to obtain the total vaklues we seek, as below.
In applying this rule, it is important to note that the closer together the ordinates are spaced, the more accurate the assumptions are, but the more curve elements there will be to calculate. Therefore, when the curve has but little change in shape, the ordinates can be widely spaced, such as in the region of a ship’s midbody sections, while in the bow and stern sections where there are significant changes in shape the ordinates need to be more closely spaced.
If you have double spacing in a vessel’s midbody, the multipliers need to be doubled up, while if you have half- spaces in the bow or stern sections, the multipliers need to be cut in half as noted below:
Half-Spaces Full Spaces
Stations: 0 ½ 1 2 3 4 5 6 7 8
Multiplier: ½ 2 ½
1 4 1
1 4 1
1 4 1
1 etc.
Summed: ½ 2 3/2 4 2 4 2 4 2 etc.
Half-Spaces Full Spaces Double Spaces
Stations: 0 ½ 1 2 3 5 7 etc.
Multiplier: ½ 2 ½
1 4 1
2 8 2 etc.
Summed ½ 2 3/2 4 3 8 etc.
In all cases, you are summing the contributions of two equally spaced area segments that are defined by three ordinates, one at either end of the segment and another in the middle. We must remember then that there are always an even number of spaces in the scheme.
In the case of waterplane offsets, where you are using only half-breadths as ordinates, it is important to remember that the answer only gives half of the water plane area, as noted above, etc.
It it important to understand that this procedure can be applied to curves that are generated for many purposes. For instance, if the curve represents areas, such as for transverse sections of a body, the integrated value represents the volume of the body. Similarly, it the curve represents the first or second moment of an ordinate, the integrated value represents the first or second moment of the area represented by the ordinates.
Using this procedure, we are able to determine the many properties of a body, such as a Waterplane Centroid, or a Waterplane Moment of Inertia. By taking into consideration the unsymmetric character of a hull form, the similar procedure can be used to determine similar characteristics.
I hope this is helpful. If further explanation is needed, just ask.