A new resolution on energy-efficiency regulation of ships was adopted at the 65th session of the Marine Environment Protection Committee (MEPC) of the International Maritime Organization (IMO),...
Using methods of the calculus, we may derive expressions for the area of the curvilinear figure and for various properties of the area. (a) Areas. Let be the area of the elementary rectangle NBPQ. Then , and the entire area under the curve, A, is given by the summation of all such elementary areas, or,
Putting this in the form of a definite integral between the limits 0 and H,
(b) Moments and Centroids. Let , be the first moment of the area of the elementary rectangle NBPQ about axis OY: Then . Hence, the moment of the entire area under the curve about axis 0Y may be written as , which may be expressed as the definite integral,
The distance of the centroid of the area from axis oY is given by the quotient of moment about 0Y divided by area or,
Let be the first moment of the elementary area NBPQ about the baseline OX Then
The moment of the entire area about the baseline becomes,
or in the form of an integral,
The distance of the centroid of the area from the baseline OX is the quotient of moment about the baseline divided by area, or,
(c) Moments of Inertia and Gyradii. Let be the second moment, or moment of inertia, of the area of the elementary rectangle NBPQ about axis OY. Then Hence the moment of inertia of the entire area under the curve about OY, is,
The gyradius r I of the area about axis OY is given by the square root of the quotient of moment of inertia divided by area, or,
If I be the longitudinal moment of inertia of the area under the curve about a transverse axis through the centroid (axis parallel to the Y-axis), we have by the parallel axis principle of mechanics,
The area under the curve AG may also be considered as comprised of many small squares such as , Figure above. Then let , be the second moment, or moment of inertia, of the area of the elementary square about the baseline OX. But . Thus the moment of inertia of the entire area under the curve about the baseline It may be written as , or,
The gyradius r t of the area about the baseline OX is given by,
In order to evaluate these integrals, naval architects again overcome the limitation that most ship lines are not represented by mathematical formulas by utilizing approximate rules of integration. A rule of integration assumes that the curve to be integrated is closely approximated by a mathematical curvethat has the same offsets (or ordinates) as the actual ship curve at a series of stations. The desired integrals are then approximated by taking the sum of products of offsets and particular multipliers developed for each rule and multiplying the sum by an integrating factor, as described in the nofollowing subsections.