Page 39 - Introduction to Naval Architecture

SHIP FORM CALCULATIONS 27

If in doubt about the multiplier to be used, a simple check can be
applied by considering the area or moment of a simple rectangle.

TCHEBYCHEFFS RULES

In arriving at Simpson's rules, equally spaced ordinates were used and
varying multipliers for the ordinates deduced. The equations con-
cerned can equally well be solved to find the spacing needed for
ordinates if the multipliers are to be unity. For simplicity the curve is
assumed to be centred upon the origin, x - 0, with the ordinates
arranged symmetrically about the origin. Thus for an odd number of
ordinates the middle one will be at the origin. Rules so derived are
known as Tchebycheff rules and they can be represented by the
equation:

Span of curve on #-axis X Sum of ordinates
A = —
Number of ordinates

Thus for a curve spanning two units, 2/i, and defined by three
ordinates:

The spacings required of the ordinates are given in Table 3.2.

Table 3,2

Number of Spacing each side of origin -r the half length
ordinates

2 0.5773
3 0 0.7071
4 0.1876 0.7947
5 0 0.3745 0.8325
6 0.2666 0.4225 0.8662
7 0 0.3239 0.5297 0.8839
8 0.1026 0.4062 0.5938 0.8974
9 0 0.1679 0.5288 0.6010 0.9116
10 0.0838 0.3127 0.5000 0.6873 0.9162