Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
These conditions make it possible to write a system of m 4
linear equations with m 4 unknown quantities
(coefficients of cubic polynomials). That system can be shown in a matrix form
5. MODEL PROPELLER SERIES A B (10)
Gauss’s method of elimination can be used to solve this system.
Solutions of the equation system are coefficients of parametric cubic polynomials, meaning that the spline function is completely determinated.
Properties of an interpolating spline function are: the curve is smooth and passes through all given points;
the level of uncertainty is not increased; and it is determined xЄ(x1,xn).
only on a known domain,
Therefore, it can be said that an interpolating spline function is the best mathematical model which describes a dependence between quantities Y and X, i.e. it best approximates a mechanical spline. This function can also be represented by the curve through the points in Figure 1.
4.3 OUTCOME
A function which expresses a dependence of the quantity Y on the quantity X can be determined in three ways: by one of the expressions (4), (6), or (10). When the expressions (4) or (6) are used, the obtained dependence between the quantities Y and X is statistical, approximate and multifold. This means that the selected functions can be several, and they more or less well describe the requested dependence, but always make an error.
When an interpolating spline function is used, the dependence between the quantities Y and X is "natural"
P Ae
Generally, a propeller series consists of a number of model propellers derived from one basic model, a parent model, by systematic change of its geometric quantities: number of blades Z, expanded area Ae and pitch P. Figure 2 shows how a series is made by changing geometric quantities.
This figure is presented by a table with quantities in a non-dimensional form, see Table 2.
Table 2: Scheme of changing Ae and P of propeller series Ае /A0
P/D
a1 ... an
p1,1 ...
pn,1
p1,2 ...
pn,2
... ... ...
p1,m
and unique. This means that there exists only one spline function, which passes through all points.
pn,m
Depending on the natural environment in which the propellers operate, there are two ways of executing the experimental tests of model propellers. If the influence of cavitation on the properties of a propeller in operation is not examined, then tests are performed in the towing tank in open-water at atmospheric pressure. If the influence of cavitation on the properties of a propeller in operation is examined, then tests are performed in a cavitation tunnel, where the pressure of the environment at which the model propeller operates can be changed. The condition of the environment is presented by the cavitation number
σ = (p0 – pv)/(ρv2/ 2)
Figure 2: Model propeller series with same number of blades (light outlines)
B-46
©2014: The Royal Institution of Naval Architects
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72 |
Page 73 |
Page 74 |
Page 75 |
Page 76 |
Page 77 |
Page 78 |
Page 79 |
Page 80 |
Page 81 |
Page 82 |
Page 83 |
Page 84 |
Page 85 |
Page 86 |
Page 87 |
Page 88