Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
same time. Results of the measurements are then entered in a table such as Table 1.
Table 1: Values of measured quantities x1 y1
x2 y2
....... .......
xi yi
....... .......
xn yn
In order to see how the quantity Y depends on the quantity X, results of the measurements are showed graphically in such a way that in a system of coordinates XOY a pair of values (xi, yi) is presented by the point T(xi, yi), where xi is its abscissa and yi is its ordinate (see Figure 1).
Y
mechanical spline and spline function y = f (x)
T1(x1,y1) Tn(x n,yn) 4. MATHEMATICAL INTERPOLATION
We often have the results of the measurements presented graphically as a curve, but we need them in the form of data in a table. In this case the curve is digitized and the results written in a table.
The emergence of computers has enabled the application of different expert software in shipbuilding practice. To design a computer program which represents a certain process, phenomenon or event, it is necessary to have a mathematical model, i.e. it has to be described by mathematical expressions.
A mathematical description of the dependence between variables Y and X gets down to a selection of a certain mathematical function that is considered to be able to most accurately express
the dependence between
variables. A mathematical function is an expression written as an equation
Y = f(X). X Figure 1: Data points in a system of coordinates
2. FINDING A DEPENDENCE The task of finding a dependence between the quantities Y and X gets down to the task of passing a curve through the points plotted in a coordinate system. That curve represents the wanted dependence. By its nature, this task is a task of interpolation.
The task can be done in two ways: graphically and mathematically.
3. GRAPHICAL INTERPOLATION
Using a long flexible wooden (or metal or plastic) strip, a curve is drawn through all points entered in the coordinate system (see Figure 1). The curve obtained in this way represents a “natural” dependence between the quantities Y and X. The value of the quantity Y for a given value of the quantity X is obtained by reading from this curve. Since the curve passes through all given points, this method does not produce any more errors and the value of uncertainty is the same.
For getting a smooth and logical line it is sometimes
necessary to move plotted points. This action is called "fairing of curves". Graphical presentation of the results of measurement and calculations, and fairing of curves are well-known and very much applied actions in shipbuilding practice.
This flexible wooden (or metal or plastic) strip is called a spline.
where i denotes the difference between measured yi and the calculated value of the function f(xi).
There are two kinds of functions: linear and nonlinear. Suppose we have a function f(x) and a real number
y = f(x) , λ R .
where denotes the error. The developed expression (2) is of the following form
yf(x ) ε 11 1 yf(x ) ε 22 2 yf(x ) ε
....................... nn n
(3) (1)
How does it work? First, X is determined, then it is put in the expression and the obtained result
is Y. If an
independent variable X takes the values xi from the table (obtained by measurement), the following will inevitably happen:
the calculated values of dependent variable Y will be different from its measured values.
Therefore, it can be said that a mathematical function cannot absolutely represent the “natural” dependence of measured quantities Y and X, but can represent it more or less accurately. Thus, the use of a mathematical function for the purpose of describing dependence between two measured quantities Y and X brings along an error. This is written as
Y f(X) ε (2)
B-44
©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