Ìàòåìàòèêà/4.Ïðèêëàäíàÿ ìàòåìàòèêà

 O.F. Ryaboshtan, Ph.D. A.M. Milenin, Ph.D. S.M. Skofenko

 

Kharkov National Technical University of Agriculture after P. Vasilenko

 

The use of transition functions for comstructing bypass gas turbine blade

 

Consider bypass arc passing through the points  and  and its derivatives , , , . Arc equation can be written as

,                           (1)

where the  - transition functions from  having a bypass nodes together with their derivatives up to the second and including zero values, except for one that ensures the implementation of the specified conditions:  .

,                                                (2)

As the transition functions, you can use algebraic polynomials, although not ruled out the possibility of using other functions.

If we consider polynomials of the same degree, then  will have the form

                                              (3)

In differentiating (1) to  have

                                (6)

                              (5)

Equations (1) to (4) and (5)  give , , . At  have , , .

Equation (1) can be written in powers 

                                          (6)

                                   (7)

Transition functions can be used to construct the arc II order of smoothness of I commit. In this case, in Equation (1) member  will be omitted, and the coefficients  you can take other

                                          (8)

                                        (9)

The meaning  a first bypass point is given from the additional conditions (e.g., convexity input), then the equation of the second derivative of the arc (8) is determined  and equation (8) is used for the next arc.

If the paint (8) in powers  obtain

(10)

To bypass the zero fixation

,                                             (11)

                                               (12)

Value  and  selecting only the first point bypass, the remaining points are calculated sequentially.

After the transformation (11), we have

.                            (13)

To bypass the I-order smoothness

                                      (14)

                               (15)

or powers                (16)

Arc bypass I order smoothness zero fixation described by the equation

,                                             (17)

where                                                                       (18)

 selected at the first point, and the rest is calculated by the equation

,                                                        (19)

Here is a set of trigonometric polynomials as transition functions to bypass II order smoothness and fixing

         (20)

You can create a similar set of contours for a smooth and fixation, but with the implementation of the necessary conditions preassigned turbine blade design.