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

 

Construction surfaces of gas turbine blades of the plurality of the envelope for the given conditions

 

Let be a set of surfaces

,                                          (1)

Surface corresponding to the given initial conditions, constructed overlay  links to the parameters and conduct of the envelope obtained  surfaces.

As initial conditions (differential geometric elements and conditions) will be considered:

- curve (surface must be incident to her);

- surface (the desired surface must be tangent to it);

- linear strip (surface must be incident to the curve carrier strip and normal equipment must be normal to the surface);

- band II order (except meet the requirements of the preceding paragraph must match the second partial derivatives of the strip and the surface).

If the initial conditions as curve

,     ,                                            (2)

It is necessary to fulfill two conditions. Conditions incidence requires that equation (2) satisfying (1), ie,

=0,                                                     (3)

On the other hand, the tangents to the curve (2) should be perpendicular to the surface normal (1)

,                                                 (4)

where the , ,   - partial derivatives  calculated under the condition (2). , ,  - derivatives on . Equation (4) is equivalent to differentiation (3) with respect to  , the ones . For surfaces  - parameter set (1) is necessary from the equations (3) and (4) eliminate the parameter and determine a constraint equation

,                                                                (5)

so that  curve gives  equations (5) of which is dependent  parameters of one and the envelope is determined.

If the initial conditions given surface

,                                                             (6)

which should cover the desired surface, the equation for the relationship between the parameters must be based on the condition that the coordinates of surface normals (1) and (6) along the line of contact, ie,

  ,                                                    (7)

where the  and  - coordinates of the normal vector.

According to the scheme of Monge equations (1) and (6) and (7) are excluded coordinates , ,  the contact point. Obtain

.                                                            (8)

It is easy to see that the surface of a variety of (1) can be provided to the touch  surface, which gives  (8) between  parameters. As before, we find the dependence  one parameter, for example,  , substitute them into (1) and define the envelope.

Suppose that the initial conditions are given as a linear strip, e.g., the curve (2) in which each point is given the normal to the desired surface.

, ,                                                 (9)

where the  and  satisfy linear strip

.                                            (10)

Linear belong lane surface if the curve - the carrier belongs to the surface that gives an equation (3), and surface normals, i.e.

, .                                                 (11)

Thus, each fitted curve gives the relationship between equations 3 and the parameters for the surface from  - parametric set (1) must have  linear strips, which is possible in the odd  .

For example. With a plurality of

                                               (12)

get a surface incident strip 

, , .                                              (13)

The solution.

1. The condition of the incidence curve  the desired surface

                                             (14)

2. Match the normal coordinates

, ,                                               (15)

, .                                               (16)

3. From (14), (15) and (16)

.                                                             (17)

4. Equation (12) with (15), (16) and (17) has the form

.                                                  (18)

5. A one-parameter set (18) fully satisfies the conditions of existence of the envelope. Therefore, differentiating (18) with respect  and are excluded from the equation and (18) the parameter  we have the final equation of the required surface

                                          (19)

Note that the initial conditions in the form of linear strips may be formulated somewhat differently: by using the set (1) to construct a surface tangent to the surface  along line

,                                                    (20)

In this case, the curve (20) is overridden in parametric form (2), and the coordinates along the normal curve determined using surface  .

Let the initial conditions are given band II order, ie, curve (2) equipped with normals (9) and second partial derivatives

, ,                                                 (21)

in equations of the second order band

, .                                          (22)

For obtaining a surface of a plurality of (1) using a procedure II band necessary incidence of band I of the order that gives the equation  and  , the differential conditions II. This means that equation (4) under the condition (2), (9) and (21) it is necessary to eliminate the parameter  using one of equations (4) or (11) obtain  .

As you can see, the job-order band II allows to make 3 constraint equation parameters ,  , so that for a plurality of surface (1) must be set bands II order that it is possible for the values

,    .                                            (23)

Similarly, we can as the initial conditions set the band III of the order, but a special reason to do so, since almost second order smoothness is sufficient for most tasks.

Note that equation (11) are integral equations (4) and the equation (1) is an integral surface (more precisely, a plurality of them) as (11), and to (4), so that the execution conditions specified differential at the solution stated technique is guaranteed.