Ryaboshtan O.F., Milenin A.M., PhD
Kharkiv Petro Vasylenko National Technical University
of Agriculture
Constructing surfaces and contours
of gas turbine blades by means of differential equations with partial
derivatives
As a result, the
solution of differential equations with partial derivatives can be associated
with constructing the surfaces of gas turbine blades with a lot of differential
geometry of the initial conditions. This raises several problems, ranging from
the ability to solve a particular equation, and ending with the need to obtain
a specific type of surface with the desired properties. Among the large variety
of equations to choose the right, giving the desired number of lines, of which
then turns the surface of the gas turbine blades, which satisfies the
requirements.
Consider the
differential equation:
(1)
The function 
is called a
solution of differential equations with partial derivatives, if she and her
partial derivatives
, (2)
satisfy the above
equation (1).
In the future, the
condition that x and y are the independent variables are
considered in a bounded region A, and
F and z are continuous and the desired number of times and continuously
differentiable.
Equation (1) can be
obtained from any two-parameter set of functions by differentiation with
respect to x and y parameters of the exception, and the resulting system of
equations.
It is known that
the solution of equation (1) will always be a two-parameter set of functions,
the differential equation which, in turn, is (1).
Always assume that
, (3)
where 
and 
- the partial
derivatives of F in (1) by u and v, respectively. Condition (3) ensures that no critical points on
the integral surface.
Equation (1) for
fixed x, y, z gives the relationship
between the parametric coordinates u and the normal to the surface, starting at
the point x, y, z.
Two-parameter set
of solutions (integral surfaces)
(4)
called a complete
integral, if in this region rank
(5)
is equal to two,
which guarantees the independence of the parameters a and b from each other.
Of the total
integral by differentiation and exclusion can be the whole set of solutions of
(1), depending on an arbitrary function. Each individual decision may be
obtained by setting an arbitrary function 
, followed by differentiation with respect to a received level of 
and the 
(building envelope). The
function b(a), we will determine the
satisfaction of the conditions of the original data.
In addition to
complete and general integrals distinguish special integrals which express the
envelope of a two-parameter set (4) and determined from the system
, 
, 
(6)
Note that the
singular integral can be obtained from the differential equation (1), making
the system
(7)
Geometric image
expressed by a particular integral is the integral of the envelope of the set
of surfaces defined by the general integrals obtained from the total.
The primary method
by which we construct the surface of gas turbine blades, will be to obtain the
complete integral of which is allocated as a result of the general satisfaction
of specified conditions.
The equation of the
first order of smoothness, regardless of the type (linear, nonlinear ...) as a
result of the decision makes one immune function, which, generally speaking,
can satisfy the initial conditions in a given curve, or the incidence of
contact to a given surface.