Ryaboshtan O.F., Milenin A.M., PhD
Kharkiv Petro Vasylenko National Technical University
of Agriculture
Construction
of portions of the surface of the second order of smoothness
Use
oftransfer functions allows us to construct surfaces of high smoothness, consisting
of quadrangles (portions) of the surface. The construction of such surfaces is
necessary and is widely used in the construction of gas turbine blades.
However, the presence of transition functions leads to an increase in the
number of calculations, enters into the equation of a set of functions of
various parameters, the joint influence on the final geometry of the surface is
difficult to predict.
Form
the equation of the surface intsindentnoy given spatial quadrilateral whose
edges have the equipment to the 2nd order inclusive.
We
consider the problem. Let the edge portions of the surface (u, 0), (u, 1), (0, v), (1, v).
).
An
equation of the line for one frame coordinates 
:
, (1)
where a
and b - a function of v.
From
(1) with u = 0 and u = 1, we obtain two equations:
(2)
Location,
; 
; 
(3)
Equation
(1) has the:
, (4)
or as a determinant:
(5)
Easy to
see that for u = 0, (0, v) = (0,
v); u = 1, (1, v) = (1, v); v = 0, (u,0)
= (u, 0); v = 1, (u, 1) = (u,
1), it
is the execution of the given boundary conditions is guaranteed.
Equation
similar to (5), for set of grid nodes of the surface was obtained and
investigated in relation to contours in 1982, consider equipping the quadrangle
first produce the norms for the border. According to the generalized algorithm
of differential-parametric method set up a system of equations similar to (2)
and reflects the condition of the cell edges intsindentnosti and differential
conditions on the boundary. Solution of the system as a determinant yields the
equation portion of the
, (6)
where 
.
Functions 
, 
, 
, 
shall meet the requirements
of compatibility of values at the nodes portions of the surface. In particular
they can be designed with the help of a polynomial, which provides non-zero
values of the mixed derivatives in the corners.
Direct
verification shows intsindentnosti resulting surface edges of the rectangular
portion of the surface. We verify that
the differential conditions. We differentiate (6) u.
(7)
If u = 0 we have 
=
; if u = 1, 
=
; for v = 0, 
=
; with v = 1, 
=
.
Similarly,
we differentiate with respect to v and we find that for v = 0 we have 
=
; for v = 1, 
=
; for u = 0, 
=
; for u = 1, 
=
.
A
further generalization of (6) leads to an equation of the surface portions,
which ensures the order of smoothness II at the junction.
, (8)
Where 
.
Function 
, 
, 
, 
must satisfy the
compatibility of values at the nodes of the bay. They can take the form (6), if
allowed in the nodal points of zero partial derivatives with respect to
parameters, or otherwise constructed with the help of a polynomial.
It is
clear that a portion of the surface (8) intsindentna given quadrilateral. We
verify that the differential conditions.
Differentiating
(8) for u. Functions in the first row of the determinant are
replaced by their derivatives with respect to u.
. (9)
Make sure that when u
= 0 we have 
=
; for u = 1, 
=
; if v = 0, 
=
; with v = 1, 
=
.
Similarly,
we can differentiate ( 8) with respect to v and to ensure
implementation of the relevant initial conditions.
Verify
the mixed derivatives, for which we differentiate (9) v.The
functions of the first column are replaced by derivatives with respect to v.
Make sure that when u = 0 we have 
=
; with u = 1, 
=
; for v = 0, 
=
; if v = 1, 
=
.
Stand the
corresponding values of u and v in the resulting
equation, we see the implementation of the second order differential terms of
the smoothness of the surface portions at the borders.