Axisymmetric thermal deformations of paraboloid mirrors, due to heating, are examined for a mirror with a optical axis oriented toward the Sun. A governing differential equation is derived using Mushtari-Donnel-Vlasov simplifications, and a solution is presented which makes it possible to determine the principal deformation characteristics

Bairamov, R.

Fokin, V. G.

Machuev, Y. I.

Nazarov, A.

Sokolov, Y. V.

Solodovnikova, L. A.

English

NASA Technical Reports Server

NASA TECHNICAL MEMORANDUM NASA TM-77935
THERMAL DEFORMATION OF CONCENTRATORS IN AN AXISYMMETRIC
TEMPERATURE FIELD
R. Bairaraov, Yu. I. Machuev, A. Nazarov, Ye. V. Sokolov,
L. A. Solodovnikova and V. G. Fokin
(NASA-TM-77935) T H E R M A L DEFOLIATION OF N86-21977
COHCENTBATORS IE AN A X I S t f H M E T E I C TEMPERATURE
FIELD (National Aeronautics aud Space
Administration) 12 p fie A02/HJ A01 CSCL 10A Onclas
05802
Translation of "Termodeformatsii kontsentratorov v osesirametrichnom
temperaturnom pole", Akademiya Nauk Turkmenskoy SSR, Izvestiya,
Seriya Fiziko-tekhnicheskikh, Khimicheskikh i Geologicheskikh Nauk,
No. 2, 1981, pp. 26-31
THIS COPYRIGHTED SOVIET WOKS IB REPRO-
DUCED AND SOLD BY NTIS UNDEB tlOENSB
FBOM VAAP, THE SOVI3T COPYRIGHT AGENCY.
NO FURTHER COPYING IS PERMITTED WITHOUT
PERMISSION FROM VAAP.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546 DECEMBER 1985
https://ntrs.nasa.gov/search.jsp?R=19860012506 2020-03-20T14:56:21+00:00Z
S T A N D A R D T I T L E
1. R«po«i No.
TM-77935
7. Go*«rn/««nt H*.
^-. ** '""THERMAL DEFORMATION OF CON-
CENTRATORS IN AN AXISYMMETRIC TEMPERA-
TURE FIELD
>. R«lpl«mVC«r«Ug Ht.
5. R«po<( O.u
December 1985
4. P«i(
7.
R. Bairamov, Yu. I. Machuev,.A. Nazarov,
Ye. V. Sokolov, L. A. Solodovnikova and
V. C! . Fokin
'g*sil«ti««
10. W«rt Unit N..
Leo Kanner Associates
Redwood City, California 94063
II. Con««Cf «« C«««l N«
NASW- 4005
National Aeronautics and Space Admini-
stration, Washington, D.C. 20546
Translation
K. Aj«n<f Col*
IS. iv;;lir-.<ntc>ir NeKi
Translation of "Termodeformatsii kontsentratorov v osesimme-
trichnom temperaturnom pole" , Akademiya Nauk Turkmenskoy SSR,
Izvestiya, Seriya Fiziko-tekhnicheskikh, Khimicheskikh i Geo-
logicheskikh Nauk, No. 2, 1981, pp. 26-31
(A81-32157) (UDC 662.977.537.22)
U.
Axisymmetric thermal deformations of paraboloid mirrors, due
to heating, are examined for a. mirror with an optical axis
oriented toward the sun. A governing differential equation
is derived using Mushtari^.Dpnnel-.Ylasov simplifications, and
a solution is presented which makes it possible to determine
the principal deformation characteristics.
17. C.r »V<4« (Stlcctco t>r Auinor(i)) It. Dlilrlbwl.on Stal«a«nt
THIS COPYRIGHTED SOVIET WORK 13 REPEO-
DUCED AND SOLD BY NTI8 UNDER LICENSE
FROM VAAP, THE SOVIET COPYRIGHT AGENCY-1
NO FtniTHER COPYING IS PERMITTED WITHOUT
PERMISSION FROM VAAP.
IT.
Unclassified Unclassified
31. «.. «( P«g««
10
37.
THERMAL DEFORMATION OF CONCENTRATORS IN AN AXISYMMETRIC
TEMPERATURE FIELD
R. Bairamov, Yu. I. Machuev, A. Nazarov, Ye. V. Sokolov,
L. A. Solodovnikova and V. G. Fokin
Paraboloid mirrors (concentrators) are widely used in solar /26'
energy technology, radioastronomy and other branches of science
and technology.
The constant improvement of designs, in order to increase
the output and efficiency of units, is accompanied by an increase
in the requirements placed on the effectiveness of all systems,
including paraboloid mirrors. One of the means of increasing the
effectiveness of mirrors and reducing the dissipation of output is
to increase the precision of manufacture of the reflecting surfaces
of the mirrors. In this connection, the problem arises of re-
taining the initial precision under operating;iconditions,. when the
design is subjected to the effect of various external factors.
One may take these factors into account by introducing corrections
into the control system, which compensate for the changes in the
parameters of the mirror, brought about by distortions of the re-
flecting surface.
The determination of the necessary corrections is possible
using measuring systems, but the effectiveness of these systems
depends largely on how accurately the behavior of the design will
be predicted >in .various positions and under, various environmental
conditions.
Units, which operate under the open sky,., are subjected to the
effects of climatic factors, including solar radiation. In this
case, the temperature field of the design, and the field of tempera-
ture deformations which corresponds to it, depend on variation in
the temperature of the surrounding air, changes in the intensity of
solar radiation, the position of the mirror relative to the direc-
*Numbers in the margin indicate pagination in the foreign text.
tion of the solar radiation, and on the wind velocity and direc-
tion.
A uniform change in the temperature of the structure causes a
similar increase or decrease in its dimensions without distortion
of its shape, with the exception of the case when the structure is
manfuactured from materials which have different coefficients of
thermal expansion.
Non-uniform heating causes temperature deformations, which lead
to distortion of the profile of the reflecting surface, a change in
the focal length, and deviation of the optical axis of the mirror.
In the given article, we examine axisymmetric temperature de-
formations of paraboloid mirrors (Fig. 1), brought about by their
heating. Such deformations take place, for example, in the case
when the optical axis of the mirror is directed towards the sun. /27
In this case, the desired magnitudes are most frequently the de-
flections and angles of rotation at individual points of the
mirror. Also of oceasioria-1 interest are the magnitudes which char-
acterize the stressed state of the mirror. These include the
forces TI, Tz/ Q and the moments MI and Ma.
Fig. 1. Paraboloid, closed at
the apex: M(V«") is a random
point on the surface of the
mirror; A is the projection of
the point M on the plane xy(z=0);
!is the polar angle.
Fig. 2. Paraboloid, open at the
apex: 1—upper edge; 2—lower
edge.
We will assume that the surface of the paraboloid mirror is
an envelope of rotation, the mean surface of which, in cylindrical
coordinates, is described by the equation
z-a, r+bt r»+c, r», (I).
where ai, bi, GI are some constants, which characterize the appear-
ance of the surface of the mirror.
The resolving differential equation of the axisymmetric tempera-
ture deformation of the envelope of rotation may be obtained on the
basis of Mushtari-Donnel-Vlasov isimplif ications , with the enlistment
of some supplementary assumptions relative to the slope of the
envelope [1], [2], [3]. This equation has the form
' i dv , r _. . . \ at i
Considering relationship (1), we obtain
r
' IfrT- +'?•- 4. |f AT./ (Ar + Br* + Cr») -\\V =
where A=ai , B=2bi , C=3ci.
The homogeneous differential equation, which corresponds to (2) ,
has the appearance
The difficulty of its solution consists of the fact that it
may not be reduced to any of the known types of differential equa-
tions with variable coefficients.
Insofar as the coef f icients of equation (3) are analytical
functions of r, one of its partial solutions, according to [4],
may be found in the form of an exponential series of the type
(4)
1-0
where n, a and £ are constants to be determined
o J
By substituting the magnitude TJ into (3) and comparing the
coefficients with identical exponents of r,. we. will have .n=l.. In
this case, we will also obtain recurrent relationships for deter-
mining the coefficients a and 3 . They will take on the form:
(5)
where a=EhY"*A, b=EhylfB, C=EhY'*C.
Without disrupting the continuity of the obtained solution,
one may assume that cto=l, Bo=0.
The second partial solution of differential equation (3),
according to [4], is found according to the. formula
- V, In r . (6)
where v and 6 are coefficients which are also to be determined.1
 s s
The recurrent relationships for determining the coefficients
Y and 6 are written in the form
7.=
I
(7)
(5 - 2)
where a, b, c are the very same magnitudes as in the system of equa-
tions (5) .
With regard for (7), one may represent dependence (6) in the
form
i- V(L-H»f)r"'f (8)
where YO=~ 2a
2
G"+b2
The general solution of the homogeneous differential equation /29
(3) is written in the form
TO-CM V,+C,.9 V,. (tf)
where C and C are complex random constants.
J. ) U Z ^  U
We would note that, with A=C=0, the functions YX and ¥2 change
into Ii(r'VT;) and YI (rjlT,) , respectively, where Ii and YI are cylin-
drical functions of the first order, of the first .and second type,
of the argument r VT.
The partial solution of the heterogeneous differential equation
(21) is found according to the general law. Thus, it may be sought
according to the method of variation of the random constants, or by
representing it in the form of a series according to increasing
powers of r.
Thus, the general solution of the heterogeneous differential
equation (2) may be represented in the following form:
Vi-C,,, V.+C,., M't+V,, (9")
where 41 is some partial solution of the differential equation (21).
Insofar as the random constants CL
 Q and €„ _ and the functions
YI, Y2 and Y may be represented in the form
ORIGINAL PAGE B
OF
 POOR QUALITY
o-C, HD,;
<f >«
the general solution is written in the following manner:
V- (C.^ -D.ti-l-CjTi-Diti+f , ) -f
).; (9)
The obtained solution contains four random constants; therefore,
two each of the boundary value conditions should be fulfilled at each
edge of the envelope, which is not closed at the apex (Fig. 2, r=a,
r=b) .
For the freely-displaced edge, we have
M,-0; r,-0. (10)
This case of the boundary value conditions corresponds, for
example, to a paraboloid mirror attached at the apex. In this case,
the conditions (10) correspond to the edge of the mirror which is
not attached by any force elements whatsoever, which hinder its
free deformation.
For a rigidly fixed edge
v,-0; f,-0. (II)
Condition (11) corresponds, for example, to fastening of the
mirror at its upper edge. Also possible are combinations of the
boundary value conditions (10) and (11).
If the envelope is closed at the apex (Fig. 1), then C2=D2=0,
insofar as
 2(0)=«> and \J)2(0)=°°. In this case, the general solution
contains only two random constants, which are determined from the /30
boundary value conditions on the remaining edge of the envelope,
which also solves the posed problem.
Thus, dependence (9), together with the boundary value con-
ditions (10) and (11), makes it possible to find the desired func-
tion ¥, via which all of the magnitudes, which are of interest to
us for practical calculations, may be expressed.
Thus , the angle of deflection of .the .random point of the. mean
surface is determined from the relationship
(12)
where Re^ is an actual part of the function ¥.
The deflection is expressed via the angle of deflection according
to the formula
. (13)
where the constant too may be placed equal to zero, since it char-
acterizes the overall displacement of the mirror as a rigid whole.
The bending moments and the intersecting force are found from
the relationships
4 ii -i-M,; (14)
(15)
dt
*-+r™>. (16)
The normal forces, which act on the mean surface, are expressed
via the function of stresses $, which is equal to
<l>--^/mV, (17)
where Im^ is an imaginary part of the function T.
In this case
7", = -y -
rfOi
*„• (19)
The radial displacement of the random point of the mean sur-
face has the form
/V<ti \ ,/. f»
- P, '<, r. (20)
The circular deformation and the change in curvature are
written in the form
(T)~nio+Mcpi
. (20
The examined case of axisymmetric temperature deformation is /31
characteristic for the paraboloid mirrors or units which operate
in a "tracking the sun" mode.
In this case, deviations of the points of the reflecting sur-
face relative to the like points of the theoretical paraboloid
cause a change in the focal length.
This change may be determined as the difference between the
focal length of the initial and the approximating surfaces.
The surface of the approximating paraboloid may be calculated
on the basis of the obtained temperature displacements of the points
of the surface of the given paraboloid, proceeding from the con-
dition of obtaining the least deviations of the points of the sur-
fact from the theoretical profile with a new focal length.
Adopted Designations
r is the radial coordinate of the random point of the mean surface
of the mirror; u is the deflection of the random point of the mean
surface of the mirror; ^u^-iy"** is a complex function; u is the
angle of deflection of the mean surface; $ is the function of
stresses on the mean surface of the mirror; y'*=2/3 (I-y3) /Eh is the
coefficient of rigidity; E, y, g are the modulus of elasticity,
the Poisson coefficient, and the coefficient of linear expansion of
the material; h is the thickness of the envelope; RI and R2 are the
main radii of curvature,
i _<n t i j di_
~K~t ~ ~<tr* ' M = r rfr :
Mi=(l+y) DBi gradt=temperature moment: gradt=At/h is the temperature
gradient according to the thickness of the envelope; At is the
difference in temperatures of the outer and inner surfaces of the
mirror; t is the temperature of the mean surface; D=Eh2/12 (1-y3)m G 3.H
is the cylindrical rigidity; TI is the normal radial force; T2 is
the normal circular force; MI is the radial bending moment; Ma is
the circular bending moment; Q is the intersecting force; u is the
radial displacement; £2 is the circular deformation; x2 is the
change in curvature of the envelope in the circular direction.
REFERENCES
1. Vlasov, V. Z., Obshchaya teoriya obolochek [General Theory of
of Envelopes], Moscow, "Gostekhizdat" Publishers, 1949.
2. Novozhilov, V. V., Teoriya tonkikh obolochek [Theory of Thin
Envelopes], Leningrad, "Sudpromgiz" Publishers, 1962.
3. Vol'mir, A. S., Ustoychivost! uprugikh sistern [Stability of
Elastic Systems], Moscow, "FM" Publishers, 1963.
4. Smirnov, V. I., Kurs vysshey matematiki [Course in Higher Mathe-
matics] , Vol. Ill, Part II, Leningrad-Moscow, "GITTL" Pub-
lishers, 1949.
10


Thermal deformation of concentrators in an axisymmetric temperature field

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