A concept for enhancing the design of control fins for supersonic tactical missiles is described. The concept makes use of aeroelastic tailoring to create fin designs (for given planforms) that limit the variations in hinge moments that can occur during maneuvers involving high load factors and high angles of attack. It combines supersonic nonlinear aerodynamic load calculations with finite-element structural modeling, static and dynamic structural analysis, and optimization. The problem definition is illustrated. The fin is at least partly made up of a composite material. The layup is fixed, and the orientations of the material principal axes are allowed to vary; these are the design variables. The objective is the magnitude of the difference between the chordwise location of the center of pressure and its desired location, calculated for a given flight condition. Three types of constraints can be imposed: upper bounds on static displacements for a given set of load conditions, lower bounds on specified natural frequencies, and upper bounds on the critical flutter damping parameter at a given set of flight speeds and altitudes. The idea is to seek designs that reduce variations in hinge moments that would otherwise occur. The block diagram describes the operation of the computer program that accomplishes these tasks. There is an option for a single analysis in addition to the optimization

Dillenius, M. F. E.

Mcintosh, S. C., Jr.

English

NASA Technical Reports Server

N89- 25169 
STATIC AEROELASTIC ANALYSIS AND TAILORING 
OF MISSILE CONTROL FINS 
S. C. McIntosh, Jr. 
McIntosh Structural Dynamics, Inc. 
Mountain View, California 
M. F. E. Dillenius 
Nielsen Engineering & Research, Inc. 
Mountain View, California 
PRECEDING PAGE BLANK NOT FILMED 
465 
https://ntrs.nasa.gov/search.jsp?R=19890015798 2020-03-20T01:33:46+00:00Z
P R 0 B LEM D EF IN IT10 N 
This paper describes a new concept for enhancing the design of control fins 
for supersonic tactical missiles. The concept makes use of aeroelastic tailoring to 
create findesigns (for given planforms) that limit the variations in hinge moments 
that can occur during maneuvers involving high load factors and high angles of 
attack. It combines supersonic nonlinear aerodynamic load calculations with finite- 
element structural modeling, static and dynamic structural analysis, and optimization. 
The problem definition is illustrated in figure 1. The fin is a t  least partly 
made up of a composite material. The layup is fixed, and the orientations of the 
material principal axes are allowed to vary; these are the design variables. The 
objective is the magnitude of the difference between the chordwise location of the 
center of pressure and its desired location, calculated for a given flight condition. 
Three types of constraints can be imposed -- upper bounds on static displacements 
for a given set of load conditions, lower bounds on specified natural frequencies, 
and upper bounds on the critical flutter damping parameter at a given set of flight 
speeds and altitudes. The idea is to seek designs that reduce variations in hinge 
moments that would otherwise occur. The block diagram a t  the left in figure 1 
describes the operation of the computer program that accomplishes these tasks. There 
is an option for a single analysis in addition to the optimization. Additional details 
concerning this work may be found in reference 1. 
Input. Initialize, Create Structural Hodel 
optimize - T ~ L ~  one Step Evaluate Objective 6 Constraints 
I Evaluate Objective 6 Constraints I No Print Results 
I I I 
Evaluate Objective L Constraints 
Evaluate Active Constraints, 
Gradients 
Objective: Dimensionless chordwise 
center of pressure offset from desired 
position, Ixcp/X - 1.01, for a given 
flight condition. 
Design Variables: Material principal 
axis directions, Bi, for a given 
stacking sequence. 
' ConsLraints: Displacements, 
z / z  - 1.0 5 0, for a given set of 
loa8 conditions 
Frequencies, 1.0 - w /o < 0 
Flutter Speeds, gr - gr 5 0, at fixed 
speed and altitude. 
CP 
r r -  
Figure 1 
466 
STRUCTURAL MODEL 
The example fin is illustrated in figure 2. It is made up of a graphite/epoxy 
composite with the stacking sequence as given in the figure. The fin is modeled with 
triangular bending elements. These are the elements described in reference 2, with 
modifications for anisotropic materials as given in reference 3. For each element, the 
orientation of the material principal axes with respect to the element local coordinate 
axes can be specified. For a given region of the fin, these angles can be specified so 
that a single angle variable governs the overall orientation. Two such regions, 
governed by design variables $1 and $2, are shown in the figure. The outer portion 
of the fin is inactive for tailoring purposes. The fin is anchored to a fixed node at 
the hinge line just inboard of the root by a beam-rod whose stiffness properties 
model the stiffness of the fin actuator and the body backup structure. 
To update the design, the fin stiffness matrix for the active regions must be 
recreated. This is a relatively simple task, since the updating affects only the rotation 
matrices that transform the element constitutive matrices from principal axes to local 
coordinate axes. Gradients of the stiffness matrices are obtained analytically by 
differentiating the expressions for these matrices with respect to the orientation angles. 
Calculations for the constraints and their gradients follow well-known procedures and 
will not be discussed here. Reference 1 can be consulted for additional information. 
Hinge line 
p 1Y Thickness, 
+45 1 2 . 5  
-45  1 2 . 5  
0 5 0 . 0  
+4 5 1 2 . 5  
-4  5 1 2 . 5  
%h 
I 
Figure 2 
467 
AERODYNAMIC LOAD CALCULATION 
The aerodynamic load prediction method (ref. 4) applies to a fin attached to 
an axisymmetric body. The missile components are represented by distributions of 
singularities derived from supersonic linear theory. The missile body is modeled with 
linearly varying supersonic line sources and line doublets. In a finned section, the 
lifting surfaces and the body portion spanned by the lifting surfaces are modeled 
with planar supersonic lifting panels. Fin thickness effects can be represented, if 
desired, by planar source panels. The panel strengths are obtained by satisfying the 
flow addition, 
the fin loads include nonlinear augmentations due to fin leading and side edge flow 
separation at high angles of attack, and (for canard fins) nonlinear loads resulting 
from vortices formed on the forebody for the proper combination of forebody length 
and angle of attack. The tangency boundary condition satisfied on the fin includes 
the changes in streamwise slope 
tangency conditions at a set of control points, one for each panel. In 
caused by elastic fin deformation. 
To compute the fin deformation, the aerodynamic loads at  the control points 
that  preserves 
general nonlinear functions 
described in detail in reference 1, is used to 
deformations and loads. The resultant chordwise center of 
is then used to calculate the objective. Gradients of the objective 
I are interpolated to  loads at the structural node points in a scheme 
overall fin load and moments. Since the fin loads are in 
of fin deflection, an iterative process, 
produce consistent fin 
pressure location 
are computed by finite differencing. 
The static aerodynamic description of the example fin is given in figure 3. 
attack of 15.4 deg, 
altitude of 30,000 ft. The fin 
is assumed, and the body has an ogive 
thickness or nonlinear effects are included in 
The following flight condition was assumed: An included angle of 
a roll angle of 0.0 deg, a Mach number of 1.6, and an 
is undeflected. A vertical plane of symmetry 
nose up to the fin leading edge. No 
this model. 
' XOXOX ' / . I  
X - - -  F l o w  t a n g e n c y  c o n t r o l  p o i n t s  a 
@ - - -  P a n e l  c e n t r o i d s  (a lso c a l l e d  
All d i m e n s i o n s  i n  i n c h e s  ( f e e t )  o x o x o x o n  
aerodynamic  c o n t r o l  p o i n t s )  
A 
0 x 0 x o x o x  
10.454 
6 . 7 3 1  
o x o x o x o  
0 x 0 x 0 x 0 x ( - 5 6 0 9 1 7 )  
v e l o c i t y  p a n e l  
I. 
I? 
N 
W 
0 
O X O X Q X O  
F. 
o x o x o x  
10.352 -4 
(0 .8627)  
2.5 
I d e a l i z e d  
Nose 
t _---_-_---- 20.4 ----- --- (1 .70)  
Figure 3 
FLUTTER CONSTRAINT 
A flutter constraint was the only one considered in this example. The results 
which presents 
ft. Structural damping of 
given by the g = 0.03 crossover 
constraint requires that the damping parameter 
of a flutter analysis at a Mach number of 1.4 are shown in figure 4, 
a velocity-damping plot at a match-point altitude of 43,500 
3% is assumed, so the match-point flutter speed is 
of the first-mode branch. The flutter 
g for this branch be less than 0.03 at this speed and altitude. 
8 
8 
Ln 
N 
8 
c3 
Z 
I 
!+ a: 
0 
0 A A '  A 
A 
A 
A * * 
0 
I X X  I 
A 
A I * 
* 0 
[I] * 
* 0 
VELOCITY, FPS 
Figure 4 
* 
469 
PARAMETRIC SURVEY 
To provide some indication of the behavior of the fin as the principal-axis 
orientations are varied, the analysis-only option of the computer program was 
exercised. The angles 81 and 82 were linked to form a single design variable. Figure 
5 displays the variation of xCp. In view of the nature of the rotation matrix that 
transforms each element constitutive matrix from principal-axis to local coordinate 
directions (see Eq. (4) of ref. l), the quasiharmonic nature of this variation is not 
surprising. The location of xcp for the same fin made of aluminum is also shown. 
0.60 
Figure 5 . 
5 0  90 
47 0 
FIN DEFORMATIONS 
Figure 6 presents perspective plots of the deformed fins for selected values of 
8. The view in these figures is outboard in the x-y plane defined in figure 2, so the 
undeformed fin would be seen as a straight line. With 8 near +45 deg or -45 deg, 
the chordwise flexibility is near maximum, which corresponds to the maximum shift 
in xcp. Contrary to what might be expected, however, the fin chordwise bending is 
concave, rather than convex. This reduces the fin loading near the leading edge, so 
the center of pressure moves aft. Since the center of pressure is always aft of the 
hinge line, the fin also has a nose-down rigid-body rotation, which also appears in 
the plots. 
When 81 and 8 2  are allowed to  be independent, the curve in figure 5 
plane with the xcp surface. This surface was 
enough analyses were performed to  suggest that  the 
shape of an egg carton, where the minima of figure 5 are the 
becomes the intersection of the 81 = 02 
not mapped extensively, but 
surface resembles the 
bottoms of valleys, and the maxima are the tops of peaks. 
(a) e, = e2 = -90 deg 
, 
(b) e1 = e2 = -45 deg 
(d) el = e2 = 4s deg 
(c) B ,  = B2 = 0 deg 
Figure 6 
47 1 
OPTIMIZATION EXAMPLE 1 
Initial Value Final Value 
Design Variable, 8 (rad) 0.700 0.0599 
' Objective, OBJ 0.202 0.0682 
Constraint Fn, G -0.0223 -0.0546 
- 
- 
- - 0 
In the first optimization example, 81 and 82 were linked to form a single 
design variable. The desired value of qp, measured from the fin leading edge at  the 
root, was set to 60% of the root chord c,. The initial value of 8 was 0.7 rad, or 
40.1 deg. The flutter constraint, fixing the critical-mode crossover at  a Mach number 
of this 
example is shown in figure 7. In five iterations, the minimum at 8 = 3.43 deg was 
found. This corresponds very well with the curve of xCp versus 8 in figure 6. 
Attempts to reach the minimum near 8 = 90 deg were not successful because the 
flutter constraint was violated. 
1.4 and an altitude of 43,500 ft, was also imposed. The iteration history for 
The optimization subroutine CONMIN (ref. 5) was used in all of the  
optimization examples. 
0.25 
0.20 
b m 
0.15 . 
$ 
Q 0.10 
4J 
0 cu 
-n 
0.05 
0.0 
2 3 4 5 6 7 0 1 
No. of Iterations 
Figure 7 
47 2 
OPTIMIZATION EXAMPLE 2 
0.15 
b 
An iteration history for the second optimization example is shown in figure 8. 
Here 81 and 82 were unlinked, and the flutter constraint was removed. The initial 
values of 81 and 82 were 0 deg and 45 deg, respectively. Convergence was achieved in 
five clearly the valley 
near the origin in design space suggested by figure 5. 
iterations with 81 = 0.334 deg and 82 = 1.53 deg. This is 
I I I 1 I I 
- - 
Initial Value Final Value 
c 
a, 
c, u 
a, 
.n 
'n 
0 
.5 0.10 
0.05 
0.0 
0 1 2 3 4 5 6 7 
No. of Iterations 
Figure 8 
473 
OPTIMIZATION EXAMPLE 3 
The third optimization example is an attempt to find a neighboring valley in 
the design space. The starting point was a t  01 = 10 deg, 02 = 70 deg, with no 
constraints other than side constraints on the design variables. The iteration history 
for this example is shown in figure 9. The minimum found here is at  01 = 2.20 
deg, 02 = 92.5 deg, a neighboring valley with a minimum not quite as low as that 
near the origin. 
Ironically, the best design -- the one with the most forward location of the 
center of pressure -- is the one initially chosen, with the "zero-deg" plies in the 
spanwise direction. From the standpoint of tailoring, this example is clearly not a 
very attractive one, since the movement of the center of pressure is not substantial, 
Different layups, and particularly those with more bending-twist coupling, would 
produce more appealing results. 
0.2t 
0.15 
b 
0 
al + 
-rl 
c, 
al 
-n 
Q 
0 
m 
. 
u 0.1c 
0.05 
0.0 
I I 1 ' I  I - 1  
I n i t i a l  Value F i n a l  Value 
Design V a r i a b l e ,  8 ( r a d )  0 . 1 7 4  0 . 0 3 8 4  
Design V a r i a b l e ,  €12(rad) 1 . 2 2  1 . 6 2  
O b j e c t i v e ,  O B J  0 .118 0 . 0 7 0 0  
- - - - - - 
0 1 2 3 4 5 6 7 
N o .  of I t e r a t i o n s  
Figure 9 
474 
REFERENCES 
1. Dillenius, M. F. E. and McIntosh, S. C., Jr.: Aeroelastic Procedure for 
Controlling Hinge Moments. AIAA Paper 88-0528, presented at the AIAA 26th 
Aerospace Sciences Meeting (Reno, Nevada), January 1988. 
2. Przemieniecki, J. S.: Theory of Matrix Structural Analysis. McGraw Hill, New 
York, 1968. 
3. Zienkiewicz, 0. C.: 
1977. 
The Finite Element Method. McGraw-Hill (UK), London, 
4. Dillenius, M. F. E.: Program LRCDM2, Improved Aerodynamics Prediction 
Program for Supersonic Canard-Tail Missiles with Axisymmetric Bodies. NASA CR- 
3883, April 1985. 
5. Vanderplaats, G. N.: CONMIN - A Fortran Program for Constrained Function 
Minimization. User's Manual. NASA TM X-62,282, August 1973. Addendum, May 
1978. 
ACKNOWLEDGEMENTS 
The work described in this paper was performed as a Phase I Small Business 
Innovation Research (SBIR) project sponsored under NAVAIR Contract N00019-86- 
C-0032 and is being continued as a Phase I1 SBIR project under NAVAIR 
Contract N00019-88-C-0071. 
475 


Static aeroelastic analysis and tailoring of missile control fins

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