The static aeroelastic performance characteristics, divergence velocity, control effectiveness and lift effectiveness are considered in obtaining an optimum weight structure. A typical swept wing structure is used with upper and lower skins, spar and rib thicknesses, and spar cap and vertical post cross-sectional areas as the design parameters. Incompressible aerodynamic strip theory is used to derive the constraint formulations, and aerodynamic load matrices. A Sequential Unconstrained Minimization Technique (SUMT) algorithm is used to optimize the wing structure to meet the desired performance constraints

Bowman, Keith B

Eastep, F. E.

Grandhi, Ramana V.

English

NASA Technical Reports Server

N89- 25171 
OPTIMUM STRUCTURAL DESIGN 
WITH STATIC AEROELASTIC CONSTRAINTS 
2 3 K. B. Bowman ! R. V. Grandhi , and F. E. Eastep 
* 
1 Aerospace Engineer, Air Force Wright Aeronautical Lab., WPAFB, Ohio 
2 Assistant Professor, Mechanical Systems Engineering, Wright State University, 
Dayton, Ohio 
J Professor, Aerospace Engineering, University of Dayton, Dayton, Ohio 
* Now the Wright Research Development Center 
497 
https://ntrs.nasa.gov/search.jsp?R=19890015800 2020-03-20T01:33:43+00:00Z
INTRODUCTION 
In this paper, the static aeroelastic performance characteristics, divergence velocity, 
control effectiveness and lift effectiveness are considered in obtaining an optimum weight 
structure. A typical swept wing structure is used with upper and lower skins, spar and rib 
thicknesses, and spar cap and vertical post cross-sectional areas as the design parameters. 
Incompressible aerodynamic strip theory is used to derive the constraint formulations, and 
aerodynamic load matrices. A Sequential Unconstrained Minimization Technique (SUMT) 
algorithm is used to optimize the wing structure to meet the desired performance 
constraints . 
MINIMUM WEIGHT DESIGN OF 
LIFI'ING SURFACES 
DIVERGENCESPEED 
C O N T R O L ~ S  
FORWARD AND AFT SWEPT WINGS 
STRIP THEORY AERODYNAMICS 
498 
STATIC AEROELASTIC ANALYSIS 
The equation of equilibrium is given in Equation 1. The divergence velocity is 
computed by setting the initial angle of attack and the flap setting to zero in Equation 1. 
The aerodynamic influence coefficient matrix [A] is computed using the strip 
aerodynamics. The lift effectiveness is the ratio of flexible to rigid lift and is computed by 
setting the flap angle in the equilibrium equation to zero, and is given in Equation 2. The 
rolling of an aircraft is affected by the raising and/or lowering of a flap located on the wing. 
Control effectiveness is the measure of the rolling moment produced by a flexible wing to 
that produced by a rigid wing at an angle of attack equal to zero, and is given in Equation 3. 
L j -  
499 
SENSITIVITY ANALYSIS 
Mathematical optimization techniques involve computation of the search direction for 
finding the optimum. This involves gradients of the constraints and the objective function 
with respect to the design variables. In the following, gradients of the aeroelastic behavior 
constraints are given. Calculation of the objective function gradient with respect to the 
design variables is straight forward. The divergence gradients are computed using the left 
and right eigenvectors. The aerodynamic matrices do not vary with the design variables. 
ds- 
dx 
dx - (h}T[Ar]{ar} 
(4 )  
500 
OPTIMIZATION PROBLEM 
The structural weight was minimized with limitations on divergence velocity, lift 
effectiveness and control effectiveness. The design variables were upper and lower skin 
thicknesses, cross-sectional areas of vertical posts and spar caps, and spar and rib 
thicknesses. Lower bounds were imposed on the design variables. The optimization 
problem was solved using quadratic extended interior penalty function method with 
Newton's method of unconstrained minimization. The computer program NEWSUMT-A 
was used. 
Minimize the structural weight, f(x) subject to 
gj(x) = Gj(x) - Gj 2 0 j = 1, 2, ..., m 
and  
( 7 )  
501 
NUMERICAL, RESULB 
The wing shown in Figure 1 is modelled with quadrilateral membrane elements for the 
upper and lower skins, shear panels for the ribs and webs, and rod elements for the spar 
caps and vertical posts. The structure consists of 66 elements, and it is made of aluminum 
with Young's modulus of 10.5~106 psi, ~ 4 . 3 ,  and a weight density of 0.1 lb/in3. The 
wing is swept through 30 degrees representing typical forward-swept wing configurations. 
The wing shown has a 180 in. semispan, a constant chord of 50 in. (i.e. untapered), and a 
symmetric airfoil. 
Y 
(SHEAR PANELS) PARS) VERTICAL POSTS 
PARS) 
Figure 1. Built-up Wing Configuration 
50 2 
Fig. 2 shows lift effectiveness and control effectiveness plotted against velocity. For 
this forward-swept configuration control reversal is higher than the divergence velocity. 
The divergence velocity is 515 ft/sec, and the control reversal (where the effectiveness goes 
to zero) is approximately at 1375 ft/sec. The typical nature of this plot is due to - > 1 as 
reported in Principles of Aeroelasticity by Bisplinghoff and Ashley. 
4R 
9D 
30 
20 
10 
400 450 50C 
J 
4 LIFT EFF. 
I + CONTROLEFF. 
550 600 650 700 
VELOCITY (FT/SEC) 
Figure 2. Control Effectiveness and Lift Effectiveness vs. Velocity 
503 
Initially the structural weight was minimized by imposing a lower limit on the 
divergence velocity. The divergence speed increased to 550.00 ft/sec from the reference 
design value of 515.06 ft/sec. A comparison of the optimum structure's divergence speed 
to NASTRAN analysis revealed less than a couple of percent difference at the initial and 
final designs. The convergence to the optimum is smooth as shown in Fig. 3. 
0 2 4 6 8 10 
ITERATION NUMBER 
Figure 3. Design Iteration History for Divergence Constraint 
504 
2 . 4  
2.2  
2 
11.8 
Ii 
ti 
1 . 6  
1 . 4  
1 . 2  
The lift effectiveness and control effectiveness were calculated at the flight speed of 
373.96 ft/sec, and were monitored as the divergence speed was increased. Several 
divergence velocities were considered for minimizing the structural weight. Fig. 4 shows 
the optimum weight vs. divergence velocity. The divergence speed lower limit was 
increased from 500 to 675 ft/sec. The optimum weight monotonically increased with the 
divergence velocity requirement. Lift effectiveness and control effectiveness both 
decreased with an increase in the divergence speed as shown in Fig. 4. 
o Weight 
0 Control eff. 
Lift eff. 
- 245 
- 220 
- 195 
4 - 
i5 
- i 7 0  ki 
- 145 
- 120 
Fig. 4 Optimum Weight, Lift Effectiveness and 
Control Effectiveness vs. Divergence Velocity 
505 
I The wing was optimized with two constraints imposed at a time to explore the opposite 
trend of the effectiveness values and divergence speeds with the increase of structural 
weight. The structure was optimized such that the divergence velocity be above 550 ft/sec 
and lift effectiveness above 2.0. The initial weight of the structure with all design variables 
set to 1.0 was 1180.80 lbs and the optimum structure had a divergence value of 560.57 
ft/sec, a lift effectiveness of 2.7 1 , and a weight of 225.46 lbs. Optimization with control 
effectiveness in place of lift effectiveness yielded a divergence of 550.03 ft/sec, a control 
effectiveness of 3.18, and a weight of 168.17 lbs. 
constraint values as mentioned above, the structure was optimized and converged to an 
optimum design after six iterations. The final weight was 249.63 lbs with a divergence 
speed of 573.36 ft/sec, a lift effectiveness of 2.53, and a control effectiveness of 2.62. 
Table 1 presents the optimum performance values obtained for different combinations of 
constraints. 
Finally all three constraints were applied to this wing concurrently. Retaining the same 
Table 1. ODtimization Results 
506 
CONCLUSIONS 
The divergence velocity of forward-swept wing configurations is the primary 
characteristic that must be improved. An increase in the structural stiffness of a wing will 
prevent a low divergence speed, but results in an increase of aircraft weight. Also an 
increase in divergence velocity affected the decrease of lift effectiveness and control 
effectiveness. Optimization of a wing for the three static aeroelastic constraints involves 
careful selection of the constraint limits. The studied wing had an initial weight of 1180.9 
lbs, and lift and control effectiveness values of 1.1 1 and 1.12 respectively. Following the 
optimization process that set constraint limits on the effectiveness values of 2.0 and 550.0 
ft/sec on divergence speed, the wing weighed 249.63 lbs, satisfying all constraints. The 
above results demonstrate the capability and feasibilty of optimizing for all three constraints 
concurrently, rather than one at a time. 
MINIMUM WEIGHT DESIGN IS PERFORMED 
FOR STATIC AEROELASTIC CONSTRAINTS 
INCREASE IN DIVERGENCE SPEED 
RESULTS IN A DECREASE OF EFFECTIVENESS 
VALUES 
CAREFUL SELECTION OF STATIC 
AEROELASTIC CONSTRAINT VALUES WILL 
RESULT IN SUCCESSFUL OPTIMIZATION 
PRESENT APPROACH USED NEWSUMT-A 
PROGRAM 
ULTIMATE GOAL IS TO DEVELOP OPTIMALITY 
TECHNIQUES 
507 
BIBLIOGRAPHY 
1. 
2. 
3. 
4. 
5. 
6. 
I 7. 
8.  
, 
9. 
10. 
Bisplinghoff, R.L, Ashley, H., Principles of Aeroelasticity, John Wiley and Sons, 
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Bisplinghoff, R.L, Ashley, H., and R. L. Halfman, Aeroelasticity, Addison- 
Wesley Publishing Company, Massachusetts, 1957, pp. 11-13,474 - 506. 
Bohlmann, J.D., Eckstrom, C.V., and Weisshaar, T.A., "Aeroelastic Tailoring for 
Oblique Wing Lateral Trim," A W A S W A S W A H S  29th Structures, Structural 
Dyn&cs and Materials Conference, Williamsburg, Virginia, AIAA-88-2263, April 
18-20, 1988. 
Eastep, F.E., "Structural Optimization With Static Aeroelastic Constraints," Flight 
Dynamics Laboratory, Wright-Patterson Air Force Base, December 1987. 
Eastep, F.E., Venkayya, V.B., and Tischler, V.A., "Divergence Speed 
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Aircrafr, Vol. 21, No. 11, November 1984, pp 921-923. 
Grandhi, R.V., Thareja, R., and Haftka, R.T., "NEWSUMT - A: A General 
Purpose Program for Constrained Optimization Using Constraint Approximations," 
ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 
Lerner, E. and MarkoWitz, J., "An Efficient Structural Resizing Procedure for 
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April 3-5, 1978, pp. 65-71. 
Schmit, L.A., and Thornton, W.A., "Synthesis of an Airfoil at Supersonic Mach 
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3087, Air Force Flight Dynamics Laboratory, June 1979. 
5 08 


Optimum structural design with static aeroelastic constraints

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