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7.3 Reduction of Trimmed Drag F. H. Lutze, Jr.
Virginia Polytechnic Institute and State University It is important at the outset to distinguish between "trim drag" and "trimmed drag." According to the USAF Stability and Control Handbook, (1) the trim drag coefficient is "the drag coefficient increment between the drag coefficient of the complete vehicle in pitch equilibrium and the drag coefficient of the wing-body- vertical tail configuration." The trimmed drag coefficient, on the otherhand, is the drag coefficient of the complete vehicle in pitch equilibrium. It is clear that our interest should be focused on reducing the trimmed drag and not on the nebulous problem of reducing the trim drag penalty. Consequently, emphasis will be placed on the complete configuration and the associated trimmed lift and drag with particular attention paid to the load distribution between the wing-body and the tail surfaces.
Aircraft Equations for Equilibrium t Balance t and Drag The equations for the total aircraft lift and pitching moment coefficient are given by (for small downwash, ,)(2), (6) CL = mwb (a - aowb ) + CL t nt St/S (I) and (2) C m = Cm0wb + Cmawb (a - aow b) - C L nt St/S lt/E t For balance in equilibrium flight, C m = 0, allowing equations (1) and (2) to be solved for the tall lift coefficient and the aircraft angle of attack: C CL m wb + awb CMow b (3) CLt = nt St/S (awb It --+ C m) c _wb
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_-_C L _ Cmow b (4) -aOW b = I t aw_ u _ + C c mawb Equations (3) and (4) govern the distribution of the required llft force between the wing and the tail and insure a zero pitching moment. Several observations can be made concerning these equations: (i) The tall contribution to the aircraft lift coefficient, (C/) = C L 1/t St/S, is a function of wing-body properties, c.g.
"t t . .
position, and lift coefficient. Consequently for a given speed and weight, (CL) t , the tail load can be adjusted by shifting the c.g. position or by changing the wing-body aerodynamic characteristics.
(ii) The expresslon in the denominator common to both equations is independent of the c.g. position.
(iii) The magnitude of CLt determined by equation (3) must be less than CLtmax.
The key to the selection of wing-body parameters and c.g. positlon is the introduction of the aircraft drag coefficient. We would like to select these para- meters to reduce or minimize the drag coefficient for a given lift coefficient. The drag coefficient for the aircraft is given by (for small downwash _):(6) (5) CD + Kwb C 2 + + Kt C 2 + c)nt St/S ] = CDowb Lwb [ (CDot Lt CL t where CLw b = awb (_- aow b) (6) ¢ = EO +j& aE (= ) +- @a a- Be - _owb ¢o The bracketed term in equation (5) is the trim drag coefficients as indicated by the definition at the beginning of the paper.
The problem of the aircraft designer then is to select the wing-body aero- dynamic parameters and c.g. position such that the trimmed drag coefficient given
by equation (5) is minimized in some sense, subjected to the equilibrium lift and zero
pitching moment constraints given by equations (3) and (4). Conventional design practices also require that certain inherent stability specifications be satisfied.
Consequently in current design practices the stability and performance characteristics of an aircraft are virtually determined independent of each other in the sense that one aspect (stability) is considered and then the other (performance). (3) The continuous improvement of digital computational equipment with respect to size, speed and reliabillty has led to the consideration of utilizing digital control systems to malntaln stabillty, reducing the number of constralnts on the selectlon of c .g. position and wing-body aerodynamic parameters to reduce the aircraft drag coefficient. (4), (5) These increased degrees of freedom present a considerable challenge to the aircraft designer leading to some of the new concepts of design associated with controlled configured vehicles. Although it is anticipated that such sophisticated control systems will not be available for general aircraft for a considerable period of tlme the advantages of such systems should nat be completely ignored.
In what follows the concept of reducing the aircraft drag coefficient by appropriate selection of the wing-body aerodynamic parameters and c.g. position, will be examined. This approach is equlvalent to finding the minimum drag for a given speed as opposed to maximum I./D for the aircraft.
C.G. Position for Minimum Drail If we ignore stability requirements it is posslble to determine the c.g.
position which minimizes the drag coefficient for a given lift coefficient. In order to accompllsh th|s, the appropriate terms in (5) are replaced by the expresslons given in (3), (4) and (6). Furthermore the c .g. position can be introduced by noting the following relatlons: t = h t - ho
(7)
C_wb = aWb (h° " hnwb) where hx is position of x in chord lengths behind the leading edge of wing and x = 0 c.g. position = t tall aerodynamic center = nwb wlng-body aerodynamic center
Thederivative of the dragcoefficient with respectto c.g. location can be
evaluated andsetequal to zero. The resulting expression can then be solved for
the c.g. position, ho, whlch provides minimum drag coefficients. The result is _)c _ _C _: _ - : [( 2kwb awb - _-_)ht+( 2k tawb- _-_)hnwb ] CL+2 [_-_ -awb (kwb+kt) ]Cmowb+C oawb (hnwb-ht)
h
!
2 [awb (kwb+kt) i)c
CL (8)
I where kt : kt/(ntSt/S ) Equatlon (8) gives the c.g. posltion for given lift coefficient for minimum drag coefficient in terms of wing-body aerodynamic parameters, tall parameters and geometry.
Several observations concerning equation (8) can be made: (i) The c.g. position for minimum drag coefficient changes with lift coefficient (speed) (il) The c.g. posltlon dictated by (8) is not restricted by stability constraints allowing the possibility of inherent static stability (iii) The c.g. location given by (8) is a function of wing-body and tail aerodynamic parameters and geometry. Consequently the c.g. location for minimum drag coefficient can be changed by judicious selection of these parameters.
Design Characteristics As indicated earlier it is undesirable to have an inherently unstable (or overly stable) aircraft when sophisticated control systems are not available for compensation purposes. Consequently it would be desirable to take advantage of observatlon (iii) and adjust the aerodynamic and geometric parameters in such a manner so that the c.g. position for minimum drag provides the desired static margin. It is possible to approach this problem several ways, two of which will be outlined below.
One method of approach is to treat the drag coefficient as a function of several aerodynamic and geometric parameters, including c.g. position and attempt to find a minimum with respect to all these parameters subject to certain specified constraints (static margin, etc.). The drawback with such a method is that a large
numberof constraintsmay have to be applied to obtain "optimal" parameters which
are realistic.
Another approach takes advantage of equation (8) and the related observa- tion (iii). Here the optimal c.g. position as a function the aerodynamic and geo- metric parameters is determined by (8). Furthermore,the drag coefficient and the neutral point position can be determined in terms of the same set of parameters.
For small changes in the parameters we can approximate the changes in drag coefficient, c.g. position for minimum drag coefficient, and neutral point by: _h o _h o Ah0 * _ +_ + from (8) _Pl APl _P2 Ap2 "'" AC D • _C D _C D _Pl _Pl + _ + from (5) (9) _P2 _P2 "'" • Bh n Bhn Ahn "_l Apl + _P2 z_p2 + "'" Consequently for a given aircraft, the "optimal" c.g. position can be selected from equation (8). Then equations (9) can be used to find the changes in the para- meters Pi required to move the c .g. and neutral point to satisfy static margin requirements and at the same time keep ACD<_ 0. In other words Ah o, Ah n and /_C D are specified and (9) solved for Pl- If there are more or less than three para- meters the solution is either nonunlque or not possible. In such a case a minimum norm. type solution is proposed.
The changes in the parameters can be incorporated by appropriate changes in the wing-body and tail geometry, another area which needs development. Again several observations can be made: (i) Clearly a necessary assumption is that the parameters can be changed independently. This assumption is better for small changes in parameters and decreases in its val idity as the magnitude of the changes increases.
(ii) The calculation of sensitivities in the above method allows an evaluation of the importance of each parameter in achieving a desired goal.
Concludin_ Remarks The above remarks were aimed at examining methods for reducing the aircraft drag coefficlent for a given aircraft llft coefficient, or speed. The emphasis was placed in determining the load distribution between the wing-body comblnation and the tail which would reduce the overall drag coefficient. Furthermore a technique was presented which would allow the determination of various aero- dynamic and geometrlc parameters which would permit the 'best' c.g. location to satisfy inherent stability requirements. Included in the method was the calculation of sensitivlty coefflcients whlch indicates the importance of various parameters in achieving speclfled goals ie. c.g. movement, drag coefficient change, etc.
Prellmlnary results indicate that such an approach is feasible. For given alrcraft parameters c.g. movement alone yields drag coefflclent reduction of the order of 1% over the nominal case for a conventionally designed alrcraft. Tentative results indicate that if the downwash angle at the tail is large enough (at zero llft) then a down load on the tail at the expense of the same additional load on the wing is desireable in reducing the overall drag coefficient. The reason is that the tail llft vector is tilted rearward by the downwash angle. If the tail llft is negative the
q
contribution to the aircraft drag is negative. Under these circumstances the optimum c.g. is forward of the nominal. The amount and direction of movement is sensltlve to thls downwash parameter.
Although the drag reduction due to c.g. movement alone is small, the inclusion of other parameter changes can improve this drag reduction significantly.
How these desired parameters changes can be obtained through wing-body and tall geometry changes still needs to be investigated. Clearly all drag reduction methods should be examined together. (7) References 1. Hoak, D.E. (Project Engineer), USAF Stability and Control DATCOM, October 1960, revised 1974. Section 4.5.3.2.
2. Etkin, B., Dynamics of Atmosphere Flight, John Wiley and Sons, Inc. New York, 1972.
3. Goldstein, S. E. and Combs, C. P., "Trimmed Drag and Maximum Flight Efficiency of Aft Tall and Canard Configurations," AIAA Paper 74-69, 12th Aerospace Sciences Meeting, Washington, D. C. 1974.
4. Lutze, F. H. and Cliff, E. M., "ControI-Conflgured General Avlatlon Aircraft," SAE paper 730303, Business Aircraft Meeting, Wichita, Kans.
April 1973.
o Hood, R.V., "Active Controls Changlng the Rules of Structural Design," Astronautlcs and Aeronautics, August 1972, pp. 50-55.
6. Hofmann, L. G. and Clement, W.F., "Vehicle Design Conslderatlons for Active Control Appllcatlon to Subsonic Transport Aircraft," NASA CR-2408, August 1974.
Redless, H.A. (Editor) "Advanced Control Technology and its Potential for Future Transport Aircraft," Preprint from NASA Symposium of same name, July, 1974.
_G LOCRTtON
Load Distribution for Various CG Positions (Speed Changing) Jt_ t0
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• 4 .6 Load Distribution for Various Speeds (CG Changing) CG /,O-- • 5 =" _- - , _ 0-0 i ' _ -3.69 L.
-.,5" 1' CG Position for Minimum Drag Coefficient WB [nduced Drag vs CG Position N0UCE0 D_A6 .50 I t It .% o TOTAl. (xlo) .r" % % I I
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Ik ./5 induced Drag vs CG Position