TRIM DRAG

Basically, trim drag is not any different from the types of drag already discussed. It arises mainly as the result of having to produce a horizontal tail load in order to balance the airplane around its pitching axis. Thus, any drag increment that can be attributed to a finite lift on the horizontal tail con­tributes to the trim drag. Such increments mainly represent changes in the induced drag of the tail. To examine this further, we again write that the sum of the lifts developed by the wing and tail must equal the aircraft’s weight.

L + LT=W

Solving for the wing lift and dividing by qS leads to

Cl.

Here, CLis the wing lift coefficient, CL is the lift coefficient based on the weight and wing area, and CLt is the horizontal tail lift coefficient. The CDi of the wing, accounting for the tail lift, thus becomes

г-* _ ^ Lw

D’w irAew

Cl 2 Cl CLt St (4.38)

u ігАе ттАе CL S

The term [CLi.(St/S)]2 has been dropped as being of higher order. Since СьІтгАе is the term normally defined as CDp it follows from Equation 4.38 that the increment in the induced drag coefficient contributed by the wing because of trim is

ДСд = -2ССі^^: (4.39)

Added to Equation 4.39 the total increment in the induced drag coefficient becomes

In order to gain further insight into the trim drag, consider the simplified configuration shown in Figure 4.28. For the airplane to be in equilibrium, it follows that

Lw + LT = W xLw = (/ – x)LT

where l is the distance from the aerodynamic center of the wing to the aerodynamic center of the tail, x is the distance of the center of gravity aft of the wing’s aerodynamic center. Soling for LT gives

LT = ±W

Thus,

Substituting Equation 4.43 into Equation 4.42 leads to

The ratio of the wingspan to tailspan is of the order of 3, while e is equal approximately to eT. With these magnitudes in mind, Figure 4.29 was pre­pared; it presents the trim drag as a fraction of the original induced drag as influenced by x/l.

Notice that the possibility of a negative trim drag exists for small, positive, center-of-gravity positions. This results from the slight reduction in wing lift, and hence its induced drag, for aft center-of-gravity positions. However, as the center of gravity moves further aft, the induced drag from the tail overrides the saving from the wing so that the net trim drag becomes more positive.

The aerodynamic moment about the airplane’s aerodynamic center was neglected in this analysis. By comparison to the moment contributed by the tail, Mac should be small. Qualitatively, the results of Figure 4.29 should be relatively unaffected by the inclusion of Mac.

The trim drag is usually small, amounting to only 1 or 2% of the total drag of an airplane for the cruise condition. Reference 4.10, for example, lists the trim drag for the Learjet Model 25 as being only 1.5% of the total drag for the cruise condition. As another example, consider the Cherokee once again at an indicated airspeed of 135 mph. At its gross weight of 2400 lb, this corresponds to a CL of 0.322. For this weight the most forward center of gravity allowed by the flight manual is 3% of the chord ahead of the quarter-chord point. With a chord of 63 in. and the distance between the wing and tail aerodynamic center of approximately 13 ft, xll has a value of -0.012. Since b — 3bT, Figure 4.29 gives

= 0.028

Figure 4.29 Effect of center-of-gravity position on trim drag.

With an effective aspect ratio of approximately 3.38, CDi will he equal to

0. 0098, so that = 0 .0003. The parasite drag coefficient is approximately

0. 0037, so that the total CD equals 0.0138. Thus, for the Cherokee in cruise, the trim drag amounts to only 2% of the total drag.