B.2.2 Estimation of Longitudinal Aerodynamic Derivatives

B.2.2 Estimation of longitudinal aerodynamic derivatives

○We have defined the following longitudinal aerodynamic derivatives:

,

.

○The values of these parameters are estimated from wind tunnel data, through formulas that are derived as follows.

(1) : The lift velocity derivative

· Lift is defined as where may also depend on . As a result, .

--- The subscript indicates that the derivative is evaluated at a constant .

--- If necessary, contribution from the horizon tail can be derived the same way.

· We often express the speed in terms of its Mach number, where is the speed of sound; hence, a better expression for will be:

.

--- Use has been made on the relation:

.

· At low subsonic region, data shows that .

We can also use the Prandtl-Glauert formula:

.

·In transonic region and supersonic region, estimate from data.

(2) : Vertical damping

· We have ; hence,

· In general, , is a function of Mach No. and of the aspect ration :

--- For high : : the Mach No. effect.

--- For low : (Slender wing theory).

(3) : Drag damping

· Again, where may also depend on ; hence,

.

· Estimation of can be made from data

--- The curve of versus closely resembling that of versus . As a result, we can ignore at low Mach number.

(4) : Drag-AOA derivative

· In general, and are related as follows:

.

--- The first term, , is the parasitic drag and the second term is the classical induce drag, for an elliptical wing in particular. For general designs using non-elliptical wing, the second term is merely an approximate formula.

--- The constant, , is called the span efficiency factor; for most conventional subsonic aircraft, lower for lower and higher flight speed.

· Then, .

(5) : Thrust-velocity derivative

· is evaluated at constant throttle positions.

· The value of depends on the kinds of power plant.

--- for propeller; for turbo jet and rocket; and for ram jet.

· In general, is not a design parameter.

· Value of is normally derived from experimental data.

(6) : Speed static stability

· We have defined with also depending on .

As a result,

· But and ; hence;

--- : Radius of gyration of the A/C about the Y-ax-s

· The speed could include effects of flow field and engine thrust , as follows:

.

--- The thrust term depends on engine type and engine location.

· A negative is usually not desired. As nosedown increases while further enhance the pitch down motion.

(7) : AOA static stability

· First of all,

--- ; if the A/C is staticaly stable

--- The value of depends strongly on the tail size and the C.G. position.

(8) : Pitching damping

· From previous discussion: .

· A constant is assumed.

--- A curved flight path results.

--- The tail feels a different angle of

attack:

--- The change in tail AOA results in

· Then, where ; hence,

.

(9) : AOA damping

· With the same token, we have

· This derivative is evaluated at constant .

--- It will result from a plunging motion.

--- In this case, change in AOA is due to change in flight path angle . Also, any change in occurs simultaneously in .

--- However, a causes the downwash to vary, and perturbation in pitching moment arises due to lag in the downwash (見下頁).

--- If the tail feels downwash from time required for air to move from wing to tail, i.e. , then the change in downwash angle due to a will be:

--- The change in pitching moment due to :

· As a result, , and we end up with

.

【Downwash deficiency in a plunging motion】

(10) : Control effectiveness

· We can write .

· We have for usual sign convention. Also, it's magnitude must be large enough to provide enough control to the aircraft.

· for an all-moving tail and for a trailing-edge flap type elevator with being the effectiveness of the elevator.

--- The later is usually a poor design at transonic and supersonic speeds.

《Longitudinal Derivatives - A Summary》

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