2.2

Preliminaries & Definitions

(2.1.1)

.(2.1.2)Remark In (2.1.1), as well as all the following equations, the angle of attack, , is relative to the zero-lift angle of attack. This is implied in (2.1.1), since the lift coefficient equals zero for . Also, (2.1.1) models lift as a linear function of . This model may be superseded by an actual graph of the relation. Definition 2.1 When the net external forces and the pitching moment are both zero, the airplane is said to be in the condition of longitudinal balance. In particular, balance requires that the pitch moment coefficient . Another term used for this balance is equilibrium.Definition 2.2 For an airplane in a longitudinally balanced (or equilibrium) condition, with corresponding angle of attack, , (where we will assume only the case ), suppose that the airplane angle of attack is changed to a value . If the resulting non-zero pitch moment, acts in a manner to tend to bring the angle of attack back toward , then the airplane is said to have positive pitch stiffness. If this moment acts in a manner to drive the angle of attack further away from , then the airplane is said to have negative pitch stiffness.Definition 2.4 The aerodynamic center of an airplane is that location (in 3-D space), about which the total longitudinal pitch moment remains unchanged as a function of the angle of attack, .Definition 2.5 The neutral point of an airplane is that x-axis location, , such that when the plane cg location, h, lies there, then the plane has zero pitch stiffness.Wing Lift and Pitching Moment: (NOTE: All equations below ignore propulsion effects)

(2.2.3)Wing/Body Lift and Pitching Moment:

.(2.2.4)

Remark 1. The authors note that one result of including the body and nacelles is .Tail Lift and Pitching Moment:

.(2.2.13a)

Remark 2. Since , (2.2.13a) can also be expressed as: .(2.2.13b)where .(2.3.12)

Airplane Lift and Pitching Moment: (2.3.16c)

where

(2.3.18)

and .(2.3.19)

(2.3.25b)where

(2.3.22a)

and

(2.3.23)

and where the plane static (stability) margin is:

(2.3.6)2.4 Longitudinal Control Using a Tail Elevator

(2.4.2a)

(2.4.2b)

(2.4.4)

Tailed Plane: ;

(2.4.8)

Tailless Plane: ;

(2.4.9)

ELEVATOR ANGLE TO TRIM .(2.4.10)

Hence:

(2.4.11a)Remark In (2.4.11) we have used the notation , as opposed to .These are one and the same angle of attack. The subscript is included in order to emphasize the fact that now the plane is in a trimmed condition. .(2.4.15)

Definition. Equation (2.4.15) is called the trimmed lift curve.

Figure 2.17 on p.37. Plots of aircraft (re: zero-lift) versus lift coefficient, for basic and trimmed conditions.

.(4a) .(4b)

where .(2c)

Remark. The term det (which, the authors note, is usually negative) does not depend on . From equations (4), we see that both the aircraft trimmed angle of attack, , and the elevator angle, , are linear functions of . .(5a)

.(5b)

This explains why the plot for (i.e. )in Figure 2.18 on p.37 is flat, and why, as increases, the slopes become increasingly steeper. VARIATION OF WITH SPEED EMBED Equation.3

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