LONGITUDINAL DYNAMIC STABILITY AND CONTROL
As an introduction to the subject of longitudinal dynamic stability and control, this chapter will treat the airplane as a rigid body having three degrees of freedom. These consist of rotation about the уaxis and translations in the x and 2 directions. In the general case, a rigid body will have six degrees of freedom: three rotations and three translations. However, because of the symmetry of an airplane, there is very little coupling between longitudinal and lateral motion, so that for most purposes the two motions can be considered independent of each other. A more formal proof of this statement can be found in advanced texts on the subject.
We cannot apply Newton’s second law of motion directly to the airplane. The forces and velocities with which we are concerned are all related to a coordinate system that is fixed to the airplane and moving with it. This is a noninertial reference frame. It is therefore necessary to perform a transformation from this moving frame of reference to one that is fixed (for our purposes, relative to the Earth).
EQUATIONS OF MOTION
Consider Figure 9.1a, which depicts the airplane coordinate system at some instant of time, t. The xaxis is aligned with the wing zero lift line, as before, and passes through the center of gravity. The zaxis is normal to the xaxis and directed downward. The resultant linear velocity of the center of gravity is denoted by V, with components of U and W along the x – and zaxes, respectively. At this instant the x – and zaxes are aligned with another set of axes, x’ and z’, which are fixed axes. Generally, the airplane is pitching upward at a rate of 0 and is accelerating both linearly and angularly.
At an increment of time, Д t, later, the picture will be as shown in Figure 9.1 b. The x – z axes have rotated through a pitch angle of 0 At relative to the fixed x’ – z’. Also, the velocity components have been incremented as shown. We now
consider the change in the components of the momentum vectors in the inertial frame of reference that occurred during the time interval of At.
If m represents the mass of the airplane, then at time t + At, the momentum in the x’ direction is given by
mom, = m[(U + AU) cos (0 At) + (W + Д W) sin (0 Д/)]
At time t, this momentum component is simply
mom, = mU
Thus the time rate of change of momentum in the x’ direction will equal
mom, (/ + Д()тот, (/)
Iim———— —————
д/»о At
From Newton’s second law of motion, the sum of the forces acting in the x’ direction, which coincides with x in the limit, is equal to this rate of momentum. Thus,
Xmg sin® = m(U+W®) (9.1)
Similarly, in the z direction,
Z + mg cos 0 = m( W – 1/0) (9.2)
X and Z represent the sum of the aerodynamic forces (including the thrust) in the jc and z directions, respectively. A dot over a quantity indicates the derivative with respect to time.
The third equation of motion can be written directly as
M = Iy®
where M is the sum of the aerodynamic moments. An alternate derivation of Equations 9.1, 9.2, and 9.3 is found at the beginning of Chapter Ten.
The righthand sides of Equations 9.1 and 9.2 can also be obtained by use of vector calculus. It is shown in Reference 8.2 that the time derivative of a vector defined in a rotating reference frame is given by
SXlSt is the apparent derivative as viewed in the moving reference system, to is the angular velocity vector for. the moving reference system.
The velocity components U and W can be expressed in terms of the resultant velocity, V, and angle of attack, a.
U = V cos a W = V sin a
Therefore,
U = V cos a – Va sin a W = V sin a + Va cos a
In terms of V and a, Equations 9.1 and 9.2 become
X — mg sin 0 = m[ V cos a – Va sin a + V® sin a] (9.5a)
Z + mg cos ® = m[ V sin a + Va cos a – V® cos a] (9.5b)
Generally, X is a function of V, a, 0, and higher derivatives of these quantities. The same is true of Z and M. In addition, X, Z, and M also depend on a control angle and, possibly, its derivatives as a function of time. If 8 is given as a function of time, and if X, Z, and M can be determined as a function of V, а, Ф, and S, then the set of nonlinear, simultaneous differential Equations 9.3 and 9.5 can be integrated numerically to determine V, a, and 0
as a function of time. The position and orientation of the airplane relative to
the fixed axes, x’ and z’, can then be determined from
x’ = x’(t = 0) + I VXcos a cos 0 + sin a sin 0) dt Jo 
(9.6) 
z’ = zt = 0) + I VXsin a cos 0 – cos a sin 0) dt Jo 
(9.7) 
0 = ©(* = 0) + f В dt Jo 
(9.8) 
The X and Z forces can be written in terms of the lift, drag, and thrust by reference to Figure 9.2.
Figure 9.2 Resolution of forces along x – and zaxes. 
X = L sin a + T cos ©r – D cos a (9.9)
Z = —L cos a – T sin @T —D sin a (9.10)
©r is the inclination of the thrust vector. The pitching moment resulting from the offset of the thrust line is included in the sum of the aerodynamic moments, M.
The lift can be written as
L = PV2S(CLa + CLfi + CLs8) 
(9.П) 
Similarly, the drag is calculated from 

D = l2pV2SCD 
(9.12) 
where 

Cd — Cd(Cl) 
The moment is determined from
M = ipV2Sc(CMa + CM. q + CMs8) + TZP (9.13)
These relationships are approximate in several ways. First, in a strict sense, the airplane CD is not a unique function of the airplane CL. One must examine the division between the wing lift and the tail lift to determine the total CD for a total Cl – For most purposes, however, it is sufficient to assume that the usual airplane drag polar holds for relating CD to CL. Second, unsteady aerodynamic effects are not included at this point. These will be incorporated in terms such as CL. and Сщ. Also, when the airplane accelerates, the air surrounding it accelerates; this leads to an effective increase in the mass of the airplane. Socalled “added mass” effects are generally negligible when considering airplane motion, but become important for lighterthanair (LTA) vehicles, where the mass of the displaced air is nearly equal to the mass of the
vehicle. This leads to terms such as Xy and Mg, which will not be considered here.
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