Linearization of the Equations

As we will see, it is quite informative to linearize the foregoing equations about a trimmed condition. In this way, the factors affecting longitudinal dynamic stability are more easily identified than by means of a numerical solution. Also, the normal modes of motion can be examined more readily and their shapes determined.

The momentum relationships previously derived will hold for any ortho­gonal axis system with its origin at the center of gravity. Thus far we have used the zero lift line of the wing to define the x-axis. For linearizing the equations, it is considerably more convenient to choose an x-axis that is aligned with the free-stream velocity in the trimmed condition. The z-axis is again directed “downward,” normal to the x-axis. As such, these axes are referred to as stability axes. Any other orthogonal sets through the center of gravity are known as body axes.

With the choice of stability axes, the U and W velocities will be denoted by

U= U0+u

W=w (9.14)

The lowercase letters indicate incremental velocities relative to their trimmed values. With the choice of axes the trimmed value of W is, of course, equal to zero. U0 represents a steady free-stream velocity. It will be assumed that

so that orders higher than the first in these terms will be neglected. The unsteady angle of attack, a, can now be written as

w

U0+u

(9.15)

Notice that a is the angle of attack of the x-axis, so the lift coefficient must now be written as

Cl = Си + CLa

From Equation 9.15, a is of the same order as u/Uo and wlU0. The X and Z forces and the moment M will be written as the sum of their trim values and small, unsteady increments.

X = X0+AX

Z = Z0 + AZ (9.17)

M = AM

There is no M0 in Equation 9.17 because M equals zero for trim.

Finally, the pitch angle is linearized by writing 0 as

0 = 0o+6> (9.18)

Here, в is assumed small and of the same order as the other incremental values.

The sine and cosine of an angle and its increment can be written as

sin (0o + в) — sin 0o+0 cos 0O cos (0O + в) — cos 0o – в sin 0O

Using these equations and substituting Equations 9.14 to 9.17 into Equations 9.1 to 9.3 results in a linearized form for the equations of motion.

AX – mg cos 0O0 = mu	(9.19a)
AZ – mg sin 0O0 = m U0(a – в)	(9.19b)
AM = Іув Also, for the trimmed condition,	(9.19b)
X0 – mg sin ©o = 0	(9.20a)
Z0 + mg cos 0O = 0	(9.20b)
О II	(9.20c)

Typically, let us consider the incremental force AZ. In coefficient form,

Z = ip[(U0 + u)2+ w2]SCz

As a dimensionless coefficient Cz would be expected to vary with a, q, and the elevator control position, 8. Thus we will assume Cz to be of the form

Cz — Cz<! + Cza + Czfl + CZs 8 Combining these two relationships results in Z = ‘2pU02SCZii + pUoSC^u

+ 2pU02S(Cz<xa + Czfl + Cz& 8) + (higher-order terms)

It follows that,

Ш5да = 2С*й+Сг-“ + С^ + Сг"8 (9.21)

AX involves the thrust, so its development is slightly different. Assuming that T lies along the jc-axis,

X = T — D cos ~a + L sin a

= T0 + AT-l2p[(U0+u)2+w2]S(CD-CLa)

CD and CL can be expressed as

Cd — Сд, + Сда Cl = Сц + Cict

For gliding flight or for a turbojet, one can assume Г to be a constant. Thus, for these cases,

X=T0- {pUjSCa, – pUouSCo, – l2pU02S(CDa – Сц)а or

<9-221

For a piston engine airplane with constant speed propellers, we will assume that the propeller efficiency remains essentially constant as и changes. Thus, the thrust power remains constant, so

(T0 + AT)(U0+u) = T0Uo

or, approximately,

AT = – T0~ (9.23)

L’O

T0 и u_ (m)pU02SU0 *U0

Thus, /or this case,

From Equation 9.20a and 9.20b,

(H2)pUo2S Сц> + Сц ta" 00

Therefore,

(H2)pUo2S (ЗСд + Сц tan ©0) ^ (Сд Сц)а

This equation of Equation 9.22 can be written in the form
where:

CXu = ~2CB[) for jets or gliding flight (9.25a)

Cxu ~ — (ЗСд, + Сц tan ©о) for piston engine airplanes (9.25b)

CXa = C, fl – CD„ (9.26)

In deriving these expressions Cx. and C4 are assumed to be negligible.

The expression for ДМ does not include a dependence on u, since Сщ = 0.

(H2)pU02Sc = См°а + См^ + Cm* 8 (9’27)

t*

In order to nondimensionalize the linearized equations of motion, a time t* is defined that is sometimes referred to as an “air second.”

Real time is then referenced with respect to t*

t

T ~ t*

Thus, for example,

du du dr

dt dr dt

2U0du c dr

Consistent with the earlier definition of q, note that a reference length of cl2 is used in the definition of t*. A dimensionless velocity й will also be used, defined by

fi=7T (9.30)

C’o

The mass of the airplane will be expressed in a dimensionless form, fi, given by

m = /*ps(§) (9.31)

Similarly, the mass moment of inertia is expressed in dimensionless form by

Iy = iypsi~j (9.32)

Equations 9.21 to 9.32, when substituted into Equation 9.19, result in a set of linearized dimensionless equations defining the longitudinal dynamic motion of an airplane about a trimmed condition. In reducing these equa­

tions, it should be noted that, because of the use of stability axes,

CZo = – Cto

Thus, from Equation 9.20b,

and, from Equation 9.20a,

— Сц tan 0q

The nondimensional, linearized equations of motion can be written finally as:

2/xM — Схй — Cxjx + Сцв = 0
2ClqH + 2/u, d — Czaot — (2/x + Cze)© + Сц tan @o0 = Czs 8
— Cm a + iy® — Cm_© = Cm5 8

It is emphasized that all derivatives in these equations are with respect to the dimensionless time r. Thus, in Equation 9.33, © is really the previously defined q.

Terms like CZa, Cz4, CXa, CMa, and CMq are referred to as stability derivatives. It may be necessary to include additional terms involving other stability derivatives in Equation 9.33, depending on the airplane configuration and its operating regime. For example, an airplane operating at transonic speeds will be sensitive to speed changes because of Mach number elfects. Hence a term will have to be added to CXu to account for the increase in CD with Mach number.

If the lift on a wing varies with time, a sheet of spanwise vorticity is shed downstream. If the rate at which the lift changes is sufficiently high, this shed vorticity can significantly change the lift, as a function of a, from the steady-state value. Except for the effect of the unsteady wing wake on the horizontal tail, the motion of an airplane is usually sufficiently slow, so that unsteady aerodynamic effects can be neglected.

One will find a fairly comprehensive discussion of stability derivatives in References 8.2 and 8.11. Following the earlier reference, an estimate of the unsteady effect of the wing on the tail can be made based on the fact that it takes a finite time for the wing wake to be convected downstream to the tail. Thus e(t) at the tail is produced by the wing a at t – At. If a is increasing at a constant rate of a,

e(t) = e„a – eaa At

where eaa would be the steady-state downwash. At is the time required for the wake to be transported from the wing to the tail. This is given ap-

proximately by Дt = 1,1 U0- Thus,

e(t) = eaa – €a6c yr Uo

or

ea = – ea^r (9.34)

The product represents a decrease in the tail angle of attack. Hence, both Z and M will vary with a, so terms – Cz<ka arid – CMaa can be added to Equations 9.33b and 9.33c, respectively. Normally, the effects embodied in ed are small, so they were neglected initially in formulating Equation 9.33.