Solution of nth-Order Linear Differential Equation with Constant Coefficients

The equations in Equation 9.33 are described as linear simultaneous differential equations with constant coefficients. The solution of these equa­tions can be written as the sum of a transient solution and a steady-state solution. These are also referred to as homogeneous and particular solutions.

The steady state, or particular solution, is the part of the total solution that satisfies exactly the differential equation, including the forcing function. The transient solution satisfies the differential equation with the forcing function set equal to zero.

A linear differential equation of the nth order can be written in a general form as


To specify the problem completely, initial conditions equal in number to the order of the equation must be given. This usually means specifying x and its derivative up to order n – 1 at t = 0.

The solution to the homogeneous equation [/(t) = 0] is of the form

The nth derivative will be

This equation substituted into Equation 9.46 for f(t) equal to zero leads to an nth degree polynomial for a known as the characteristic equation.

Cn<rn + Cn-itrn 1 + • • • + Сгсг2 + Ci<r + Co= 0

There are n roots or values of a that will satisfy this polynomial. Thus, the general solution for x will be

x = Axe‘r, t + A2e<T2‘ + • ■ ■ + Anea(9.49)

where 2,… <rn are the roots of Equation 9.48. These roots may be positive or negative real numbers or complex numbers with positive or negative real parts. If a complex root exists, its conjugate must also exist. Complex roots always appear in pairs of the form

и = a ±ib

Au A2,…, A„ are real or complex constants to be determined from the initial conditions.

Let us examine the behavior of a term containing a particular type of root. Consider the following items.

1. ea’.

2 e-at.

3 g^a+ibi‘

4. el-a+ib)t’

If a and b are positive real quantities, it is obvious that item 1 will become infinitely large as f approaches This is an unstable transient motion. Conversely, item 2 is stable. Item 3 can be written as



e‘bt = cos bt + і sin bt

the complex root with a positive real part represents an oscillatory motion having an amplitude that increases without limit as t approaches 00. This situation is also unstable. Conversely, item 4 is a stable oscillatory root. Thus, to investigate the stability of the transient solution, one need only consider the roots of the polynomial given by Equation 9.57.

The particular solution is referred to as the steady-state solution since, for a stable system, it remains while the transient solution vanishes as t approaches °°. The particular solution is of the same form as fit) if /(f) can be expressed as a polynomial in t or in the form of

fit) = Aeы or

fit) = A cos a>t + В sin at For example, suppose /(f) were given by

/(f) = At3 + ВеШІ + C cos bit

Then the steady-state solution for x would take the form x = at3 + bt2 + ct + d + fеш‘ + g cos <ot

Notice that when f(t) is a polynomial, x must include all orders of t up to the highest in f(t), even though some of the lower-order terms may be missing in fit).

As a fairly simple (and well-used) example of the foregoing, consider the second-order damped system governed by the equation

x + 2 £wnx + w„2x = f(t) (9.50)

The characteristic equation becomes

<t2 + 2£ып(т + Ып = 0

The roots of this equation are

<гіл = Шп(-£± V<T2-1 (9.51)

C is referred to as the critical damping ratio for reasons that are now obvious. When £ is positive but less than unity, <r will be complex and x will be a damped oscillation. When £ is greater than unity, a will be a negative real number, and the motion will be aperiodic.

Suppose f(t) is given by

fit) = A + Веш (9.52)

The particular solution will be of this same form.

x = Хп+Х, еш

Substituting this into the original equation gives

Xi<i>n — o)2 + 2 £(ti(oni) є‘,>1 + o)nXо= A + Be*0*

In order for this equation to be satisfied for all t, the coefficients of like terms involving t must identically satisfy the equation. Thus,


(tiff — со2 + 2 £(tiO)ni

x – A л0 — 2 (tin

It is instructive to clear the denominator of X of the complex number, so

v _ о w<* ~ ы2 — 2g(ti(ti„i

‘ (шп2 – (ti2)2 + 4^i(tii(ti„2

If we further denote the frequency ratio а>Ішп by r, Xt can be written as

В 1 – r2 – 2£ri 1 w7(l – r2)2 + 4£2r2

Bla>n is simply the static value of X, for a> = 0. Thus the ratio of Xx to its static value can be written as

o>„2Xi _ 1 – r2 2£r

В “ (1 – r2)2 + 4£2r2 ‘ (l-r2)2 + 4£2r2

The complete solution to Equation 9.50, forced in the manner given by Equation 9.52, is thus

= 4- ———з—і

(X’e +Х2Є ) + W +4 r1 x, and x2 are constants to be determined by the values of x and x at t = 0.

The complex form for x and the forcing function may seem somewhat mysterious at first if one is not used to this type of mathematical treatment. Certainly there is nothing imaginary about the longitudinal motion of an airplane. The complex notation is simply a convenient mathematical device to show the amplitude and phase relationships between various terms in the system.

For example, consider Equation 9.53. This can be written in the form
£^=C,(r,£)-«’C2(r, fl

so that the steady-state solution becomes

x = —2 + —2 [С,(г, О – iC2(r, £)]«" (9.55)

(On (On

The complex quantity in the brackets can be further expressed as

[Ci(r, o-iC2(r, D] = Re~i*


b = Vc,2 + c22

ф = tan 1 ~r

Thus Equation 9.55 becomes

x = A + ^ (9.56)

(On (On

If Веш is pictured as a force vector of magnitude В rotating around the origin in the complex plane, the time-dependent part of x, x – A/w„2, can be pictured as a displacement vector of magnitude BRIa>n2 rotating at the same



rate of o) but lagging behind the force vector by an angle of ф. This is illustrated in Figure 9.3.

іo„ is known as the undamped natural frequency, since this is the frequency at which the force-free system will oscillate when perturbed and with £ = 0. It is interesting to examine Equation 9.56 when the system is forced at its natural frequency. For this case r=1.0, so C,(r, () = 0 and Сг(г, О = (2£)_1 and ф therefore equals тг/2 or 90°. Thus the displacement of a second-order damped system, when forced at its natural frequency, will always lag the force by 90°, regardless of the damping.


X Derivatives and Parameters

Equation 9.33a contains the terms p, CXu, CXa, and Сц. Given the wing area, mean aerodynamic chord, air mass density, and airplane mass, the dimensionless “mass,” p, can be calculated from Equation 9.31. Сц is simply the trim lift coefficient and can be determined from

n – mg Сц QoS

where q0 = ipU02.

Expressions for the derivative CXu are presented in Equation 9.25. These expressions involve the trim drag coefficient Cq, and the trim pitch angle 0O. Since we are using stability axes, ©0 is simply the climb angle for the trim conditions. Сц, is a function of Сц and is given by

с Л+£к

Cd° S irAe

We will again use the Cherokee 180 pictured in Figure 3.62 as an example. For the trim condition, straight and level flight at a true airspeed of 50m/s (111.8mph) at a standard altitude of 1500 m (4921ft) is selected. For this airplane,

A = 5.625

Iy = 1693 kg – m2 (1249 slug-ft2)

W = 10,680.0 N (2400 lb)

S= 14.86 m2 (160 ft2) б = 1.60 m (5.25 ft)

An / of 0.5 m2 (5.38 ft2) (see discussion of Table 4.4) and an e of 0.6 (see discussion following Equation 4.33) will be assumed.

From Figure 2.3, the standard mass density at 1500 m equals 1.058 kg/m2. Thus,

q0 = j(1.058)(50)2 = 1323.0 N/m2

Thus the trim lift and drag coefficients become

Сц = 0.543 Сц, = 0.0615

Since @o is zero, CXu for this propeller-driven airplane becomes

Cxu = -0.185

From Equation 9.26,

Cx„ – Cl,, – Cn

= c^-c,„


CLa was calculated previously for the Cherokee wing-tail combination as equal to 4.50/rad. Using Equation 8.71, an increment to CLa due to the fuselage is estimated to equal 0.13/rad. The propeller contribution to CLa can be estimated using Figure 8.22a together with the estimated CDo. Assuming a C, of 0.6 and a / of 1.0,


-jr — 0.8/rad

ACla = 0.8 CDo
= 0.05

The total CLa is equal to the sum of these separate contributions.

CLa = 4.68/rad CXa = 0.0637/rad

The gross weight corresponds to a mass of 1089 kg. Thus, for the given gross weight and moment of inertia,

fi = 86.6 /v = 210.0

Using the foregoing numerical constants, Equation 9.33a becomes

173.0Й + 0.185м -0.0637a +0.5430 =0 (9.36)

Z Derivatives and Parameters

In addition to some of the terms just considered, Equation 9.33b contains the terms Cz„, Cz„-, and CZs – To these we will add the derivative C/a.

The derivative Cz„, because of the use of stability axes, is simply given by

Cza — — CLa (9.37)

Cz. can be obtained by using Equation 8.49. At the trimmed condition,

AZ — r),q0S, a,

Based on the experimental data of Reference 8.9, an tj, of 1.0 appears reasonable. At the cruise thrust coefficient, tj, is slightly greater than unity at low a values and less than unity at the higher a values. .The horizontal tail volume for the Cherokee is approximately 0.392. Thus,

Cz,, — — 2.88

The increment in the Z force resulting from d can be determined from Equation 9.34. At the trim condition,

AZ = r),q0STaT(~ e„^-)d


In Equation 9.39, d is with respect to the dimensionless time t, whereas in the preceding equation for AZ, the derivative is with respect to real time.

ea is estimated in Chapter Eight for the Cherokee to equal 0.447. Thus, Cz for the Cherokee is estimated to be

CZa = – 1.29/rad-air sec

The increment in Z resulting from an elevator deflection will be

AZ = – r),q0S, a,T Se

For a stabilator configuration, the effective lift curve slope is greater than a, (with the tab fixed) because of the linked tab. This increased slope was given earlier as

Thus, Equation 9.40, for the stabilator configuration, becomes

CZs = – Vl^at(l-Tk’) (9.41)

For the Cherokee, kt = —1.5. The effective т was estimated previously to equal 0.44. Thus,

CZt = – 0.934/rad

With the foregoing constants, Equation 9.33b for the Cherokee 180 becomes (including a term)

1.09й + 175.0d + 4.68a – 170.00 = – 0.9345 (9.42)

M Derivatives and Parameters

The equation of motion governing pitching, Equation 9.33 c, requires the values of Сщ, См^ and Сщ. In addition, we will include the term Сщ.

Сщ can be estimated on the basis of Equation 8.8 for a wing-tail combination. An increment to CK for the fuselage is obtained from Equation 8.72 and for a propeller from Figure 8.22a and 8.22c.

For the Cherokee these components of Сщ are estimated to equal:

ДСМ„ = -0.963/rad (wing-tail)

Д Сщ = +0.072/rad (propeller)

Д CM„ = +0.150/rad (fuselage)

Thus, for the total,

Cm. = -0.741/rad

The damping moment derivative, Сщ, is given by Equation 8.54. For the Cherokee 180,

Сщ = -7.42

CM. is equal to CZa multiplied by the tail length referenced to c.

Cm& = — 2rj, Унєа 4 a, (9.43)

For the Cherokee 180,

См„ = – 3.32/rad-air sec

Сщ is equal to Czs multiplied by the dimensionless tail length. Thus, for

the stabilator configuration,

Сщ = – 2.40/rad

Using the preceding stability derivatives for the Cherokee 180 results in the following for Equation 9.33c.

0.741a + 3.32a + 210.00 + 7.420 = – 2.40S

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




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


AZ – mg sin 0O0 = m U0(a – в)


AM = Іув

Also, for the trimmed condition,


X0 – mg sin ©o = 0


Z0 + mg cos 0O = 0





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


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)


T0 и u_

(m)pU02SU0 *U0

Thus, /or this case,

From Equation 9.20a and 9.20b,

(H2)pUo2S Сц> + Сц ta" 00


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

This equation of Equation 9.22 can be written in the form

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)


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*

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)


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


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.


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 у-axis 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).


Consider Figure 9.1a, which depicts the airplane coordinate system at some instant of time, t. The x-axis is aligned with the wing zero lift line, as before, and passes through the center of gravity. The z-axis is normal to the x-axis 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 z-axes, respectively. At this instant the x – and z-axes 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,

X-mg 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 right-hand 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


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


z’ = zt = 0) + I VXsin a cos 0 – cos a sin 0) dt Jo


0 = ©(* = 0) + f В dt



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 z-axes.

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)


Similarly, the drag is calculated from

D = l2pV2SCD



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. So-called “added mass” effects are generally negligible when considering airplane motion, but become important for lighter-than-air (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.

Yawing Moment with Boll Rate

Referring again to Figure 8.35, it is seen that on the right wing, which is moving down, the lift vector is inclined forward through the angle PylV. On the left wing, the inclination is to the rear. The components of these tilted lift vectors in the x direction give rise to a differential increment in the yawing moment equal to

dN = -2 yqcCL^ydy

Integrating from 0 to b/2, the total yawing moment increment in coefficient form for a linearly tapered wing becomes

с»~¥(їтг) «-«И

Rolling Moment with Sideslip Angle—Dihedral Effect

The rate of change of rolling moment with sideslip angle, С((,, is important to the handling qualities of an airplane. Generally, a small negative value of Cip is desirable, but too much dihedral effect makes an airplane uncomfortable to fly. The principal factors affecting С/э are sweepback, placement of the wing on the fuselage, and the dihedral angle of the wing, Г.

The primary control over Clf) is exercised through the dihedral angle, Г, shown in Figure 8.38. Although the sweepback angle also affects Clf) significantly, Л is normally determined by considerations other than C/r From Figure 8.38, it can be seen that a positive sideslip results in an upward velocity component along the right wing and a downward component along

the left wing. Added to the free-stream velocity, this results in an increase in the angle of attack over the right wing equal to

Да = (ЗГ

Over the left wing, an opposite change in a occurs. This differential increment in a results in a differential rolling moment, given by

dl = 2qcajiYy dy


For a linearly tapered wing, this reduces to


The derivation of Equation 8.108 has neglected induced effects that are appreciable for low aspect ratios. Its primary value lies in disclosing the linear relationship between Г and C/0.

It is recommended that the contribution to C[fS from Г be estimated using Figure 8.39. This figure represents the departure of Q0 from the following normal case and is based on graphs presented in Reference 8.11 or 5.5.

A =0.5 A = 6.0

Д1/2 = 0

C, f = -0.00021 r/deg(T in deg)

Figure 8.39 Effect of wing dihedral angle on CNote that factors represent relative effects of varying each parameter independently from the normal case. Л = 0.5, Л = 6.0, Лі/2 = 0.

where the functions kA, kA, and kA are obtained from Figure 8.39 as functions of A, A, and Лід, respectively.

The effect of sweepback on Cis determined with the help of Figure 8.40. A swept wing is shown operating at a positive sideslip angle /3. From the geometry, the velocity component normal to the leading edge of the right wing is given by

V cos (Л – /3)

The corresponding velocity on the left wing is

V cos (Л + /3)

If Cin is the section lift coefficient based on the normal velocity and “normal chord,” then the differential lift on the right and left wings will be

dLR = q cos2 (Л – /3)c cos ЛС(, ds

dLL = q cos2 (Л + /3)c cos ACln ds

The differential rolling moment is therefore

dl = qcCiny [cos2(A + j8) – cos2 (A – /3)] cos A ds But

у = s cos A
dy = ds cos A



l = qCln[cos2(A + /3) – cos2 (A – /3)] І су dy


The total lift (for /3 = 0) is given by


L = 2q cos2 АС/ I c dy

" Jo

= qSCtn cos2 A

Thus the wing CL and the normal section C, n are related by

Cj = Cin cos2A

and the rolling moment coefficient becomes



I Jo_____


If this equation is differentiated with respect to /3 and evaluated at /3 = 0, the following results.

For a linearly tapered wing, Equation 8.110 reduces to

Again, this result is only qualitatively correct. Generally,

Ci0 = -/(A, X)CL tan Л (8.112)

Figure 8.41 (based on Ref. 5.5) presents CtJCL as a function of Л for a range of aspect ratios. The variation with tan Л is seen to hold only for the higher aspect ratios. This figure can be used with Equation 8.111 to estimate Ci0 for other taper ratios.

Observe that wing sweep can contribute significantly to dihedral effect. In order to avoid an excessive dihedral effect on aircraft with highly swept wings, it is frequently necessary to employ a negative dihedral angle on the wing, particularly if the wing is mounted high on the fuselage.

The effect that the wing placement on the fuselage has on Ctj) is seen by reference to Figure 8.42. In a plane across the top of the fuselage, the cross-flow around the fuselage is seen to go up on the right side and down on the left. Thus, for a high wing, this flow increases the angle of attack of the right wing while decreasing a on the left wing. This results in a negative rolling moment comparable to a positive dihedral effect. For a low wing, the effect is just the opposite. This is the reason, as you may have observed, that

Figure 8.42 Rear view showing the cross-flow around a fuselage.

the dihedral angle for unswept low wings is generally greater than for high wings. Many high-wing airplanes do not have any dihedral angle at all.

As a rule, it is recommended that the following be added to ClfS to account for the fuselage cross-flow (Ref. 8.6).

High-wing Д Clfj = -0.00016/deg

Midwing AC/f) = 0

Low-wing AC/fj = 0.00016/deg

This information on C/0 and the other stability derivatives is intended only as an introduction to the subject. For more complete information on these quantities, see Reference 5.5 and 8.3.

Rolling Moment with Rudder

If the center of pressure of the fin-rudder combination lies a distance of Zv above the longitudinal axis passing through the center of gravity, an increment, ALV, in the lift (side force) will produce an increment in the rolling moment, given by


In coefficient form this becomes

Zv A Lv b qS

= тгСг. в,

The side force derivative CYs is obtained from

Rolling Moment with Yaw Rate

Figure 8.37 illustrates a wing having a tapered planform that is yawing at the rate of R rad/sec. The wing is operating at a lift coefficient of CL. It will be assumed that the section lift coefficients are constant and equal to CL. Because – of the rotational velocity R, a section on the right side located at у experiences a local velocity equal to V – Ry while the corresponding section on the left wing has a velocity of V + Ry. Thus a differential rolling moment is produced equal to

dL = уpcCL[{V + Ryf-{V-Ryf] dy

Expanding and integrating from 0 to Ы2 results in

, pcCJtVb3 12

=qSb? CL

As an exercise, show that, for a linearly tapered wing,

^ Cl/1 + 3A

C, f 6 u + A j

The vertical tail can also contribute to Ctf Because of the yaw rate, its angle of sideslip is decreased by RIJV. Thus, an incremental side force on the tail is generated, given by

A V c ^

Д Y = – y-

Flgure 8.37 A wing yawing and translating.

In coefficient form,

CY = 2r),av Vvr (8.104)

This incremental side force acting above the center of gravity produces an increment in the rolling moment, given by

Д L = r)tqSvaK Zv

or, in coefficient form,

Clf = Cyrf (8.105)

= 2т

The total C(, equals


Coupling Effects

There are several interactions or couplings involving lateral-directional controls and motions that are important considerations in providing satis­factory handling or riding qualities for an airplane. We have already seen one of these, a coupling between yaw and aileron displacement. Similarly, the rudder deflection can produce a rolling moment, since the center of pressure

generally lies above the center of denoted in coefficient form by:


Other couplings include:

c, r



These represent, respectively, a rolling moment due to yawing, a yawing moment due to rolling, and a rolling moment resulting from sideslip! This last coupling is an important and well-known one that is called “dihedral effect.”

Steady Rolling Motion

A measure of aileron control power is afforded by the steady roll rate, P, produced by the aileron. This rate is such that the rolling moment from the ailerons equals the damping moment resulting from P. The origin of this damping moment can be seen from Figure 8.35.

A section of the right wing located a distance у from the centerline will experience an increment in its angle of attack because of P, given by

Neglecting induced effects, Да results in a differential moment given by


dL = – yqcao-ydy

The total rolling moment is obtained by integrating this equation from 0 to b/2 and multiplying by two to account for the left wing. In coefficient form this becomes

c–¥dv)/.’ (§)*’* <8-98)

The quantity (Pb/2V) will be denoted by p and is comparable to the dimen­sionless pitch rate, q. p can be interpreted geometrically as the tangent of the helix angle prescribed by the wing tip as the rolling airplane advances through the air.

For a linearly tapered wing, Equation 8.98 integrates to

_ a0p 1 + 3A


n — ао 1 + ЗА 12 1 + A

The derivation of Equation 8.99 provides some insight into the origin of the roll damping but, otherwise, is of little value because of induced effects that were neglected. Correcting a0 for aspect ratio improves the accuracy somewhat.

Calculated values of Clf based on lifting surface theory are presented in Figure 8.36 (taken from Ref. 8.11). The original source of these calculations is Reference 5.5. These graphs are all for zero sweep angle of the midchord line. Results for other sweep angles can be found in the references. Generally, C((S is insensitive to А ід up to approximately ±20°. This range increases as the aspect ratio decreases.

Calculations of Ct. for a complete airplane configuration must also include contributions from the horizontal and vertical tails. The horizontal tail’s contribution can be determined on the basis of Figure 8.36 multiplied by Stb2/Sb2. The vertical tail’s contribution can also be determined on the same basis by visualizing the tail to be one-half of a wing. Thus the geometric aspect ratio of the vertical tail is doubled to enter Figure 8.36. The resulting value of Ci – is multiplied by Svb2ISb2 and then halved. If the horizontal stabilizer is mounted on top of the vertical tail, the value of Av must be increased, possibly by as much as 20%, to account for the end-plate effect of the horizontal tail on the vertical tail.

In a steady roll, the aileron-produced rolling moment and the damping moment will be equal in magnitude but of opposite sign. To find the steady

0 2 4 6 8 10


Figure 8.36 Roll damping coefficient derivative. Note that Cif should be corrected by ratio of section lift curve slope to 2тг.

rolling velocity as a function of aileron angle, we set the sum of the aileron and damping moments to zero.

Ch 5a + C,/ = 0 or


Thus, the dimensionless roll rate is directly proportional to the aileron deflection angle. PbtlV will typically range from approximately 0.05 to 0.10, depending on the maneuverability requirements imposed on the airplane.

Past practice was to specify values of p but both civil and military requirements (Refs. 8.1 and 8.12) now state specific requirements on P or, to be more specific, on a displacement of ф in a given time. For example, FAR Part 23 requires that an airplane weighing less than 6000 lb on the approach must have sufficient lateral control to roll it from 30° in one direction to 30° in the other in 4 sec. In landing at 20% above the stalling speed, this number is increased to 5 sec. Mil-F-8785 В requires an air-to-air fighter to be rolled 360° in 2.8 sec, an interceptor aircraft 90° in 1.3 sec, a transport or heavy bomber 30° in 1.5 sec, and a light utility aircraft 60° in 1.4 sec.

Aileron Reversal

Before leaving the subject of ailerons, mention should be made of a behavior known as aileron reversal. As the name suggests, it is a behavior whereby the action of an aileron, as related to its deflection, is opposite to what one would normally expect. This anomalous behavior is primarily associated with airplanes that have thin, flexible, sweptback wings that operate at high speeds. As a result of the aerodynamic moment, such a wing will tend to twist as the aileron is deflected. For example, an up aileron, which is intended to drop the wing, may instead twist the wing nose up sufficiently to cause a net increase in the wing lift.

To pursue this behavior, consider Figure 8.34, which illustrates the situation with a simplified model. A symmetrical airfoil section is shown mounted on a torsional spring with a spring constant of ke. When the flap is deflected through a positive angle 8, an aerodynamic moment is produced that causes the airfoil to rotate through an angle a (shown positively). This results in an opposing moment generated by the spring that, at some a, balances the aerodynamic moment. Thus,

kff(x cjc Cm& 8


C, = (C, a + Ch 8)

Figure 8.34 A wing section mounted on a torsional spring to illustrate aileron reversal.


Since CMs is negative, it is obvious that, for a given ke, there is a q above which the flap deflection will produce a decrease in the lift.

Aileron reversal can be avoided by the obvious means of limiting the operational q or by stiffening the wing; either method can impose operational penalties on the airplane. A less obvious method is used to lock out con­ventional outboard ailerons at high speeds and employ spoiler control instead.

Roll Control by the Use of Spoilers

A spoiler is a device on the upper surface of an airfoil that, when extended into the flow, causes the flow to separate, resulting in a loss of lift. The effect is illustrated in Figure 8.32, which also illustrates several types of spoilers.

Roll control can be accomplished by the use of spoilers installed on the outboard extent of each wing in place of conventional ailerons. If a roll to the right is desired, the spoiler on that side is raised, reducing the lift on the right wing and causing a positive rolling moment. The drag will also be increased on the right wing, producing a favorable yawing moment.

Spoilers for roll control are currently used on several military airplanes and most jet transports to augment the primary aileron controls. They are also used on at least one general aviation airplane, the twin-turboprop Mitsubishi MU-2, as the sole means of roll control. In this case, the use of spoilers only for roll control permits the use of full-span Fowler flaps with a higher wing loading. According to Reference 8.10, “As the requirements increase for higher wing loading to improve ride and performance, better handling quali-

-8 -7 -6 -5 -4 -3 -2 -1 0

Aileron projection, % chord

Figure 8.33 Performance of retractable spoiler ailerons. A = 4.13, x, = 0.4, x2 = 1.0; rectangular planform.

ties, and improved high lift devices, spoilers will probably be applied more extensively to light aircraft in the future.”

Figure 8.33 (taken from Ref. 8.8), presents some rolling moment data typical of the retractable spoiler aileron of the type pictured in Figure 8.32. Note that for aileron projections in excess of 2% of the chord, the variation of Ci with the projection is quite linear. However, for projections less than this amount, the graphs are highly nonlinear. In fact, little or no rolling moment is produced until the projection exceeds % of the chord. This “dead” region would not produce a very satisfactory feel to the pilot.

The configurations of Figure 8.32b and 8.32e are intended to remove the dead region at low spoiler deflections. As soon as the gap is opened, its influence is felt in producing separated flow. The configuration of Figure 8.32e is the type used on the MU-2.

In the past, spoilers have not been used as the sole primary source of roll control because of their nonlinear characteristics. At high angles of attack their effectiveness can be reduced or even reversed. This nonlinear behavior is also reflected in the control forces that result from nonlinear hinge moments. In addition, a lag can occur in reestablishing the attached flow pattern when the spoiler is retracted. Objections to spoiler ailerons have also been based on the fact that the rolling moment is produced by the loss of lift on one side of the wing. This results in an accompanying loss in altitude.

It appears that by careful design, these objections can be overcome. Also, actual flight experience and simulation studies indicate a negligible

difference in the roll dynamics between conventional and spoiler-type ailerons. Thus the loss in altitude associated with spoiler control does not seem to be a problem.