Category Dynamics of Flight

5.1 General Remarks

We saw in Chap. 4 how the aerodynamic actions on the airplane can be represented approximately by means of stability derivatives (or more exactly by aerodynamic transfer functions). Indeed, all the aerodynamics involved in airplane dynamics is concentrated in this section of the subject: i. e., in the determination of these deriva­tives (or transfer functions). Each of the stability derivatives contained in the equa­tions of motion is discussed in the following sections. Wherever possible, formulas for them are given in terms of the more elementary parameters used in static stability and performance. Where this is not feasible, it is shown in a qualitative way how the particular force or moment is related to the relevant perturbation quantity. No data for estimation are contained in this chapter; these are all in Appendix B.

EXPRESSIONS FOR Cx AND Cz

For convenience, we shall want the derivatives of Cx and Cz expressed in terms of lift, drag, and thrust coefficients. The relevant forces are shown in Fig. 5.1. As shown, the thrust line does not necessarily lie on the x axis. However, the angle between them is generally small, and we shall assume it be zero. With this assumption, and for small ax, we get1

Cx — CT + CLax Cn C – — ~(CL + CDax)

where CT is the coefficient of thrust, T!pV2S.

Although the inertia terms of the previous equations, for example, (4.9,18) and (4.9,19), remain unchanged to first order by the presence of elastic motions, the elas­tic and rigid-body motions are not nevertheless entirely uncoupled.

The deformations of the structure in general cause perturbations in the aerody­namic forces and moments. These may be introduced into the linearized equations of motion by the addition of appropriate derivatives to the expressions for the aerody­namic forces given by (4.9,17). For example, the added terms in the pitching moment associated with the nth elastic degree of freedom would be

Meen + Mten + Mten (4.12,2)

Similar expressions appear for each of the added degrees of freedom, and in each of the aerodynamic force and moment equations. An example of the elastic stability de­rivatives is given in Sec. 5.10. Alternatively, the aerodynamic forces may be formu­lated in the form of transfer functions.

THE ADDITIONAL EQUATIONS OF MOTION

The additional equations are most conveniently found by using Newton’s laws as ex­pressed by Lagrange’s equations of motion (Synge and Griffith, 1942) with the en as

‘The eigenfunctions of the linear vibration problem.

generalized coordinates. The appropriate form of Lagrange’s equation for this appli­cation is

d дТ ЭТ dU _ ^

dt дєп дєп дє„ "

where Т is the kinetic energy of the elastic motion relative to FB, U is the elastic strain energy, and SFn is the generalized external force. Since the coordinates are mea­sured in the frame FB, which is non-Newtonian by virtue of its general motion, an ap­propriate modification must be made to the external force field acting on the system when calculating the generalized force. This consists of adding to each element of mass 8m an inertial body force equal to — a’ 8m where a’ is that part of the total ac­celeration of 8m that arises from the acceleration and rotation of FB (see Appendix A.6).

Since normal modes have been chosen as the degrees of freedom, then the indi­vidual equations of motion are independent of one another insofar as elastic and iner­tia forces are concerned (this is a property of the normal modes), although the equa­tions will be coupled through the aerodynamic contributions to the SFs. The lack of elastic and inertia coupling permits the left-hand side of (4.12,3) to be evaluated by considering only a single elastic degree of freedom to be excited. Let its generalized coordinate be en. The kinetic energy is given by

T = J (x’2 + y’2 + z’2) dm

where the integration is over all elements of mass of the body. From (4.12,1) this be­comes (with only e„ excited)

F = hi2 j (f2 + g2 + h2) dm

The integral is the generalized inertia in the nth mode, and is denoted by

In = j(f2n + g2„ + h2n)dm (4.12,4)

so that

T = HA (4.12,5)

The first term of (4.12,3) is therefore I„en, and the second term is zero.

The strain-energy term is conveniently evaluated in terms of the natural fre­quency of the nth mode by applying Rayleigh’s method. This uses the fact that, when the system vibrates in an undamped normal mode, the maximum strain energy occurs when all elements are simultaneously at the extreme position, and the kinetic energy is zero. This maximum strain energy must be equal to the maximum kinetic energy that occurs when all elements pass simultaneously through their equilibrium position, where the strain energy is zero. Hence, if e„ = a sin wnt, then the maximum kinetic energy is, from (4.12,5)

= Lola2

Since the stress-strain relation is assumed to be linear, the strain energy8 is a qua­dratic function of e„; that is, U = ke^r Hence

tfma* = 2 ka2 = 7max = hlnco2na2

It follows that к = Ina>2, and that

U =

and hence dU/den = Inu>len. The left side of (4.12,3) is therefore as follows:

hK + 1 nofen = 3%, (4.12,6)

When structural damping is present, this simple form of uncoupled equation is not exact but the changes in frequency and mode shape for small damping are not large. Hence damping can be allowed for approximately by adding a damping term to

(4.12,6) , that is,

К + 2£>ne„ + w2„en = 9JIn (4.12,7)

without changing wn or the mode-shape functions. The value of £ is ordinarily less than 0.1, and usually must be found by an experimental measurement on the actual structure.

EVALUATION OF

The generalized force is calculated from the work done during a virtual displacement,

dW

3%, = 3- (4.12,8)

de„

where IV is the work done by all the external forces, including the inertia forces asso­ciated with nonuniform motion of the frame of reference. The inertia force field is given by

dt,= -(u’)dm (4.12,9)

where the components of the r. h.s. in FB are given by (A.6,8) without the terms (x, y, z). The work done by these forces in a virtual displacement of the structure is

SWi = I (8x dfx. + 8y dfy. + 8z dfz.)

where the integration is over the whole body. Introducing (4.12,1) this becomes

і = X Sc, J (fn dfx, + Sn dfy, + К dfz)

When the inertia-force expressions are linearized to small disturbances, and substi­tuted into (4.12,10), all the remaining first-order terms contain integrals of the fol­lowing types:

‘For example, in a spring of stiffness к and stretch x, the strain energy is U = kx2.

f fn dm, J (yh„ – Zgn) dm

The first of these is zero because the origin is the mass center, and the second is zero because the angular momentum associated with the elastic mode vanishes. The net result is that 3F„. = 0. This result simply verifies what was stated above; that is, there is no inertial coupling between the elastic and rigid-body degrees of freedom.

The remaining contribution to??,, is that of the aerodynamic forces. Let the local normal-pressure perturbation at an element dS of the airplane’s surface be p(x0, y0, Zq), and let the local outward normal be n(nx, ny, nz). Then the work done by the aero­dynamic forces in a virtual displacement is

SWa = – j pn-(r – r0) dS

where the integral is over the whole surface of the airplane, and (r – r0) is the vector displacement at dS. It is given by

oo

Г – 1*0 = z (*/» + j8n + кhn) 8en

swa = ~ X 8e" P(nJn + «v?» + nzK) dS n= 1

and

‘dfn = = – f p(nj„ + nyg„ + nzhn) dS (4.12,11)

ОЄп

Each of the variables inside the integral is a function of (jc0, y0, z()), i. e., of position on the surface, and moreover, p is in the most general case a function of all the general­ized coordinates, of their derivatives, and of the control-surface angles. The result is that 3Fn is a linear function of all these variables, which may be expressed in terms of a set of generalized aerodynamic derivatives (or alternatively aerodynamic transfer functions), namely,

3%, = AnuAu + Апйй + • ■ • + Anpp + • • • + AnS8r + ■■■

+ X anmem + X bnm^m + X Cnntm (4.12,12)

m—I m=1 m= 1

In application, only the important derivatives would be retained in any given case. The values of the derivatives kept would be computed by application of (4.12,11). An example of this computation is given in Sec. 5.10.

RESUME

The effects of structural dynamics on the stability and control equations can be incor­porated by adding structural degrees of freedom based on free normal modes. For an exact representation, an infinite number of such modes are required; however, in

practice only a few of the lowest modes need be employed. The six rigid-body equa­tions are altered only to the extent of additional aerodynamic terms of the type given in (4.12,2). One additional equation is required for each elastic degree of freedom

(4.12,7) . The generalized forces appearing in the added equations contain only aero­dynamic contributions, which are computed from (4.12,11) and expressed as in (4.12,12).

Many of the important effects of distortion can be accounted for simply by altering the aerodynamic derivatives. The assumption is made that the changes in aerody­namic loading take place so slowly that the structure is at all times in static equilib­rium. (This is equivalent to assuming that the natural frequencies of vibration of the structure are much higher than the frequencies of the rigid-body motions.) Thus a change in load produces a proportional change in the shape of the vehicle, which in turn influences the load. Examples of this kind of analysis are given in Sec. 3.5 (ef­fect of fuselage bending on the location of the neutral point), and Secs. 5.3 and 5.10.

THE METHOD OF NORMAL MODES

When the separation in frequency between the elastic degrees of freedom and the rigid-body motions is not large, then significant inertial coupling can occur between the two. In that case a dynamic analysis is required, which takes account of the time dependence of the elastic motions.

The method that is described here for accomplishing this purpose is based upon the representation of the deformation of the elastic vehicle in terms of its normal modes of free vibration. Imagine that the vehicle is at rest under the action of no ex­ternal forces, aerodynamic, gravity, or other, and that a frame of reference with origin at the mass center, but otherwise arbitrary, is attached to it. The position of mass ele­ment dm is then (x0, y0, z0). Now let the structure be deformed by a self-equilibrating set of external forces and couples, so that it takes a new form, stationary with respect to the coordinate system. Upon instantaneous release of this force system, a free vi­bration ensues, that is, one in which external forces play no part, and in which the po­sition of Sm at time t is (x, y, z). Since there is zero net force, and zero net moment, the linear and angular momenta of the elastic motion must vanish, whatever the ini­tial distorted shape. In particular this is true for each and every undamped normal mode of free vibration. Any small arbitrary elastic motion of the vehicle can, there­fore, relative to the chosen axes (transients as well as steady oscillations), be repre­sented by a superposition of free undamped normal modes as follows:

oo

x'{t) = X /„(*о. Jo, Zo)e„(t)

1

oo

y’it) = X 8n(*o> Уо, z0)e„(0

1

00

z'(t) = X hn(x0, y0, Zo)€„(t)

where (x’, y’, z!) are the elastic displacements, (x — x0) etc., (/„, gn, h„) are the mode shape functions,7 and e„(f) are the generalized coordinates giving the magnitudes of the modal displacements.

We have specified idealized undamped modes, as opposed to the true modes of a real physical structure with internal and external damping, because the latter may not be “simple” modes with fixed nodes, describable by a single set of three functions. More generally they each consist of a superposition of two “submodes” 90° out of phase. Because of this, the equations of motion for the elastic degrees of freedom of the real structure are not perfectly uncoupled from one another, but contain intercou­pling damping terms that would usually be negligible in practical applications.

The use of the free undamped normal modes is seen to ensure that the linear and angular momenta of the distortional motion vanishes. Consequently the elastic mo­tions have no inertial coupling with the rigid-body motions except through the mo­ments and products of inertia. However, it can be shown that this coupling is second – order and negligible in the small-perturbation theory. The determination of the shapes and frequencies con of the normal modes is a major task, and the methods for finding them are beyond the scope of this text. For treatises on this subject the reader should refer to (Bisplinghoff et al., 1955; Fung, 1955). As indicated in (4.12,1), there are ac­tually an infinite number of normal modes of vehicle structures. In practice, of course, only a finite number N of those at the low-frequency end of the set need be retained, and the summations in (4.12,1) are approximated by finite series of N terms. Some judgment and experience is needed to decide just how many modes are needed in any application, but a general rule that is helpful is to discard those whose frequen­cies are substantially higher than the significant ones present in the spectral represen­tation of inputs arising from control action or atmospheric turbulence.

We now need expressions for the derivatives that appear in (4.9,18) and (4.9,19) in terms of the nondimensional derivatives. A few examples of these are derived as fol­lows to illustrate the procedure, and the whole set needed for (4.9,17) is displayed in Tables 4.4 and 4.5. Derivatives with respect to v or /3 are usually negligible and are not included.

THE Z DERIVATIVES

THE M DERIVATIVES

These are also found in a manner similar to the Z derivatives. In this case we start with M = CmpV2Sc and note from (4.9,6) that Cmo = 0.

THE L DERIVATIVES

From Table 4.1, L = CtpV2S —. Hence

PU()S ^ ch

Also

~~ ^ pu»b2SC/p

Similarly

THE N DERIVATIVES These are found in a manner similar to the L derivatives.

THE Y DERIVATIVES These are also found in a manner similar to the L derivatives. In this case we start with Y = CypV2S.

4.12 Elastic Degrees of Freedom In the preceding sections we have presented the “main” equations of motion, that is, those associated with the six rigid-body degrees of freedom. Now it is known that the stability and control characteristics of flight vehicles may be profoundly influenced by the elastic distortions of the structure under aerodynamic load (AGARD, 1970; Milne, 1964; McLaughlin, 1956; Rodden, 1956). Additionally, there are phenomena not primarily related to stability and control, but rather to structural integrity, in which elastic deformation is a primary element—i. e., structural divergence and flut­ter. In order to understand and analyze all these effects, one needs the equations that govern the elastic deformations, and as well the changes that such deformation intro­duces into the six main dynamical equations. A full treatment of this branch of flight mechanics—aeroelasticity and structural vibration—is beyond the scope of this text, and the reader is referred to (Bispling – hoff, 1962 and Dowell, 1994) for comprehensive treatises on it. Here we content our­selves with presenting the framework of the analysis, but omit most of the structural

and aerodynamic details. Enough material is given, however, to show how the static and dynamic deformations are integrated into the preceding mathematical model of the “gross” vehicle motion.

The deformation analysis is almost invariably treated by a linear theory, even when the rigid-body motion is not. We shall therefore assume at the outset that the distortional motions are “small” and that all the associated aerodynamic forces are linear functions.

The nondimensional stability derivatives are the partial derivatives of the force and moment coefficients in lines 1, 3, and 4 of Table 4.1 with respect to the nondimen­sional motion variables in lines 5, 6, and 7. The notation for these is displayed in Ta­bles 4.2 and 4.3. Each entry in the tables represents the derivative of the column heading with respect to the row variable.

Since ax differs from a only by a constant (the angle between the zero-lift line and the x axis), then Дax = Да, Э/Эал = Э/Эа, and no distinction need be made be­tween these two derivatives.

NONDIMENSIONAL EQUATIONS

It is possible with the definitions given in Tables 4.1—4.3 to make the equations of motion entirely nondimensional, and such equations have been widely used in the past, especially for analytical work (see Etkin, 1972 and 1982). The prevailing cur­rent practise in design and research, however, is to use the dimensional equations and program them for calculation on a digital computer. We are therefore not including the nondimensional equations in this book. There is no real loss in so doing however, since any analytical results that are obtained with the dimensional equations can sub­sequently be expressed, for maximum generality, in nondimensional form. Examples of this are contained in Chaps. 6 and 7.

The reader will already be familiar with the great advantage of using nondimensional coefficients for aerodynamic forces and moments such as lift, drag, and pitching mo­ment. In this way the major effects of speed, size, and air density are automatically accounted for. Similarly we need nondimensional coefficients for the many deriva­tives—Xu and so on—that occur in (4.9,18 and 4.9,19). Unfortunately, there is no universally accepted standard for these coefficients, although attempts have been made to devise one (e. g., ANSI/AIAA, 1992). The student, and indeed the practising engineer, should be sure to note carefully the exact notation and definitions employed in any reference material or data sources being used. The notation and definitions used in this book are essentially the NASA system, which is widely used.

Before presenting this system, we digress briefly to a dimensional analysis of the general flight dynamics problem. This helps to provide insight into what the true un­derlying variables are, and provides a basis for what follows. Imagine a class of geo­metrically similar airplanes of various sizes and masses in steady unaccelerated flight at various heights and speeds. Suppose that one of these airplanes is subjected to a disturbance. After the disturbance, some typical nondimensional variable 77 varies with time. For example, тг may be the angle of yaw, the load factor, or the helix angle in roll. Thus, for this one airplane, under one particular set of conditions we shall have

77=/(f) (4.10,1)

Let it be assumed that this equation can be generalized to cover the whole class of airplanes, under all flight conditions. That is, we shall assume that 77 is a function not of t alone, but also of

n0, p, m, l, g, M, RN

where m is the airplane mass and / is a characteristic length. Instead of (4.10,1), then, we write

7Г = f(u0, p, m, /, g, M, RN, t) (4.10,2)

Buckingham’s 77 theorem (Langhaar, 1951) tells us that, since there are nine quanti­ties in (4.10,2) containing three fundamental dimensions, L, M, and T, then there are 9-3 = 6 independent dimensionless combinations of the nine quantities. These six so-called 77 functions are to be regarded as the meaningful physical variables of the equation, instead of the original nine. Two systems of the same class are dynamically similar when all the 77 functions of one are numerically equal to those of the other. By inspection, we can easily form the following six independent nondimensional combinations:

m

tt, M,RN, —r, Pi

Following the 77 theorem, we write as the symbolic solution to our problem

The effects of the six variables m, p, l, g, u0, and t are thus seen to be compressed into the three combinations: m/pl3, u^/lg, and u0t/l. We replace l3 by SI, where S is a char­acteristic area, without changing its dimensions, and denote the resulting nondimen­sional quantity m/pSl by рь. The quantity l/u0 has the dimensions of time and is de­noted t*. The quantity u20llg is the Froude number (FN). Equation 4.10,3 then becomes

гг = /(M, RN, FN, p,, t/t*) (4.10,4)

The significance of (4.10,4) is that it shows 77 to be a function of only five vari­ables, instead of the original eight. The result is of sufficient importance that it is cus­tomary to elaborate on it still further. Since pt is the ratio of the airplane mass to the mass of a volume SI of air, it is called the relative mass parameter or relative density parameter. It is smallest at sea level and increases with altitude.

The main symbols for which nondimensional forms are wanted are listed in Table 4.1. The nondimensional item in column 3 is obtained by dividing the corre­sponding dimensional item of column 1 by the divisor in column 2. In the small-dis­turbance case, v and w are aerodynamic angles, for then

V = [(«о + Дм)2 + v2 + w2)]1/2

From (1.6,4)

ax = tan-‘(w/m) = tan~'[w/(M0 + Дм)]

To first order in v, w, and Дм these are

V — u0 + Au
w

ar— — = w
u0

and similarly

, v v

ft = sin — = —— = V

V u0

When (4.9,17) are substituted into (4.9,7 to 4.9,10) two of the equations [(4.9,7c) and (4.9,86)], contain w terms on the right-hand side. In order to retain the desired form with first derivatives of the dependent variables on the left, we have to solve these two equations simultaneously for w and q. The result is presented in matrix form in (4.9,18 and 4.9,19). Here the equations are divided into two groups, termed longitu­dinal and lateral, for reasons that are explained as follows.

As a consequence of the simplifying assumptions made in their derivation, the pre­ceding equations have the exceedingly useful property of splitting into two indepen­dent groups. Suppose that ф, v, p, г, ДYc, ALC and ANC are identically zero. Then (4.9,19) are all identically satisfied. The remaining equations (4.9,18) form a com­plete set for the six homogeneous variables Am, w, q, A6, AxE, AzE. Thus we may conclude that modes of motion are possible in which only these variables differ from zero. Such motions are called longitudinal or symmetric, and the corresponding equa­tions and variables are likewise named. Conversely, if the longitudinal variables are set equal to zero, the remaining six equations (4.9,19) form a complete set for the de­termination of the variables ф, ф, v, p, r, yE. These are known as the lateral variables, the corresponding equations and motions being likewise named.

It is worthwhile recording here the specific assumptions upon which this separa­tion depends. A study of the various steps that have led to the final equations reveals these facts—the existence of the pure longitudinal motions depends on only two as­sumptions:

1. The existence of a plane of symmetry.

2. The absence of rotor gyroscopic effects.

The existence of the pure lateral motions, however, depends on more restrictive ap­proximations; namely

1. The linearization of the equations.

2. The absence of rotor gyroscopic effects.

3. The neglect of all aerodynamic cross-coupling (approximation 1 p. 110).

If the equations were not linearized, then there would be inertial cross-coupling between the longitudinal and lateral modes, as evidenced by terms such as mpvE in (4.7,1c) and rp{lx – /,) in (4.1,2b). That is, motion in the lateral modes would induce longitudinal motion.

Equations (4.9,18 and 4.9,19) are both in the desired first-order form, commonly referred to as state vector form, conventionally written in vector/matrix notation as

x = Ax + Be (4.9,20)

Here x is the state vector, c is the control vector, and A and В are system matrices. The state vectors for the longitudinal and lateral systems are, respectively:

x = [Am w q Ав]т

x = [v p г ф]г (4.9,21)

and the matrices A for the two cases can be inferred from the full equations. The de­pendent variables xE, yE, zE and ф are not included in the state vectors because they do not appear on the right-hand side of the equations. The matrix В will be discussed later when we come to the analysis of controlled motions.

At the heart of the subject of atmospheric flight mechanics lies the problem of deter­mining and describing the aerodynamic forces and moments that act on a given body in arbitrary motion. It is primarily this aerodynamic ingredient that distinguishes it from other branches of mechanics. Aerodynamic forces and moments are strictly speaking functionals of the state variables. Consider for example the time-dependent lift L(t) on a wing with variable angle of attack a(t). Because the wing leaves behind it a vortex wake that in general generates an induced velocity field at the wing, and because hysteresis is present in flow separation processes, the aerodynamic field that fixes the lift at any given moment is actually dependent not only on the instantaneous value of a but strictly speaking on its entire past history. This functional relation is expressed by

L(t) = Т[а(т)] —oo < r<t (4.9,11)

When a(r) can be expressed as a convergent Taylor series around t, i. e.

a(f) = a{t) + (t — t)a(t) + {t — t)[9] [10]a(t) + ••• (4.9,12)

then the infinite series a(t), a(t), a(t) ••• can replace a(r) in (4.9,11), i. e.

L{t) = L(a, a, a ■ ■ ■) (4.9,13)

where a, a ■ ■ ■ are values at time t. Thus the lift at time t is in this case determined by a and all its derivatives at time t. A further series expansion of the right-hand side of (4.9,13) around t = 0 yields

A L(t) = LaAa + L(jAa)2 + – + La Да + hLaa(Aa)2 + – (4.9,14)

in which all the products and powers of Act, A a ••• appear, and where

dL

La = — etc. (4.9,15)

• da a=0

The classical assumption of linear aerodynamic theory, due to Bryan (1911) is to ac­cept the linear reduction of (4.9,14) as a representation of the aerodynamic force, even when Aa(t) is not an analytic function as implied by (4.9,12), i. e.

AL(t) = LaAa + Ьы Ad + L& Да + ••• (4.9,16)

Derivatives such as La in (4.9,16) are known as the stability derivatives, or more gen­erally as aerodynamic derivatives. For most forces and state variables, only the first term of (4.9,16) is kept, but in some cases, terms up to the second derivative must be retained for sufficient accuracy. This assumption has been found to work extremely well over a wide range of practical applications. Occasionally the addition of nonlin­ear terms such as Laa(Aa)2 = Ьаг(Аа)2 can extend the useful range considerably. Another way of including nonlinear effects is to treat the derivatives as functions of the variables, for example, La = La(a).

A major fraction of the total effort in aerodynamic research in the past has been devoted to the determination, by theoretical and experimental means, of the aerody­namic derivatives needed for application to flight mechanics. A great mass of infor­mation about these parameters has now been accumulated and Chap. 5 is devoted to this topic.

For a truly symmetric configuration, it is evident that the side force Y, the rolling moment L, and the yawing moment N will all be exactly zero in any condition of symmetric flight, that is, when the plane of symmetry remains in a fixed vertical plane. In that case, /3, p, г, ф, ф, and yE are all identically zero. Thus the derivatives of the asymmetric or lateral forces and moments, Y, L, N with respect to the symmet­ric or longitudinal motion variables u, w, q are zero. In writing out the complete lin­ear expression for the aerodynamic forces and moments, we use this fact, and in ad­dition make the further approximations:

3. The derivative Xq is also negligibly small.

4. The density of the atmosphere is assumed not to vary with altitude (see Sec. 6.5).

It should be emphasized that none of these assumptions is basically necessary for the solution of airplane dynamics problems. They are made as a matter of experience and convenience. When it appears necessary to do so, any of the terms dropped can be restored into the equations. With these assumptions, however, the linear forces and moments are:

Aerodynamic Transfer Functions

The preceding equations are subject to the theoretical objection (not of great practical importance) that the Bryan formulation for the aerodynamics is not quite sound even within the restriction of linearity. This is readily illustrated by considering the lift on a wing following a step change in angle of attack. Let Да be given by Дa(t) = a0l(f) where a0 is a constant. For t > 0, the Bryan formula (4.9,16) gives

A L(t) = Laa0 = const

whereas in fact the lift undergoes a transient approach to the asymptote Laa0, the de­tails of which depend on the wing shape and the Mach number. Equation (4.9,16) fails in this case because Да is not an analytic function, having a discontinuity at t = 0. Now the transient process is often well approximated as a linear one, and as such is subject to exact representation by linear mathematics in the form of an indi – cial function (Tobak, 1954), or an aerodynamic transfer function (Etkin, 1956). The implementation of these alternative representations of aerodynamic force is described in Etkin (1972, Sec. 5.11).

If all the disturbance quantities are set equal to zero in the foregoing equations, then they apply to the reference flight condition. When this is done, we get the following relations, which may be used to eliminate all the reference forces and moments from the equations:

X0 — mg sin 90 = 0 Fo = 0

Z0 + mg cos в0 = 0 (4.9,6)

L0 = M0 = N0 = 0

xEo = n0 cos 60, >>,, = 0, zEo = – U0 sin e{)

We further postulate that in the reference steady state the aileron and rudder angles are zero. When (4.9,6) are substituted into (4.9,2 to 4.9,5), (4.9,3) are solved for p and r, and the equations are rearranged, the result is (4.9,7 to 4.9,10).

The small-disturbance equations will be slightly restricted by the adoption of two more assumptions, which correspond to current practice. These are

1. The effects of spinning rotors are negligible. This is the case when the air­plane is in gliding flight with power off, when the symmetrical engines have opposite rotation, or when the rotor angular momentum is small.

2. The wind velocity is zero, so that Vе = V

LINEARIZATION

When the small-disturbance notation is introduced into the equations of Sec. 4.7, the additional assumptions noted above are incorporated, and only the first-order terms in disturbance quantities are kept, then the following linear equations are obtained.