1

Aircraft Stability

The topic of aircraft stability involves the study of the equations of motion without forcing terms (arising from the controls). It tells us the damping and natural frequencies of the free modes of motion of the aircraft, whose properties determine its stability. The course outline is as follows:

• aircraft equations of motion

• free modes, derivation of approximate forms to show the influence of design on the modal characteristics

• “static” stability and its relation to the modes

The application of control forces to the aircraft will be dealt with in the second part of the course. Here we shall see that the properties of the modes are also fundamental to the controlled behaviour of the aircraft.

The theory of aircraft stability involves a large number of dimensionless coefficients, most of which will be introduced where necessary. However, those associated with the lift, drag, and aerodynamic pitching moment are fundamental to the course, and you should be familiar with them before continuing. They are discussed in Appendix 1.

2

1. Aircraft Equations of Motion 1.1 Restrictions and Assumptions

The theory will be developed under the following assumptions:

• the aircraft is rigid—structural deflections are neglected

• the aircraft motion can be characterised in terms of small departures from an equilibrium state, so linearised equations can be used

• the equilibrium state is straight and level flight

These restrictions can be relaxed (with more or less difficulty!), but doing so complicates the theory without significantly changing the fundamental conclusions. 1.2 Axis System and Nomenclature

It is most convenient to use axes fixed in the aircraft (these are known as “stability axes”): Equilibrium state:

Perturbations from the equilibrium state are characterised as follows:

3

Perturbed state: Rotation Translation

NB! Note link between angle of attack and heave velocity:

4

Aerodynamic forces and moments Again, we consider perturbations from the equilibrium state. These are characterised by 3 forces and 3 moments:

The nomenclature introduced here is summarised in Appendix 2, which also gives the convention for moments of inertia. 1.3 Lateral/Longitudinal Decoupling

As the aircraft has 6 degrees of freedom (3 velocities, 3 rotations), there should be 6 coupled equations of motion. Fortunately, things are a bit simpler than this.

• Symmetry implies that the symmetric (or “longitudinal”) motions (u, w, q) cannot cause contributions to the anti-symmetric aerodynamic forces (Y, L, N).

• Linearity implies that any contributions to the symmetric aerodynamic forces (X, Z, M) from the anti-symmetric (or “lateral”) motions (v, p, r) are negligible. For example, consider the normal force caused by right and left sideslip:

5

Thus if Z(v) is expanded as a series in powers of v, it must take the form All of these terms are negligible in a linear theory.

Hence there is no coupling between longitudinal and lateral aircraft motions, and our 6 equations of motion consist of 3 coupled longitudinal equations, and 3 coupled lateral equations. This is a considerable simplification. 1.4 The Equations of Motion

1.4.1 Linear accelerations

The aircraft velocity vector is given by:

!

V = (U + u)ex

+ vey

+ wez

Following the rules for differentiating a rotating vector (remember that the stability axes are fixed in the aircraft), the acceleration is given by

!

a =dV

dt=

!

˙ u ex

+ ˙ v ey

+ ˙ w ez

6

where Hence the components of the acceleration are:

!

ax

=

!

ay

=

!

az

=

1.4.2 Rate of change of angular momentum

The general expression for the angular momentum of a three dimensional body is H = I! , where I is a matrix. If we assume that the stability axes effectively coincide with the principal axes of the aircraft, then this expression simplifies to

!

H = Ixpe

x+ I

yqe

y+ I

zre

z

(This assumption becomes problematic at high angles of attack, but the general form of the equations we derive in 1.4.7 and 1.4.8 can still be retained by some cunning redefinition of terms.) The rate of change of angular momentum is thus

!

dH

dt=

7

1.4.3 Aerodynamic forces

Characterising the aerodynamic forces in unsteady motion is not straightforward, because they depend on the present and the past state of the aircraft.

eg Y(t) depends on the integrated effect of v(t’), p(t’) and r(t’), –∞ < t’ < t. (If this concept seems rather abstract, note that a linear system with impulse response h(t) has exactly the same property; its response s(t) to an input u(t) is given by the convolution integral

!

s(t) = h(t " # t )u( # t )d # t "$

t

% .)

To find a representation suitable for the stability analysis, we assume that v, p and r may be expanded as Taylor series, eg v(t’) = Then the integration over t’ will give Y(t) = which depends only on the current state. In linearised form, this is The terms Yv, Yp ... are called stability derivatives. There is obviously a limit to how many can be included—in practice only keeping derivatives with respect to

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linear and angular velocities has been found to give sufficient accuracy. There is one exception: the pitching moment, which is written as

1.4.4 Control moments

The main effect of the control surfaces on an aircraft is to provide moments; the associated forces are relatively small. The disturbance moments associated with the control surface deflections are: Motion: Roll Pitch Yaw Control surface: Linearised relation: Secondary effects:

9

1.4.5 Engine thrust

We assume that the engine thrust acts in the x direction, and is a function of forward speed only. Hence it can be included in the derivative Xu.

1.4.6 Weight

In the disturbed state, gravity has components along all three axes:

The disturbance forces are: in the x direction in the y direction

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1.4.7 The longitudinal equations of motion

We can now relate the accelerations of 1.4.1 and 1.4.2 to the forces and moments of 1.4.3–1.4.6. For symmetrical motions, Newton II gives: surge: heave: pitch: (1.1)

1.4.8 The lateral equations of motion

The corresponding equations for anti-symmetric motions are: sideslip: roll: yaw: (1.2)

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2. The Modes of Motion

The modes of the aircraft are motions which are possible without any control inputs. The response of the aircraft to a disturbance will consist of components in one or more of the modes, so the damping and natural frequencies of the modes tell us about the stability of the aircraft.

2.1 The Longitudinal Modes

2.1.1 The characteristic equation As there is no control forcing, we set and look for solutions of the form Equations (1.1) become

!

m" # Xu

#Xw

# Xq" #mg( )

#Zu

m" # Zw

#"(mU + Zq)

#Mu

#(M ˙ w " + M

w) "(I

y" # M

q)

$

%

& & &

'

(

) ) )

u0

w0

*0

$

%

& & &

'

(

) ) )

=

0

0

0

$

%

& & &

'

(

) ) )

(2.1)

matrix equation: solutions:

12

The condition detA = 0 leads to a quartic equation in λ:

!

"4

+ B"3

+ C"2

+ D" + E = 0 (2.2)

This is known as the characteristic equation. The coefficients B–E are listed in Appendix 3.

The 4 solutions for λ (the eigenvalues of the equations of motion) give the time dependence of the modes. Each solution has an associated relationship between u0, w0 and q0 (the eigenvector) which determines the mode “shape”.

In general, the solutions to the characteristic equation are two complex conjugate pairs, corresponding to decaying oscillations:

!

" = "r

± i"i

The oscillations are:

• the phugoid mode, a low frequency, lightly damped motion

• the short period oscillation (SPO), a high frequency motion with better damping than the phugoid

More information about these modes can be obtained from an approximate analysis of the stability equations. In order to carry out this analysis, we first need to rewrite the equations in non-dimensional form.

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2.1.2 The non-dimensional equations These equations are obtained by non-dimensionalising the component terms in equation (2.1) on appropriate dimensional quantities—see Appendix 4. Starting from the dimensional equations:

!

m" # Xu

#Xw

# Xq" #mg( )

#Zu

m" # Zw

#"(mU + Zq)

#Mu

#(M ˙ w " + M

w) "(I

y" # M

q)

$

%

& & &

'

(

) ) )

u0

w0

*0

$

%

& & &

'

(

) ) )

=

0

0

0

$

%

& & &

'

(

) ) )

we find the dimensionless set

!

" # xu

#xw

# xq" µ

1#C

L( )#z

u" # z

w#"(1+ z

qµ

1)

#mu

#(m ˙ w " µ

1+ m

w) "(i

y" #m

q) µ

1

$

%

& & &

'

(

) ) )

u0

U

w0

U

*0

$

%

& & &

'

(

) ) )

=

0

0

0

$

%

& & &

'

(

) ) )

(2.3)

The non-dimensional quantities Λ, CL and µ1 arise automatically, given the definitions in Appendix 4. They are defined by Λ = (2.4) CL = µ1 = (2.5)

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Before proceeding to the analysis of (2.3), some further points should be noted: (1) mu = 0 for subsonic aircraft Proof:

!

Cm

= Cm(",#)

Equilibrium state: Consider disturbed state with (2) Having chosen non-dimensionalising variables appropriate to the stability derivatives and moments of inertia, we expect all these terms to be roughly the same size, around about unity. The same applies to the lift coefficient, CL. However, Λ and µ1 could be either large or small.

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(3) We can estimate µ1 for a given aircraft. eg an Airbus A320 cruising at 10km altitude has m = 70000kg ρ = 0.41 kg/m3 S = 120m2 c = 3.6m We can thus try to find approximate solutions to (2.3) as series in inverse powers of µ1, eg

!

" # µ1

2c0

+ c1µ1

$1 + c2µ1

$2 +K( ) Here the powers of µ1 are integer, but this need not always be so. The basic variation of ! is like

!

µ1

2, and we write this briefly as

2.1.3 The phugoid approximation To illustrate the method, we look for a small or O(1) solution for Λ in (2.3): The other variables (or, more strictly, their ratios) will also depend on µ1 in some way: Our first job is to find the coefficients a, b and c. Consider the orders of magnitude of the largest (leading order) terms in (2.3):

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!

" # xu( )u0 U

!

"xww0U

!

" xq# µ

1"C

L( )$0 = 0 (1)

!

"zuu0U

!

+ " # zw( )w0

U

!

"# 1+ zq

µ1( )$0 = 0 (2)

0

!

" m˙ w # µ

1+ m

w( ) w0

U

!

+" iy" #m

q( )$0 µ1 = 0 (3)

If the terms on the LHS of each equation sum to zero, then in particular the leading order terms must cancel. This puts conditions on our unknown powers of µ1: (3) ==> (2) ==> (1) ==> In summary:

!

" ~ µ1

0, u

0U ~ #

0, w

0U ~ #

0µ1

This tells us that the phugoid motion consists mainly of speed (u0) and attitude (θ0) changes at a more-or-less constant angle of attack (

!

w0U ).

17

We can now go on to derive an approximate characteristic equation, by retaining only the leading order terms in (1)–(3) above:

!

" # xu

0 CL

#zu

0 #"

0 #mw

" iy" #m

q( ) µ1

= 0

(2.6) Solutions: (2.7) To see what this implies about the natural frequency and damping of the phugoid mode, we need to look more closely at the stability derivatives xu and zu. First consider xu (assuming constant thrust):

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equilibrium state perturbed state

!

X = ("D) " ("D0) = " 1

2#(U + u)

2SC

D($

0,%

0) + 1

2#U 2

SCD($

0,%

0)

This tells us that the phugoid damping is associated with the equilibrium drag coefficient, which is typically small for a well-designed aircraft. In fact, it can even become negative if engine thrust effects are significant, making the phugoid mode unstable. For zu, the analysis is similar, but with lift replacing drag. We find: zu = –2CL The frequency of the phugoid then follows from (2.7):

19

!

Im(") =1

2#US

m($z

u)C

L$ x

u

24 =

The period of the phugoid is given by

!

T = 2" Im(#), ie For high speed cruise, with U = 200m/s, this gives T ≈ 90 seconds—a very long period motion. Note that, at this level of approximation, the phugoid period depends only on the flight speed, and is independent of the properties of the aircraft. Later, we shall derive a more accurate approximation, which is influenced by the aircraft stability derivatives. The phugoid mode was first identified and named by Lanchester, an early 20th century pioneer in aerodynamics. He derived the expression for the period by assuming flight at constant angle of attack with thrust equal to drag, so that energy is conserved. For this reason, the phugoid motion is often described as a balance between kinetic and potential energy (high speed = low height and vice versa). Unfortunately Lanchester’s expertise did not extend to Greek—he chose the name phugoid for this flying motion without realising that the Greek word he had used was associated with flying in the sense of running away (eg fugitive).

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2.1.4 The SPO approximation We now look for a solution to (2.3) with Λ large: and Similar (though more involved) arguments to those above then give: These values can be confirmed by checking the magnitudes of the leading order terms in (2.3):

!

" # xu( )u0 U

!

"xww0U

!

" xq# µ

1"C

L( )$0 = 0

!

"zuu0U

!

+ " # zw( )w0

U

!

"# 1+ zq

µ1( )$0 = 0

0

!

" m˙ w # µ

1+ m

w( ) w0

U

!

+" iy" #m

q( )$0 µ1 = 0

This tells us that the SPO is an oscillation largely in pitch and angle of attack, at more or less constant forward speed. As before, we can now obtain the approximate characteristic equation by discarding negligible terms in (2.3)—see above. Note that here we keep the first two important orders of magnitude in each equation, giving us

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!

" # xu

#xw

# xq" µ

1#C

L( )0 " # z

w#"

0 # m˙ w " µ

1+ m

w( ) " iy" #m

q( ) µ1

= 0

ie

!

(" # xu)" (" # z

w)i

y" #m

q

µ1

#m ˙ w

"

µ1

#mw

$

% &

'

( ) = 0

As we are looking for a solution with Λ ~

!

µ1

1/2, the Λ = 0 and xu roots are not applicable. This leaves us with the following characteristic equation: (2.8) Thus the damping of the SPO is associated with the coefficients zw, mq and

!

m˙ w , and

its natural frequency with zw, mq and mw. We now consider these coefficients in more detail: zw:

The z axis perturbation force due to a heave velocity w, with u = θ = 0, is:

!

Z = " 12#U 2

S CL($

0+ w U)cos(w U) "C

L($

0) + C

D($

0+ w U )sin(w U)[ ]

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mq is mainly due to the tailplane:

The change in tailplane angle of attack is ql/U, and the associated moment perturbation is thus

!

M = "l # 1

2$U 2

ST%CLT

%&T

ql

U

23

!

m˙ w is the only stability derivative that cannot be dealt with in a quasi-steady manner.

It is partly due to unsteady lift effects, but the main contribution is from ‘downwash lag’:

The instantaneous tail angle of attack is given by:

!

"T

="(t) #$(t # l U ) The contribution to the (dimensional) stability derivative is

!

M ˙ w = "l

#

# ˙ w

12$U

2S

TC

LT(%

T,&)[ ]

!

= "l 1

2#U

2S

T

$CLT

$%T

$%T

$ ˙ w

24

In summary, zw, mq and

!

m˙ w are all negative, so the Λ coefficient in (2.8) will be

positive, implying positive damping for the SPO. None of the stability derivatives are particularly small, so the magnitude of the coefficient usually corresponds to acceptable damping. The natural frequency of the SPO is given by the square root of the ‘stiffness’ term in (2.8),

!

mqzw"µ

1m

w[ ] iy . Our results show that mqzw will be positive, but the second

term is, as yet, an unknown quantity. We shall consider mw later; for the moment we just note that it is usually negative, giving a positive contribution to the natural frequency. The µ1 factor in this term ties in with our order of magnitude estimate for Λ, and shows that the SPO is a high frequency mode in comparison to the phugoid.

2.2 The Lateral Modes

2.2.1 The characteristic equation The characteristic equation is derived in exactly the same way as for the longitudinal modes. Thus, in (1.2) we set and look for solutions of the form

25

Equations (1.2) become

!

m" #Yv

#(Yp" + mg) mU #Y

r

#Lv

"(Ix" # L

p) #L

r

#Nv

#Np" I

z" # N

r

$

%

& & &

'

(

) ) )

v0

*0

r0

$

%

& & &

'

(

) ) )

=

0

0

0

$

%

& & &

'

(

) ) )

(2.9)

and, as before, the only way there can be non-zero solutions is if the determinant of the matrix is zero. This leads to a characteristic equation

!

"4

+ B"3

+ C"2

+ D" + E = 0 (2.10) (The coefficients B–E are given in Appendix 3.) The solutions to (2.10) usually take the following form:

• a complex conjugate pair (the Dutch roll)

• a negative real root (the roll subsidence)

• a negative or positive real root (the spiral mode)

The Dutch roll, like the phugoid and the SPO, is a damped oscillation. The roll subsidence and spiral modes behave like

!

e"t , with λ real, and are thus non-oscillatory.

The roll subsidence always decays with time, but the spiral mode may either grow or decay.

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2.2.2 Non-dimensional equations Equations (2.9) are non-dimensionalised in a similar way to their longitudinal counterparts. However the relevant length scale is now the wing semi-span, rather than the chord. The resulting scheme is described in Appendix 4. On applying it, we obtain

!

" # yv #(yp" µ2

+ CL ) µ2# yr

#lv "(ix" # l p ) µ2

#lr#nv #np " µ

2iz" # nr

$

%

& & &

'

(

) ) )

v0U

*0

r0s U

$

%

& & &

'

(

) ) )

=

0

0

0

$

%

& & &

'

(

) ) )

(2.11)

where µ2 is the counterpart to µ1 in the longitudinal equations: µ2 is smaller than µ1, but is still large—for the Airbus A320 example used previously it is around 160. We can thus use the same technique as before to find approximate solutions for the modes of motion.

2.2.3 The spiral mode and roll subsidence approximations If we start by looking for a small or O(1) Λ solution to (2.11), we find that a consistent set of order of magnitude solutions is These results can be justified by checking the leading order terms in (2.11):

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!

" # yv( )v0 U

!

" yp# µ

2+ C

L( )$0

!

+ µ2" y

r( ) r0s U( ) = 0

!

"lvv0U

!

+" ix" # l p( )$0 µ2

!

"lrr0s U( ) = 0

!

"nvv0U

!

"np#$

0µ2

!

+ iz" # n

r( ) r0s U( ) = 0 To characterise the motion, we need to compare the sideslip and roll angles, v0/Ueλt and φ0eλt, with the yaw angle ψ0eλt. ψ0 is related to the yaw rate amplitude r0 by

!

"0

=r0

#= (2.12)

which is of the same order as φ0. We are thus looking at motions mainly in roll and yaw, with little sideslip. Retaining the leading order terms above then leads to the characteristic equation

!

0 "CL µ2

"lv #(ix# " l p ) µ2

"lr

"nv "np # µ2

iz# " nr

= 0

(2.13) Expanding the determinant gives

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!

CL "lv (iz# " nr ) " lrnv[ ] + # lvnp + nv (ix# " l p )[ ] = 0

ie (2.14) This quadratic turns out to have two real roots, corresponding to the roll subsidence and spiral modes. In principle, we could find expressions for these roots with the equation as it stands, but they would be unpleasantly complicated. It turns out that we can simplify things by looking at the relative sizes of the stability derivatives in (2.14). Without a detailed knowledge of the aircraft geometry, it is not possible to obtain “exact” estimates along the lines of the longitudinal derivatives. However, we can determine the proportional dependence of the lateral derivatives on the aircraft lift and drag coefficients, and this will be enough for us to compare their relative sizes. In doing this, we will need to bear in mind the following rough values: CD ≈ CL ≈

!

dCD

d"#

!

dCL

d"#

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lp: rolling moment due to roll

The normal force contribution at distance y along the wing is given by

!

dZ = " 12#U 2

c(y)dy( )$cl (y)

$%+ cd (y)

&

' ( )

* + py

U

(cf the analysis for zw). This gives a rolling moment contribution dL = y dZ = This expression cannot be integrated without knowledge of the specific wing geometry and its aerodynamic properties. However, we can get a feel for the result by assuming that both chord and section are fixed, with

!

"cl (y) "# $ dCLW d# and

!

cd (y) " CDW, giving

L =

!

l p =(L p)12"USs 2

=

In general, then, we expect

!

l p "

30

np: yawing moment due to roll The motion is the same as we have just considered, but now it is the forwards force component which is important:

Taking moments and integrating gives us lr: rolling moment due to yaw

The wing angle of attack is unchanged, but its speed is altered (cf the analysis for zu). The change in lift at position y is proportional to

!

12" (U # ry)2 #U 2[ ]CLW

($0)

31

Hence nr: yawing moment due to yaw

The contributions from wing drag and the fin force give: wing: fin force: In addition, the motion is opposed by fuselage drag. Thus, overall:

32

lv: rolling moment due to sideslip This derivative arises from several sources; dihedral, sweep and fuselage effects Dihedral:

The increase in angle of attack of the right hand wing is thus vγ/U, while the left hand wing experiences an equal decrease. This means that the lift on the right hand wing goes up by an amount proportional to and the lift on the left hand wing decreases correspondingly, giving rise to a negative rolling moment. Integrating and non-dimensionalising gives Sweep:

33

The lift on the section shown is thus proportional to

!

(U cos")2CLW

#0cos"( ) =

When the aircraft sideslips, the sweep angles of the wings change:

This gives rise to a rolling moment proportional to

!

CLWcos " + v U( ) # cos " # v U( )[ ] =

and hence, due to sweep, we have

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Fuselage effects: low wing high wing

Δα is: ie For stability reasons (see later) it is desirable to have lv negative—so low wing aircraft are designed so that the effects of sweep and dihedral outweigh the fuselage contribution. However, an excessively stable aircraft is difficult to fly, so high wing aircraft often have anhedral (negative dihedral) to counteract the stabilising fuselage effect. nv: yawing moment due to sideslip

35

The contribution to the yawing moment is thus proportional to ie This is the only significant contribution to nv—it is almost entirely determined by the size of the fin. This completes our survey of the stability derivatives that appear in (2.14). The important point to note is that they are all relatively small, with the exception of lp. This means that (2.14) can be further simplified; from

!

ixnv"2 + lv np # izCL( ) # l pnv[ ]" + CL lvnr # lrnv[ ] = 0

to This kind of quadratic equation has well-separated roots. For the bigger, Λb say, the first two terms dominate, and the last can be neglected, giving On the other hand, substituting in the little root, Λl, leads to a negligible first term, and

36

(If this seems too much like sleight of hand, write down the exact solutions for the two roots and then use the relative sizes of the coefficients to approximate them. You should be able to re-derive the above results.) Now consider the types of motion described by the two solutions for Λ. Remember that we already know that sideslip is negligible in comparison to roll and yaw. We can relate the yaw and roll angles through (2.12) and the first line of (2.13):

!

"0

=µ2

#

r0s

U and

!

r0s

U=C

L

µ2

"0

ie The Λ = Λb solution thus describes a motion where roll dominates yaw. Furthermore, as it is relatively large and negative, the motion decays rapidly in time. This is the roll subsidence. (Note that its characteristic frequency corresponds exactly to the result we would obtain if we assumed a priori that the aircraft oscillates purely in roll for this mode.) Conversely, the Λ = Λl solution describes a motion which is mainly in yaw. This is the spiral mode. As Λl is relatively small, it is a slow motion, and is stable if

37

We can interpret these competing effects as follows; consider a drop in the right hand wing: +ve N +ve L -ve N -ve L If these factors balance, then the mode is neutrally stable, ie a ‘displacement’ will not decay or grow. This situation corresponds to a steady turn at constant roll angle, yaw rate and (small) sideslip. To see why, first note that this state, if achievable, must have zero net rolling and yawing moments, ie Eliminating v and r from these equations gives which is the condition for neutral spiral mode stability. An aircraft with this property can be turned by simply applying aileron to roll it to a given bank angle, then centring the stick and relaxing. Aircraft designers thus aim for this desirable state of

38

affairs. However, it’s hard to achieve perfectly, and the spiral mode usually has a very small decay or growth rate. An unstable spiral mode is not usually a problem—it’s so slow that it is easily controlled by the pilot. If visibility is poor, though, its growth is not easily detected and this can result in the disastrous and aptly named “graveyard spiral”.

2.2.4 The Dutch roll approximation If we now look for a large Λ solution to equations (2.11), we find that the following orders of magnitude form a consistent set: Again, we check this statement by looking at the leading order terms:

!

" # yv( )v0 U

!

" yp# µ

2+ C

L( )$0

!

+ µ2" y

r( ) r0s U( ) = 0

!

"lvv0U

!

+" ix" # l p( )$0 µ2

!

"lrr0s U( ) = 0

!

"nvv0U

!

"np#$

0µ2

!

+ iz" # n

r( ) r0s U( ) = 0

39

To characterise the motion, we recall (2.12) to find the yaw angle amplitude:

!

"0

=µ2

#

r0s

U~

which is of the same order of magnitude as φ0 and v0/U; ie the motion we are considering involves yaw, roll and sideslip in roughly equal amounts. This is the Dutch roll. The approximate characteristic equation follows from retaining leading order terms only, ie

!

" 0 µ2

#lv

ix"2 µ

20

#nv

0 iz"

= 0 (2.15)

ie (2.16) with solutions This implies that the Dutch roll is a pure oscillation, with no damping; in practice the damping is small, which can cause unpleasant flying qualities. The frequency of the Dutch roll is determined by the yawing moment of inertia and the stability derivative nv which, as we shall see later, represents the yaw stiffness of the aircraft. For the moment, we just note that this expression can also be obtained by considering a single-degree-of-freedom model where the aircraft is constrained to oscillate about the yaw axis only. Unlike the single-degree-of-freedom approximation to the roll subsidence, however, one cannot justify this as a good representation of the motion.

40

To obtain an estimate of the damping, we could retain higher order terms in our approximation. Fortunately there is an alternative, less painful route. Consider the characteristic equation in factorised form:

!

"4

+ B"3

+K = (" # "DR)(" # "

DR

*)(" # "

SM)(" # "

RS)

Since the coefficient B gives us the sum of the roots, we can use it to estimate the Dutch roll damping contribution. In dimensionless terms:

from (2.14):

!

" #RS + # SM( ) =lv np " izCL( ) " l pnv

ixnv

and (A3.7)

!

"m

1

2#US

B = yv +l p

ix+nr

iz

so (2.17)

41

There are two points worth noting here:

• the stability derivatives yv and nr contribute to the damping, as we would expect, since they represent the resistance of the aircraft to sideslip and yaw. However, the corresponding derivative for roll, lp, does not, even though rolling motions are involved in the mode.

• to reduce the influence of the detrimental final term, we should decrease the magnitude of lv and increase nv. This could be done by reducing dihedral and enlarging the fin, with the latter change also having a beneficial effect on yv and nr. However, these modifications will tend to destabilise the spiral mode. Thus the extent to which Dutch roll damping can be increased is constrained, and it is typically unsatisfactorily small.

42

3. Longitudinal Static Stability

3.1 Static and Dynamic Stability Consider the response of a system to a disturbance. Initially, there are two possibilities: • divergence: • initial return towards equilibrium: If, in addition to static stability, the response also displays • long term return towards equilibrium: then the system is said to be ‘dynamically stable’. The conditions for static and dynamic stability depend on the order of the differential equation describing the system: System Condition(s) Stability Order Equation Static Dynamic

!

˙ x + ax = 0

!

m˙ ̇ x + l ˙ x + kx = 0

43

Hence longitudinal static stability is associated with the stiffnesses (or equivalently natural frequencies) of the phugoid mode and the SPO. 3.2 Overall Static Stability 3.2.1 Stability condition We write the characteristic equation,

!

"4

+ B"3

+ C"2

+ D" + E = 0, as a product of the phugoid and SPO quadratic factors, each expressed in terms of the damping factor (c) and natural frequency (ω):

!

"2 + 2c

SPO#

SPO" +#

SPO

2( ) "2 + 2cph#

ph" +#

ph

2( ) = 0

The product of the natural frequencies is the coefficient E, ie from (A3.5), with Mu = 0:

!

"SPO

2" ph

2=gMwZu

mIy

or, in terms of the dimensionless equivalents

!

"SPO

2 "ph

2=

m1

2#US

$

% &

'

( )

4

*SPO

2 *ph

2= (3.1)

For overall static stability, we require that the stiffnesses of both modes are positive, ie

44

Recalling that zu = –2CL, (3.1) tells us that a necessary condition for static stability is (3.2)

Now, from (2.8):

!

"SPO

2 =m

qzw#µ

1m

w

iy

(3.3)

Hence (3.2) guarantees positive stiffnesses for the two modes, and is also a sufficient condition for overall longitudinal static stability. Note that

!

"SPO

2 remains positive when mw = 0, which implies that the overall static stability must be linked to that of the phugoid. This is not clear from (2.6), which implies that the phugoid natural frequency is independent of mw. However, this estimate can be improved via (3.1) and (3.3), to give

45

Interpretation of (3.2): equilibrium state pitch disturbance

Hence mw < 0 implies that a change in pitch attitude induces a restoring moment, ie the aircraft has positive pitch stiffness.

3.2.2 The neutral point The static stability condition (3.2) can alternatively be written in terms of the pitching moment coefficient, because

!

mw

=1

1

2"USc

#M

#w=

This quantity, in turn, is linked to the variation of aircraft lift with incidence, as follows: α = 0 α ≠ 0

46

(3.4) The point where the additional lift acts, defined by the distance xn, is known as the neutral point. It is fixed (in the linear regime), and the static stability condition, mw < 0, implies that the neutral point must be aft of the CG for static stability. The degree of stability is determined by xn/c, which is known as the static margin or CG margin.

3.2.3 Effect of the tail on the neutral point It follows from the definition of the neutral point that the aerodynamic moment coefficient taken about here is constant, and hence that the aerodynamic loads can be represented as:

If we now insist that the CG should be forward of the neutral point (for static stability), it is immediately clear that moment equilibrium is impossible when M0 is negative (ie in the pitch-down direction). Unfortunately, that is exactly the sign we expect if the lift is generated by a single wing with positive camber. This is why conventional aircraft have a second lift-generating surface: the horizontal tailplane. To see how the tailplane helps matters, we consider how adding it changes the neutral point location.

47

The analysis proceeds with the following assumptions:

• drag contributions to the pitching moment are negligible

• variations in the position of the tail lift force are negligible

• the lift contribution from the rest of the aircraft (mainly from the wing) has its neutral point at x = xnw, ie (from (3.4))

Moments about the CG give us

!

M = M wing " (xnw + l) 12#U 2

STC

LT

Differentiating with respect to α:

!

"Cm

"#= $

xnw

c

dCLW

d#$xnw

+ l

c

ST

S

"CLT

"#

48

(3.5) This tells us that the tail has a stabilising effect. Its magnitude depends on:

•

!

lST

cS= V

•

!

dCL

d"= a

•

!

"CLT

"#

a) The stick-fixed neutral point In this case, we consider the controls to be held fixed, so that (3.5) becomes Recall that the wing induces downwash on the tail, so its incidence αT is reduced compared to α, and

49

which shows how the downwash diminishes the tail’s influence. In the linear region, the dependence of the tail lift coefficient on the angle of attack and the elevator is of the form CLT = a0 + a1!T + a2" so the shift in the neutral point may be written concisely as: Typical values for the terms in this expression are: ie the aircraft would be statically unstable in the absence of the tail, and has a relatively small stability margin with the tail on. There isn’t much room for manoeuvre in varying the CG position (think about the required tail lift if it’s too far forward), which is why considerable care is required in loading up aircraft. b) The stick-free neutral point If the controls are released, the variation of tail lift with overall angle of attack is changed, and the neutral point moves. The stick-free neutral point is important for smaller aircraft, whose elevators are directly connected to the pilot controls, rather than servo-driven. To derive an expression for it, we need to formulate the tail lift as a function of αT and control force, rather than deflection.

50

Consider the tail and elevator:

In the linear region:

!

CLT

= a0

+ a1"T

+ a2# and

!

Cmh

= b0

+ b1"T

+ b2#

ie (3.6) In steady flight, the hinge moment Mh is provided by a trimming mechanism (to save pilot effort). So releasing the stick implies that Cmh is constant, and (3.5) becomes

!

xn" x

nw

c=lS

T

cS

#CLT#$

dCLd$

=

51

The coefficients a1 and a2 are positive, but the ratio b1/b2 depends on the elevator design:

• if b1/b2 > 0, the elevator is convergent, and freeing the controls decreases the static margin

• if b1/b2 < 0, the elevator is divergent, and freeing the controls increases the static margin

It is generally desirable to keep b2 small, so the aircraft controls are light. However, this can lead to unpleasantly large contributions from the stick-free modifying factor unless care is taken in the elevator design. 3.2.4 Measurement of the stick-fixed neutral point The importance of the neutral point to the aircraft stability means that we need to be able to measure its position in flight tests. However, this cannot be done directly. To see why, recall the definition of the neutral point, equation (3.4):

!

"Cm

"#= $

xn

c

dCL

d#

Now, for a fixed elevator angle, Cm is a function of α only, and we can say

52

or, graphically:

This graph illustrates the ‘trimming’ role of the elevator; to maintain equilibrium flight as CL is varied, the elevator angle must be altered. Since flight tests necessarily take place at equilibrium conditions, what we can measure is the variation of elevator angle with lift coefficient,

!

d" dCL Cm=0

. To link this to the static margin, some

further analysis is required. We start from equilibrium flight at speed U1, angle of attack α, and elevator angle η. Now imagine making a small change of speed to U2, and adjusting the angle of attack and elevator setting to maintain equilibrium flight. There are three contributions to the additional lift force: the first is associated with the change in speed, and acts at the CG (since Mu = 0); the second is associated with the change in angle of attack, and acts at the stick-fixed NP; the third is associated with the change in elevator angle, and acts at the tail:

53

The condition that moment equilibrium is maintained gives ie (3.7) Now so ie NB The same analysis carried out exactly gives which shows that the final error associated with our approximations is indeed small.

54

On any given flight, we can obtain dη/dCL from measurements at varying speeds. If data from several flights with varying CG positions are plotted, we then have

3.2.5 Measurement of the stick-free neutral point A similar analysis applies for the stick-free neutral point, but now we split the additional lift into the component associated with Δα for constant hinge moment, and the component associated with the change in hinge moment. Equation (3.7) then takes the form: (since (3.6) tells us that !CLT !Cmh = a2 b2 ). We again make the approximation that aΔα = ΔCL, to obtain and the stick-free neutral point is thus found by plotting stick force data from several flights with different CG positions:

55

3.3 SPO Static Stability

3.3.1 The manoeuvre point The stiffness term in the SPO characteristic equation, (2.8), is

!

mqzw"µ

1m

w

iy

=

where ‘Manoeuvre Point’

3.3.2 Measurement of the manoeuvre point Since the manoeuvre point is directly linked to the SPO stiffness, it is measured from the SPO steady-state response to a change in elevator angle. We analyse this response by using the SPO approximation to simplify equations (1.1). This means that we are ignoring any long-term phugoid response to the elevator, which is valid in the initial stages of the motion.

56

SPO approximation:

!

u " 0 (1.1) heave:

!

m( ˙ w "Uq) # Zww + Z

qq

(1.1) pitch:

!

Iy˙ q = M

ww + M

qq + M

˙ w ˙ w + M"E

"E

and we also neglect Zq (=mUzq/µ1) in comparison with mU. Then the equation for the steady-state response ( ˙ w = ˙ q = 0 ) is

!

Zw

mU

Mw

Mq

"

# $

%

& ' w

q

"

# $ %

& ' = (

0

M)E)E

"

# $

%

& '

(3.8) Thus the position of the manoeuvre point determines the magnitude of the pitch rate and change in angle of attack in response to elevator motion, ie the manoeuvrability of the aircraft.

57

The usual flight test measurement taken is the normal acceleration: Now, from (3.8): (3.9) For a given flight test, we can find

!

d" dnz (the ‘elevator angle per g’) from a set of

measurements at different elevator angles. The manoeuvre point can then be located by plotting data from several flight tests with varying CG positions:

This gives us the stick-fixed manoeuvre point, because the stability derivatives in (3.8) are all for fixed elevator angle. The measurement used here,

!

d" dnz, is the

counterpart of the ‘elevator angle to trim’,

!

d" dCL

, which is used to locate the stick-fixed neutral point.

58

3.3.3 The stick-free manoeuvre point The stick-free manoeuvre point is similarly related to the stiffness of the stick-free SPO. The stick-free modes arise from the equations of motion in control force form: A similar analysis to 3.3.2 then leads to the stick-free equivalent of (3.9): The ‘stick force per g’ can be found from a set of measurements of stick force and normal acceleration on a given flight, and data from several flights with varying CG positions can then be plotted to locate the stick-free manoeuvre point:

59

Just as the elevator angle per g was the manoeuvre point counterpart of the elevator angle to trim in the neutral point analysis, so the stick force per g is the analogue of the stick force to trim,

!

dCmh

dCL

. 3.4 Summary In this section, we have considered the static stability of the longitudinal motions of the aircraft. We have identified two key quantities:

• the pitch stiffness,

!

"mw

=xn

c

dCL

d#

• the manoeuvre stiffness,

!

"mw +mqzw

µ1

=xmp

c

dCL

d#

The pitch stiffness is linked to the static stability of the phugoid mode. Positive pitch stiffness implies the static stability of both the phugoid and the SPO, and hence overall static stability for longitudinal motions. The manoeuvre stiffness describes the static stability of the SPO alone, and determines the sensitivity of the short-term aircraft response to control motions.

60

4 Lateral-Directional Static Stability We have seen that the longitudinal static stability of the aircraft is associated with the existence of a restoring moment following a disturbance in pitch. By analogy, ‘lateral-directional’ static stability considers the restoring moments in roll and yaw associated with angular displacements about these axes. Note that (confusingly) this is not necessarily the same thing as overall static stability in anti-symmetric motions. This is determined by the stability of the lateral modes of motion and, as we shall see, is not guaranteed by lateral-directional static stability.

4.1 Directional Static Stability An aircraft is said to have directional static stability if Interpretation: consider a displacement in yaw:

61

Implication: recall the approximate characteristic equation for Dutch roll, (2.16):

!

iz"2 + µ

2nv

= 0 ie Directional static stability <==> We can now see why the classical aircraft configuration has a fin; it ensures that nv is positive and thereby gives the aircraft directional stability. Remember, though, that increasing nv has an adverse effect on spiral mode stability.

4.2 Lateral Static Stability An aircraft is said to have lateral static stability if Interpretation: consider a displacement in roll:

The condition lv < 0 thus implies that a roll-induced sideslip will result in a restoring rolling moment. This is not roll stiffness, but it does imply a tendency to oppose disturbances in roll.

62

Implication: recall that the spiral mode root in the characteristic equation is given by

!

" # $CL

lvnr $ lrnv

$l pnv

ie Lateral static stability does not guarantee spiral mode stability but is a necessary condition for spiral mode stability In fact, as we have already noted, the spiral mode should be close to neutral, rather than strongly stable, if the aircraft is to handle well. This criterion, however, still requires lv < 0; ie lateral static stability is also a necessary condition for a neutrally stable spiral mode.

4.3 Measurement of Lateral-Directional Static Stability The lateral-directional static stability of an aircraft is assessed by measuring the control surface deflections necessary to maintain a steady sideslip at a constant bank angle:

63

The equations of motion, (1.2), then simplify to (with control roll/yaw coupling terms neglected). The convention for measuring aileron and rudder angles means that

!

L"A,N"R

< 0 , so our lateral-directional static

stability conditions become: lv < 0:

nv > 0:

These results tell us about the controls-fixed lateral-directional static stability (since the derivatives lv and nv are for fixed aileron and rudder angles). For assessment of controls-free lateral-directional static stability, the aileron and rudder control forces would replace δA and δR (as with the longitudinal controls-free static stability).