Flight dynamics is the science of
air and
space vehicle
orientation and control in three dimensions. The three critical
flight dynamics parameters are the angles of rotation in three
dimensions about the vehicle's
center of mass, known as
pitch,
roll and
yaw (quite different from their use as
TaitBryan angles).
Aerospace engineers develop
control systems for a vehicle's
orientation (
attitude) about its
center of mass. The control systems include actuators, which exert
forces in various directions, and generate rotational forces or
moment about the
aerodynamic center of the aircraft, and
thus rotate the aircraft in pitch, roll, or yaw. For example, a
pitching moment is a vertical force
applied at a distance forward or aft from the aerodynamic center of
the aircraft, causing the aircraft to pitch up or down.
Roll, pitch and yaw refer to rotations about the respective axes
starting from a defined equilibrium state. The equilibrium roll
angle is known as wings level or zero bank angle, equivalent to a
level
heeling angle on a ship. Yaw
is known as 'heading'. The equilibrium pitch angle in submarine and
airship parlance is known as 'trim', but in aircraft, this usually
refers to
angle of attack, rather
than orientation. However, common usage ignores this distinction
between equilibrium and dynamic cases.
The most common aeronautical convention defines the roll as acting
about the longitudinal axis, positive with the starboard(right)
wing down. The yaw is about the vertical body axis, positive with
the nose to starboard. Pitch is about an axis perpendicular to the
longitudinal plane of symmetry, positive nose up.
A
fixedwing aircraft increases
or decreases the lift generated by the wings when it pitches nose
up or down by increasing or decreasing the
angle of attack (AOA). The roll angle is
also known as bank angle on a fixed wing aircraft, which usually
"banks" to change the horizontal direction of flight. An aircraft
is usually streamlined from nose to tail to reduce
drag making it typically advantageous to keep
the sideslip angle near zero, though there are instances when an
aircraft may be deliberately "sideslipped" for example a
slip in a fixed wing aircraft.
Coordinate systems
The position (and hence motion) of an aircraft is generally defined
relative to one of 3 sets of coordinate systems:
 Wind Axes
 X Axis  Positive in the direction of the oncoming air
(relative wind)
 Y Axis  Positive to Right of X Axis, perpendicular to X Axis
 Z Axis  Positive downwards, perpendicular to XY plane
 Inertial Axes (or Body Axes)  based about aircraft CG
 X Axis  Positive forward, through nose of aircraft
 Y Axis  Positive to Right of X Axis, perpendicular to X Axis
 Z Axis  Positive downwards, perpendicular to XY plane
 Earth Axes
 X Axis  Positive in the direction of North
 Y Axis  Positive in the direction of East (perpendicular to X
Axis)
 Z Axis  Positive towards the centre of Earth (perpendicular to
XY Plane)
For flight dynamics applications the Earth Axes are generally of
minimal use, and hence will be ignored. The motions relevant to
dynamic stability are usually too short in duration for the motion
of the Earth itself to be considered relevant for aircraft.
In flight dynamics, pitch, roll and yaw angles measure both the
absolute attitude angles (relative to the horizon/North) and
changes in attitude angles, relative to the equilibrium
orientation of the
vehicle. These are
defined as:
 Pitch  Angle of X Body Axis (nose) relative to horizon. Also a
positive (nose up) rotation about Y Body Axis
 Roll  Angle of Y Body Axis (wing) relative to horizon. Also a
positive (right wing down) rotation about X Body Axis
 Yaw  Angle of X Body Axis (nose) relative to North. Also a
positive (nose right) rotation about Z Body axis
In analysing the dynamics, we are concerned both with rotation and
translation of this axis set with respect to a fixed inertial
frame. For all practical purposes a local Earth axis set is used,
this has X and Y axis in the local horizontal plane, usually with
the xaxis coinciding with the projection of the velocity vector at
the start of the motion, on to this plane. The z axis is vertical,
pointing generally towards the Earth's centre, completing an
orthogonal set.
In general, the body axes are not aligned with the Earth axes. The
body orientation may be defined by three
Euler angles, the
TaitBryan rotations, a
quaternion, or a direction cosine matrix
(
rotation matrix). A rotation matrix
is particularly convenient for converting velocity, force,
angular velocity, and
torque vectors between body and Earth coordinate
frames.
Body axes tend to be used with missile and rocket configurations.
Aircraft stability uses wind axes in which the xaxis points along
the velocity vector. For straight and level flight this is found
from body axes by rotating nose down through the
angle of attack.
Stability deals with small perturbations in angular displacements
about the orientation at the start of the motion. This consists of
two components; rotation about each axis, and angular displacements
due change in orientation of each axis. The latter term is of
second order for the purpose of stability analysis, and is
ignored.
Design cases
In analysing the stability of an aircraft, it is usual to consider
perturbations about a nominal equilibrium position. So the analysis
would be applied, for example, assuming:
 :Steady level flight
 :Turn at constant speed
 :Approach and landing
 :Take off
The speed, height and trim angle of attack are different for each
flight condition, in addition, the aircraft will be configured
differently, e.g. at low speed
flaps
may be deployed and the
undercarriage
may be down.
Except for asymmetric designs (or symmetric designs at significant
sideslip), the longitudinal equations of motion (involving pitch
and lift forces) may be treated independently of the lateral motion
(involving roll and yaw).
The following considers perturbations about a nominal straight and
level flight path.
To keep the analysis (relatively) simple, the control surfaces are
assumed fixed throughout the motion, this is stickfixed stability.
Stickfree analysis requires the further complication of taking the
motion of the control surfaces into account.
Furthermore, the flight is assumed to take place in still air, and
the aircraft is treated as a
rigid
body.
Spacecraft
Unless designed to conduct part of the mission within a planetary
atmosphere, a
spacecraft would generally have no discernible
front or side, and no bottom unless designed to land on a surface,
so reference to a 'nose' or 'wing' or even 'down' is arbitrary. On
a manned spacecraft, the axes must be oriented relative to the
pilot's physical orientation at the flight control station.
Unmanned spacecraft may need to maintain orientation of
solar cells toward the Sun, antennas toward the
Earth, or cameras toward a target, so the axes will typically be
chosen relative to these functions.
Longitudinal modes
It is common practice to derive a fourth order
characteristic equation to describe
the longitudinal motion, and then factorise it approximately into a
high frequency mode and a low frequency mode. This requires a level
of algebraic manipulation which most readers will doubtless find
tedious, and adds little to the understanding of aircraft dynamics.
The approach adopted here is to use our qualitative knowledge of
aircraft behaviour to simplify the equations from the outset,
reaching the same result by a more accessible route.
The two longitudinal motions (modes) are called the
short period pitch oscillation (SSPO), and the
phugoid.
Shortperiod pitch oscillation
A short input (in
control systems
terminology an
impulse) in pitch (generally
via the elevator in a standard configuration fixed wing aircraft)
will generally lead to overshoots about the trimmed condition. The
transition is characterised by a damped
simple harmonic motion about the new
trim. There is very little change in the trajectory over the time
it takes for the oscillation to damp out.
Generally this oscillation is high frequency (hence short period)
and is damped over a period of a few seconds. A realworld example
would involve a pilot selecting a new climb attitude, for example
5ยบ nose up from the original attitude. A short, sharp pull back on
the control column may be used, and will generally lead to
oscillations about the new trim condition. If the oscillations are
poorly damped the aircraft will take a long period of time to
settle at the new condition, potentially leading to
Pilotinduced oscillation. If the
short period mode is unstable it will generally be impossible for
the pilot to safely control the aircraft for any period of
time.
This
damped harmonic motion is called the
short period pitch oscillation, it
arises from the tendency of a stable aircraft to point in the
general direction of flight. It is very similar in nature to the
weathercock mode of missile or rocket
configurations. The motion involves mainly the pitch attitude
\theta (theta) and incidence \alpha (alpha). The direction of the
velocity vector, relative to inertial axes is \theta\alpha. The
velocity vector is:
 :u_f=U\cos(\theta\alpha)
 :w_f=U\sin(\theta\alpha)
where u_f, w_f are the inertial axes components of velocity.
According to
Newton's Second
Law, the
accelerations are
proportional to the
forces, so the forces in
inertial axes are:

:X_f=m\frac{du_f}{dt}=m\frac{dU}{dt}\cos(\theta\alpha)mU\frac{d(\theta\alpha)}{dt}\sin(\theta\alpha)

:Z_f=m\frac{dw_f}{dt}=m\frac{dU}{dt}\sin(\theta\alpha)+mU\frac{d(\theta\alpha)}{dt}\cos(\theta\alpha)
where m is the
mass.By the nature of the
motion, the speed variation m\frac{dU}{dt} is negligible over the
period of the oscillation, so:
 :X_f= mU\frac{d(\theta\alpha)}{dt}\sin(\theta\alpha)
 :Z_f=mU\frac{d(\theta\alpha)}{dt}\cos(\theta\alpha)
But the forces are generated by the
pressure distribution on the body, and are referred
to the velocity vector. But the velocity (wind) axes set is not an
inertial frame so we must resolve the fixed
axes forces into wind axes. Also, we are only concerned with the
force along the zaxis:
 :Z=Z_f\cos(\theta\alpha)+X_f\sin(\theta\alpha)
Or:
 :Z=mU\frac{d(\theta\alpha)}{dt}
In words, the wind axes force is equal to the
centripetal acceleration.
The moment equation is the time derivative of the
angular momentum:
 :M=B\frac{d^2 \theta}{dt^2}
where M is the pitching moment, and B is the
moment of inertia about the pitch
axis.Let: \frac{d\theta}{dt}=q, the pitch rate.The equations of
motion, with all forces and moments referred to wind axes are,
therefore:
 :\frac{d\alpha}{dt}=q+\frac{Z}{mU}
 :\frac{dq}{dt}=\frac{M}{B}
We are only concerned with perturbations in forces and moments, due
to perturbations in the states \alpha and q, and their time
derivatives. These are characterised by
stability derivatives determined from
the flight condition. The possible stability derivatives are:
 ::Z_\alpha Lift due to incidence, this is negative because the
zaxis is downwards whilst positive incidence causes an upwards
force.
 ::Z_q Lift due to pitch rate, arises from the increase in tail
incidence, hence is also negative, but small compared with
Z_\alpha.
 ::M_\alpha Pitching moment due
to incidence  the static stability term. Static stability requires this
to be negative.
 ::M_q Pitching moment due to pitch rate  the pitch damping
term, this is always negative.
Since the tail is operating in the flowfield of the wing, changes
in the wing incidence cause changes in the downwash, but there is a
delay for the change in wing flowfield to affect the tail lift,
this is represented as a moment proportional to the rate of change
of incidence:
 ::M_\dot\alpha
Increasing the wing incidence without increasing the tail incidence
produces a nose up moment, so M_\dot\alpha is expected to be
positive.
The equations of motion, with small perturbation forces and moments
become:

:\frac{d\alpha}{dt}=\left(1+\frac{Z_q}{mU}\right)q+\frac{Z_\alpha}{mU}\alpha

:\frac{dq}{dt}=\frac{M_q}{B}q+\frac{M_\alpha}{B}\alpha+\frac{M_\dot\alpha}{B}\dot\alpha
These may be manipulated to yield as second order linear
differential equation in \alpha:

:\frac{d^2\alpha}{dt^2}\left(\frac{Z_\alpha}{mU}+\frac{M_q}{B}+(1+\frac{Z_q}{mU})\frac{M_\dot\alpha}{B}\right)\frac{d\alpha}{dt}+\left(\frac{Z_\alpha}{mU}\frac{M_q}{B}\frac{M_\alpha}{B}(1+\frac{Z_q}{mU})\right)\alpha=0
This represents a
damped simple harmonic motion.
We should expect \frac{Z_q}{mU} to be small compared with unity, so
the coefficient of \alpha (the 'stiffness' term) will be positive,
provided
M_\alpha<\FRAC{Z_\ALPHA}{MU}M_Q<></\FRAC{Z_\ALPHA}{MU}M_Q<>math>.
This expression is dominated by M_\alpha, which defines the
longitudinal static
stability of the aircraft, it must be negative for stability.
The damping term is reduced by the downwash effect, and it is
difficult to design an aircraft with both rapid natural response
and heavy damping. Usually, the response is underdamped but
stable.
Phugoid
If the stick is held fixed, the aircraft will not maintain straight
and level flight, but will start to dive, level out and climb
again. It will repeat this cycle until the pilot intervenes. This
long period oscillation in speed and height is called the
phugoid mode. This is analysed by assuming that the
SSPO performs its proper function and
maintains the angle of attack near its nominal value. The two
states which are mainly affected are the climb angle \gamma (gamma)
and speed. The small perturbation equations of motion are:
 :mU\frac{d\gamma}{dt}=Z
which means the centripetal force is equal to the perturbation in
lift force.
For the speed, resolving along the trajectory:
 :m\frac{du}{dt}=Xmg\gamma
where g is the
acceleration due to
gravity at the earths surface. The acceleration along the
trajectory is equal to the net xwise force minus the component of
weight. We should not expect significant aerodynamic derivatives to
depend on the climb angle, so only X_u and Z_u need be considered.
X_u is the drag increment with increased speed, it is negative,
likewise Z_u is the lift increment due to speed increment, it is
also negative because lift acts in the opposite sense to the
zaxis.
The equations of motion become:
 : mU\frac{d\gamma}{dt}=Z_u u
 : m\frac{du}{dt}=X_u u mg\gamma
These may be expressed as a second order equation in climb angle or
speed perturbation:

:\frac{d^2u}{dt^2}\frac{X_u}{m}\frac{du}{dt}\frac{Z_ug}{mU}u=0
Now lift is very nearly equal to weight:
 :Z=\frac{1}{2}\rho U^2 c_L S_w=W
where \rho is the air density, S_w is the wing area, W the weight
and c_L is the lift coefficient (assumed constant because the
incidence is constant), we have, approximately:
 :Z_u=\frac{2W}{U}=\frac{2mg}{U}
The period of the phugoid, T, is obtained from the coefficient of
u:
 :\frac{2\pi}{T}=\sqrt{\frac{2g^2}{U^2}}
Or:
 :T=\frac{2\pi U}{\sqrt{2}g}
Since the lift is very much greater than the drag, the phugoid is
at best lightly damped. A
propeller with
fixed speed would help. Heavy damping of the pitch rotation or a
large
rotational inertia increase
the coupling between short period and phugoid modes, so that these
will modify the phugoid.
Lateral modes
With a symmetrical rocket or missile, the
directional stability in yaw is the
same as the pitch stability; it resembles the short period pitch
oscillation, with yaw plane equivalents to the pitch plane
stability derivatives. For this reason pitch and yaw directional
stability are collectively known as the 'weathercock' stability of
the missile.
Aircraft lack the symmetry between pitch and yaw, so that
directional stability in yaw is derived from a different set of
stability derivatives, The yaw plane equivalent to the short period
pitch oscillation, which describes yaw plane directional stability
is called Dutch roll. Unlike pitch plane motions, the lateral modes
involve both roll and yaw motion.
Dutch roll
It is customary to derive the equations of motion by formal
manipulation in what, to the engineer, amounts to a piece of
mathematical sleight of hand. The current approach follows the
pitch plane analysis in formulating the equations in terms of
concepts which are reasonably familiar.
Applying an impulse via the rudder pedals should induce
Dutch roll, which is the oscillation in roll and
yaw, with the roll motion lagging yaw by a quarter cycle, so that
the wing tips follow elliptical paths with respect to the
aircraft.
The yaw plane translational equation, as in the pitch plane,
equates the centripetal acceleration to the side force.
 :\frac{d\beta}{dt}=\frac{Y}{mU}r
where \beta (beta) is the
sideslip
angle, Y the side force and r the yaw rate.
The moment equations are a bit trickier. The trim condition is with
the aircraft at an angle of attack with respect to the airflow, The
body xaxis does not align with the velocity vector, which is the
reference direction for wind axes. In other words, wind axes are
not
principal axes (the mass is not
distributed symmetrically about the yaw and roll axes). Consider
the motion of an element of mass in position z,x in the direction
of the yaxis, i.e. into the plane of the paper.
If the roll rate is p, the velocity of the particle is:
 ::v=pz+xr
Made up of two terms, the force on this particle is first the
proportional to rate of v change, the second is due to the change
in direction of this component of velocity as the body moves. The
latter terms gives rise to cross products of small quantities
(pq,pr,qr), which are later discarded. In this analysis, they are
discarded from the outset for the sake of clarity. In effect, we
assume that the direction of the velocity of the particle due to
the simultaneous roll and yaw rates does not change significantly
throughout the motion. With this simplifying assumption, the
acceleration of the particle becomes:
 ::\frac{dv}{dt}=\frac{dp}{dt}z+\frac{dr}{dt}x
The yawing moment is given by:
 ::\delta m x \frac{dv}{dt}=\frac{dp}{dt}xz\delta m +
\frac{dr}{dt}x^2\delta m
There is an additional yawing moment due to the offset of the
particle in the y direction:\frac{dr}{dt}y^2\delta m
The yawing moment is found by summing over all particles of the
body:
 ::N=\frac{dp}{dt}\int xz dm +\frac{dr}{dt}\int x^2 + y^2 dm
=E\frac{dp}{dt}+C\frac{dr}{dt}
where N is the yawing moment, E is a product of inertia, and C is
the moment of inertia about the
yaw axis.A
similar reasoning yields the roll equation:
 ::L=A\frac{dp}{dt}E\frac{dr}{dt}
where L is the rolling moment and A the roll moment of
inertia.
Lateral and longitudinal stability derivatives
The states are \beta (sideslip),r (yaw rate) and p (roll rate),
with moments N (yaw) and L (roll), and force Y (sideways). There
are nine stability derivatives relevant to this motion, the
following explains how they originate. However a better intuitive
understanding is to be gained by simply playing with a model
aeroplane, and considering how the forces on each component are
affected by changes in sideslip and angular velocity:
 ::Y_\beta Side force due to side slip (in absence of yaw).
Sideslip generates a sideforce from the fin and the fuselage. In
addition, if the wing has dihedral, side slip at a positive roll
angle increases incidence on the starboard wing and reduces it on
the port side, resulting in a net force component directly opposite
to the sideslip direction. Sweep back of the wings has the same
effect on incidence, but since the wings are not inclined in the
vertical plane, backsweep alone does not affect Y_\beta. However,
anhedral may be used with high backsweep angles in high performance
aircraft to offset the wing incidence effects of sideslip. Oddly
enough this does not reverse the sign of the wing configuration's
contribution to Y_\beta (compared to the dihedral case).
 ::Y_p Side force due to roll rate.
Roll rate causes incidence at the fin, which generates a
corresponding side force. Also, positive roll (starboard wing down)
increases the lift on the starboard wing and reduces it on the
port. If the wing has dihedral, this will result in a side force
momentarily opposing the resultant sideslip tendency. Anhedral wing
and or stabiliser configurations can cause the sign of the side
force to invert if the fin effect is swamped.
 ::Y_r Side force due to yaw rate.
Yawing generates side forces due to incidence at the rudder, fin
and fuselage.
 ::N_\beta Yawing moment due to sideslip forces.
Sideslip in the absence of rudder input causes incidence on the
fuselage and
empennage, thus creating a
yawing moment counteracted only by the directional stiffness which
would tend to point the aircraft's nose back into the wind in
horizontal flight conditions. Under sideslip conditions at a given
roll angle N_\beta will tend to point the nose into the sideslip
direction even without rudder input, causing a downward spiralling
flight.
 ::N_p Yawing moment due to roll rate.
Roll rate generates fin lift causing a yawing moment and also
differentially alters the lift on the wings, thus affecting the
induced drag contribution of each wing, causing a (small) yawing
moment contribution. Positive roll generally causes positive N_p
values unless the
empennage is anhedral or
fin is below the roll axis. Lateral force components resulting from
dihedral or anhedral wing lift differences has little effect on N_p
because the wing axis is normally closely aligned with the centre
of gravity.
 ::N_r Yawing moment due to yaw rate.
Yaw rate input at any roll angle generates rudder, fin and fuselage
force vectors which dominate the resultant yawing moment. Yawing
also increases the speed of the outboard wing whilst slowing down
the inboard wing, with corresponding changes in drag causing a
(small) opposing yaw moment. N_r opposes the inherent directional
stiffness which tends to point the aircraft's nose back into the
wind and always matches the sign of the yaw rate input.
 ::L_\beta Rolling moment due to sideslip.
A positive sideslip angle generates empennage incidence which can
cause positive or negative roll moment depending on its
configuration. For any nonzero sideslip angle dihedral wings
causes a rolling moment which tends to return the aircraft to the
horizontal, as does back swept wings. With highly swept wings the
resultant rolling moment may be excessive for all stability
requirements and anhedral could be used to offset the effect of
wing sweep induced rolling moment.
 ::L_r Rolling moment due to yaw rate.
Yaw increases the speed of the outboard wing whilst reducing speed
of the inboard one, causing a rolling moment to the inboard side.
The contribution of the fin normally supports this inward rolling
effect unless offset by anhedral stabiliser above the roll axis (or
dihedral below the roll axis).
 ::L_p Rolling moment due to roll rate.
Roll creates counter rotational forces on both starboard and port
wings whilst also generating such forces at the empennage. These
opposing rolling moment effects have to be overcome by the aileron
input in order to sustain the roll rate. If the roll is stopped at
a nonzero roll angle the L_\beta
upward rolling moment
induced by the ensueing sideslip should return the aircraft to the
horizontal unless exceeded in turn by the
downward L_r
rolling moment resulting from sideslip induced yaw rate.
Longitudinal stability could be ensured or improved by minimizing
the latter effect.
Equations of motion
Since
Dutch roll is a handling mode,
analogous to the short period pitch oscillation, we shall ignore
any effect it might have on the trajectory. The body rate r is made
up of the rate of change of sideslip angle and the rate of turn.
Taking the latter as zero, because we assume no effect on the
trajectory, we have, for the limited purpose of studying the Dutch
roll:
 ::\frac{d\beta}{dt}= r
The yaw and roll equations, with the stability derivatives
become:
 :C\frac{dr}{dt}E\frac{dp}{dt}=N_\beta \beta  N_r
\frac{d\beta}{dt} + N_p p (yaw)
 :A\frac{dp}{dt}E\frac{dr}{dt}=L_\beta \beta  L_r
\frac{d\beta}{dt} + L_p p (roll)
The inertial moment due to the roll acceleration is considered
small compared with the aerodynamic terms, so the equations
become:
 :C\frac{d^2\beta}{dt^2} = N_\beta \beta  N_r
\frac{d\beta}{dt} + N_p p
 :E\frac{d^2\beta}{dt^2} = L_\beta \beta  L_r \frac{d\beta}{dt}
+ L_p p
This becomes a second order equation governing either roll rate or
sideslip:

:\left(\frac{N_p}{C}\frac{E}{A}\frac{L_p}{A}\right)\frac{d^2\beta}{dt^2}+
\left(\frac{L_p}{A}\frac{N_r}{C}\frac{N_p}{C}\frac{L_r}{A}\right)\frac{d\beta}{dt}\left(\frac{L_p}{A}\frac{N_\beta}{C}\frac{L_\beta}{A}\frac{N_p}{C}\right)\beta
= 0
The equation for roll rate is identical. But the roll angle, \phi
(phi)is given by:
 ::\frac{d\phi}{dt}=p
If p is a damped simple harmonic motion, so is \phi, but the roll
must be in
quadrature with the roll
rate, and hence also with the sideslip. The motion consists of
oscillations in roll and yaw, with the roll motion lagging 90
degrees behind the yaw. The wing tips trace out elliptical
paths.
Stability requires the '
stiffness' and
'damping' terms to be positive. These are:

::\frac{\frac{L_p}{A}\frac{N_r}{C}\frac{N_p}{C}\frac{L_r}{A}}
{\frac{N_p}{C}\frac{E}{A}\frac{L_p}{A}} (damping)

::\frac{\frac{L_\beta}{A}\frac{N_p}{C}\frac{L_p}{A}\frac{N_\beta}{C}}
{\frac{N_p}{C}\frac{E}{A}\frac{L_p}{A}} (stiffness)
The denominator is dominated by L_p, the roll damping derivative,
which is always negative, so the denominators of these two
expressions will be positive.
Considering the 'stiffness' term: L_p N_\beta will be positive
because L_p is always negative and N_\beta is positive by design.
L_\beta is usually negative, whilst N_p is positive. Excessive
dihedral can destabilise the Dutch roll, so configurations with
highly swept wings require anhedral to offset the wing sweep
contribution to L_\beta.
The damping term is dominated by the product of the roll damping
and the yaw damping derivatives, these are both negative, so their
product is positive. The Dutch roll should therefore be
damped.
The motion is accompanied by slight lateral motion of the centre of
gravity and a more 'exact' analysis will introduce terms in Y_\beta
etc. In view of the accuracy with which stability derivatives can
be calculated, this is an unnecessary pedantry, which serves to
obscure the relationship between aircraft geometry and handling,
which is the fundamental objective of this article.
Roll subsidence
Jerking the stick sideways and returning it to centre causes a net
change in roll orientation.
The roll motion is characterized by an absence of natural
stability, there are no stability derivatives which generate
moments in response to the inertial roll angle. A roll disturbance
induces a roll rate which is only cancelled by pilot or
autopilot intervention. This takes place with
insignificant changes in sideslip or yaw rate, so the equation of
motion reduces to:
 :A\frac{dp}{dt}=L_p p.
L_p is negative, so the roll rate will decay with time. The roll
rate reduces to zero, but there is no direct control over the roll
angle.
Spiral mode
Simply holding the stick still, when starting with the wings near
level, an aircraft will usually have a tendency to gradually veer
off to one side of the straight flightpath. This is the (slightly
unstable)
spiral mode. The opposite holds for a
stable spiral mode. The spiral mode is sonamed because when it is
slightly unstable, and the controls are not moved, the aircraft
will tend to increase its bank angle slowly at first, then ever
faster. The resulting path through the air is a continuously
tightening and ever more rapidly descending
spiral. An
unstable spiral mode is common to most aircraft. It is not
dangerous because the times to double the bank angle are large
compared to the the pilot's ability to respond and correct errors
with aileron inputs.
When the spiral mode is stable, it behaves in a way opposite to the
exponential divergence of the unstable mode. The stable spiral
mode, when starting with the wings at a moderate bank angle, will
return to near wings level, first quickly, then more slowly. When
the spiral mode is stable and starting at a moderate bank angle,
the spiral nature of the flight path is not as obvious. This is
because usually only a fraction of a turn is made while the wings
are not fully level. The turning starts out (relatively) tight,
then becomes less and less so as the wings become more level.
The divergence rate of the
unstable spiral mode will be
roughly proportional to the roll angle itself (i.e. roughly
exponential growth). The
convergence rate of the
stable spiral mode will be roughly proportional to the
roll angle itself (i.e. roughly exponential
decay).
Spiral mode trajectory
In studying the trajectory, it is the direction of the velocity
vector, rather than that of the body, which is of interest. The
direction of the velocity vector when projected on to the
horizontal will be called the track, denoted \mu (mu). The body
orientation is called the heading, denoted \psi (psi). The force
equation of motion includes a component of weight:
 :\frac{d\mu}{dt}=\frac{Y}{mU} + \frac{g}{U}\phi
where g is the gravitational acceleration, and U is the
speed.
Including the stability derivatives:
 :\frac{d\mu}{dt}=\frac{Y_\beta}{mU}\beta + \frac {Y_r}{mU}r +
\frac{Y_p}{mU}p + \frac{g}{U}\phi
Roll rates and yaw rates are expected to be small, so the
contributions of Y_r and Y_p will be ignored.
The sideslip and roll rate vary gradually, so their time
derivatives are ignored. The yaw and roll
equations reduce to:
 :N_\beta \beta + N_r\frac{d\mu}{dt} + N_p p = 0 (yaw)
 :L_\beta \beta + L_r\frac{d\mu}{dt} + L_p p = 0 (roll)
Solving for \beta and p:
 ::\beta=\frac{(L_r N_p  L_p N_r)}{(L_p N_\beta  N_p
L_\beta)}\frac{d\mu}{dt}
 ::p=\frac{(L_\beta N_r  L_r N_\beta)}{(L_p N_\beta  N_p
L_\beta)}\frac{d\mu}{dt}
Substituting for sideslip and roll rate in the force equation
results in a first order equation in roll angle:
 ::\frac{d\phi}{dt}=mg\frac{(L_\beta N_r  N_\beta L_r)}{mU(L_p
N_\beta  N_p L_\beta)Y_\beta(L_r N_p  L_p N_r)}\phi
This is an
exponential growth or decay,
depending on whether the coefficient of \phi is positive or
negative. The denominator is usually negative, which requires
L_\beta N_r > N_\beta L_r (both products are positive). This is
in direct conflict with the Dutch roll stability requirement, and
it is difficult to design an aircraft for which both the Dutch roll
and spiral mode are inherently stable.
Since the
spiral mode has a long
time constant, the pilot can intervene to effectively stabilise it,
but an aircraft with an unstable Dutch roll would be difficult to
fly. It is usual to design the aircraft with a stable Dutch roll
mode, but slightly unstable spiral mode.
See also
References
Footnotes
External links