GYROMECHANICS Rotational Motion

1
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion A very
important dynamics topic is rotational motion
especially for aerospace engineers - our vehicles
are free in both translation and rotation - 6
degrees of freedom.  Let us look at a very
simple system. An AXISYMMETRIC body, freely
rotating about its symmetry axis e3 with angular
velocity ? , possesses total angular momentum
If no external forces act on the body, then
 If a small force F is applied to the axis at r,
then a torque r F results, and the equation of
motion is then r F dJ/dt. This means that
the spin axis will change direction, the body
acquiring a component of angular velocity
perpendicular to that it already possesses . If
the torque is small enough, the angular velocity
with which the axis moves will be small cf. the
angular velocity around the axis, and we can then
make the simplifying assumption that changes in
the angular momentum components perpendicular to
the spin axis may be neglected to 1st order.
2
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion r F
dJ/dt
3
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion Since the
disturbing torque is perpendicular to J (r r
.e3, so r ll J, and r F perpendicular to r),
then the magnitude of J is unchanged, but its
direction does change, moving in the direction
of r F , perpendicular to applied force! This
seems counter-intuitive, but is an exact
parallel to the tranverse force provided
by tension in a string on a swung mass, or the
Lorentz force on a charged particle moving in a
magnetic field - as in these cases, the result
is a rotation of the (angular) velocity vector
with no change in magnitude.
4
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion If F remains
constant in magnitude and direction w.r.t. the
spinning body, then the result is PRECESSION of
the rotation axis the angular momentum vector
moves on the surface of a cone with angular
velocity ? . If this condition is not
satisfied (equivalent to our simplifying
assumption being invalid) then a full analysis
using Lagrangian mechanics to investigate energy
changes to the disturbed system is necessary,
and results in a further motion, NUTATION or
nodding, where the spin axis not only rotates on
the surface of a cone in space, but oscillates
toward and away from the axis of the cone.
This is also the case for the transient start up
and slow down situations.
5
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion Further
complications, also requiring a full Lagrangian
treatment, arise in considering real bodies,
which depart from the axisymmetric, so that we
cannot regard their rotational inertia as being
characterised by a 3 component vector
, of which I3 is a component, but need to
involve a full tensor treatment where   J I
.? or explicitly The principle moments I1,
I2, I3 have become the 3 on-diagonal terms I11,
I22, I33 of the inertia tensor (33 matrix) The
presence of the non-zero off diagonal terms Imn
in the inertia tensor I will mean that, in
general, J is not parallel to ?, so that even in
the unperturbed state (J conserved), there may be
circular motions present about the J vector
(coning). This sounds complicated, but in
everyday terms, a wheel that is not balanced will
wobble! We make it not wobble by adding balance
weights, which alter its mass properties such
that the off diagonal terms in the inertia matrix
approximate to zero.
6
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion STABILITY OF
SPINNING BODIES   We have glanced at rotation
about a principal axis, e3. This, again
revealed by a Lagrangian analysis, will only be
stable if the chosen rotation axis e3 has the
largest of the 3 principal moments (I3 gtI2 , I2
gtI1 ). It is especially unstable if I3 is the
middle principal moment.  The physical argument
is based on the conservation of angular momentum
. For a given angular momentum L, the kinetic
energy is given by . Clearly, the lowest
energy is given by the largest principle moment.  
7
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion STABILITY OF
SPINNING BODIES
If there is any dissipative system which can lose
energy of rotation (e.g. flexible lossy joints or
structure, aerodynamic drag, liquid reservoirs
etc.), then a stable rotation about the smallest
principal moment (a high energy state) becomes
unstable, as a route for it to lose energy to a
stable rotation about the largest principal
moment (a low energy state) has become available.

This problem occurred on the first US satellite,
Explorer 1, which was a slim cylinder, intended
to be spin-stabilised about its long axis.
After a relatively short time it was tumbling
uncontrollably, due to lossy structure.
8
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion Example-Applicat
ion of gyroscopic principles to spacecraft   The
effective stiffness of systems possessing
rotational inertia J is of great use in
spacecraft Attitude and Orbit Control Systems (
AOCS). If SPIN STABILISATION can be applied
without impairing mission performance, then the
effect of disturbing torques is reduced on the
pointing accuracy. The attitude control problem
is much reduced, from one of full 3-axis control,
to coping with PRECESSION and NUTATION, and
ensuring rotational stability. This is so
useful, that the design constraints on spacecraft
mass properties, to ensure I3 maximum, and I
symmetric, and the difficulties imposed in other
areas - e.g. power systems, may be tolerated for
the sake of AOCS simplicity.
9
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion For some
missions, where most of the instrumentation
sensors and payload may be boresighted on or near
the spin axis, or the spin may be utilised to
provide "free" scanning for the payload, there is
no problem in adopting this design solution.
For other missions, e.g. communications
satellites, where the payload requires steering
to specific directions, the solution is more
problematic, but spin stabilisation may be
sufficiently attractive to regard the additional
complexity of a DESPUN payload platform as
acceptable. This leads to the DUAL SPIN
configuration. Since we now have a s/c with 2
separate but coupled rotational inertias, the
control problem is non-trivial. PRECESSION is
relatively easy to cope with - the periodic
nature of NUTATION means that the addition of a
third rotating mass, with a loss mechanism
included, can be used as a NUTATION DAMPER.
10
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion For most
demanding missions from an attitude control point
of view, a 3-axis design is generally used.
11
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion Calculation
Example - Changing the spin axis vector (and
hence the attitude) of a spin stabilised
satellite. A spin stabilised satellite of mass 2
tonnes, and cylindrical radius 0.50 m, rotates
at 15 rpm. 2 x 3 n thrusters mounted on the
periphery fire in the z direction while in the
x-z plane for 0.1 sec. What is the resultant
change in spacecraft attitude?
12
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion Calculation
Example Disturbing torque rF 3 x 1 Nm 3
Nm Assume uniform mass distribution
(Why?) Principle moment of inertia about spin
axis I3 Mr2/2 2000 x 0.52/2 250
kgm2 Rotation rate ?15 rpm 2? x 15/60 ?/2
rad/s Precession rate while torque is applied
This is applied for 0.1 sec (short cf.
rotation period of 4 sec) Total precession 0.1
x 0.437? 0.0437? 60 x 0.0437 2.62, in y-z
plane. The spin axis rotates 2.62 in y-z plane
13
Flight Mechanics 2 GYROMECHANICS RTM2003
USES OF ROTATIONAL INERTIA   Direction reference
- could use a moving (translating) mass -
unfortunately, does not stay in same place - but
a ROTATING mass does.   The GYROSCOPE - a
rotating mass with minimum perturbing torques
will act as an attitude or direction reference.
E.g. Gyrocompass (constrained to rotate in
HORIZONTAL plane)Turn slip (constrained to
rotate in ROLL plane)Artificial horizon
(quasi-free)subject to drift rate - precession,
see example.
14
Flight Mechanics 2 GYROMECHANICS RTM2003
RATE GYRO The classical gyro gives an output
which is the angle between vehicle axes and some
absolute reference. Sometimes what we want (in
FCS, for example) is not angle, but rate of
change of angle - this is achieved with a rate
gyro. If we impose an angular rotation
rate on a gyro, along an axis perpendicular to
its rotation axis, and the system is constrained
in rotation about the third axis by a torsion
spring, then the torsion spring displacement will
be proportional to the imposed angular rate,
because the achieved angular rate can only be
forced by an applied torque from the torsion
spring. This is precession backwards, so to
speak! The gyro can only be forced to rotate
about the rate axis at the imposed rotation rate,
by a disturbing torque provided by the action of
the torsion spring. The spring displacement is
proportional to forcing torque and therefore also
proportional to angular (precession) velocity,
and may therefore be used as a measure of angular
rotation rate.
15
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion EFFECT OF
ROTATIONAL INERTIA ON VEHICLE BEHAVIOUR We have
looked at initially at a spinning satellite - it
is the simplest example as the only forces acting
are those providing the disturbing torque, and
Newton's laws work perfectly - no ifs, no buts.
What else can we look at? The classic example is
the bicycle - in forward motion it possesses
considerable rotational inertia which dominates
its dynamics - as we lean (roll axis) the bike
over we force precession in the yaw axis, and the
bike turns into the direction of the lean!
Motion in one axis causes motion in a cross axis
- physics is working with us for once. Sometimes
the vehicle contains unwanted gyroscopic effects
- a successful scheme for an electric vehicle
(gyrobus) is to store energy in the mechanical KE
of a spinning vertical axis flywheel, using it to
power the vehicle between termini, and
recharging it by spinning up with an electric
motor powered by the mains supply when at the
termini. What happens when the gyrobus turns a
corner? What happens when the gyrobus starts to
go uphill? What happens when the gyrobus rolls
with the road camber?
16
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion  Now translate
these ideas to a 3-d vehicle - the aeroplane.
Think about the effects on the control system -
human pilot or computer.   Schempp-Hirth
Discus What happens to a sailplane (high aspect
ratio) manoeuvring in 3-d space?
17
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion  Now translate
these ideas to a 3-d vehicle - the aeroplane.
Think about the effects on the control system -
human pilot or computer.   Sopwith
Camel What happens to a WW1 fighter, with a light
compact airframe and a rotary engine (Sopwith
Camel, Fokker Triplane) manoeuvring in 3-d space?
18
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion  Now translate
these ideas to a 3-d vehicle - the aeroplane.
Think about the effects on the control system -
human pilot or computer.   Supermarine
Spitfire What happens to a WW2 fighter with a
narrow track undercarriage and a powerful piston
engine (Spitfire, Me109) on its takeoff run?
19
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion  Now translate
these ideas to a 3-d vehicle - the aeroplane.
Think about the effects on the control system -
human pilot or computer.   Hawker
Hunter What happens to a fast jet manoeuvring in
3-d space?
20
Flight Mechanics 2 GYROMECHANICS RTM2003
GYROMECHANICS Rotational Motion  Now translate
these ideas to a 3-d vehicle - the aeroplane.
Think about the effects on the control system -
human pilot or computer.   BAe Sea
Harrier What happens to a VSTOL fighter with a
large turbofan engine (Harrier) manoeuvring in
3-d space at low or zero forward speed?
21
Flight Mechanics 2 GYROMECHANICS RTM2003

• GYROMECHANICS Rotational Motion
•  Now translate these ideas to a 3-d vehicle
• the aeroplane.
• Think about the effects on the control system
• - human pilot or computer.
•  
• AgustaWestland EH101 Merlin
• What happens to a helicopter manoeuvring in 3-d
  space?