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@TecQulpment Ltd 1999No part of this publication may be reproduced or transmitted inany form or by any means, electronic or mechanical, includingphotocopy, recording or any information storage and retrievalsystem without the express permission of TecQuipment limited.
All due care has been taken to ensure that the contents of thismanual are accwrate and up to date. However, if any errors arediscovered please inform TecQuipment so the problem may berectified.
A Packing Contents list is supplied with the equipment.Carefully check the contents of the package(s) against the list. Ifany items are missing or damaged, contact your localTecQuipment agent or TecQuipment Ltd immediately.
EducationalPRODUCTS
KEEP FOR FUTURE REFHKBNCE
Safety and Operation Information
PRODUCT: TMI04 GYROSCOPE APPARATUS
In compliance with the EC directive on Safety of Machinery, the followinginformation should be noted:
This equipment is only to be used in accordance with instructions in themanual. Students using the equipment must be adequately supervised. Localregulations regarding the use of electricity, gasoline, diesel oil, kerosene,mercury must be observed in using this apparatus.
Foreseen use of apparatusDemonstration of gyroscopic effects
Installation and assembly instructionsThe apparatus is supplied fully assembled. Before use, remove the perspexcover and take off the red transit bracket which holds the torque arm in place.Replace the cover.
The apparatus is designed to operate with the following TecQuipmentsupplied units:
2xE67 Speed Control UnitslxE64 Tachometer UnitRefer to separate literature for safety and operation information about theseunits.
Operating Ins tru cti 0 osThe perspex cover of the apparatus is fitted with an interlock so that neitherof the two motors will operate unless the cover is fitted securely. Connect thetwo E67 units and the EM to the electrical supply. Connect the outputterminals on the E67s to the input terminals on the TMI04 using 4mmconnectors. Connect the tachometer output on the TMI04 to the input on theEM using the single lead provided. Switch on the electrical supply. The twoE67s control the speed of rotor rotation and the precession speed. Thetachometer measures rotor rotation speed. A stopwatch is necessary to timethe speed of precession of the apparatus.
The equipment must be used within its operating limits (see operatingconditions).
Maintenance and inspectionPeriodically inspect elecbical leads and connections for wear. Periodicallycheck that all warning labels are in position and legible.
Handling instructionsNet weight 22kg.Ensure the correct procedures for handling the above weight are used whenmoving this apparatus.
Operating Conditions
~hoDi1cr TMI04- See serial number plate
+SOC to +4OOC~ to 9s-.!o (In\~ens~)
~
~ Safe opera~ relative humidity ran~
Noise LevelThe measured sound pressure level of this apparatus is less than 70 dB(A).
SparesRefer to Packing Contents List for any consumables to cover the warrantyperiod supplied with the apparatus. Refer to manufacturer or importer forany other spares required.
-~~
1. INTRmUCTION
Gyroscopic action occurs whenever the axis of a rotating body is made to
change its direction. The angular momentum of a rotating body causes the
axis of rotation to remain in the same direction so long as no external
couple acts on the system. However, if a turning couple is applied to
the axis, a torque reaction is produced which tends to turn the axis in
a plane at right angles to the plane in which the applied couple acts.
This torque reaction, or "gyroscopic couple" as it is called, results
from attempting to alter the direction of angular momentum of the body.
The study of gyroscopic action is particularly important in the field of
vehicle engineering. The gyroscopic couple produced by rotating com-
ponents can often lead to undesirable effects which affect the stability
of vehicles. for example, when a road vehicle travels round a bend, the
gyroscopic couple produced by turning the axes of the wheels tends to
overturn the vehicle. In the case of an aircraft changing direction,
the gyroscopic couple due to the rotating components of the engine
causes the aircraft to pitch up or down. In a similar way, the couple
produced by a turbine rotor in a pitching ship tends to make the ship
swing sideways. A knowlege of gyroscopic action can enable the designer
to calculate the torque reactions and thus allow for any undesirable
effects.
Gyroscopic effects can also be used to advantage as in the case of gyro-
stabilisers and gyroscopic instruments. If mounted in a suitable
position, a gyroscope consisting of a rotating disc can be used to re-
sist undesirable motion and so provide a means of stabilisation.
Successful use of gyroscopic stabilisation has been achieved in ships
where significant reductions in rolling amplitudes have been obtained.
In the gyro-compass, the effect of gravity is used to produce rotation
of the axis of a gyroscope in the horizontal plane such that the axis is
always aligned in a north-south direction, irrespective of its position
on the earth's surface.
TecQuipmentscopic effects
couple and the
TMl04 Gyroscope is designed to demonstrate the gyro-
and to enable the relationship between the gyroscopic
direction of rotation or "precession" of the gyroscope
axis to be determined
-3-
2. THEORY
Gyroscopic Couple
If we have a stationary flywheel, of moment of inertia I, on a shaft
mounted in a trunion frame such that it is supported but free to rotate
about an axis, then any couple applied to the system will cause the
shaft to move in the plane of application of the couple.
Now case shown in rig 2.1, where the flywheel disc is
spinn J angular velocity w, and the axis of spin simultaneouslyrotating in the horizontal plane lOX with angluar velocity w. The
pangular momentum of the disc can be represented by the vector Oa at one
instant, and Ob after a short interval of time ct. The momentum vector
lies along the axis of rotation, in a direction such that the rotation
is clockwise when viewed in the direction of the vector (right hand
screw rule). rrom rig 2.1 it is clear that there is a change in angular
momentum, as represented by the vector ab. This change in momentum must
be produced by the action of a couple in the disc. The applied couple
is equal to the rate of change of angular momentum, so the torque is
given by:
consider
ina with
the
tS(lw)~ t
or =
The change of angular momentum is represented by the vector ab, so we
can write 6 (100) = ab = Oa.60 where 66 is the angle through which the
axis of spin rotates in time 6t.
cSe"&
6e~
Os Ioo,. = =
In the limit when o~~t:'..+
deIw "dt"T !(A)-.OOP
= . . . . 2.1=
where: w is the "precession" velocity in rad/spw is the angular velocity of the disc in rad/s
I is the moment of inertia of the disc in kgm
-5-
From Fig 2.1 the vector ab lies in the XOZ plane and in the limit when
06 is very small, its direction is perpendicular to Oa, that is to say,
perpendicular to the XOV plane. The direction of the vector lies along
the axis about which the couple acts, so the applied couple must there-
fore act in the XOV plane. To conform with the right hand screw rule,
its sense must be clockwise when viewed in the direction abn, that is
when viewed in the direction OZ.
The applied couple represents the couple required to keep the axis of
the disc rotating in the XOZ plane. By rotating the axis of the disc,
the disc therefore produces a couple which acts in the opposite direct-
ion to the applied couple, that is to say, anti-clockwise about the OZ
axis. This is termed the gyroscopic couple. Thus, if no restoring
couple were applied, any attempt to rotate the axis of the disc in the
XOZ plane would result in the axis tipping in the anti-clockwise direct-
ion about the OZ axis.
2.2 Moment of Inertia
In order to investigate the validity of equation 2.1 it is necessary to
determine the moment of inertia of the gyroscope rotor. In the experi-
ment, this is done by suspending the rotor on two wires as shown in rig
2.2. If the rotor is of mass M and the wires are of length L and dis-
tance d apart, then the tension in each wire is Mg/2. If the rotor is
rotated through a small angleeabout its vertical axis, then an angular
displacement ~ is produced at the wires. If both angles are small, we
may write L~ = del2. The restoring force due to the tension in each wire
is:
Mg sin ~/2 Mg #2 (for small ~ )=
Substituting for ~ = dti/2L we obtain the restoring force as,
~4L
The restoring couple is thus:
Mgde.d- 4l
-6-
The equation of motion is therefore:
~~- 4LI~ =
Re-arranging thia equation we have:
e+~ e4IL
0=
This represents
gi ven by,fijF;T 2 4IL
= 1fMg"'dT
simple harmonic motion in which the periodic time T is
The moment of inertia I is therefore:
~g~16'"' LI . . . . 2.2
-7-
APPARATUS3.
The apparatus (see fig 3.1 consiats of a rotor disc (A) mounted on the
shaft of a small variable speed motor (B) which is carried in a gimbal
frame (C). This assembly can be rotated about the vertical axis by a
variable speed geared motor housed inside the base of the app-
Attached to one end of the rotor motor is a torque arm which
a mass (0) at its end to balance the motor and rotor disc. A
! balance weight (H) is also fitted to the torque srm to balsnce
the assembly in its static unloaded condition. The motor pivots in the
gimbal frame such that it csn rotate in the vertical plsne. A retsining
plate (E) is fitted over the torque arm to limit the angular movement of
the motor assembly. Additional masses (f) can be attached to the end of
the torque arm to bslance the gyroscopic couple produced when the rotor
disc is spinning and the gyroscope is being rotated (precessed) about
the vertical axis. A removable, but electrically interlocked trans-
parant safety cover is fitted over the complete rotating assembly. Re-
moving this cover automatically stops both motors.
second
aratus.
carries
movable
The rotor motor assembly is fitted with an opt{cal pick up (G) which
picks up from the four strips of reflective tape on the rotor disc (A).
The distance between thia disc and the optical pick up is pre-set at the
factory and must not be adjuated. Power is supplied to the rotor motor
via a slip ring unit (J) mounted at the base of the gimbal frame, which
also provides a signal path from the optical sensor to the E64
Electronic Tachometer. -
The apparatua is designed to operate from two E67 units, which allow
independant control over the rotor motor and the geared precession motor
mQunted in the base.,' When in operstion; the rotor speed, vsrisble
between 0 and 3000 rpm, is measured on the E64 Electronic Tachometer,
whilst the precession rate, vsriable between 0 and 40 rpm, is measured
using a stop-watch.
An additional gyroscope rotor and armature assembly is provided together
with a aimple fold-out bifilar suspension arm mounted on the base of the
apparatus. The moment of inertia of the rotor assembly can be deter-
mined by suspending it from the arm and timing the torsional
oscillations
~ o. 5 M~ ~ 2.0~M
8. View of Main Gyroscope Components
b. View of Gyroscope with Trsnsit Brscket in Position -rig 3.1 Gyroscope Assembly
-9-
4. EXPERIMENTAL PROCEDURE AND RESULTS
Installation
Remove the safety cover by pushing the cover sideways until the rim is
free of one of the retaining blocks and lift the cover away. Remove
the red transit bracket (Fig 3.lb, item K), which holds the torque arm
rigidly to the frame. The apparatus requires two 12 V d.c. variable
voltage supplies which are provided by the TQ [67 Speed Control Units.
These operate from a mains input and two 4mm terminals are provided for
connection to the respective inputs on the TM104 unit. .
Make the following connections:
1. The mains input supply to the [64 Tachometer and the [67 Speed
Control ~its.
The output terminals of the [67 units to the input terminals on the
TMlO4 unit.
The tachometer output on the TMl04 to the [64 Tachometer using the
signal lead provided.
2.
J.
is now wired up and ready for operation.
Switch on all the units.
Re-fit theThe apparatus
safety cover.
Investigstion of Gyroscopic Couple Direction
Ensure that the power supplies to the unit are switched OFF. Remove the
safety cover and check that the rotor assembly is balanced, so that with
no weights added to the mass at the end of the torque arm, the torque
arm lies between the marks on the retaining plate. If necessary, slack-
en the knurled retaining screw on the torque arm balance weight and
position the balance weight to obtain the balanced condition. When the
rotor assembly is satisfactorily balanced, re-tighten the knurled re-
taining screw and re-fit the safety cover.
Check that the cover is correctly in position, then set the rotor and
precession motors running. Note the direction of rotation of the rotor,
the direction of precession of the gyroscope and whether the torque arm
-11-
rises or falls. By interchanging the motor input connections on the
front panel, determine the direction of the gyroscope couple for each
combination of rotor and precession directions.
The results should be ss shown in rig 4.1. It will be seen that the
gyroscopic couple produced by precessing the gyroscope slways acts about
an axis which is perpendicular to both the gyroscope rotor and precess-
ion axes. The couple direction depends on the directions of precession
and rotor rotation. Notice that the couple always acts to tip the gyro-
scope in the same direction as the leading edge of the rotor. ror
exsmple, in disgram (s) the leading edge is moving downwsrds and the
gyroscopic couple scts to tip the rotor downwards.
petermination of Moment on I~tia
Lift the spare armature and rotor aasembly from its clips and fold out
the bifilar support arm. Hang the assembly from the arm as shown in Fig
4.2, then twist it about the vertical axis by about loa and release it.
Use a stop-watch to time, say, 50 oscillations of the rotor assembly.
Measure the length L of the wires and the distance d between them. The
mass of the rotor assembly is nominally 1.09 kg. You can check this
value by weighing the assembly on suitable scales if you wish to do so.
These measurements provide the information required to calculate the
moment of inertia I from equation 2.2. Check that you have recorded the
correct readings, then fold the arm away and replace the rotor assembly
in its clips.
Typical resulta are a8 follows
0.53 m0.073 m1.09 kg
47.5 8
0.95 8
l
d
M
t
T
=Length of wires
Distance between wires
~ss of rotor
Time for 50 oscillations
Periodic time =
Substituting valuea in equation 2.2 givea the moment of inertiathese
8S:1.09 x 9.81 x 0.0731 X 0.952I 16 x-"- xO.53
6.14 x 10-" kg m2Therefore I =
-l~-
Magnitude of Gyroscopic Couple
The object of this psrt of the experiment is to investigate the
relationship between the gyroscopic couple, the angulsr velocity of the
rotor and the precession velocity. The results provide the dsta
necessary to check the validity of equation 2.1, that is
Gyroscopic couple, T Iw w=p
Ideally the tests would involve measuring the gyroscopic couple for two
sets of conditions:- (i) varying rotor veocity at a constant precession
velocity, and (ii) varying precession at a constant rotor velocity.
However, the gyroscopic couple can only be set at a limited number of
discrete values (as determined by the balance mass on the torque ar~),
so a slightly different procedure has to be used. This involves measur-
ing the precession velocity for a range of rotor speeds and values of
balance mass. The procedure is as follows:
1. Screw a g mass onto the end of the torque arm and replace the
safety dome. Connect the rotor and precession motor supplies so
that the gyroscopic couple will raise the torque arm. Use the
results obtained under section 4.2 to determine the correct
electrical connections to achieve this.
50
2. Vary the precession velocity until the torque arm rises
to a level at which the scribed line on the arm lines
up with the indicating strips on the bracket (shown by
'E' in Figure 3.1 page 8). This is the point of balance
at which the gyroscopic couple is equal to the moment
produced by the mass on the torque arm.
.J. At this condition, measure the precession speed by
timing a suitable number of revolutions of the assembly
using a stop-watch. The number of revolutions you will
need to time depends on the test condition. To obtain
good accuracy, always use a timed period of at least 30seconds. Record the exact value of rotor speed.
-15-
4. Decreaae the rotor speed in steps of 500 rev/min and determine the
precession speed at the balance point for each different rotor
speed down to 500 rev/min.
5. Add additional masses to the torque arm and obtain similar sets of
results for each value of mass. It is best to obtain results for
50g increments in mass, so giving a total of 7 sets of results up
to a value of 350g. However, if time is limited it is sufficient
to use values of 50,150,250 and 350g.
A typical set of results is given in Table 4.1 for four different values
of balance mass. The gyroscopic couple is calculated from the mass and
the length of the torque arm (O.l4m). for example, for the firat
result, the couple is:
0.05 x 9.81 x 0.14
0.0687 ~
T =
The equation for gyroscopic couple suggests that graphs of the recip-
rocal of precession velocity (l/w ) against rotor velocity (w) should bep
straight lines. The results are plotted in this way in Fig 4.3, from
which it can be seen that the results do indeed lie on straight lines
which pass through the origin.
Two further graphs can be constructed from the reaults shown in Fig 4.3,
which demonstrates the independent influence of preceasion velocity and
rotor velocity. Fig 4.4 shows values of gyroscopic couple plotted
against precession velocity for a constant rotor velocity of 300 rad/s-
As expected, the results lie on a straight line through the origin.
Theoretical values of the couple can be calculated uaing the moment of
inertial found in the previous section. For example, at a precession
velocity of 2.5 rad/a:-
6.14 x 10~"
0.4608 ~
T x 300 x 2.5
The experimental
to within about 3~.
values are shown to agree with the theoretical values
Recommended