Chapter 13

Gyroscope and its Applications

13.1 Introduction
Whenever a rotating body changes its axis of rotation, a couple is applied on the rotating body
(shaft). This couple is known as gyroscopic couple. This couple acts on the bearings which support
the rotating shaft.
For example, let us consider a disc rotating in anticlock
wise direction at an angular velocity of w rad/s when seen from
the front as shown in Fig.13.1.The spinning of disc takes place
about the axis OX. Let the axis OX is turned in the plane
Assume the
XOZ' through an angle 0 to the position OX.
w rad/s. The
angular velocity of the disc at this position also
Disc

OX itself rotating
disc is rotating about the OX and the axis simultaneously. Direction of

about the axis OY. Two motions take place viewing

known as gyroscopic
Now a couple is acting on the disc and is Fig. 13.1 Gyroscopic couple
couple.
Applications: instrument or toy known as gyroscope.
1. The gyroscopic principle is used in an
principle is used for stabilising seaborne ships by minimising effects of waves.
2. This
It is used in directional control. eg., in the gyro compass used on airplane, and in inertial
3. space travel.
gujdance controlsystems for missiles and
bearing reactions in crank shafts of automobiles as the automobile negotiates a
4. It increases
Curve.

Motion
13.2 Precessional Angular
angular acceleration is the rate of change of angular velocity. It is a vector quantity
We know that
and can be
represented both in magnitude and direction with the help of right hand screw rule.
12. 13.2 Dynamics of Machinery
M Abody rotating in a plane about afixed axis may be subjected to angular acceleration but
while rotating if its axis also undergoes an angular displacement (known as precession), the body
will be subjected to both the angular acceleration and another acceleration is known as gyro.
scopic acceleration i.e. the total angular acceleration of the rotating body is the sum of angular
acceleration and gyroscopic acceleration.
Illustration
Consider a disc, as shown in Fig.13.2(a), rotating (or spinning) about the axis of spin 0X in
anticlockwise direction when seen from the front, with an angular velocity win a plane at right
angles to the paper.
spin
of
axis (0Q+
New
Disc
)S
Direction of Viewing

Axis of spin
(a) (b)

Fig. 13.2 Precessional Angular Motion

Let the axis of spin move to a new position OX' in short interval of time St such that the
angular displacement of the axis of spin is Qe and the angular velocity of the disc is increased to
(wtów). Using the right hand screw rule, the initial angular velocity of the disc (w) is represented
by vector oz and the final angular velocity (w + ou) is represented by or' as shown in Fig.I13.2(6).
The vector zr represents the change of angular velocity in time ôt. ie. the angular acceleration
of the disc.
Resolving vector Tr into two mutually perpendicular components, we get component rT n
the direction of oz and ra perpendicular to it.
Component of angular accelerationin the direction of ox
oc' cos 80 - oC
St St St
(wt ow) cos 80 -w wCOs O0 + ow cOs O8 -w
St 8t
Since 89 is very small, therefore substituting cos S8 = 1, we 'have
w+ ow-w Jw
St St
In the limit, when St ’0,
Ow dw
8t’0 St dt
 Gyroscope and its Applications 13.3

Component of angular acceleration in the directlon perpendicular lo ox

ra' or' sin S0 (w+ 6w) sin 50 W
sin50+ õw sin 60
dt dt dt dt

Since S in very small, therefore substituting sin 60 = 60, we have,

woo + bw 60 woO (Neglecting bw.60being very small)
St
In the limit when &t ’0,

ç = Lt
St0 St
wSO
= W
X
St
= WWp (-)
Total angular acceleration of the disc
Therefore, total angular acceleration of the disc,
of a; and ac
a= vector æ= vector sum
(13.1)
dw
+ wwp
dt
gyroscopic acceleration
Qe = WWp is known as
where
angular velocity of precession or
wp = d0/dt is known as
precessional angular velocity
of spin of the disc
= Angular acceleration
dw
dt

precession.
O Note: turms, is known as axis of
The axis about which the axis of spin
about the axis of precession is known as precession al
1, spin
angular motion of the axis of
The
2. angular constant in magni-
motion.
changes its axis of rotation but remains
angular velocity ofthe disc (w) the disefrom equation (13.1) is given by,
3. Ifthe acceleration of
tude, then totalangular (::=0)
a = wwp = aC dt
total angular acceler.
acceleration a, is known as
gyroscopic acceleration. ie. the
This angular body is equal to its
gyroscopicacceleration ony.
ation of the rotating

13.3 Gyroscopic Couple

rad/s about the axis of spin OX, in anti-
velocity of w
disc spinning with an angular shown in fig. 13.3(a). The axes OX, OY and 02
Consider a
direction when seen from the
front, as
clockwise
mutually
perpendicularto one another.
are
13.4 Dynamics of Machinery

Since the plane in which the disc is rotating is parallel to the plane YOZ,therefore it is
rotates in
known as plane of spinning. The plane XOZ' is a horizontal plane and the axis ofofspin
axis spin is said to
aplane parallel to the horizontalplane about an axis OY. In otherwords, theplane XOZ' is known
precessing about axisOY at an angular velocity w, rad/s. This horizontal
as plane of precessionand OY is the axis ofprecession.
and
Let I=Mass moment of inertia of the disc about0 X(kg.m)
w = Angular velocity of the disc (rad/s)
Angular momentum of the disc,
= Iw
quantity. In the
Procedure for vector diagram: We know that angular momentum is a vectory
always. Looking from the
vector diagram 13.3(b), O as the fixed point which represents front end
diagram, always look at
front end, the disc is rotating in anticlockwise direction. (To draw vector
in the diagram.) Therefore
from the front irrespective of view point and mention that point as o
and it is parallel to the axis
vector oa will be from left to right side (as per right hand screw rule)
through a smallangle õ0
OX. Let the axis OX is turned in the plane XOZ' to the position OX'
anticlockwise when seen from
radians in time St seconds (The axis of spin OX is also rotating
be constant. Now the angular
the top about the axis OY). Assuming the angular velocity w to Fig.13.3(b).
in
momentum is represented by vector ob parallel to the axis OX'
Derivation: Let oç = Initial angular momentum.
ob = Angular momentum after time St seconds.
ab = Change in angular momentum in time St seconds.
Change in angular momentum,
= ob -0d = aó= od.6e (in the direction of ab)
= lw: 60

Rate of change of angular momentum,

80
= lw:
St
application of a couple to
Since the rate of change of angular momentum is resulted by the
couple or gyroscopic torque or
the disc which is known as active gyroscopic couple or the active
precession,
active torque. Therefore the couple applied to the disc causing
80 =Wp
C= Lt Iw x = lw X = lwwp (13.2) dt
O0 St dt
the speed
where Wp = angular velocity of precession of the axis of the spin or
of rotation of the axis of the spin about the axis of precession OY.
 13.5
Gyroscope and its Applications

rPlane of actlve
gyroscoplo
Plane of couple
spinning
Axis of active
gyroscoplc Direction of change in
angular momenturn
couple
"Plane of Reactive gyorscopic
Axis of |precession COuple
Precession
X 8ense
Direction
of
| Axis of spin viewing active gyroscopic Sense of active
COuple
gyroscopic couple
Disc

Axis of reactive
gyroscapic couple
-Plane of reactive
gyorscopilc couple

(b) (c)
(a)
Fig. 13.3 Gyroscopic couple
units, the unit of couple C is N.m where I is in kg.m². It is noticed that
In SI
direction of ab (representing the change in angular momentum), the couple lwwp
1. In the of spin is made to rotate with angular
velocity
applied on the disc when the axis
has to be as active gyroscopic couple. The
of precession. This couple is known
about the axis For very smalldisplacement 80, abwill
X0Z' or the horizontal plane.
ab lies in the plane The plane which is perpendicular toactive
plane XOY.
be perpendicular to the vertical active gyroscopic couple (XOY). The axis 0Z'
gyroscopic couple is called the plane of
about which the couple acts, is called the axis of activegyroscopic
perpendicular to XOY,
couple. couple
OX moves with an angular velocity wp, the disc is subjected to
2. When the axis of spin opposite in direction to that of active couple.
Iwwy) but
whose magnitude is same (i.e. This is represented by ba'shown in
reactive gyroscopic couple.
This couple is known as
Fig.13.3(b).
Tofind the direction of active couple: precession X0Z (i.e. horizontally)
direction ab along the plane of
() Keep the thumb in the
as shown in Fig.13.3(c). the clockwise sense which represents sense
of
curling of fingers is in
(i) Now the direction of reactive gyroscopic couple is anticlockwise.
Thus the sense of
active gyroscopic couple.
 Gyroscope and its Applications 13.11

Example 13.4: Adisc supported between two bearings on ashaft offnegligible weight has a
mass of 80 kg and aradius of gyration of 300 mm. The distances of the disc from the bearings
are 300 mm to the right from the left hand bearing and 450 mm to the left from the right hand
bearing. The bearings are supported by thin vertical cords. When the disc rotates at 100 radls
in the clockwise direction looking fromthe left-hand bearing, thecord supporting the left hand
side bearing breaks. Find the angular velocity of precession at theinstant the cord is cut and
discuss the motion of the disc.
Given data: m = 80 kg; k=300 mm = 0.3 m; w= 100 rad/s.
To find: Angular velocity of precession (wp)
Solution:
We know that mass moment of inertia of the disc, P

I= mk² = 80x (0.3) = 7.2 kg.m h=300 4=450

When the cord at A is cut, couple due to mass
cord
disc (or applied couple) shaft

CA =W x lg = mg x lB
A W
= 80 x 9.81 x 0.45 = 353.16N.m Disc
Bearing Bearing
View from A
We know that gyroscopic couple,
C= Iwwp =7.2 x 100 x wp = 720 wp N.m Fig. 13.7
For equilibrium, applied couple = gyroscopiccouple
353.16=720 wp Or wp =0.4905 rad/s Ans.

Discussion on motion of the disc

When the cord atAis cut, a couple due to weight of the disc (known as applied couple) acts in
anticlockwise at bearing B. Therefore, gyroscopic couple should act in clockwise as the system is
hunting for equilibrium. In this case,
In this case,
-?
i) View point - Left end (i.e. rear) (ii) Spinning - clockwise (iii) Precession
In order to have clockwise gyroscopic couple, the precession of the disc must be clockwise
Ans. (refer to case (2) of table 13.1)

13.4 Effect of Gyroscopic Couple onan Aeroplane

The front view and top view of an aeroplane are shown in fig.13.8(a) and (b). Let us consider the
propeller or engine rotates in the clockwise direction when seen from the rear end of the aeroplane
and the aeroplane takes left turn.
 Dyna1nics of Machinery.
13.12
2

Terminology
aeroplane - Nose
Front end of Nose
aeroplane - Tail
Rear end of (rad/s), Tail
Angular velocity oftheengine (a) Front view
w=
Let
of the engine and propeller (kg),
m = Mass
gyration ofengine and JLeft turn
k= Radius of Direction
propeller(m), of
engine viewing Nose
of inertiaof the
I= Mass moment
mk (kgm)
and the propeller =
Rear
or Propeller
Linear veiocity of the aeroplane (m/s), tailend
V= Wings
curvature(m), and
R= Radius of
precession = v/R (rad/s)
(b) Top view
velocity of
lwy = Angular aeroplane
the gyroscopiccouple
act Fig. 13.8 Views of
discussed already,
As we
ing on the aeroplane,
C= lwwp

Couple on Aeroplane
Effect of Gyroscopic
()Taking Left Turn turns left.
shows the vector diagram when the aeroplane
Fig.13.9(a)
before turning
oç = Angular momentum
turning
ob = Angular momentum after
momentum or active gyroscopic couple or applied couple
ab = Rate of change of angular
gyroscopic couple or simply gyroscopic couple or reaction couple
ba' = Reactive
of ö0, ab is perpendicular to oá i.e. ab is perpendicular to plane of active gy
For small value diagram for this case are shown in Fig.13.9(b)
diagram and plane
roscopic couple XOY. The axis
and (c) respectively.
hand thumb rule with the thumb representing ab (i.e. change in angular
Applying the right
the plane of precession XOZ, the curling of fingers will be in clockwise di
momentum) along
clockwise sense. Therefore the reactive gyroscople
rection. Hence active gyroscopic couple acts in
reactive gyroscopic couple acts 1n u
couple acts in anticlockwise sense. The force due to this
plane of reactive gyroscopic couple) and tends to raise the nose and dip tne
vertical plane (i.e.
tail of theaeroplane as explained in Fig.13.9 (d).
 Gyroscope and its Applications 13.13

Active Plane of
9yroscoplc couple spinning Plane of active
b
gyroscopilc couple
Reactive gyroscoplc
cOuple
Plane of
precession
(a) Vector diagram

Y
(c) Plane diagram
Axis of precession

ZAxis of active
9yroscopic couple Nose
Ta
View point Axis of spin
X
Reactive gyroscopic couple
Axis of reactive
gyroscopic couple
(b) Axis diagram (d) Effect gyroscoplc couple

Fig. 13.9 Left turnof an Aeroplane

(i) Taking Right Turn

(BAxis of precession
Tail
Active
gyroscopic Axis of active
Nose
çouple gyroscopic couple
a a'
Reactive X
gyroscopic F Reactive gyroscopic
b b' couple Axis of reactive
gyroscopic couple couple

(a) Vector diagram (b)Axis diagram (c) Effect gyroscopic couple

Fig. 13.10 Right turn of an Aeroplane

Keepingthe other parameters unaltered, when the aeroplane takes right turn, the reactive gy-
roscopic couple actsin clockwise sense as shown in Fig.13.10(a). The force due this reactive
gyroscopic couple acts in the vertical plane and tends to dip the nose and raise the tail as ex-
13.10(c).
plainedin Fig.
 Gyroscope and its Applications 13.15
13.4.1 Gyroscopic Effect Chart for the
Aeroplanes
For predicting the effect of gyroscopic couple on aeroplanes, one has to
three parameters:
consider the following
1. View point - Nose end or tail end,
2. Direction of rotation of propeller - clockwise or
anticlockwise.
3. Turn - Left or right. (viewed from the above).
Four different cases and their gyroscopic effects are shown in table 13.2.

Example 13.5: The rotor of turbojet engine has a mass of 200 kg and a radius of gyration
250mm. The speed of the engine is 10000 r.p.m in the clockwise direction when viewed from
the front of the aeroplane. The aeroplane while flying at 1000 kmhr turns with a radius of 2
km to the right. Compute the gyroscopic moment exerted by the rotor on the plane structure.
Also determine whether the nose of the plane tends to rise or fallwhen the plane turns.
Given data: m = 200 kg: k= 250 mm =0.25 m; N =10000 r.p.m. ; R=2km =2000 m;
U= 1000 km/hr = (1000 x 1000)/3600 = 277.78 m/s
To find: Gyroscopicmoment (C)and its effect
Solution:
We know that angular velocity of propeller,
w= (2rN)/60= 2T x 10000/60 = 1047.2rad/s
Angular velocity of precession,
Wp = v/R= 277.78/2000 = 0.1389rad/s
Mass moment of inertia of the rotor,
I= mk = 200 x 0.252 = 12.5kg.m?
structure.
Gyroscopic moment exerted by the rotor on the plane
C= Iwwy = 12.5 x 1047.2 x 0.1389 = 1818.2 N.m.

Gyroscopic Effect
front
1. View point-
a

rotation -clockwise
2.Direction of
Fig. 13.11
Turn - right
3.The vector diagram for this case are shown in Fig.13.11. The reactive gyroscopic couple acts
anticlockwise sense andit tends to raise the nose and depress the tail.
inthe
1: 13.16 Dynamics of Machinery.
500 kg and the radius of gyration450
BExample 13.6: The engine and the propeller of mass clockwise direction when viewed from
mm.The propeller of theengine rotates at 3000 rp.m. in
quarter of a cycle of radius 90 m while
the rear. ð the aeroplane turns towards left and makes
fAying at 240 km/hr.
and state its effect.
1. Determine the gyroscopic couple on the aircraft
2. In what way is the effect changed when
() the aeroplane turns towards right. end) and the aeroplane
(ü) che engine rotates clockwise when viewedfrom the front(nose
turns (a) left(b) right.
N= 3000 r.p.m. or w= 2T x 3000/60
Given data: m =500 kg. : k= 450 mm=0.45 m;
= 66.67m/s
314.16 rad/s; R= 90 m; v= 240 km/hr =240 x 1000/3600
To find: Gyroscopiccouple and its effec.
Solution:
Gyroscopiceffect when the aeroplane turns left
We know that mass moment of inertia of the engine and propeller,
I= mk? 500x (0.45) = 101.25kg.m
Angular velocity of the precession,
wy = v/R= 66.67/90 = 0.7407rad/s
We know that gyration couple acting on the aircraft,
C= lwwp = 101.25 x 314.16 x 0.7407
=23563.18 N.m or 23.56 kN.m Ans
Reactive gy.
couple
b
A &
0. a'a
a

b6 bb
(a) case (0) (b) case 2 (0) (c) case 2 (1) (a) (d)case 2() (b)
Fig. 13.12
Case 1: 1. View point-rear; 2. Direction of rotation-clockwise; 3. Turn-left
The vector diagram for this case is shown in Fig.13. 12(a). The reactive gyroscopic couple acts
in anticlockwise sense. This tends to raise the nose and depress the tail.
Case 2 (i) : Whenthe aeroplane turns right
1. View point -rear; 2. Direction of rotation -clockwise; 3. Turn -right.
The vector diagram for this case is shown in Fig. 13.12(b).The reactive gyroscopic couple a
inclockwise sense.This tends to depress the nose and raise the tail.
 13.17
Gyroscope and its Applications

Case 2(ii) a: 1. Viewpoint -front; 2. Direction of rotation -clockwise; 3. Turn -left

The vector diagram for this case is shown in Fig. 13.12(c). The reactive gyroscopic couple acts
inclockwise sense. This tends to depress the nose and raise the tail.
Case 2 (ii) b: 1. View point - front; 2. Direction of rotation - clockwise; 3. Turn - right
The vector diagram for this case is shown in Fig. 13.12(d). The reactive gyroscopic couple acts
in anticlockwise sense. This tends to raise the nose and depress the tail.

13.5 Gyroscopic Effect on Naval Ships (or Sea Vessels)

Refore going to discuss the effect of gyroscopic couple on naval ships, let us first study about the
terms used in naval ship.
Y
.Port Transverse
13.5.1 Terminology axis

naval ship are shown in

The top and front view of a Direction
Left

Fig. 13.13(a) and (b) respectively. of viewing

terms used: Longitudinal axis
The following arethe Stern Bow X
end of the ship. (Rear end) (Fore-end)
1. Bow - Itis the fore or front Rotor
aft - It is the rear or back end of the
2. Stern or
Star board
ship.
isthe left hand side of the ship looking (a) Top view
3. Port - It end.
from the rear
A Starboard - It is the rignthand side of the shin
end.
looking from the rear Bearings
(b) Front view Propeller
Types ofMovements
Aship may have the following three movements: Fig. 13.13
Seoering : ltS Lug O a ship in a curve either to the