Unit 1: Gyroscopic Effect

Gyroscopic effect is a three dimensional effect which consist of (1) Axis of

spin [X-axis] (2) Axis of gyroscopic couple [Z-axis] and (3) Axis of precession [Y
axis], which are mutually perpendicular to each other.
Considera disc spinning with an angular velocity 'w' r/s about X-axis in
anti-clockwise direction when seen from front as shown in Fig.1. Plane YOZ is
called plane of spinning as the disc is parallel to plane and axis OX is axis of
spinning. Plane ZOX is plane of precession and axis OY is axis of precession. Plane
XOY is plane of gyroscopiccouple and axis 0Z is axis of gyroscopic couple. If the
disc is allowed to turn in clockwise or anti-clockwise direction when seen
from top
in ZOX i.e. about OY axis, then reactive gyroscopic
couples setup in either
clockwise or anti-clockwise direction due to which the system goes up or down.

Sin do =

ot yie)

Q. Derive the equation for gyroscopic couple C (Refe 6g. 2)

Change in angular momentum = xr
= I(w+dw)sin(d0)
= lwd + ldwde

= lwd

Rate of Change of angular momentum =Gyroscopic couple C= lwd0/dt

:.C= lwuwp
Where 'C is gyroscopic couple in Nm, ' is mass moment of inertia in kg-m²,'w' is
angular velocity of disc (spin) in rad/s, 'wp'is angular velocity of precession in
rad/s.
 Ingyroscope, always there are two inputs which are as below:
When two inputs are Spinning and Precession,then we get output due to
Reactive
gyroscopic couple.
In gyroscope two principles are used namely
[1] Conservation of angular momentum,and
[2] Newton's Law.
The direction are simply regulated by right hand screw rule and there can
be following four situations in order toobtain the effect of reactive
couple.
gyroscopic

Qu. Discuss the effect of reactive gyroscopic couple on aircraft (S-17, W-16]
The effect of reactive gyroscopic couple on an aeroplane are as below:
When the engine or propeller rotates in anticlockwise direction when seen
from front or nose end and the aeroplane takes a left turn then the effect of
reactive gyroscopic couple is to raise the nose and dip the tail or rear end of the
aeroplane. IRels fu.)
12) When the engine or propeller rotates in anticlockwise direction when seen
from front or nose end and the aeroplane takes right turn then the
effect
of
reactive gyroscopic couple is to dip the nose and raise the tail or rear end of the
aeroplane. [Aees Aig 2)
3] When the engine or propeller rotates in clockwise direction when seen
from the front or nose end and the aeroplane takes a left turn then the
effect of
reactive gyroscopic couple is to dip nose and raise the tail of the aeroplane.[Refer Fig3/
[4) When the engine or propeller rotates in clockwise direction when seen
from the front or nose end and the aeroplane takes right turn then the
effect of
reactive gyroscopic couple i_ to raise nose and dip the tail of the aroplane. [Res Fig.e
Left Ríght
c 4 acy
acil obse ve
Psecession A
Dike chen
of vey
Qc
ac
Reactive qyhosepc
couple (4) (3)
 1]A f\ywheel having amass of 20 kg and aradius of gyration of 300 mm is given a
spin of 500 rpm about its axis which is horizontal. The flywheel is suspended at a
point that is 250 mm from the plane of rotation of flywheel. Find the rate of
precession of the flywheel ? (S-16]
(2]An aeroplane makes a complete half circle of radius 50 mtowards left when
fiying at 200 km/hr. The rotary engine and the propeller of the plane has a massof
400 kg with radius of gyration 300 mm. The engine runs at 2400 rpm clockwise
when viewed from rear end. Find the gyroscopiccouple on aircraft and state the
effect of reactive gyroscopic couple on it. What willbe the effect if the aeroplane
turn to its right instead of left ? (S-18]
The terminology used in a naval ship or vessel are as below:

[1] The front end of ship is called as bow or fore.

[2] The rear end of ship is called as stern or aft.
[3] The left hand side of ship when viewed from stern is called as port.
[4] The right hand side of ship when viewed from stern is called as star board.
{A} The effect of reactive gyroscopic couple on a naval ship during steering is as
below:
[1] When the rotor of a ship rotates in anticlockwise direction when viewed
from front end or bow and the ship is steering to the left, then the effect of
reactive gyroscopic couple is to raise the bow and lower the stern
(2] When the rotor ofaship rotates in anticlockwise direction when viewed
from front end or bow and ship steers to right then the effect of reactive
gyroscopic couple is to lower bow and raise the stern.

(3] When the rotor of a ship rotates in clockwise direction when viewed from
front endor bow and the ship is steering to the left, then the effect of reactive
gyroscopiccouple is to lower bow and raise the stern.

[4) When the rotor ofa ship rotates in clockwise direction when viewed from
front end or bow and the ship steers to right then the effect of
reactive gyroscopic
couple is to raise bowand lower the stern.
 naval ship during pitching
gyroscopic couple on a
{B} Effect of reactive
down in vertical plane (XOY)
and
movement of a ship up of ship is
assumed to
Pitching is the precession. The pitching
i.e. axis of
about transverse axis Harmonic Motion (SHM)
Simple
take place with couple is
effect of the reactive gyroScopic
When the pitching isupward, the
[1) towards star-board.
to move the ship
couple
effect of the reactive gyroscopic
When the pitching is downward, the
[2]
towards port-side.
isto move the ship
on a naval ship during rolling:
{C) Effect ofgyroscopiccouple
of spin, hence there is
the axis of precession is parallel to axis
During rolling
reactive gyroscopic couple during rolling.
no effect of

ship is 10 tonnes and has a radius of gyration

rotor of a
(3] The mass of a turbine when looking from stern. Determine the
rpm clockwise
0.8 m. It rotates at 1600
- [S-19,W-16,S-17,W-18]
magnitude and gyroscopic effect in
of 80 m radius.
ship is travelling at 120 km/hr steers to left in a curve
() If the

and the bow is descending with the maximum velocity.

(i) If the ship is pitching being 20 sec. and total angular movement
The pitch is SHM, the periodic time
deg.
between the extreme positions is 10

instant has an angular velocity of 0.04

(ii) If the ship is rolling and at a certain
r/s clockwise when looking from stern.

(iv) Also find maximum angular acceleration during pitching.

 automobile is travelling along a tract of 100 m mean radius. Each
|4] Arear engine 2.5 kg-m^2 and aneffective diameter
moment of inertia of
of the four wheels has moment of inertia 1.2 kg-m^2. The
rotating parts of engine have
of 0.6 m. The crankshaft rotates in the same sense
as
to rear axle and the
engine axis is parallel The
wheels. The ratio of engine speed to back axle speed is3:1.
the road above the
a mass of 1600 kg and has its center of gravity 0.5 m
automobile has Determine the limiting speed of
track of vehicle is 1.5 m.
road level.The width of wheels to maintain contact with road
for all the four
the vehicle around the curve
surface. [W-17]
motor-cycle is of 600 mm in diameter and has a moment of
5]Each wheelof a is 180 kg
kg-m^2. The total mass of the motor cycle and the rider
inertiaof 1.2 above the ground level when the
center of mass is 580 mm
and the combined rotating parts of engine is 0.2 kg
moment of inertia of
motor cycle is upright. The the wheels and is in the same
speed is 5 times the speed of
m^2. The engine turn
Determine the angle of heel necessary when the motor cycle takes a
sense.
of 35 m radius at a speed of 54 kmph.
(S-17]
Qu. Explain gyroscopic analysis of gyro-crusher.
effect deliberately introduced in
In gyro-crusher or grinding mill, gyroscopic
gyro-crusher as shown in figure 1. It
order to get more crushing force. Consider a
which is hinged to central
consist of roller inapan and is free to rotate on shaft
around the pan and
driving shaft. When the driving shaft rotates the roller move
to weight of
crush material placed within it. The crushing takes place not only due
roller but also because of extra crushing force developed by gyroscopic action

R/mg =1 + (wwp)/(mgr)
Thus, R/mg > 1
where is radius of roller
 Assignment No.1
|1]A rour-wheeled trolley car of mass 2500 ko runs on rails which are 1.5 ape
and travels around a curve of 30 m left. The rails are at
radius at 24 kmph towards
Same level. Each wheel of trolley is 0 75 m in diameter and each of the two dxie
isdriven by a motor running in same direction to that of wheels at aspeed or nve
imes the speed of rotationof wheel The moment of inertia of each axle and twO
wheels is 18 kg-m^2.Each motor has a moment of inertia of 12 kg-m^2. ne
Center oT gravity of car is 0.9 m above rail level, Determine the vertical forces
exerted byeach wheel on the rails.
travels on rails of 1.6 m gauge
(2|A Tour wheeled trolley car of total mass 2000 kg
8 de8. Tne
rounds acurve of 30 m radius at 54kmph. The track is banked at
each pair with axle has a mass Or
Wneels having an external diameter of 0.7 m and gravity
The radius of gyration for each pair is 0.3 m. The height of center of
200 Kg.
Determine allowing for centrifugal force and
OT car above the wheel base is 1 m. [W-16]
8yroscopic actions,the pressure on each rail.
uniform disc of 20 cm diameter and mass 20 g is mounted on the shaft that
B] A left
bearings with a span of 2.5 m. The disc is mounted at 1 m from
rotates in two are bolted
horizontal plane. The bearings
hand bearing and shaft is maintained in at
horizontal platform which itself rotates in horizontal plane. The disC rotates
to a
viewed from left hand bearing and the
2000 rpm in clockwise direction when
viewed from top (precess).
platform rotates in counter clockwise direction when
Atwhat limiting speed the platform carrying
the shaft can be rotated so that the
Consider both weight of disc
maximum bearing reaction do not eXceed 1500 N.
and gyroscopiceffect. Neglect other effect. [W-17]
 Gyroscopic Effect ofa four wheel drive moving in acurved path:
Considera four wheelcar taking left turn on acurved path as shown in fig.I.
Wheels Aand C are inner wheels, whereas B and D are outer wheels if thecar is
taking left turn.
Let,
m =mass of car in kg
rw=radius of wheel in m
R =Radius of curvature in m
h = distance of center of gravity of car from road surface in m
X = width of track in m
Iw = mass moment of inertia of one wheel of car in kg-m
le =mass moment of inertia of rotating parts of engine in
kg-m
OW = angular velocity of wheel in r/s
0e = angular velocity of rotating parts of engine in r/s
G =gear ratio(s peed kato)
G = oe/ow
V =linear velocity of car in m/s (or km/hr)
V=(0w)(rw)
op = angular velocity of precession of car in rs
op = v/R
Fc = Centrifugal force in N
Fc = mv²/R
delt

Dis.
View
TV
cw
64 h

FV
R
 ]Weight W=mg of car is distributed equally over four wheels. Hence road
reaction on each wheel = W/4 --- (1)
[2] Effect of gyroscopic couple C:
C=4(lw)(ow)(op) + (leXG)(ow\op) when wheel and engine rotate in same dir.
C= 4(lw)(ow)(op) -(le)\G)(ow)\(op) when wheel &engine rotate in opposite dir.
Px =C.:.P=C'x .. PI2 =C/2x -- --- (2)
P ía N

Due to gyroscopic couple C, verticalreaction on road surface willbe produced.

The reaction (P/2)will be vertically upward on outer wheel and the reactions are
vertically downward on inner wheels. when cat s taking eft tus 9.
[3] Effect due to Centrifugal couple Co:
The effect of centrifugal force Fc is act outwardly at center of gravity and its effect
is also to overturn the car. Couple tending to overturn the car is
Co = (Fc)(h) =mv'h/R
This overturning couple is balanced by vertical reactions (Q/2) which are vertically
upward on outer wheels and vertically downwards on inner wheels Cooy let ty
Qx=Co :Q=Co/x :.Q/2 = Co/2x .Q2 =mv'h/2xR -(3)
14] Total vertical reaction at each outer wheel Ro in N
Ro = W/4 + PI/2 + O/2

[5]Total vertical reaction at each inner wheel Ri io N

Ri= W/4 PI2Q/2
[6] In order toavoid overturning and to have always contact of wheels of car with
road surface, vertical reaction at each inner wheel (Ri) should not be negative
.. P/2+ Q/2 = W/4 Ri-Ve
From above equation limiting speed of car on curved path can be calculated.
Gyroscopic effect on a twowheeler moving on a curved path :
Consider a two wheeler taking a left turn on a curved path as shown in iig.2
Let,
m =mass oftwo wheeler alongwith rider in kg
h =height of center of gravity of vehicle in m
rw =radius of wheel of two wheeler in m
R = Radius of curved path in m
Iw = mass moment of inertia of each wheel
le =mass moment of inertia of engine
G =Gear ratio(sped hoeko)
G = oe/ow
V =linear velocity of two wheeler in m/s (or km/hr)
0 =Angle of heel with respect to vertical for equilibrium
Cb Balancing couple in N-m
C=Gyroscopic couple in N-m
Co- Centrifugal couple in N-m

cw

h
hcast

hsinb

Fiy. 2
 1] Effectot gyroscopic couple :
c=[2w)\ow)(op) +t (Ie)(G)(ow)(op)lcoso when wheel &engine rotate in same
direction.

c-[2\w)(ow (op)-(le)(GX(ow)\(op)]cos0 when wheel &engine rotate in

opposite direction.
Gyroscopic couple acts in clockwise direction when two wheeler is taking left turn
[2] Effect of Centrifugal couple:
Co-fmv'hRos9
[3] Balancing couple =Gyroscopic couple +Centrifugal couple
inclined by an angie
When a twowheeler is taking a left turn the vehicle has to be
0 ie. angle of heel w.r.t. vertical for equilibrium
Cb =C+ Co
Cb =mghsine
(le)(G)\(ow)\op) +mv'h/R]cos®
:.mghsin = [2(Iw)(ow)\(op) +
when wheel & engine rotate in same direction.
=[2(lw)\ow)(op) + (le)(GXow)(op) + mv'h/R]/ mgh
.:.tan
avoid over turning of two
heel can be calculated to
From above equation angle of
wheeler.