Gyroscope Ppt 1.pdf

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CHAPTER 5
GYROSCOPE
DYNAMICS OF MACHINERY

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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.
The couple is applied on the bearings which support
the rotating shaft. The reaction of this couple will be
equal and opposite on each bearing. The couple
makes a change in the direction of angular velocity,
but it does not change the magnitude of angular
velocity i.e. it remains constant.

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Application of gyroscopes
Aeroplane while taking a turn, steering, pitching,
Rolling of ships,
An automobile rounding a curve etc…
DYNAMICS OF MACHINERY

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Derive expression for gyroscopic couple
Gyroscopic couple,

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DYNAMICS OF MACHINERY
Motion of aeroplanes

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1. The propeller shaft of an aeroplane has a speed of 2400
RPM. The direction of rotation is clockwise when looking
from the tail end of aeorplane. The rotary engine of the
aircraft has a mass of 410 kg. Determine gyroscopic couple
acting on the aeroplane when it travels @ a speed of 240
KMPH and takes a turn to the left along a circular path of
70m radius.
Explain the effect of gyroscopic couple on the aircraft
Take radius of gyration of rotating parts = 310mm.
DYNAMICS OF MACHINERY

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2. Rotary engine of an aircraft weighs 3800 N & its
radius of gyration is 300 mm when flying @ speed
250 KMPH. The aircraft takes a turn towards the right
along a circular path of 55m radius. Calculate the
gyroscopic couple acting on the aircraft & its effects.
Assume engine rotates clockwise when viewed from
rear end with speed of 1500 RPM

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Motion of Ships

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DYNAMICS OF MACHINERY
Expression for angular acceleration of ship:
Angular acceleration,
Gyroscopic effect due to Rolling:
During rolling since the axis of rolling and axis of the
turbine are same, there will not be any precession of
spin axis and hence there will not be any gyroscopic
couple and effect on the ship.

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Problem 1. A boat is operated by a steam turbine which
rotates at 3100 RPM in the clockwise direction when looking
from the bow end. What will be the magnitude and effect of
gyroscopic couple acting on the boat when the boat travels
along a circular path making one complete revolution in 15
seconds? The moment of inertia of rotating parts of the
turbine is 515 Kg-m2.
DYNAMICS OF MACHINERY

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Problem 2. The steam urbine of a ship has a rotor of mass 1000 kg
and runs at a speed of 3200 RPM. The rotor has a radius of
gyration = 0.6 m The rotor rotates in the clockwise direction
when looking from the rear end of the ship.
Calculate the gyroscopic couple and explain its effect on the ship in
the following cases.
a) The ship p itches 60 above and 60 below the horizontal
position. The bow is ascending with its maximum velocity.
The motion due to pitching is S. H. M and periodic time is 20 s.
b) The ship rolls and at a certain instant it has an angular velocity
of 0.015 rad/s in clockwise direction when looking from stern.
DYNAMICS OF MACHINERY

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Stability of an automobile

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Gyroscopic couple,
Centrifugal couple acting on the
automobile =
𝒘
𝒈
𝒗𝟐
𝒓

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Problem 1. A rear engine automobile is traveling along a track of
mean 5Om radius. Each of the 4 wheels has a moment of inertia
of 18 Kg-m2 and an effective diameter of 580 mm. The rotating
parts of the engine have moment of inertia of 9 Kg-m2. The engine
axis is parallel to the rear axle and the crankshaft rotates in the
same directions or sense as that of road wheels. Gear ratio of
engine to back axle is 4:1. The weight of the automobile = 16000
N. Center of gravity (c.g) is 0.6 m above the road level. Width of
the track is 1.42 m. Determine the limiting speed of the vehicle
around the circle for all the wheels to maintain contact with road
surface if the surface of the r ad is horizontal.

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Stability of a two wheeler taking a turn

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Equating the overturning couple with the restoring couple we get,
W = Weight of the vehicle and its rider.
h = Height of C. G. of the vehicle and the rider,
rw = Wheel radius,
R - Track radius,
Iw = Moment of inertia of each wheel,
ωw= Angular velocity of wheels,
ωe= Angular velocity of engine rotating parts,
G = Gear ratio = ωe/ ωw.
V = linear velocity of the vehicle = rWωW
θ = angle of heel or inclination of vehicle to the vertical.

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Problem 1. A motor cycle with the rider has weight of 1800 N. The c. g. of
the motor cycle and its rider combined is 0.60m above the ground level
when the motor cycle is standing upright. Each wheel has a moment of
inertia of 12 kg-m2 and a rolling diameter of 0.575m. The engine rotates at
6 times the speed of wheel and in the same sense. Moment of inertia of
rotating parts of the engine is 1.7 Kg-m2. Determine the angle of heel of the
motor cycle if the rider is traveling at a speed of 52 Km/h in a circle of 35 m
radius.

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Stabilization of ships