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CO
No.
Course Outcomes (COs):
CO1 Calculate the gyroscopic couple acting on applications like ships,
aero planes, two and four wheeler
CO2 Determine the dimensions of flywheel based on the turning
moment diagram and calculate the characteristics of governors.
CO3 Determine balancing force for rotating and reciprocating masses
using static and dynamic balancing method
CO4 Calculate natural frequency for a single degree of freedom free
vibratory systems
CO5 Determine response to forced vibration due to harmonic
excitation, base excitation and excitation due to unbalance
forces.
CO6 Determine vibration and noise measurement parameters for
industrial / real life applications

Gyroscopic couple and its effect on spinning bodies,
Gyroscopic effect on naval ships during steering,
pitching and rolling, Ship stabilization with gyroscopic
effect. Two wheeler and four wheeler on curved path-
effect of gyroscopic and centrifugal couples, maximum
permissible speeds on curve paths, Gyroscopic effect
due to lateral misalignment of rigid disc mounted on
shaft, gyroscopic effects in airplanes, Introduction to
MEMS based gyroscopes
Dynamics of Machines (ME 403)
Gyroscope

Gyroscope, Gyroscopic Effect, Bike, ship, Planes

Gyroscopic motion is the tendency of a rotating
Disc to maintain the orientation of its rotation.
Rotating object possesses angular momentum
Angular Momentum of Disc:
Disc resists any change in its axis of rotation ( precession. So
change in orientation results in a change in angular momentum
ob oa
ab
Iw


 
 
 
Rate of change of change in angular momentum
Gyroscopic Couple Derivation
During Precession change in orientation
oa Iw
 
Change in angular momentum
p
C Iw
dt
C Iww





Terminology
Gyroscopic Effect
Rotor Spins(W) about Spin axis Anticlockwise seen from LH
Rotor Precesses about vertical axis towards LH side
Due to this precession (Wp) of the spin axis, the gyroscopic
couple is applied on rotor
RH Rule
Effect: Rotor tilts outside
Input: Spin(W), Rotor Precesses (Wp)
Output: Gyroscopic Couple (C)
C: Gyroscopic Couple

Rotor with different orientation
e.g.: Now W is in same direction but
Wp goes down (Precession to Right) So
Couple (C) will change its direction
IMPORTANT Note: Out of three vectors
(W, Wp & C) any Two will change the
directions at a time and third will be same.
Effect: Rotor tilts Inside (To Left)
W, WP, C are Mutually
Perpendicular to each
other
Case 2: Rotor Precesses (Wp) DOWN or Right Turn

Gyroscopic Effect when Couple Input
Output: Rotor Precesses (Wp)
Perecesses to Left Side
Input:
Spin(W)
Gyroscopic Couple (C)
Application/Example:
1) Pitching movement in Ship/ Aero plane (Couple input)
2) Gyroscopic motion can cause catastrophic failure in wind turbines
due to Couple input (Wind Air direction change)
For Couple input Standard Diagram Direction of C is opposite

Case A
Input:
Disc Spinning, w
Disc Precession, wp
Outut:
Gyro Couple C
Case B
Input:
Disc Spinning, w
Gyro Couple, C
Outut:
Disc Precession, wp
IMPORTANT
Precession Input Couple Input

Gyroscopic Effect in Ship due to Engine Rotor
Bow or Fore: It is the fore or front end of the ship.
Stern or aft: It is the rear end of the ship.
Port:
It is the left hand side of the ship when looking
from the rear end.
Star board: It is the right hand side of the ship when looking from rear end.
Port: an opening in the side of a ship
for loading and unloading cargo -

Ship Terminology
Rolling is a rotation of ship around a longitudinal axis,
Tilting motion of the ship from Starboard to port
Types of Movements
Gyroscopic Effect on Naval Ship
Pitching is a rotation of ship around the transverse axis
Pitch describes the up and down motion of a vessel. This is characterized by the
rising and falling of bow and stern.
Steering(yawing) is a rotation of ship around the vertical axis.

Effect of Steering to Left(Yaw)
Input:
• Spinning of Engine Rotor, clockwise seen from
rear (Forward w),
• Steer to Left, seen from Top,(Upward Wp)
Output, C: Effect: Bow UP, Stern
DOWN
Input:
• Spinning of Engine Rotor, clockwise seen from
rear (Forward W),
• Couple Input as Port and Starboard up and down
Output:
As C & W are in Same Line (Not in Perpendicular
to each other)
Effect: So Third will not Produce
NIL
Effect of Rolling

Effect of Pitching (Stern up) CR: Input
Input:
• Spinning of Engine Rotor, clockwise
seen from rear (Forward W),
• Couple Input as Stern up
Output:
Wp Upward
Effect: Turn to Left Towards Port
Couple Input

A ship turbine rotor has a mass of 200Kg and rotates at N=3000 RPM,
clockwise seen from stern. Its radius of gyration 0.5 Determine the
gyroscopic couple and effect if the ship is steering to Right in a curve of
100 m radius and speed of 16.1 knots.( 1 Knot=1855 m.hr)
Input:
• Spinning of Turbine Rotor, clockwise seen from
rear (Forward W),
• Turn to Right: Wp Down
Effect: Stern UP
p
C I w w
  
2
314.158
60
N
w RPS

 
2 2 2
2000(0.5) 500
I mk kg m
   
16.1 1855
5.295 /
3600
0.08295
p
v m s
v
w RPS
R

 
 
13031.28
p
C I w w Nm
   
Ans
Ans

When a Single-Engine aero plane turns when the turn is to the Left,
the gyroscopic effect tends to raise the nose
Gyroscopic Effect on Aeroplane Effect of Turns/ Steering to Left
Input:
• Spinning of Engine Rotor, clockwise seen from rear
(Forward W),
• Steer to Left, seen from Top,(Upward Wp)
Output:
C: Vector Towards Starboard
Effect: Nose tends to RAISE

An aeroplane makes a complete half circle of 50 metres radius,
towards left, when flying at 200 km per hr. The rotary engine and the
propeller of the plane has a mass of 400 kg and a radius of gyration of
0.3 m. The engine rotates at 2400 r.p.m. clockwise when viewed from
the rear. Find the gyroscopic couple on the aircraft and state its effect
on it.
Input: Spinning of Engine Rotor, clockwise seen from rear
(Forward W), Steer to Left, seen from Top,(Upward Wp)
Output:C: Vector Towards RIGHT
Effect: Nose tends to UPWARD
Tail:Down

If you lean left, you turn left
A rider leaning left will produce
a Couple(C) which will cause the bicycle
wheel to Precess (Steer) counterclockwise
as seen from above, turning the bicycle left
Gyroscopic Effect : Bike
Angle of Heel: This is a requisite angle i.e
vehicle tilting inside to counter effect of
Gyro couple, which may tend to tilt vehicle
outside

= Angle of heel=?
R = radius of the curve=30m
r = radius of the wheel=0.31m
G = engine to wheel speed ratio=5.5
V= velocity of the two wheeler=55KM/h=15.278m/sec
M = mass=250Kg
h = height of CG above ground in vertical position=0.6m
Iw= 1 Kg-m2
Ie= 1 Kg-m2
A motorcycle and its rider together have a mass of 250Kg and their
center of gravity is 0.6m above the ground level. Each road wheel is of
0.62m diameter and has a moment of inertia 1 Kg-m2. The rotating parts
of the engine have a moment of inertia 0.18Kg-m2. The engine moves at
5.5 times. The speed of the road wheels and in the same sense.
Calculate the angle of heel require when the motor cycle is rounding a
curve of 30m. Radius at a speed of 55Km/hr.

0.50 /
p
V
W Rad Sec
R
 
Precession Velocity of Bike

2
1750.62
mV
CF N
R
 
Centrifugal Force(CF)
50.92 /
w
V
W Rad Sec
r
 
0.50 /
p
V
W Rad Sec
R
 
5.5(50.92)
280.093 /
e w
W G W
Rad Sec
   
Spin Velocity of Wheel
Spin Velocity of Engine
Precession Velocity of Bike

( )
CA Cw Ce
Same direction
 
cos( )
CF
C CF h 
 
sin( )
weight
C Mg h 
 
2( cos( ) )
cos( )
w w
e e
CA I W Wp
I W Wp


   
 
Couple due to Self weight (Mg)
Couple due to CF
Couple due to Gyroscopic effect(Wheel & Engine)
Balance Equation CF Weight
CA C C
 
0
40.402
 
On solving we get Angle of Heel

2
2
250
1
0.3
5
0.6
90 /
25 / sec
0.3
50
w
e
e w
M Kg
I Kgm
I Kgm
w w
h m
V Km hr
V m
r m
R m



 





Stability of two wheeler

83.33 / sec
5 416.66 / sec
w
e w
V
w rad
r
w w rad
 
 
0.5 / sec
p
V
w rad
R
 
2
3125
MV
CF N
R
 
Spinning velocities of Wheels and Engine Precession Velocity
G CF Wt
Couple Balance
C C C
 
Centrifugal Force
Upright Vehicle Angle of heel of Vehicle Couple Balance
1
2
3 4
5
6
7

2( cos( ) )
( cos( ) )
w w w p
e e e p
C I w w
C I w w




2(1 83.33 cos( )0.5) 83.33 cos( )
0.3 416.66 cos( ) 0.5 62.499 cos( )
145.829 cos( )
w
e
G w e
C N m
C N m
C C C N m
 
 

    
     
   
p
C Iww

G CF Wt
Couple Balance
C C C
 
cos( )
CF
C CF h 
 
3125 0.6 cos( )
1875cos( )
CF
C 

  
sin( )
Wt
C Mg h 
 
250 9.81 0.6 sin( )
1471.5sin( )
Wt
C 

   
Couple due to CF Couple due to Weight
Couple due to Gyroscopic effect
8 9
10
11
12

G CF Wt
Couple Balance
C C C
 
145.829 cos( )
G
C 
 1875cos( )
CF
C 
 1471.5sin( )
Wt
C 

2020.82 cos( ) 1471.5sin( )
 

2020.82
tan( )
1471.5
 
0
53.94
 
Angle of Heel to maintain stability
13
15
16
17

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Gyroscope, Gyroscopic Effect, Bike, ship, Planes

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Gyroscope, Gyroscopic Effect, Bike, ship, Planes