Dynamics of Machinery U5MEA19

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Dynamics of MachineryU5MEA19

• Prepared by
• Mr.Shaik Shabbeer Mr.Vennishmuthu.V
• Assistant Professor, Mechanical Department
• VelTech Dr.RR Dr.SR Technical University

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• UNIT I Force Analysis
• Rigid Body dynamics in general plane motion
  Equations of motion - Dynamic force analysis -
  Inertia force and Inertia torque DAlemberts
  principle - The principle of superposition -
  Dynamic Analysis in Reciprocating Engines Gas
  Forces - Equivalent masses - Bearing loads -
  Crank shaft Torque - Turning moment diagrams -
  Fly wheels Engine shaking Forces - Cam dynamics
  - Unbalance, Spring, Surge and Windup.
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•  

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• Static force analysis.
• If components of a machine accelerate, inertia
  is produced due to their masses. However, the
  magnitudes of these forces are small compares to
  the externally applied loads. Hence inertia
  effect due to masses are neglected. Such an
  analysis is known as static force analysis
• What is inertia?
• The property of matter offering resistance to any
  change of its state of rest or of uniform motion
  in a straight line is known as inertia.
•  

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• conditions for a body to be in static and
  dynamic equilibrium? 
• Necessary and sufficient conditions for static
  and dynamic equilibrium are
• Vector sum of all forces acting on a body is zero
• The vector sum of the moments of all forces
  acting about any arbitrary point or axis is zero.

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• Static force analysis and dynamic force analysis.
• If components of a machine accelerate, inertia
  forces are produced due to their masses. If the
  magnitude of these forces are small compared to
  the externally applied loads, they can be
  neglected while analysing the mechanism. Such an
  analysis is known as static force analysis.
• If the inertia effect due to the mass of the
  component is also considered, it is called
  dynamic force analysis.

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• DAlemberts principle. 
• DAlemberts principle states that the inertia
  forces and torques, and the external forces and
  torques acting on a body together result in
  statical equilibrium. 
• In other words, the vector sum of all external
  forces and inertia forces acting upon a system of
  rigid bodies is zero. The vector sum of all
  external moments and inertia torques acting upon
  a system of rigid bodies is also separately zero.

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• The principle of super position states that for
  linear systems the individual responses to
  several disturbances or driving functions can be
  superposed on each other to obtain the total
  response of the system.
• The velocity and acceleration of various parts of
  reciprocating mechanism can be determined , both
  analytically and graphically.

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• Dynamic Analysis in Reciprocating Engines-Gas
  Forces
• Piston efforts (Fp) Net force applied on the
  piston , along the line of stroke In horizontal
  reciprocating engines.It is also known as
  effective driving force (or) net load on the
  gudgeon pin.
• crank-pin effort.
• The component of FQ perpendicular to the crank is
  known as crank-pin effort.
• crank effort or turning movement on the crank
  shaft?
• It is the product of the crank-pin effort (FT)and
  crank pin radius(r).
•  

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• Forces acting on the connecting rod 
• Inertia force of the reciprocating parts (F1)
  acting along the line of stroke.
• The side thrust between the cross head and the
  guide bars acting at right angles to line of
  stroke.
• Weight of the connecting rod.
• Inertia force of the connecting rod (FC)
• The radial force (FR) parallel to crank and
• The tangential force (FT) acting perpendicular to
  crank

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• Determination of Equivalent Dynamical System of
  Two Masses by Graphical Method
• Consider a body of mass m, acting at G as
• shown in fig 15.15. This mass m, may be replaced
• by two masses m1 and m2 so that the system
  becomes dynamical equivalent. The position of
  mass m1 may be fixed arbitrarily at A. Now draw
  perpendicular CG at G, equal in length of the
  radius of gyration of the body, kG .Then join AC
  and draw CB perpendicular to AC intersecting AG
  produced in
• B. The point B now fixes the position of the
  second
• mass m2. The triangles ACG and BCG are similar.
  Therefore,

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• Turning movement diagram or crank effort diagram?
• It is the graphical representation of the turning
  movement or crank effort for various position of
  the crank.
• In turning moment diagram, the turning movement
  is taken as the ordinate (Y-axis) and crank angle
  as abscissa (X axis).

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UNIT II Balancing Static and dynamic
balancing - Balancing of rotating masses
Balancing reciprocating masses- Balancing a
single cylinder Engine - Balancing Multi-cylinder
Engines, Balancing V-engines, - Partial
balancing in locomotive Engines-Balancing
machines.
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STATIC AND DYNAMIC BALANCING
When man invented the wheel, he very quickly
learnt that if it wasnt completely round and if
it didnt rotate evenly about its central axis,
then he had a problem! What the problem he
had? The wheel would vibrate causing damage to
itself and its support mechanism and in severe
cases, is unusable. A method had to be found to
minimize the problem. The mass had to be evenly
distributed about the rotating centerline so that
the resultant vibration was at a minimum.
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UNBALANCE
The condition which exists in a rotor when
vibratory force or motion is imparted to its
bearings as a result of centrifugal forces is
called unbalance or the uneven distribution of
mass about a rotors rotating centreline.
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BALANCING
Balancing is the technique of correcting or
eliminating unwanted inertia forces or moments in
rotating or reciprocating masses and is achieved
by changing the location of the mass centres. The
objectives of balancing an engine are to
ensure 1. That the centre of gravity of the
system remains stationery during a complete
revolution of the crank shaft and 2. That the
couples involved in acceleration of the different
moving parts balance each other.
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Types of balancing
a) Static Balancing i) Static balancing is a
balance of forces due to action of gravity. ii) A
body is said to be in static balance when its
centre of gravity is in the axis of rotation. b)
Dynamic balancing i) Dynamic balance is a
balance due to the action of inertia forces. ii)
A body is said to be in dynamic balance when the
resultant moments or couples, which involved in
the acceleration of different moving parts is
equal to zero. iii) The conditions of dynamic
balance are met, the conditions of static balance
are also met.
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BALANCING OF ROTATING MASSES
When a mass moves along a circular path, it
experiences a centripetal acceleration and a
force is required to produce it. An equal and
opposite force called centrifugal force acts
radially outwards and is a disturbing force on
the axis of rotation. The magnitude of this
remains constant but the direction changes with
the rotation of the mass.
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In a revolving rotor, the centrifugal force
remains balanced as long as the centre of the
mass of rotor lies on the axis of rotation of the
shaft. When this does not happen, there is an
eccentricity and an unbalance force is produced.
This type of unbalance is common in steam turbine
rotors, engine crankshafts, rotors of
compressors, centrifugal pumps etc.
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The unbalance forces exerted on machine members
are time varying, impart vibratory motion and
noise, there are human discomfort, performance of
the machine deteriorate and detrimental effect on
the structural integrity of the machine
foundation.
Balancing involves redistributing the mass which
may be carried out by addition or removal of mass
from various machine members. Balancing of
rotating masses can be of 1. Balancing of a
single rotating mass by a single mass rotating in
the same plane. 2. Balancing of a single rotating
mass by two masses rotating in different
planes. 3. Balancing of several masses rotating
in the same plane 4. Balancing of several masses
rotating in different planes
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BALANCING OF A SINGLE ROTATING MASS BY A
SINGLE MASS ROTATING IN THE SAME PLANE
Consider a disturbing mass m1 which is attached
to a shaft rotating at rad/s.
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r radius of rotation of the mass m The
centrifugal force exerted by mass m1 on the shaft
is given by, F m r c 1 1 This force acts
radially outwards and produces bending moment on
the shaft. In order to counteract the effect of
this force Fc1 , a balancing mass m2 may be
attached in the same plane of rotation of the
disturbing mass m1 such that the centrifugal
forces due to the two masses are equal and
opposite.
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BALANCING OF A SINGLE ROTATING MASS BY TWO MASSES
ROTATING
There are two possibilities while attaching two
balancing masses 1. The plane of the disturbing
mass may be in between the planes of the two
balancing masses. 2. The plane of the disturbing
mass may be on the left or right side of two
planes containing the balancing masses. In order
to balance a single rotating mass by two masses
rotating in different planes which are parallel
to the plane of rotation of the disturbing mass
i) the net dynamic force acting on the shaft must
be equal to zero, i.e. the centre of the masses
of the system must lie on the axis of rotation
and this is the condition for static balancing
ii) the net couple due to the dynamic forces
acting on the shaft must be equal to zero, i.e.
the algebraic sum of the moments about any point
in the plane must be zero. The conditions i) and
ii) together give dynamic balancing.
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Balancing Multi-cylinder Engines, Balancing
V-engines
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Problem 1. Four masses A, B, C and D are attached
to a shaft and revolve in the same plane. The
masses are 12 kg, 10 kg, 18 kg and 15 kg
respectively and their radii of rotations are 40
mm, 50 mm, 60 mm and 30 mm. The angular position
of the masses B, C and D are 60 , 135 and 270
from mass A. Find the magnitude and position of
the balancing mass at a radius of 100 mm.
Problem 2 The four masses A, B, C and D are 100
kg, 150 kg, 120 kg and 130 kg attached to a shaft
and revolve in the same plane. The corresponding
radii of rotations are 22.5 cm, 17.5 cm, 25 cm
and 30 cm and the angles measured from A are 45,
120 and 255. Find the position and magnitude of
the balancing mass, if the radius of rotation is
60 cm.
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UNIT III Free Vibration Basic features of
vibratory systems - idealized models - Basic
elements and lumping of parameters - Degrees of
freedom - Single degree of freedom - Free
vibration - Equations of motion - natural
frequency - Types of Damping - Damped vibration
critical speeds of simple shaft - Torsional
systems Natural frequency of two and three rotor
systems
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Introduction

• Mechanical vibration is the motion of a particle
  or body which oscillates about a position of
  equilibrium. Most vibrations in machines and
  structures are undesirable due to increased
  stresses and energy losses.

• Time interval required for a system to complete a
  full cycle of the motion is the period of the
  vibration.

• Number of cycles per unit time defines the
  frequency of the vibrations.

• Maximum displacement of the system from the
  equilibrium position is the amplitude of the
  vibration.

• When the motion is maintained by the restoring
  forces only, the vibration is described as free
  vibration. When a periodic force is applied to
  the system, the motion is described as forced
  vibration.

• When the frictional dissipation of energy is
  neglected, the motion is said to be undamped.
  Actually, all vibrations are damped to some
  degree.

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Free Vibrations of Particles. Simple Harmonic
Motion

• x is a periodic function and wn is the natural
  circular frequency of the motion.

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Free Vibrations of Particles. Simple Harmonic
Motion
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Free Vibrations of Particles. Simple Harmonic
Motion
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Simple Pendulum (Approximate Solution)

• Results obtained for the spring-mass system can
  be applied whenever the resultant force on a
  particle is proportional to the displacement and
  directed towards the equilibrium position.

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Simple Pendulum (Exact Solution)
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Sample Problem

• For each spring arrangement, determine the spring
  constant for a single equivalent spring.

• Apply the approximate relations for the harmonic
  motion of a spring-mass system.

A 50-kg block moves between vertical guides as
shown. The block is pulled 40mm down from its
equilibrium position and released. For each
spring arrangement, determine a) the period of
the vibration, b) the maximum velocity of the
block, and c) the maximum acceleration of the
block.
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Sample Problem

• Springs in parallel
• determine the spring constant for equivalent
  spring

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Sample Problem

• Springs in series
• determine the spring constant for equivalent
  spring

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Free Vibrations of Rigid Bodies

• Analysis objective is to determine wn.

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Sample Problem

• From the kinematics of the system, relate the
  linear displacement and acceleration to the
  rotation of the cylinder.

• Based on a free-body-diagram equation for the
  equivalence of the external and effective forces,
  write the equation of motion.

A cylinder of weight W is suspended as
shown. Determine the period and natural frequency
of vibrations of the cylinder.

• Substitute the kinematic relations to arrive at
  an equation involving only the angular
  displacement and acceleration.

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Sample Problem
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Sample Problem

• Using the free-body-diagram equation for the
  equivalence of the external and effective
  moments, write the equation of motion for the
  disk/gear and wire.

• With the natural frequency and moment of inertia
  for the disk known, calculate the torsional
  spring constant.

The disk and gear undergo torsional vibration
with the periods shown. Assume that the moment
exerted by the wire is proportional to the twist
angle. Determine a) the wire torsional spring
constant, b) the centroidal moment of inertia of
the gear, and c) the maximum angular velocity of
the gear if rotated through 90o and released.

• With natural frequency and spring constant known,
  calculate the moment of inertia for the gear.

• Apply the relations for simple harmonic motion to
  calculate the maximum gear velocity.

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Sample Problem
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Sample Problem
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Principle of Conservation of Energy
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Sample Problem

• Apply the principle of conservation of energy
  between the positions of maximum and minimum
  potential energy.

• Solve the energy equation for the natural
  frequency of the oscillations.

Determine the period of small oscillations of a
cylinder which rolls without slipping inside a
curved surface.
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Sample Problem
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Sample Problem
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Forced Vibrations
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Forced Vibrations
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Sample Problem

• The resonant frequency is equal to the natural
  frequency of the system.

• Evaluate the magnitude of the periodic force due
  to the motor unbalance. Determine the vibration
  amplitude from the frequency ratio at 1200 rpm.

A motor weighing 350 lb is supported by four
springs, each having a constant 750 lb/in. The
unbalance of the motor is equivalent to a weight
of 1 oz located 6 in. from the axis of rotation.
Determine a) speed in rpm at which resonance
will occur, and b) amplitude of the vibration at
1200 rpm.
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Sample Problem

• The resonant frequency is equal to the natural
  frequency of the system.

W 350 lb k 4(350 lb/in)
Resonance speed 549 rpm
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Sample Problem

• Evaluate the magnitude of the periodic force due
  to the motor unbalance. Determine the vibration
  amplitude from the frequency ratio at 1200 rpm.

W 350 lb k 4(350 lb/in)
xm 0.001352 in. (out of phase)
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Damped Free Vibrations

• All vibrations are damped to some degree by
  forces due to dry friction, fluid friction, or
  internal friction.

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Damped Free Vibrations
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Damped Forced Vibrations
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Electrical Analogues

• Oscillations of the electrical system are
  analogous to damped forced vibrations of a
  mechanical system.

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Electrical Analogues

• The analogy between electrical and mechanical
  systems also applies to transient as well as
  steady-state oscillations.

• With a charge q q0 on the capacitor, closing
  the switch is analogous to releasing the mass of
  the mechanical system with no initial velocity at
  x x0.

• If the circuit includes a battery with constant
  voltage E, closing the switch is analogous to
  suddenly applying a force of constant magnitude P
  to the mass of the mechanical system.

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Electrical Analogues

• The electrical system analogy provides a means of
  experimentally determining the characteristics of
  a given mechanical system.

• The governing equations are equivalent. The
  characteristics of the vibrations of the
  mechanical system may be inferred from the
  oscillations of the electrical system.

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UNIT IV Forced Vibration Response to
periodic forcing - Harmonic Forcing - Forcing
caused by unbalance - Support motion Force
transmissibility and amplitude transmissibility -
Vibration isolation.
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Damping

• a process whereby energy is taken from the
  vibrating system and is being absorbed by the
  surroundings.
• Examples of damping forces
• internal forces of a spring,
• viscous force in a fluid,
• electromagnetic damping in galvanometers,
• shock absorber in a car.

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Damped Vibration (1)

• The oscillating system is opposed by dissipative
  forces.
• The system does positive work on the
  surroundings.
• Examples
• a mass oscillates under water
• oscillation of a metal plate in the magnetic field

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Damped Vibration (2)

• Total energy of the oscillator decreases with
  time
• The rate of loss of energy depends on the
  instantaneous velocity
• Resistive force ? instantaneous velocity
• i.e. F -bv where b damping coefficient
• Frequency of damped vibration lt Frequency of
  undamped vibration

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Types of Damped Oscillations (1)

• Slight damping (underdamping)
• Characteristics
• - oscillations with reducing amplitudes
• - amplitude decays exponentially with time
• - period is slightly longer
• - Figure
• -

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Types of Damped Oscillations (2)

• Critical damping
• No real oscillation
• Time taken for the displacement to become
  effective zero is a minimum

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Types of Damped Oscillations (3)

• Heavy damping (Overdamping)
• Resistive forces exceed those of critical damping
• The system returns very slowly to the equilibrium
  position

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Example moving coil galvanometer

• the deflection of the pointer is critically damped

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Example moving coil galvanometer

• Damping is due to induced currents flowing in the
  metal frame
• The opposing couple setting up causes the coil to
  come to rest quickly

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Forced Oscillation

• The system is made to oscillate by periodic
  impulses from an external driving agent
• Experimental setup

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Characteristics of Forced Oscillation

• Same frequency as the driver system
• Constant amplitude
• Transient oscillations at the beginning which
  eventually settle down to vibrate with a constant
  amplitude (steady state)

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Characteristics of Forced Oscillation

• In steady state, the system vibrates at the
  frequency of the driving force

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Energy

• Amplitude of vibration is fixed for a specific
  driving frequency
• Driving force does work on the system at the same
  rate as the system loses energy by doing work
  against dissipative forces
• Power of the driver is controlled by damping

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Amplitude

• Amplitude of vibration depends on
• the relative values of the natural frequency of
  free oscillation
• the frequency of the driving force
• the extent to which the system is damped

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Effects of Damping

• Driving frequency for maximum amplitude becomes
  slightly less than the natural frequency
• Reduces the response of the forced system

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Phase (1)

• The forced vibration takes on the frequency of
  the driving force with its phase lagging behind
• If F F0 cos ?t, then
• x A cos (?t - ?)
• where ? is the phase lag of x behind F

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Phase (2)

• Figure
• 1. As f ? 0, ? ? 0
• 2. As f ? ?, ? ? ?
• 3. As f ? f0, ? ? ?/2
• Explanation
• When x 0, it has no tendency to move. ?maximum
  force should be applied to the oscillator

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Phase (3)

• When oscillator moves away from the centre, the
  driving force should be reduced gradually so that
  the oscillator can decelerate under its own
  restoring force
• At the maximum displacement, the driving force
  becomes zero so that the oscillator is not pushed
  any further
• Thereafter, F reverses in direction so that the
  oscillator is pushed back to the centre

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Phase (4)

• On reaching the centre, F is a maximum in the
  opposite direction
• Hence, if F is applied 1/4 cycle earlier than x,
  energy is supplied to the oscillator at the
  correct moment. The oscillator then responds
  with maximum amplitude.

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Forced Vibration

• Adjust the position of the load on the driving
  pendulum so that it oscillates exactly at a
  frequency of 1 Hz
• Couple the oscillator to the driving pendulum by
  the given elastic cord
• Set the driving pendulum going and note the
  response of the blade

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Forced Vibration

• In steady state, measure the amplitude of forced
  vibration
• Measure the time taken for the blade to perform
  10 free oscillations
• Adjust the position of the tuning mass to change
  the natural frequency of free vibration and
  repeat the experiment

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Forced Vibration

• Plot a graph of the amplitude of vibration at
  different natural frequencies of the oscillator
• Change the magnitude of damping by rotating the
  card through different angles
• Plot a series of resonance curves

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Resonance (1)

• Resonance occurs when an oscillator is acted upon
  by a second driving oscillator whose frequency
  equals the natural frequency of the system
• The amplitude of reaches a maximum
• The energy of the system becomes a maximum
• The phase of the displacement of the driver leads
  that of the oscillator by 90?

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Resonance (2)

• Examples
• Mechanics
• Oscillations of a childs swing
• Destruction of the Tacoma Bridge
• Sound
• An opera singer shatters a wine glass
• Resonance tube
• Kundts tube

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Resonance

• Electricity
• Radio tuning
• Light
• Maximum absorption of infrared waves by a NaCl
  crystal

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Resonant System

• There is only one value of the driving frequency
  for resonance, e.g. spring-mass system
• There are several driving frequencies which give
  resonance, e.g. resonance tube

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Resonance undesirable

• The body of an aircraft should not resonate with
  the propeller
• The springs supporting the body of a car should
  not resonate with the engine

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Demonstration of Resonance

• Resonance tube
• Place a vibrating tuning fork above the mouth of
  the measuring cylinder
• Vary the length of the air column by pouring
  water into the cylinder until a loud sound is
  heard
• The resonant frequency of the air column is then
  equal to the frequency of the tuning fork

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Demonstration of Resonance

• Sonometer
• Press the stem of a vibrating tuning fork against
  the bridge of a sonometer wire
• Adjust the length of the wire until a strong
  vibration is set up in it
• The vibration is great enough to throw off paper
  riders mounted along its length

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Oscillation of a metal plate in the magnetic field
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Slight Damping
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Critical Damping
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Heavy Damping
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Amplitude
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Phase
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Bartons Pendulum
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Damped Vibration
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Resonance Curves
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Resonance Tube
A glass tube has a variable water level and a
speaker at its upper end
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UNIT V GOVERNORS AND GYROSCOPES Governors -
Types - Centrifugal governors - Gravity
controlled and spring controlled centrifugal
governors Characteristics - Effect of friction -
Controlling Force . Gyroscopes - Gyroscopic
forces and Torques - Gyroscopic stabilization -
Gyroscopic effects in Automobiles, ships and
airplanes
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Governors

• Engine Speed control
• This presentation is from Virginia Tech
  and has not been edited by Georgia Curriculum
  Office.

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Governors

• Governors serve three basic purposes
• Maintain a speed selected by the operator which
  is within the range of the governor.
• Prevent over-speed which may cause engine damage.
• Limit both high and low speeds.

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Governors

• Generally governors are used to maintain a fixed
  speed not readily adjustable by the operator or
  to maintain a speed selected by means of a
  throttle control lever.
• In either case, the governor protects against
  overspeeding.

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How does it work?

• If the load is removed on an operating engine,
  the governor immediately closes the throttle.
• If the engine load is increased, the throttle
  will be opened to prevent engine speed form being
  reduced.

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Example

• The governor on your lawnmower maintains the
  selected engine speed even when you mow through a
  clump of high grass or when you mow over no grass
  at all.

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Pneumatic Governors

• Sometimes called air-vane governors, they are
  operated by the stream of air flow created by the
  cooling fins of the flywheel.

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Air-Vane Governor

• When the engine experiences sudden increases in
  load, the flywheel slows causing the governor to
  open the throttle to maintain the desired speed.
• The same is true when the engine experiences a
  decrease in load. The governor compensates and
  closes the throttle to prevent overspeeding.

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Centrifugal Governor

• Sometimes referred to as a mechanical governor,
  it uses pivoted flyweights that are attached to a
  revolving shaft or gear driven by the engine.

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Mechanical Governor

• With this system, governor rpm is always directly
  proportional to engine rpm.

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Mechanical Governor

• If the engine is subjected to a sudden load that
  reduces rpm, the reduction in speed lessens
  centrifugal force on the flyweights.
• The weights move inward and lower the spool and
  governor lever, thus opening the throttle to
  maintain engine speed.

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Vacuum Governors

• Located between the carburetor and the intake
  manifold.
• It senses changes in intake manifold pressure
  (vacuum).

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Vacuum Governors

• As engine speed increases or decreases the
  governor closes or opens the throttle
  respectively to control engine speed.

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Hunting

• Hunting is a condition whereby the engine speed
  fluctuate or is erratic usually when first
  started.
• The engine speeds up and slows down over and over
  as the governor tries to regulate the engine
  speed.
• This is usually caused by an improperly adjusted
  carburetor.

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Stability

• Stability is the ability to maintain a desired
  engine speed without fluctuating.
• Instability results in hunting or oscillating due
  to over correction.
• Excessive stability results in a dead-beat
  governor or one that does not correct
  sufficiently for load changes.

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Sensitivity

• Sensitivity is the percent of speed change
  required to produce a corrective movement of the
  fuel control mechanism.
• High governor sensitivity will help keep the
  engine operating at a constant speed.

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Summary

• Small engine governors are used to
• Maintain selected engine speed.
• Prevent over-speeding.
• Limit high and low speeds.

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Summary

• Governors are usually of the following types
• Air-vane (pneumatic)
• Mechanical (centrifugal)
• Vacuum

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Summary

• The governor must have stability and sensitivity
  in order to regulate speeds properly. This will
  prevent hunting or erratic engine speed changes
  depending upon load changes.

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Gyroscope
A gyroscope consists of a rotor mounted in the
inner gimbal. The inner gimbal is mounted in the
outer gimbal which itself is mounted on a fixed
frame as shown in Fig. When the rotor spins about
X-axis with angular velocity ? rad/s and the
inner gimbal precesses (rotates) about Y-axis,
the spatial mechanism is forced to turn about
Z-axis other than its own axis of rotation, and
the gyroscopic effect is thus setup. The
resistance to this motion is called gyroscopic
effect.
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GYROSCOPIC COUPLE
Consider a rotary body of mass m having radius of
gyration k mounted on the shaft supported at two
bearings. Let the rotor spins (rotates) about
X-axis with constant angular velocity rad/s. The
X-axis is, therefore, called spin axis, Y-axis,
precession axis and Z-axis, the couple or torque
axis .
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GYROSCOPIC EFFECT ON SHIP
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THANK YOU