FORCED VIBRATION

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FORCED VIBRATION DAMPING
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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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Free Vibration

• Vibrate in the absence of damping and external
  force
• Characteristics
• the system oscillates with constant frequency and
  amplitude
• the system oscillates with its natural frequency
• the total energy of the oscillator remains
  constant

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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
• Figure

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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
• Figure
• Computer simulation

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Example moving coil galvanometer (1)

• the deflection of the pointer is critically damped

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Example moving coil galvanometer (2)

• 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 (1)

• 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 (2)

• 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
• Figure

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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
• Figure

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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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Bartons Pendulum (1)

• The paper cones vibrate with nearly the same
  frequency which is the same as that of the
  driving bob
• Cones vibrate with different amplitudes

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Bartons Pendulum (2)

• Cone 3 shows the greatest amplitude of swing
  because its natural frequency is the same as that
  of the driving bob
• Cone 3 is almost 1/4 of cycle behind D. (Phase
  difference ?/2 )
• Cone 1 is nearly in phase with D. (Phase
  difference 0)
• Cone 6 is roughly 1/2 of a cycle behind D. (Phase
  difference ?)

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Hacksaw Blade Oscillator (1)
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Hacksaw Blade Oscillator (2)

• Damped vibration
• The card is positioned in such a way as to
  produce maximum damping
• The blade is then bent to one side. The initial
  position of the pointer is read from the attached
  scale
• The blade is then released and the amplitude of
  the successive oscillation is noted
• Repeat the experiment several times
• Results

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Forced Vibration (1)

• 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 (2)

• 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 (3)

• 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 (3)

• 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 (1)

• 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 (2)

• 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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EXAMPLES
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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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Swing
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Tacoma Bridge
Video
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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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Kundts Tube
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Sonometer