Transcript forced vibration & damping

FORCED VIBRATION &
DAMPING
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.
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
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
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 < Frequency
of undamped vibration
Types of Damped Oscillations (1)
 Slight damping (underdamping)
 Characteristics:
 - oscillations with reducing amplitudes
 - amplitude decays exponentially with time
 - period is slightly longer
 - Figure
a1 a2 a3


 .......  a constant
 a2 a3 a4
Types of Damped Oscillations (2)
 Critical damping
 No real oscillation
 Time taken for the displacement to become
effective zero is a minimum
 Figure
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
Example: moving coil galvanometer
(1)
 the deflection of the pointer is critically
damped
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
Forced Oscillation
 The system is made to oscillate by periodic
impulses from an external driving agent
 Experimental setup:
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)
Characteristics of Forced Oscillation
(2)
 In steady state, the system vibrates at the
frequency of the driving force
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
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
Effects of Damping
 Driving frequency for maximum amplitude
becomes slightly less than the natural
frequency
 Reduces the response of the forced system
 Figure
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
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
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
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.
Barton’s 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
Barton’s 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 = )
Previous page
Hacksaw Blade Oscillator (1)
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
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
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
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
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
Resonance (2)
 Examples
 Mechanics:
 Oscillations of a child’s swing
 Destruction of the Tacoma Bridge
 Sound:
 An opera singer shatters a wine glass
 Resonance tube
 Kundt’s tube
Resonance (3)
 Electricity
 Radio tuning
 Light
 Maximum absorption of infrared waves by a NaCl
crystal
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
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
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
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
Oscillation of a metal plate in the magnetic
field
Slight Damping
Critical Damping
Heavy Damping
Amplitude
Phase
Barton’s Pendulum
Damped Vibration
Resonance Curves
Swing
Tacoma Bridge
Video
Resonance Tube
A glass tube has a
variable water level and
a speaker at its upper
end
Kundt’s Tube
Sonometer