vibrat damp
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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 coeff
icient 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 - constant a.......
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3
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Critical damping No real oscillation Time taken for the displacement to become
effective zero is a minimum Figure
Types of Damped Oscillations (2)
Heavy damping (Overdamping) Resistive forces exceed those of critical da
mping The system returns very slowly to the equili
brium position Figure Computer simulation
Types of Damped Oscillations (3)
the deflection of the pointer is critically damped
Example: moving coil galvanometer (1)
Damping is due to induced currents flowing in the metal frame
The opposing couple setting up causes the coil to come to rest quickly
Example: moving coil galvanometer (2)
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)
In steady state, the system vibrates at the frequency of the driving force
Characteristics of Forced Oscillation (2)
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
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 (3)
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.
Phase (4)
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
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 = )
Barton’s Pendulum (2)
Previous page
Hacksaw Blade Oscillator (1)
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
Hacksaw Blade Oscillator (2)
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
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 (2)
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
Forced Vibration (3)
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
Electricity Radio tuning
Light Maximum absorption of infrared waves by a NaCl cryst
al
Resonance (3)
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
Sonometer Press the stem of a vibrating tuning fork against th
e bridge of a sonometer wire Adjust the length of the wire until a strong vibratio
n is set up in it The vibration is great enough to throw off paper ri
ders mounted along its length
Demonstration of Resonance (2)
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
Resonance Tube
A glass tube has a variable water level and a speaker at its upper end
Kundt’s Tube
Sonometer