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.......4

3

3

2

2

1 a

a

a

a

a

a

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)

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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

Tacoma Bridge

Video

Resonance Tube

A glass tube has a variable water level and a speaker at its upper end

Kundt’s Tube

Sonometer