# 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 ins tantaneous velocity Resistive force instantaneous velocity i.e. F = -bv where b = damping coeffici

ent 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 da mping

The system returns very slowly to the equili brium 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 d

riving force should be reduced gradually so that th e oscillator can decelerate under its own restoring force At the maximum displacement, the driving force b ecomes zero so that the oscillator is not pushed an y further Thereafter, F reverses in direction so that the oscill

ator is pushed back to the centre Phase (4) On reaching the centre, F is a maximum in t he 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 respon ds with maximum amplitude. 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 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 = ) 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 childs swing

Destruction of the Tacoma Bridge Sound: An opera singer shatters a wine glass Resonance tube Kundts tube

Resonance (3) Electricity Radio tuning Light Maximum absorption of infrared waves by a NaCl cryst al

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 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 Oscillation of a metal plate in the magnetic field

Slight Damping Critical Damping Heavy Damping Amplitude

Phase Bartons 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

Kundts Tube Sonometer