Force Damped Vibrations

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

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

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

Slight damping (underdamping) Characteristics: - oscillations with reducing

amplitudes - amplitude decays

exponentially with time - period is slightly longer.

constant a.......4

3

3

2

2

1 aa

aa

aa

Critical damping No real oscillation Time taken for the displacement to become

effective zero is a minimum.

Types of Damped Oscillations

Heavy damping (Overdamping) Resistive forces exceed those of critical damping The system returns very slowly to the

equilibrium position.

Types of Damped Oscillations

the deflection of the pointer is critically damped

Example: moving coil galvanometer

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)

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

Effects of Damping

Driving frequency for maximum amplitude becomes slightly less than the natural frequency

Reduces the response of the forced system

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 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

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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