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### Damped Simple Harmonic Oscillator

Chapter – 2

Mrs. Rama Arora

Assoc. Professor

Deptt. Of Physics

PGGCG-11

Chandigarh

Damping is the mechanism that results in dissipation of the energy of an oscillator.

In case of damped simple harmonic vibrations amplitude goes on decreasing with the passage of time and ultimately the body comes to rest.

The damping factor for the mechanical oscillator is a force.

Units of damping constant ‘r’ is kg/s or Ns/m.

In electrical oscillator damping factor is e.m.f.

The unit of R is ohms.

In mechanical oscillator the damping force is due to (i) viscous damping (ii) friction damping (iii) structure damping

In electrical oscillator the damping is due to resistance in the circuit.

Damping force is neither a constant nor depends upon displacement or acceleration but depends upon velocity only.

Differential equation of Simple Damped Harmonic motion

Restoring force is always proportional to the displacement of the body.

Damping force which is proportional to the velocity of the body.

Differential equation of damped SHM oscillator is:

The solution is

Term is an exponentially decreasing term with increasing time i.e. amplitude goes on decreasing with time.

For heavy damping so damping makes the system non oscillatory.

For critical damping b2 = w2 . The displacement decays to zero exponentially and the system returns to the initial state in the minimum possible time.

For light damping b2 < w2; amplitude of the damped oscillations reduces exponentially to zero. The oscillations cease almost during the same time in which oscillator returns to initial state.

The frequency of damped oscillations is given by

Comparison between undamped and damped oscillations

Undamped oscillation

Damped oscillations

Motion is strictly periodic and simple harmonic.

Amplitude is constant.

Frequency is determined by inertia and elastic properties.

No dissipation of energy occurs.

Oscillations continue indefinitely.

Motion is not strictly periodic or simple harmonic.

Amplitude decreases with time.

Frequency is determined by inertia, elastic properties and the damping constant.

Energy is dissipated continuously.

Oscillations cease after some time.

Damped Electrical Oscillator

In the damped electrical oscillator, the dissipation of energy occurs in the resistance of the circuit which may be distributed due to connecting wires and inductor.

Equation of damped electrical oscillator is

The solution of equation of damped electrical oscillator is

Heavy damping b2 > w2, the charge on the capacitor decays to zero in the minimum possible time.

Critical damping b2 = w2; the discharge is non-oscillatory.

The behaviors of the oscillator is said to be dead beat.

Light damping b2 < w2

The discharge is oscillatory and frequency of the damped oscillations is given by

Logarithmic decrement of a damped oscillator is the natural logarithm of the ratio of amplitude of oscillation at any instant and that one time after it.

For mechanical oscillator

For electrical oscillator

Relaxation time is defined as the time interval during which the amplitude of damped oscillator decays to 1/e times its initial value.

For mechanical oscillator

Relaxation time is inversely proportional to the damping constant.

For electrical oscillator

Thus, the relaxation time is inversely proportional to the resistance of the oscillatory circuit.

Quality factor gives the rate of decay of energy of the damped oscillator and is a number equal to 2 times the ratio of the instantaneous energy of the oscillator to the energy lost during one time period after that instant.

i.e.

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