CHAPTER - 3

FORCED OSCILLATOR

Mrs. Rama AroraAssoc. ProfessorDeptt. Of Physics

PGGCG-11Chandigarh

Free oscillations are the vibrations when a system oscillates under the action of forces inherent in the system itself.

Forced oscillations are those vibrations which take place under the excitation of external forces.

Resonant oscillations are the forced oscillations of a system such that the amplitude of the oscillations has the maximum possible value.

Free oscillations occur at a definite frequency called the natural frequency of the oscillator.

Resonant oscillations occur at a definite frequency called the resonant frequency of the oscillator.

Forced oscillations can occur at any frequency.

Forced oscillator is an oscillator to which continuous excitation is provided by some external agency.

Driving agency of mechanical system is the source of mechanical force and in case of electrical system is a source of electromotive force.

Driven system is generally a damped oscillator.

A Mechanical forced oscillator

A forced oscillator is driven continuously by an external periodic force to maintain its oscillation.

Transient State: The natural oscillations of the oscillator persist only in the beginning and the behavior of the oscillator during this short interval of time is the transient state.

Steady State: When the transient phase is over, the oscillator starts oscillating in the steady state with the frequency of external periodic force.

Solution of differential equation of a forced oscillator

The solution is

Where Zm is the mechanical impedence of the oscillator. It is complex quantity with two parts.

i. Resistive part – r mechanical resistanceii. Reactive part (mw – s/m) is mechanical

reactance

Behavior of displacement amplitude with driving the force frequency

Amplitude of the oscillationsi. When ω is very small, ω -> 0, then the

oscillator is stiffness controlled.A = Fo/S

Amplitude is independent of frequency.ii. When ω is very very large, say ω->∞, then

the oscillator is inertia controlled.A = Fo/ω2m

Since ω->∞, therefore A->0

At resonance when the amplitude is maximum, ωZm to be minimum then the frequency of the oscillator is

where ω = √(s/m) = natural frequency of the undamped oscillator

The maximum displacement amplitude of the oscillations or resonance occurs at a frequency slightly less than the natural frequency of undamped oscillator.

Variation of phase difference between displacement and driving frequency

The displacement of the oscillator at any time t

Driving Force

Phase difference between displacement and the force is

Displacement lags behind the driving force

a) When ω -> 0: Phase difference is zerob) When ω->∞: Phase difference is –π rad;

displacement lags behind the force by π radians.

c) At resonating frequency: Phase diference between the displacement and the driving frequency is -π/2Displacement x lags behind the force by π/2 radians.