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**Damped Oscillations**

Oscillations with a decreasing amplitude with time are called damped oscillations.

The displacement of the damped oscillator at an instant t is given by

x = x_{o}e^{– bt / 2m} cos (ω’ t + φ)

where x_{o}e^{– bt / 2m} is the amplitude of oscillator which decreases continuously with time t and ω’.

The mechanical energy E of the damped oscillator at an instant t is given by

E = 1 / 2 kx^{2}_{o}e^{– bt / 2m}

**Un-damped Oscillations**

Oscillations with a constant amplitude with time are called un-damped oscillations.

**Forced Oscillations**

Oscillations of any object with a frequency different from its natural frequency under a periodic external force are called forced oscillations.

**Resonant Oscillations**

When an external force is applied on a body whose frequency is an integer multiple of the natural frequency of the body, then its amplitude of oscillation increases and these oscillations are called resonant oscillations.

**Lissajous’ Figures**

If two SHMs are acting in mutually perpendicular directions, then due to then: superpositions the resultant motion, in general, is a curve loop. The shape of the curve depends on the frequency ratio of two SHMs and initial phase difference between them. Such figures are called Lissajous’ figures.

1. Let two SHMs be of same frequency (e.g., x = a_{1} sinωt and y = a_{2} sin (omega;t + φ), then the general equation of resultant motion is found to be

x^{2} / a^{2}_{1} + y^{2} / b^{2}_{2} – 2xy / a_{1}a_{2/sub> cos φ = sin2 φ}

The equation represents an ellipse. However, if φ = O° or π or nπ, then the resultant curve is a straight inclined line.

2. Let two SHMs be having frequencies in the ratio 1 : 2, then, in general, the Lissajous figure is a figure of eight (8).

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