FREE DAMPED AND FORCED OSCILLATIONS

When a body is displaced from its mean position, it oscillates with its natural frequency. Such vibrations/oscillations are called free oscillations since no external force acts on the body. In this case, total energy is conserved. Amplitude of vibrations remains constant with time. In this ideal situation, displacement traces an infinite sine curve if the motion is SHM.

FREE DAMPED OSCILLATIONS:

A free oscillation is an ideal case since every motion is exposed to a resistive/frictional force as a result of which its amplitude decreases with time. The total energy is not conserved.

External friction in the form of air resistance is present in the case of an oscillating pendulum. Both internal and external friction is present in vibrating tuning fork.

Such forces are termed as damping forces as they dissipate energy of the system.

EQUATION OF MOTION FOR FREE DAMPED SHM:

In free damped oscillations, body is acted upon by following forces:

• Restoring force (F) is proportional to the displacement of the body from its position. If it is x, then F_r= -Kx, where K is the elastic constant.
• We assume that damping force is proportional to the instantaneous velocity of the body.

EFFECT OF DAMPING:

LOGARITHMIC DECREMENT

FORCED VIBRATIONS:

A free vibration on account of frictional resistance loses its energy, its amplitude decreases continuously. If the body is to be maintained at constant vibration, energy needs to be supplied externally.

When a body vibrates under the effect of an external force, its frequency is different from that of its natural frequency. Such vibrations are called forced vibrations and its frequency is ?_f.

Equation of motion is given by:

Transient state: the complementary function gives the solution of this state.

Transient state dies away and a steady state is reached. Then body vibrates with frequency of periodic force, this is a condition of forced oscillations.