How is the restoring force acting in damped vibrations ?

Dear student,

When the motion of an oscillator reduces due to an external force, the oscillator and its motion are damped. These periodic motions of gradually decreasing amplitude are damped simple harmonic motion. An example of damped simple harmonic motion is a simple pendulum.

The damping force depends on the nature of the surrounding medium. When we immerse the block in a liquid, the magnitude of damping will be much greater and the dissipation energy is much faster. Thus, the damping force is proportional to the velocity of the bob and acts opposite to the direction of the velocity. If the damping force is F_d, we have,

F_d = -bυ                              (I)

where the constant b depends on the properties of the medium(viscosity, for example) and size and shape of the block. Let’s say O is the equilibrium position where the block settles after releasing it. Now, if we pull down or push the block a little, the restoring force on the block due to spring is F_s = -kx, where x is the displacement of the mass from its equilibrium position. Therefore, the total force acting on the mass at any time t is, F = -kx -bυ.

Now, if a(t) is the acceleration of mass m at time t, then by Newton’s Law of Motion along the direction of motion, we have

ma(t) = -kx(t) – bυ(t)                        (II)

The above equation takes both damping and restoring force.
Regards