Question

can someone tell me the differences between “Free Undamped Motion”, “Free Damped Motion”, “Driven Undamped Motion”, or “Driven Damped Motion”

sample drawings will be helpful.

Answer 1

a mechanical system is displaced from its equilibrium position and then released, the restoring force will bring about return towards the equilibrium position. This is referred to as “Free Vibration”.

This type of vibration arises from an initial impact energy that is continually changing from potentialto kinetic form. In a free vibration, the system is said to vibrate at its natural frequency. However,due to various causes there will be some dissipation of mechanical energy during each cycle of vibration and this effect is called “Damping.” (Ryder and Bennett, 1990).

Theoretically, an un-damped free vibration system continues vibrating once it is started.Vibration can be classified in several ways, the important ones includes:

Free and Forced Vibration: in this type of vibration, no external force acts on the system asthe system is left to vibrate on its own after an initial disturbance. E.g.: the oscillation of a simple pendulum. Similarly, when the system is subjected to an external force (often, arepeating type of force. E.g.: the oscillation of machines like diesel engines

Undamped and Damped Vibration: if no energy is lost or dissipated in friction or resistanceduring oscillation, the vibration is known as undamped vibration. Should any energy be lostin its way, it is called damped vibration. (S.S Rao. 2011)

DRIVEN OR FORCED HARMONIC MOTION

as we know due to resistance oscillation eventually die out. to maintain the oscillation one need driving force.

so it is called driven undamped motion if we didnot use any damping element like viscous force or other damping element

if we use damping elemnt with external force than it is called driven damped montion

mx"(t) + kx(t) = f(t)