Free vibration is initiated by disturbing the system from its static equilibrium position by imparting the mass some displacement u(0) and velocity u’(0) at time t=0.
There are two cases in free vibration

1. undamped free vibration
   
2. damped free vibration
   

Undamped free vibration : The governing equation for undamped free vibration is

where,
m = mass ,u'' = acceleration, k = stiffness, u = displacement.
The solution to homogeneous equation is :

where,
u(0) = initial displacement ,u'(0) = initial velocity.
Here in this solution we can observe that the system will vibrate only if initial displacement and/or initial velocity is given.

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