Fig-1 (Over damped system)
We know that the characteristic equation of the damped free vibration system is,
This is a quadratic equation having two roots and ;
In order to convert the whole equation in the form of , we will use two parameters, critical damping coefficient '' and damping factor ''. So the roots and can be written as follows;
where, = natural frequency of undamped free vibration = rad/s
So we can write roots and and as;
Overdamped system (ξ>1)
If the damping factor ‘’ is greater than one or the damping coefficient ‘’ is greater than critical damping coefficient ‘’, then the system is said to be over-damped.
In overdamped system, the roots are given by;
For , we get and as real and negative so we get,
Now differentiating equation (1) with respect to ‘t’, we get;
Now, let at
Substituting this value in equation (1) we get;
Substituting this value in equation (2) we get;
From equation (3), and putting value of B in (4);
Putting , in equation (4);
Now putting equation (5) and (6) in equation (1), we get;
Above equation represents the equation of motion for overdamped system.
The motion obtained by above equation is aperiodic (aperiodic motion motions are those motions in which the motion does not repeats after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.
This type of system does not show much damping, so this systems are used very rarely.