Fig.1(Critically Damped System)
- Critically damped system(ξ=1): If the damping factor
ξ is equal to one, or the damping coefficient
c is equal to critical damping coefficient "cc", then the system is said to be a critically damped system.
- Two roots for critically damped system are given by S1 and S2 as below:
- For ;
Here both the roots are real and equal, so the solution to the differential equation can be given by
- Now differentiating equation (1) with respect to ‘t’, we get:
Now, let at
Substituting these values in equation (1):
- Same way, from equation (2), we get
- Now putting the values of A and B in equation (1), we get:
From above equation (5), it is seen that as time
t increases, the displacement
x decreases exponentially.
The motion of a critically damped system is aperiodic (aperiodic motion motions are those motions in which the motion does not repeat after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.
For critically damped systems, if a system is displaced from its initial position, it will try to reach its mean position in a very short time.
Critically damped systems are generally seen in hydraulic doors closer as it is necessary for the door to come to its initial position in a very short time.