Lets start with a scenario…..
Suppose there are 3 persons P1, P2 and P3 as marked in the figure. All have to reach the center of the blue ring ( Steady State Value). Now to complete the errand all three get into 3 different airplanes : Over damped (O), Critically damped (C) and Under damped(U). Now I would like you to pay attention to both the time and displacement axis considering the displacement axis as the vertical distance of planes from ground.
Over Damped: The O airplane takes off very slowly and hence has a very high value of Rise Time as compared to other two systems. Rise time being defined as time taken to reach the level of center of the blue denoted by dotted line in the figure. Also due to some navigational anomalies it never reaches its destination and stays below the required height.
Critically Damped: The C plane is better than the O and almost (asymptotically) reaches the target and that too at a better speed and less time than O.
Under Damped: The U plane pilot is very enthusiastic and in its excitement goes at very high speed hence goes pass the target and that too in very less time (Rise Time) as compared to other two planes. Releasing his mistake it again comes back but again enthusiasm kicks in and he again passes the destination this time in opposite direction. This process continues until a specific time until he is settled. We call this time as the Settling Time and after this time he finally reaches his destination.
So in our little story the three paths takes by the planes represents the 3 time responses of Second order system when subjected to unit step response. Main difference lies in their Rise Time and Settling time and final steady state value achieved which dictate various design parameters in control systems.
Mathematical Explanation:
Any general second order equation can be represented as:
with general solutions as:
Importance: Many complex higher order systems can be reduced to second order systems and analysis can be performed.
Applications: Analysis of active suspension in performed by studying various parameters of systems like stiffness, damping coefficient settling time etc. They form the science for which we pay ridiculous amount of money while buying a Mercedes or BMW.
I hope it helped….
Cheers!!!