Module 8 :  Free Vibration with Viscous Damping; Critical Damping and Aperiodic
                        Motion; Logarithmic Decrement;Systems with Coulomb Damping.
Lecture 1 :   Free Vibration with Viscous Damping

We define Damping factor as


8.1.4
8.1.5
  From equations 8.1.4 and 8.1.5 , we can write the two roots and as follows:
Thus mainly three cases arise depending on the value of
Whenthe system undergoes aperiodically decaying motion and hence such systems are said to be Overdamped Systems. An example of such a system is a door damper – when we open a door and enter a room, we want the door to gradually close rather than exhibit oscillatory motion and bang into the person entering the room behind us! So the damper is designed such that
Critically damped motion ( a hypothetical borderline case separating oscillatory decay from aperiodic decay) is the fastest decaying aperiodic motion.
 
When “< 1”, x(t) is a damped sinusoid and the system exhibits a vibratory motion whose amplitude keeps diminishing. This is the most common vibration case and we will spend most of our time studying such systems. These are referred to as Underdamped systems.
   
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