Module 8 : Free Vibration with Viscous Damping; Critical Damping and Aperiodic
                      Motion; Logarithmic Decrement; Systems with Coulomb Damping.
Lecture 3 : Logarithmic Decrement
 Logarithmic Decrement:
  Consider the two peaks A and B as shown in fig 8.3.1. The amplitude at A and B are and at time and respectively.The periodic displacement from to represents a cycle. The time period for this complete cycle is given by:
 

8.3.1
  This is the time period of damped oscillations and is damped natural frequency.
  The amplitude of damped oscillations is given by the expression:
 

8.3.2
  which is the envelope of maximum of displacement -time curve. Therefore at t= and the amplitudes are given by:
 

8.3.3
 
8.3.4
 

Therefore
 


8.3.5
 
Therefore                                                                              
8.3.6
 
 

This is called the Logarithmic Decrement denoted by

 
8.3.7
 
if <<1
  This shows that the ratio of any two succesive amplitudes for an underdamped system, vibrating freely , is constant and is a function of the damping only.
 

Sometimes, in experiments, it is more convenient/accurate to measure the amplitudes after say "n" peaks rather than two succesive peaks (because if the damping is very small, the difference between the succesive peaks may not be significant). The lograthmic decrement can then be given by the equation

 
8.3.8
     
1