Solution of characteristic equation of motion for the free hanging pendulum

          

Assume a solution of the form  


          

          

          

           Let  

                     


 = 
 
 = 
 
 = 
 
 = 
 
 = 
 

Here , , and B are constants dependant upon initial conditions. In general the response    will be damped sinusoidal oscillations with a frequency   and damping envelope of 
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