Single Degree Support Motion

Now to derive the equation of motion, from the free body diagram of the mass as shown figure

                                                                           
let z = x-y   

                                                                                                   
solution of equation of the above equation can be written as

                                                                                           
and            

If the absolute motion x of the mass is required, we can solve for x = z + y. Using the exponential form of harmonic motion

                                                                                     
Substituting equation we obtain

         
The steady state amplitude and Phase from this equation are

and

                       

       Amplitude ratio ~ frequency ratio plot for system with support motion

It is clear that when the frequency of support motion nearly equal to the natural frequency of the system, resonance occurs in the system. This resonant amplitude decreases with increase in damping ratio for  .
 
At , irrespective of damping factor, the mass vibrate with an amplitude equal to that of the support For , amplitude ratio becomes less than 1, indicating that the mass will vibrate with an amplitude less than  support motion.

But with increase in damping, in this case, the amplitude of vibration of the mass will increase.

So in order to reduce the vibration of the mass, one should operate the system at a frequency very much greater than times the natural frequency of the system. This is the principle of vibration isolation.

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