Theory: transmissibility (of excitation force)

To avoid the mathematics becoming too involved it is better to use the exponential representation of harmonic excitation. Thus the equation of motion is,

and the response will be

So that substituting in the equation of motion,

and rearranging gives,

................ (1)

From the diagram the force on the abutment is,

For sinusoidal excitation this becomes

Rearrnaging and substituting for X from (1),

Thus the transmissibility is given by

This is identical to the equation for X/Xo for abutment excitation.