# Relationship between velocity ratio and dynamic displacement in sinusoidual vibration

## Homework Statement

I am asked to provide an anti vibrational support mount for equipment mounted on turbo-prop aircraft. The equipment has a mass of 40kg.
the vibrational response of the equipment to the environmental disturbances will need ot satisfy a safety limit prescribed by the customer.
the performance of the vibration isolation system will be tested using transducers that measure velocity of the vibration as shown:

## Homework Equations

amplitude of Velocity of equipment/amplitude of Velocity of support structure = k,

where k must lie below a certain limit for a given frequency of operation.

I have been given an equation that relates

amplitude of Displacement of equipment(X)/ amplitude of displacement of support structure(Y)

which is the displacement transmissibility Td,
where the equation is Td = (k + jwc)/(k-mw^2 + jwc)
where k is stiffness, c is the damping constant, m is the mass and w is the frequency of operation and j is the complex component.

## The Attempt at a Solution

i would deem this question to be a Single Degree of Freedom with Base Excitation, as we would only consider motion in vertical direction here (given in question)

i just need help to affirm that the relationship of the ratio of the amplitudes of the displacements is the SAME as that as the ratio of the amplitude of velocities.

by considering y(t) to be base excitation and x(t) to be the response, both y(t) and x(t) have the same frequency and a phase lag in between.

using complex algebra to represent this ,

we have x(t) =Xe^jwt , xdot(t) = jwXe^jwt
and y(t) = Ye^jwt, ydot(t) = jwYe^jwt

hence from here, we can tell that the ratios of displacements and velocities are the same.

Am i right here? or have i over simplified things and i have missed out something?

You're well on the right track there...

Uni of manchester student by any chance? =P

You're well on the right track there...

Uni of manchester student by any chance? =P

HAHAHA!!! This is the first time I come across something like this here :rofl:

Thank you lecturer!

lol, is this considered cheating if i ask such a question over here?

well, yea, i am studying in university of manchester now, do you happen to be my Dynamics lecturer?? Now this is pretty amazing =))