# Homework Help: Frequency Response Function - Random Vibration

1. Apr 30, 2010

### Dunc26

Hi looking for some help on the below I'm a little bit stuck.

The effect of an earthquake on an elevated water tank is to be investigated. The water tank has mass m=2x10^6 kg and natural frequency wn=10.6 rad/s and a damping coefficient of 12% critical damping.
Ground acceleration can be approximated by constant spectrum So=0.0075m2/s3
Obtain:
i) The frequency response function for the displacement of tank mass relative to the ground
ii) Mean square response of displacement
iii) Standard deviation of displacement

Frequency Response = Output / Input

mz"+cz'+kz=-my" ........................................Eq 1

Where z is the relative displacement
and y is input

I've done some work on something similar in the past where I have used the Force input to the system to get the frequency response function and obtain the equation:

H(w) = 1 / (k-(mw^2)+icw

This gives the frequency response function of the elevated mass displacement relative to the input force?

Using the equation: mz"+cz'+kz=-my"

If I take the input displacement
y = Ae^iwt (where A is a constant then)
y' = iwtAe^iwt
y"= -w^2.Ae^iwt

The output would then be
z = Be^iwt
z' = iwtBe^iwt
z" = -w^2.Be^iwt

subtituting into Eq 1

(k+iwc-mw^2)B.e^iwt = (mw^2)A.e^iwt

Therefore response is equal to B/A
= (mw^2) / (k+iwc-mw^2)

Is this along the correct lines??

Last edited: Apr 30, 2010
2. Apr 25, 2012

### msc_af

could anyone shed some light on this question? i have the same problem (saves me writing a new thread etc.)