# Accelerometer and Displacement

• makethings
In summary, the conversation discusses using an accelerometer to measure acceleration in a single degree of freedom vibration with harmonic support motion. The frequency of support motion is swept through to find the resonant natural frequency, and data is collected in a table with frequency and acceleration values. The question is how to find displacement from this data, and it is suggested to differentiate the position equation and use the acceleration data to find maximum displacement. However, it is also pointed out that acceleration is a function of both frequency and time, so a series of sampled acceleration values would be needed to integrate and find displacement accurately.

## Homework Statement

In Single Degree of Freedom vibration with harmonic support motion, I have used an accelerometer to measure acceleration of the vibrating system. The frequency of the support motion is sweeped through in order to find the resonant natural frequency of the system. So in tabular form I have frequency data and acceleration data. How do I find displacement from this data?

## Homework Equations

I figured you have to integrate twice...but time is not a variable. so i don't know what to integrate.

## The Attempt at a Solution

What is the equation for the position of your measurement device as a function of time? It's a sinusoid with an amplitude A, correct? What do you get when you differentiate that x(t) equation once? What comes out of the sin() function when you do that first differentiation? And then when you take the derivative of that v(t) equation, what comes out of the sin() term again? Now do you have enough info to solve the problem?

but there are no time values... my data makes it look like acceleration is a function of excitation frequency

Acceleration is a function of frequency, given the same amplitude. Write out the equations and you will see that.

You can use the acceleration data to find maximum displacement quite easily with no math, if that's all youre looking for

If x: displacement of the mass, w: excitation frequency of the support motion, A: amplitude, t: time
response is (ignoring phase):
x = A sin (wt)
x' = A w cos (wt)
x" = -A w^2 sin (wt)
If I were to differentiate wrt to frequency
x' = A t cos (wt)
x" = -A t^2 sin (wt)
I have the value of x", and w...but again... how do I take care of that time variable..

Freq (Hz) Acceleration(g)
15.01472 0.3581786
15.02945 0.3570644
15.0442 0.3566944
15.05897 0.3563249
15.07374 0.3559486
15.08854 0.3558424
15.10334 0.3557363
...

That's some of the data I have.. and I want to be able to plot a displacement vs freq. graph.

Do you know how you measured the acceleration values?
Clearly acceleration is a function of both excitation frequency and time. Obviously, at each frequency, you took some measurement of the acceleration of the object, but how did you make the measurements? If you just took one measurement at each frequency value, how do you expect to find displacement? You need to hold the frequency steady (instead of sweeping it) for some time, record a sequence of sampled acceleration values, and then you can integrate, then you move on to the next point in frequency.