Vibration of beam with bump test

[Lisa Jones]

Hi, I need some help with an experiment:

I have a cantilevered steel beam, fixed at one end and free at the other.

When I excite the beam with forced vibration at the fixed end, resonant frequencies are evident. The first is at 10 Hz and the second is at 60Hz.

Performing analytic calculations, the first two natural frequencies are indeed 10Hz and 60Hz.

When I do a bump test (give the beam an initial displacement and use an accelerometer to measure its subsequent natural vibrations) the peak vibration frequencies occur at 7Hz, 20Hz and 33Hz (I view the accelerometer data in the frequency domain)

Why is there this difference? Is it becuase of what the accelerometer is measuring? Is it because of where the accelerometer is positioned on the beam? Is it because of the way the Fourier transform is performed?

Any help ASAP would be much appreciated.

 

[JolileChat]

Hi Lisa.

How did you adjust the sampling rate and the length of the time domain acquisition?

 

[Studiot]

Free and forced resonances occur at near frequencies, rather than the same or multiples thereof.

This is because the differential equation for forced vibration has an extra term. Are you familiar with these?

 

[Lisa Jones]

Hi,

HARMONICS!

I have discovered that the frequencies are at exactly 1 times, 3 times and 5 times the 7Hz signal. There is also a small bump at 7 times!

This leads me to think the mass of the accelerometer has reduced the natural frequency from 10Hz to 7Hz, and then the other peaks are harmonics? This works very well with exact numbers.

Question: Why are only the odd harmonics significant? Why not the even harmonics?
Question: Why is there no peak at ~60 Hz? - maybe the amplitude is just so low compared to the fundamental freq and its harmionics?

Thanks again for your help!

 

[Studiot]

Question: Why are only the odd harmonics significant? Why not the even harmonics?
Question: Why is there no peak at ~60 Hz? - maybe the amplitude is just so low compared to the fundamental freq and its harmionics?

Think about the wave shapes the cantilever can make in vibration.

Look again at my first answer.

 

[JolileChat]

Free and forced resonances occur at near frequencies, rather than the same or multiples thereof.

This is because the differential equation for forced vibration has an extra term. Are you familiar with these?

Hi, Studiot.

The extra term is the forcing term, and it does not change the natural frequency of the beam.

Natural frequency and damping are properties of the system, therefore it does not matter if we are talking about the forced response or the response to initial conditions.

 

[JolileChat]

Hi,

HARMONICS!

I have discovered that the frequencies are at exactly 1 times, 3 times and 5 times the 7Hz signal. There is also a small bump at 7 times!

This leads me to think the mass of the accelerometer has reduced the natural frequency from 10Hz to 7Hz, and then the other peaks are harmonics? This works very well with exact numbers.

Question: Why are only the odd harmonics significant? Why not the even harmonics?
Question: Why is there no peak at ~60 Hz? - maybe the amplitude is just so low compared to the fundamental freq and its harmionics?

Thanks again for your help!

Lisa, please let me know what is exactly the bump test. You "bump" the beam only one time or you keep bumping the beam at a fixed frequency?

Now let's talk about the initial condition response.

When you impose a displacement to the tip of your beam, it will assume a shape that is very close to the shape of the first mode of the beam.

When the beam is released, it will vibrate with a motion that can be described as a weighted sum of all the mode shapes. Superposition in action here.

Since the initial condition is a shape that looks closely to the first mode, it is reasonable to expect that the first mode participation in this weighted sum is dominant. Therefore, you should find a clear peak in the FFT of the accelerometer signal. For the other modes, you should find smaller peaks.

It is important to look the acquisition parameters. To capture the 60Hz mode, the sampling rate should be at least 120 Hz. Have you checked it?

Other important thing is the amplitude of the initial displacement. Are you are imposing a large displacement? And how is the beam fixed? The beam suport is really fixing the beam? Or it vibrates with the beam? The other frequencies you are finding might come from the structure at which the beam is fixed.

Let's make sure that these harmonics are not associated to the fixing structure or to signal processing issues. Harmonics might even indicate nonlinear phenomena in you experiment. But I'm almost sure that your objective was not to explore nonlinear dynamics.

 

[Lisa Jones]

Thanks for your help.

The bump test was performed by displacing the free end of the beam by hand, and then letting it oscillate naturally. The fixed end is properly fixed.

The sampling rate was very high - no aliasing problems that I can see.

The peaks are perfect multiples of the 6.75 Hz fundamental frequency (1 times, 3 times, 5 times and 7 times). It therefore seems to me that they must be harmonics.

Is there a reason why only the odd harmonics show significant peaks?

 

[Studiot]

Hi, Studiot.

The extra term is the forcing term, and it does not change the natural frequency of the beam.

Natural frequency and damping are properties of the system, therefore it does not matter if we are talking about the forced response or the response to initial conditions.

Yes I could have phrased it better, but real life and the mechanical equations of vibration, I'm afraid, is more complicated than the above.

@Lisa

I asked if you understood the differential equations of vibration, you did imply you had calculated the free resonant frequency.
Above this frequency a mechanical system will see inharmonic response peaks, just as you have. The reason is best discussed in terms of the diff equation, it shows whether the system is mass, stiffness or resistance controlled.

As to why the responses correspond to odd multiples, again I say look at the wave shape. The boundary conditions are such that the cantilever has one fixed and one free end. It vibrates in standing wave mode and must have a node at the fixed and and an antinode at the free.

Does this help?

 

[JolileChat]

6.75 Hz doesn't seem to be related with the frequencies you have found after the modal analysis (10 Hz and 60 Hz). By the way, 60 Hz is a suspect frequency, be sure it is not coming from the AC electric power.

How much is the initial displacement? And the length of the beam?

Can you repeat your experiment with a different initial displacement? With an initial displacement much smaller, the smallest you can impose, we could have a better idea of what is happening with the cantilever beam.

Thanks for your help.

The bump test was performed by displacing the free end of the beam by hand, and then letting it oscillate naturally. The fixed end is properly fixed.

The sampling rate was very high - no aliasing problems that I can see.

The peaks are perfect multiples of the 6.75 Hz fundamental frequency (1 times, 3 times, 5 times and 7 times). It therefore seems to me that they must be harmonics.

Is there a reason why only the odd harmonics show significant peaks?