Sampling frequency and Nyquist theorem - Data acquisition

[Mari121311]

Hi all, 

I have a rectangular steel beam which will be impacted with a weight of 100 kg and I would like to look for any modules able to sample the signal correctly. 

The Nyquist theorem states that if half sampling frequency is higher than the input signal, it will be recorded correctly. 

What shall I consider before I buy a data acquisition module to find out the signal of the rectangular steel beam? Shall I perform a finite element analysis model using elastic properties or plastic properties? Is the natural frequency of the structure associated with the input signal?

Thank you,

Mari

[mikeporter · Accepted solution]

Using whatever technique is appropriate, determine what the highest frequency component is of interest in your signal. Set your sample rate to twice that value. In addition, to protect the data, construct a hardware antisliasing filter there attenuates all energy above the highest frequency of interest.

Mike...

[Mari121311]

Hi Mike, 

Thanks for your answer. Can you please correct my thinking?

I got the first ten frequency values of the structure (Nyquist theorem, for industrial applications). I have then set sample rate to twice that value. The ninth and tenth frequency mode are 1100 Hz and 1400 Hz and therefore the sample rate can be set to 2800 Hz. 

1) If the sample rate is fixed to 2500 Hz, it is then within the range of the maximum frequency that can be captured with 2500 Hz. Is there going to have any effect in the overall response for the tenth frequency vibration mode? The Nyquist theorem for industrial applications states that the max. Frequency is between 5 - 10 higher. 

2) In the attached picture, there is a displacement - time graph with its relevant power spectrum. There is only a first peak and then the graph drops down. As can be seen, there is nothing after 400 Hz. Can we then say that the figure suggest that a minimum data sampling frequency in the order of 400 Hz is required to ensure accurate representation of the frequency contented of the measure response?

Regards, 

Mari

[Bob_Schor · Accepted solution]

Here's an easy way to understand the Nyquist criterion.  Suppose you had a 60Hz signal (very easy to get -- we usually call it "A/C noise").  To make it interesting, assume it is a pure 60Hz sine wave.  Sample it at 60 Hz.  What signal will you get?  Why, a DC signal, since you will always be sampling at the same point in the sinusoid.  [Note that if you take three measurements, you'll get three DC values, but they will almost certainly be three different values ...].  Now sample at 120 Hz.  You will get (depending on where you start your samples) alternating high and low values (but, again, different samples will probaly give you different peak and trough values).  

Mathematically, you can't get information about signals greater than half the sampling frequency.  When dealing with "real-world" (as opposed to ideal mathematical) signals, many engineers would say "a rule of thumb is to sample at 10 times the maximum frequency you want to study".  Because of "aliasing" (where a signal frequency higher than the sampling frequency will give the same values as a suitable frequency below the sampling frequency), you also want to remove higher frequency signals with an appropriate "anti-aliasing filter", as was mentioned.

Bob (it's just Math) Schor

[johnsold · Accepted solution]

Mari,

1) If you sample at 2500 Hz any energy in the tenth mode (1400 Hz) will not be correctly sampled. It will produce what is called an "alias." That aliased signal will be mixed in with the other data and CANNOT be separated out later by any mathematical procedure. 

If you must sample at 2500 Hz due to other limits on your system and there is energy in the 1400 Hz component, then you must insert a suitable HARDWARE anti-aliasing filter in the analog signal path before the signal reaches the A/D converter. This filtering cannot be done in software.  The design of the filter may not be simple because a very sharp filter will be required to pass the 1100 Hz component while suppressing the 1400 Hz component. Such filters may have significant delay and phase shift which varies with frequency (even in the passband) and that could make interpretation of the results much more challenging.

How much harmonic energy do you expect to exist at those frequencies?  Is there significant energy in higher harmonics, even if you are not interested in them?

My recommendation - without knowing all the details - would be to include a simple anti-aliasing filter with a cutoff frequency around 2 kHz and set the sampling rate to 10 kHz or higher.

2) It is hard to tell from a linear scale graph just how much energy is at the higher frequencies.  The right edge of the graph might be at ~1% of the peak value.  That is only 20 dB down.  I like to see about 60 dB between the highest peak and the lowest peak of interest. Try plotting the spectral curve with a logarithmic amplitude scale.  

Sampling at 400 Hz only gives you 7 harmonics of the 28 Hz peak. That might be OK but again I would prefer to sample at higher rates to make sure no important information is lost.  It is easy to discard data you do not need but impossible to reconstruct it if you did not acquire it in the first place.

It may also be useful to remove the steady state displacement (~18 mm) from the data before doing the spectral analysis. Just subtract the mean value of the data set before calculating the spectrum.

Lynn

[Matthew_Kelton · Accepted solution]

To expand on Bob's statement as to why Nyquist is not really enough, imagine you hit the crossing point (or very near it) of that 60Hz signal with your 120Hz sampling rate.  Your sampled signal will look to be zero (or a constant DC value if there is an offset to the sine wave).  Designing to meet Nyquist is a bad idea.  As Bob said, a good general rule of thumb is ten times your frequency.

[Mari121311]

Hi all, 

Thank to everyone for your answers. So, I might then increase the length of the beam in order to obtain lower frequencies able to satisfy the Nyquist theorem right?

Regards, 

Mari

[johnsold · Accepted solution]

That may help. Remember that the impact itself is a high frequency event, so even though the natural frequency of the beam may be low, there could still be energy at higher frequencies.

Lynn

[Mari121311]

Thanks Lynn

[Mari121311]

I have reduced the beam dimensions and it seems to satisfy the theorem. So I guess that I can now use the acquisition without making use of any kind of fibreboard on the beam to attenuate and slow down the rate of the signal right?

I have another question: If I had an acquisition exciding the theorem, how should I be identified that the signal contains aliasing? And how much this added to the signal?

Regards,

Mari