I'm studying for my class of physics laboratory and I need help with something:

Let's say I need to deduce the constant of elasticity of a spring and I will do it using a dual-range force sensor, throught dynamic and estatic measurements. So it is necessary to stablish the frecuency sample-rate according of the period that it will last my experiment. Considering that $F=n° of samples/T$ where $F$ is the frecuency, and the number of samples is one and $T$ is the period that last my experiment.

What do you recommend me to establish the period and frequency without really knowing a lot about the spring and its own characterstics? It should be through a simple and not necessarily precise way the estimation of the frecuency sample-rate.

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    $\begingroup$ Per Nyquist, you need at least 2 samples per period (not 1 like you proposed) of whatever phenomenon you're measuring. Typically you want some higher multiple ("oversampling") like 4, 8, or 16x to conveniently be able to reconstruct the original signal. $\endgroup$ – The Photon Jun 25 '17 at 16:50
  • $\begingroup$ Let's say I estimate the $T$ of the spring bouncing, and that is $1.2 s$ and I decide to get $30$ samples for second, then I will get a frecuency sample-rate of $25Hz$, don't you think is too little to get 25 points per second? I heard some friends used like $200Hz$. How can I justify mathematically and experimentally it is necessary to use that frecuency, for example? $\endgroup$ – Neisy Sofía Vadori Jun 25 '17 at 17:01
  • $\begingroup$ Depends on the analysis you're going to do. I don't know what parameters you're going to try to estimate on the signal or what math you plan to apply to get there, so I can't give a complete answer. $\endgroup$ – The Photon Jun 25 '17 at 17:06
  • $\begingroup$ I will estimate the constant of elasticity throught two different methods: one is dynamic, which means, I have to make the spring bouncing with a mass added to the extreme, and other, is static, whitout making the spring bouncing. And what I will do is measure the force in both cases with the instrument I specified, and then process the data in the origin. $\endgroup$ – Neisy Sofía Vadori Jun 25 '17 at 17:08

According to Nyquist's theorem, you cannot accurately measure the frequency of a signal unless you sample at least twice per period. For instance, if you are expecting a $\rm1\, Hz$ frequency, you must measure at least twice per second. However, if you are only measuring twice per second, a $\rm1.2\,Hz$ signal will be aliased and appear in your data like a $\rm0.8\,Hz$ signal. So in general you want to sample your signal more frequently than the minimum rate required by Nyquist's theorem.

How much more frequently is a judgement call to be made by the experimenter, depending on the equipment you have at hand and your goals. For instance if your question is not merely "what is the frequency of my spring's motion" but also "is the simple harmonic oscillator a good model for my spring", you would want to sample frequently enough to see several of the harmonics of your expected frequency.

Now that we live in a future when data storage and processing are extremely cheap, the usual method for choosing a sampling frequency is to sample as fast as your sensor will permit, reducing the sampling rate if you find yourself running out of disk space or your data taking too long to process. You can always effectively reduce your sampling rate by throwing away some of your data, but you can never go back and re-measure something you didn't think of initially.

  • $\begingroup$ Do you think if I measure one oscillation of my spring (having stablished an specific length of elongation that I will mantain the whole process) and I observe in the software Motion DAQ how much aproximally last one period, in this case let's suppose: $1.2$ seconds and I arbitrary decide I want to get $200$ samples each period, so I will use a sample rate of $166Hz$ and I justify this by saying that I stablished that amount of samples because I wanted to reconstruct the original signal and reduce the noise properly, will be enough as justification? $\endgroup$ – Neisy Sofía Vadori Jun 25 '17 at 18:13
  • $\begingroup$ In general if you want to measure a frequency, you want to measure many oscillations. If I saw three experiments which measured a 0.8 Hz signal with sampling frequencies 150 Hz, 166 Hz, 200 Hz, I would consider them identical experiments. I don't know what you mean by "enough justification"; that depends on whom you are trying to impress. $\endgroup$ – rob Jun 25 '17 at 18:19
  • $\begingroup$ @NeisySofíaVadori Not sure what you're doing exactly, but if you measure a 1-way crossing at a fixed amplitude, your result will be biased: as the oscillation decays, the phase of the signal at the fixed amplitude changes. $\endgroup$ – JEB Jun 27 '18 at 3:18

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