Vibration Data and a Few Techniques to Analyze It

What is a Vibration Signal?

Vibration is a mechanical event in which oscillations occur about an equilibrium point, and the time series that carries the information of those oscillations, is called vibration signal. These oscillations from the equilibrium point need to be acquired at a high sampling rate. A vibration signal can be generally denoted by a sine wave, as shown in the figure below, that has a few important properties which should be remembered while analyzing it (described below).

• Amplitude or peak: provides shock events detail, but it doesn’t consider the time duration and thus the energy in the event.
• Peak-to-peak: Provides the maximum excursion of the wave, which is useful when looking at displacement information
• RMS (root mean square): Most useful because it is directly related to the vibration signal’s energy content and thus the vibration’s destructive capability because it takes the time history of the waveform into account.

The vibration signals are nonlinear and non-stationary (mostly) and it is a challenging task to analyze the non-stationary signals.

What are stationary and non-stationary signals?

A stationary signal can be denoted by a sine-wave with a constant time period, whereas a non-stationary signal would have a sine wave with varying time periods. Or in other words, we can call a signal stationary if its frequency or spectral contents are not changing with respect to time because the frequency in the non-stationary signal varies with time. One example of the stationary and non-stationary signal is given below in the figure, along with their frequency distributions.

What are the techniques to analyze the vibration signals?

Empirical Mode Decomposition (EMD)

EMD is an adaptive method to decompose nonlinear and non-stationary signals into several intrinsic mode functions (IMFs) and a residue. EMD algorithm is based on a sifting process that ends when residual remains either a constant, a monotonic slope, or a function with only one extreme.

Specific frequency band separation using Hilbert-Huang transform and EMD

Empirical mode decomposition (EMD) can disintegrate vibration signal S[n] into narrowband IMFs. The two essential conditions are satisfied by each IMF:

• The counting of extrema and the counting of zero crossings in the data must be the same or are allowed to differ by utmost one, and
• As per their definitions, the average value of the envelopes obtained from the local maxima and the local minima are zero in instantaneous ways.

After decomposing vibration signal into narrow bands, apply Hilbert transform (HT) on IMFs to extract the instantaneous frequencies. For example, five frequency range-based bands of the vibration signal have been shown in the following figure, which are separated from the vibration signal decomposed by EMD with the help of instantaneous frequency, which is extracted from every IMF using HT that determines a single frequency value for every sample.

Using the mentioned method, multiple rhythms of the vibration signals can be extracted, which are helpful for various aspects.

A few popular methods to perform time-frequency analysis on vibration signals are listed below:

1. Short-time Fourier Transform (STFT)
2. Wavelet Transform (Continuous/Discrete WT )
3. Stockwell-Transform (ST)
4. Wigner-Ville distribution (WVD)
5. Smooth Pseudo Wigner-Ville distribution (SPWVD)

We will try to understand their time-frequency representations with a very simple example of a vibration signal (containing low and high frequencies) which is shown in the figure below.

Now, let’s see the time-frequency representation (TFR) of the given signal using various methods.

Here, the purpose of a transform is to present discriminative time and frequency features visually; thus, only the magnitudes are shown. Through this blog, one can understand how the spectrum computed from various methods discriminatively presents a vibration signal. Selection of the technique depends on the signal properties and/or the further use or subsequent processing of the results. ​The important aspects are desired resolution in time and frequency as well as the tolerability of artifacts. For example, if we want to detect the cross-term artifacts, we should consider WVD while ignoring that artifact SPWVD is a better option.

Another example for the selection of method could be the figure above (STFT with short and long window) contains the spectrogram by short and long windows, where we see the high and low resolution of the frequency, respectively. Depending upon the application, we can select the window size. However, with the increase of a high-resolution frequency, the time resolution decreases. At the same time, we can not have high resolution in both, but the window size allows us to trade-off between time and frequency resolution, and one can optimize the window size according to the requirement.

There are a few following points regarding the methods that have been included in this blog, which might be helpful to select an appropriate method as per the requirements.

• STFT – Easy to interpret; Fast implementations using fast Fourier transform; But the limited and fixed resolution
• WT – Resolution not fixed; Resolution depends on the frequency (multi-scale property); Generally, lower frequency components have more satisfactory frequency resolution and coarser time resolution and reverse for higher frequencies
• ST – Tends to emphasize higher frequency content.
• WVD – Overcomes the limited resolution (Calculates the frequency content for each time step), Strong artifacts, not fast implementation

A schematics of the resolutions of the transforms, including the time domain representation and the standard Fourier spectrum for comparison, is given below.

This blog presents a brief introduction to vibration signals and their important features. Further in the blog, we discuss the stationary and non-stationary properties of a time series. We also discuss a few popular time-frequency analysis techniques which are used for analyzing vibration signals. At the end of the blog, we talk about the pros cons of the techniques and how to opt for a technique for vibration signal analysis.

References

1. Puente Guillen, P., 2016. Predicting Sleepiness from Driving Behaviour (Doctoral dissertation, University of Leeds).
2. Rai, K., Bajaj, V. and Kumar, A., 2015. Features extraction for classification of focal and non-focal EEG signals. In Information Science and Applications (pp. 599–605). Springer, Berlin, Heidelberg.
3. Bajaj, V., Rai, K., Kumar, A., Sharma, D. and Singh, G.K., 2017. Rhythm-based features for classification of focal and non-focal EEG signals. IET Signal Processing, 11(6), pp.743–748.
4. Scholl, S., 2021. Fourier, Gabor, Morlet or Wigner: Comparison of Time-Frequency Transforms. arXiv preprint arXiv:2101.06707.