Preserving physics of vibration data using post-Nyquist data-driven techniques
Imparting physics in end-to-end data-driven learning
“If you want to find the secrets of the universe, think in terms of energy, frequency, and vibration.”
― Nikola Tesla
Vibration is one of the most common sensor modalities used for measuring a variety of real-world phenomena. It is great at capturing the ‘signature’ of underlying phenomena of interest. For example:
(i) testing of cars / satellites / rockets pre-production
(ii) vibration isolation for precision machines for making semiconductors
(iii) bearing faults or shaft unbalance in a rotating machine like engine / motor / turbine
(iv) testing vibration levels in car interiors and testing for modal frequencies
The sampling frequency of vibration data depends on the application, and ranges from a few Hertz to several Kilohertz (KHz) depending on what is being monitored. For example, 100 Hz might be good enough for human activity recognition in wearables, but we may need to sample at 48 KHz for studying bearing faults, given the higher-order non-linear dynamics in play.
Conventionally, vibration data has been analyzed in time and frequency domains depending on the application. Time-domain features such as mean, max, min, kurtosis, skewness, etc. are sufficient in some cases. However, in others involving rotational dynamics, periodicity, or recurrence, frequency domain analysis is used. Fourier analysis has been the backbone of scientific computing for 50+ years. The periodic structure in such vibration data is efficiently modelled using trigonometric functions and computed in the form of Fourier co-efficients, Wavelet transforms, spectrograms, etc.
Let us look at 3 examples showing the need of frequency domain analysis for high-frequency industrial monitoring applications. These measurements were made using a tri-axial Piezoelectric accelerometer (fs = 4096 Hz) for a machinery shaft unbalance detection use-case.
Piezo 25 Hz
The following figure shows a time domain and frequency domain plot for shaft balance and unbalance, measured at a rotational speed of 25 Hz. Note that in the time domain plot, the mean vibration of the test rig is higher for the shaft balance condition. In the Fourier domain also, the overall energy of the system is higher in the balanced shaft case. However, the Fourier domain indicates a higher amplitude at 25 Hz for the unbalanced shaft, which is the signature (1x rotational frequency) for shaft unbalance. In the FFTs, we can see that the unbalance shaft leads to a higher peak at 25 Hz.
So, this is an example wherein a type of fault / signature can only be seen in the frequency domain.
Piezo 17 Hz
For a rotational speed of 17 Hz, the overall vibration energy is higher for the unbalanced shaft case, as seen in both time and frequency domains. However, as seen in the previous example, this was not the case. Or in other words, the frequency domain always maps to the fault correctly, whereas time domain features necessarily do not.
MEMS 17 Hz
In this case, we used a tri-axial MEMS accelerometer to observe the same fault at a rotational speed of 17 Hz. In the time domain, the mean line is slightly higher for the balanced shaft case, whereas in the Frequency domain, the amplitude at 17 Hz is higher for the shaft unbalance case, again showing the superiority of the frequency domain for identifying fault signatures.
Both time and frequency domain analyses are based on the Shannon-Nyquist sampling theorem, which governs the data collection rate. As we move towards ubiquitous sensing and intelligence in the physical world, the amount of sensors and data generation is skyrocketing. Understanding the fact that there is a lot of structure and redundancy in sensor data, are there ways to exploit this to develop efficient algorithms? Let us approach this via three questions:
1. How do conventional techniques preserve sensor data physics?
2. What physics do we need to preserve?
3. What minimum data can we collect while preserving physics?
Let us address each of these questions one-by-one:
1. How do conventional techniques preserve sensor data physics?
By preserving the time-frequency information at multiple scales.
This involves deciding the window length depending on the sampling frequency and stationarity observed in the data. Techniques such as Fourier transforms, spectrograms, and wavelets are used for multi-scale time-frequency decomposition problems.
2. What physics do we need to preserve?
We need to preserve the information carried in each time window independently as we consider a window length which is appropriate to represent the stationarity of a signal.
Moreover, we need to make sure the techniques are robust to frequency perturbances, meaning that they are not sensitive to low nor high frequencies.
The way to observe these metrics might be different in data-driven approaches as we don’t have closed form solutions. Rather, we may have to depend on things like class activation maps, spectral decomposition, sensitivity heatmaps, etc. to understand the time-frequency information captured by the data-driven models.
For example, the following spectral decomposition shows that vision transformers are more robust than CNNs to spectral disturbances. The top-left/bottom-right quadrants denote low-frequency/high-frequency regions. As can be seen, low frequency perturbance can disrupt CNN representations, whereas ViTs are not selectively sensitive to low nor high frequency disturbances.
As can be seen in the following sensitivity heatmap, the CNN loses its consistency in the high frequency regions, whereas ViT are not selectively sensitive to low or high frequencies.
3. What minimum data can we collect while preserving physics?
We can sample according to the information present in a signal, or based on a generalization of the Shannon-Nyquist sampling theorem. Compressed sensing introduced in 2006 presents some foundational theory on the amount of data to be collected based on the information content in a signal. The undersampled data can be used to reconstruct the raw data, meaning that the time-frequency information is still preserved while undersampling. More details are available here and here. Compressed sensing however is computationally expensive, takes a lot of processing time, and works signals with inherent sparsity.
Now that we know what physics-based properties we need to preserve in sensor data, we can develop purely data-driven techniques that work with undersampling. We need these techniques to evaluate each window independently and be robust against low or high frequency disturbances.
In summary, we can see that preserving the physics of sensor data involves preserving multi-scale time-frequency information. There are ways to preserve these physics-based properties even when working undersampled data, which offer significant benefits in terms of the compute, power, storage, transmission, and latency required to process raw sensor data. You can learn more how this is being used today here.