Understanding importance of positive and negative signs of optimized weights used in the sum of weighted normalized Fourier spectrum/envelope spectrum for machine condition monitoring

Machine condition monitoring is an emerging research domain to use monitoring data to monitor machine conditions and prevent unexpected machine failures. In our previous study, the sum of weighted normalized square envelope was proposed as a generalized framework of some wellknown sparsity measures including kurtosis, negative entropy, smoothness index, and Gini index. This framework revealed that a main difference among these sparsity measures is that they use different weights. Consequently, the design of new weights can generate new sparsity measures. Our previous study also showed that a convex-optimization problem could be formulated to automatically design weights by a data-driven way. One attractive experimental finding was that solving the sum of weighted normalized Fourier spectrum/envelope spectrum results in informative frequency bands/fault characteristic frequencies. However, this finding was only experimentally observed based on positive optimized weights and omitted the importance of negative optimized weights. In this short communication, we revisit this work and provide insightful investigations for signs of optimized weights and mathematically prove that both positive and negative optimized weights are extremely important to distinguish fundamental frequency components and fault-generated frequency components. Three new propositions are proposed to show the ability of optimized weights for determining informative frequency bands/fault characteristic frequencies. Experimental results are provided to verify the effectiveness of our new propositions given in this short communication. The significance of this short communication is that it is helpful for engineers and scholars to quickly and effectively identify all vibration contributions from fundamental frequency components and fault-generated frequency components using positive and negative signs of optimized weights in our previous framework. Moreover, fully interpretable weights would be helpful for designing physics-informed machine learning methods for machine condition monitoring.

Automated summary

This article discusses methods for machine condition monitoring using monitoring data to prevent machine failures. The research found that different weights can generate different sparsity measures, and optimized weights can determine informative frequency bands/fault characteristic frequencies. The effectiveness of these methods is verified through experimental results.