The basic signal unit representing events in machining condition
Minimum Sufficient Unit
Jiduo Zhang1*, Robert Heinemann1, Otto Jan Bakker1, Siqi Li2, Xiaoyu Xiao3, Yixian Ding4
1 Department of Mechanical and Aerospace Engineering, The University of Manchester
2 Department of Mathematics, The University of Manchester
3 Department of Electrical and Electronic Engineering, The University of Manchester
4 School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University
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Signal in frequency domain
The detrend signal is transformed into frequency domain using Fast Fourier Transform (FFT) algorithm.
Harmonic
Strike by the tool engagement frequency, the signal exhibits a significant harmonic structures, as shown in the peaks at 67, 133, 267 Hz.
Attenuation
Due to damping property of mechanical structure, the amplitude of harmonics decays notably along with the multiples of frequency.
Consistent
The frequency components in thrust and torque is relatively stable, thereby enabling a robust recognition despite the condition change.
4x
Up to four times the harmonics components are highly informative in the frequency domain.
Minimum Sufficient Signal Condition
Boundary condition derived from harmonic structure according to Shannon-Nyquist theorem
Frequency
Anti-aliasing To protect the informative information from aliasing artifacts, according to Shannon-Nyquist sampling theorem, the sampling frequency should surpass twice the frequency of informative components in signal.
In the following charts, when the sampling frequency is below 133 Hz (twice the harmonics frequency, 67 Hz), because of the severe aliasing artifacts, nearly all harmonics features are lost.
Duration
Bin resolution To protect the informative information from cross-talking in frequency spectrum, it is neccessary to keep the harmonic components separated and distinguishable.
Time domain To cover the harmonic components of the lowest frequency, the sampling duration must exceed its period (0.015 s in accordance to 67 Hz).
Frequency domain In the following charts, when the sampling duration is below 0.030 s (twice reciprocal of harmonics frequency, 67 Hz), the bin resolution could not satisfy the recognition requirements to distinguish harmonic peaks according to Rayleigh Criterion. Even in this condition, since only one point lies between the harmonic peaks, the peaks structure is less significant and may impede model from recognising the events.
Phase
Stationary In stationary signal, as long as at least one period is covered, the differences in phase have no influence on the appearance of signal.
Conclusion Phase will not have an impact on signal's representation of and deep learning model recognise the events.
Summary
- In frequency domain, both thrust and torque signal exhibit decaying harmonic structures with fundamental frequency of spindle rate in drilling process while the spectrum of y-axis acceleration and AE are less structural.
- The first four order harmonics and mean value of signal play a dominant role in signal’s representation and model’s classification for process incidence.
- To prevent harmonic components from aliasing, the sampling frequency should surpass the eight times the spindle rate.
- To resolve bin frequency to prevent harmonic components from being indistinguishable, the sample duration should be longer than twice the reciprocal of spindle rate, which is the minimum sufficient duration for signal.
Model Response
The classification accuracy performance of ResNet in identifying process incidence under different combination of sampling condition.
Saturated
With the increase in sample frequency, the classification accuracy firstly increases and gets saturated once it exceeds the threshold, called saturated frequency, despite the change in sample duration.
CEA theorem
Because of the same native accuracy, the accuracy response regarding sample duration when it exceeding the MSD follows the continuous equivalent accuracy equation.
Homologous
Any consecutive signal segment fulfilling the minimum sufficient condition for frequency and duration, also the minimum sufficient unit, is homologous to represent process incidence with the same native accuracy regardless of initial phase and location in the signal.
Additional Conclusions
- To ensure sample diversity in the training process, the frequency of the master from which training data is generated should surpass twice the sample frequency.
- To train deep learning model to identify process incidence with lossless accuracy, the sampling frequency in the signal acquisition procedure should exceed sixteen times the spindle rate.
Jiduo would like to acknowledge the continuing contribution to this research by China scholarship council.
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