Composite test inclusive of Benford’s law, noise reduction and 0–1 test for effective detection of chaos in rotor–stator rub

Aman K Srivastava,Akhilendra Singh,Mayank Tiwari

doi:10.1007/s11071-021-06844-y

Abstract

Segregating noise from chaos in dynamic systems has been one of the challenging work for the researchers across the globe due to their seemingly similar statistical properties. Even the most used tools such 0-1 test and Lyapunov exponents fail to distinguish chaos when signal is mixed with noise. This paper addresses the issue of segregating the dynamics in a rotor-stator rub system when the vibrations are subjected to different levels of noise. First, the limitation of 0-1 test in segregating chaos from signal mixed with noise has been established. Second, the underexplored Benfords Law and its application to the vibratory dynamical rotor-stator rub system has been introduced for the first time. Using the Benfords Law Compliance Test (BLCT), successful segregation of not only noise from chaos but also very low Signal to Noise Ratio (SNR) signals which are mainly stochastic has been achieved. The Euclidean Distance concept has been used to explore the scale-invariant probability distribution of systems that comply with Benfords Law to separate chaos from noise. Moreover, for moderate bands of noise in signals, we have shown that the Schreibers Nonlinear Noise Reduction technique works effectively in reducing the noise without damaging the dynamic properties of the system. Combining these individual layers (0-1 Test, BLCT and Noise reduction) on a rotor system, a Decision Tree based method to effectively segregate noise from chaos and identify the correct dynamics of any system with time series data set has been proposed.

Highlights

Over the years, the researchers have tried to understand the nonlinear phenomena in rotating machinery which are responsible for its failure
We have explored the possibility of identifying the correct dynamics, namely regular and chaotic dynamics in a rotor-stator dynamical system for all bands of Signal to Noise Ratio (SNR)
The different types of vibrations such as periodic, quasi-periodic and chaotic Vibration Data will be fed to the 0-1 test and the output will be observed for ‘0’ or ‘1’ value to differentiate regular dynamics from chaotic dynamics

Summary

Introduction

The researchers have tried to understand the nonlinear phenomena in rotating machinery which are responsible for its failure. The presence of chaotic vibrations in rotor-stator rub was studied in [1] through experimental, numerical and analytical methods. With the presence of chaos in almost all kinds of nonlinear rotating systems, it became significantly important for researches to identify tools to detect chaos. It was observed that the presence of noise caused significant deviation in the calculation of Lyapunov exponents [8]. This led to the development of ScaleDependent Lyapunov Exponents (SDLE) to distinguish noise-mixed chaos from chaos for short time series [9]. The calculation of Lyapunov Exponent was not very cost effective computationally and this led to development of statistical approaches to identify chaos. It is difficult to segregate pure noise from chaos due to their seemingly similarity and effort to reduce noise distorts the dynamics data and makes chaos undetectable

Methods

Results

Conclusion