An algorithm for the simulation of faulted bearings in non-stationary conditions

Abstract

In the field of condition monitoring the availability of a real test-bench is not so common. Furthermore, the early validation of a new diagnostic technique on a proper simulated signal is crucial and a fundamental step in order to provide a feedback to the researcher and to increase the chances of getting a positive result in the real case. In this context, the aim of this paper is to detail a step-by-step analytical model of faulted bearing that the reader could freely and immediately use to simulate different faults and different operating conditions. The vision of the project is a set of tools accepted by the community of researchers on condition monitoring, for the preliminary validation of new diagnostics techniques. The tool proposed in this paper is focused on ball bearing, and it is based on the well-known model published by Antoni in 2007. The features available are the following: selection of the location of the fault, stage of the fault, cyclostationarity of the signal, random contributions, deterministic contributions, effects of resonances in the machine and working conditions (stationary and non-stationary). The script is provided for the open-source Octave environment. The output signal is finally analysed to prove the expected features.

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Abbreviations

B(t):

Function which takes into account the purely cyclostationary content

\(COV\{ \cdot \}\) :

Covariance

D :

Pitch circle diameter

\(E\{\cdot \}\) :

Expectation operator

F :

Amplitude of the force exciting the SDOF system

L :

Vector length

SNR :

Signal-to-noise ratio

\(P_{noise}\) :

Noise poser

\(P_{signal}\) :

Signal power without noise

T :

Inter-arrival time between two consecutive impacts

d :

Bearing roller diameter

\(f_c\) :

Carrier component of the rotation frequency

\(f_d\) :

Frequency deviation of the rotation frequency

\(f_m\) :

Frequency modulation of the rotation frequency

\(f_r(\theta )\) :

Angular dependent rotation frequency

\(f_s\) :

Sample frequency

h(t):

Impulse response to a single impact measured by the sensor

k :

SDOF system stiffness

l :

Vector index

m :

SDOF system mass

p(t):

Function which takes into account periodic component

\(p_{rot}(\theta )\) :

Deterministic part related to the rotation speed in the angular domain

\(p_{stiff}(\theta )\) :

Deterministic part related to the stiffness variation in the angular domain

q(t):

Function which takes into account load distribution, bearing unbalance and periodic changes in the impulse response

\(q_{rot}\) :

Positive number which weight the amplitude of \(p_{rot}(\theta )\)

\(q_{stiff}\) :

Positive number which weights the amplitude of \(p_{stiff}(\theta )\)

\(q_{Fault}\) :

Positive number governing the amplitude of the modulating function related to distributed fault

n(t):

Background noise

\(n_r\) :

Number of rolling elements

x(t):

Simulated vibration signal

\(x_{SDOF}(t)\) :

Time response of a SDOF system to unit impulse

\(\beta\) :

Contact angle

\(\delta\) :

Kronecker’s symbol

\(\Delta \theta _{imp}\) :

Angular position of a series of equispaced impulses

\(\Delta T_i\) :

ith inter-arrival time

\(\Delta \theta _i\) :

ith angle between two consecutive impulses

\(\varepsilon\) :

Error term

\(\omega _n\) :

Natural frequency of the SDOF system

\(\omega _d\) :

Damped natural frequency of the SDOF system

\(\sigma ^2\) :

Standard deviation

\(\tau _i\) :

Inter-arrival time jitters of the ith impact

\(\tau _{stiff}\) :

Geometrical bearing parameter related to the stiffness variation

\(\tau _{Fault}\) :

Geometrical bearing parameter related to the fault

\(\theta\) :

Angular variable

\(\zeta\) :

Damping coefficient of the SDOF system

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Acknowledgements

Acknowledgement is made for the measurements used in this work provided through data-acoustics.com Database. In particular, the authors thank Prof. Gareth Forbes at Department of Mechanical Engineering of Curtin University (Australia), who provided the experimental data through Creative Commons Attribution 4.0 International License.

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Correspondence to Marco Cocconcelli.

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The authors declare that they have no conflict of interest.

Appendix: Octave code

Appendix: Octave code

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D’Elia, G., Cocconcelli, M. & Mucchi, E. An algorithm for the simulation of faulted bearings in non-stationary conditions. Meccanica 53, 1147–1166 (2018). https://doi.org/10.1007/s11012-017-0767-1

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Keywords

  • Ball bearings
  • Simulation
  • Algorithm