Deterministic-random separation in nonstationary regime

TLDR: In this article, a generalized synchronous average (GSA) was proposed to extract the deterministic part of a cyclo-non-stationary vibration signal, i.e. the analog of the periodic part of cyclostationary signals.

About: This article is published in Journal of Sound and Vibration.The article was published on 2016-02-03 and is currently open access. It has received 50 citations till now. The article focuses on the topics: Cyclostationary process & Vibration.

Figures

Fig. 5. Academic example illustrating the limitation of the SA when applied to a single dof system excited by a linear chirp. (a) Reference instantaneous frequency, (b) constant amplitude linear chirp synchronous with the 15th order of the reference instantaneous frequency, (c) amplitude and (d) phase spectrum of the single dof linear system, (e) angular resampled response, (f) response order spectrum, (g) SA, (h) SA order spectrum, (i) residual signal, and (j) residue order spectrum. Note that the SA energy comprises only 13.94% of the response energy.

Fig. 10. (a) Acceleration signals acquired at operating speeds (a) 650 rev/min, (b) 950 rev/min, (c) 1250 rev/min, (d) 1550 rev/min and (e) 1850 rev/min.

Fig. 19. SES of the (a) resampled signal, (b) the SA residue and (c) the GSA residue applied on signal Acc2 in runup.

Fig. 2. Trajectory of a CNS1 signal conditioned to a given speed profile ω(θ). Each horizontal slice in this field associated with a fixed speed value ω represents a CS process with specific statistics. A CNS process is then a trajectory in this field defined by the speed profile.

Table 1 Characteristic of the rolling element bearing (Deep groove ball 6820).

Fig. 9. The test rig located at CETIM.

Frequently asked questions

Q1. What are the contributions in "Deterministic-random separation in nonstationary regime" ?

A first object of this paper is to investigate the nature of the nonstationarity induced by the response of a linear time-invariant system subjected to speed varying excitation. For this purpose, the concept of a cyclo-non-stationary signal is introduced, which extends the class of cyclostationary signals to speed-varying regimes. A brief statistical study of it is performed, aiming to provide the user with confidence intervals that reflect the `` quality '' of the estimator according to the SNR and the estimated speed. 

Q2. How many samples are enlarged to obtain a SNR142?

a white Gaussian noise is added on the response to obtain a SNR¼2 and the signal duration is enlarged 10 times (i.e. one million samples) while keeping the same speed variation. 

Q3. What is the efficient indicator for the assessment of CS2 sources in rotating machine?

In vibration analysis of rotating machine, the squared envelope spectrum (SES) is one of the most efficient indicators for the assessment of CS2 sources which are typical symptoms of damage in rolling element bearing faults [35]. 

Q4. What is the reason for the high error in the second?

According to Eq. (20), the high error in the former is due to the low number of cycles associated with the regime, while the high error in the second is due to the high mean instantaneous power. 

Q5. What is the flow-chart of this method?

The flow-chart of this method is provided in Fig. 4.Since all signals are finite-length in practice, the asymptotic conditions in the GSA definition (see Eq. (15) cannot be met, resulting in a bias and variance of the raw estimator. 

Q6. What is the Fourier coefficient of the response at a given instant?

In this case, the Fourier coefficients of the response at a given instant are principally dependent on the operating speed at that instant, as well as past and future instances of the speed profile. 

Q7. What is the KDE method for smoothing the GSA?

In this context, the kernel density estimation (KDE) method—also termed the Parzen–Rosenblatt window method—provides a non-parametric solution for an efficient smoothing operation [29,30]. 

Q8. What are the two parameters that govern the bias of the estimator?

there are two parameters that govern the bias, namely the speed resolution, δω, and the second derivative of the actual GSA with respect to speed. 

Q9. How is the effect of the transmission path in rotating machines modeled?

A natural way to model the effect of the transmission path in rotating machines is by exciting a single degree-of-freedom (dof) system by a constant-amplitude chirp. 

Q10. What is the speed profile of a finite-length CNS signal?

In details, the speed profile is divided into a predefined set of speed intervals called regimes defined by their central frequency ωr and the speed resolution δω. 

Q11. How can the GSA be relaxed in the general case?

In particular, it has been assumed that the speed variations are slower than the signal cycle; although this condition seems reasonable in several applications, it can easily be relaxed in the general case by averaging portions of signal smaller than cycles. 

Q12. What is the heaviside function used to enforce the system causality?

with A¼ 10; ξ¼ 0:05; ωn ¼ 2π 200, ωd ¼ωnffiffiffiffiffiffiffiffiffiffiffiffi1 ξ2q, and ~U tð Þ the Heaviside function used to enforce the system causality.