Deterministic-random separation in nonstationary regime

TL;DR: 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.

Summary

1. Introduction

• [2] The Eastern Alps result from the consumption of the small Penninic Ocean and the subsequent collision of the European plate with the Adriatic plate, an off-split from Africa.
• The lack of seismic reflection data in the Eastern Alps was the motivation for TRANSALP which is a European multidisciplinary research programme for investigating orogenic processes by partner institutions from Italy, Austria and Germany [TRANSALP Working Group, 2001].
• Seismic profiling forms the core of the study, which is accompanied by a number of specific geoscientific projects.
• The project gathered for the first time a continuous, approx.

2. Data Acquisition and Processing

• Seismic reflection data were acquired using the Vibroseis and the explosion technique during three field campaigns between autumn 1998 and autumn 1999 and a small complementary explosion experiment in summer 2001.
• The table lists the most important acquisition parameters.
• The seismic reflection profiling was accompanied by passive cross-line recording (7 cross lines of about 20 km length each) for 3-D control.
• The velocity model required for depth migration was obtained by analysis of the stacking velocities in the layered structure of the Molasse areas and by tomographic inversion of the first arrivals of Vibroseis and explosive data recorded in the wide-aperture stationary network (offsets up to 80 km resulted in depth penetration to about 15 km) in the Alpine areas.

3. The New Seismic Sections and Their Interpretation

• The interpretation of the TRANSALP data is based upon a variety of different versions at appropriate scales (1:50000, 1:100000).
• Both migration techniques were reasonably successfull.
• This indicates that the NCA continue to the South where they are overthrust by the Greywacke Zone (GWZ), their former basement.
• From the north to the central part of the section the base of the reflective lower crust can be followed with a slight dip from 30 km depth in the north to 55–60 km depth beneath and south of the Tauern Window.
• This thick (or double) lower crust, confirmed further south by high velocities determined by previous seismic refraction profiling [Scarascia and Cassinis, 1997], might be of pre-Alpine (Hercynian or older) age due to crustal stacking [Castellarin and Vai, 1981].

4. Two Alternative Models

• Figure 3 shows two alternative models which are presently in discussion.
• A common feature of both models is a major crustal shear zone acting as a ramp, dipping from the Inn valley southward (‘Sub-Tauern Ramp’, Figure 3) over 80 km into the mountain root.
• According to this model, the edge of the initially cooler Tauern Window crust, supposed to represent the former European passive plate boundary, wedges deep ( 50 km) into the approaching Adriatic plate splitting-up its upper and lower crust at the level of the brittle/ductile transition and thereby displacing the Dolomite block to the South.
• In contrast to model A, it presumes that the steep northward dip of the PF observed near the surface continues to greater depth.
• The authors thank two anonymous reviewers for their helpful constructive comments.

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.

Fig. 16. (a) GSA of signal Acc1 in runup speed regime estimated with 20 non-overlapping intervals, (b) its amplitude order spectrum and (c) its zoom around the 1st meshing order. (d) GSA estimate of signal Acc2 in random speed regime estimated with 20 non-overlapping intervals, (e) its amplitude order spectrum and (f) its zoom around the 1st meshing order.

Fig. 15. Classical SA (a) and GSA applied on signal Acc1 in runup speed profile for the speeds (b) 650 rev/min (c), 950 rev/min (d), 1250 rev/min (e), 1550 rev/min and (f) 1850 rev/min. The same unit scale is used to evidence the differences.

Fig. 8. Percentage evolution of the GSA energy (with respect to the energy of the actual response) with the number of regimes for 10 s time record.

Fig. 7. (a) Order spectrum of the actual response and GSA estimates obtained with different speed resolutions and (b) percentage evolution of the GSA energy (with respect to the energy of the actual response) with the numbers of regimes for 10 s time record.

Fig. 3. Flow-chart of the GSA raw estimator.

Fig. 17. Amplitude order spectrum of the SA and GSA residues of Acc2 in runup speed regime, close-ups around (a) the 1st meshing harmonic, (b) the 2nd meshing harmonic and (c) the 3rd meshing harmonic. Amplitude order spectrum of the SA and GSA residues of Acc1 signal in random speed-varying regime, close-ups around (d) the 1st meshing harmonic, (e) the 2nd meshing harmonic and (f) the 3rd meshing harmonic.

Fig. 14. Comparison between the GSA of signal Acc1 acquired at runup and the GSA of the same accelerometer acquired at random variable speed: (a) 650 rev/min, (b) 950 rev/min, (c) 1250 rev/min, (d) 1550 rev/min, (e) 1850 rev/min; (f)–(j) are the close-ups between 0 and 0.3 rev of plots (a)– (e) respectively.

Fig. 13. Comparison between the SA of signal Acc1 acquired at constant speeds and the GSA of the same accelerometer acquired in random variable speed: (a) 650 rev/min, (b) 950 rev/min, (c) 1250 rev/min, (d) 1550 rev/min, (e) 1850 rev/min; (f)–(j) are the absolute differences of the plots in (a)–(e) respectively and their corresponding threshold.

Fig. 6. (a) Actual deterministic response and GSA estimate together with their (b) order spectra. The estimation comprises 92.43% of the energy of the actual response.

Fig. 4. Flow-chart of the proposed method.

Fig. 18. Comparison of the SES of residue, the SES of SA residue and the SES of the GSA residue.

Fig. 1. Transmission path effect on cyclostationary excitations.

Fig. 11. (a) Runup speed profile and (b) its corresponding acceleration signals (Acc1 and Acc2). (c) Random speed profile and (d) its corresponding acceleration signals (Acc1 and Acc2).

Fig. 12. Comparison between the SA of Acc1 signal acquired at constant speeds and the GSA of the same accelerometer acquired in runup: (a) 650 rev/min, (b) 950 rev/min, (c) 1250 rev/min, (d) 1550 rev/min, (e) 1850 rev/min; (f)–(j) are the absolute differences of the plots in (a)–(e) respectively and their corresponding threshold.

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.