Parametric time-domain methods for non-stationary random vibration modelling and analysis — A critical survey and comparison

doi:10.1016/j.ymssp.2005.10.003

Mechanical Systems and Signal Processing

Volume 20, Issue 4, May 2006, Pages 763-816

https://doi.org/10.1016/j.ymssp.2005.10.003 Get rights and content

Abstract

A critical survey and comparison of parametric time-domain methods for non-stationary random vibration modelling and analysis based upon a single vibration signal realization is presented. The considered methods are based upon time-dependent autoregressive moving average (TARMA) representations, and may be classified as unstructured parameter evolution, stochastic parameter evolution, and deterministic parameter evolution. The main methods within each class are presented, and model “structure” selection is discussed. The methods are compared, via a Monte Carlo study, in terms of achievable model parsimony, prediction accuracy, power spectral density and modal parameter accuracy and tracking, computational simplicity, and ease of use. Comparisons with basic non-parametric methods are also made. The results of the study confirm the advantages and high performance characteristics of parametric methods. They also confirm the increased accuracy and performance characteristics of the deterministic, as well as stochastic, parameter evolution methods over those of their unstructured parameter evolution counterparts.

Section snippets

Acronyms

AIC	Akaike information criterion	PE	prediction error (method)
AR	autoregressive	RELS	recursive extended least squares (method)
ARMA	autoregressive moving average	RML	recursive maximum likelihood (method)
BIC	Bayesian information criterion	RML-TARMA	RML-estimated TARMA (model)
FS-TAR	functional series TAR (model)	RSS	residual sum of squares
FS-TARMA	functional series TARMA (model)	SP-TARMA	smoothness priors TARMA (model)
KF	Kalman filter	ST-ARMA	short-time ARMA
NID	normally independently distributed	STFT	short-time Fourier

Important conventions and symbols

A functional argument in parentheses designates function of a real variable; for instance $x (t)$ is a function of analog time $t \in R$ .

A functional argument in brackets designates function of an integer variable; for instance $x [t]$ is a function of normalized discrete time $(t = 1, 2, \dots)$ . The conversion from discrete normalized time to analog time is based upon $(t - 1) T_{s}$ , with $T_{s}$ standing for the sampling period.

A time instant used as superscript to a function indicates the set of values of the function up to

Non-stationary signal representations

Non-stationary signal representations may be of the parameterized or non-parameterized type. Parameterized representations attempt to model a non-stationary signal via a description that may be specified via a more or less limited number of parameters. As already indicated, they are based upon conceptual extensions of the ARMA representations of the stationary case [17], [18], and, being in the focus of this paper, are presented in detail in the next subsection.

Non-parameterized representations

Non-stationary TARMA model identification

Given a single, N-sample long, non-stationary signal record (realization) $x^{N} ≜ {x [1] \dots x [N]}$ and a selected representation class (unstructured, stochastic, or deterministic parameter evolution), the TARMA identification problem may be stated as the problem of selecting the corresponding model “structure”, the model AR and MA parameters $a_{i} [t]$ and $c_{i} [t]$ , respectively, and the innovations variance $σ_{e}^{2} [t]$ that “best” fit the available measurements. Model “fitness” may be understood in various ways, a

Model-based analysis

Once a TARMA representation has been obtained, model-based analysis may be performed. This includes the computation of non-parameterized representations (such as the model's impulse response function, autocovariance function, appropriate time-frequency distributions) which are now obtained based upon the TARMA representation, as well as “frozen” modal quantities.

Application of the methods to non-stationary vibration modelling and analysis

The parametric modelling methods of Section 3 are presently applied to the problem of modelling and analysis of a non-stationary random vibration signal. Their various performance characteristics, including the attained accuracy, are studied via Monte Carlo experiments. Comparisons with the non-parametric spectrogram and Wigner–Ville distribution methods are also made.

Concluding remarks

A critical overview and comparison of parametric methods for non-stationary random vibration modelling and analysis, based upon a single vibration signal realization, was presented. The methods were all based upon time-dependent ARMA representations, which, according to the form of time-dependence, were classified as unstructured parameter evolution, stochastic parameter evolution and deterministic parameter evolution. As implied by its name, the first class imposes no “structure” on the signal

Acknowledgements

The authors wish to acknowledge the partial financial support of this study by the VolkswagenStiftung (Grant No. I/76938). They also wish to thank Professor S.G. Braun of Technion (Israel) for his encouragement on this undertaking, and one anonymous referee for his constructive comments that led to improvements in the manuscript.

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^☆: Research supported by the VolkswagenStiftung — Grant no I/76938.

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Invited SurveyParametric time-domain methods for non-stationary random vibration modelling and analysis — A critical survey and comparison☆

Abstract

Section snippets

Acronyms

Important conventions and symbols

Non-stationary signal representations

Non-stationary TARMA model identification

Model-based analysis

Application of the methods to non-stationary vibration modelling and analysis

Concluding remarks

Acknowledgements

Journal of Sound and Vibration

Mechanical Systems and Signal Processing

Journal of Sound and Vibration

Mechanical Systems and Signal Processing

Signal Processing

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Signal Processing

Journal of Sound and Vibration

Engineering Structures

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Probabilistic Engineering Mechanics

Journal of Sound and Vibration

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing

Journal of Sound and Vibration

Signal Processing

Signal Processing

Mechanical Systems and Signal Processing

Linear Algebra and its Applications

Non-linear and Non-stationary Time Series Analysis

Random Vibration and Statistical Linearization

Random Data: Analysis and Measurement Procedures

Random Vibration and Spectral and Wavelet Analysis

Random Vibration and Spectral Analysis

Smoothness Priors Analysis of Time Series

Non-stationary vibration modelling and analysis via functional series TARMA models

Identification and model updating of a non-stationary vibrating structure

Probabilistic Methods in Structural Engineering

System Identification: Theory for the User

System Identification

Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation

Spectral Analysis and Time Series

Time Series Analysis Forecasting and Control

Vibration-based on-line fault detection in non-stationary structural systems via a statistical model based method

Wavelet analysis of vibration, parts 1 and 2

Journal of Vibration and Acoustics

Identification of nonstationary systems using time-frequency methods

Spectral analysis for non-stationary signals from mechanical measurements: a parametric approach

Mechanical Systems and Signal Processing

Functional series TARMA modelling and simulation of earthquake ground motion

Earthquake Engineering and Structural Dynamics

Non-stationary random vibration modelling and analysis via functional series TARMAX models

Invited Survey
Parametric time-domain methods for non-stationary random vibration modelling and analysis — A critical survey and comparison☆