迭代广义短时Fourier变换在行星齿轮箱故障诊断中的应用

短时Fourier变换(STFT)在分析非平稳信号的过程中,受调制系数的影响,时频分布图中的能量扩散至主导频率的周围,降低了时频分布的可读性.运用STFT分析瞬时频率缓变或恒定的信号时,调制系数的影响较小甚至可以忽略不计,而得到能量聚集程度很高的时频分布.根据这一特点,提出了迭代广义短时Fourier变换(IG-STFT),该方法有效改善了时频图的可读性.首先运用迭代广义解调分离出频率恒定的单分量成分,然后运用STFT分析信号的时频分布,最后依据STFT的分析结果和相位函数得到原信号的时频分布.通过行星齿轮箱仿真信号和实验信号分析,验证了该方法在分析非平稳信号中的有效性,准确诊断了齿轮故障.     

Evolutionary Spectra Estimation Using Wavelets

A wavelets-based method is developed to estimate the evolutionary power spectral density (EPSD) of nonstationary stochastic processes. The method relies on the property that the continuous wavelet transform of a nonstationary process can be treated as a stochastic process with EPSD given in terms of the EPSD of the process in a closed form. This yields an equation in the frequency domain relating the instantaneous mean-square value of the wavelet transform to the EPSD of the process. A number of these equations are considered, each related to a certain scale of the wavelet transform, in conjunction with representing the target EPSD as a sum of time-independent shape functions modulated by time-dependent coefficients; the squared moduli of the Fourier transforms of the wavelets associated with the selected scales are taken as shape functions. This leads to a linear system of equations which is solved to determine the unknown time-dependent coefficients; the same system matrix applies for all time instances. Numerical examples demonstrate the accuracy and computational efficiency of the proposed method.     

Polynomial Chaos Decomposition for the Simulation of Non-Gaussian Nonstationary Stochastic Processes

A method is developed for representing and synthesizing random processes that have been specified by their two-point correlation function and their nonstationary marginal probability density functions. The target process is represented as a polynomial transformation of an appropriate Gaussian process. The target correlation structure is decomposed according to the Karhunen–Loève expansion of the underlying Gaussian process. A sequence of polynomial transformations in this process is then used to match the one-point marginal probability density functions. The method results in a representation of a stochastic process that is particularly well suited for implementation with the spectral stochastic finite element method as well as for general purpose simulation of realizations of these processes.     

Simulation of Homogeneous and Partially Isotropic Random Fields

A rigorous methodology for the simulation of homogeneous and partially isotropic multidimensional random fields is introduced. The property of partial isotropy of the random field is explicitly incorporated in the derivation of the algorithm. This consideration reduces significantly the computational effort associated with the generation of sample functions, as compared with the case when only the homogeneity in the field is taken into account. The approach is based on the spectral representation method, utilizes the fast Fourier transform, and generates simulations with random variability in both their amplitudes and phases, or in their phases only. Spatially variable seismic ground motions experiencing loss of coherence are generated as an example application of the developed approach.     

Simulation of Highly Skewed Non-Gaussian Stochastic Processes

In this paper, a simulation methodology is proposed to generate sample functions of a stationary, non-Gaussian stochastic process with prescribed spectral density function and prescribed marginal probability distribution. The proposed methodology is a modified version of the Yamazaki and Shinozuka iterative algorithm that has certain difficulties matching the prescribed marginal probability distribution. Although these difficulties are usually sufficiently small when simulating non-Gaussian stochastic processes with slightly skewed marginal probability distributions, they become more pronounced for highly skewed probability distributions (especially at the tails of such distributions). Two major modifications are introduced in the original Yamazaki and Shinozuka iterative algorithm to ensure a practically perfect match of the prescribed marginal probability distribution regardless of the skewness of the distribution considered. First, since the underlying “Gaussian” stochastic process from which the desired non-Gaussian process is obtained as a translation process becomes non-Gaussian after the first iteration, the empirical (non-Gaussian) marginal probability distribution of the underlying stochastic process is calculated at each iteration. This empirical non-Gaussian distribution is then used instead of the Gaussian to perform the nonlinear mapping of the underlying stochastic process to the desired non-Gaussian process. This modification ensures that at the end of the iterative scheme every generated non-Gaussian sample function will have the exact prescribed non-Gaussian marginal probability distribution. Second, before the start of the iterative scheme, a procedure named “spectral preconditioning” is carried out to check the compatibility between the prescribed spectral density function and prescribed marginal probability distribution. If these two quantities are found to be incompatible, then the spectral density function can be slightly modified to make it compatible with the prescribed marginal probability distribution. Finally, numerical examples (including a stochastic process with a highly skewed marginal probability distribution) are provided to demonstrate the capabilities of the proposed algorithm.     

Simulation of Multivariate Stationary Gaussian Stochastic Processes: Hybrid Spectral Representation and Proper Orthogonal Decomposition Approach

By observing that the optimal basis for the proper orthogonal decomposition (POD) can be obtained from the cross power spectral density (XPSD) matrix of a multivariate stationary Gaussian stochastic process, the computational efficiency, in both time and memory consumption, of simulations of this process is improved by using a hybrid spectral representation and POD approach with negligible loss of accuracy. This hybrid approach actually simulates another multivariate process with many fewer variables in an optimal subspace obtained by the POD. This approach is straightforward, effective, and does not place any conditions on the XPSD matrices. Furthermore, the error induced by the reduction of variables is predictable and controllable prior to the simulation procedure. The spectral representation method (SRM) is discussed in a heuristic way. In this paper, a specific POD theorem is formally stated, proved, and related to XPSD matrices. A numerical example is given to demonstrate the effectiveness of this hybrid approach. This approach may also have potential applications for simulations of nonstationary non-Gaussian processes.     

Simulation of Stochastic Wind Velocity Field on Long-Span Bridges

An improved algorithm is introduced in this paper for digital simulation of the stochastic wind velocity field on long-span bridges, when the cross-spectral density matrix of the field is given. The target wind velocity field is assumed to be a one-dimensional, multivariate, homogeneous stochastic process. The basic method of simulation used is the spectral representation method. It is improved by explicitly expressing Cholesky's decomposition of the cross-spectral density matrix in the form of algebraic formulas, then cutting off as many as possible of the cosine terms, so long as the accuracy of results is not affected. The fast Fourier transform technique is used to enhance the efficiency of computation. A numerical example of simulation for buffeting analysis is included in this paper to illustrate the improved method introduced. It is demonstrated that deviations between the simulated correlation functions and the target are sufficiently small and that the simulated power spectra are close to the target.     

Conditional Simulation of a Class of Nonstationary Space-Time Random Fields

This paper first addresses the problem of conditionally simulating stationary, space-time, Gaussian random fields. A method developed for quadrant-symmetric, stationary, space-time fields is extended to account for the imaginary part of the complex cross-spectrum. The case of amplitude- and frequency-modulated (AFM) nonstationary space-time fields is studied next. The evolutionary spectral density and cross-correlation structure of such a class of nonstationary random fields are analyzed in terms of the envelope and frequency modulation functions. A method for the conditional simulation of AFM space-time fields is advanced. The AFM nonstationary fields are mapped to a domain where the conditional simulation is performed as for stationary, space-time fields. Examples are given to illustrate the use and capabilities of the method including applications to the simulation of earthquake ground motions.     

Error Assessment for Spectral Representation Method in Wind Velocity Field Simulation

Stochastic wind velocity fields are usually simulated as N-variate stationary Gaussian processes by using the spectral representation method (SRM). However, the temporal statistics estimated from one SRM-simulated sample wind process cannot coincide with the target characteristics; the disagreements can be described by the bias and stochastic errors. For controlling the errors efficiently, this paper assesses the errors produced by both the Cholesky decomposition-based SRM (CSRM) and the eigendecomposition-based SRM (ESRM). The SRM is revisited first, followed by computing the temporal mean value, correlation function, power spectral density (PSD), and standard deviation of the SRM-simulated wind process. It is shown that the temporal correlation function and standard deviation are Gaussian, while the temporal PSD is non-Gaussian. Further, as mathematical expectations and standard deviations of the corresponding temporal estimations, the bias errors and stochastic errors of all the first- and second-order statistics are obtained in closed form for both the CSRM and the ESRM; the closed-form solutions are then verified in the numerical example. More importantly, this example is employed for taking a clear look at and making a comparison between the stochastic errors produced by the CSRM and by the ESRM; observations suggest that (1) in sum, the ESRM produces smaller stochastic errors than the CSRM and (2) if the ESRM is employed, stochastic errors will be distributed to each component of the wind process in a more uniform pattern. Moreover, some practical approaches are proposed to control the stochastic errors.     

Nonstationary Random Critical Excitation for Acceleration Response

The critical excitation method is promising as a robust method for accounting for inherent uncertainties in predicting forthcoming earthquake events and for constructing design earthquake ground motions in a reasonable way. Most of the proposed theories are based on deterministic approaches and deal with displacement responses. A stochastic acceleration response index is treated here as the objective function to be maximized. The power (area of power spectral density function) and the intensity (magnitude of power spectral density function) are fixed and the critical excitation is found under these restrictions. It is shown that the original idea for stationary random inputs can be utilized effectively in the procedure for finding a critical excitation for nonstationary acceleration responses of nonproportionally damped structural systems. Several numerical examples are presented to demonstrate the characteristics of generalized time-varying frequency response functions for models with various stiffness and damping distributions.     

Tuned Mass Damper Design for Optimally Minimizing Fatigue Damage

Traditionally the usage of a tuned mass damper (TMD) is to improve the survivability of the primary structure under an extraordinary loading environment while the design loading condition is often described by a harmonic function, or sometimes by a stationary random process that can be fully characterized by a power spectral density (PSD) function. In contrast, this paper considers the environmental loading to be a long-term nonstationary stochastic process characterized by a probabilistic PSD function. One engineering motivation to design a TMD under a long-term random loading condition is for prolonging the fatigue life of the primary structure. The primary contribution of this study is to provide the theoretical framework for designing a TMD that can optimally minimize structural fatigue damage.     

Hilbert-Huang Transform Analysis of Dynamic and Earthquake Motion Recordings

This study examines the rationale of Hilbert-Huang transform (HHT) for analyzing dynamic and earthquake motion recordings in studies of seismology and engineering. In particular, this paper first provides the fundamentals of the HHT method, which consist of the empirical mode decomposition (EMD) and the Hilbert spectral analysis. It then uses the HHT to analyze recordings of hypothetical and real wave motion, the results of which are compared with the results obtained by the Fourier data processing technique. The analysis of the two recordings indicates that the HHT method is able to extract some motion characteristics useful in studies of seismology and engineering, which might not be exposed effectively and efficiently by Fourier data processing technique. Specifically, the study indicates that the decomposed components in EMD of HHT, namely, the intrinsic mode function (IMF) components, contain observable, physical information inherent to the original data. It also shows that the grouped IMF components, namely, the EMD-based low- and high-frequency components, can faithfully capture low-frequency pulse-like as well as high-frequency wave signals. Finally, the study illustrates that the HHT-based Hilbert spectra are able to reveal the temporal-frequency energy distribution for motion recordings precisely and clearly.     

Selective discrete Fourier transform algorithm for time-frequency analysis: method and application on simulated and cardiovascular signals

The Selective Discrete Fourier transform (DFT) Algorithm [SDA] method for the calculation and display of time-frequency distribution has been developed and validated. For each time and frequency, the algorithm selects the shortest required trace length and calculates the corresponding spectral component by means of DFT. This approach can be extended to any cardiovascular related signal and provides time-dependent power spectra which are intuitively easy to consider, due to their close relation to the classical spectral analysis approach. The optimal parameters of the SDA for cardiovascular-like signals were chosen. The SDA perform standard spectral analysis on stationary simulated signals as well as reliably detect abrupt changes in the frequency content of nonstationary signals. The SDA applied during a stimulated respiration experiment, accurately detected the changes in the frequency location and amplitude of the respiratory peak in the heart rate (HR) spectrum. It also detected and quantified the expected increase in vagal tone during vagal stimuli. Furthermore, the HR time-dependent power spectrum displayed the increase in sympathetic activity and the vagal withdrawal on standing. Such transient changes in HR control would have been smeared out by standard heart rate variability (HRV), which requires consideration of long trace lengths. The SDA provides a reliable tool for the evaluation and quantification of the control exerted by the Central Nervous System, during clinical and experimental procedures resulting in nonstationary signals.     

Nonlinear Signal Analysis: Time-Frequency Perspectives

Recently, there has been growing utilization of time-frequency transformations for the analysis and interpretation of nonlinear and nonstationary signals in a broad spectrum of science and engineering applications. The continuous wavelet transform and empirical mode decomposition in tandem with Hilbert transform have been commonly utilized in such applications, with varying success. This study evaluates the performance of the two approaches in the analysis of a variety of classical nonlinear signals, underscoring a fundamental difference between the two approaches: the instantaneous frequency derived from the Hilbert transform characterizes subcyclic and supercyclic nonlinearities simultaneously, while wavelet-based instantaneous frequency captures supercyclic nonlinearities with a complementary measure of instantaneous bandwidth characterizing subcyclic nonlinearities. This study demonstrates that not only is the spectral content of the wavelet instantaneous bandwidth measure consistent with that of the Hilbert instantaneous frequency, but in the case of the R?ssler system, produces identical oscillatory signature.     

Control of epilepsy associated with cerebral arteriovenous malformations after radiosurgery

The autocorrelation function pertaining to spatial distributions of ultrasonic scatterers in soft tissue is believed to contain useful information related to tissue morphology. A simple processing method applied to radio-frequency echo signals estimates this function for a sample having isotropic scattering conditions. It utilizes backscattered echo signals from the sample and echo signals from a reference object having defined scattering properties. The ratio of the echo signal power spectrum from the sample to the echo signal power spectrum from the reference object is obtained, and corrected for attenuation differences between the two media. This yields a "form factor" for the sample, whose inverse Fourier transform is the autocorrelation function. The method was tested using tissue-mimicking samples for which spatial autocorrelation functions could be modeled from the dimensions of embedded scatterers. The shapes of the measured autocorrelation functions were in reasonable agreement with those estimated, although measured functions overestimated the function at small lag distances. Scatterer diameters estimated from the zeros of the autocorrelation function agreed to within 6% of expected values when the measurement system bandwidth satisfied minimal criteria.     

Inferring pattern and process: maximum-likelihood implementation of a nonhomogeneous model of DNA sequence evolution for phylogenetic analysis

A nonhomogeneous, nonstationary stochastic model of DNA sequence evolution allowing varying equilibrium G + C contents among lineages is devised in order to deal with sequences of unequal base compositions. A maximum-likelihood implementation of this model for phylogenetic analyses allows handling of a reasonable number of sequences. The relevance of the model and the accuracy of parameter estimates are theoretically and empirically assessed, using real or simulated data sets. Overall, a significant amount of information about past evolutionary modes can be extracted from DNA sequences, suggesting that process (rates of distinct kinds of nucleotide substitutions) and pattern (the evolutionary tree) can be simultaneously inferred. G + C contents at ancestral nodes are quite accurately estimated. The new method appears to be useful for phylogenetic reconstruction when base composition varies among compared sequences. It may also be suitable for molecular evolution studies.