Nonstationary Random Processes Research Articles

Wavelet-based generation of fully non-stationary random processes with application to seismic ground motions

Assessing the performance of earthquake-resistant structural and geotechnical systems is crucial for achieving a desired reliability level and enhancing the resilience of the built environment in seismic-prone regions. Nonlinear dynamic analyses are widely used to quantify structural and geotechnical performance. Still, they require an accurate representation of nonlinear behaviours and proper modelling of the expected seismic events. Stochastic approaches are popular strategies for modelling dynamic actions to account for the uncertain nature of ground shaking. In this framework, joint time–frequency signal representations of seismic records are powerful tools to analyze signals’ time-varying amplitude and frequency content. This paper presents a new method for the stochastic generation of artificial accelerograms using the circular harmonic wavelet transform, which possesses joint time–frequency localization capabilities and offers the engineers a clear and transparent interpretation of the results. The proposed approach adopts a new exponential auto-correlation structure for generating the random phases in the “child (generated) signals” starting from the deterministic ones in the “parent record”. The effects of the correlation structure and different subdivisions of earthquake records in frequency bands are investigated and discussed, leading to practical considerations for identifying an effective trade-off between localization in time and frequency domains. The method can be used for seismic assessment and design purposes, and numerical applications illustrate its potency.

Influence of Ground Motion Non-Gaussianity on Seismic Performance of Buildings

The non-Gaussian feature of seismic ground motion has been reported in some works. However, there remains a lack of research on the influence of the ground motion non-Gaussianity on the seismic performance of buildings, which motivates this study. By employing a non-Gaussian non-stationary random process simulation method previously proposed by the authors, 40,000 ground motion acceleration signals are efficiently generated, including 20,000 Gaussian and 20,000 non-Gaussian records. As computational examples, a four-story frame building and a three-tower super-tall building are selected. The generated acceleration signals serve as external excitations for the two buildings, allowing for a comparison of the differences in seismic structural responses caused by the Gaussian and non-Gaussian earthquake groups. Probability analysis is performed using top-layer displacement and maximum inter-story drift ratio as damage indicators. The results show that the structural responses induced by both Gaussian and non-Gaussian earthquake groups have identical first- and second-order moments but different higher-order moments. The responses from non-Gaussian earthquakes display distinct non-Gaussian traits, with their distribution of extreme values exhibiting a longer tail compared to the Gaussian counterparts. This leads to a notably larger value of non-Gaussian responses under high crossing probabilities, with an amplification that can surpass 18%.

Methodology for detecting and removing outliers in statistical studies

The paper presents a calculation method for detecting and eliminating outlying values. It is shown that its effectiveness depends on the amount of a priori information on the examined process. The proposed method is used for cases whereas the process is stationary and has a Gaussian probability density law. When analysing non-stationary random processes, the existing methods and algorithms rely on the fact that the outlying component is additive and the characteristics of the outlying values are known a priori. The work used the statistical decisions theory that allows formalising the verification algorithms and selecting a criterion for detecting outlying values. Both parametric and non-parametric methods were proposed. In the first case, it is required to have a priori information both on the function of the useful component and on the distribution law of the outlying component of the process, as well as its parameters. It is postulated that the use of non-parametric processing methods requires significantly less a priori information, but their effectiveness is defined by the processing parameters that, in turn, depend on the function of the useful and the distribution law of the outlying components of the process. It is noted that an outlier may prove to be one of the extreme values of the probability distribution of a random variable. The authors outline the problems of ambiguity of input data in case of classical computing. The paper examines the way the external factors affect the dependability and the degree to which such factors are taken into consideration in the existing methods. Methods for assessing the life of the examined items are presented, among which control chart-based methods hold a prominent place. It is shown that the range proves to be a more convenient measure for data dispersion calculation than the standard deviation. Plotting the range of sample on a control chart along with the expectation makes it easier to notice an anomaly. The range is a rough measure of the rate of change of the monitored variable and its value may exceed the control limits on the range chart and inform of an anomaly much earlier than the change in the mean that may still be within the specified control limits.

Ensuring Seismic Resistance of Reinforced Concrete Buildings

There are a large number of works on a comprehensive assessment of the seismic resistance of buildings and structures. However, these studies, as a rule, do not take into account the random nature of the seismic impact, which is a pronounced non-stationary random process. An adequate assessment of the seismic resistance of buildings and structures is possible only on the basis of methods that allow taking into account the large variability of seismic impact parameters. The article presents a probabilistic method for calculating multi-storey reinforced concrete buildings designed in seismic regions, taking into account the interaction of a building with a non-linearly deformable foundation. The developed technique makes it possible to provide the required level of seismic resistance for the designed buildings based on the non-collapse criterion. As an ex-ample, the calculation of a multi-storey reinforced concrete building is considered. External seismic action is represented as a non-stationary random process. The external seismic action is considered as a non-stationary random process, which is obtained by multiplying the stationary random process by a deterministic envelope function. The parameters necessary for constructing the envelope and the stationary random process were obtained from the results of processing the available database of intense earthquakes. The stationary random process was generated by the shaping filter method. The impact parameters are based on the results of processing the available database of intense earthquakes. When modeling reinforced concrete structures, a concrete model is used with the function of damage accumulation under cyclic loads, as well as taking into ac-count the degradation of the strength and stiffness of the material during an intense earthquake. Accounting for the interaction of the building with the soil base is implemented using the SSI interface (Soil Structure Interaction). To prevent the influence of waves reflected from the boundaries of a limited ground massif, a PML layer (Perfectly Matched Layer) is used. The calculation was carried out using explicit methods for integrating the equations of motion on a computing cluster using parallel computing technology. The presented technique makes it possible to investigate the nature of the destruction of reinforced concrete structures during intense earthquakes and to identify zones with a deficiency in bearing capacity. The proposed probabilistic approach to modeling seismic impact as an implementation of a non-stationary random process with given parameters, together with taking into account the nonlinear deformation of the reinforced concrete structures of the building and foundation, allows you to control the level of reliability and design buildings with a given seismic resistance.

Online Detection Method for Capacitor Voltage Transformer Measurement Error Adapted to Non-Stationary Random Processes

Online Detection Method for Capacitor Voltage Transformer Measurement Error Adapted to Non-Stationary Random Processes

ON A CLASS OF NONSTATIONARY CURVES IN HILBERT SPACE

Stationary random processes have been studied quite well over recent years starting with the works of A. N. Kolmogorov. The possibility of building nonstationary random process correlation theory was considered in the monographs by M. S. Livshits, A. A. Yantsevich, V. A. Zolotarev and others. Some classes of nonstationary curves were investigated by V. E. Katsnelson. In this paper nonstationary random processes are represented as curves in Hilbert space which "slightly deviate" from random processes with the correlation function of special kind. The infinitesimal correlation function has been introduced; in essence, this function characterizes the deviation from the correlation process with the given correlation function. The paper discusses the cases of nonstationary random processes, the operator of which has one‑dimensional imaginary component. Cases of a dissipative operator with descrete spectrum are also considered in this work. It is shown that the nonstationarity of the random process is closely related to the deviation of the operator from its conjugated operator. Using the triangle and universal models of non‑self‑ajoint operators it is possible to obtain the representation for the correlation function in the case of nonstationary process which replaces the Bochner – Khinchin representation for stationary random processes. The expresson for the infinitesimal correlation function was obtained for different cases of operator spectrum: for the descrete spectrum placed in the upper half‑plane and for the contrast‑free spectrum at zero. In the case of dissipative operator with descrete spectrum the infinitesimal function can be found in terms of special lambda function. For Lebesque spaces of compex‑valued squared integratable functions the expresson of infinitesimal function was found in terms of special zero order modified Bessel function. It was shown that a similar approach can be applied for the evolutionarily represented sequences in Hilbert spaces.

Advantages of periodic non-stationary random process model in vibration signal processing

The use of two different techniques for the analysis of vibration signals, whose carrier harmo¬nics are modulated by high-frequency narrow-band random processes is discussed. Periodically non-stationary random processes (PNRP) are suitable models for description of vibration signals of damaged mechanism. A proposed processing technique can be considered as an alternative to squared envelope analysis, kurtosis techniques, squared envelope spectrum (SES) and its use in the analysis of a vibration signal is discussed. It is shown that the spectral estimates obtained by the envelope square method are biased and inconsistent. The possibility of obtaining of the unbiased estimates by the PNVP method even for a signal/noise ratio equal to 0.07 has been demonstrated.

Methods of periodically non-stationary random processes for vibrations monitoring of rolling bearing with damaged outer race

The vibrations of a rolling bearing with a damaged outer race are analyzed using a method based on periodically non-stationary random processes (PNRPs). PNRPs can be considered as a particular case of poly-periodically non-stationary random processes (PPNRs), which describe the interactions of vibration stochastic rhythms with different periods that are determined based on the characteristic bearing frequencies. A PNRP model for these vibrations is verified using methods for discovering the first- and second-order periodicities. By applying the least squares technique, it is shown that the mean function and variance are time periodic functions, and their basic frequencies are determined. This enables us to calculate the amplitude spectra for the deterministic oscillations and the time changes in the power for the stochastic part. It is shown that the dependencies of the Fourier coefficients for the covariance function on lag have the forms of slowly damped groups. Vibration covariance structures are compared for different fault widths; it is found that high-frequency modulations of the PNRP carrier modulations are narrow-band, and Rice representations are used to model them. The PRNP model of vibrations is then represented in the form of a superposition of the stationary high-frequency components that are jointly periodically non-stationary. The mutual periodical non-stationarity of these components results in variance time changes. Band-pass filtering of the vibration signal is conducted, and the influence of the filter bandwidth on the number and values of the amplitude of the variance harmonics is analyzed. It is shown that PNRP methods for time series processing enable fault detection in cases where the squared envelope and other familiar techniques are ineffective.

PEM-based random vibration analysis under service-braking conditions on an irregular track

PurposeAccording to relevant research, non-uniform speed has a significant impact on the vehicle-track systems. Up to now, research work on it is still very limited. In this paper, the PEM is adopted to further transform it into a deterministic process to solve the vehicle’s problem of running at a non-uniform speed.Design/methodology/approachThe multi-body vehicle model has 10 degrees of freedom and the track is regarded as a finite long beam supported by lumped sleepers and ballast blocks. They are connected via linear Hertz springs. The vertical track irregularity is a Gaussian stationary process in the space domain. It is transformed into a uniformly modulated nonstationary random process in the time domain with respect to the non-uniform vehicle speed. By solving the equation of motion of the coupled vehicle-track system with the pseudo-excitation method, the pseudo-response and consequently the power spectral density and the standard deviation of the structural response can be obtained.FindingsTwo kinds of vehicle braking programs are taken in the numerical example and some beneficial conclusions are drawn.Originality/valueThe pseudo-excitation method (PEM) was used to perform the random vibration analysis of a coupled non-uniform speed vehicle-track system. Transforming the track irregularity into a uniformly modulated nonstationary random process in time domain with respect to the non-uniform vehicle speed was undertaken. The pseudo-response of the coupled system is solved by applying the Newmark algorithm with constant space integral steps. The random vibration transfer mechanism of the coupled system is fully discussed.

Site response analysis: Uncertain motions propagating through uncertain elastoplastic soil

Presented is a stochastic site response analysis of uncertain seismic motions propagating through uncertain elastoplastic soil. Time domain stochastic bedrock motions are simulated from GMPE of Fourier amplitude spectra and Fourier phase derivative. The uncertain motions are directly modeled as a non-stationary random process in the time domain, capturing all the important seismic characteristics. Uncertain heterogeneous parameters for the elastoplastic behavior of soils are also considered and modeled as random fields. Both uncertain motions and uncertain material parameters are discretized in the probabilistic space using orthogonal Hermite polynomial chaos basis. The probabilistic site response is efficiently solved using Galerkin stochastic elastoplastic finite element method, circumventing Monte Carlo simulations. The implication of the presented methodology for site-specific probabilistic seismic hazard analysis is discussed as well.

Determination of the Characteristics of Non-Stationary Random Processes by Non-Parametric Methods of Decision Theory

This article is devoted to methods of processing random processes. This task becomes particularly relevant in cases where the random process is broadband and non-stationary; then, the measurement of a random process can be associated with an assessment of its probabilistic characteristics. Very often, a non-stationary broadband random process is represented by a single implementation with a priori uncertainty about the type of distribution function. Such random processes occur in information and measuring communication systems in which information is transmitted at a real-time pace (for example, radio telemetry systems in spacecraft). The use of methods of traditional mathematical statistics, for example, maximum likelihood methods, to determine probability characteristics in this case is not possible. In addition, the on-board computing systems of spacecraft operate under conditions of restrictions on mass-dimensional characteristics and energy consumption. Therefore, there is a need to apply accelerated methods of processing measured random processes. This article discusses a method of processing non-stationary broadband random processes based on the use of non-parametric methods of decision theory. An algorithm for dividing the observation interval into stationary intervals using non-parametric Kendall’s statistics is considered, as are methods for estimating probabilistic characteristics on the stationary interval using ordinal statistics. This article presents the results of statistical modeling using the Mathcad program.

Hilbert transform for covariance analysis of periodically nonstationary random signals with high-frequency modulation

We discuss the use of the Hilbert transform for the analysis of periodically non-stationary random signals (PNRSs), whose carrier harmonics are modulated by jointly stationary high-frequency narrow-band random processes. PNRS of this type are suitable models for numerous natural and man-made phenomena, including the vibration of a damaged mechanism. We show that the auto-covariance function of the signal and its Hilbert transform are the same, and that their cross-covariance functions differ only in their sign, meaning that the sum of squares of the signal and its Hilbert transform cannot be considered a ‘squared envelope’ and no new information is contained compared with the variance of the raw signal. A representation of the signal in the form of a superposition of high-frequency components is obtained and it is shown that these components are jointly periodically non-stationary random processes. The properties of the band-pass filtered signals are examined, and it is shown that band-pass filtering can reduce both the number of signal variance cyclic harmonics and their amplitudes. We show that it is possible to extract the quadratures of narrow-band high-frequency modulation processes using the Hilbert transform. The results obtained here theoretically substantiate the use of the Hilbert transform for the analysis of high-frequency modulation which occurs when a fault appears. They offer a new way to consider the traditional approach to vibration diagnosis. A processing technique that can be considered an alternative to envelope analysis is described, and its use in the analysis of a vibration signal is discussed.

Choice of the Parameters of an EMI Monitoring System for an AC Traction Network

The problem of ensuring the electromagnetic compatibility of the traction network with railway signaling lines is of practical interest, since electromagnetic interference from the traction network can cause malfunctions in the signaling systems. In order to ensure the electromagnetic compatibility of traction current with track circuits, which are widely used as track vacancy sensors in front of a moving train, electromagnetic interference in rails is periodically monitored in accordance with their maintenance schedule, which makes it possible to detect excessive increases in the level of interference in the rails in a timely manner and prevent possible failures in signaling systems. The problem of developing embedded systems for continuous monitoring of harmonics in the traction network lies in the need to ensure the necessary measurement accuracy of the interference monitoring system for a traction network in accordance with the strict requirements of standards. The correct choice of control system parameters is difficult because the traction current is a non-stationary random process, and the alternating traction current, in addition, has a large dynamic range of harmonic values, their proximity to each other and to a powerful fundamental harmonic, as well as the limitations inherent in the discrete Fourier transform, associated with the impossibility of simultaneously improving the accuracy of measurements in the time and frequency domains, which leads to the need for compromises when choosing the parameters of the monitoring system. The work is devoted to the practice of applying the theory of analog-to-digital conversion and the windowed Fourier transform to develop a technique for correctly choosing the parameters of an electromagnetic interference monitoring system for traction lines.

An Adaptive Extrapolator of Nonstationary Random Processes

An Adaptive Extrapolator of Nonstationary Random Processes

Time-Dependent Reliability Analysis of RC Bridges Considering Shrinkage, Creep, Resistance Degradation, and Vehicle Load Flows

Due to concrete shrinkage, concrete creep, resistance degradation, and vehicle load flows, the long-term performance of reinforced concrete bridges exhibits time-varying characteristics. The vehicle-borne process is a nonstationary random process, which interacts with concrete shrinkage, concrete creep, and resistance degradation to cause constant changes in bridge internal force and deformation, further, bridge dynamic reliability occurs. This study presented a reasonable method for calculating the time-varying reliability of simply supported girder bridges under the effects of traffic loads and shrinkage creep. Stochastic truck-load models were simulated based on site-specific weigh-in-motion measurements, then the internal forces of the bridge were calculated when the mean value of vehicle-mounted effects increases by 1% per year. Considering the resistance degradation and load effect increase, the reliability of the RC bridge in serviceability limit state and capacity-carrying ultimate state were analyzed using first-passage probability method and Monte Carlo method, respectively. The accuracy of the proposed method was verified by comparison. The evaluation results show that the failure probability including shrinkage and creep has increased significantly; creep is affected by the bridge span and the number of main girders, and the reliability assessment results are safer for bridges with smaller spans.

Efficient ground motion conditional simulation method for nonstationary multivariate processes based on Kriging method and orthogonal discrete wavelet transform

Conditional ground motion simulation methods preserve more physical features of actual ground motions than unconditional methods by including actual earthquake records. This study developed a conditional multivariate nonstationary process simulation method based on the combination of the Kriging method and the orthogonal discrete wavelet transform (DWT). The Kriging method is used to ensure the spectral coherences between simulations as prescribed and is extended to nonstationary random processes by applying it to the average power spectral density (APSD) of ground motions. The ground motions' DWT wavelet coefficients are assumed to be uniformly time-modulated stationary processes whose time-modulating functions are the envelopes of the seed record's wavelet coefficients. The APSD and the envelopes together outline the energy distribution of the random field in both the time and frequency domain features. The power spectral densities of the stationary random processes' DWT coefficients are derived and used to estimate the APSD of the ground motions' DWT coefficients. Approaches to realizing uniformly time-modulated stationary processes with prescribed APSDs and envelope functions are developed. Monte Carlo simulations were conducted to verify the new method's effectiveness. Results show that both the target APSD and coherences can be well approximated, but precise approximation cannot always be realized in the high-frequency range. The new approach possesses a simple structure, involves simple algorithms, and applies to time-independent spectral coherence models, making it an efficient method compared with existing methods.

Random Dynamic Analysis and Running Safety Assessment of Long-Span Cable-Stayed Bridge Under High-Speed Trains Subjected to Earthquake Excitations and Track Irregularities

Running trains on a long-span bridge under earthquake is an inevitable problem for the high-speed railway in seismic regions. This paper aims to investigate the seismically random dynamic behavior of train–cable stayed bridges interaction system using the probability density evolution method (PDEM). The system simultaneously involves the multi-effect of random multi-point earthquake excitations with traveling waves and random track irregularity. The multi-point earthquake excitations, composed of lateral and vertical positions, are generated by stochastic harmful functions (SHFs) and uniformly modulated into nonstationary random processes. The motion equation of such a system is established by coupling the 38-degree vehicles and bridges through the refined random wheel/rail contact relationship and accounting for the phase-lags of seismic travelling waves between pier excitations. PDEM is proven to be applicable to such time-dependent systems and is then used to transform the random excitations into a series of deterministic representative excitations with initial probability. By solving for the corresponding deterministic probability responses, various nonstationary random responses, including the time-dependent probability density evolution functions of random responses, mean values curves and standard deviations, can be obtained easily. A case study based on the cable-stayed bridge is then presented, and the results show the effectiveness and accuracy of the proposed method by comparison with Monte Carlo Simulation. Additionally, the influence of train speed and seismic intensity on the safety and reliability of trains is discussed.

Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates.

The onboard atomic frequency standard (AFS) is a crucial element of Global Navigation Satellite System (GNSS) satellites. However, it is widely accepted that periodic variations can influence the onboard AFS. The presence of non-stationary random processes in AFS signals can lead to inaccurate separation of the periodic and stochastic components of satellite AFS clock data when using least squares and Fourier transform methods. In this paper, we characterize the periodic variations of AFS using Allan and Hadamard variances and demonstrate that the Allan and Hadamard variances of the periodics are independent of the variances of the stochastic component. The proposed model is tested against simulated and real clock data, revealing that our approach provides more precise characterization of periodic variations compared to the least squares method. Additionally, we observe that overfitting periodic variations can improve the precision of GPS clock bias prediction, as indicated by a comparison of fitting and prediction errors of satellite clock bias.

First Passage Dynamic Reliability Analysis of Non-Stationary and Non-Gaussian Buffeting Random Dynamic Responses of Long-Span Suspension Bridge

In the complex mountain wind environment, especially in the strong wind, fluctuating wind speed is a non-stationary and non-Gaussian random process. With the increase of suspension bridges’ span, they are more sensitive to wind-induced vibration response. In this paper, a non-stationary and non-Gaussian buffeting reliability analysis method for long-span bridges is presented. Firstly, the non-stationary wind speed is simulated by the modulation function based on the stationary wind speed model. Secondly, the load is obtained by simulated wind speed, the obtained loads are loaded on the finite element model by ANSYS and the whole bridge time-domain analysis is performed, the time history of displacement response is obtained. Thirdly, based on the first excursion failure criterion of random vibration, the samples of the non-stationary and non-Gaussian displacement time history are transformed into a standard Gaussian process through a modified Fleishman approximation method, and then the non-stationary Poisson distribution method is used for structural reliability analysis. Finally, a mountainous long-span suspension bridge is used as the engineering background, and the proposed reliability analysis method is applied to analyze the reliability of the bridge. The influence of different wind attack angles and modulation functions on the dynamic reliability of the bridge is further studied. The results indicate that the displacement response obtained by the transformation of non-stationary and non-Gaussian random process is less reliable than that obtained by traditional analysis method. Its dynamic reliability is maximum at 1[Formula: see text] wind attack angle, the reliability of negative wind attack angle is lower than that of positive wind attack angle, and decreases with the increase of wind attack angle. The reliability obtained by using different modulation functions is lower than that of traditional method. If the traditional analysis method is still used for reliability analysis, it will produce unsafe consequences and reduce the engineering safety reserve.

INNOVATIVE METHOD FOR STUDYING THE DYNAMICS OF POLLUTION AND HUMIDITY CHANGES IN SPORTS FACILITIES: FITNESS ROOMS AND SWIMMING POOLS

It is necessary to take into account the reaction time of the system to changes and provide for changes in parameters when using modern automation systems and indoor microclimate control with a dynamic change in the intake of harmful substances. The change in carbon dioxide (CO2) pollution and air humidity in sports facilities, such as fitness rooms and swimming pools, is a non-stationary random process with a complex type of non-stationarity and can be described by statistical characteristics. Obtaining these statistical characteristics is complicated by the probabilistic nature of the whole process and the presence of random systematic errors in measuring pollution (CO2) and air humidity of a particular sports facility, the specifics of the functioning of heat and mass transfer processes, moisture exchange in the premises of sports facilities (swimming pools, fitness halls, gyms, etc.). etc.) and has not been studied in modern scientific literature. Therefore, this study was carried out precisely in order to improve the situation and bring it out of the impasse (uncertainty) regarding sports facilities and premises for mass physical culture, recreation, and leisure. The paper proposes a comprehensive method for studying the dynamics of changes in pollution (CO2) and air humidity in a sports facility (fitness rooms, swimming pools, etc.). It allows you to determine the statistical characteristics of the process and identify it with an accuracy that is satisfactory for practical purposes (the error is no more than 10%). In the process of research, it is necessary to use daily diagrams of the operation of CO2 air pollution meters and moisture meters in fitness rooms and swimming pools for a long period of time (70 days). An optimal flow meter of air pollution with carbon dioxide (CO2) or a moisture meter should be built according to a discrete circuit with digital processing of information that comes from a sensor (an artificial intelligence system for controlling air gas exchange processes in a sports facility), according to an algorithm for determining a moving average over a period of 10 ... 20 minutes averaging of the signal and the interval for issuing results 2 ... 5 min. This is what will make it possible to reduce random measurement errors and ensure accurate reproduction of the dynamics of the processes of changing Ac(t) or Wp(t) (which, by the way, can be quite fleeting).