Evolutionary Power Spectral Density Research Articles

Modeling evolutionary power spectral density functions of strong earthquakes via copulas

This paper proposes a new approach for analyzing seismic accelerograms using the evolutionary Power Spectral Density function (EPSD). The accelerogram of an earthquake can be accurately modeled and simulated from its spectrogram, based on the oscillatory stochastic processes theory. To adequately characterize a spectrogram that is consistent with the response spectra, a parametric model with 16 parameters is proposed. This model describes the square of the amplitude spectrum, an envelope of the square of the accelerogram, and a copula that constructs a time–frequency model from the time and frequency marginals. The use of copulas to model a bivariate probability distribution is a common practice in statistics, particularly when the marginal distributions are known. The periodogram can be viewed as an unnormalized probability density function, where the total energy serves as the normalization constant, since the total energy of a seismic motion is always finite. Additionally, a reduced model consisting of only 10 parameters is presented, which may be especially valuable when only shear wave effects are relevant.

Non-stationary modeling and simulation of strong winds

Wind velocity is usually assumed to obey a stationary stochastic process in wind engineering, and this may cause significant bias in describing extremely severe strong wind such as typhoons and thunderstorms. To take into account the non-stationary characteristics of extreme wind, a novel evolutionary power spectral density (EPSD) model is proposed, and the spectral representation method (SRM) is introduced to simulate the whole process of strong winds. Firstly, the wavelet transform (WT) method is adopted to capture the three-dimensional time-varying properties of the low-frequency mean winds, and the associated turbulence features, including turbulent intensity, gust factor, probability density function, and power spectrum, are analyzed in depth. Secondly, the measured horizontal EPSD of strong winds are estimated. Thirdly, the performance of the proposed EPSD model is validated. Finally, the whole process of non-stationary strong winds are simulated and discussed. The results show that the proposed EPSD models are in good agreement with the measured EPSD, and the time-frequency features of the power spectrum of the simulated winds are well reproduced, which provides a powerful tool for large eddy simulation and wind engineering studies under non-stationary extreme wind climate.

Null Phase Assumption-Based Technique for Constructing the Target Model of Seismic Irregularity on High-Speed Railways

High-speed railways are “lifeline projects” that shoulder the heavy responsibility of transporting relief supplies and medical forces for the first time after earthquakes. To ensure the train’s safety after earthquakes, it is of great urgency to ascertain a post-earthquake speed threshold. To that end, a target model for seismic irregularity emerges as a key parameter. In this paper, a null phase assumption-based technique for the mutual conversion between evolutionary power spectral density and non-stationary signal was proposed. Taking a high-speed railway track-bridge system as the research object, the target model of seismic irregularities was constructed based on the proposed technique. The rationality of the target model of seismic irregularities was verified, and the construction parameter settings were discussed. Moreover, a simplified frequency-domain fitting method for the target model of seismic irregularities was proposed based on the spectral decomposition theory. According to the research findings, the null phase assumption-based technique is capable of performing interconversion between seismic irregularity and its evolutionary power spectral density with satisfactory accuracy. It is recommended to set the minimum number of seismic irregularities, spatial sampling interval, hop size, and length of window function as 50, 0.25[Formula: see text]m, 1, and 40 to 100, respectively.

Time-frequency analysis of nonlinear structures under nonstationary excitation by directly solving structural dynamic equation through absolute displacement

The time-frequency evolution of seismic ground motion energy under strong earthquakes, which can be described by the evolutionary power spectral density (EPSD) matrix, will amplify the most unfavorable responses of members in nonlinear structural systems. However, the existing general finite element software fails to reduce the dimension of the EPSD matrix and then decouple time from frequency in frequency domain analysis. This failure greatly limits the simulation and calculation of multi-dimensional nonlinear structures with multiple degrees of freedom by the stochastic vibration theory. In this study, the pseudo excitation method (PEM), which can direct solve structural dynamic equations using absolute displacement, is used to derive the self-spectral and cross-spectral densities of seismic ground motions in one-dimensional and multi-dimensional cases. In MATLAB, the EPSD matrix is transformed into a four-dimensional uniform modulation excitation matrix independent of the time variable. Nonlinear iteration is conducted and the precise integration method (PIM) is employed through a transient analysis module. Furthermore, a simple one-dimensional example is presented to verify the correctness of the proposed algorithm herein. Finally, as an example, we select important parameters of the coherence model at different supports of a half-through three-span concrete-filled steel tubular (CFST) tied-arch bridge in view of real engineering conditions, and calculate the time-varying power spectrum and variance of the responses of arch ribs, arch feet and the bridge deck system. The results show that the theoretical derivation and the algorithm constructed in this paper can simulate the time-delay effect of the nonlinear structural system. This proves the feasibility of general finite element software in solving and analyzing random responses to multi-dimensional and multi-support non-stationary excitation of nonlinear structures.

Nonstationary elastoplastic response analysis of curved beam bridges under spatial variability of earthquake ground motion using absolute displacement method

Current studies indicate that curved beam bridges are more sensitive to the spatial variability of earthquake ground motion (SVEGM) than straight bridges. However, research methods used currently typically disregard the nonlinear characteristics of bridges or use only deterministic excitations for analysis. Furthermore, the sensitivity of curved beam bridges with different curvature radii to the SVEGM has not been investigated comprehensively. Hence, in this study, a nonlinear finite element model of curved beam bridges is established using the ANSYS platform and the MATLAB is then employed to reduce and decouple the non-stationary seismic evolutionary power spectral density (EPSD) matrix. The absolute displacement method is employed for the multidimensional and multipoint nonlinear time-history analysis of the bridges. Considering different wave velocities, site conditions, and coherence types, this study comprehensively analyses the random response and frequency domain characteristics of curved continuous beam bridges under different SVEGMs. The analysis includes a verification of curved beam bridges with different curvature radii based on 48 cases (six SVEGMs with eight curvature radii). The results indicate that the presence of curvature renders curved beam bridges more sensitive to the SVEGM. The SVEGM significantly affects the random response and response frequency domain distribution of curved beam bridges, with smaller curvature radii contributing more significantly to the response. Therefore, the SVEGM must be considered in the seismic analysis of curved beam bridges, particularly those with smaller radii, to avoid an inaccurate estimation of the seismic performance of the bridges. The findings of this study can improve the seismic design and evaluation of curved beam bridges as well as enhance their seismic performance and reliability.

Relaxed evolutionary power spectral density functions: A probabilistic approach to model uncertainties of non-stationary stochastic signals

The identification of patterns and underlying characteristics of natural or engineering time-varying phenomena poses a challenging task, especially in the scope of simulation models and accompanying stochastic models. Because of their complex nature, time-varying processes such as wind speed, seismic ground motion, or vibrations of machinery in the presence of degradation oftentimes lack a closed-form description of their underlying Evolutionary Power Spectral Density (EPSD) function. To overcome this issue, a wide range of measurements exist for these types of processes. This opens up the path to a data-driven stochastic representation of EPSD functions. Rather than solely relying on time–frequency transform methods like the familiar short-time Fourier transform or wavelet transform for EPSD estimation, a probabilistic representation of the EPSD can provide valuable insights into the epistemic uncertainty associated with these processes. To address this problem, the evolutionary EPSD function is relaxed based on multiple similar data to account for these uncertainties and to provide a realistic representation of the time data in the time–frequency domain. This results is the so-called Relaxed Evolutionary Power Spectral Density (REPSD) function, which serves as a modular probabilistic representation of the time–frequency content of stochastic signals. For this purpose, truncated normal distributions and kernel density estimates are used to determine a probability density function for each time–frequency component. The REPSD function enables the sampling of individual EPSD functions, facilitating their direct application to the simulation model through stochastic simulation techniques like Monte Carlo simulation or other advanced methods. Even though the accuracy is highly dependant on the data available and the time–frequency transformation method used, the REPSD representation offers a stochastic representation of characteristics used to describe stochastic signals and can reduce epistemic uncertainty during the modelling of such time-varying processes. The method is illustrated by numerical examples involving the analysis of dynamic behaviour under random loads. The results show that the method can be successfully employed to account for uncertainties in the estimation of the EPSD function and represent the accuracy of the time–frequency transformation used.

Nonstationary time-frequency characteristics of vehicle suspension systems based on evolutionary power spectral density

Improving the ride comfort for vehicles that travel at varying speeds is a nonstationary problem and has been an increasingly important topic for suspension system research. Accurate identification of frequencies associated with the resonant responses is needed for effective design of active suspension control strategies. However, traditional time-frequency (TF) methods have resolution limitations in either the time domain or the frequency domain. This paper proposes a new methodology to employ the evolutionary power spectral density (EPSD) to describe the responses of vehicles in the time-frequency domain. The key to calculate the responses in nonstationary cases is to derive the nonuniform modulation function which is obtained by the pseudo excitation method. To illustrate the applicability of the proposed method, an example of a seven-degree-of-freedom vehicle model is investigated in which the nonstationary excitations acting on the wheels are computed by the nonuniform modulation function and a precise integration method. Results show that resolutions obtained by the EPSD are more precise than the traditional TF methods with less than 0.2 Hz of bandwidth in both the lower frequency range at the first natural frequency of 1 Hz and in the higher frequency around 10 Hz. Moreover, the changing of the resonant frequency with varying speeds provides the insights, from the perspective of human body’s sensitivity to frequencies of vibration, to improve the ride comfort for vehicles under nonstationary driving conditions.

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

Determination of Nonstationary Stochastic Response of Linear Oscillators With Fractional Derivative Elements of Rational Order

Abstract In this paper, a technique is developed for determining the nonstationary response statistics of linear oscillators endowed with fractional derivative elements. Notably, fractional operators are particularly effective in modeling solid mechanics problems as they offer the option of influencing both the elasticity and the energy dissipation capacity of the system. In this paper, particular attention is devoted to the case of fractional derivatives of rational order that approximates reasonably well any real order model. The oscillators are subjected to stationary stochastic excitations, and the pertinent nonstationary response statistical moments are determined by first introducing a finite number of oscillator response related states; this is afforded by the rational number order of the fractional operator. Next, the technique involves proceeding to treating the problem in the Laplace transform domain. This leads to multiple convolution integrals determined by representing the transfer function of the oscillator in a partial fraction form by a pole-residue formulation. In this manner, the response evolutionary power spectral density of the fractional oscillator is derived in a closed form, while nonstationary second-order statistics can be obtained by mundane numerical integration in the frequency domain. Applications to oscillators comprising one or two fractional derivative elements are presented, considering the case of a white noise excitation and of a random process possessing the classical Kanai–Tajimi spectrum. Reliability of the developed technique is assessed by juxtaposing its analytical results with pertinent Monte Carlo simulation data.

Exploring frequency-domain nonstationary of seismic ground motions using evolutionary power spectrum and Archimedean Copula functions

In this paper, the frequency-domain nonstationarity of seismic ground motions is investigated using Archimedean Copula functions. To measure the nonstationarity of the ground motions, a group of nonstationarity indexes is introduced based on the evolutionary power spectral density. The values of the proposed nonstationarity indexes of 1034 seismic ground motion records are calculated and their distributions are obtained. The Gumbel, Clayton, and Frank Copula functions among the nonstationarity indexes and intensity measures of the ground motions are estimated from the Kendall rank correlation coefficients. The best-fit Copula functions are determined by minimizing these three estimated Copula functions and the empirical Copula function. The correlation among the nonstationarity indexes and intensity measures from the Copula functions indicates that the average frequency and bandwidth of the seismic ground motion tend to decrease with time increasing, and the ground motion with long duration tends to have remarkable low-frequency components. It is suggested that the nonstationarity of the ground motions may increase the seismic risk of civil engineering structures and could not be ignored in the structural design.

Multi-taper S-transform method for estimating Wigner-Ville and Loève spectra of quasi-stationary harmonizable processes

Current non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to overestimation and ambiguity of structural responses. The quasi-stationary harmonizable process with its Wigner-Ville spectrum (WVS) and Loève spectrum, which do not suffer from the deficiencies of EPSD, is suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, the multi-taper S-transform (MTST) method for estimating WVS and Loève spectrum of multi-variate quasi-stationary harmonizable processes is presented. The analytical biases and variances of the WVS, Loève spectrum, and time-invariant and time-varying coherence estimators from the MTST method are provided under the assumption that the target multi-variate harmonizable process is Gaussian. Using a numerical case of a bivariate harmonizable wind speed process, the superiority and reliability of the MTST method are demonstrated through comparisons with several existing methods for the WVS and Loève spectrum estimations. Finally, the MTST method is applied to two pieces of ground motion acceleration records measured during the Turkey earthquake in 2023.

Evolutionary power spectral density study of the earthquake-induced dynamic irregularity based on short-time Fourier transform

The stiffness degradation of the track-bridge system under earthquake is the main reason for the deterioration of track dynamic smoothness. In order to investigate the dynamic smoothness performance of the track surface of the track-bridge system after earthquake. In this paper, a nonlinear time analysis of ballastless track-simply supported bridge system under transverse random earthquakes is carried out, and the degradation mode and degradation degree of each key component of the track-bridge system after transverse earthquakes are investigated. Based on the post-earthquake stiffness of each component, a train-track-bridge system considering earthquake-induced stiffness degradation is established. The sample library of earthquake-induced dynamic irregularity is constructed by additional deformation of rails, and the power spectral density of the earthquake-induced dynamic irregularity is studied. Based on the short-time Fourier transform irregularity inversion technique, the earthquake-induced dynamic irregularity characteristic curve based on the probability guarantee rate is proposed, and the effects of train operating speed and ground vibration intensity on the earthquake-induced dynamic irregularity characteristic curve are investigated. The research results show that: The transverse stiffness degradation of the bearings and piers of the track-bridge system is the most obvious during an earthquake, and the stiffness degradation degree of the middle piers is greater than that of the side piers. The stiffness degradation degree of the track-bridge system rises significantly with increasing PGA, while the variability of stiffness degradation of the middle and side piers drops. The most significant earthquake-induced dynamic irregularity sample is the track alignment irregularity. Low-frequency components dominate the evolutionary power spectral density corresponding to track dynamic irregularity, and the amplitude of the power spectral density reduces considerably with increasing spatial frequency. The vehicle speed has little effect on the earthquake-induced track dynamic irregularity. The irregularity amplitude grows significantly with increasing damage degree. The results indicated that the stiffness degradation damage degree of the track-bridge system under earthquake can be quantified by constructive earthquake-induced track dynamic irregularity.

Characteristics of Fluctuating Wind Speed Spectra of Moving Vehicles under the Non-Stationary Wind Field

To promote energy saving, emission reduction, and sustainable development of high-speed trains, as well as achieve low-carbon operation of these trains. It is necessary to establish a fluctuating wind speed spectra model that can accurately describe the characteristics of the fluctuating wind speed field of the moving vehicle. This will help explore the effects of strong winds on the running resistance, energy consumption, safety, and comfort of trains. In this paper, based on Priestley’s evolutionary power spectral density (EPSD) theory, an efficient method was developed for generating the fluctuating wind speeds at the moving point under the non-stationary wind field. On such basis, the effects of different mean wind speeds, ground clearances, temporal modulation function parameters, and vehicle’s moving speeds on the time-varying correlation function ratio of fluctuating wind speed at fixed and moving points were analyzed. Subsequently, the relationship between the time-varying correlation functions of fluctuating wind speed at the fixed and moving points was established by analyzing the sensitivity of the above parameters, and a theoretical model of fluctuating wind speed spectra of the moving point under the non-stationary wind field was proposed. In addition, the relational expression of fluctuating wind speed spectra of the moving point under stationary and non-stationary wind fields was established, which was further validated using the fluctuating wind speed spectra model at the fixed points with different modulation function forms. The results demonstrated that the direct generation method can avoid n times of POD decomposition and Ns∑j=1nNqjj times of FFT calculation, improve the calculation speed, and save memory. The proposed fluctuating wind speed spectra model at the moving point under the non-stationary wind field is in good agreement with the corresponding target one, indicating the high accuracy of the proposed model. Meanwhile, it is also noted that the fluctuating wind speed spectra at the moving point under the non-stationary wind field can be obtained by modulating the spectra under the stationary wind field using temporal modulation function, which is the same as that of the fluctuating wind speed spectra at fixed points under the non-stationary wind field.

A novel shock life model for BGA solder joints based on evolutionary power spectral density

Multiple shock damage of Ball grid array (BGA) solder joint is one of the important reasons for space mission failure. A novel shock life model is proposed for BGA solder joints to evaluate its life under shock loads based on the evolutionary power spectral density (EPSD). Considering the bandwidth and non-Gaussian features of the stress signals during different stages, the shock stress response was treated as a non-stationary and non-Gaussian stochastic process. EPSD of the shock stress response is derived in matrix form based on wavelet transforms. The weight coefficient and non-Gaussian damage factor are proposed to describe the influence of the bandwidth and non-Gaussian characteristics of shock response on fatigue damage, respectively. The damage values of different stress stages were calculated by weight coefficient and corrected using non-Gaussian damage factor. Then the shock life model is proposed based on damage accumulation theory. Tests for the BGA solder joints shock life were conducted, and the result shows that the proposed model is feasible. Therefore, the developed method in the paper is expected to provide reference for the prediction of BGA solder joint shock life in aerospace engineering.

Design sensitivity analysis of structural systems with damping devices subjected to fully non-stationary stochastic seismic excitations

In the framework of dynamic excitations to be considered during the design phase of structures, the most crucial one is the ground motion acceleration. To increase the structural performances against seismic actions, one of the most effective design criteria is to introduce damping devices.An efficient approach to define the optimal parameters of the damping system is based on the design sensitivity analysis, which provides a quantitative estimate of desirable design change, by relating the available design variables.In this paper a method to evaluate the sensitivities of stochastic response characteristics of structural systems with damping devices subjected to seismic excitations, modelled as fully non-stationary Gaussian stochastic processes, is proposed. The main steps are: i) to define the time–frequency varying response (TFR) function for non-classically damped systems; ii) to evaluate closed form solutions for the first-order derivatives of the TFR function as well as of the one-sided evolutionary power spectral density function of the structural response, with respect to damping parameters of devices; iii) to perform a design sensitivity analysis selecting as performance measure function the non-geometric spectral moments of nodal displacements.A numerical application demonstrates how the proposed approach is suitable to cope with practical problems of engineering interest.

Study on the EPSD of Wind-Induced Responses of the Sutong Bridge Using Harmonic Wavelets

Many long-span bridges are located at typhoon prone regions. With the continuous increase of the bridge span, the typhoon-induced buffeting becomes more and more prominent. In this study, based on the structural health monitoring system installed in the Sutong Bridge, the recorded buffeting responses of the main girder during typhoons Damrey and Haikui were analyzed. The run test method demonstrated that the recorded acceleration responses can be regarded as zero-mean non-stationary random processes. Hence, to capture the energy distribution of the recorded data in the time-frequency domain, the evolutionary power spectral density (EPSD) estimation was conducted using efficient generalized harmonic wavelet (GHW) and filtered harmonic wavelet (FHW), respectively. Compared with the GHW, narrower wavelet bandwidth is required by the FHW to yield a compromise between the time and frequency resolution. For the FHW-based method, the power spectral density amplitudes of the averaging EPSDs are slightly larger for certain major frequency components than those obtained by the Pwelch method. Results show that the non-stationary features of the buffeting of long-span bridges during Typhoon events should be considered. This study can also provide references for non-stationary buffeting analysis of other long-span bridges during extreme wind events.

On the estimation of the evolutionary power spectral density

Two popular spectral-based approaches for estimating the evolutionary power spectral density (EPSD) function from the samples of the evolutionary process are based on the short-time Fourier transform (STFT) and the continuous wavelet transform. Both rely on the concept of slowly varying modulation or EPSD function, although the quantification of the effect of the ‘‘slow’’ variation in the estimated EPSD is elusive.We propose, in the present study, to use the derivatives of the EPSD function to quantify the smoothness of the EPSD function in the context of estimating the EPSD function. We derive equations for estimating EPSD by using the S-transform and continuous wavelet transform. These equations are as simple to use as that derived based on STFT. We also derive the corresponding equations for assessing the residual for the estimated EPSD by using these transforms, including STFT. The residual provides an approach for identifying or quantifying, in the context of its estimation, the “slow” variation of the EPSD function. The derived equations and numerical results indicate that the residual depends on both the derivatives of the EPSD function with respect to time and frequency as well as the adopted transform.

Effects of stochastic generation on the elastic and inelastic spectra of fully non-stationary accelerograms

Nonlinear dynamic analyses are a state-of-the-art tool to assess the performance of earthquake-resistant structures. Inevitably, the validity of the predicted seismic response depends on the fidelity of the computational model to the actual structural behavior and the representativeness of the time histories of ground acceleration as realizations of the seismic hazard for the site under consideration. The generation of artificial time histories is generally allowed by international seismic codes and represents a valid alternative to recorded accelerograms, provided that the key features in the expected seismic input are preserved in the generated signals. Different stochastic generation methods of fully non-stationary accelerograms have been proposed in the literature. Two alternative randomization strategies are compared in this paper, based on (i) wavelets analysis and (ii) evolutionary power spectral density (PSD) functions. The analyses are focused on the aleatory variability observed in the generated elastic and inelastic response spectra in relation to different modeling choices, offering qualitative and quantitative information to designers using stochastically generated accelerograms.

Direct estimation of evolutionary power spectral density of nonstationary winds: A unified formulation

Estimating the evolutionary power spectral density (EPSD) of non-stationary winds (e.g., tropical storms and downbursts) is necessary to predict the response of structures under such extreme winds. Following the review of the existing direct estimation methods of EPSD, this paper offers a two-step unified formulation, i.e., raw estimation and associated error reduction. The raw estimation is expressed in terms of a time-frequency transform with a general kernel function. It is shown that if the kernel function is described by a time-frequency analysis tool such as the short-time Fourier transform, the wavelet transform, and the S-transform, the generalized raw EPSD estimation becomes a particular case of the existing methods. The unified estimation method presented here can be viewed as a filter bank with adjustable time and frequency resolution. The analysis of error in the raw estimation is carried out on the bias and random errors accounting for the approximation in both the time and frequency domains. Various techniques for reducing such errors are then summarized and recast in the unified formulation, including series expansion, short-time window smoothing, and multi-tapering. Based on the unified perspective, a discussion and some prospects of EPSD estimating are provided.

Field measurement analysis of wind parameters and nonstationary characteristics in mountainous terrain: Focusing on cooling windstorms

Cold fronts passing through the mountain areas may produce continuous windstorm and cooling processes, potentially impacting the mountain areas' infrastructure. Cooling windstorms belong to the dynamically-driven wind in the mountain wind system. To accurately estimate the wind field parameters of these storms and provide a basis for structural wind resistance design, cooling windstorms should be extracted and analyzed separately from the wind field observation data. Based on the long-term field measurement in complex mountain terrain, a framework for identifying cooling windstorms is established according to temperature and wind speed features. The mean and fluctuating wind parameters of these intense winds are then obtained. Strong nonstationary is detected for cooling windstorms. This paper focuses on the time-varying regularity of fluctuating parameters during intense wind with cooling based on the evolutionary power spectral density. From the perspective of turbulent energy, the basis for selecting the optimal wavelet decomposition levels is put forward when extracting the time-varying mean wind through discrete wavelet transform in the nonstationary wind speed model. This investigation can provide a reference for studying the nonstationary wind field.