Power Spectrum Density Function Research Articles

IDENTIFICATION OF ELECTROMAGNETIC ULTRA LOW FREQUENCY (ULF) SIGNAL ANOMALIES AS PRECURSORS OF A MAGNITUDE 5 EARTHQUAKE IN THE SULAWESI REGION

Sulawesi is situated at the intersection of three tectonic plates the Indo-Australian, Eurasian, and Philippine plates. The Palu Koro fault is one of the primary causes of seismic activity in the Sulawesi region, making it vulnerable to earthquakes on this Island. As an initial step in disaster mitigation, the information provided is useful in determining the initial signs before the occurrence of an earthquake, it is necessary to identify Ultra Low Frequency (ULF) Signal Anomalies as earthquake precursors. The methods employed are those of power spectrum density (PSD) and single-station transfer function (SSTF). The analysis of anomalies suspected of being precursors to 5 earthquake events revealed that in the February 27, 2023 earthquake, the anomaly appeared twice with a lead time of 17 days. In the February 28, 2023 earthquake, the anomaly appeared twice. In the March 7, 2023 earthquake, the anomaly appeared 3 times with a lead time of 20 days. The March 28, 2023 earthquake had 6 anomalies with a lead time of 24 days. The April 6, 2023 earthquake had 5 anomalies with a lead time of 29 days. anomalies were detected with lead times ranging from 17 to 30 days. The anomaly source azimuth direction was also determined.

Failure Probability and Economic Loss Assessment of a High-Rise Frame Structure under Synthetic Multi-Dimensional Long-Period Ground Motions

Multiple research studies and seismic data analyses have shown that multi-directional long-period ground motion affects crucial and intricate large-scale structures like oil storage containers, long-span bridges, and high-rise buildings. Seismic damage data show a 3–55% chance of long-period ground motion. To clarify, the chance of occurrence is 3% in hard soil and 83% in soft soil. Due of the above characteristics, the aseismic engineering field requires a realistic stochastic model that accounts for long-period multi-directional ground motion. A weighted average seismic amplification coefficient selected NGA database multi-directional long-period ground motion recordings for this study. Due to the significant low-frequency component in the long-period ground motion, this research uses empirical mode decomposition (EMD) to efficiently decompose it into a composite structure with high- and low-frequency components. Given the above, further investigation is needed on the evolutionary power spectrum density (EPSD) functions of high- and low-frequency components. Analyzing the recorded data will reveal these functions and their corresponding parameters. Proper orthogonal decomposition (POD) is needed to simulate samples of high- and low-frequency components in different directions. These samples can be combined to illustrate multi-directional long-period ground motion. Representative samples exhibit the seismic characteristics of long-period multi-directional ground motion, as shown by numerical examples. This proves the method’s engineering accuracy and usefulness. Moreover, this study used incremental dynamic analysis (IDA) to apply seismic vulnerability theory. This study investigated whether long-period ground motions in both x and multi-directional directions could enhance the seismic response of a high-rise frame structure. By using this method, a comprehensive seismic economic loss rate curve was created, making economic loss assessment clearer. This study shows that multi-directional impacts should be included when studying seismic events and calculating structure economic damages.

Uncertainty propagation of turbulence parameters for typhoon and Non-typhoon winds in buffeting analysis of Long-span bridges

Wind turbulence-induced buffeting vibration affects the serviceability of long-span bridges and lead to fatigue problems. The randomness of turbulence parameters in the power spectrum density function is always disregarded, failing to well reproduce the dispersion of structural buffeting responses and achieve the extreme vibration amplitude. This study proposes a probabilistic buffeting analysis approach based on multi-point estimation, dimension reduction techniques, and the principle of maximum entropy accounting for the uncertainty of turbulence parameters. The uncertainties in turbulence power spectral density functions are quantified based on the long-term real observational wind data captured by the sensors installed at Xihoumen Bridge. These uncertainties are propagated into the buffeting response of the bridge using the proposed method. The results are compared with those of Monte Carlo simulation, suggesting a good agreement with substantially lower computational costs can be attained by the proposed method. Then, the probability density evolution of buffeting response versus wind speed is calculated. Results reveal that typhoon-induced buffeting response is greater than non-typhoon’s under low mean wind speeds (e.g., 25 m/s), whereas it is the opposite at high wind speeds (e.g., 30 m/s). The predicted probability density evolution of buffeting response exhibits a great consistency with real observations. By introducing the mean wind speed hazard curves for both typhoon and non-typhoon winds, the exceeding probability of vibration within a given period can be easily attained. It allows the estimation of return periods of structural responses and development of the uniform-risk approach for structure design rather than the conventional uniform-hazard design method.

Reliability Evaluation of a Nonlinear Frame Structure under Explosive Ground Motions Generated by Dimension-Reduction Method

In the present study, a stochastic model of explosive ground motions applying the dimension-reduction method is proposed, and the reliability evaluation of a nonlinear frame structure under such excitations is realized by means of the probability density evolution method and an equivalent extreme-value-based reliability evaluation strategy. Firstly, the evolutionary power spectrum density function of the explosive ground motions is modeled by respectively identifying the normalized total energy distribution function and the frequency total energy distribution function on the basis of the measured motion records. In addition, an exponential model is constructed to forecast the seismic characteristics of the explosive ground motions based on the given distance to the explosive source and the charge quantity. Then, the representative samples of the explosive ground motions are simulated using the dimension-reduction method. The simulation results show that the generated acceleration samples have significant seismic characteristics of the explosive ground motions, and the accuracy is verified by comparing the second-order statistics with the sample set and the corresponding targets. Due to the fact that the probabilities of the representative samples simulated by the dimension-reduction method can compose a comprehensive probability set, it contributes to the refined dynamic response analysis and reliability evaluation of complex structures combining with the probability density evolution method. The accurate dynamic response analysis and reliability evaluation of a nonlinear frame structure illustrates the effectiveness of the proposed model and the dimension-reduction method for simulating the explosive ground motions. The numerical results demonstrate that the explosive ground motions have a substantial effect on the nonlinear behavior and the security of engineering structures.

Analytical Study on the Random Seismic Responses of an Asymmetrical Suspension Structure

An asymmetrical suspension structure, without vertical column support and without supplying the flexibility of spatial arrangement, is more sensitive to ground movement. The structural responses of an asymmetrical suspension structure subjected to Clough–Penzien spectrum excitation were analytically investigated in this study. First, the governing equation was decoupled into an independent state equation in generalized coordinates through the real mode decomposition method and by creatively combining it with finite element methods to acquire modal coefficients. Through the pseudo excitation method (PEM), the frequent domain solution of the dynamic response was acquired, and its power spectrum density function was then quadratically decomposed to obtain its corresponding 0–2-order spectral moments. A practical case study was performed to verify the high accuracy and computational efficiency of the proposed closed-form solution comparative to the traditional PEM. Finally, an extended analysis of the effect of the suspended span and comparisons to a normal framed structure and symmetrical suspension structures were carried out. The analysis results indicate that the larger suspended span could consume more seismic energy and result in smaller horizontal displacement and acceleration. Moreover, the comparison results also point out that the existence of the suspension part showed better seismic energy dissipation capacity compared to the normal framed structure, and two symmetrical suspension parts also performed better than a single asymmetrical part in seismic energy dissipation.

Reference Values of the Forearm Tremor Power Spectra for Youth Athletes.

The aim of the study was to determine reference graphs of power spectral density functions of forearm physiological tremor and to compare their parameters in the male and female population of young athletes from various sports. One hundred fifty-nine (159) female (15.7 ± 2.1 years, 59.8 ± 8.1 kg, 169.1 ± 7.5 cm) and 276 male (16.4 ± 1.9 years 72.7 ± 10.3 kg and 180.9 ± 8.7 cm) youth athletes participated in the study. Forearm tremor was measured accelerometrically in a sitting position. Power spectrum density (PSD) function was calculated for each individual tremor waveform. Because of right skewness of power distribution, the PSD functions were subjected to logarithmic transformation. Average log-powers in low (2-4 Hz) and high (8-14 Hz) frequency ranges and mean frequencies in those ranges were analyzed. Tremor log-powers for male were greater than for female athletes (p < 0.001), while frequencies of spectrum maxima did not differ from each other. Frequencies of spectrum maxima correlated (p < 0.001) with age (r = 0.277 and 0.326 for males and females, respectively). The obtained reference functions may be utilized in order to quantify and assess tremor size and its changes evoked by stress and fatigue, which can be applied for selection and training monitoring in sports, but also in medicine for detection and diagnosis of pathologic tremor in young individuals.

Efficient simulation of multivariate non-stationary ground motions based on a virtual continuous process and EOLE

Simulation of multivariate non-stationary ground motions (MNSGM) is of paramount importance to seismic reliability evaluation of large-span structures, e.g., bridge and dome structures. In this paper, the expansion optimal linear estimation (EOLE) method is extended to the simulation of MNSGM with high efficiency. A virtual continuous process is proposed to connect the multivariate ground motions to formulate a 1D−1V process. In that regard, the main work of this study is to obtain the expanded covariance/correlation matrix for the virtual 1D−1V process including the auto- and cross-correlation information, which will be determined by using inverse Fourier transform from the corresponding auto- and cross-evolutionary power spectrum density (EPSD) functions. Then, the EOLE method is applied to generate the 1D−1V virtual continuous process by decomposing the expanded covariance/correlation matrix. Finally, the MNSGM can be easily obtained by splitting the 1D−1V virtual continuous process correspondingly. The most attractive feature of the proposed method is that it only needs to decompose the covariance/correlation matrix once, which effectively circumvents the computational challenge of classic spectral representation method (SRM) for simulating MNSGM. Besides, two types of modulation functions for non-stationarity, i.e., the time- and time–frequency-modulations are considered. Two examples, involving the simulation of two types of MNSGM are presented to verify the effectiveness of this approach.

PIV Measurement and Proper Orthogonal Decomposition Analysis of Annular Gap Flow of a Hydraulic Machine

The fluid stress or flow-induced vibration of annular gap flow always has some influence on the stable working conditions of a hydraulic machine. A time-averaged analysis of flow may not have to explicitly acknowledge these factors. Accordingly, a finite-axial-length annular gap was measured via particle image velocimetry (PIV), with inner boundary motion and a stable outer boundary. As a statistic result regarding the fluid stress, the Reynolds stresses soared in the first region, were sustained in the middle region, but decreased at last. The flow had a higher convective transportation intensity in the radial direction than in other directions. Flow diagnostics were also performed by proper orthogonal decomposition (POD). As a result, the coherent structures were found. Then, the power spectrum density (PSD) functions were also calculated for finding the flow-induced vibration characteristics; the functions had high amplitude in the low-frequency domain and low amplitude in the high-frequency domain, with an order of magnitude between the two amplitudes of 10−1 to 10−2. In addition, the frequency was higher at a smaller gap width in the middle-frequency domain, but the condition was the opposite in the high-frequency domain. In conclusion, the fluid stresses were changeable and uneven along the flow direction, and flow-induced vibration obviously existed. Remarkably, the turbulence characteristics of the annular gap flow were not “laminar approximating,” while the diameter ratio of the gap was 0.6 to 0.8.

Stochastic ground motion simulation and seismic damage performance assessment of a 3-D subway station structure based on stochastic dynamic and probabilistic analysis

In practical engineering of underground structures, it is well-known that the randomness of earthquake generation and a structure's seismic response need to be considered for the safety of the project. The dynamic response and reliability level of a subway station are analyzed from the perspective of random vibration and the effect of stochastic ground motions on seismic responses is considered in this paper. This research proposes a model that can generate completely nonstationary ground motions and combines it with the probability density evolution method (PDEM) to analyze the seismic response and reliability level of a subway station. First, combining random function spectrum representation and evolutionary power spectrum density function, a probability model of nonstationary ground motion is established. According to the code (Gb 50909-2014), the model parameters of 5 sites are suggested, and the efficiency and accuracy are verified. A series of intensity–frequency nonstationary stochastic earthquakes are generated for the seismic response analysis. Then, a 3-D finite element analysis is performed for the deterministic dynamic response of the Daikai subway station, and the second-order statistics of three evaluation indicators (interstory drift, relative settlement, damage area ratio) are attained. Finally, the stochastic dynamic response and reliability level of the station structure are obtained by processing the data with the PDEM. The limit state related to the three evaluation indicators is established through probability density function (PDF) and cumulative distribution function (CDF). In addition, this probability evaluation system is more intuitive and effective than conventional methods. The statistical and probabilistic results show that the characteristics of earthquakes have a significant influence on the seismic response of subway stations, furthermore, the stochastic earthquakes generated by the intensity–frequency nonstationary ground motion model match well with the target value and the PDEM can comprehensively evaluate the seismic performance and reliability level of subway stations. Therefore, combing the stochastic earthquake and PDEM is an effective way to analyze the seismic response of subway stations.

A POD-based methodology for structural finite element model updating

This paper presents an efficient Proper Orthogonal Decomposition (POD) based methodology for structural finite element model updating problem. In the proposed methodology, the vibrational responses of the system are projected into proper modal coordinates and then the Power Spectrum Density (PSD) Functions of the projected data are used to establish the objective function. Since the new objective function integrates the effects of frequencies and damping of the structure in addition to its Proper Orthogonal Modes (POMs), it performs better than other POD-based objective functions, especially in cases of using ambient vibration data. Also, an effective algorithm is introduced to decide how many and which of the POMs should be selected for defining the objective function to minimize noise effects. Unlike the conventional energy criterion that has been widely used in previous research, the new algorithm provides a good view of noise distribution between POMs. Therefore, in severe noisy conditions, it can be used as a useful tool to detect affected POMs and remove them from the calculations. Several numerical case studies and a full-scale experimental model were used to evaluate the effectiveness and capability of the proposed methodology. The case studies showed that the POD-based methodology gives encouraging results even in cases of using ambient vibration or high noisy vibration data.

Direct numerical simulation-based characterization of pseudo-random roughness in minimal channels

Direct numerical simulations (DNS) are used to systematically investigate the applicability of the minimal-channel approach (Chung et al., J. Fluid Mech., vol. 773, 2015, pp. 418–431) for the characterization of roughness-induced drag on irregular rough surfaces. Roughness is generated mathematically using a random algorithm, in which the power spectrum (PS) and probability density function (p.d.f.) of the surface height can be prescribed. Twelve different combinations of PS and p.d.f. are examined, and both transitionally and fully rough regimes are investigated (roughness height varies in the range $k^+ = 25$ –100). It is demonstrated that both the roughness function ( ${\rm \Delta} U^+$ ) and the zero-plane displacement can be predicted with ${\pm }5\,\%$ accuracy using DNS in properly sized minimal channels. Notably, when reducing the domain size, the predictions remain accurate as long as 90 % of the roughness height variance is retained. Additionally, examining the results obtained from different random realizations of roughness shows that a fixed combination of p.d.f. and PS leads to a nearly unique ${\rm \Delta} U^+$ for deterministically different surface topographies. In addition to the global flow properties, the distribution of time-averaged surface force exerted by the roughness onto the fluid is calculated. It is shown that patterns of surface force distribution over irregular roughness can be well captured when the sheltering effect is taken into account. This is made possible by applying the sheltering model of Yang et al. (J. Fluid Mech., vol. 789, 2016, pp. 127–165) to each specific roughness topography. Furthermore, an analysis of the coherence function between the roughness height and the surface force distributions reveals that the coherence drops at larger streamwise wavelengths, which can be an indication that very large horizontal scales contribute less to the skin-friction drag.

Assessment of the Current Collection Quality of Pantograph–Catenary With Contact Line Height Variability in Electric Railways

The numerical tools are widely used to facilitate the design of the railway catenary. Only radical results are obtained as the current assessment is generally performed without any practical errors. In this article, the most common error, the contact line height variability, is appropriately included in evaluating the current collection quality. Based on the real-life contact line height data collected from a high-speed network, the heights of the key points in the contact line are extracted, of which the pointwise stochastics is represented by the power spectrum density (PSD) function. Random samples are generated by the inverse Fourier transform based on the Monte Carlo method. An advanced initialization procedure is presented to properly include the contact line height variability in the initial configuration. A stochastic analysis with a high cutoff frequency is performed to evaluate a double pantograph–catenary system’s current collection quality. Unlike the deterministic result obtained by the perfect design data, the contact line height variability causes uncertainty. The present analysis method indicates some risks of safety accident and contact loss that the traditional simulation cannot capture with ideal design data or low cutoff frequency.

Measurement and Comparison Study of Road Roughness Conditions by using Two Compact Road Profiler for Evaluating Ride Quality

More recently, it is strongly required that road management involves a wide range of topics over the world and it must be considered from a perspective of customer-oriented levels of service and road administrators are requested to present a visual map of road network in terms of road network monitoring among others. To fulfill these requirements, firstly, this paper introduces new topics, which is related to two Compact Road Profiler (CRP) to monitor the road roughness conditions including localized distresses. In addition, we analyze road profile data considering two wheel paths by the survey vehicle, known as inner and outer wheel paths, and using a sophisticated driving simulator (DS) to evaluate ride quality conditions based on Power Spectrum Density (PSD) function to identify the sensitive range of ride quality. Finally, we calculate the Root Mean Square (R.M.S) and Crest Factor (CF) using vertical acceleration data of the passenger seat in DS. Consequently, it is shown that advanced evaluation of the ride quality under two wheel paths is performed. As regards standard for ride evaluation, ISO 2631 standard is adopted.

Generalized fractional Gaussian noise and its application to traffic modeling

The highlights in this paper are in two aspects. First, we introduce a type of novel fractional noise termed generalized fractional Gaussian noise (gfGn). Its autocorrelation function, power spectrum density function, and the fractal dimension are given. The second aspect is in the case study using gfGn for modeling real traffic traces to exhibit that the gfGn model is more accurate than the conventional fractional Gaussian noise (fGn) one in traffic modeling.

Sensitivity and Hessian matrix analysis of power spectrum density function for non-classically damped systems subject to stationary stochastic excitations

Calculating the first and second derivatives of Power Spectrum Density (PSD) function with respect to various design variables is a prerequisite for random responses when gradient-based algorithms are adopted. This paper presented two numerical methods to capture the sensitivity and Hessian matrix of the PSD function for non-classically damped systems subject to stationary stochastic excitations. The direct differentiate method (DDM) is adopted to develop the design sensitivity analysis (DSA). By using Pseudo-Excitation Method (PEM), the governing equations of the non-classically damped system subject to stationary stochastic excitations are transformed into a corresponding deterministic harmonic response problem. Then, the first and second derivatives of the PSD function are given in detail using the DDM. Two numerical methods, namely PEM-modal displacement method (MDM) and PEM-hybrid expansion method (HEM), are proposed to compute the sensitivity and Hessian matrix of the PSD function. The computational accuracy and efficiency of both methods are discussed and compared theoretically and numerically by two illustrations. The results indicate that both methods are valid for the DSA of the PSD function for non-classically damped systems and the PEM-HEM is more suitable than the PEM-MDM with all computational considerations.

Simulation of airfoil surface pressure due to incident turbulence using realizations of uncorrelated wall plane waves.

A numerical technique is proposed for synthesizing realizations of airfoil surface pressure induced by incoming turbulence. In this approach, realization of the surface pressure field is expressed as a set of uncorrelated wall plane waves. The amplitude of these plane waves is determined from the power spectrum density function of the incoming upwash velocity fluctuation and the airfoil aeroacoustic transfer function. The auto-spectrum of the surface pressure is obtained from an ensemble average of different realizations. The numerical technique is computationally efficient as it rapidly converges using a relatively small number of realizations. The surface pressures for different airfoils excited by incoming turbulence are numerically predicted, and the results are compared with experimental data in the literature. Further, the unsteady force exerted on an airfoil due to the airfoil-turbulence interaction is also computed, and it is shown to be in very good agreement with analytical results.

A torsional-axial vibration analysis of drill string endowed with kinematic coupling and stochastic approach

In this paper, a lumped mass model is proposed to investigate the torsional-axial vibration behavior of drill string. This model exclusively involves two statuses of axial vibration, which are the contacted status and the lifted status, and two statuses of torsional vibration, which are the slip status and the stick status. The criteria of identifying these vibration statuses are proposed. The lumped mass model concentrates the masses along the drill string and calibrates the properties of formation rock using a mass-spring-dashpot system. Instead of utilizing force coupling approaches on the bit-rock interaction boundary, a kinematic coupling approach is proposed to correlate the axial displacement of the bit to the rotation angle of the bit. The kinematic coupling function is expressed as a sinusoidal function of rotation angle with certain frequency and amplitude. The drill string responses at different drilling parameters are discussed, including the weight on bit, the rotary velocity, the borehole depth, the rock property, etc. Finally, the drill string responses are proposed in a stochastic sense by setting the kinematic coupling function be white noise. The power spectrum density functions of drilling parameters are discussed. The model in this paper kinematically couples torsional and axial vibration in stochastic sense and casts light on analyzing drill string responses in a different perspective.

Entropy Measure of Generating Random Rough Surface for Numerical Simulation of Wave Scattering

Numerical simulation of random rough surface finds wide applications in scientific disciplines, e.g., radar remote sensing of terrain and sea. In scattering simulation of rough surface, not only energy conservation must be ensured, but also, perhaps equally important, the surface inherent properties must be preserved. However, the proper choice of surface and grid sizes that are statistically representative poses a problematic issue. This study applied the entropy measure to determine such parameter settings by examining the relative error of sample entropy associated with roughness parameters and by noticing the fact that a rough surface with certain roughness parameters, including power spectrum density function, must have unique sample entropy. It is found that if the two criteria are met, proper choice of surface length and grid size is attainable to warrant minimum uncertainties of rough surfaces and maximum information content for different roughness spectra density functions under different correlation lengths. The feasibility and superiority of the proposed entropy-based method are validated in terms of minimum error of roughness parameters and also the energy conservation in bistatic scattering coefficients of rough surfaces generated using obtained simulation parameters.

Proposal of frequency‐based decomposition approach for minimization of errors in surface texture parameter calculation

In this paper, the frequency‐based decomposition approach (FBDA) for preprocessing of surface texture parameters calculation/assessment was proposed. Two types of textures were analysed: turned or grinded details. They were measured with interferometric method. For detection/minimization of high‐frequency measuring errors (HFME) the wavelets, Haar, Daubechies, Coiflet, biorthogonal or reverse‐biorthogonal approaches were proposed. Moreover, in some cases, the power spectrum density and/or autocorrelation function analysis provided an additional and/or valuable information about surface texture/features as well. It was assumed that application of various extraction methods (such as FBDA) can provide reasonable results for noise extraction in accordance to the commonly used algorithms, such as median filter, moving average (arithmetic mean) filter or other denoising procedures (e.g., Gaussian denoising filter) proposed in commercial software. The influence of proposed procedure on calculation (error minimization) of surface topography parameters (from ISO 25178 standard) was also presented. Some errors (especially those caused by false estimation of noise‐removal procedure) in surface topography parameter calculation were comprehensively studied with profile analysis; for areal HFME decomposition, the noise surface was defined. For some turned details, the HFME was observed in profiles analysis mostly. The biggest high frequencies from removed noise surface (created by noise extraction with FBDA) contained, the better results (of proposed approach) were obtained. It was also assumed that FBDA for preprocessing of both grinded and turned surfaces with various decomposition filters (in some cases the RBW) gave more ‘direct results’ that commonly used denoising approaches.

An Analytical Method for High-Frequency Electromagnetic Scattering of Arbitrary Undulating Rough Surfaces

This article presents a new approach to analytically compute high-frequency (HF) electromagnetic (EM) scattering from an arbitrary undulating rough surface according to the statistical characteristics of its geometric fluctuation. The integral expression of the scattering coefficient is derived from Kirchhoff approximation (KA) and extended to arbitrary spectra. Two methods are presented to calculate the integral: 1) the discrete Fourier transforming method based on the autocorrelation function (ACF) (KAC) and 2) the convolution method based on the power spectrum density function (KAW). For some analytically expressed spectra, the scattering coefficients can be calculated directly by the KAW and KAC methods according to its power spectrum density (PSD). The analytical expressions of classical KA for Gaussian and exponential spectrum are used to validate the proposed methods. For the case of unknown undulation height spectra of rough surfaces, the ACF is statistically calculated first, and then the KAC and KAW methods are used to calculate the average scattering coefficients numerically. Moreover, the numerical results show that the KAW method takes up more memory to retain enough energy of the ACF for accurate scattering coefficients. The two new methods are effective in the range of $\theta _{i} $ from 0° to 45° and have broad application prospects for real scenarios.