Cross Power Spectral Density Functions Research Articles

Discrimination and Experimental Study of HSCCC Operational Modal based on LSCF

The dynamic characteristics of high-speed counter-current chromatography (HSCCC) have a significant impact on its separation efficiency and equipment stability. The accurate discrimination of the dynamic characteristic during the separation process constitutes the prerequisite for clarifying the separation mechanism and performance improvement. This study establishes the HSCCC operational modal discrimination model using the least-square complex frequency domain method (LSCF) and then derives the discrimination function based on the cross-power spectral density (CPSD) function and cross-half power spectral density (PSD) function. Then an operational modal acquisition and analysis system was built for the operational modal in three directions at six common speeds. Finally, the inherent frequencies and mode of vibration of the experimental modes were verified by the Modal Assurance Criterion (MAC), Mean Phase Deviation (MPD), Modal Phase Collinearity (MPC), and Mode Indicator Functions (MIF) in this study. The results prove the feasibility of the proposed method. It shows that the CPSD function is more applicable to the HSCCC dynamic performance analysis, which should be considered in the actual separation process analysis.

Assessment of Mode Shape in Power System Using TVF-EMD and Spectral Analysis

This paper presents a robust dynamic approach for the monitoring and estimation of electromechanical oscillatory modes in the power system in real-time with less computational burden. Extensive implementation of phasor measurement units (PMU) and the utilization of advanced signal processing techniques help in identifying the dynamic behaviors of oscillatory modes. Conventional nonstationary analysis techniques are computationally weak to handle a larger quantity of data. This research utilizes the time-varying filter based empirical mode decomposition method for signal decomposition, which is highly tolerant to noise and computationally more robust. Low frequency modes are estimated by analyzing the power spectral density of the most suitable decomposed mode, the selection of which is done using correlation analysis. Instantaneous mode shapes of the signals are determined using cross-power spectral density functions, which will give the operator much information about the nature of oscillations and provide proactive steps to improve the operation of the power system. The proposed approach has been tested using signals obtained from two areas Kundur system and actual PMU data recorded from Power System Operation Corporation (POSOCO) Limited of the Indian power grid. The results confirm the superior viability and adaptability of the proposed approach in estimating the electromechanical modes. The instantaneous mode shapes are analyzed accurately with less computational complexity compared to the existing nonstationary strategies which are used for power system mode estimation.

Effect of spatial variability of earthquake ground motions on the reliability of road system

This paper presents an approach for assessing the failure probability of a road system due to its interaction with landslides induced by the earthquake. The landslide occurrence and failure probability of the road system is estimated by considering the spatial variation of earthquake ground motion. Since significant incoherence in earthquake ground motions is usually observed within the spatial dimensions of large horizontally expanded structures like roads, bridges and dams, it is important to consider the uncertainty due to spatially varying ground motion. The stochastic nature of seismic loading is incorporated in the study by generating a series of earthquake ground motions while the spatial variation of ground motion is accounted for by using cross power spectral density function. Two phase material point method (MPM) is used to simulate the dynamic behaviour of saturated slopes. MPM is able to capture the failure mechanism of slopes under seismic loading and evaluates the travel distance of sliding mass.

Cross-spectral recognition method of bridge deck aerodynamic admittance function

This study proposes a new identification algorithm about the admittance function, which can estimate the full set of six aerodynamic admittance functions considering cross power spectral density functions about the forces and the turbulence components. The method was first numerically validated through Monte Carlo simulations, and then adopted to estimate the aerodynamic admittance of a streamlined bridge deck. The identification method was further validated through a comparison between the numerical calculation and wind tunnel tests on a moving bridge section.

Analysis of propagating temperature perturbations in the primary circuit of PWRs

Temperature perturbations circulating with the primary coolant are investigated using noise data measured at nominal power of a VVER-440 reactor. An average, one-loop model is introduced to improve the through-core correlation between the inlet and outlet temperature measurements. The sources, frequency dependence and life cycle of the perturbations are determined using a kind of spectrum decomposition method. The investigation is based on the analyses of auto and cross power spectral density functions between the temperature noise measured in the hot and cold legs of the primary circuit as well as at the core outlet. Transfer properties of the loop investigated in previous papers of the authors were also used. The RMS (Root Mean Square) ratio and APSDs (Auto Power Spectral Densities) were found not to change during a fuel cycle in operation at nominal power. The feedback through the circulation of the primary coolant was found to be about 8%, it should therefore be considered in the evaluation of real measurement data. The main sources of the noise arising in the primary circuit are the steam generator, the reactor core and the upper plenum.

Subspace-based spectrum estimation in frequency-domain by regularized nuclear norm minimization

Subspace-based methods have been effectively used to estimate multi-input/multi-output, discrete-time, linear-time-invariant systems from noisy spectrum samples. In these methods, a critical step is splitting of two invariant subspaces associated with causal and non-causal eigenvalues of some structured matrices built from spectrum measurements via singular-value decomposition in order to determine model order. Mirror image symmetry with respect to the unit circle between the eigenvalue sets of the two invariant spaces, required by the subspace algorithms, is lost due to low signal-to-noise ratio, unmodeled dynamics, and insufficient amount of data. Consequently, the choice of model order is not straightforward. In this paper, we propose a new model order selection scheme that is insensitive to noise and undermodeling and based on the regularized nuclear norm optimization in combination with a recently developed subspace-based spectrum estimation algorithm which uses non-uniformly spaced, in frequencies, spectrum measurements. A detailed simulation study shows the effectiveness of the proposed scheme to large amplitude noise over short data records. Examples illustrating application of the proposed scheme to real-life problems are also presented. The proposed scheme can be readily integrated into frequency-domain instrumental variable subspace algorithms to estimate auto-power spectral density or cross-power spectral density function matrices from non-uniformly spaced, in frequencies, spectrum measurements.

Vibrations of an aramid anchor cable subjected to turbulent wind

The vibration response of an initially pre-stressed anchor cable made of parallel-lay aramid fibres excited by a measured and artificially simulated spatial turbulent wind field is presented in the paper. Results of the analyses of in situ measured wind records are described. For selected data set statistical characteristics and power spectral density functions of the measured wind velocity components are calculated. The wind stochastic velocity fluctuation is modelled as a one-variate bi-dimensional random field. Cross-power spectral density functions, at different point locations are introduced. The combination of the weighted amplitude wave superposition method (WAWS) with the Shinozuka–Deodatis method is used for the analyzed problem. A time-dependent behaviour of the synthetic cable is investigated which is subjected to turbulent wind with large expected oscillations that arise as a result of slackening due to the relaxation effects. A nonlinear transient dynamic analysis is used in conjunction with the finite element method to determine the dynamic response of the cable subjected to turbulent wind at its initially prestressed state and in the selected times after the relaxation effect. The constitutive equation of the relaxation of the aramid cable follows an experimentally obtained law of the logarithmic type. To monitor the dependences of the individual quantities of cable vibration in the phase space, attractors and Poincaré maps are created by sampling the cable’s displacement and velocity at periods of relevant frequencies. Interesting findings based on the response of the cable with rheological properties to turbulent wind are presented.

Modeling of wall pressure fluctuations for finite element structural analysis

This paper investigates a modeling technique of wall pressure fluctuations (WPF) due to turbulent boundary layer flows on a surface for finite element structural analysis. This study is motivated by critical issues of structural vibrations due to turbulent WPF over the surface of a body. The WPFs are characterized with random behavior in time and space. The temporal and spatial random behavior of the WPFs is mathematically expressed in the form of the auto- and cross-power spectral density functions (PSDF) in the frequency domain (e.g., Corcos Model). For finite element modeling of the random distributed fluctuations, the cross-PSDF is properly modeled over the finite element structural mesh. The quality of modeling of the cross-PSDF is directly affected by the structural mesh size. We first examine the maximum mesh size required for reliable finite element analysis. The reliability of the FEA results is confirmed by the theoretical results. It is found that the maximum mesh size should be determined under consideration of the spatial distribution of the cross-PSDF in addition to the representation of dynamic behavior of the structure in the frequency range of interest. It is also recognized that the requirement of the maximum mesh size is unrealistic in many practical cases. We then investigate practical modeling schemes under a realistic mesh size condition. We found that the WPF can be modeled without the exact consideration of cross-correlation if the power due to the wall pressure fluctuation can be properly compensated. This is owing to the feature of decreasing the cross-correlation of WPFs at high frequencies and the fact that the WPF does weakly couple into the structural modes at high wavenumbers such that 2 π f / U c < k max . The wall pressure fluctuations can therefore be modeled as uncorrelated loadings with power compensations.

Property of critical excitation for moment-resisting frames subjected to horizontal and vertical simultaneous ground motions

It has often been reported that, when building structures are subjected to near-fault earthquake ground motions, horizontal and vertical impulsive inputs may cause critical damage during the first few seconds. In practical design of building structures, however, the safety check, taking into account the effect of multi-component ground motions, is hardly conducted except the design of important structures such as high-rise buildings and nuclear power plants. Furthermore, it is not clear how the correlation of multi-component ground motions influences the actual safety of structures. In this paper, the detailed property of critical excitation is discussed in association with the relationship between the characteristics of ground motions and those of structures. The properties of various auto power spectral density (PSD) functions of the horizontal and vertical ground motions are investigated, and those of the critical cross PSD function of these two-directional ground motions are found by a devised algorithm in a feasible complex plane. A closed-form expression is derived from the critical relation of the auto PSD functions of the simultaneous inputs. This critical excitation method provides us with a new approach for earthquake-resistant design against the possible future earthquake which causes the critical damages to buildings.

Critical correlation of bi-directional horizontal ground motions

A stochastic model is treated of bi-directional horizontal ground motions (2DGM). It is shown that, in comparison with the Penzien–Watabe model (1975) [3], the cross power spectral density (PSD) function between 2DGM along the building structural axes can be treated in a more general manner by using an extended Penzien–Watabe model introduced in this paper. The auto PSD functions of 2DGM along the building structural axes are assumed to be given and the cross PSD function between these 2DGM is treated as a complex unknown function. A critical excitation problem is then considered for a one-story one-span moment resisting three-dimensional frame. The corner-fiber stress at the column-end is taken as the objective function and the worst cross PSD function of the 2DGM is determined so that the maximum corner-fiber stress at the column-end is maximized. It is shown that the real part (co-spectrum) and the imaginary part (quad-spectrum) of the worst cross PSD function can be obtained by a devised algorithm including the interchange of the double maximization procedure in the time and frequency domains.

Wind-induced fatigue of structures under neutral and non-neutral atmospheric conditions

The authors of this paper have carried out a wide research project aimed at determining the wind-induced fatigue of structures, hypothesising the wind field as neutrally stratified, as it is typical of evaluations addressed to ultimate loading conditions. As wind-induced fatigue is sensitive to moderate wind velocities, for which stable or unstable atmospheric conditions can occur, this paper extends the procedures previously established to non-neutral thermal stratifications. The mean wind velocity profile and the cross-power spectral density functions of the atmospheric turbulence are expressed in terms of the Monin–Obukhov length. The occurrence of stable, unstable and neutral conditions is taken into account by means of the joint probability distribution of the mean wind velocity and direction and of the Monin–Obukhov length. The effects of non-neutral stratifications on the structural response and on the mean fatigue life are preliminary discussed with reference to an example concerning a steel chimney.

Monte Carlo simulation for the response analysis of long-span suspended cables under wind loads

This paper presents a time-domain approach for analyzing nonlinear random vibrations of long-span suspended cables under transversal wind. A consistent continuous model of the cable, fully accounting for geometrical nonlinearities inherent in cable behavior, is adopted. The effects of spatial correlation are properly included by modeling wind velocity fluctuation as a random function of time and of a single spatial variable ranging over cable span, namely as a one-variate bi-dimensional (1V-2D) random field. Within the context of a Galerkin`s discretization of the equations governing cable motion, a very efficient Monte Carlo-based technique for second-order analysis of the response is proposed. This procedure starts by generating sample functions of the generalized aerodynamic loads by using the spectral decomposition of the cross-power spectral density function of wind turbulence field. Relying on the physical meaning of both the spectral properties of wind velocity fluctuation and the mode shapes of the vibrating cable, the computational efficiency is greatly enhanced by applying a truncation procedure according to which just the first few significant loading and structural modal contributions are retained.

USING THE CROSS-CORRELATION TECHNIQUE TO EXTRACT MODAL PARAMETERS ON RESPONSE-ONLY DATA

Modal parameter identification is used to identify those parameters of the model which describe the dynamic properties of a vibration system. Classical modal parameter extractions usually require measurements of both the input force and the resulting response in laboratory conditions. However, when large-scale operational structures are subjected to random and unmeasured forces such as wind, waves, or aerodynamics, modal parameters estimation must base itself on response-only data. Over the past years, many time-domain modal parameter identification techniques from output-only have been proposed. Among them, the natural excitation technique (NExT) has been a very powerful tool for the modal analysis of structures excited in operating environment. This issue reviews the theoretical development of natural excitation technique (NExT), which uses the cross-correlation functions of measured responses coupling with conventional time-domain parameter extraction under the assumption of white-noise random inputs. Then a frequency-domain poly reference modal identification scheme by coupling the cross-correlation technique with conventional frequency-domain poly reference modal parameter extraction is presented. It uses cross-power spectral density functions instead of frequency response functions and auto- and cross-correlation functions instead of impulse response functions to estimate modal parameters from response-only data. An experiment using an airplane model is performed to investigate the effectiveness of the cross-correlation technique coupled with frequency-domain poly reference modal identification scheme.

On cross-correlation function evaluation from a periodically time-varying digital filter's output

Concise mathematical models are established for one-dimensional, linear, periodically time-varying digital filters. The three representations considered include linear difference equations, Green's function, and state-space structures. Linear time-invariant equivalent descriptions are then formulated, and from these, for the particular case of a white noise input, various analytical expressions are derived for both the cross-power spectral density functions and cross-correlation functions of the filter's subsampled output. These results are compared and contrasted with similar formulas for a linear time-invariant system.

Critical cross power spectral density functions and the highest response of multi-supported structures subjected to multi-component earthquake excitations: A. Sarkar & C. S. Manohar, Earthquake Engineering & Structural Dynamics, 25(3), 1996, pp 303–315

Critical cross power spectral density functions and the highest response of multi-supported structures subjected to multi-component earthquake excitations: A. Sarkar & C. S. Manohar, Earthquake Engineering & Structural Dynamics, 25(3), 1996, pp 303–315

CRITICAL CROSS POWER SPECTRAL DENSITY FUNCTIONS AND THE HIGHEST RESPONSE OF MULTI-SUPPORTED STRUCTURES SUBJECTED TO MULTI-COMPONENT EARTHQUAKE EXCITATIONS

The highest response of multi-supported structures subjected to partially specified multi-component earthquake support motions is considered. The seismic inputs are modelled as incompletely specified vector Gaussian random processes with known autospectral density functions but unknown cross spectral densities and these unknown functions are determined such that the steady state response variance of a given linear system is maximized. The resulting cross power spectral density functions are shown to be dependent on the system properties, autospectra of excitation and the response variable chosen for maximization. It emerges that the highest system response is associated neither with fully correlated support motions, nor with independent motions, but, instead, specific forms of cross power spectral density functions are shown to exist which produce bounds on the response of a given structure. Application of the proposed results is demonstrated by examples on a ground based extended structure, namely, a 1578 m long, three span, suspension cable bridge and a secondary system, namely, an idealized piping structure of a nuclear power plant.

The mapping of vortex fields around single and dual bluff bodies

The development and use of a system to measure important parameters of the vortex field in a coordinate grid around several single bluff bodies and dual bluff body combinations are described. Two hot wire sensors were used: a reference sensor at a fixed position and a moving sensor with position coordinates adjusted by using computer control. The strength, frequency and regularity of the vortex shedding are found from the auto-power spectral density of the moving sensor signal. The phase difference between moving and reference sensor signals is found from the cross-power spectral density function between the signals. The results are presented as maps of vortex parameters plotted as a function of sensor position coordinates. The main features of these maps, including the enhancement of vortex shedding from certain dual bluff body combinations, is then discussed.

Asymptotic Frequency Behavior of the Two-Detector Cross Power Spectral Density Function for a Propagating Perturbation of Moderator Density

A number of investigators have discussed the use of two-detector cross power spectral density (CPSD) measurements to obtain the velocity of an axially propagating perturbation of the moderator dens...

Dynamic loading of containment during blowdown: review of experimental data from marviken and brunsbüttel

This paper deals with the problem of the dynamic loading of the pressure suppression type containment due to pressure fluctuations which are induced during blowdown as a result of steam condensation in the suppression pool. It reviews the experimental data on pressure fluctuations and on the corresponding dynamic response of the structures submerged in the water charge. Two series of containment response experiments were performed on real pressure suppression systems. One series of measurements was performed within the framework of relief valve vent tests in the Brunsbüttel Power Plant (Federal Republic of Germany) in 1974. Another series of measurements was carried out within the framework of the blowdown tests performed at the Marviken Power Plant (Sweden) in 1972–1973. For the measurements of pressure fluctuations in the suppression pool strain gage-based pressure transducers were used. The dynamic response of the loaded structure was measured by strain gages or by piezoresistive (in some cases piezoelectric) accelerometers. The signal was amplified by carrier-frequency amplifiers (or loading amplifiers). The amplified signals were recorded on magnetic tape (FM recording). A two-step evaluation procedure was used. In the first step the time history plots of the variables directly measured (such as pressure, bending strain and acceleration) were plotted and interpreted. In the second evaluation step the cross-power spectral density function of different signal pairs was calculated, plotted and interpreted. For evaluation of the fluctuating pressure field in the suppression pool the combination was chosen of a reference pressure signal (measured at a fixed position called ‘reference position’) with a second pressure signal (measured at a different, arbitrary position) in most cases to perform the cross-spectra analysis. The set of cross-spectral density functions obtained by analyzing all signal combinations chosen was used to determine the power spectra, RMS amplitudes and phases of the corresponding pressure waves impinging on the structure submerged in the suppression pool. For the evaluation of pressure field—structural response relations one pressure signal was combined with one response signal (bending strain or acceleration) to give the input for cross-spectral analysis. By the combination of two response signals (bending strain—bending strain or acceleration—acceleration) for cross-spectral analysis the modal shapes of the responding structure was determined.

地動が基礎の各点で異なる構造物の応答解析 : その 1. 入力地動について

When the structures are subjected to multi-input seismic waves, two methods which are spectral analysis and time history analysis, are considered. In this paper the data which are needed in the analyses are researched. The theoretical correlation functions and spectral density functions between acceleration, velocity and displacement in three cases, are obtained. Those three cases are (a) the same seismic waves, (b) the seismic waves which have the same wave and the time lag and (c) the seismic waves which have only different Fourier amplitude spectra. From mean values of several power spectral density functions in accelerations, one equation is made, and mean square power and cross power spectral density functions are obtained. As the time history analysis, the method (a) by Fourier trans-form and the method (b) that the simulation of the seismogragh is used, are studied. The method (a) is the way that the lower frequency part of the Fourier amplitude spectra is cut off. The relations of the maximum values of the result's acceleration, velocity and displacement, are given with the cut-off frequency ω_c. The best result is given when the cut-off frequency ω_c equals to 1.0. The method (b) is the way that uses the filter which is disital simulation of simple pendulum. The velocity is obtained by the filter of which the period is too long, and the displacement is obtained by the filter of which the period and the damping coefficient are respectively 2.0sec. and 0.8, and the displacement's amplification factor equals to 2.7.