Response Power Spectral Density Research Articles

Calculation of Evolutionary Power Spectral Density of Bridge Response to Earthquakes by Convolution Summation

This paper presents a method for calculating the evolutionary power spectral density (EPSD) of the seismic response of bridges using the convolution summation. With zero initial values, a formula for the dynamic component of the response of bridges to spatially varying seismic ground motions is derived as the convolution summation, by assuming the seismic acceleration to vary linearly between two adjacent time stations. The convolution summation is used for calculating the convolution integral of the dynamic component response factor in the EPSD, in which the constant coefficients are independent of the harmonically modulated excitation. The constant coefficients are obtained by the time-history analysis using triangular unit impulse acceleration excitations. The computational cost of the EPSD depends mainly on the amount of degrees-of-freedom (DOFs) numbers of bridge supports in contact. The corresponding computational scheme is proposed, and its validity is indirectly verified with a single DOF system, by comparing the results obtained with those of the existing methods. Finally, a three-span continuous rigid-frame bridge is taken as a case study to illustrate the applicability and effectiveness of the proposed scheme.

Random Vibration Analysis of Coupled Three‐Dimensional Maglev Vehicle‐Bridge System

This paper presents a framework for the linear random vibration analysis of the coupled three‐dimensional (3D) maglev vehicle‐bridge system. Except for assembling the equation of motion of vehicle only via the principle of virtual work, the fully computerized approach is further expanded to assemble the governing equation of fluctuating current via the equilibrium relation. A state‐space equation couples the equation of motion of the vehicle and the governing equation of fluctuating current. The equation of motion of a real three‐span space continuous girder bridge is established by using finite element methods. A separated iteration method based on the precise integration method and the Newmark method is introduced to solve the state‐space equation for the maglev vehicle and the equation of motion for the bridge. Moreover, a new scheme to application of the pseudoexcitation method (PEM) in random vibration analysis is proposed to maximize the computational efficiency of the random vibration analysis of the maglev vehicle‐bridge system. Finally, the numerical simulation demonstrates that the proposed framework can efficiently obtain the mean value, root mean square (RMS), standard deviation (SD), and power spectral density (PSD) of dynamic response for the coupled 3D maglev vehicle‐bridge system.

A new modeling framework for fatigue damage of structural components under complex random spectrum

Time and frequency domains-based fatigue damage prediction approaches have been developed over past decades to predict fatigue performance of mechanical structures subjected to random loads. Frequency domain approaches are increasingly being adapted to provide fatigue assessment of mechanical components subjected to random loads due to computational efficiency and cost savings. Current frequency domain damage models only deal with stationary random loadings where Power Spectral Density (PSD) of random loadings does not change in time. However, many machine components, such as jet engines and tracked vehicles are subjected to evolutionary PSD i.e. random-on-random loadings under real service loads. A new fatigue damage modeling framework is proposed to predict fatigue damage of structures under complex evolutionary PSD where the topology of PSD function changes with time. The proposed modeling approach is based on the underlying concept that the evolutionary PSD response of a structure can be decomposed into a finite number of discrete PSDs. Each PSD can be split into narrow frequency bands so that each of narrowbands can be associated with Rayleigh distribution of stress cycles. Fatigue damage can then be predicted by summing up damages for each individual band and each discrete PSD function on the basis of a damage accumulation rule. The proposed modeling approach is numerically and experimentally validated by a finite element method and experiments using three simplified structures made of 5052-H32 aluminum alloy. The proposed approach provides a more efficient and accurate modeling technique, and account for complex random loadings of structural components.

Power Spectral Density Response of Bridge‐Like Structures Loaded by Stochastic Moving Forces

In this study, simple and manageable closed form expressions are obtained for the mean value, the spectral density function, and the standard deviation of the deflection induced by stochastic moving loads on bridge‐like structures. As a basic case, a simply supported beam is considered, loaded by a sequence of concentrated forces moving in the same direction, with random instants of arrival, constant random crossing speeds, and constant random amplitudes. The loads are described by three stochastic processes, representing an idealization of vehicular traffic on a bridge in case of negligible inertial coupling effects between moving masses and structure. System’s responses are analytically determined in terms of mean values and power spectral density functions, yielding standard deviations, with the possibility to easily extend the results to more refined models of single span bridge‐like structures. Potential applications regard structural analysis, vibration control, and condition monitoring of traffic excited bridges.

Multi-exciter stationary non-Gaussian random vibration test with time domain randomization

Abstract This paper presents a new control method for multi-input multi-output stationary non-Gaussian random vibration test using time domain randomization. The control objectives are composed of response skewnesses, kurtoses and power spectral densities. The generation process of stationary and coupled reference non-Gaussian signals by specified reference skewnesses, kurtoses and spectra is analyzed. The reference non-Gaussian signals combined with system frequency response functions are then utilized to obtain the desired drive signals for dynamic inputs, in which the inverse system method in the frequency domain is employed. The primary advantages of the proposed methods are the high computational efficiency and simultaneous control of the time-frequency characteristics of response signals. In consideration of system cross coupling characteristics manifested in coherence and phase coefficients, the skewness and kurtosis tuning steps for each control channel are formulated by using a sequential phase modification method. The relationships between reference skewnesses, kurtoses and spectra are discussed and they reveal that the reference spectra have an influence on the settings of reference skewnesses and kurtoses, which implies that proper settings of reference skewnesses, kurtoses and spectra are necessary. A numerical example and a triaxial vibration test are provided and the results show the validity and feasibility of the proposed method.

Research on non-stationary wind-induced effects and the working mechanism of full scale super-large cooling tower based on field measurement

Recent measurement results indicate that the wind-induced response of super-large cooling tower (include super-large cooling tower in wet-cooling-systems and super-large cooling tower in dry-cooling-systems) exhibit remarkable non-stationary feature. In this study, wind-induced response signals of a typical super-large cooling tower under real Reynolds number and turbulent conditions were collected based on field measurement. Meanwhile, some other super-large cooling towers were measured to verify the research in this paper. On these bases, non-stationary and non-Gaussian features of wind-induced response of super-large cooling tower were studied. And evolutionary power spectral density of wind-induced response of super-large cooling tower were analyzed. Furthermore, the resonance effect were discussed by studying proportion of resonance component in wind-induced response. Finally, the extreme response and damping ratios considering non-stationary feature of wind-induced effect were obtained in this paper. The results show that the wind-induced response of super-large cooling tower is characterized by stable frequency evolution characteristics and non-stationary evolution characteristics in intensity aspect. Some detailed research conclusions can provide references for wind resistance safety design of super-large cooling tower and disaster prevention study of similar tall thin-walled structure.

Monte Carlo investigation of noise and high-order harmonic extraction in graphene

High-order harmonic generation is an interesting option to reach the THz range for generation and detection applications. In this work the time-domain and spectral behavior of carrier velocity fluctuations in free-standing graphene under the action of a high frequency alternating electric field are studied by means of an ensemble Monte Carlo simulator. The distortion of the velocity response is explained in terms of the carrier wavevector distribution, average energies and scattering activity. The intensity of the harmonics is analyzed for several frequencies and electric field amplitudes. Noise and regular contributions to the power spectral density of the velocity response are discerned, and the harmonic modes available for several bandwidths are discussed. Results show that it is possible to reach the THz range with conditions comparable to those found for III–V materials, at the expense of narrow bandwidths.

Elastic perfectly plastic oscillator under random loads: Linearization and response power spectral density

The randomly-excited elastic-perfectly-plastic oscillator—hysteretic bilinear oscillator with zero secondary stiffness—has been extensively researched. The vast majority of the work on that system has investigated the time-domaine statistics of the response. No studies have focused on the power spectral densities. This study specifically examines the system's velocity power spectral density—the system's displacement is nonstationary—under wide-band random excitations by means of a statistical linearization/stochastic averaging technique that is developed in one existing and two new procedures, one with constant, the other with amplitude-dependent parameters. The three procedures are evaluated against Monte Carlo simulation and classical Gaussian linearization in terms of the velocity average power, the peak frequency of the power spectral density, its peak value, its bandwidth, and its overall shape around the main frequency. The best predictions were yielded by the new procedure with amplitude-dependent parameters, which combines an extended amplitude-phase transformation, a linearization with random parameters into a Maxwell system, a correlation-function-based criterion, the conditional power spectral density concept, and a power conservation correction step. The new procedures can be adapted to apply to other hysteretic systems.

Dynamic response of tank trains to random track irregularities

This paper presents a dynamic analytical model for tank train vibrations. The train is considered as a system of 27 degrees of freedom consisting of lateral, roll, yaw, vertical, and pitch motions for the vehicle body and its two bogies and lateral, roll and vertical motions for the four wheel-sets. Liquid sloshing in the tank is modeled using an equivalent mechanical mass-spring model. Coupling between the vehicle system and the railway track is realized through the interaction forces between the train and the rail, where the vertical and lateral irregularity profiles of the track are regarded as stationary ergodic Gaussian random processes and simulated by polynomial functions. Random vibration theory is used to obtain the response power spectral densities. Finally, numerical results for a typical test case including natural frequencies of a coupled system, frequency response functions, and output power spectral densities are presented.

Nonstationary seismic response analysis of long-span structures by frequency domain method considering wave passage effect

In this paper, a frequency domain method is proposed for the nonstationary seismic analysis of long-span structures subjected to random ground motions considering the wave passage effect. Based on the correlation analysis theory and fast Fourier transform (FFT), a semi-analytical solution is derived for the evolutionary power spectral density of the random response of long-span structures in the frequency domain. The expression of this solution indicates that the evolutionary property of nonstationary random responses can be determined completely by the modulation function of random ground motions, and hence the solution has clear physical interpretations. For slowly varying modulation functions, the FFT can be implemented with a small sampling frequency, so the present method is very efficient within a given accuracy. In numerical examples, nonstationary random responses of a long-span cable stayed bridge to random ground motions with the wave passage effect are studied by the present method, and comparisons are made with those of the pseudo excitation method (PEM) to verify the present method. Then the accuracy and efficiency of the present method with different sampling frequencies are compared and discussed. Finally, the influences of the apparent velocity of the seismic waves on nonstationary random responses are investigated.

Effect of interlaminar flaw on composite panel behaviour under acoustic loading

A design of airframe structures has to be accompanied by material properties testing. Special focus has to be laid on high frequency fatigue behaviour. The acoustic fatigue test equipment for grazing wave’s incidence was designed based on the FE analyses. Flat composite panels were designed and manufactured using the Hexply 8552/AGP193-PW prepreg with different fiber lay-ups and with simulation of production imperfections or operational damage. Dynamic behaviour of panels was tested using three regimes of acoustic loading: white noise spectrum, engine noise spectrum and discrete harmonic frequencies loading. Rayleigh damping coefficients were evaluated for later use in FE models. Panel health was checked periodically through a power spectral density response on white noise signal. The ultrasonic NDT instruments were used for monitoring of relevant delamination increment. No increments of delamination were detected. The available acoustic power was below the fatigue limit of the tested structure.

湍流边界层作用下薄板随机振动声辐射的辛方法

The random sound radiation of thin plates subjected to turbulent boundary layer (TBL) excitations was studied in the symplectic duality system. Firstly, the cross power spectral density of the TBL was represented by a Fourier series, and the problem of the random sound radiation of structures excited by a random field was reduced to solve the deterministic response function, i.e. the structural response to a spatial and temporal harmonic pressure of unit magnitude. Secondly, the free vibration analysis of thin plates was introduced to the symplectic duality system, then a symplectic eigenproblem was formed with the method of separation of variables. Finally, the decoupled governing equations were derived through expansion of the response and excitation vectors in the symplectic space, to reduce the difficulty of solving the equations, and the symplectic analytical solution was obtained. In contrast to the modal decomposition method (MDM), the presented method is formulated in the symplectic duality system and does not need modal truncation, hence the computations are of high precision. In the numerical examples, the harmonic response functions for the thin plate were studied, and a comparison was made with the MDM to verify the effectiveness of the presented method. Thereafter, the sound pressure levels (SPL) of the power spectral density of the sound pressure response to the TBL were obtained, the convergence induced by the Fourier series expansions was examined, and the directivity functions of the radiation sound field were extensively investigated.

Probability distributions control for multi-input multi-output stationary non-Gaussian random vibration test

An inverse system method for multi-input multi-output stationary non-Gaussian random vibration test is proposed in the paper to control the response characteristics both in time-domain and frequency-domain simultaneously. Hence, the control objectives are not only the traditional response power spectral densities but also probability distributions. The control of the probability distributions is more comprehensive and reasonable than the control of kurtoses in order to duplicate the time-domain characteristics of the measured data because kurtoses provide only partial behaviors of the probability distributions. The nonlinear transformation process is introduced to generate non-Gaussian random signals with specified probability distributions. To obtain the desired vibration environments for the test, the reference response signals are synthesized first by the reference spectra and probability distributions, and then the coupled drive signals are generated via the inverse system in the time domain. The close loop correction algorithms are implemented to update the drive signals, actually to update the reference response signals according to the deviations between the references and the measured responses. Finally, a numerical example by a cantilever beam and a biaxial vibration test are carried out and the results demonstrate the effectiveness and feasibility of the proposed method.

Strong wind characteristics and dynamic response of a long-span suspension bridge during a storm

As Storm Tor struck the western coast of Norway, wind speeds and bridge deck accelerations along the Hardanger Bridge girder were recorded by the monitoring system installed on the bridge. Using 13.5 h of data, mean wind speed, turbulence intensities, gust factor, turbulence length scales, angle-of-attack, and one-point and two-point turbulence spectra are studied using 10-minute stationary averaging intervals. Using the measured turbulence statistics as inputs, the buffeting response of the bridge deck is calculated in the frequency domain. The calculated response is compared with the measured response in terms of the root-mean-square (RMS) of acceleration and displacement components and the power spectral density of the acceleration response. Significant discrepancies are found in the case of the vertical response. Predicting the spectral response is found to be more difficult than predicting the RMS response, in particular for high-frequency responses. Considering the spanwise non-uniformity of turbulence statistics did not affect the predictions significantly.

On the dynamics of non-local fractional viscoelastic beams under stochastic agencies

Abstract Non-local viscoelasticity is a subject of great interest in the context of non-local theories. In a recent study, the authors have proposed a non-local fractional beam model where non-local effects are represented as viscoelastic long-range volume forces and moments, exchanged by non-adjacent beam segments depending on their relative motion, while local effects are modelled by elastic classical stress resultants. Long-range interactions have been given a fractional constitutive law, involving the Caputo's fractional derivative. This paper introduces a comprehensive numerical approach to calculate the stochastic response of the non-local fractional beam model under Gaussian white noise. The approach combines a finite-element discretization with a fractional-order state-variable expansion and a complex modal transformation to decouple the discretized equations of motion. While closed-form expressions are derived for the finite-element matrices associated with elastic and fractional terms, fractional calculus is used to solve the decoupled equations of motion, in both time and frequency domain. Remarkably, closed-form expressions are obtained for the power spectral density, cross power spectral density, variance and covariance of the beam response along the whole axis. Time-domain solutions are obtained by time-step numerical integration methods involving analytical expressions of impulse response functions. Numerical examples show versatility of the non-local fractional beam model as well as computational advantages of the proposed solution procedure.

Nonlinear random vibrations of plates endowed with fractional derivative elements

This paper deals with the problem of determining the nonlinear response of a plate endowed with fractional derivative elements and exposed to random loads. It shows that an approximate solution of the nonlinear fractional partial differential equation governing the plate vibrations can be obtained via a statistical linearization based approach. The approach is implemented by considering a time-dependent representation of the response involving the eigen-functions of the linear problem. This representation allows deriving a nonlinear fractional ordinary differential equation governing the variation of the time-dependent part of the response, which is linearized in a mean square sense. Then, an iterative procedure provides the response statistics and power spectral density functions.Next, a Boundary Element Method is proposed for conducting relevant Monte Carlo data. The method is developed in conjunction with a Newmark integration scheme for estimating the response in the time domain given spectrum compatible realizations of the excitation. Monte Carlo data and statistical linearization solutions are calculated for square plates with simply supported stress-free edges, but problems involving other boundary conditions can be solved by the proposed approach.

DAMAGE QUANTIFICATION OF BEAMS USING FREQUENCY SIGNATURE

A rapid method for determining the damage severity sustained by a beam proved to bechallenging due to either limited studies conducted on the subject or alternative methods require highlysophisticated and costly equipment to perform. In this research, the unique frequency signature emitted by abeam when excited by an external force was utilized in order to determine the changes in the properties of thebeam. Experiments were performed using a roving accelerometer hammer impact test on a beam with agrounded configuration to test the changes occurring as the controlled damage sustained by the beam increases.The acceleration response of the beam obtained from the experiment is then processed using softwareincorporating Kalman Filter and structural dynamics. Results show that the dominant frequency obtained in boththe Fast Fourier Transform and Power Spectral Density of the acceleration response of the beam decreases as thedamage incurred by the beam increases. The results also show that regardless of the position of theaccelerometer, dominant frequencies tend to converge to a value depending on the damage sustained in the beam.Damping ratio of the beam also decreased as the damage sustained by the beam increased. Inversely, the increasein damage of the beam corresponds to an increase in the dissipation rate of the beam. The study was able toachieve its goal of quantifying damage in a beam through the use of frequency signature by identifying thechanges in its dominant frequencies and the damping ratio and dissipation rate.

Fatigue life prediction of Package-on-Package stacking assembly under random vibration loading

This paper presents a general methodology to predict the fatigue life of the Package-on-Package (PoP) under random vibration loading by means of vibration tests and finite element (FE) simulation. The behavior of the critical solder joints of the PoP under vibration loading was accurately described by FE model using ANSYS software and confirmed by modal analysis and linear sweep tests. The stress-life (S-N) curve of the PoP solder joints was obtained by the sinusoidal fatigue vibration tests and FE simulations to characterize the fatigue properties of the PoP under vibration. The comparison of the S-N curves of the SAC305 solder joints in different structures indicates the S-N curve of the SAC305 solder joints depends on the package structure. With the same fatigue cycles, the stress levels for the current 3D package is the smallest compared with the other 2D package due to the stacked structure of the PoP. Spectrum analysis for the random vibration tests was performed by the FE simulations to acquire the stress response power spectral density (PSD) of the critical solder joint. A specific frequency domain approach based on the Palmgren-Miner's rule was established to predict the fatigue life of the PoP under random vibration. Results show that simulated fatigue life matches well with the random vibration test results, which implies that this approach could be a potential method for the predication of fatigue life of the 3D packaging under random vibration.

Damage detection by a FE model updating method using power spectral density: Numerical and experimental investigation

This paper investigates the viability of damage detection using power spectral density (PSD) of structural response both numerically and experimentally. The paper establishes a sensitivity based damage detection method to use PSD. The advantages of PSD as a model updating metric are explained and its challenges are addressed. An approximate frequency response function of damaged model is used to redeem the method for the effect of incomplete measurement. The robust solution of the developed sensitivity equation is achieved through a least-squares error minimization scheme, and the challenging issues are discussed. The ability of the method in localizing and quantifying the damage and its robustness against measurement and modeling errors is investigated by a numerical example. Experimental vibration test data of a laboratory concrete beam with various level of distributed damage is used to probe the method in practical conditions. The results show that PSD of response can be used to detect damages in lower frequency ranges with acceptable accuracy.

Extraction of damage-sensitive eigen-parameters for supervised SHM

These last decades have seen an exponential increase in the amount of research related to structural health monitoring (SHM) due to its potential for significant life-safety and economic benefits. However, the success of this powerful tool strongly depends on the implemented damage identification strategy. Reliable and efficient damage identification algorithms enable to detect faults that lie beneath the surface of the structure and to spot system’s vulnerabilities at a very early-stage. This allows to adopt appropriate remedial measures in a timely fashion thereby minimizing the risk of unexpected collapses.The present paper describes a spectrum-driven damage identification method that investigates three levels of damage, i.e. detection, localisation and assessment. Peculiarity of the method is the use of spectral frequency -dependent Eigen-parameters estimated from the response Power Spectral Density (PSD) matrix, which is demonstrated to be very sensitive to damage-induced changes. The approach is detailed, including initial assumptions, scientific formulation of the problem and derivation of the algorithm. Finally, the effectiveness of the method is validated through a numerical simulation and verified on a case-study structure.