Response Power Spectral Density Research Articles

A perspective on conditional spectrum-based determination of response statistics of nonlinear systems

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system's displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Random Vibration Analysis of Double Deck Rocking Self-Centering Pier Under Seismic Excitation

This paper investigates nonlinear random vibration of double deck rocking self-centering pier structure subjected to seismic excitation. First of all, a novel type of double deck rocking self-centering pier that offers better resilience is designed. The stochastic dynamical model of the system is then established, in which the seismic excitation is viewed as Kanai–Tajimi filtered white noise and the self-centering restoring force is characterized by classical flag-shaped hysteretic model. Subsequently, the self-centering restoring force is decomposed into equivalent quasi-linear elastic and damping forces by adopting the harmonic balance (HB) scheme. Following this, the stochastic averaging (SA) technique is implemented to derive the averaged Itô equation. The steady-state response probability density with respect to the amplitudes is solved from the averaged Fokker–Planck–Kolmogorov (FPK) equation. Finally, the effects of system parameters on the stochastic response of the double deck rocking self-centering pier structure are performed, and validated by the Monte Carlo simulation (MCS). In addition, with the help of the Laplace transform, the transition function as well as conditional power spectral density are achieved, which are combined with the steady-state probability density to obtain the power spectral density response. This research will contribute to the optimal seismic design of double-deck rocking self-centering piers.

Research on control parameters of high-speed maglev train under stochastic track irregularities

During operational phases, maglev trains encounter track irregularities, presenting a formidable challenge to the integrity of their control systems owing to the inherently stochastic nature, a challenge accentuated in the context of high-speed maglev trains. This study aims to comprehensively examine control parameters within the context of stochastic excitation resulting from track irregularities. Through the development of a rigorous stochastic response analysis model utilizing the pseudo excitation method, the investigation delves into the dynamics of an EMS-type high-speed maglev train-rigid track system. Evaluation indices, derived from the power spectral density of the maglev train's response, are established to quantify both control stability and operational stability. Subsequently, based on these indices, a recommended range of PD control parameters is proposed, accounting for the requirements stipulated by the evaluation metrics. The study elucidates that variation in PD control parameters exert discernible effects on the system's stability, primarily altering the nature frequency and damping ratio. Specifically, control stability demonstrates a positive correlation with increasing PD control parameters, while operational stability exhibits a nuanced relationship—initially bolstered by escalating proportional coefficients but subsequently tempered, albeit gradually, with heightened derivative coefficients. The delineated range of values for PD control parameters is meticulously determined, considering pertinent physical parameters of the electromagnetic system, such as mass, static suspension current, and static suspension gap, alongside factors like the Sperling indicator and suspension gap variation.

Vibration analysis of an open cylindrical thin shell under random excitation

This paper introduces an analytical approach to the random vibration of a cylindrical thin shell structure with an opening angle of 180°. The approach involves employing the exact analytical solution for the natural vibration of a thin shell structure with an opening, grounded in a continuum model, and a pseudo-excitation method to derive the exact analytical solution of the random response power spectral density for the open cylindrical thin shell structure. Simultaneously, a comparison is made between the results obtained through the analytical method and those obtained by using the finite element program, confirming the effectiveness of the analytical model constructed in this study. Finally, an analysis is conducted on the impact of thickness on the characteristics of the vibration response of the open cylindrical thin shell.

Transmissibility-based operational modal analysis: A unified scheme and uncertainty quantification

The transmissibility-based Operational Modal Analysis (OMA) technique has attracted substantial research interest over the past two decades due to its capability to extract structural modal parameters in the presence of non-white excitation. The transmissibility function used in OMA can be categorized as Response Transmissibility (RT) and Power Spectral Density Transmissibility (PSDT). Modal parameters can be identified by combining RTs from multiple load cases or PSDTs from multiple transfer outputs. Nevertheless, the mathematical formulations and implementation procedures of the various RT-based and PSDT-based methods exhibit significant differences. Furthermore, the uncertainty associated with modal parameter estimates is rarely assessed in these methods. This work aims to propose a scheme that unifies various transmissibility-based OMA algorithms through a consistent parametric formulation. Within this scheme, a universal parametric model and a standardized implementation procedure are developed to accommodate two types of transmissibility in both time and frequency domains and six OMA algorithms; additionally, a generalized framework for quantifying the uncertainties associated with modal parameter estimates is provided. The proposed scheme is finally validated using a planar truss structure.

A New Approach to Structural Damage Identification Based on Power Spectral Density and Convolutional Neural Network

In the field of structural health monitoring, vibration-based damage identification remains a formidable challenge. Key to this challenge is the establishment of a reliable association between observed vibration characteristics and the actual state of structural damage (e.g. stiffness reduction). This association not only accurately indicates the presence of damage, but also the location and severity of the damage. To solve this complex pattern identification problem, a large number of approaches, including deep learning, have emerged in recent years. In this paper, we propose a new structural damage identification method that utilizes the vibration information of the structure and a convolutional neural network based on Alex NET improvement. The method consists of calculating the acceleration response power spectral density of damaged and undamaged structures under impact loading separately, and then making a difference between the two power spectral data, and subsequently introducing these power spectral difference data into the convolutional neural network for training. The use of power spectral density analysis as a preprocessing step converts the time-domain signals into frequency-domain signals, and this conversion allows the convolutional neural network to capture and learn from the specific frequency characteristics of the data, thus facilitating the learning process of the neural network model. In this paper, the effectiveness of the method is critically evaluated through numerical simulation and experimental validation, and 3% and 5% noise are added to the numerical study to test the robustness of the method. During the convolution neural network training process, the optimal training mean squared error (MSE) is [Formula: see text] in the case of no noise addition; the optimal training MSE is [Formula: see text] in the case of noise addition. Both the results of simulations and experiments confirm the high accuracy and good robustness of the method in localizing structural damage.

Drill string vibrational analysis and parametric optimization for a portable water well rig development

The vibration of a drill string is a key topic of study in drilling and mining engineering. Excessive Drill String (DS) vibrations can lead to early failure of bits due to fatigue, Drill Pipe (DP) fractures, bearing damage, and a decrease in the Rate of Penetration (ROP). This occurs when the drill string rotates at its natural frequency, which is known as resonance. A modal and harmonic vibrational analysis of a drill string up to 60-65 meters long was conducted in a vertical water well through parametric studies. This project investigates the vibration of a drill string under various parameters, including drill string lengths, Weight on Bit (WOB), and drill string rotational speeds. The Rotational Speed (RS) in the drill string induces lateral vibrations and stick-slip in torsional vibrations. Finite element modeling of the drill string was performed using ANSYS Workbench (Mechanical). Modal and harmonic frequencies were determined to obtain the optimal drill string length and rotational speeds, using Campbell diagrams and varying weights on bits. Different weights on the bit were assumed to simulate the drilling environment. One of the modern methods used to simulate dynamic behavior in drilling operations is random vibrational analysis. A random vibrational analysis of the drill string was performed in a vertical water well using ANSYS Workbench 2020. This analysis aimed to identify the frequency response of the drill string, with the probability of deformation for each sigma under random vibration being evaluated. The analysis indicated a higher probability approximately 99.7% of failure along the axis of the supplied PSD acceleration. The response power spectral density (RPSD) was analyzed in the x, y, and z directions, focusing on any point in the bottom hole assembly, primarily the bit. The response PSD results indicated that as the Drill Pipe (DP) increased, resonance spikes rose across all axes, leading to lateral and longitudinal vibrations in the drill string. These findings highlight lateral movement as a primary cause of vibration, offering insights for optimizing drilling parameters.

Analytical Study on Response Power Spectral Density of Structures Subjected to Nonstationary Excitation Considering Transient Responses

Analytical Study on Response Power Spectral Density of Structures Subjected to Nonstationary Excitation Considering Transient Responses

Reliability of simplified wind fields and structure models for the response simulation of parked offshore wind turbines

Over the past few decades, offshore wind turbines (OWTs) have been popular in the wind engineering and renewable energy fields. Numerous studies have investigated the dynamic responses of OWTs by numerical simulation. However, the different numerical models gave different results, making it difficult to compare and select models for different design and research purposes. In this study, six models including the main wind field and structure model were built based on the National Renewable Energy Laboratory 5-MW OWT, and the dynamic responses were simulated to compare and quantify the effects of the models on the response of the parked OWT. The results show that a uniform distribution of the wind field is too conservative for estimating the dynamic responses of the parked OWT. The structural response calculated using a uniformly distributed wind field is twice as large as those with the spatial distribution and coherence of the wind speed. The wind profile significantly affected the response of the parked OWT, particularly on the standard deviation of the displacement. When estimating the dynamic response of the tower of a parked OWT, the simplified model, in which the blades and wind load are simplified as a concentrated mass and load, is relatively reliable. The maximum displacement at the tower top of the simplified model is close to that of the detailed model. However, a detailed model is necessary to obtain the blade responses and accurate power spectral density of the structural response. In addition, the wind-field model significantly affected the spectral characteristics of the dynamic response of the blades.

Random Vibration Analysis of an Axially Moving Viscoelastic Beams based on Complex mode and Pseudo Excitation

Axially Moving Beams can serve as simplified models for many engineering systems, such as aerial cable trams, power conveyor belts, magnetic tapes, textile fibers, aerospace vehicles, etc. They often work in open environments in the wild, with random excitations. In engineering practice, viscoelastic materials are widely used in various engineering fields. This article studies the lateral random response of axially moving viscoelastic beams under various random excitations. Under the random external excitation of concentrated and distributed loading, analytical expressions of the mean function, correlation function, power spectral density, and mean square function of the random response of an axially moving viscoelastic beam are derived. They prove the correctness and simplicity of the Pseudo-excitation method for solving the power spectral density of the random displacement response of axially moving viscoelastic beams.

Data analysis of simulated fuel-loaded sea transportation tests under normal conditions of transport

In this study, to evaluate the shock and vibration load characteristics of used fuel, a sea transportation test was conducted using simulated fuel assemblies under normal transport conditions. An overall test data analysis was performed based on the measured strain and acceleration data obtained from cruise, rotation, acceleration, braking, depth of water, and rolling tests. In addition, shock response spectrum and power spectral densities were obtained for each test case. Amplification and attenuation characteristics were investigated based on the load path. The load was amplified as it passed from the overpack to the simulated used fuel-assembly. As a result of the RMS trend analysis, the fuel-loading position of the transportation package affected the measured strain in the fuel rod, and the maximum strains were obtained at the spans with large spacing. However, even these maximum strains were very small compared to the fatigue strength and the cladding yield strength. Moreover, the fuel rods located on the side exhibited a larger strain value than those at the center.

Extended harmonic input method for response spectrum determination of vehicle–bridge systems

The calculation of response power spectral densities (PSDs) from the excitation PSDs of vehicle–bridge (VB) systems is crucial for the safety evaluation of operating vehicles, bridge fatigue analysis, and vibration control optimization. However, calculations using the available methods result in considerable computational errors when applied to evaluate the response PSDs of VB systems. Hence, an extended harmonic input method (HIM) is proposed to compute the time-varying response spectrum in VB systems. First, the unique characteristics of VB systems that may cause the inapplicability of conventional methods are systematically analyzed. Subsequently, an extended version of the HIM is mathematically established to calculate the time-varying response spectrum of VB systems via the representation of inputs and decomposition of responses. Finally, a numerical example of an actual high-speed railway bridge subjected to track irregularities or seismic excitations is presented to illustrate the efficacy of the novel method.

A Volterra-PEM approach for random vibration spectrum analysis of nonlinear systems

In this paper, the Volterra series and the pseudo-excitation method (PEM) are combined to establish a frequency domain method for the power spectral density (PSD) analysis of random vibration of nonlinear systems. The explicit expression of the multi-dimensional power spectral density (MPSD) of the random vibration response is derived analytically. Furthermore, a fast calculation strategy from MPSD to physical PSD is given. The PSD characteristics analysis of the random vibration response of nonlinear systems is effectively achieved. First, within the framework of Volterra series theory, an improved PEM is established for MPSD analysis of nonlinear systems. As a generalized PEM for nonlinear random vibration analysis, the Volterra-PEM is used to analyse the response MPSD, which also has a very concise expression. Second, in the case of computation difficulties with multi-dimensional integration from MPSD to PSD, the computational efficiency is improved by converting the multi-dimensional integral into a matrix operation. Finally, as numerical examples, the Volterra-PEM is used to estimate the response PSD for stationary random vibration of a nonlinear spring-damped oscillator and a non-ideal boundary beam with geometrical nonlinearity. Compared with Monte Carlo simulation, the results show that by constructing generalized pseudo-excitation and matrix operation methods, Volterra-PEM can be used for input PSD with arbitrary energy distribution, not only restricted to broadband white noise excitation, and accurately predict the secondary resonance phenomenon of the random vibration response of nonlinear systems in the frequency domain.

Fatigue Life Prediction for the Skin Structures of Aircraft Sensor Pod Under Acoustic Load with Mean Stress

The skin structure of sensor pod mounted on the exterior of aircraft can be exposed to the acoustic dynamic load and static load such as aerodynamic pressure and inertial load during flight. Fatigue life prediction of structural model under acoustic load should be performed and also differential stiffness of model modified by static load should be considered. The acoustic noise test spectrum of MIL-STD-810G was applied to the structural model and the stress response power spectral density (PSD) was calculated. The frequency response analysis was performed with or without prestress induced by inplane static load, and the response spectrum was compared. Time series data was generated using the calculated PSD, and the time and frequency domain fatigue life were predicted and compared. The variation of stress response spectrum due to static load and predicted fatigue life according to the different structural model considering mean stress were examined and decreasing fatigue life was observed in the model with prestress of compressive static load.

A Background/Resonant decomposition based method to predict the behavior of 2-dof aeroelastic oscillators

Flutter is known as an important issue when designing flexible structures under environmental loading. The frequency-domain approach is one way to carry out this analysis under aeroelastic and buffeting forces, and consists in the determination of the variance of the response for all wind speeds of the design envelope. Each variance is classically obtained by numerical integration of the response power spectral densities (PSD), using a large number of integration points to capture the sharpness of their resonant peaks. This operation is particularly resource consuming, and considerably restricts the speed of the analysis. This paper presents an alternative semi-analytical method for integrating the PSD of a two degree-of-freedom aeroelastic model, based on the Background/Resonant decomposition of the response spectra. It focuses particularly on the integration of the covariances, that model the intermodal coupling effects in the response. The proposed method is compared to other existing Background/Resonant decomposition formulations, and recovers the Davenport’s approximation when neglecting all aeroelastic effects. It is illustrated on an application under the form of a parametric study, and its accuracy is compared to classical numerical methods. The errors committed on the exact nodal variances are of order of 1%.

Random vibration analysis of nonlinear structure with pounding tuned mass damper

To mitigate the undesired vibrations of the structure, a cost-effective approach is to set up the nonlinear dynamic vibration absorber (NLDVA) on the structure. As an alternative NLDVA, the Pounding Tuned Mass Damper (PTMD) gathered extensively attention in recent years. However, the random vibration analysis of structure coupled with PTMD still poses a challenge. Besides, there remains an urgent need to construct a more comprehensive recovery coefficient (RC) for the two-stage damping mechanism of PTMD. To fill in this gap, this work involves the nonlinear random vibration of a type of two degree-of-freedom system composing of a main structure (MS) coupled to a PTMD device, in which the collision process of the system is portrayed by a segmented recovery coefficient (RC) model. Specifically, an equivalent system of structure-PTMD is firstly constructed by coordinate transformation. Then, the approximate probability density function (PDF) of equivalent system is performed by combining non-smooth conversion and stochastic averaging method (SAM). Further, the approximate PDF solution of the structure-PTMD can be determined by resorting to Jacobi transformation and verified numerically. Remarkably, the approximate PDF solutions of the system can also be applied to estimate the power spectral density of the system response. Being analytical, the analysis in this work yields results in the closed-form expressions that enable one to acquire new insight into the dynamic features of the structure-PTMD system. Moreover, the proposed scheme will promote the design and application of PTMD in practical projects from a theoretical point of view.

Analysis on Vibroacoustic Fatigue Life of Empennage in a Hypersonic Vehicle

In this study, the fatigue life prediction in hypersonic vehicle noise environment is researched under the structural vibration caused by the engine jet noise and the aerodynamic noise. A finite element analysis model is first established, and then natural frequency and vibration mode are obtained. According to the modal analysis, the vibroacoustic coupling calculation is finished in the dynamic analysis software. Modal participation factor, modal displacement, and the power spectral density of the stress response on the coupled surface are acquired. The finite element, border component, and vibrating stress analytical techniques are also employed in this research to predict the acoustical fatigue performance of the tail fins of a specific type of spacecraft. On the basis of vibration fatigue theory, identified material S–N curve, and linear cumulative damage theory, the Dirlik model in the frequency domain method for vibration fatigue is used to predict the fatigue life of empennage. According to the characteristics of the acoustic load, effects of the load characteristics, and damping ratios on the fatigue conditions, the design of the antifatigue design is put forward.

Model parameter estimations of the multi-channel turbulence response from flutter flight tests based on autoregressive spectra

Signal processing of flutter flight test data enables verification of aircraft flutter design, and the signal from a flutter flight test excited by atmospheric turbulence is a particularly important form of the flutter test. Owing to the randomness of atmospheric turbulence excitation, multi-channel analysis of turbulence responses at various positions in the same component can improve the analytical accuracy of flutter signal processing. The relationship between the maximum singular value of the multi-channel turbulence response power spectral density matrix and the system self-power spectral density function is elucidated herein using a frequency domain decomposition method. However, there is a contradiction in the power spectral density function between the spectral line density and the spectral line smoothing calculated based on the periodogram of the frequency domain decomposition. By applying an autoregressive spectral model, the power spectral density function of the turbulence response is calculated to achieve spectral line smoothing and sufficient spectral line density. Additionally, the power spectral density function is then used to construct the power spectral density function matrix of the multi-channel turbulence response, and the maximum singular value curve is calculated based on the singular value decomposition of each spectral pin. Finally, the modal parameters of the turbulence response signal are estimated via multi-modal frequency domain fitting. The developed approach is validated based on simulations and flutter flight test turbulence response signals.

Frequency-domain method for non-stationary stochastic vibrations of train-bridge coupled system with time-varying characteristics

When a train passes over a bridge, the vibrations of the vehicles and the bridge are the result of non-stationary stochastic processes due to the time-dependent characteristics of the coupled vehicle-bridge system. The aim of this study is to generalize the frequency domain method to investigate the non-stationary random vibration of the coupled system subjected to the excitation of track irregularities with consideration of time-dependent characteristics. To illustrate the method, a three-span simply supported bridge traversed by a single railway vehicle is adopted as an example. The time-dependent frequency response function (FRF) of the coupled system is theoretically derived through solving ordinary differential equations with variable complex coefficients, and the perturbation method is adopted to improve the calculation efficiency. By combining this with Priestley’s Evolutionary Spectra theory, the evolutionary power spectral density (PSD) of the non-stationary random response of the system is then derived. The transitions will occur when the wheels cross the joints between each bridge span and between the bridge and the adjacent roadway. By adopting mode shapes of the full structure, the change of states of the vehicle crossing multiple bridge spans and moving onto the roadway can be solved as a continuous process without separation. The proposed method is validated by comparisons with the Monte Carlo method, showing higher accuracy and efficiency when calculating the time-varying standard deviation of the response. It is found that the vibration of the vehicle is approximately stationary but with large variance due to the random track irregularities, while the bridge vibration follows a strongly non-stationary process with small randomness and is more related to the moving mass effect.

Preliminary data analysis of surrogate fuel-loaded road transportation tests under normal conditions of transport

In this study, road transportation tests were conducted with surrogate fuel assemblies under normal conditions of transport to evaluate the vibration and shock load characteristics of spent nuclear fuel (SNF). The overall test data analysis was conducted based on the measured acceleration and strain data obtained from the speed bump, lane-change, deceleration, obstacle avoidance, and circular tests. Furthermore, representative shock response spectrums and power spectral densities of each test mode were acquired. Amplification or attenuation characteristics were investigated according to the load transfer path. The load attenuated significantly as it transferred from the trailer to the cask. By contrast, the load amplified as it transferred from the cask to the surrogate SNF assembly. The fuel loading location on the cask disk assembly did not exhibit a significant influence on the strain measured from the fuel rods. The principal strain was in the vertical direction, and relatively large strain values were obtained in spans with large spacing between spacer grids. The influence of the lateral location of fuel rods was also investigated. The fuel rods located at the side exhibited relatively large strain values than those located at the center. Based on the strain data obtained from the test results, a hypothetical road transportation scenario was established. A fatigue evaluation of the SNF rod was performed based on this scenario. The evaluation results indicate that no fatigue damage occurred on the fuel rods.