Stationary Random Vibration Research Articles

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.

Exact Solutions of Fully Nonstationary Random Vibration for Rectangular Kirchhoff Plates Using Discrete Analytical Method

This paper proposes the discrete analytical method (DAM) to determine exactly and efficiently the fully nonstationary random responses of rectangular Kirchhoff plates under temporally and spectrally nonstationary acceleration excitation of earthquake ground motions. First, the fully nonstationary power spectral density (PSD) model is suggested by replacing the filtered frequency and damping of Gaussian filtered white-noise model with the time-variant ones. The exact solutions of free vibration of thin plates with two opposite edges simply supported boundary conditions are introduced. Then, the full analytical procedure for random vibration analysis of the plate is established by using a pseudo excitation method (PEM) that can consider all modal auto-correlation and cross-correlation terms. Owing to involving a series of Duhamel time integrals of single degree of freedom systems, it is difficult to fully analytically evaluate the PSD of time-variant responses such as the transverse deflection, velocity, acceleration and stress components. Thus, DAM that combines the PEM with precise integration technique is developed to enhance the computational efficiency. Finally, comparison of the results by the DAM with Monte Carlo simulations and the analytical stationary random vibration analysis demonstrates the high efficiency and accuracy of DAM. Moreover, the fully nonstationary excitation imposes a remarkable effect on the response PSD of rectangular Kirchhoff plates.

Synthesis of Sine-on-Random vibration profiles for accelerated life tests based on fatigue damage spectrum equivalence

Abstract In many real life environments, mechanical and electronic systems are subjected to vibrations that may induce dynamic loads and potentially lead to an early failure due to fatigue damage. Thus, qualification tests by means of shakers are advisable for the most critical components in order to verify their durability throughout the entire life cycle. Nowadays the trend is to tailor the qualification tests according to the specific application of the tested component, considering the measured field data as reference to set up the experimental campaign, for example through the so called “Mission Synthesis” methodology. One of the main issues is to define the excitation profiles for the tests, that must have, besides the (potentially scaled) frequency content, also the same damage potential of the field data despite being applied for a limited duration. With this target, the current procedures generally provide the test profile as a stationary random vibration specified by a Power Spectral Density (PSD). In certain applications this output may prove inadequate to represent the nature of the reference signal, and the procedure could result in an unrealistic qualification test. For instance when a rotating part is present in the system the component under analysis may be subjected to Sine-on-Random (SoR) vibrations, namely excitations composed of sinusoidal contributions superimposed to random vibrations. In this case, the synthesized test profile should preserve not only the induced fatigue damage but also the deterministic components of the environmental vibration. In this work, the potential advantages of a novel procedure to synthesize SoR profiles instead of PSDs for qualification tests are presented and supported by the results of an experimental campaign.

Benchmark solutions of stationary random vibration for rectangular thin plate based on discrete analytical method

This paper aims to accurately and efficiently achieve the benchmark solutions of stationary stochastic responses for rectangular thin plate. Firstly, the exact solutions of free vibration for thin plate with SSSS, SSSC, SCSC, SFSF, SSSF and SCSF boundary conditions are introduced to random vibration analysis. Based on pseudo excitation method (PEM), the analytical power spectral density (PSD) functions of the transverse deflection, velocity, acceleration and stress responses for thin plate under random base acceleration excitation are derived. Subsequently, to enhance computational efficiency, the discrete analytical method (DAM) that realizes the discretization for the modal coordinates and frequency domain is proposed. Finally, the efficiency of DAM and the accuracy of benchmark solutions are scrutinized by comparison with the analytical solutions and finite element solutions.

Adaptive operational modal identification for slow linear time-varying structures based on frozen-in coefficient method and limited memory recursive principal component analysis

Abstract To adaptively identify the transient modal parameters for linear weakly damped structures with slow time-varying characteristics under unmeasured stationary random ambient loads, this paper proposes a novel operational modal analysis (OMA) method based on the frozen-in coefficient method and limited memory recursive principal component analysis (LMRPCA). In the modal coordinate, the random vibration response signals of mechanical weakly damped structures can be decomposed into the inner product of modal shapes and modal responses, from which the natural frequencies and damping ratios can be well acquired by single-degree-of-freedom (SDOF) identification approach such as FFT. Hence, for the OMA method based on principal component analysis (PCA), it becomes very crucial to examine the relation between the transformational matrix and the modal shapes matrix, to find the association between the principal components (PCs) matrix and the modal responses matrix, and to turn the operational modal parameter identification problem into PCA of the stationary random vibration response signals of weakly damped mechanical structures. Based on the theory of “time-freezing”, the method of frozen-in coefficient, and the assumption of “short time invariant” and “quasistationary”, the non-stationary random response signals of the weakly damped and slow linear time-varying structures (LTV) can approximately be seen as the stationary random response time series of weakly damped and linear time invariant structures (LTI) in a short interval. Thus, the adaptive identification of time-varying operational modal parameters is turned into decompositing the PCs of stationary random vibration response signals subsection of weakly damped mechanical structures after choosing an appropriate limited memory window. Finally, a three-degree-of-freedom (DOF) structure with weakly damped and slow time-varying mass is presented to illustrate this method of identification. Results show that the LMRPCA algorithm, which is robustness to Gauss measurement noise disturbances, can well identify the transient modal parameters (transient natural frequencies & transient modal shapes) for weakly damped and slow LTV structures online only from non-stationary random response signals.

Vibration Control Design for a Plate Structure with Electrorheological ATVA Using Interval Type-2 Fuzzy System

This study presents vibration control using actively tunable vibration absorbers (ATVA) to suppress vibration of a thin plate. The ATVA is made of a sandwich hollow structure embedded with electrorheological fluid (ERF). ERF is considered to be one of the most important smart fluids and it is suitable to be embedded in a smart structure due to its controllable rheological property. ERF’s apparent viscosity can be controlled in response to the electric field and the change is reversible in 10 microseconds. Therefore, the physical properties of the ERF-embedded smart structure, such as the stiffness and damping coefficient, can be changed in response to the applied electric field. A mathematical model is difficult to be obtained to describe the exact characteristics of the ERF embedded ATVA because of the nonlinearity of ERF’s viscosity. Therefore, a fuzzy modeling and experimental validations of ERF-based ATVA from stationary random vibrations of thin plates are presented in this study. Because Type-2 fuzzy sets generalize Type-1 fuzzy sets so that more modeling uncertainties can be handled, a semi-active vibration controller is proposed based on Type-2 fuzzy sets. To investigate the different performances by using different types of fuzzy controllers, the experimental measurements employing both type-1 fuzzy and interval type-2 fuzzy controllers are implemented by the Compact RIO embedded system. The fuzzy modeling framework and solution methods presented in this work can be used for design, performance analysis, and optimization of ATVA from varying harmonic vibration of thin plates.

Stationary random vibration analysis of vehicle-track coupled system with nonlinear stiffness of railpads

According to time-temperature superposition (TTS) and WLF formula, the temperature-dependent and frequency-dependent kinetic parameters of the railpads of WJ-7 fasteners were predicted based on the experimental results at a certain frequency and under the different temperatures. By combining the pseudo excitation method and symplectic mathematics scheme, the impact of railpads frequency-dependent and temperature-dependent stiffness on the random variation of the coupled vehicle-track system was investigated efficiently. The results suggest that, within the range of environment temperature and frequency-dependent extent of railpads in the railway, the PSD peak of the vertical random vibration acceleration of the wheelsets and rail gets the largest increase of 3.3 times at least, and the increment of their 1st dominant frequency also reaches up to 20.6 Hz or more. Therefore, in order to accurately analyze the random vibration responses of the vehicle-track coupled system, experiments must be conducted to obtain the accurate kinetic parameters of the polymer materials like railpads.

A response spectrum method for peak floor acceleration demands in earthquake excited structures

This paper addresses the prediction of the median peak floor acceleration (PFA) demand of elastic structures subjected to seismic excitation by means of an adapted response spectrum method. Modal combination is based on a complete quadratic combination (CQC) rule. In contrast to previous studies, in the present contribution closed form solutions for the correlation coefficients and peak factors entering the CQC rule are derived using concepts of normal stationary random vibration theory. A ground motion set, which matches the design response spectrum for a specific site and a target dispersion, is used to define the stochastic base excitation. The response spectrum method is tested for various planar and spatial generic high-rise structures subjected to this particular ground motion set. A comparison of the outcomes with the results of computationally more expensive response history analyses shows the applicability and accuracy of the proposed simplified method.

Random vibration energy harvesting on thin plates using multiple piezopatches

Vibrational energy harvesting using piezoelectric cantilever beams has received significant attention over the past decade. When compared to piezoelectric cantilever-based harvesters, piezopatch energy harvesters integrated on plate-like thin structures can be a more efficient and compact option to supply electrical power for wireless structural health and condition monitoring systems. In this article, electroelastic modeling, analytical and numerical solutions, and experimental validations of piezopatch-based energy harvesting from stationary random vibrations of thin plates are presented. Electroelastic models for the series and parallel connected multiple piezopatches are given based on a distributed-parameter modeling approach for a thin host plate excited by a transverse point force. The analytical and numerical solutions for the mean power output and the mean-square shunted vibration response are then derived. The experimental measurements are carried out by employing a fully clamped thin plate with three piezopatches connected in series. It is shown that the analytical and numerical model predictions for the mean power output and the mean-square velocity response are in very good agreement with the experimental measurements. The electroelastic modeling framework and solution methods presented in this work can be used for design, performance analysis, and optimization of piezoelectric energy harvesting from stationary random vibration of thin plates.

Harmonic wavelets based response evolutionary power spectrum determination of linear and non-linear oscillators with fractional derivative elements

A harmonic wavelets based approximate analytical technique for determining the response evolutionary power spectrum of linear and non-linear (time-variant) oscillators endowed with fractional derivative elements is developed. Specifically, time- and frequency-dependent harmonic wavelets based frequency response functions are defined based on the localization properties of harmonic wavelets. This leads to a closed form harmonic wavelets based excitation-response relationship which can be viewed as a natural generalization of the celebrated Wiener–Khinchin spectral relationship of the linear stationary random vibration theory to account for fully non-stationary in time and frequency stochastic processes. Further, relying on the orthogonality properties of harmonic wavelets an extension via statistical linearization of the excitation-response relationship for the case of non-linear systems is developed. This involves the novel concept of determining optimal equivalent linear elements which are both time- and frequency-dependent. Several linear and non-linear oscillators with fractional derivative elements are studied as numerical examples. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the technique.

Response evolutionary power spectrum determination of chain-like MDOF non-linear structural systems via harmonic wavelets

A harmonic wavelets based technique is developed for determining the evolutionary power spectrum (EPS) matrix of the response of non-linear chain-like MDOF structural systems subject to a multi-component non-stationary stochastic excitation. Specifically, first a relationship between the evolutionary power spectrum matrices of the excitation and of the system response is derived by using a recently proposed locally stationary wavelet based representation of non-stationary stochastic processes. The relationship can be construed as a direct extension of the celebrated spectral input–output relationship of the linear stationary random vibration theory. Further, a harmonic wavelets based statistical linearization technique is proposed for the case of MDOF non-linear systems with chain-like architecture systems and hysteretic non-linearities. Numerical examples include MDOF non-linear systems comprising the versatile Bouc–Wen hysteretic model. Pertinent Monte Carlo simulations demonstrate the reliability of the technique.

Efficient Sensitivity Analysis of Structures with Local Modifications. II: Transfer Functions and Spectral Densities

The deterministic frequency response analysis and the stationary random vibration of large, linear structural models are often computationally intensive tasks. The computational expense is only exacerbated when stiffness and damping variabilities are present in the system’s description. This paper proposes an approach for the highly computationally efficient and accurate computation of the sensitivities, to changes in design parameters that are spatially localized within some small portion of the computational model, of the transfer functions (TFs), power spectral densities, and mean-square responses of a structural system. The proposed method is derived, first for system TFs and then their sensitivities; the corresponding spectral densities are then derived using the TF approach. Finally, the proposed method is applied to two examples, one simple model where exact solutions can be computed, and a second, more realistic, building example with a high-order model that causes significant difficulties for standard state-space (SS) approaches in MATLAB as a result of numerical inaccuracies in the computations for very high-order SS models. The proposed method is shown to be highly accurate, with relative errors on the order of 10−10, but is 4–6 orders of magnitude faster than conventional approaches.

Modeling and Simulation of Linear and Nonlinear MEMS Scale Electromagnetic Energy Harvesters for Random Vibration Environments

The simulation results for electromagnetic energy harvesters (EMEHs) under broad band stationary Gaussian random excitations indicate the importance of both a high transformation factor and a high mechanical quality factor to achieve favourable mean power, mean square load voltage, and output spectral density. The optimum load is different for random vibrations and for sinusoidal vibration. Reducing the total damping ratio under band-limited random excitation yields a higher mean square load voltage. Reduced bandwidth resulting from decreased mechanical damping can be compensated by increasing the electrical damping (transformation factor) leading to a higher mean square load voltage and power. Nonlinear EMEHs with a Duffing spring and with linear plus cubic damping are modeled using the method of statistical linearization. These nonlinear EMEHs exhibit approximately linear behaviour under low levels of broadband stationary Gaussian random vibration; however, at higher levels of such excitation the central (resonant) frequency of the spectral density of the output voltage shifts due to the increased nonlinear stiffness and the bandwidth broadens slightly. Nonlinear EMEHs exhibit lower maximum output voltage and central frequency of the spectral density with nonlinear damping compared to linear damping. Stronger nonlinear damping yields broader bandwidths at stable resonant frequency.

A Practical Response Spectrum Method in Considering the Input Angle of Seismic Excitation

To the asymmetric-plan structures, the torsion model is obvious, and the influence of input angle of excitation on structural response is sensitive, so a practical response spectrum method for analyzing the behaviors of this kind of structure is studied. Based on the achievements about the multi-components accelerations power spectra matrix, a rational formula, considering the input angle of excitation, is deduced by using stationary random vibration principle. A practical formula is proposed by introducing displacement response spectrum as equally as to considering the non-stationarity of excitation.

Stationary and non-stationary random vibration modelling and analysis for an operating wind turbine

The problem of stationary and non-stationary random vibration modelling and analysis for an operating wind turbine is considered by using an acceleration vibration signal measured on a NegMicon NM52/900 fixed speed wind a turbine tower. Three stationary methods, namely Welch spectral estimation, autoregressive, and autoregressive moving average modelling, are considered, along with five non-stationary methods, namely non-parametric Wigner–Ville and spectral correlation, and parametric modelling by means of smoothness priors (SPs) time-dependent autoregressive modelling, functional series (FS) time-dependent autoregressive modelling, and adaptable functional series (AFS) time-dependent autoregressive modelling. The need for non-stationary modelling is mainly dictated by blade rotation and the wind dynamics. The results of the study confirm the cyclo-stationary and broader non-stationary nature of the random vibration, dictating the need for corresponding methods for accurate modelling and analysis. The capabilities and facets of the various methods in terms of modelling and analysing operating wind turbine random vibration are presented.

Random Response Analysis of a Large-Size Cooling Tower Subjected to the Stationary Earthquake Motion

The response of a large-size cooling tower with 250m high subjected to the seismic action are investigated by both random vibration theory and response spectrum method. Shell element is taken to model the tower body, and beam element is used for the circular foundation and supporting columns. The earthquake motion input is a colored filtered white noise model and mode superposition method is adopted to analyze the random response of the large-size cooling tower. The paper presents the power spectrum density functions (PDF) and standard deviation of the displacement of the top and characteristic node, and the analysis results indicate that the results of the stationary random vibration theory and the response spectrum method are the same order of magnitude. The power spectrum density function of the bottom node stress is obviously bigger than the one at the top and the throat, and the random response of meridonal stress is dominated at the top. In addition, the peak frequency position of the power spectrum density function is different from the corresponding stress.

Stochastic hydroelastic analysis of pontoon-type very large floating structures considering directional wave spectrum

The hydroelastic response of pontoon-type very large floating structures (VLFS) is obtained by resolving the interaction between the surface waves and the floating elastic body. We carry out the analysis in the frequency domain, assuming that the surface waves can be described by a directional wave spectrum. The response spectra can then be computed by application of stationary random vibration analysis. Applying the modal expansion method, we obtain a discrete representation of the required transfer matrices for a finite number of frequencies, while the influence of the wave direction is obtained by numerical integration of the directional components of the spectrum. Moreover, assuming a Gaussian input, we can apply well known approximations to obtain the distribution of extremes. The method is applied to an example VLFS and the effect of different mean wave angles on the stochastic response is investigated.

Suppression of random vibration in flexible structures using a hybrid vibration absorber

A design method is proposed to suppress stationary random vibration in flexible structures using a hybrid vibration absorber (HVA). While the traditional vibration absorber can damp down the vibration mainly at the pre-tuned mode of the primary structure, active damping is generated by the proposed HVA to damp down all resonant modes of interest of the vibrating structure and the spatial average mean square motion of the vibrating structure can be minimized. Only one absorber and one feedback signal are required to achieve global vibration suppression of a flexible structure under stationary random excitation. A special pole-placement controller is designed such that all vibration modes of the flexible structures become critically damped. It is proved analytically that the proposed HVA damps the vibration of the entire structure instead of just the attachment point of the absorber. The proposed optimized HVA is tested on a beam structure and it shows a superior performance on global suppression of broadband vibration in comparison to other published designs of passive and hybrid vibration absorbers.

The Vehicle Dynamic Load Indentification under the Excitation of Random Road Surface

In this paper, inverse pseudo excitation method for vehicle dynamic load identification is used. When the vehicle is in stationary random vibration, the vehicle dynamic load spectrum recognition problem is solved by using the deterministic method. The auto-PSD and cross-PSD of vehicle vibration response is known. Base on the reversing the power spectrum of road excitation, Vehicle dynamic load PSD is obtained. The results show that, Inverse pseudo excitation method for solving the vehicle dynamic load spectrum has good solution accuracy, put forward new ideas and methods for vehicle engineering practice, and has broad application prospects.

Sensitivity Analysis for Structures Subjected to Stationary Random Excitations

A method for sensitivity analysis of stationary random vibration is proposed within the framework of pseudo-excitation method (PEM). As the stationary random vibration analysis is transformed into harmonic vibration analysis, therefore the sensitivity analysis of stationary random responses is transformed into sensitivity analysis of pseudo harmonic responses of structures subjected to pseudo harmonic excitations. Using the PEM principle once again, the formulae for calculating the sensitivities of power spectrum density (PSD) and variances of random responses are derived. Based on the above work, the study of statistical characteristics of stationary random responses for structures with uncertain parameters is carried out.