Non-stationary Random Vibration Research Articles

Modified time domain randomization technique for multi-shaker non-stationary non-Gaussian random vibration control

Traditional random vibration environmental tests simulate stationary Gaussian random vibration environments. In this work, the generation and closed-loop control of multi-shaker non-stationary non-Gaussian random vibration signals with the modified time domain randomization technique are studied. The amplitude modulation function is introduced to control the non-stationary and non-Gaussian characteristics that the root mean square value of the generated signal changes with time. The kurtosis relationship between the one frame pseudo-random signal and consecutive true random signal is derived theoretically. It is found that the kurtosis of the consecutive non-stationary random signal is determined by the amplitude modulation function and kurtosis of the pseudo-random signal. The standard deviation of the time-varying root mean square reflects the non-stationary characteristics of the generated signal. A control strategy for multi-shaker non-stationary non-Gaussian random vibration environmental test is proposed. The non-stationary and non-Gaussian characteristics are extracted from the measured response signals so that the power spectral densities, kurtoses and non-stationary characteristics can be controlled simultaneously. The results from the simulation examples and multi-shaker vibration experiments prove the feasibility and practicability of the proposed non-stationary non-Gaussian random signal generation method and control strategy.

Retracted: Research on Numerical Simulation Method of Nonstationary Random Vibration Signal Sensor in Railway Transportation

Retracted: Research on Numerical Simulation Method of Nonstationary Random Vibration Signal Sensor in Railway Transportation

Non-stationary non-Gaussian random vibration analysis of Duffing systems based on explicit time-domain method

Non-stationary non-Gaussian random vibration problems of structures are challenging and drawing increasing attention. In the present study, firstly, an explicit time-domain method (ETDM) is proposed to determine the higher-order response statistics of linear systems subjected to non-stationary non-Gaussian random excitations, in which the first four orders of cumulants of dynamic responses are directly formulated through the cumulant operation rule based on the explicit expressions of responses. Secondly, an equivalent linearization – explicit time-domain method (EL-ETDM) is further developed to solve the non-stationary non-Gaussian random vibration problems of Duffing systems, in which the equivalent linear system is derived discarding the assumption of Gaussian response, and the corresponding higher-order cumulant analyses of the linearized system are accomplished by the efficient ETDM. The present approach can account for non-Gaussian random excitations with arbitrary forms, and two specific applications to the Poisson white noise and the square form of Gaussian random process are investigated. Four numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Extended statistical linearization approach for estimating non-stationary response statistics of systems comprising fractional derivative elements

An efficient technique to analyze non-stationary nonlinear random vibrations of dynamic systems endowed with a fractional derivative term is presented. The technique itself represents an extension of the concept of statistical linearization to this kind of systems, and it is applicable for both analytic and hysteretic system nonlinearities. The technique first resorts to harmonic balancing in deriving response-amplitude dependent equivalent damping and stiffness. This enables representation of the fractional derivative term as a linear combination of the system response displacement and velocity with amplitude dependent coefficients. Then, the expected values of these parameters are considered in proceeding to formulate a statistical linearization solution scheme. In this context, the solution procedure is completed by integrating in time the covariance Lyapunov equation associated with the derived equivalent linear system. The reliability of the proposed technique is tested by a series of germane Monte Carlo studies. This juxtaposition is also used to elucidate salient features of the technique, by varying the order of the fractional derivative term, and of the degree of the nonlinearity in the system. It also points out the versatility of the technique in determining the non-stationary values of auto-correlation and cross-correlations response parameters involving even the fractional derivative term.

Prediction method of non-stationary random vibration fatigue reliability of turbine runner blade based on transfer learning

In order to solve the problems such as lack of fault information, sample variation with time and expensive calculation in the estimation of the vibration fatigue reliability of the turbine runner blade under the non-stationary hydraulic excitation. A prediction method of non-stationary random vibration fatigue reliability of the turbine runner blade based on transfer learning is proposed in this paper. Firstly, the dynamics model of the cracked turbine runner blade under the non-stationary hydraulic excitation is established to analyze the characteristics of the non-stationary random vibration fatigue of the turbine runner blade. Secondly, the transformation matrix between the source domain and target domain in the hidden space is found by the transfer learning method of balanced distribution adaptation (BDA). The adaptation of active learning and Kriging-based system reliability method (AK-SYSi) is applied to estimate the non-stationary random vibration fatigue reliability of the turbine runner blade with multi-failure-mode. Finally, an example is analyzed, and the Monte Carlo simulation (MCS) is used to verify the correctness of the proposed method. The results show that the method proposed in this paper can effectively predict the failure probability of the non-stationary vibration fatigue of the turbine runner blade in future time.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

AbstractBased on polynomial dimensional decomposition (PDD), a novel method for analysing nonstationary random vibration of structures with uncertain parameters is proposed in this paper. The uncertain structural parameters are assumed to be independent random variables, with certain types of probability distributions; thus, the nonstationary random vibration responses of the structures become stochastic functions of these uncertain parameters. A dimensional decomposition and Fourier expansion were applied to calculate the statistical characteristics of the nonstationary random vibration responses of the structures. To calculate the conditional power spectral density (CPSD) and conditional standard deviation (CSD) of the nonstationary random vibration responses, CPSD and CSD were explicitly expressed using expansion coefficients and polynomial basis. Subsequently, the probability distributions of the structural responses can be effectively calculated in terms of these explicit expressions. Dimension‐reduction integration and Gauss numerical integration were applied to calculate expansion coefficients. The validity of the proposed method was verified by numerical examples, including a single‐degree‐of‐freedom system and a multi‐degree‐of‐freedom system (reinforced concrete shear wall). The results indicate that the proposed method is as accurate as Monte Carlo simulation (MCS), whereas the proposed method requires considerably smaller number of nonstationary random vibration analyses than MCS. Moreover, the nonstationary random vibration analysis of a long‐span cable‐stayed bridge demonstrates that the proposed method is feasible for quantifying the uncertainties of the nonstationary random vibration responses of complex engineering structures with uncertain parameters.

Vibration Characteristics of Corn Combine Harvester with the Time-Varying Mass System under Non-Stationary Random Vibration

In field harvesting conditions, the non-stationary random vibration characteristics of the harvester are rarely considered, and the results of vibration frequency calculated by different time–frequency transformation methods are different. In this paper, the harvester’s vibration characteristics under the time-varying mass were studied, and the correlation between vibration frequency and modal frequency was analyzed. Firstly, under the conditions of time-varying mass (field harvesting conditions) and non-time-varying mass (empty running condition), the non-stationarity characteristics of vibration signals at 16 measurement points of a combined corn harvester frame were studied. Then, fast Fourier transform (FFT), short-time Fourier transform (STFT), and continuous wavelet transform (CWT) were used to calculate the vibration frequency distribution characteristics of the corn harvester. Finally, based on the EFDD (enhanced frequency domain decomposition) algorithm, the correlation between the primary vibration frequency and the operating mode frequency is studied. The results show that the mean, variance, and maximum difference of the vibration amplitude under harvesting conditions (mass time-varying system) are 0.10, 26.5, and 1.0, respectively, at different harvesting periods (0~10 s, 10~20 s, 20~30 s). The harvesting conditions’ vibration signals conform to the characteristics of non-stationary randomness. The FFT algorithm is used to obtain more dense vibration frequencies, while the frequencies based on STFT and CWT algorithms are sparse. The correlation between the FFT method and the EFDD algorithm is 0.98, and the correlation between the STFT, CWT, and the EFDD algorithm is 0.99 and 0.98. Therefore, the primary frequency of the STFT methods is closer to the modal frequency. Our research laid the foundation for further study and application of mass time-varying combined harvester system non-stationary random vibration modal frequency identification and vibration control.

Random vibration study of functionally graded porous curved beams with elastically restrained ends

Firstly, the random vibration characteristics of functionally graded porous (FGP) curved beams with elastically restrained ends are studied. An efficient model is presented by the Timoshenko beam theory and spectral-Chebyshev method. In this paper, three different types of porous distribution are considered, and the relationship between porosity coefficient and material parameters is determined according to the typical mechanical properties of open cell foam metal. Four types of curved beams with different curvatures are selected for the study, which are elliptical beam, parabolic beam, hyperbolic beam and circular beam. The one-dimensional admissible displacement functions of the FGP curved beam are constructed by Chebyshev polynomials of the first kind with Gauss-Lobatto sampling points discretization. Three artificial boundary springs are used to impose elastic boundary constraints at the ends of the curved beam. The pseudo excitation method is used to apply stationary and non-stationary random excitations, including point excitation and base acceleration excitation. The stationary random vibration responses of the FGP curved beam with different boundary conditions, involving the power spectral density (PSD) and root mean square (RMS) values of displacement, velocity and acceleration, are calculated and agreed well with the finite element method (FEM). At last, the RMS values of the non-stationary random vibration response of the FGP curved beam are given.

A Modal Frequency Estimation Method of Non-Stationary Signal under Mass Time-Varying Condition Based on EMD Algorithm

A method to estimate modal frequency based on empirical mode decomposition (EMD) and ensemble empirical mode decomposition (EEMD) is proposed. This method can decrease the difficulties in identifying modal frequency of combine harvesters. First, we used 16 acceleration sensors installed at different test points to collect vibration signals of a corn combine harvester under operating conditions (mass time-varying conditions). Second, we calculated mean value, variance and root mean square (RMS) value of the vibration signals, and analyzed its stationarity of vibration signals. Third, the main frequencies of the 16 points were extracted using the EMD and EEMD methods. Finally, we considered modal frequencies identified by the SSI algorithm as standard, and calculated the fitting degrees of the EMD and EEMD methods. The results show that in different time periods (0~60 s and 60~120 s), the maximum differences of the mean value, variance and RMS value of signals were 0.8633, 171.1629 and 11.3767, and the vibration signal under the operating condition of field harvesting can be regarded as a typical non-stationary random vibration signal. The EMD method had more modal aliasing than EEMD, and when we obtained the fitting equations of EMD, EEMD and SSI methods, the value of the Euler distance between the EMD fitting equation and the SSI fitting equation was 446.7883, while that for EEMD and SSI was 417.2845. The vibration frequencies calculated by the EEMD method is closer to the modal frequencies identified by SSI algorithm. The proposed method provides a reference for modal frequency identification and vibration control in a complex working environment.

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.

Research on Numerical Simulation Method of Nonstationary Random Vibration Signal Sensor in Railway Transportation

During railway transportation, due to various factors such as road conditions and operating conditions and produced vibrations and shocks, this kind of vibration environment may cause fatigue damage to on-board equipment and transported goods. The authors propose a research on the numerical simulation method of the nonstationary random vibration signal sensor of railway transportation; first, they establish the mathematical model of the railway nonstationary random vibration signal sensor and then introduce the method of reconstructing the railway nonstationary random vibration signal sensor. For railway nonstationary non-Gaussian random vibration reconstruction signal, compare the time-domain characteristics of the sampled signal, and for railway nonstationary non-Gaussian random vibration reconstruction signal, compare the frequency domain characteristics of the sampled signal. The results show that the relative error of the RMSM function is within 6%, the relative error of the sliding bias function is within 10%, and the relative error of the sliding kurtosis function is within 8%. The energy distribution of the edge Hilbert amplitude spectrum is very similar, with absolute error less than 6%. The energy fluctuations are similar in each band, with absolute error rates less than 4% in most bands. The method proposed in this article, suitable for reconstruction of railway nonstationary Gaussian random vibration and nonstationary non-Gaussian vibration signal sensor, verifies the effectiveness and feasibility of the signal reconstruction method. The model and signal reconstruction method proposed in this paper are applied to the railway nonstationary Gaussian and nonstationary non-Gaussian random vibration sampling signals.

Auxiliary harmonic excitation generalized method for random vibration analysis of linear structures under non-stationary Gaussian excitation

Frequency-domain methods are the most fundamental and effective methods for non-stationary random vibration analysis. However, the existing frequency-domain methods often involve truncation for degree of mode or decomposition of the power spectrum in multi-correlation conditions, which may have impact on the computational accuracy and efficiency of the methods. To avoid the truncation for degree of mode and the decomposition of the power spectrum, an accurate and efficient auxiliary harmonic excitation generalized method is proposed for random vibration analysis of linear structures under non-stationary Gaussian excitation. First, several existing frequency-domain analysis methods are investigated and discussed. Second, the generalized impulse response function, the generalized frequency response function, and the evolutionary generalized frequency response function are proposed. Meanwhile, the physical meaning of the evolutionary frequency response function is re-investigation, in which the response for the evolutionary harmonic excitation and the evolutionary frequency response function proves equivalent. Subsequently, according to the physical meaning, a novel and easy-to-implement algorithm of the response PSD is proposed. The response for the evolutionary auxiliary harmonic excitation can be quickly evaluated based on the time-domain explicit formulation method. Finally, two examples are investigated to verify the accuracy and efficiency of the proposed method.

Nonstationary random vibration analysis of fractionally-damped systems by numerical explicit time-domain method

Random vibration analysis of fractionally-damped systems has received considerable attentions over the past two decades. In this paper, an efficient explicit time-domain approach is developed to determine the response statistics of multi-degrees-of-freedom linear systems endowed with fractional derivative terms subjected to nonstationary random excitations. The explicit time-domain expression of the state vector is constructed for the fractionally-damped system, and on this basis, the statistical moments of responses are directly formulated with the moment operation rule or efficiently calculated with the Monte-Carlo simulation. Dimension-reduced statistical analysis can be easily conducted just focusing on certain responses of interest with the use of the explicit expressions of dynamic responses. The present approach can account for nonstationary random excitations with arbitrary evolutionary power spectrum forms, even of the non-separable kind. A 20-degrees-of-freedom shear-type fractionally-damped system subjected to fully nonstationary random excitation is analysed to illustrate the accuracy and efficiency of the proposed method.

The Theoretical Impact Factor Spectrum for Highway Beam Bridges

The impact effect of vehicle loads is an important and challenging problem in bridge engineering. In the last few decades, experimental and numerical researches on this problem have been extensively carried out, but a satisfactory formula that can take various factors into account has not been obtained. The vehicle–bridge interaction (VBI) under the excitation of road roughness is essentially a nonstationary random vibration problem. In this paper, the random load of a bridge is expanded into an expression in both the time and frequency domains using Fourier series; the dynamic response of the bridge under the moving random load is solved by the modal superposition method; an analytical solution of the second-order statistical characteristics of the vibration response of the bridge in the form of series is obtained; and on these bases, a theoretical formula of the impact factor (IM) spectrum is obtained. This approach is more scientific and reasonable in theory than the traditional deterministic approach because the theoretical IM spectrum is based on the second-order statistical characteristics of bridge dynamic responses. Moreover, the theoretical IM spectrum can more comprehensively reflect the various influencing factors of vehicle impact effects on the bridge.

Research on probability statistics of train running safety indices based on pseudo-excitation method

In order to explore the random nature of high-speed railway train operation safety indices, the pseudo-excitation method, extreme value theory, and non-stationary harmonic superposition theory are used in this paper to study the statistics of train operation safety indices. The pseudo-excitation load formulation for track irregularity is obtained by the pseudo-excitation method, and the resulting non-stationary random vibration problem is transformed into a deterministic time history problem. The pseudo-excitation method is used to establish the dynamic equations of motion, and the separation iteration method is used to solve the equations, so as to obtain the power spectral density of the wheel-rail interaction forces. The wheel-rail interaction forces are obtained by using a modulation function and the harmonic superposition method. By fitting an extreme value distribution, the maximum values of the train running safety indices are explored. The proposed numerical approach is validated experimentally using the data from a 24.6 m long simply supported concrete bridge by studying the extreme value distributions of driving safety indices. Additional numerical simulation are conducted for varying train speeds and bridge spans. The results show that the Gumbel distribution can fit the extreme value of driving safety parameters for different speeds and different bridge span lengths. It is observed that the higher the speed, the sharper the extreme value distribution of train running safety indices, and the larger the train running safety index values corresponding to 99.87% confidence level. The corresponding extreme values at the 99.87% confidence level are greater than the maximum value of each time-domain sample.

Exact benchmark solutions of random vibration responses for thin-walled orthotropic cylindrical shells

The exact solutions of random vibration responses of orthotropic thin cylindrical shells are imperative for verifying numerical approaches. This study proposes the discrete analytical method (DAM) and achieves the exact benchmark responses of thin-walled orthotropic cylindrical shells under stationary and nonstationary random excitations for the first time. Firstly, by revisiting the governing motion equations of elastic orthotropic shells based on the Flügge shell theory, the closed-form modal shape functions of orthotropic cylindrical shell with shear diaphragm boundary condition are introduced into stationary and nonstationary random vibration analyses. The power spectral density functions and root mean squares (RMSs) for stochastic responses of thin shell subjected to various random excitations, including the point, surface, base acceleration and moving excitations, are analytically derived by using the pseudo excitation method. Moreover, DAM is proposed to enhance the computational efficiency by implementing the discretization of the modal coordinates, frequency and temporal domains. Finally, the exact benchmark solutions of stationary and nonstationary stochastic responses are achieved, and the results indicate high accuracy and efficiency of proposed DAM. The excitation bandwidth and spatial distribution have significantly different effects on various types of responses, and maximum response RMS declines with increasing of acceleration of moving load.

Efficient non-stationary random vibration analysis of vehicle-bridge system based on an improved explicit time-domain method

Based on the recently developed explicit time-domain method (ETDM), this paper presents a new efficient computational framework for non-stationary random vibration analysis of the vehicle-bridge system (VBS) subjected to random track irregularity excitation. In this framework, the time-varying VBS equation is discretized using the time-stepping integration method, leading to an explicit expression for the stochastic system response in terms of the track irregularity excitation. Based on this expression, the response moments of the VBS can be easily evaluated in a direct way; the response probability density can also be obtained by using the Gaussian characteristic of the irregularity excitation. To further improve the computational efficiency, an efficient spectral representation-based formulation is proposed to calculate the excitation covariance matrix required in the analysis, which avoids tedious Fourier integrals required in the traditional ETDM. In the numerical application, the accuracy and efficiency of the proposed method are validated by the solutions obtained using the Monte Carlo and pseudo-excitation methods; the effects of several parameters on the stochastic responses of the VBS are investigated.

Frequency-domain characterization of varying random vibration loading by a non-stationarity matrix

Spectral analysis constitutes an essential technique for random vibration fatigue. The power spectral density (PSD) provides an efficient, statistically unique characterization of stationary Gaussian loading which can be effectively processed via linear systems theory and load-spectrum estimators such as the Dirlik formulation. The PSD represents vibration loading by its averaged intensity for frequency. However, if the intensity varies throughout a dataset, i.e. the loading is non-stationary, the PSD has a fundamental flaw – it conceals any information about its evolution. Consequently, the PSD is not qualified to describe non-stationary loading and thus cannot represent instantaneous highly damaging events in a fatigue assessment. Therefore, this article proposes a statistical characterization particularly formulated for the specifications of non-stationary random vibration fatigue. We introduce a non-stationarity matrix defined as the auto-correlation matrix of short-time Fourier transforms, which describes the average variation and interaction of intensity by frequency. This characterization can directly be related to fourth-order statistics (kurtosis, trispectrum, etc.) and allows to be processed via linear systems theory. Thereby the influence of a loading’s non-stationarity structure on structural responses, and thus fatigue damage, can be assessed. Furthermore, this article provides the basis for an accompanying article in which the non-stationarity matrix finds application to efficiently perform fatigue analyses for non-stationary loading via a statistical approach using load-spectrum estimators.

An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

The sensitivity analysis of hysteretic systems under nonstationary random excitations is of great concern to the stochastic optimal design and control of structures. The equivalent linear equation of motion is first constructed for the hysteretic system by the equivalent linearization method (ELM), and the sensitivity equation of the equivalent linear system is then derived by the direct differentiation technique. These two equations are further combined into an overall equivalent linear equation, which depends on the statistical moments and sensitivities of responses of the system and should be solved on an iterative basis. The analysis problem is thus transformed to a series of nonstationary random vibration analyses of the iterative linearized systems, which can be solved with the explicit time-domain method (ETDM) recently proposed for linear random vibration problems. The ETDM has high computational efficiency owing to the explicit formulations of statistical moments of responses. Therefore, a numerical algorithm is developed by combining ELM and ETDM for efficient stochastic sensitivity analysis of hysteretic systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.