Random vibration analysis of FGM plates subjected to moving load using a refined stochastic finite element method

The article introduces a finite element procedure using the bilinear quadrilateral element or four-node rectangular element(namely Q4 element) based on a refined first-order shear deformation theory(rFSDT) and Monte Carlo simulation(MCS), so-called refined stochastic finite element method to investigate the random vibration of functionally graded material(FGM) plates subjected to the moving load.The advantage of the proposed method is to use r-FSDT to improve the accuracy of classical FSDT, satisfy the stress-free condition at the plate boundaries, and combine with MCS to analyze the vibration of the FGM plate when the parameter inputs are random quantities following a normal distribution. The obtained results show that the distribution characteristics of the vibration response of the FGM plate depend on the standard deviation of the input parameters and the velocity of the moving load.Furthermore, the numerical results in this study are expected to contribute to improving the understanding of FGM plates subjected to moving loads with uncertain input parameters.

Random vibration of hysteretic systems under Poisson white noise excitations

Hysteresis widely exists in civil structures,and dissipates the mechanical energy of systems.Research on the random vibration of hysteretic systems,however,is still insufficient,particularly when the excitation is non-Gaussian.In this paper,the radial basis function(RBF)neural network(RBF-NN)method is adopted as a numerical method to investigate the random vibration of the Bouc-Wen hysteretic system under the Poisson white noise excitations.The solution to the reduced generalized Fokker-PlanckKolmogorov(GFPK)equation is expressed in terms of the RBF-NNs with the Gaussian activation functions,whose weights are determined by minimizing the loss function of the reduced GFPK equation residual and constraint associated with the normalization condition.A steel fiber reinforced ceramsite concrete(SFRCC)column loaded by the Poisson white noise is studied as an example to illustrate the solution process.The effects of several important parameters of both the system and the excitation on the stochastic response are evaluated,and the obtained results are compared with those obtained by the Monte Carlo simulations(MCSs).The numerical results show that the RBF-NN method can accurately predict the stationary response with a considerable high computational efficiency.

Adaptive inverse control of random vibration based on the filtered-X LMS algorithm

Random vibration control is aimed at reproducing the power spectral density （PSD） at specified control points. The classical frequency-spectrum equalization algorithm needs to compute the average of the multiple frequency response functions （FRFs）, which lengthens the control loop time in the equalization process. Likewise, the feedback control algorithm has a very slow convergence rate due to the small value of the feedback gain parameter to ensure stability of the system. To overcome these limitations, an adaptive inverse control of random vibrations based on the filtered-X least mean-square （LMS） algorithm is proposed. Furthermore, according to the description and iteration characteristics of random vibration tests in the frequency domain, the frequency domain LMS algorithm is adopted to refine the inverse characteristics of the FRF instead of the traditional time domain LMS algorithm. This inverse characteristic, which is called the impedance function of the system under control, is used to update the drive PSD directly. The test results indicated that in addition to successfully avoiding the instability problem that occurs during the iteration process, the adaptive control strategy minimizes the amount of time needed to obtain a short control loop and achieve equalization.

SEISMIC RANDOM VIBRATION ANALYSIS OF LOCALLY NONLINEAR STRUCTURES

A nonlinear seismic analysis method for complex frame structures subjected to sta- tionary random ground excitations is proposed.The nonlinear elasto-plastic behaviors may take place only on a small part of the structure.The Bouc-Wen differential equation model is used to model the hysteretic characteristics of the nonlinear components.The Pseudo Excitation Method (PEM)is used in solving the linearized random differential equations to replace the solution of the less efficient Lyapunov equation.Numerical results of a real bridge show that the method proposed is effective for practical engineering analysis.

SEISMIC RANDOM VIBRATION ANALYSIS OF STOCHASTIC STRUCTURES USING RANDOM FACTOR METHOD

Seismic random vibration analysis of stochastic truss structures is presented. A new method called random factor method is used for dynamic analysis of structures with uncertain parameters, due to variability in their material properties and geometry. Using the random factor method, the natural frequencies and modeshapes of a stochastic structure can be respectively described by the product of two parts, corresponding to the random factors of the structural parameters with uncertainty, and deterministic values of the natural frequencies and modeshapes obtained by conventional finite element analysis. The stochastic truss structure is subjected to stationary or non-stationary random earthquake excitation. Computational expressions for the mean and standard deviation of the mean square displacement and mean square stress are developed by means of the random variable＇s functional moment method and the algebra synthesis method. An antenna and a truss bridge are used as practical engineering examples to illustrate the application of the random factor method in the seismic response analysis of random structures under stationary or non-stationary random earthquake excitation.

Parametric characteristic of the random vibration response of nonlinear systems

Volterra series is a powerful mathematical tool for nonlinear system analysis,and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model.In the present study,the random vibration of nonlinear systems is investigated using Volterra series.Analytical expressions were derived for the calculation of the output power spectral density（PSD） and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation.Based on these expressions,it was revealed that both the output PSD and the input-output crossPSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity.Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship.The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.

DIRECT SPECTURM ANALYSIS OF THE STATE SPACE FOR RANDOM VIBRATION OF MDOF LINEAR DAMPED SYSTEM

This paper proceeds from the general case of the unsymmetric linearized multi-degrees of free- dom(MDOF)systems.By adopting the general complex modal theory of the state space,the response analysis for a sys- tem subjected to random excitation of the same source is carried out using as a kind of direct spectrum analysis method in frequency domain.With the input of power spectral density function given,the explicit expression of the power spectral density function matrix of the output response can be obtained.By taking Fourier inverse transform,the integrated expres- sions of the correlation function matrix and of the spectrum moment matrix are obtained.Comparing with the time domain method,this method enjoys the merit of visualization and avoids the procedure of transformation from the obtained re- sponse correlation function to be solved for the output spectrum utilizing Fourier transform.This paper has extended the application range of the traditional frequency domain analysis method.The mean square values and variety of statistical val- ues can be obtained conveniently.This method and the time domain method are different in approach but equally satisfac- tory in their results.

Analysis on Pseudo Excitation of Random Vibration for Structure of Time Flight Counter

Traditional computing method is inefficient for getting key dynamical parameters of complicated structure.Pseudo Excitation Method（PEM）is an effective method for calculation of random vibration.Due to complicated and coupling random vibration in rocket or shuttle launching,the new staging white noise mathematical model is deduced according to the practical launch environment.This deduced model is applied for PEM to calculate the specific structure of Time of Flight Counter（ToFC）.The responses of power spectral density and the relevant dynamic characteristic parameters of ToFC are obtained in terms of the flight acceptance test level.Considering stiffness of fixture structure,the random vibration experiments are conducted in three directions to compare with the revised PEM.The experimental results show the structure can bear the random vibration caused by launch without any damage and key dynamical parameters of ToFC are obtained.The revised PEM is similar with random vibration experiment in dynamical parameters and responses are proved by comparative results.The maximum error is within 9%.The reasons of errors are analyzed to improve reliability of calculation.This research provides an effective method for solutions of computing dynamical characteristic parameters of complicated structure in the process of rocket or shuttle launching.

Seismic safety assessment with non-Gaussian random processes for train-bridge coupled systems

Extensive high-speed railway(HSR)network resembled the intricate vascular system of the human body,crisscrossing mainlands.Seismic events,known for their unpredictability,pose a significant threat to both trains and bridges,given the HSR’s extended operational duration.Therefore,ensuring the running safety of train-bridge coupled(TBC)system,primarily composed of simply supported beam bridges,is paramount.Traditional methods like the Monte Carlo method fall short in analyzing this intricate system efficiently.Instead,efficient algorithm like the new point estimate method combined with moment expansion approximation(NPEM-MEA)is applied to study random responses of numerical simulation TBC systems.Validation of the NPEM-MEA’s feasibility is conducted using the Monte Carlo method.Comparative analysis confirms the accuracy and efficiency of the method,with a recommended truncation order of four to six for the NPEM-MEA.Additionally,the influences of seismic magnitude and epicentral distance are discussed based on the random dynamic responses in the TBC system.This methodology not only facilitates seismic safety assessments for TBC systems but also contributes to standard-setting for these systems under earthquake conditions.

Gray bootstrap method for estimating frequency-varying random vibration signals with small samples

During environment testing, the estimation of random vibration signals （RVS） is an important technique for the airborne platform safety and reliability. However, the available meth- ods including extreme value envelope method （EVEM）, statistical tolerances method （STM） and improved statistical tolerance method （ISTM） require large samples and typical probability distri- bution. Moreover, the frequency-varying characteristic of RVS is usually not taken into account. Gray bootstrap method （GBM） is proposed to solve the problem of estimating frequency-varying RVS with small samples. Firstly, the estimated indexes are obtained including the estimated inter- val, the estimated uncertainty, the estimated value, the estimated error and estimated reliability. In addition, GBM is applied to estimating the single flight testing of certain aircraft. At last, in order to evaluate the estimated performance, GBM is compared with bootstrap method （BM） and gray method （GM） in testing analysis. The result shows that GBM has superiority for estimating dynamic signals with small samples and estimated reliability is proved to be 100% at the given confidence level.

Matrix Power Control Algorithm for Multi-input Multi-output Random Vibration Test

Both auto-power spectrum and cross-power spectrum need to be controlled in multi-input multi-output （MIMO） random vibration test. During the control process with the difference control algorithm （DCA）, a lower triangular matrix is derived from Cholesky decomposition of a reference spectrum matrix. The diagonal elements of the lower triangular matrix （DELTM） may become negative. These negative values have no meaning in physical significance and can cause divergence of auto-power spectrum control. A proportional root mean square control algorithm （PRMSCA） provides another method to avoid the divergence caused by negative values of DELTM, but PRMSCA cannot control the cross-power spectrum. A new control algorithm named matrix power control algorithm （MPCA） is proposed in the paper. MPCA can guarantee that DELTM is always positive in the auto-power spectrum control. MPCA can also control the cross-power spectrum. After these three control algorithms are analyzed, three-input three-output random vibration control tests are implemented on a three-axis vibration shaker. The results show the validity of the proposed MPCA.

Control method for multi-input multi-output non-Gaussian random vibration test with cross spectra consideration

A control method for Multi-Input Multi-Output（MIMO） non-Gaussian random vibration test with cross spectra consideration is proposed in the paper. The aim of the proposed control method is to replicate the specified references composed of auto spectral densities, cross spectral densities and kurtoses on the test article in the laboratory. It is found that the cross spectral densities will bring intractable coupling problems and induce difficulty for the control of the multioutput kurtoses. Hence, a sequential phase modification method is put forward to solve the coupling problems in multi-input multi-output non-Gaussian random vibration test. To achieve the specified responses, an improved zero memory nonlinear transformation is utilized first to modify the Fourier phases of the signals with sequential phase modification method to obtain one frame reference response signals which satisfy the reference spectra and reference kurtoses. Then, an inverse system method is used in frequency domain to obtain the continuous stationary drive signals. At the same time, the matrix power control algorithm is utilized to control the spectra and kurtoses of the response signals further. At the end of the paper, a simulation example with a cantilever beam and a vibration shaker test are implemented and the results support the proposed method very well.

Fuzzy norm method for evaluating random vibration of airborne platform from limited PSD data

For random vibration of airborne platform, the accurate evaluation is a key indicator to ensure normal operation of airborne equipment in flight. However, only limited power spectral density（PSD） data can be obtained at the stage of flight test. Thus, those conventional evaluation methods cannot be employed when the distribution characteristics and priori information are unknown. In this paper, the fuzzy norm method（FNM） is proposed which combines the advantages of fuzzy theory and norm theory. The proposed method can deeply dig system information from limited data, which probability distribution is not taken into account. Firstly, the FNM is employed to evaluate variable interval and expanded uncertainty from limited PSD data, and the performance of FNM is demonstrated by confidence level, reliability and computing accuracy of expanded uncertainty. In addition, the optimal fuzzy parameters are discussed to meet the requirements of aviation standards and metrological practice. Finally, computer simulation is used to prove the adaptability of FNM. Compared with statistical methods, FNM has superiority for evaluating expanded uncertainty from limited data. The results show that the reliability of calculation and evaluation is superior to 95%.

Non-random vibration analysis for general viscous damping systems

The authors recently developed a kind of non-probabilistic analysis method, named as‘non-random vibration analysis’, to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to be an effective supplement of the traditional random vibration theory. In this paper, we further extend the nonrandom vibration analysis into the general viscous damping system, and formulate a method to calculate the dynamic response bounds of a viscous damping vibration system under uncertain excitations. Firstly, the unit impulse response matrix of the system is obtained by using a complex mode superposition method. Secondly, an analytic formulation of the system dynamic response middle point and radius under uncertain excitations is derived based on the Duhamel’s integral, and thus the upper and lower response bounds of the system can be obtained. Finally, two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Random Vibration Control of Laminated Composite Plates with Piezoelectric Fiber Reinforced Composites

This paper presents an analysis of the active control of random vibration for lami- nated composite plates using piezoelectric fiber reinforced composites （PFRC）. With Hamilton＇s principle and the Rayleigh-Ritz method, the equation of motion for the resulting electromechani- cal coupling system is derived. A velocity feedback control rule is employed to obtain an effective active damping in the suppression of random vibration. The power spectral density and mean- square displacements of the random vibration for laminated plates under different control gains are simulated and the validity of the present control strategy is confirmed. The effect of piezoelec- tric fiber orientation in the PFRC layer on the random vibration suppression is also investigated. The analytical methodology can be expanded to other kinds of random vibration.

In-plane Auto-Parametric Vibration of Inclined Cables under Random Transverse Excitation

In-plane auto-parametric stochastic vibration of inclined cables subjected to Gaussian white noise in transverse bridge orientation is investigated. Based on Newton＇s laws of motion and Galerkin＇s modal truncation principle, the influences of geometry nonlinearity induced by sag and large displacement of cables and the initial equilibrium state are taken into account. Meanwhile, the three-dimensional non-linear differential equations of inclined cables for coupling vibration are deduced, equivalent stochastic linearization method is applied to derive the 14-dimensional first-order nonlinear differential equations of state vectors, and the Runge-Kutta integration method is utilized to obtain the root mean square （RMS） response. Results show that when the transverse random excitation imposed on the stayed cable exceeds a critical value, the in-plane transverse vibration of the cable are excited due to tim auto-parametric nonlinear coupling, and the critical value of random excitation increases with the damping ratio. In this motion, the cable response possesses non-stationary characteristics, even though the loading keeps stationary.

Performance improvement of the stochastic-resonance-based tri-stableenergy harvester under random rotational vibration

In this paper,the stochastic-resonance-based tri-stable energy harvester(TEH)is proposed to enhance harvesting performance under random rotational vibration.An electromechanical coupled system interfaced with a standard rectifier circuit driven by colored noise is considered.The stationary probability density function(SPDF)of the harvester is obtained by the improved stochastic averaging.Then,with the adiabatic approximation theory,the analytical expression of signal-to-noise ratio(SNR)for the TEH is deduced to characterize stochastic resonance(SR).To enhance direct current(DC)power delivery from a rotational TEH,the influences of system parameters on SR is discussed.The obtained results suggest that there are damping-induced resonance and noise-intensity-induced SR in the tri-stable system.The TEH has higher harvesting performance under the optimal SR.That is,the optimal parameter combinations can induce optimal SR and maximize harvesting performance.Thus,the stochastic-resonance-based TEH can be optimized to enhance energy harvesting through choosing the optimal parameter.

Frequency Domain Fatigue Evaluation on SCR Girth-Weld Based on Structural Stress

The Steel Catenary Riser(SCR)is a vital component for transporting oil and gas from the seabed to the floating platform.The harsh environmental conditions and complex platform motion make the SCR’s girth-weld prone to fatigue failure.The structural stress fatigue theory and Master S-N curve method provide accurate predictions for the fatigue damage on the welded joints,which demonstrate significant potential and compatibility in multi-axial and random fatigue evaluation.Here,we propose a new frequency fatigue model subjected to welded joints of SCR under multiaxial stress,which fully integrates the mesh-insensitive structural stress and frequency domain random process and transforms the conventional welding fatigue technique of SCR into a spectrum analysis technique utilizing structural stress.Besides,a full-scale FE model of SCR with welds is established to obtain the modal structural stress of the girth weld and the frequency response function(FRF)of modal coordinate,and a biaxial fatigue evaluation about the girth weld of the SCR can be achieved by taking the effects of multi-load correlation and pipe-soil interaction into account.The research results indicate that the frequency-domain fatigue results are aligned with the time-domain results,meeting the fatigue evaluation requirements of the SCR.

Seismic spatial effects on long-span bridge response in nonstationary inhomogeneous random fields

The long-span bridge response to nonstationary multiple seismic random excitations is investigated using the PEM (pseudo excitation method). This method transforms the nonstationary random response analysis into ordinary direct dynamic analysis, and therefore, the analysis can be solved conveniently using the Newmark, Wilson-9 schemes or the precise integration method. Numerical results of the seismic response for an actual long-span bridge using the proposed PEM are given and compared with the results based on the conventional stationary analysis. From the numerical comparisons, it was found that both the seismic spatial effect and the nonstationary effect are quite important, and that both stationary and nonstationary seismic analysis should pay special attention to the wave passage effect.