Random Vibration Problems Research Articles

Nonstationary random vibration analysis of hysteretic systems with fractional derivatives by FFT-based frequency domain method

Nonstationary random vibration problems of nonlinear fractional systems are challenging and drawing increasing attention. This study presents an efficient numerical method for the random vibration analysis of large-scale nonlinear fractional systems subjected to fully nonstationary excitations. Firstly, the fast Fourier transform (FFT)-based frequency domain method originally proposed for the nonstationary random vibration analysis of linear integer-order systems is extended to linear fractional systems, in which the explicit expression of dynamic responses of fractional systems is derived based on the load superposition principle and Newmark method. Secondly, an equivalent linearization analysis framework is constructed to solve the nonstationary responses of hysteretic systems with fractional derivatives, in which the nonstationary response analyses of equivalent linear systems are accomplished by the FFT-based frequency domain method. The presented method can account for nonstationary excitations with arbitrary forms of evolutionary power spectrum, even of the non-separable one. Two numerical examples are used to confirm its superior performance in terms of accuracy and computational efficiency.

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.

Efficient Method for Approximating the Joint Extreme Value Distribution of Multivariate Stationary Gaussian Processes

The joint extreme value distribution (JEVD) of multivariate random processes is important for evaluating the system reliability of a structure subjected to random vibrations. There are very limited analytical approaches for predicting the JEVD, and these approaches are only computationally viable for problems up to three dimensions. This paper presents an efficient approximate method, which is not limited to low-dimensional problems, for estimating the JEVD of multivariate stationary Gaussian processes. The proposed method modifies an existing method by using the Gauss-Legendre quadrature to evaluate the extreme value correlation coefficients under the bivariate Poisson assumption. Then, the JEVD is approximated using the Nataf transformation. The effectiveness of the proposed method is demonstrated via two numerical examples. The first example concerns the airgap problem of an offshore structure subjected to random waves, in which the extreme wave elevations are evaluated at six locations, and failure is defined as threshold exceedance for any location. The second example is a random vibration problem comprising a three degrees-of-freedom system. Previous studies focused on series system reliability; here three system reliabilities are examined, including series, parallel, and hybrid systems. The proposed method solves the problems efficiently, and it is found to provide an accurate prediction of the JEVD by comparison with Monte Carlo simulation.

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.

Transient response prediction of randomly excited vibro-impact systems via RBF neural networks

Systems exposed to random excitations may fail long before stationarity is achieved, which necessitates consideration of the transient responses of the system. On the other hand, the transient response prediction of non-smooth systems may face more obstacles than that of smooth systems. One of the foremost challenges lies in the handling of vector fields with singularities in non-smooth systems. In this work, the radial basis function neural networks (RBFNN) is utilized for the first time to analyze the transient response of randomly excited vibro-impact systems (VIS), a class of typical non-smooth systems. The solution of the system response is expressed in terms of a serious of Gaussian activation functions (GAFs) with time-dependent weights. These time-dependent weights are determined by minimizing a loss function, which involves the residual of the differential equations and constraint conditions. To avoid the singularity of the initial condition being a Dirac delta function, a strategy of short-time Gaussian approximation is presented to obtain the initial weights. Two typical VISs exposed to stochastic excitations are presented to verify the suggested scheme. Appropriate comparisons to the data obtained by digital simulation show that the method yields reliable results even for strongly nonlinear systems. With the merits of high computational accuracy and satisfactory efficiency, this approach is expected to be an effective method for solution of random vibration problems of high-dimensional complex non-smooth systems.

Position-guided ptychography for vibration suppression with the aid of a laser interferometer

As a variant of the coherent X-ray diffraction imaging (CDI) method, ptychography provides an ultrahigh resolution of sub-10 nm for extended samples through iterative phasing of diffraction patterns; this advantage makes ptychography have great application potential in nanoscience, materials, biology and magnetics. However, irregular vibrations of motion stages in ptychography experiment system can induce incoherent blurring of diffraction patterns, thereby resulting in degradation of recovered image quality. Herein, we propose a new ptychography approach, termed position-guided vibration separation, to improve ptychography reconstruction in the presence of sample random vibration relative to the illumination by utilizing the instantaneous position data recorded by a laser interferometer (LIM) during exposures of the pixel detector. This approach provides a solution to the sample-stage random vibration problem by mapping the object mixed states onto the LIM-measured vibrational positions. Both the simulation and experimental results demonstrate the effectiveness of this approach in suppressing the random vibration effects on ptychography. Therefore, this new approach can significantly decrease the mechanical stability requirements on ptychography experiment instruments.

A simplified weak simulation method for the probabilistic response analysis of nonlinear random vibration problems

Numerical simulation methods are among the most used techniques to evaluate the stochastic response of engineering systems subjected to random excitations. In many practical situations the system should be integrated over long time intervals and a large number of sample trajectories need to be generated to assess the quantities of interest. In this circumstance conventional methods usually show exploding behavior or their implementation is computationally very demanding. In this paper we propose a new numerical simulation method for the probabilistic response analysis of nonlinear random vibration problems. We devise a simplified weak method having the simplicity of basically needing only three-point distributed random variable per time step for its implementation. Also, we prove that it reproduces the same statistical properties that characterize the exact solution to the linearized equation corresponding to the underlying system. This makes the method an efficient tool for stable long term simulations. Numerical experiments are carry out to illustrate the practical performance of the method.

Finite dimensional (FD) Slepian models for non-Gaussian processes

Conditions under which samples of continuous stochastic processes X(t) on bounded time intervals [0,τ] can be represented by samples of finite dimensional (FD) processes Xd(t) are augmented such that samples of Slepian models Sd,a(t) of Xd(t) can be used as surrogates for samples of Slepian models Sa(t) of X(t). FD processes are deterministic functions of time and d<∞ random variables. The discrepancy between target and FD samples is quantified by the metric of the space C[0,τ] of continuous functions. The numerical illustrations, which include Gaussian/non-Gaussian FD processes and solutions of linear/nonlinear random vibration problems, are consistent with the theoretical findings in the paper.

Equivalent linearization method improved by higher order statistics in modal space for geometrically nonlinear vibrations

In this paper, an equivalent linearization method considering higher order statistics based on nonlinear reduced order modeling techniques is proposed for the geometrically nonlinear random vibration problems of complex structures. Nonlinear reduced order models of the structures are constructed by leveraging the nonlinear analysis capabilities of commercial finite element codes, and an improvement to the Stiffness Evaluation Procedure method for determining the stiffness coefficients is achieved through the equivalence relation, leading to a notable reduction in computational cost with no further requirements for commercial finite element codes. The nonlinear terms are twice regulated equivalent, and then a regulated form of the stiffness coefficients is derived to introduce higher order statistics into the equations of motion. Then a linearized system obtained by the criterion of force error minimization is used to predict the random response of the original nonlinear system. The nonlinear problems are solved by the linearized system in the modal space, increasing computational efficiency significantly. Higher order statistical information of the response is introduced to improve the accuracy. Typical examples are used to verify the effectiveness of the proposed method, while its applicability is further demonstrated via the analysis of turbulence-excited composite laminates.

Higher-order analysis for random vibration of stochastic structures under Gaussian stationary excitation

The stochastic structural parameters may lead to non-Gaussianity of response even under Gaussian stationary excitation. To deal with this hybrid random vibration problem, a new method by combining conditional expectation, pseudo excitation method and mixed-degree cubature formula is proposed. First, the hybrid random vibration problem is converted into a series of conventional random vibration ones of deterministic structures by deriving the conditional expectations of the moment spectra of response. Second, formulas of the power spectral density (PSD) and higher-order spectrum of response of the conditional deterministic structure are derived in terms of the PSD of the Gaussian excitation. Third, a more efficient and practical formula of the higher-order spectrum of response of the conditional deterministic structure is proposed based on pseudo excitation method. Fourth, the mixed-degree cubature formula is introduced to evaluate the integration of the conditional expectation and obtained the PSD and higher-order spectrum of the hybrid random vibration response. Finally, two examples are illustrated to verify the precision 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.

Parallel Calculation of Random Vibration for Reentry Vehicles in Fluctuating Pressure Environment

The reentry vehicle is in an unconstrained free-flight state during the reentry process. The pressure of air fluctuations acts on the external surface of the vehicle and induces a random vibration environment with a wide-frequency range, which makes great influences on the structural dynamic properties of vehicles. This paper aims at the parallel calculation of the wide-frequency random vibration problems for the free body under multipoint fluctuating pressure loads. The calculation method is based on the self-developed JAUMIN framework and PANDA platform. The novel algorithm based on the modal superposition method reduces the computational complexity, realizes the parallel calculation, and achieves the maximum numerical simulation calculation capacity of 1.045 billion degrees of freedom (DOFs) while improving the calculation efficiency and the accuracy of the results. Finally, this article also uses typical examples to verify the correctness and parallel scalability of algorithms and programs.

Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

This paper analytically and experimentally investigates the random vibration problem of pretensioned rectangular membrane structures under heavy rainfall. An analytical model is firstly proposed. In this model, the heavy rainfall is modeled as a series of discretized pulse load whose amplitude follows the non-Gaussian distribution; then, the nonlinear governing equation and corresponding Fokker Planck Kolmogorov (FPK) equation of membrane structures are derived and solved by the perturbation method, considering both the geometrical nonlinearity of structure and the non-Gaussian type of load; consequently, the analytical solutions of the probability density function (PDF), mean and standard deviation of displacement response can be obtained. An artificial heavy rainfall experimental system is then developed. This system is used to (i) verify the analytical model and (ii) evaluate the effects of rainfall intensity, pretension level, and material properties. The results show that the displacement follows the significant non-Gaussian distribution and proves strong nonlinear characteristics with increasing rainfall intensity and pretension. The proposed analytical model can guide engineers to design membrane structures from the perspective of probability.

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.

Random Response of Multi-Segment Beam May Exceed Response of Homogeneous Counterparts by Order of Magnitude

This paper investigates the dynamic properties of an inhomogeneous, Bernoulli–Euler multi-segment beam composed of different materials. To the best of knowledge of the authors, the problem of random vibrations of beams composing of different chunks of the beams, namely, strong and weak parts, has not been studied in the literature. In this paper, exact solution of the natural frequencies and associated mode shapes of the multi-segment Bernoulli–Euler beam are obtained using Krylov–Duncan functions, followed by free, forced, and random vibration analyses using the normal mode method. Special emphasis is placed on two special configurations of multi-segment beam, namely, the ‘rigid-soft-rigid beam’ (RSR beam) and ‘soft-rigid-soft beam’ (SRS beam) as simplest manifestations of the multi-chunked structures. Some remarkable properties exhibited by the dynamic response of multi-segment beam are demonstrated through this work, which may be of considerable engineering significance, and could not have been anticipated in advance, especially quantitatively.

Analytical stochastic responses of thin cylindrical shells under various stationary excitations

This study proposed an efficient analytical framework to achieve exact stochastic responses of circular cylindrical shells under various stationary excitations. By introducing the exact modal shapes of closed and open cylindrical shells based on the Flügge theory, the analytical power spectral densities of responses for the two shells were derived via the pseudo excitation method (PEM), where the calculus operation in the spatial domain and the integral in frequency domain were involved. To take full advantage of PEM for matrix operation and improve computational efficiency, the discrete analytical method (DAM), implementing analytically the calculus operation in the spatial domain and discretizing the frequency domain, was proposed. Furthermore, the exact stationary stochastic responses of three shell examples under point, surface, base acceleration and spacecraft specification spectrum excitations were achieved. By comparing the root mean squares of responses, especially twisting moment and membrane shear force with those by finite element analysis software, the high accuracy and efficiency of DAM were demonstrated. The remarkable effect of frequency bandwidth of excitation on stochastic responses was also revealed. The exact stochastic responses calculated by DAM can be taken as benchmark solutions to verify the numerical methods for the high-frequency wideband random vibration problem.

On the functional model–based method for vibration-based robust damage detection: versions and experimental assessment

The problem of random vibration–based robust damage detection for structures operating under varying and non-measurable environmental and operating conditions is considered via a novel unsupervised functional model–based method. Two versions of the method are employed based on either the residual variance or uncorrelatedness (whiteness) of a proper functional model that incorporates the varying environmental and operating conditions in a scheduling vector. This article constitutes a proof-of-concept study in which a comprehensive laboratory assessment of the functional model–based method is undertaken using hundreds of experiments with a composite tail structure of an unmanned aerial vehicle and two early-stage damages under a considerable number of different environmental and operating conditions. Comparisons with two alternative state-of-the-art statistical time series type methods, that is, a multiple model–based method and a principal component analysis–based method, are also performed. The results indicate ideal detection performance for the functional model–based and multiple model–based methods, with the true positive rate reaching 100% at 0% false positive rate, but degraded performance for the pricipal component analysis–based method.

Nonstationary vibration responses of a three-dimensional tunnel-soil system excited by moving stochastic loads

Abstract This study investigates the nonstationary vibration problem of a three-dimensional tunnel-soil system under moving stochastic loads. A semi-analytical model combing an integral-transform method (ITM) with the pseudo-excitation method (PEM) is proposed to calculate the stochastic responses of the tunnel-soil system. Due to the load movement and randomness, the dynamic responses of the tunnel and soil remain nonstationary at a fixed location. With the aid of the PEM, the nonstationary random vibration problem can be transformed into a moving pseudo harmonic load analysis. To verify the accuracy and efficiency of the proposed method, it is compared with the Monte Carlo simulation. The nonstationary power spectral density (PSD) function and its spatial distribution of the dynamic responses of soil are then systematically analyzed. High frequency stochastic loads play an important role in shaping the PSD response. The stochastic load can cause a more apparent oscillation of wave propagation around the tunnel. Finally, a parametric study is conducted on the effects of the load velocity, soil and tunnel lining parameters on the nonstationary stochastic responses. The maximum RMS of radial stress and pore-water pressure are shown to increase, whereas the radial displacement is demonstrated to decrease with an increase in soil stiffness.

A spectrum-driven damage identification by minimum constitutive relation error and sparse regularization

Abstract This paper proposes a novel model-based damage identification strategy based on minimum constitutive relation error and sparse regularization using the power spectrum density data. Firstly, the stationary random vibration problem is transformed into a series of harmonic vibrations by the pseudo excitation method and the error in constitutive relation is established by the admissible stress field and admissible displacement field. A much more general and simpler strategy so as to build the admissible stress field is addressed by requiring only an extra decomposition of the stiffness matrix. Then, the sparse regularization is added to the original constitutive relation error objective function to circumvent the ill-posedness of the inverse problem. Finally, the solution of this nonlinear optimization problem is solved by the alternating minimization method. The proposed method has the advantage that only measurement power spectrum density data from few limited sensors are needed in the inverse analysis. Numerical and experimental results show the effectiveness and robustness of this approach.