Stationary Random Vibration Research Articles

Ritz Solutions of Stationary Random Vibrations for GPL-Reinforced FG Rectangular Plate

This paper aims to establish a comprehensive model for evaluating the response characteristics of functionally graded graphene platelet-reinforced composite (FG-GPLRC) rectangular plates under stationary random acceleration excitation. Utilizing the first-order shear deformation theory (FSDT) and Halpin–Tsai assumptions, a dynamic model for the rectangular plate is derived, followed by obtaining the variational solution using the Rayleigh–Ritz method. To describe displacement components associated with dynamic equations and boundary conditions, a spectral geometry method (SGM) is employed. Additionally, a pseudo-excitation method (PEM) is utilized for the analysis of stationary random vibration. The proposed calculation method is validated through convergence analysis and comparative evaluation with existing literature and finite element models. Furthermore, the study investigates the influence of various factors, such as boundary conditions, the number and size of functionally graded rectangular plate layers, GPL distribution types, and material parameters, on the stationary random response attributes of FG rectangular plates. This research contributes to a deeper understanding of the dynamic response of functionally graded materials in structural configurations and offers a valuable analytical approach for researchers in this field.

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.

Influence of pile length on dynamic performance of monopile wind turbine

The influence of pile configuration on safety of wind turbines has been attracting large amounts of attention recently. On the one side, the cost for the foundation accounts for 35 % of the total cost of monopile supported wind turbines (MWTs). On the other side, the working environment of wind turbines brings complex loads to monopiles. In this paper, a series of wind tunnel tests are performed with scaled model under different ground and operation conditions. The test results show that pile length factor (PLF) affect nacelle displacement and maximum pile moment negatively. Top acceleration confirmed a stationary random vibration. The natural frequency of model MWTs increases 11.6 % (dense ground) and 10.1 % (loose ground) when PLF increase from 0.264 to 0.362. While damping ratio decreases 1.447 % (dense ground) and 1.493 % (loose ground). These test results can be up scaled to estimate prototype's dynamic properties. According to test results, some numerical fitting surfaces including d-Ω-PLF, f-Ω-PLF and ξ-Ω-PLF have been established to check unfavorable working conditions, which can be used to guide the design of piles to support MWTs.

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

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

Investigation of planar translational and rotational stationary non-Gaussian random vibration test

Multi-shaker random vibration control with coordinate transformation technique is an advanced mechanical environment test method that translational and rotational vibrations of structure can be controlled simultaneously. This work presents theoretical and experimental investigations of multi-shaker planar translational and rotational non-Gaussian random vibration tests. First, the coordinate transformation technique is used to obtain the translational and rotational vibration responses of the test structure. Then the control procedure is formulated theoretically with matrix presentation in which a new nonlinear transformation method for generating non-Gaussian random signals is proposed. The correlation between translational and rotational vibrations is explored briefly from the perspective of common square control and transformation control. The influence of elastic characteristics of the test structure on the response spectral densities is investigated and a level definition method for translational and rotational vibrations is proposed. Finally, a planar translational and rotational non-Gaussian random vibration test is carried out to verify the validity of the proposed methods.

浮体式洋上風車の風と波による応答および構造損傷発生確率評価に関する研究

An evaluation method of structural damage probability of floating offshore wind turbines under wind and wave actions is presented. The structural damage probability of the floating wind turbines is analyzed by using the first order reliability method in the structural reliability evaluation. In the evaluation of the probability, the response analysis results of the floating wind turbine for the wind and wave actions are used. The response for the action by the mean wind speed is analyzed by using the statically equation of equilibrium. On the other hand, the response for the dynamic actions by the variable wind speed and the wave is analyzed by using a mode superposition method and a stationary random vibration theory. Numerical model of the floating wind turbine is discretized by finite elements, whereas interface of the floating wind turbine and the sea water is discretized by boundary elements. Moreover, variation of the damage probability of the floating wind turbine for different wind speed is discussed by using the proposed method.

Stationary Random Vibration Analysis of Composite Laminated Shell Structures of Revolution in Thermal Environment

Currently, few studies are focused on the stationary random vibration for composite laminated shell structures of revolution (CLSSR), including composite laminated cylindrical shell (CLCY), composite laminated conical shells (CLCO), and composite laminated annular plates (CLAP). To fill this void corresponding to the above research in the literatures, a combination of the spectro-geometric method (SGM) and pseudo-excitation method (PEM) was developed to construct the theoretical model within the first-order shear deformation theory (FSDT). The different boundary restraints and coupling conditions were achieved by taking the appropriate stiffness values of artificial springs, and the thermal effect induced by thermal load was considered. Moreover, the Rayleigh–Ritz method was employed to deduce the governing differential equation. Further, the solution accuracy of the established model was assessed by comparing the obtained results with those from the literatures and the finite element method (FEM). Finally, the effect of specific parameters (i.e. fiber angle, temperature value and ply number) on the stationary random response of CLSSR was explored. According to the results, the proposed method proved effective for predicting the stationary random response characteristics of CLCY, CLCO, and CLAP in a thermal environment.

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.

Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method

With the increase in train speeds on high-speed railways, the excitation frequency of track irregularities increases, which has a negative impact on train comfort and opposes the passengers’ desire for high ride comfort. In addition, the uncertainty of train comfort results from the stationary randomness of track irregularities and the different zonal distribution in the car body. Therefore, the application of the stationary random vibration method to analyze the zonal distribution of train comfort and the relevant influencing factors is important to guarantee the passengers’ comfortable experience in each ride and to provide a theoretical basis for comfort optimization. First, the train was modeled using eight independent vehicle elements. Second, the pseudo-excitation method was applied to obtain the theoretical zonal distribution of the Sperling index, an indicator of comfort, via the linearity of the power spectrum density of train acceleration. Third, considering various factors affecting train comfort, the results were compared with those calculated using the Monte Carlo method. It was found that the most comfortable area was located slightly in the front of the center of the car body. Improving track irregularities and reasonably controlling the speed of a train will increase the train’s comfort, while it will deteriorate with a loss in car-body mass and damage to the secondary suspension system.

Spectral moment segmentation and spectrogram synthesis for simulation of road vehicle vibrations

Vibration testing procedures relying solely on a Fourier profile can introduce only stationary random vibrations. It is in contrast with the non-stationary and non-Gaussian nature of the Road vehicle vibrations (RVV). The following procedure segments the first four spectral moments of the time–frequency domain of RVV, constructs probability density arrays per frequency bins, and perform simulations according to random segment lengths and -root mean squares, yielding more realistic representations of RVV. The distribution of time- and frequency domain moments and normalized spectral entropies are confronted. The Probability-based spectrogram synthesis (PBSS) offers a data-driven stochastic modelling framework for simulating non-stationary RVV.

A hybrid method for analysing stationary random vibration of structures with uncertain parameters

This paper presents a new hybrid polynomial dimensional decomposition-pseudo excitation method (PDD-PEM) for analysing stationary random vibration of structures with uncertain parameters. The power spectral density (PSD) of the structural random response is expressed as a function of uncertain parameters using the PEM and a finite hierarchical expansion is performed using the component functions of the PDD. The component functions are then represented by approximate Fourier-polynomial expansions with orthonormal polynomial bases and expressions are given for calculating the first two moments of the PSD of random structural responses, using the coefficients of the expansion. Considering that calculating the coefficients requires difficult multi-dimensional integration, a dimension-reduction integration method and a Gaussian numerical integration method are introduced to effectively reduce the computational efforts of the calculation of the coefficients. The results of five numerical examples indicate that the proposed method is nearly as accurate as but more efficient than the Monte Carlo simulation method.

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.

Peak Factor–Based Modal Combination Rule of Response-Spectrum Method for Peak Floor Accelerations

The seismic safety of acceleration-sensitive secondary elements rigidly connected to the primary structure can be ensured in the case of ordinary structures by simply and accurately estimating the largest peaks in the floor accelerations consistent with a given seismic hazard. A new modal combination rule is developed in this study to estimate the peak floor accelerations in the case of lumped mass, classically damped, multistoried structures. The formulation is based on stationary random vibration theory and the use of nonstationary peak factors together with the amplitudes of the specified design spectra. The proposed formulation is supported by the development of (1) an expression for estimating the peak factor for floor acceleration from the peak factors for modal displacement responses and peak modal contributions to the peak floor response, wherein both floor acceleration and modal peak factors are normalized by the (nonstationary) peak factor for input ground acceleration, and (2) a two-parameter power function for estimating the normalized modal peak factors for relative velocity from the oscillator period, together with the scaling equations for the two parameters in terms of ground-motion characteristics. A numerical study using five example buildings and six example ground motions shows that the proposed modal combination rule works well, with the absolute error averaged over all floors of the structure being less than 10% for 90% of the cases. Also, the proposed rule works better compared to when various peak factors are taken to be identical.

Continuous convolution and nonlinear transformation for multi-shaker non-Gaussian random vibration control

This study proposes a continuous convolution method combined with memoryless nonlinear transformation for multi-input multi-output stationary non-Gaussian random vibration tests. The challenge of the multi-shaker non-Gaussian random vibration test lies in the coupling problems that are manifested in the inherent physical system and in the existence of cross-spectral densities. In the presented method, the independent stationary Gaussian random signals pass through a designed finite impulse response filter with a convolution manipulation first, and then the resulting signals are transformed to the non-Gaussian random signals by the memoryless nonlinear transformation method. The desired drive signals are obtained by the input–output relationship in the frequency domain. The finite impulse response filter is constructed by the frequency sampling technique in which the amplitude characteristics of the filter are determined by the predefined reference power spectral densities. A new monotonic nonlinear transformation function with an approximate kurtosis solution is provided. It only contains one parameter for kurtosis control both in sub-Gaussian and super-Gaussian cases. The memoryless nonlinear transformation is used to maintain the cross-spectral densities, although some distortions are introduced to the power spectra during the transformation process. The inverse system method is used to overcome the coupling problem caused by the inherent physical system. A simulation example and a triaxial vibration test are carried out, and the results indicate the validity and feasibility of the proposed method.

A time-domain procedure for non-Gaussian stationary environmental testing using zero-memory nonlinear transformation

This article proposes a time-domain procedure for a non-Gaussian stationary random vibration test with prescribed power spectral densities. Previous procedures for generating non-Gaussianity suffered from certain defects. For example, zero-memory nonlinear transformation, an algorithm frequently applied to transform Gaussian signals into non-Gaussian signals, often produces changes in both auto-power spectral densities and cross-power spectral densities, which might result in control instability under certain circumstances. In this article, the authors propose a different approach for the zero-memory nonlinear function. First, a time-domain procedure for a non-Gaussian random test is introduced. Second, a rescaling method is applied to correct the magnitude amplification on the auto-power spectral density because of zero-memory nonlinear transformation. We offer experience formulas in this method to adjust the auto-power spectral density of both super-Gaussian and sub-Gaussian responses. Third, a control strategy using a finite impulse response filter is proposed to further improve the auto-power spectral density if the shape of the auto-power spectral density is distorted. The kurtosis loss induced by the filtering process is also analysed and a corresponding solution is put forward to ease the reduction. Numerical test and a biaxial shaker table test are conducted to validate the availability and superiority of the proposed procedure.

Analysis of low damping ratios in multi-exciter stationary non-Gaussian random vibration control

This article investigates the influence of low damping ratios on the performance of the multi-exciter stationary non-Gaussian random vibration control system. The basic theory of the multi-exciter stationary non-Gaussian random vibration method is reviewed first, and then the influences of low damping ratios on multi-output spectra and kurtoses are analyzed. The low damping ratios cause an ill-conditioned problem which will make the drive spectral matrix solution inaccurate; thus, some spectral lines located at resonance peaks in the response spectra cannot be modified within the preset tolerances by the control algorithms. The regularization method is used to alleviate the calculation error. The output kurtoses are dependent not only on the characteristics of the system but also on the input signals. It is found that the kurtosis control will be intractable if the damping ratios are very low. A two-input two-output cantilever beam simulation example is described to illustrate the analysis results.

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.

Stationary random vibration of a viscoelastic Timoshenko cantilever beam under diverse random processes

In this paper, the stochastic properties of a uniform Timoshenko cantilever beam are investigated systematically. Based on the external viscous damping and Kelvin–Voigt viscoelastic damping, the partial differential equations of the Timoshenko beam subjected to random excitation are derived. The applied load is the concentrated force, and the excitation related to includes the ideal white noise, the band-limited white noise, and the exponential noise. Expressions are obtained for the space–time correlation functions and the space–frequency power spectral density functions of the transverse displacement response. The evident improvement is that the infinite integral and the definite integration in the mean square responses are worked out by means of the residue integral method and the integration by partial fraction, and the exact solutions of the mean square response are obtained in the form of an infinite series finally. This improvement provides a basis for both the mode truncation and the modal cross-spectral densities whether which can be ignored. Providing the numerical example, the numerical results obtained show the effectiveness of the theoretical analysis.

Multi-exciter stationary non-Gaussian random vibration test with time domain randomization

Abstract This paper presents a new control method for multi-input multi-output stationary non-Gaussian random vibration test using time domain randomization. The control objectives are composed of response skewnesses, kurtoses and power spectral densities. The generation process of stationary and coupled reference non-Gaussian signals by specified reference skewnesses, kurtoses and spectra is analyzed. The reference non-Gaussian signals combined with system frequency response functions are then utilized to obtain the desired drive signals for dynamic inputs, in which the inverse system method in the frequency domain is employed. The primary advantages of the proposed methods are the high computational efficiency and simultaneous control of the time-frequency characteristics of response signals. In consideration of system cross coupling characteristics manifested in coherence and phase coefficients, the skewness and kurtosis tuning steps for each control channel are formulated by using a sequential phase modification method. The relationships between reference skewnesses, kurtoses and spectra are discussed and they reveal that the reference spectra have an influence on the settings of reference skewnesses and kurtoses, which implies that proper settings of reference skewnesses, kurtoses and spectra are necessary. A numerical example and a triaxial vibration test are provided and the results show the validity and feasibility of the proposed method.

Symplectic analysis of stationary random vibration of vehicle–track coupled system with rail-borne electromagnetic generator

In this article, the authors establish the vehicle-track coupled model of smart railway track with rail-borne electromagnetic generator and conduct a symplectic analysis in frequency domain. A vehicle-track model considering vehicle traveling load is constructed, and the pseudo-excitation method is used. The frequency-dependent stiffness of rail pad is involved in the calculation model to get accurate results of rail natural frequency. The calculated results indicate that the smart railway track with electromagnetic generator will not affect the random vibration response of vehicle and bogie, but it could have certain influences on the wheelset and under-rail structure, especially the vibration response of rail, whose acceleration Power Spectrum Density difference reaches 18.6% with frequency shift of 54 Hz at basic frequency of 730.2 Hz, whereas the low frequency (below 200 Hz) rail vibration is not affected. The methods and results could serve as a guideline for the safety checking and dynamics calculation of smart railway track with rail-borne electromagnetic generator.