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The U-shaped beam, a novel beam type, is extensively used in urban rail transit due to its ability to block radiation noise from wheel–rail interactions, providing effective noise control at the source. This study aims to investigate the dynamic response and structure-borne noise of a 25m concrete simply supported U-shaped girder at a vehicle speed of 80km/h, and to assess the effectiveness of novel track vibration damping devices. Grounded in stochastic vibration theory, this study employs pseudo excitation methods (PEM) to obtain the frequency domain response of a finite element model of the U-shaped girder. The structure-borne noise is then analyzed using the acoustic indirect boundary element method (BEM), with frequency-domain responses from PEM as input conditions. The eastern extension section of Nanjing Metro Line 2’s simply supported U-shaped beam is used as a case study for practical and theoretical analyses of dynamic characteristics and structural radiation noise. This study reveals that the sound pressure level of the mid-span section’s bottom plate is higher directly below and above it, while it is lower on either side of the web. Considering the ground reflection effects, sound pressure levels are highly closer to ground reflections. The damping device significantly attenuates vibration noise in the frequency range of 50–200Hz, with the bottom plate exhibiting superior vibration reduction effects compared to the web. The predictive method aligns closely with field test results, validating the feasibility of the approach. This study demonstrates that the U-shaped beam effectively controls rail transit noise, and the novel track vibration damping devices significantly reduce vibration noise, particularly in the bottom plate. This research provides valuable insights for noise control in urban rail transit systems.

The high-speed maglev train is a potential innovative and convenient transportation. Its stability and vibration performances while moving on bridges are still the fundamental considerations to be determined. The stable control condition of the high-speed maglev train moving on an irregulated track is analyzed theoretically employing a simplified moving electromagnet model at first. The Routh–Hurwitz stability criterion is introduced to determine the limiting values of the electromagnetic control parameters. It is interesting that the obtained stable critical values of the control parameters are not sensitive to the moving speed and the bridge parameters. The stable critical value of the electromagnetic control parameters is dominated by the negative stiffness and negative damping mechanism. The coupled vibration system of the high-speed maglev train–bridge considering the track irregularity is then established. The explicit time-domain integration method based on spectral decomposition is applied to solve the random vibration of the system, while the classical Newmark-$\beta$ method is used to solve the deterministic responses. The numerical results are compared and validated with the Monte Carlo simulation and the measurement data. The statistical response characteristics of the high-speed maglev train and the bridge under random track irregularity are then analyzed. The vibration of the train fluctuates obviously during the suspension process with a great standard derivation. Like the comment wheel rail train on bridge, responses increase obviously with the increase of train speed and the deterioration of the track irregularity.

In order to enhance the efficiency of stochastic vibration analysis for train–bridge coupling systems, this paper proposes a novel approach based on the parallel adaptive enhanced (PAE)-surrogate model. First, an initial surrogate model is established to predict the extreme values of dynamic responses in the train–bridge coupling system using a small number of training samples. Second, a multipoint adaptive sampling method is employed to determine a set of new samples that provide more information. The theoretical extreme values of the dynamic responses corresponding to the new samples are calculated using parallel computing technology. Third, the surrogate model is optimized by incorporating the set of new samples and their corresponding theoretical extreme values. Finally, new samples are continuously added, and the surrogate model is enhanced until the number of training samples reaches the preset requirement. To validate the effectiveness of the proposed method, two examples are examined, encompassing analytical functions and the analysis of the wheel load reduction rate (WLRR) for trains on the bridge. The results show that the proposed PAE-surrogate model can select samples containing valuable information, significantly improving the prediction accuracy of the surrogate model without increasing the number of training samples. Additionally, the proposed method can fully exploit computational resources, thereby decreasing the number of iterations needed and increasing training efficiency. By considering a four-car CHR2 train passing through a three-span simply supported girder bridge as an example, the proposed method achieves 2.62times higher training efficiency compared to the nonparallel method.

In this work a new method for the optimal design of generic elastic structures, subject to random dynamic loads, is proposed. Elastic structures are described as deterministic multi-degree of freedom systems in which the structural failure probability, referred to as a first crossing passage problem, is minimized. The proposed method is here applied to optimize the shape of a vertical column with an extra-mass located at the free top end, subject to a base acceleration modeled as a Gaussian, stationary, filtered stochastic process. The elastic threshold crossing probabilities are determined in a finite number of column sections and the objective function is assumed to be a measure of these probabilities. Finally, this measure is minimized under a constant weight constraint (or even under more general conditions).

Fatigue life, stability and performance of majority of the structures and systems depend significantly on dynamic loadings applied on them. In many engineering cases, the dynamic loading is random vibration and the structure is a plate-like system. Examples could be printed circuit boards or jet impingement cooling systems subjected to random vibrations in harsh military environments. In this study, the response of thin rectangular plates to random boundary excitation is analytically formulated and analyzed. In the presented method, closed-form mode shapes are used and some of the assumptions in previous studies are eliminated; hence it is simpler and reduces the computational load. In addition, the effects of different boundary conditions, modal damping and excitation frequency range on dynamic random response of the system are studied. The results show that increasing both the modal damping ratio and the excitation frequency range will decrease the root mean square acceleration and the maximum deflection of the plate.

In this paper, all the parameters affecting the nonlinear behavior of S-FGM plates under lateral stochastic white noise excitations are investigated. First the governing equation for a general S-FGM plate is derived for the assumed problem. Then it is rewritten by introducing some nondimensional parameters such that the results are applicable for a wide range of plates. Without loss of generality and using an example, the effective parameters on the instability and bifurcation of the transverse vibration of plates are studied, including the mean value of the lateral load with the in-plane forces and the material property. Especially, the role of material property is highlighted and some analogical figures are drawn to compare the behavior of the FGM plates with the homogenous ones. It is shown that the material property can affect the behavior of instability and bifurcation of the plate and its occurrence.

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

This paper presents a non-stationary random vibration analysis of railway bridges under moving heavy-haul trains by the pseudo-excitation method (PEM) considering the train-track-bridge coupling dynamics. The train and the ballasted track-bridge are modeled by the multibody dynamics and finite element (FE) method, respectively. Based on the linearized wheel-rail interaction model, the equations of motion of the train-ballasted track-bridge coupling system are then derived. Meanwhile, the excitations between the rails and wheels caused by the random track irregularity are transformed into a series of deterministic pseudo-harmonic excitation vectors by the PEM. Then, the random vibration responses of the coupling system are obtained using a step-by-step integration method and the maximum responses are estimated using the 3$\sigma$ rule for the Gaussian stochastic process. The proposed method is validated by the field measurement data collected from a simply-supported girder bridge (SSB) for heavy-haul trains in China. Finally, the effects of train speed, grade of track irregularity, and train type on the random dynamic behavior of six girder bridges for heavy-haul railways are investigated. The results show that the vertical acceleration and dynamic amplification factor (DAF) of the midspan of the SSB girders are influenced significantly by the train speed and track irregularity. With the increase in the vehicle axle-load, the vertical deflection-to-span ratio ($\gamma$) of the girders increases approximately linearly, but the DAF and vertical acceleration fail to show clear trend.

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.

The pseudo excitation method (PEM) is improved for its efficiency by incorporating the self-adaptive Gauss integration (SGI) technology as a new combining integration. The PEM can transform the random rail irregularities into some pseudo harmonic excitation, which is a mature approach to deal with the random excitation for vehicle–bridge systems. The SGI was used to distinguish the significant from the insignificant parts of an integral section for the random excitation frequency on the stochastic response of the system, thereby reducing the computational effort required for the random vibration analysis of the system. Also, the SGI can intelligently handle the recognized integral section, by subdividing the important sections into several necessary frequency points, making rough decomposition, and allowing the unimportant regions to be eliminated. Based on selected frequency points, the deterministic pseudo harmonic excitations were generated, and then the standard deviation (SD) of the time history for the system was calculated by the PEM. The vehicle subsystem was simulated as a 23-degree of freedom model, and the bridge subsystem as a three-dimensional finite element model. The time-varying power spectral density (PSD) plots of the system were presented. Besides, the cumulative distribution function (CDF) of the response was calculated using Poisson’s crossing assumption. The random characteristics for the vehicle–bridge vibrations for different speeds and rail irregularities were calculated.

The high-speed maglev vehicle-guideway coupled system (MVGCS) is a complex system, whose random vibration characteristics have not been well studied due to a limited number of examples. To address this issue, a new efficient approach is proposed for the random vibration analysis of the MVGCS, which combines the probability density evolution method and multi-time step method with multiple random loads considered. The random model established for 10-degree-of-freedom maglev vehicles and guideway is time-dependent, considering two different supporting conditions. The Monte Carlo method is used to assess the accuracy and efficiency of the proposed approximate approach, and the random model is verified through comparison with available results. The stochastic dynamic responses of the vehicles, guideway, and electromagnetic levitation forces, including the mean values and standard deviations, are determined in a case study. The results show that the proposed method is feasible for the dynamic analysis of maglev systems with a reasonably good efficiency in computation. Furthermore, critical parametric analyses involving vehicle speed, irregularity, and cut-off wavelength are performed with the results discussed.

A time–frequency random approach is proposed in this paper for the prediction of subway train-induced tunnel and ground vibrations. This is a development of the random approach previously proposed by the authors, which takes the discrete track support, singular track defects, etc., into consideration. The proposed approach is developed using a two-step method. First, the pseudo-excitation method (PEM) and the two-dimensional multibody system/finite element method model are effectively combined to derive the track–tunnel pseudo-interaction forces by employing the power spectral density of track irregularity. Second, the random vibrations of the tunnel–soil system are obtained via the PEM in the wavenumber–frequency domain. To improve the computational efficiency, a fast-computing strategy is proposed based on the multipoint synchronous algorithm. Using numerical examples, the proposed time–frequency hybrid modeling process is verified by comparing it with the fully coupled time-dependent three-dimensional approach. Furthermore, the influence of the discrete track support on the random vibrations of the tunnel and ground is discussed by comparing the results predicted by the proposed approach with those predicted by the previously developed approach.

Seismic action and wind excitation are the main sources of excitation to civil engineering structures. The analytical structural responses are similar for both cases, but the simplified formula in design codes on the displacement response under these excitations is quite different. This paper re-visits this difference from under stationary random excitation. The power spectrum density function of the above excitations contains several parameters which define the excitations in frequency domain. The simplified formulas of the displacement variance under different excitations are derived by adjusting these parameters. The responses from these formulas always include the resonance component of the response, whereas the presence of the background component depends on the ratio between the predominant frequency of excitation and the natural frequency of the structure. The influence of this ratio on the displacement covariance and the modal combination rules is then further discussed.

The paper presents a parametric optimization of Tuned Liquid Column Damper (TLCD) to control the structural vibration of 5MW NREL offshore wind turbine (OWT) subject to the wind and waves random forces. In this work, the fluid–structure interaction of monopile with seawater are modeled as an added mass and the soil-structure interaction of monopile foundation is considered through simplified model of discrete coupled springs model. Using a proprietary genetic algorithm, the TLCD on nacelle has its parameters optimized to reduce the standard deviation or roots mean square (RMS) dynamic response on tower top. The displacement results are compared to the ones obtained by exhaustive search methods via a response map. From the stochastic analysis of the structure response, the ideal tuning ratio and damping ratio are determined for the liquid column damper to present its highest efficiency, i.e. the lowest RMS displacement at the tower top. The damper parameters achieved by both optimization methods show a significant agreement between them. In addition, the use of the genetic algorithm, a parametric optimization of the TLCD installed in OWT is carried out considering the random excitation of the wind, wave, and rotor forces (Kaimal, JONSWAP, and rotational spectra, respectively). In possession of the TLCD optimal parameters determined by the parametric study, the mitigation of displacements at the standard wind turbine top is analyzed. The fore-aft vibration at tower top with a TLCD attached shows a significant reduction for actions dynamic components. Finally, TLDC optimal parameters have a direct relation with the considered force spectrum and the turbine transfer function, and both must be considered for the damper optimization process.

The focus of this paper is to examine the dynamic factor of the suburban railway by utilizing the random vibration approach. Breaking through the previous methods of relying on huge amounts of measured data, herein, the dynamic factor essentially results from two major parts: the dynamic effect caused by moving train loads and that generated by track irregularity, which has clear physical significance. As the internal excitation of the vehicle–bridge system, track irregularity has strong randomness. Based on the dimension reduction method, the spatial domain power spectral density (PSD) of the track irregularity is transformed into the time-domain PSD. Therefore, the randomness of the random process is reduced by exploiting the constraint form of a random function, and then, the typical samples of the track irregularity considering randomness are constructed. Using the vehicle–bridge coupled vibration model, the standard deviation of the dynamic factor is evaluated accounting for the random track irregularity and 99.7% guarantee rate. Finally, the impact coefficient of the track irregularity on the bridge is methodically obtained. The sensitivity of the standard deviation of the dynamic factor to vehicle speed and bridge frequency is analyzed. The given solution methodology can fully take into account the randomness of the track irregularity. Thereby, it provides the dynamic factor formulas as a reference for the dynamic performance evaluation of suburban railway bridges and possible revision of current design specifications.

A meshless model is proposed to carry out the free and stationary stochastic vibration analysis of the functionally graded combined rectangular and cylindrical shells (FG-CRCS) structure under the aerodynamic and thermal environment. The FG-CRCS structure contains three combined structure types including rectangular–cylindrical shell (RC-type), cylindrical–rectangular–cylindrical shell (CRC-type), and closed rectangular–semicylindrical shell (CRS-type). The effects of aerodynamic and thermal loads on the dynamic behaviors of the FG-CRCS structure with temperature-dependency of material properties are investigated by introducing the supersonic piston theory and thermo-elastic theory. Furthermore, the pseudo-excitation method (PEM) is adopted to simulate the random loads applied to the FG-CRCS structure. The dynamic equations of the FG-CRCS structure are established in the theoretical frame of the first-order shear deformation theory (FSDT), whose general boundary conditions and coupling relationship are regulated by the artificial springs. Then, the reasonableness of this meshless model to predict free and random vibrations in aerodynamic and thermal environments is verified by comparing it with published literature and FEM results. On this basis, the contribution of essential parameters (including the aerodynamic load, thermal load, and boundary condition) on the free and random vibration behaviors of the FG-CRCS structure is presented, which may serve as guidance for the design of the plate–shell coupled structures in aerospace.

Impellers of centrifugal compressors are generally loaded by fluctuating aerodynamic pressure in operations. Excessive vibration of the impellers can be induced by unsteady airflows and lead to severe fatigue failures. Traditional transient stress analyses implemented in time domain generally require multiple load-step, very time-consuming computations using input of temporal pneumatic force previously obtained from Computational fluid dynamics (CFD) analyses. For quick evaluation of structural integrity of impellers, it is necessary to develop random vibration models and solution approaches defined in frequency domain. In this paper, the Pseudo-Excitation Method (PEM) is used to obtain power spectral density of three-dimensional, dynamic displacement and stress of impellers. A finite element model of an unshrouded impeller of a centrifugal compressor is generated based on the result of unsteady CFD analysis. Compared with the direct transient stress analyses in time domain, the pseudo-excitation method provides accurate and fast estimation of dynamic response of the impeller, making it an applicable and efficient method for analyzing random vibration of impellers.

This paper presents a new approach named optimization-oriented exponential–polynomial-closure (OEPC) to study the behavior of stochastic nonlinear oscillators under both displacement-multiplicative and additive excitations that are Gaussian white noise. In addition to the original projection exponential–polynomial-closure (PEPC) solution procedure, the OEPC method provides an alternative procedure for solving the Fokker–Planck–Kolmogorov (FPK) equation. The OEPC method is formulated by constructing an objective function with the residue of FPK equation. By minimizing the objective function with a gradient-based method, the parameters in the exponential polynomial can be determined and the approximate PDF solution of the nonlinear random oscillator can be obtained. The solutions obtained through the OEPC approach are highly consistent with the available true solutions in special case or the Monte Carlo simulations (MCS). They are much more accurate than those obtained using the Gaussian closure method when the nonlinearity is strong. In addition, the OEPC method is computationally more efficient than MCS. The difference of the results from the OEPC and PEPC is compared and the advantage of the OEPC over PEPC is also shown when the system nonlinearity is strong and complicated.

Engineering structures under external loadings are generally modeled as multi-degree-of-freedom (MDOF) nonlinear dynamical systems under random excitations. Based on the response Markov assumption, MDOF dynamical system with $N$-dimensional state variables generally results in high-dimensional Fokker–Planck (FP) equation with $2N$ spatial dimensions and one time dimension. It has been challenging to solve high-dimensional FP equation, especially the one with time dimension. In this paper, the previously proposed technique state-space-split-exponential-polynomial closure (SSS-EPC) method is further generalized to consider probability density evolution process and applied for time-dependent FP equation. In the proposed solution procedure, after spatial dimensionality reduction treatment of high-dimensional FP equation, the approximate solution is generalized as a time-dependent function with only interested state variables. The herein developed generalization and enhancement make the objective of determining time-dependent non-stationary response PDF available. Meanwhile, Monte Carlo simulation (MCS) is conducted to validate the proposed solution procedure. Three examples of ten-degree-of-freedom and eight-degree-of-freedom MDOF nonlinear systems under random external and/or parametric excitations are investigated. It is found that the obtained results agree well with the simulated ones at both the transient and the stationary stages. Moreover, the computation time taken by the proposed procedure is only a fraction of the one taken by the simulated method.

In a recent publication [H. Waubke and C. Kasess, Gaussian closure technique applied to the hysteretic Bouc model with nonzero mean white noise excitation, J. Sound Vibr.382 (2016) 258–273], the response of a single-degree-of-freedom (SDOF) system under Gaussian white noise and a constant dead load is presented. The system has a hysteresis described by Bouc [R. Bouc, Forced vibration of mechanical systems with hysteresis, in Proc. Fourth Conference on Nonlinear Oscillation (Prague, 1967), p. 315]. New is the usage of a slowly time-varying deterministic load added to the Gaussian white noise process. The transient solution is calculated using the Gaussian closure technique together with an explicit time step procedure. All moments in the Gaussian closure technique are evaluated analytically. The results of the Gaussian closure technique are in good agreement with the results from the Monte-Carlo method.