Pseudo-excitation Method Research Articles

Free vibration and random dynamic analyses for the composite cabin-like combined structure in aero-thermal environment

This work presents the numerical investigations on the modal behaviors and stochastic dynamics of a composite cabin-like combined structure considering the aero-thermal factor and various random loads based on the meshless approach. Following the thermo-elastic theory, the influence of the aero-thermal factor on the dynamic behaviors of the combined structure is included. The cabin-like combined structure is disassembled into the open elliptical, conical and cylindrical shells, and coupling conditions between adjacent subshells are imposed by the massless connection springs, in which the Hamilton's principle is adopted to establish free and stochastic vibration models of each subshell. The dynamic solutions for the combined structure system are calculated by employing the meshless technique combined with the pseudo excitation method (PEM). The numerical validity concerning meshless model is presented by performing several case studies related to the modal frequencies and stochastic responses. Additionally, some physical insights into the influences of aero-thermal effect, type of random loads, and geometric dimensions, etc., on the free as well as stochastic vibration properties of the composite cabin-like combined structure are provided.

A Symplectic-Spectral Element Method for High Frequency Vibration Analysis and Its Application in Structure-Borne Noise Prediction

The frequency-domain vibration model of the vehicle-track structure is established based on the symplectic-spectral element method (S-SEM). To this end, the vehicle is considered as 10-degree-of-freedom, and the rail is modeled by an infinite Timoshenko beam supported by discrete sprung fasteners. Further, the track slab and base plate are regarded as two-dimensional Timoshenko beams supported by uniformly distributed sprung elements. Using the pseudo excitation method (PEM), the track irregularity is viewed as the pseudo load. The vibration response and the internal force of the single-layer or multi-layer beam models are obtained by solving the corresponding S-SEM-based equations. The spectral forces are also applied in frequency bands to the finite element (FE) and statistical energy analysis (SEA) acoustic models. The steel-concrete composite bridge deck is simulated with composite plate materials. The effectiveness of the proposed method is verified by comparing the predicted results with those obtained from the field test. The achieved results reveal that multi-layer beam structures effectively consume energy, and elastic fasteners and steel springs are capable of effectively reducing bridge noise.

Random vibration analysis of an uncertain vehicle-track coupled system based on a polynomial dimensional decomposition

ABSTRACT In this paper, the uncertainty propagation of an uncertain vehicle-track coupled system (VTCS) subjected to track irregularity is quantified using a polynomial-dimensional decomposition (PDD). Firstly, the conditional power spectral density (PSD) of response related to uncertain parameters is derived using the pseudo-excitation method. Secondly, the PDD surrogate model that can describe the probabilistic characters of the uncertainty propagation is established by conducting the dimensional decomposition to the conditional PSD with component functions and performing the Fourier expansion to the component functions. Finally, the dimensional reduction integration and Gauss integration are introduced to overcome the difficulty of high-dimensional integration when calculating the expansion coefficients. In numerical example, the proposed method is applied to the uncertainty quantification of vertical vibration response of an uncertain VTCS, the accuracy and efficiency of the proposed method are verified by comparing the results of the PDD method with that of Monte-Carlo simulation.

Research on vibration fatigue of sensor bracket in a metro bogie based on the pseudo excitation method

Aiming at investigating on fatigue failure of sensor bracket in a metro bogie, a vibration fatigue assessment method based on the pseudo excitation was proposed in this paper. The fatigue failure mechanism of sensor bracket was primarily investigated through a comprehensive field test, consisting of measuring the frequency response function, acceleration, and dynamic stress of the sensor bracket, as well as the roughness for both rail and wheel. The experimental results showed that the vibration modal near 61 Hz of sensor bracket can be excited by the rail corrugation-induced impact, thereby significantly contributes to stress concentration and increased dynamic stress near the bolt hole of the sensor bracket. To better understand the fatigue failure mechanism and evaluate the fatigue lifetime of the sensor bracket, a rigid-flexible coupled dynamic model of sensor bracket incorporating with the pseudo excitation method was developed to reproduce the vibration level of sensor bracket based on the measured acceleration. This enables us to obtain the full field stress state of sensor bracket, rather than the locations where the strain gauge installed alone. The results indicate that the parent material near the bolt hole of sensor bracket shows the highest damage, which is quite comparable with those observed in the failed sensor bracket.

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.

Stochastic Analysis of a Large-Span Continuous Girder High-Speed Railway Bridge under Fully Non-Stationary Earthquake

The main objective of the present work is to evaluate the stochastic seismic response of a large-span continuous girder high-speed railway (CGHSR) bridge subjected to fully non-stationary seismic excitation. The stochastic ground motion is determined by a fully non-stationary ground acceleration model with both the time-varying intensity and frequency content. The stochastic seismic analysis of the large-span CGHSR bridge under a tri-directional earthquake, taking into consideration the wave passage effect, is performed using the pseudo excitation method (PEM) in frequency domain and the Monte Carlo sampling (MCS) method in time domain. A framework is proposed to determine the pseudo static displacement for the PEM under fully non-stationary seismic excitation, using the proper orthogonal decomposition (POD). A comparative study on the stochastic seismic response of the bridge is conducted using both the fully and modulated non-stationary ground acceleration. The results show that the structural nodal displacements estimated by the PEM agree well with those obtained by the MCS method. The proposed POD-based method works well in estimating the structural quasi-static displacement. Ignoring the frequency non-stationarity of the earthquake may underestimate the quasi-static displacement caused by the surface waves.

Meshless analysis for modal properties and stochastic responses of heated laminated rectangular/sectorial plates in supersonic airflow

The object of this work is to develop a general meshless vibration model for analyzing the modal properties and stochastic responsesof laminated rectangular and sectorial plates subjected to stationary/nonstationary base acceleration random excitation in an aero-thermal environments. The supersonic piston theory is combined with the thermo-elastic theory to consider the thermal and aerodynamic effects of the laminated plate, and the Hamilton's principle is employed for formulating the complete governing equations of the structure system. The displacement components related to dynamic equations and the boundary conditions are characterized by a set of Chebyshev meshless shape functions with orthogonal properties. Then, the vibration behaviors for laminated rectangular/sectorial plates subjected to various boundary constraints are solved, where the pseudo excitation method (PEM) is adopted for performing the stationary/non-stationary stochastic analysis. The accuracy and stability of the present meshless solutions are validated by implementing some convergence checks and verification studies, in which the available results of FEM simulation models as well as the archived literature are provided as reference data. Also, physical insights into the effects of boundary stiffness, thermal/aerodynamic loads and different random excitations on the vibration mechanisms of laminated rectangular/sectorial plates are given, which might serve as the helpful guides for structural dynamic design.

Seismic responses of structure with six-element generalized model viscoelastic dampers

In this paper, the closed-form solution of seismic response of the structure with six-element generalized model (SGM) viscoelastic damper is analytically investigated based on complex model method (CMM) and pseudo excitation method (PEM), named as CM-PEM. Firstly, the damping characteristic of SGM viscoelastic damper is combined into structural dynamic equation which was decoupled into linear combination of first order equations by CMM. And the closed-form solutions of the structural dynamic responses are obtained as follow. Based on PEM, the power spectral density function (PSDF) of the structure is divided into the product of PSDF of the ground excitation and frequency response eigenvalue function. Through mathematical transformation, PSDF of the structural dynamic responses is quadratically orthogonalized, which can be integrated to obtain the closed-form expression of spectral moments. A 20-storey reinforced concrete structure is taken as an example study for verifying the accuracy and efficiency of the proposed CM-PEM method. A comparatively study between increasing column lateral stiffness and installation of SGM damper is finally conducted, and the results indicate installation of SGM damper shows better energy dissipation capacity.

Benchmark solutions for stochastic dynamic responses of rectangular Mindlin plates

There exist some researches on the benchmark responses of moderately thick plates based on Mindlin theory under static loads. In practice, rectangular Mindlin plates are often subjected to various random excitations. However, the benchmark solutions of analytically stochastic dynamic responses for rectangular Mindlin plates under random excitations were not investigated. In this paper, analytical power spectral density functions and root mean squares for stationary and nonstationary stochastic responses of rectangular Mindlin plates are derived, which provide benchmark solutions for other numerical methods and experimental design. Firstly, based on the closed-form free vibration analyses of rectangular Mindlin plates with two opposite simply supported edges, exact natural frequencies and modal shapes are obtained. Then, the pseudo excitation method is introduced into the decoupled differential equations of motion. By combining the exact solutions of free vibration, the analytical method for random vibration analysis of elastic Mindlin plate is proposed, and the analytical stochastic responses are derived. Due to the low efficiency of the AM based on symbolic operation, the efficient discrete analytical method, which can ensure that the results are exact in the spatial domain, is further developed. Finally, the performance of the proposed approaches is verified by comparing with the commercial software, Monte Carlo simulation, and literature. The remarkable effects of spatial distributions of excitations on the stochastic responses of Mindlin plates are illustrated. The important influences of parameters including relative thickness, elastic modulus, and boundary constraint degree on responses are revealed.

Experimental and numerical study on vibration and structure-borne noise of composite box-girder railway bridges

ABSTRACT This paper adopts a proficient approach to evaluate vibration and structure-borne noise and makes up for the steel-concrete composite (SCC) box girder vacancy in the SCC structure. First, the dynamic equation of the train-rail system is established, the pseudo-response of the system in the frequency domain is appropriately assessed by pseudo excitation method (PEM), and the frequency spectra of the wheel-rail force as well as the forces transmitted to the bridge are obtained. Next, applying the force to the finite element method (FEM), boundary element method (BEM), and statistical energy analysis (SEA) model. The vibration and structure-borne noise of the SCC box girder are calculated in the frequency domain. Then, the effectiveness of the method and model is verified through the comparison of the field experimental results. Finally, the effect of the ground reflection in the SEA model is considered using the sound ray-tracing method for the first time.

Probability distribution of derailment factor using identical probability distribution evolution method

To better understand the randomness of the derailment factor of the high-speed trains, its probability distribution function (PDF) and extreme value PDF are studied using the identical probability distribution evolution method (IPDEM). The standard deviation and mean of wheel-rail contact forces are obtained by pseudo-excitation method (PEM) and time history simulations. Then, based on the mean and standard deviation, the PDF and extreme value PDF of the derailment factor are obtained by IPDEM. The PDF and extreme value PDF of the derailment factor are validated by Monte Carlo method (MCM) simulations. Using a 6-car Pioneer EMU train passing over a 3-span 32-m long simply supported bridge as a case study, the applicability of the PDF and extreme value PDF of the derailment factor under varying train speeds, different track irregularities and different bridge deflection/span ratios are discussed. Finally, experimental data obtained from a field test are also used to verify the accuracy of the theoretical approaches. The results show that the PDF and extreme value PDF of the derailment factor proposed in this paper can provide a foundation for statistical evaluations.

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.

Equivalent linearization and parameter optimization of isochronous-pendulum tuned mass damper

The isochronous-pendulum tuned mass damper (IPTMD) with curved supports was proposed to eliminate the loss of vibration control performance induced by large swing angles in our previous study. However, the nonlinearity caused by the IPTMD leads to difficulty in its parameter optimization. To overcome this issue, an approach based on the equivalent linearization method (ELM) and genetic algorithm (GA) is proposed in this paper to facilitate the optimal design of the IPTMD. First, based on the pseudo excitation method, the nonlinear IPTMD system is equivalent to a linear one using the ELM. The numerical results demonstrate that the equivalent linearization model of the IPTMD is fairly precise to approximate the original nonlinear IPTMD. Then the optimal design based on the H∞ norm and GA is performed to compute the design parameters of the IPTMD, which can minimize the peak response of the main structure. Additionally, to evaluate the effectiveness of the proposed approach, the optimal parameters are applied to compute the dynamic response of a tall building installed with the IPTMD under different earthquakes. The numerical results reveal that the optimized IPTMD can reduce the displacement of the tall building effectively.

An analysis method for fatigue life of rail vehicle axles considering rotation effect of wheelset

In this paper, the discrepancy between the calculated and actual fatigue life distribution trend of the axles of railway vehicle structures is corrected. The calculations were made using the traditional fixed-axle model in which the wheelset rotation is neglected in random vibration analysis. A correction method is proposed, which is based on the original methods of fatigue life analysis of the axle structure of railway vehicles (i.e. the pseudo-excitation method and finite element model) and according to the characteristics of the symmetrical axle structure and the actual rotational operation state of the railway vehicle wheelset. Taking a locomotive system as an example, the minimum fatigue life calculated by the fixed-axle model is only 0.81 million km. In contrast, the minimum fatigue life calculated by the proposed method is 5.85 million km, which is consistent with the actual fatigue life of the axle. The position of the minimum fatigue life is also consistent with the actual position. The reliability of the proposed method for calculating the random fatigue life of the axle structure is illustrated. The influence and contribution of the prestress generated by axle press-fitting and vehicle serving weight of the railway vehicle acting on the axle on the axle fatigue life were studied. The results show that the prestress has a different influence on the fatigue life reduction of the wheelset. The minimum fatigue life reduction can reach 16.68% when all the prestresses are considered. Therefore, the influence of prestress should be considered in the design of railway vehicle axle structures. The idea of considering prestress should be proposed when revising axle standards.

Investigation on Excitation Equivalence Technique for Thin-Walled Panel Based on Pseudo-excitation Method

Investigation on Excitation Equivalence Technique for Thin-Walled Panel Based on Pseudo-excitation Method

Stochastic vibration suppression of composite laminated plates based on negative capacitance piezoelectric shunt damping

This paper studies the stochastic vibration suppression of the composite laminated plates by using the negative capacitance piezoelectric shunt damping. The energy method and Hamilton’s principle are employed to formulate the dynamic model of the composite laminated plate with negative capacitance piezoelectric shunt damping circuit. The random dynamic response of the coupled system subjected to nonstationary stochastic excitations is further obtained based on the modified Ritz method and pseudo excitation method (PEM). The accuracy of the proposed method is validated by comparing the obtained random results with those from the published literature and satisfactory agreements can be observed between them. Furthermore, the vibration suppression performance of the proposed piezoelectric shunt damping with series negative capacitor or parallel negative capacitor is discussed, respectively. Finally, the effects of key parameters of the piezoelectric shunt damping on the vibration suppression performance are discussed in detail.

Random Vibration Analysis of a Vehicle–Bridge Interaction System Subjected to Traveling Seismic Ground Motions Using Pseudo-excitation Method

Random Vibration Analysis of a Vehicle–Bridge Interaction System Subjected to Traveling Seismic Ground Motions Using Pseudo-excitation Method

A two-step weighting regularization method for stochastic excitation identification under multi-source uncertainties based on response superposition-decomposition principle

Excitation identification has received considerable attention because of its importance in safety assessment and structural design. This paper proposed a power spectral density (PSD) identification method for stationary stochastic excitations considering multi-source uncertainties in load fluctuations, material dispersions and measurement noises. Based on the traditional inverse pseudo-excitation method, a two-step weighting regularization strategy is creatively developed to reduce the amplification effects of the uncertainties in the transfer matrix and measurements on reconstructed results near natural frequencies. Especially, to enhance the generalizability of regularization operations, a weighting matrix is defined based on the interval-quantized deviation analysis of pseudo excitations and then an improved Tikhonov regularizing operator is defined given the features of the weighting transfer matrix and pseudo responses. Next, the response superposition-decomposition principle is performed to determine the boundaries of excitation PSD and two uncertainty propagation methods are developed. To guarantee the accuracy and efficiency of uncertainty analysis, the adaptive reduced-dimension Chebyshev model is adopted to characterize the nonlinear response-parameter relationships, and the first-order Taylor series approximation is used to describe the linear response-excitation relationships. Eventually, two numerical examples and one experimental example are discussed to demonstrate the feasibility of the developed approach. The results suggest its promising applications in complicated structures and loading conditions.

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.

Improved state augmentation method for buffeting analysis of structures subjected to non-stationary wind

Extreme winds such as hurricanes and thunderstorms often present non-stationary characteristics, having time-varying mean wind speeds and non-stationary wind fluctuations. When concerning the wind-induced vibrations under non-stationary wind, the excitation will be a non-stationary process, and the wind-structure coupled system can be represented by a linear time-varying (LTV) system. The aim of this study is to present a state augmentation method to investigate the non-stationary buffeting of a model bridge tower subjected to non-stationary wind with consideration of the aeroelastic damping. Based on the theory of stochastic differential equations and Itô’s lemma, the statistical moments of the non-stationary buffeting response are derived through solving a first-order ordinary differential equation system. The proposed method is validated by comparisons with the Monte Carlo method and the pseudo excitation method. The result shows that the state augmentation method has higher accuracy and efficiency than the well-accepted time–frequency techniques.