Non-stationary Response Research Articles

Random Dynamic Analysis and Running Safety Assessment of Long-Span Cable-Stayed Bridge Under High-Speed Trains Subjected to Earthquake Excitations and Track Irregularities

Running trains on a long-span bridge under earthquake is an inevitable problem for the high-speed railway in seismic regions. This paper aims to investigate the seismically random dynamic behavior of train–cable stayed bridges interaction system using the probability density evolution method (PDEM). The system simultaneously involves the multi-effect of random multi-point earthquake excitations with traveling waves and random track irregularity. The multi-point earthquake excitations, composed of lateral and vertical positions, are generated by stochastic harmful functions (SHFs) and uniformly modulated into nonstationary random processes. The motion equation of such a system is established by coupling the 38-degree vehicles and bridges through the refined random wheel/rail contact relationship and accounting for the phase-lags of seismic travelling waves between pier excitations. PDEM is proven to be applicable to such time-dependent systems and is then used to transform the random excitations into a series of deterministic representative excitations with initial probability. By solving for the corresponding deterministic probability responses, various nonstationary random responses, including the time-dependent probability density evolution functions of random responses, mean values curves and standard deviations, can be obtained easily. A case study based on the cable-stayed bridge is then presented, and the results show the effectiveness and accuracy of the proposed method by comparison with Monte Carlo Simulation. Additionally, the influence of train speed and seismic intensity on the safety and reliability of trains is discussed.

Spatially non-stationary response of habitat quality to land use activities in World's protected areas over 20 years

Land use activities play an essential role in habitat degradation and restoration. As priority zones for global habitat conservation, studying the impact of land use activities on habitat quality (HQ) in the World Protected Areas (WDPA) has great importance. However, insights into land use activities and HQ in the WDPA remain scarce and their relationship is still unclear. Here, we assessed land use activities and HQ in the WDPA over the past 20 years and analyzed the impact of land use activities on HQ. The results showed that forestland was the dominant land use type in the WDPA and that the area of forestland, bare land, and water area decreased while the area of other land use types increased during the study period. Land use change in protected areas in different continents shows great differences. Land use intensity (LUI) in the WDPA increased over the 20 years, but HQ remained largely unchanged. LUI increased but HQ decreased was the main change type of grid cells in the WDPA. The bivariate spatial autocorrelation results showed that High-Low (high HQ and low LUI) clusters were the dominant clustering type in the WDPA, while Low-High (low HQ and high LUI) clusters were the least represented clustering type. The regression results showed that LUI tended to have a negative effect on HQ in forested areas and a positive effect on HQ in desert, permafrost areas. These findings can provide a reference for effective land use differentiation control measures and biodiversity conservation measures in the WDPA.

Structural Modal Parameters Identification Under Ambient Excitation Using Optimized Symplectic Geometry Mode Decomposition

In the process of structural modal parameters identification under environmental excitation, the employed measured dynamic response signal is usually non-stationary and contains noise. As a novel signal analyzed method, symplectic geometry mode decomposition (SGMD) has been proven to be effective for dealing with non-stationary and noisy signals. However, the traditional SGMD may treat noise as modal information, which inevitably undermines the accuracy of modal identification. To overcome this problem, this paper proposes an optimized SGMD algorithm for structural modal parameter identification. First, the SGMD algorithm is refined with the Hankel matrix, Kurtosis theory, Pearson correlation coefficient, and energy entropy theory. Then, the natural frequencies and damping ratios are identified using the proposed method, which consists of the optimized SGMD algorithm, natural excitation technique (NExT), direct interpolating method, and curve-fitting function. Finally, the applicability of the proposed method under environmental excitation is investigated by two examples: a two-story frame structure and an 88-story office tower. The results demonstrate that the proposed method is effective for identifying the civil structural modal parameters from non-stationary dynamic responses.

An Interpretable Method for Operational Modal Analysis in Time-Frequency Representation and Its Applications to Railway Sleepers

Operational modal analysis (OMA) enables the identification of modal characteristics under operational loads and conditions. Traditional frequency-domain methods cannot directly capture modal changes over time, while existing time-frequency representations are not sufficiently interpretable due to spurious modes and implicit parameter design. This paper develops a new OMA method in time-frequency representation based on frequency-domain decomposition (FDD). Short-time FDD and a convolution-based strategy are proposed to obtain singular values and local mode shape similarity, respectively, which are further fused into mode indicators by a fuzzy-based strategy mimicking the modal assurance criterion. The method provides not only a global view of the modal characteristics over time and frequency but also estimates of the modal parameters. It is applicable to strongly nonstationary responses under time-varying loads and conditions. All the parameters explicitly affect the time-frequency representation, and the interpretability is enhanced by including physical information from the user’s prior knowledge in selecting parameters and peak bands. The proposed method is validated based on a study of railway sleepers under train passage. The rigid-body motions and bending modes are identified at frequencies up to 6,500 Hz in laboratory tests and 4,500 Hz in field tests at speeds up to 200 km/h. The identified natural frequencies and mode shapes agree with experimental modal analysis (EMA). The proposed method outperforms EMA in terms of broad frequency range and low measurement cost and can be potentially applied to structural health monitoring under operational conditions.

Nonlinear dynamics of time-varying curvature balance rope coupled with time-varying length hoisting rope in friction hoisting system

Mine friction hoisting system is a typical multi-body system composed of traction sheave, conveyance, hoisting rope, and balance rope. Hoisting rope is of large-scale and flexible tension, leading to large amplitude elastic vibration and huge tension fluctuation during hoisting movement. Balance rope is of small tension amplitude and bending modulus, undergoing significant curvature changes, large displacement, and large deformation. Considering the complex coupling vibrations between the moving hoisting rope and balance rope in the ultra-kilometer-deep shaft, a coupling modeling method of the variable-length hoisting rope with the variable-curvature balance rope is proposed. Based on the absolute nodal coordinate formulation(ANCF), the balance rope is modeled by non-equal-length and variable-number elements, and the hoisting ropes on both sides are modeled by variable-length and variable-number elements. A flexible rope vibration experiment platform is built in the laboratory to verify correctness of the proposed theoretical model.The experimental results are in good agreement with those in theoretical simulation. Besides, the nonlinear coupling dynamic behaviors of hoisting rope and balance rope in an ultra-kilometer-deep well are studied. It reveals the non-stationary phenomenon of hoisting rope and balance rope with a long-time range. Both sides of flat balance rope undergo large bending deformation and non-stationary response when transient resonance occurs in hoisting ropes.

Seismic Responses of an Isolated Long-span Bridge using Frequency Domain and Time Dependent Procedures

Seismic behavior of an isolated bridge is analyzed in the frequency domain under the effects of nonstationary ground motions. For dynamic solutions, different ground environments are considered by simulating nonstationary quakes that can be represented from bedrock to soft ground level. In the simulations, power spectral functions and filtered white noise model are adopted for spectral densities of the earthquake excitations. Various computer algorithms have been developed for earthquake simulations, establishing the bridge finite element model and stochastic solutions. Twenty simulated ground motions are used for each soil profile and the parameters of Rayleigh dispersion are estimated by evaluating the system responses for each ensemble. A number of peak response factors dependent on soil conditions are presented for seismic responses. In addition, extreme value distributions of the responses are shown with the probability of exceeding functions and tables. The responses are discussed for the specific exceedance level of probabilities used in probabilistic design process. The stochastic analyses generally yielded responses consistent with time domain solutions. Exceedance probability functions of the peak responses were obtained in a close relationship. However, the probability distributions of the responses decomposed for the soft soil case and they displayed a wider dispersion even for low exceedance levels. The peak responses are expressed with some exceedance probabilities. In the estimation of response variations, this study showed the practicality of the frequency domain method and the results revealed higher peak response factors and variances for softer soil conditions. Furthermore, this study indicated that the frequency domain procedure is an effective tool in the obtaining of nonstationary seismic responses.

Vibration Response Law of Aircraft Taxiing under Random Roughness Excitation

Understanding the vibration response patterns of aircraft taxiing under runway roughness excitation is crucial for aircraft design and runway performance evaluation. In this paper, we establish pavement roughness models for both asphalt and concrete surfaces, taking into account their unique structural characteristics. We construct a six-degrees-of-freedom aircraft model using multi-rigid-body system dynamics theory and employ the pseudo-excitation method to examine the influence of pavement roughness types on the steady vibration response of aircraft taxiing at constant speeds. Furthermore, we analyze the non-stationary vibration response patterns of aircraft during takeoff and landing taxiing using the method of instantaneous frequency response function with space frequency. Lastly, we explore the effect of stochastic structural parameters on aircraft vibration response using the Monte Carlo method. Our findings reveal that the roughness power spectrum differs between asphalt and concrete pavements, and the established roughness models in this paper demonstrate a strong fit (R2 > 0.95). The type of pavement roughness has a relatively minor impact on the power spectral density distribution of the aircraft vibration response, suggesting that the same roughness model can be used for both asphalt and concrete pavements when high accuracy is not required. The power spectral density distribution of aircraft vibration response varies across different motion attitudes, with the vibration response during landing being significantly larger than that during takeoff. Among the aircraft structural parameters, the randomness of the sprung mass has the most substantial effect on the aircraft vibration response, potentially causing the variation coefficient of dynamic load on the front landing gear to exceed 0.11. Tire stiffness comes next, which can lead to the variation coefficient of dynamic load on the main landing gear reaching 0.07. The results have a guiding role in optimizing aircraft structure design and ensuring pavement performance.

On non-stationary response of cracked thin rectangular plates acted upon by a moving random force

On non-stationary response of cracked thin rectangular plates acted upon by a moving random force

Extended statistical linearization approach for estimating non-stationary response statistics of systems comprising fractional derivative elements

An efficient technique to analyze non-stationary nonlinear random vibrations of dynamic systems endowed with a fractional derivative term is presented. The technique itself represents an extension of the concept of statistical linearization to this kind of systems, and it is applicable for both analytic and hysteretic system nonlinearities. The technique first resorts to harmonic balancing in deriving response-amplitude dependent equivalent damping and stiffness. This enables representation of the fractional derivative term as a linear combination of the system response displacement and velocity with amplitude dependent coefficients. Then, the expected values of these parameters are considered in proceeding to formulate a statistical linearization solution scheme. In this context, the solution procedure is completed by integrating in time the covariance Lyapunov equation associated with the derived equivalent linear system. The reliability of the proposed technique is tested by a series of germane Monte Carlo studies. This juxtaposition is also used to elucidate salient features of the technique, by varying the order of the fractional derivative term, and of the degree of the nonlinearity in the system. It also points out the versatility of the technique in determining the non-stationary values of auto-correlation and cross-correlations response parameters involving even the fractional derivative term.

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

A multi-sensor mapping Bi-LSTM model of bridge monitoring data based on spatial-temporal attention mechanism

The premise of intelligent structural health monitoring is to build a digital benchmark model representing the multi-sensor mapping relationship. Previously, researchers have used machine learning methods such as Long-and-Short-Term-Memory (LSTM) networks for modeling multi-sensor mapping relationships. However, the typical LSTM networks treat multi-dimensional input data equally, ignoring the possible correlations of different periods and sensor placements. Consequently, they can hardly obtain an accurate model for long-time series and multidimensional datasets. To address the issues, by introducing a two-layer attention mechanism in the temporal and spatial dimensions, an improved Bi-direction LSTM neural network model embedding the attention mechanism is proposed. The model focuses on grasping the non-stationary response process and the spatial correlation of multi-sensors. The accuracy and efficiency of the proposed approach are verified through two case studies. In addition, based on the visualization of attention weight assignments, the spatial–temporal attention distribution is studied to give an engineering interpretation.

Quantifying the spatial nonstationary response of influencing factors on ecosystem health based on the geographical weighted regression (GWR) model: an example in Inner Mongolia, China, from 1995 to 2020

The identification of ecosystem health and its influencing factors is crucial to the sustainable management of ecosystems and ecosystem restoration. Although numerous studies on ecosystem health have been carried out from different perspectives, few studies have systematically investigated the spatiotemporal heterogeneity between ecosystem health and its influencing factors. Considering this gap, the spatial relationships between ecosystem health and its factors concerning climate, socioeconomic, and natural resource endowment at the county level were estimated based on a geographically weighted regression (GWR) model. The spatiotemporal distribution pattern and driving mechanism of ecosystem health were systematically analysed. The results showed the following: (1) the ecosystem health level in Inner Mongolia spatially increases from northwest to southeast, displaying notable global spatial autocorrelation and local spatial aggregation. (2) The factors influencing ecosystem health exhibit significant spatial heterogeneity. Annual average precipitation (AMP) and biodiversity (BI) are positively correlated with ecosystem health, and annual average temperature (AMT) and land use intensity (LUI) are estimated to be negatively correlated with ecosystem health. (3) Annual average precipitation (AMP) significantly improves ecosystem health, whereas annual average temperature (AMT) significantly worsens eco-health in the eastern and northern regions. LUI negatively impacts ecosystem health in western counties (such as Alxa, Ordos, and Baynnur). This study contributes to extending our understanding of ecosystem health depending on spatial scale and can inform decision-makers about how to control various influencing factors to improve the local ecology under local conditions. Finally, this study also proposes some relevant policy suggestions and provides effective ecosystem preservation and management support in Inner Mongolia.

Efficient buffeting analysis of long-span bridges under non-stationary winds: A 2D interpolation enhanced approach

The buffeting analysis of long-span bridges under non-stationary winds is a crucial consideration in the context of exceptional winds recurring in frequent and violent patterns. The time–frequency representation is widely used for the solution of structural non-stationary buffeting responses due to its popularity and simplicity. However, the traditional approach consumes considerable computational resources to achieve high accuracy, which renders the approach computationally inefficient. In such a case, a 2D interpolation enhanced approach is developed for the efficient buffeting analysis of long-span bridges under non-stationary wind actions. Central to this approach is the 2D interpolation applied to the reconstruction of structural buffeting responses, accompanied by the well-designed non-uniformly distributed interpolation knots. For the enhanced approach, the computations in solving governing equations are only required at limited interpolation knots instead of at each frequency and time instant, which yields a dramatic reduction of computational resources and efforts. By taking a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of interpolation intervals on the simulation efficiency and accuracy, and the preferred suggestions are made accordingly. Then, the effectiveness of the enhanced approach is further verified via a case study by focusing on the evolutionary spectra and non-stationary buffeting responses. The analytical results indicate that the enhanced approach is particularly efficient in simulating non-stationary buffeting responses of long-span bridges with few losses of accuracy.

Nonstationary response of floating structures to random waves

Even if ocean waves are treated as a stationary random process, dynamic responses of floating structures to random waves at the transient state are always nonstationary. When nonstationary response statistics is desired, a common technique is to apply Monte Carlo simulations; however, its implementation is costly in computational time. Analytically, this article develops an efficient method for computing nonstationary response statistics, including evolutionary power spectrum and time-varying mean-square values. Assuming a hydrodynamic software has been employed to get various frequency response functions, a prerequisite of the proposed method is to get the elevation-to-motion transfer function formulated in its pole-residue form. The proposed method is applicable to arbitrary wave spectrum and has been based on pole-residue operations implemented in the Laplace domain to obtain closed-form solutions for the response evolutionary power spectrum. Numerical examples choose a single-degree-of-freedom Spar model and a six-degree-of-freedom Floating Production Storage and Offloading model to a Pierson–Moskowitz wave spectrum, and the correctness of the computed mean-square values is verified by Monte Carlo simulations.

Nonstationary Stochastic Response of Hysteretic Systems Endowed With Fractional Derivative Elements

AbstractIn this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.

NON-STATIONARY RESPONSE ANALYSIS OF A HIGH-RISE BUILDING IN HAIKOU DURING TYPHOONS

From 2014 to 2016, several wind resistant field measurements were conducted to the high-rise building in Haikou. Based on these measurements, the present paper disclosed the characteristics of the time-history responses of axial acceleration on different floors during four typhoons, including the Rammasun, Kalmaegi, Mujigae and Sarika typhoons. The modal parameters of the measured building were identified by Morlet time-frequency wavelet transform methods, and the amplitude-dependent modal damping ratios and frequencies along translational directions were investigated. The results show that the variation trend of modal frequency with acceleration amplitude identified by the Morlet wavelet is the same as that recognized by time-domain method, while it is scattered with the interval bar (min-average-max) due to the nonstationary response of typhoon. Meanwhile, the larger the amplitude of acceleration response of high-rise buildings under strong wind, the greater the time-varying fluctuation of modal parameters identified by wavelet transform, and the bigger the difference between the interval bar (min-average-max). The full-scale study is expected to provide useful information on the wind-resistant design of high-rise buildings in typhoon-prone regions.

A hybrid deep belief network-based label distribution learning system for seismic damage estimation of liquid storage tanks

Liquid storage tanks play a vital role in the modern chemical process industry (CPI). The strong ground motion caused by large-scale earthquakes may easily impose severe structural damage on liquid storage tanks, leading to a series of catastrophic cascaded events. The seismic damage estimation of liquid storage tanks is a challenging problem, as the fluid-structure interaction exhibits extremely complicated and non-stationary response behavior. This study develops a novel data-driven methodology to estimate the seismic damage state probability distribution of liquid storage tanks in the contexts of label ambiguity and data imbalance. With the support of the advanced deep learning framework, synthetic oversampling methods, and label enhancement techniques, a hybrid deep belief network-based label distribution learning system (HDBN-LDLS) is proposed for probability distribution learning. The proposed HDBN-LDLS is evaluated on the widely used ALA database. Simulation results indicate that HDBN-LDLS can achieve a balanced estimation for all damage states while maintaining sufficient robustness to cope with label ambiguity. The reliability of the obtained data-driven model is validated by a damaged tank in the 2006 Silakhor earthquake. For practical applications, a more natural way to estimate a seismic damaged tank is to assign a membership degree to each possible damage state. The proposed methodology can quickly obtain the seismic damage state probability curves of a specific liquid storage tank, which can be used to support quantitative risk assessment and seismic design.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

An Alternating Efficient Approach for Determination of the Non-Stationary Responses of Strongly Nonlinear Systems Driven by Random Excitations

Abstract An alternating efficient approach for predicting non-stationary response of randomly excited nonlinear systems is proposed by a combination of radial basis function neural network (RBFNN) and stochastic averaging method (SAM). First, the n-degree-of-freedom quasi-non-integrable-Hamiltonian (QNIH) system is reduced to a one-dimensional averaged Itô differential equation within the framework of SAM for QNIH. Subsequently, the associated Fokker–Planck–Kolmogorov (FPK) equation is solved with the RBFNN. Specifically, the solution of the associated FPK equation is expressed in a linear combination of a series of basis functions with time-correlation weights. These time-depended weights are solved by minimizing a loss function, which involves the residual of the differential equations and the constraint conditions. Three typical nonlinear systems are studied to verify the applicability of the developed scheme. Comparisons to the data generated by simulation technique indicate that the approach yields reliable results with high efficiency.

Influence of uncertainty of structural parameters on the stochastic dynamic response of asphaltic lining dam-foundation systems to non-stationary random seismic excitation

The present work aims toward the effect of uncertain structural parameters on the stochastic dynamic response of an asphaltic lining dam-foundation system subjected to stationary as well as non-stationary random excitation. Uncertain structural parameters of interest are shear modulus and mass density, modeled using the lognormal distribution. The stochastic seismic response of the dam-foundation system to the random loads with uncertain structural parameters is carried out with the Monte Carlo simulation method. The spatial variability of ground motion is considered with incoherence and wave passage effects as stationary as well as non-stationary random excitation. A time-dependent frequency response function is used throughout the study for non-stationary responses. Obtained results indicate that the variability of shear modulus and mass density can be neglected in stochastic dynamic analysis of an asphaltic lining dam-foundation system. Also, it is seen from the results that stationarity is a reasonable assumption for asphaltic lining dams to typical durations of strong shaking.