Nonstationary Excitation Research Articles

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Online damage detection of earthquake-excited structure based on near real-time envelope extraction

A robust real-time damage detection technique of earthquake-excited structures based on a new demodulation technique for nonlinear and non-stationary vibration signals through the identification of signal envelopes in real time is presented. In the present work, the need for the detection of envelope in a vibration signal in real time is addressed by reformulating the concept of Hermitian interpolation functions to a recursive Hermitian polynomial, which is a key entitlement of the present work. Once, the near real-time demodulation is achieved, the proposed framework proceeds to the newly developed error-adapted framework by addressing the errors accrued between the standard and generalized eigen perturbation formulation in the context of real-time estimation of proper orthogonal modes and linear normal modes. In the adaptive framework, the error is modeled as a feedback, which is constructed to account for the truncation in the order of eigen perturbation. In addition to the improved accuracy due to the envelope extraction, the proposed error-adapted eigen perturbation further improves the detectability over the currently available eigen perturbation–based real-time algorithms. To ensure robustness of the proposed algorithm, a new real-time damage indicator based on the maximum of principal eigenvector of the evolving transformed covariance matrix is proposed. The proposed modules together not only improve the detectability of the damage detection in real-time but also enhance the overall performance in presence of non-stationary excitation, that often mask the damage information in the higher energy zones of the amplitude and frequency-modulated response. Simulations for the proposed framework is performed by considering a 5 degrees-of-freedom linear and base-isolated nonlinear structural system driven by non-stationary stochastic excitations with damage simulated at intermediate floor at a particular time instant. Evidence of the near real-time demodulation and/or envelope removal from the signal and improved damage identification is also provided. An examination of the proposed framework using experimental data further validates the robustness of the proposed scheme.

Fatigue assessment of non-stationary random loading in the frequency domain by a quasi-stationary Gaussian approximation

The power spectral density (PSD) is a fundamental technique of random vibration fatigue providing an effective statistical characterization that can be processed by linear systems theory and load spectrum estimators. This lays the basis for a statistical-based fatigue assessment. While the PSD assembles a full stochastic characterization of stationary Gaussian loading, for loading subjected to changing operational, environmental, and excitational conditions, it provides no means of a fluctuating spectral density. Therefore, the PSD neither qualifies to characterize varying loads, nor reproduces comparable stress amplitudes to a referencing non-stationary excitation following a statistical-based stress analysis. Consequently, this paper employs the non-stationarity matrix to characterize the varying evolution of realistic loading and proposes a system of equations that decomposes this characterization into stationary Gaussian portions. The fundamental idea is to approximate realistic loading by abstracting a series of stationary segments, whose assembly in return embodies a full statistical characterization. The resulting quasi-stationary load definition better reflects the fatigue damage potential of realistic, non-stationary loading and allows to implement load spectrum estimators, ensuing computationally efficient and statistically robust structural lifetime predictions. Further, quasi-stationary load definitions are utilized to advance the concept of damage-equivalent statistical load definitions to be independent of a specific Miner exponent.

Stochastic response of MDOF system to non-stationary random excitation

An approximate scheme is developed for efficiently determining non-stationary response PDF of multi-degree-of-freedom (MDOF) dynamical system to time-modulated random excitation. It further generalizes state-space-split exponential polynomial closure (SSS-EPC) technique and enhances its application for determining non-stationary response probability density function (PDF). To this aim, time variable is introduced into the approximate solution by setting polynomial coefficients as time dependent ones. With the proposed procedure, non-stationary solution to high-dimensional FPK equation can be obtained. Meanwhile, Monte Carlo simulation (MCS) method is conducted to assess the reliability of the proposed method. Three typical examples of linear and nonlinear MDOF dynamical systems to different modulated Gaussian white noise excitations are considered. Comparisons with the simulated results and computational time of the MCS method demonstrate the relatively high efficiency and accuracy of the proposed procedure.

Machine‐learning‐enhanced tail end prediction of structural response statistics in earthquake engineering

AbstractEvaluating the response statistics of nonlinear structures constitutes a key issue in engineering design. Hereby, the Monte Carlo method has proven useful, although the computational cost turns out to be considerably high. In particular, around the design point of the system near structural failure, a reliable estimation of the statistics is unfeasible for complex high‐dimensional systems. Thus, in this paper, we develop a machine‐learning‐enhanced Monte Carlo simulation strategy for nonlinear behaving engineering structures. A neural network learns the response behavior of the structure subjected to an initial nonstationary ground excitation subset, which is generated based on the spectral properties of a chosen ground acceleration record. Then using the superior computational efficiency of the neural network, it is possible to predict the response statistics of the full sample set, which is considerably larger than the initial training sample set. To ensure a reliable neural network response prediction in case of rare events near structural failure, we propose to extend the initial training sample set increasing the variance of the intensity. We show that using this extended initial sample set enables a reliable prediction of the response statistics, even in the tail end of the distribution.HIGHLIGHTSA new Monte Carlo method is developed that provides the response statistics of a nonlinear system in the tail end of the distribution.A nonstationary filter is applied to a Gaussian white noise to generate realistic artificial earthquake records.An extended training strategy using neural networks is proposed to improve the reliability of the method in the tail end of the distribution.The new strategy reveals a significant speedup as well as a prediction of the response statistics in the tail end with high confidence.

Short-time fatigue-life estimation for non-stationary processes considering structural dynamics

Structural dynamics relates the excitation profile to the dynamic loads at a particular location of the structure. Spectral methods fatigue-life estimation is based on the assumptions of stationarity and the Gaussianity. This research proposes a method for a fatigue-life estimation for non-stationary excitation. The method is based on the short-time spectral narrowband method, where the short-time width was found to be related to the structural dynamics (i.e., natural frequency, damping) of the excited structure. A detailed numerical and a real experiment shows a significant accuracy increase for fatigue-life estimation at non-stationary and non-Gaussian excitation.

Structural response and fatigue assessment of a small vertical axis wind turbine under stationary and non-stationary excitation

The paper discusses the response and fatigue damage of the supporting tower of a small vertical axis wind turbine subject to stationary and non-stationary excitation due to wind, turbine rotation, emergency stop and start. Using in-field measurements and the Hot Spot Stress Approach, the study analyses a number of aspects that are usually disregarded in design verifications. The results highlight the importance of the detail modeling, the fundamental role played by the non-stationary conditions and the errors committed when using conventional models of the load. The emphasis of this paper is put on two main aspects: the use of full scale data and the detection of isolated events of non-stationary vibrations, the effects of which can be disregarded when using stress time-histories of 10 min - 1 h.

Optimal design of nonlinear energy sinks for mitigation of seismic response on structural systems

This paper proposes an approach for the optimal design of nonlinear energy sinks (NES) installed in structures subjected to stochastic excitation. The response of these nonlinear systems subjected to stationary or non-stationary stochastic excitations has been characterized by the time-history of its covariance matrix. However, time-invariant equivalent linearization methods of NES equipped structures will not always converge due to its essential nonlinearity. A solution strategy is proposed using equivalent linearization at each time step of the governing matrix differential equation, obtained through a sequence of Lyapunov equations. The adequacy of this method for optimization problems is evaluated by comparison with Monte Carlo simulations of two representative NES examples. Moreover, the influence of the Gaussian assumption of the equivalent system response is studied in these examples. An optimization formulation to design systems equipped with NES devices is proposed, in which the performance function is defined in terms of the maximum variance over the time-history. This result is obtained by using the proposed method with a Gaussian assumption of the equivalent linear system. To illustrate the proposed methodology, parametric optimization of an NES installed at the top of a shear building structure is performed.

ИССЛЕДОВАНИЕ АСИНХРОННОЙ ЭЛЕКТРИЧЕСКОЙ МАШИНЫ С КОРОТКОЗАМКНУТЫМ РОТОРОМ ДЛЯ ВОЗОБНОВЛЯЕМОЙ ЭНЕРГЕТИКИ ПРИ НЕСТАЦИОНАРНОМ КОНДЕНСАТОРНОМ ВОЗБУЖДЕНИИ

Актуальность. Доля электроэнергии, вырабатываемой установками на основе возобновляемой энергии, постоянно растет, в связи с чем потребность в развитии систем питания и автоматического управления электрическими машинами, лежащими в основе ветро- и гидрогенераторов, не теряет актуальности. В составе таких генераторных установок переменного тока применяют синхронные электрические машины, асинхронные машины с фазным и короткозамкнутым ротором. Преобразователи частоты, устанавливаемые в статорные и роторные цепи асинхронных машин переменного тока, позволяют управлять процессами их возбуждения, однако для начала генерации требуется использовать дополнительные внешние источники питания. Благодаря остаточному намагничиванию в магнитопроводе можно обеспечить процесс гарантированного самовозбуждения асинхронной машины с помощью подключения батарей конденсаторов к ее статорным обмоткам без применения дополнительного внешнего источника питания. Предложенный способ нестационарного конденсаторного возбуждения позволяет обеспечить адаптацию генераторной установки к изменению режимов работы в условиях децентрализованного электроснабжения. Цель: исследовать предложенную систему стабилизации напряжения асинхронной машины с короткозамкнутым ротором с варьируемым конденсаторным возбуждением в составе источника возобновляемой энергии. Методы: теоретические – теория дифференциальных уравнений, методы численного решения обыкновенных дифференциальных уравнений, теория электропривода, теория электрических машин, численные методы аппроксимации данных, и экспериментальные – проведение испытаний асинхронной электрической машины с нестационарным конденсаторным возбуждением на разработанном испытательном стенде с целью получения нагрузочных характеристик и осциллограмм напряжений на обмотках статора асинхронных машин с короткозамкнутым ротором в различных режимах работы, методы исследования. Результаты. Разработана и изготовлена оригинальная экспериментальная установка с узлом, имитирующим работу турбины. Электромеханический преобразователь энергии выполнен в виде асинхронной машины с короткозамкнутым ротором с нестационарным конденсаторным возбуждением. Блок управления установкой выполнен в виде интегрированного с силовым блоком гальванически развязанного модуля во влагостойком исполнении с применением беспроводного интерфейса связи Bluetooth. Описан и протестирован способ коммутации батарей конденсаторов с применением тиристоров в качестве управляемого ключа с двухсторонней проводимостью. Анализ полученных нагрузочных характеристик показывает принципиальную возможность обеспечить гарантированную выработку электроэнергии со стабилизацией напряжения в допустимых пределах изменения мощности нагрузки. Примененная в составе экспериментальной установки система автоматической коммутации конденсаторов с варьируемой в зависимости от потребляемой мощности емкостью позволила обеспечить приемлемое время динамической реакции на возмущающее воздействие при нестационарной нагрузке.

A shape memory alloy-tuned mass damper inerter system for passive control of linked-SDOF structural systems under seismic excitation

This work introduces a passive control device, shape memory alloy tuned mass damper inerter (SMA-TMDI), for the control of linked-single degree of freedom (SDOF) systems subjected to base excitation. The adjacent SDOF systems are connected through a device called inerter with force proportional to the relative acceleration of the individual SDOF oscillators of the linked-SDOF systems. The SMA element of SMA-TMDI dissipates the energy of primary oscillator through the hysteretic phase transformation, while, the mass-amplification effect of the inerter is utilized to reduce displacement of the secondary oscillator of the linked-SDOF systems. The mean square displacement responses of both the oscillators of the linked-SDOF systems subjected to white noise base excitation are derived based on stochastic equivalent linear parameters of the SMA spring through an iterative process. Parametric studies under white noise excitation are conducted and based on the results obtained, a multi-objective optimization is performed considering displacement variances of both the SDOF oscillators as the objective function. Under white noise excitation, the optimal performances of SMA-TMDI and TMDI systems are analyzed in terms of displacement mean square responses and the root mean square control forces transferred to the linked-SDOF systems are also examined. Further, the performance comparison of the SMA-TMDI and TMDI passive control devices are carried out under non-stationary Kanai-Tajimi excitation based on an Ito-Taylor formulation of the mean square stochastic differential equations. Based on the results obtained for both white noise and ground motion base excitation cases, it can be observed that the SMA-TMDI system performs better in comparison to the TMDI system with significantly lesser requirement on total damper mass and inertance.

Time-dependent pounding probability analysis between adjacent decks of bridges under non-stationary stochastic seismic excitations

This paper proposes a time-dependent probability method to predict the time-variant pounding risk, owing to this reason that the required separation gap width fails to accommodate the relative movement between two adjacent decks, of bridge systems subjected to non-stationary stochastic seismic excitations. Firstly the pseudo excitation method in random theory is utilized to transform the motion equation of system under the non-stationary stochastic seismic excitations into the transient response equation under deterministic external loads. And then the high precision direct integration method (HPDIM) is also employed to reduce the large number of transient analysis into two times for overcoming the challenges of strenuous computing effort. The time-dependent pounding system probability of bridges is derived by the combination of extreme value distribution theory and matrix-based system probability theory. Finally based on the proposed time-dependent probability method the numerical analysis of a real two span simply-supported beam bridge subjected to spatial and non-stationary ground motions is conducted herein for the pounding risk assessment. The results of this study provides in-depth insight into the time-dependent pounding probability of bridge systems subjected to spatially varying and non-stationary ground motions.

An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

The sensitivity analysis of hysteretic systems under nonstationary random excitations is of great concern to the stochastic optimal design and control of structures. The equivalent linear equation of motion is first constructed for the hysteretic system by the equivalent linearization method (ELM), and the sensitivity equation of the equivalent linear system is then derived by the direct differentiation technique. These two equations are further combined into an overall equivalent linear equation, which depends on the statistical moments and sensitivities of responses of the system and should be solved on an iterative basis. The analysis problem is thus transformed to a series of nonstationary random vibration analyses of the iterative linearized systems, which can be solved with the explicit time-domain method (ETDM) recently proposed for linear random vibration problems. The ETDM has high computational efficiency owing to the explicit formulations of statistical moments of responses. Therefore, a numerical algorithm is developed by combining ELM and ETDM for efficient stochastic sensitivity analysis of hysteretic systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.

Response and first passage probability of linear elastic SDOF systems subjected to nonstationary stochastic excitation modelled through S-transform

The S-transform is a special type of Fourier transform with the time localization and frequency dependent window. It is viewed as a hybrid of continuous wavelet transform and short-time Fourier transform. In the present study, we propose an S-transform-based random vibration theory to evaluate the response of the linear elastic single-degree-of-freedom (SDOF) system subjected to nonstationary stochastic excitation. We show that within the established framework, the time–frequency power spectral density (TFPSD) function of the structural response could be represented as the multiplication of the square of the mechanical admittance function and the TFPSD function of the nonstationary stochastic excitation. This simple relation facilitates the computation and statistical characterization of the structural responses subjected to nonstationary stochastic excitation that is modelled based on the S-transform, which implicitly includes the time-varying amplitude and frequency characteristics. We illustrate the use of the TFPSD function to estimate the statistics of the responses and first passage probability of SDOF systems subjected to stochastic earthquake or wind excitations, where the stochastic excitations are represented using the TFPSD functions obtained from the S-transform.

Exact Solutions of Random Vibration Responses for Orthotropic Rectangular Kirchhoff Plates

AbstractThis paper efficiently determined exact stochastic responses of orthotropic rectangular Kirchhoff plates under stationary and nonstationary random excitations in a semianalytical manner. Th...

Identification of structural element mass and stiffness changes using partial acceleration responses of chain-like systems under ambient excitations

It is important to conduct structural damage detection using only output time histories of structures under ambient excitations. Recently, an approach has been proposed for structural damage detection under ambient excitations using the synthesis of cross-correlation functions of partial structural responses and the extended Kalman filter (EKF). However, structural mass is assumed known in this approach. Although some methodologies have been presented for detecting damage due to relative structural mass and stiffness changes in chain-like systems, measurements of structural responses including acceleration, velocity, and displacement at each degree of freedom are required. This greatly limits the application of these methodologies. In this paper, without structural mass information, an algorithm is proposed to identify structural element mass and stiffness changes using only partial acceleration responses of chain-like systems under ambient excitations. First, based on the equations for cross-correlation functions of structural responses under ambient stationary excitations modeled as independent stationary white noise processes, EKF is utilized for identifying the structural element stiffness-mass coupled coefficients using only partial acceleration responses. Then, these coefficients are decoupled by cluster analysis and the least squares estimation to obtain structural element stiffness and mass changes. Furthermore, the proposed algorithm is extended for identifying chain-like structural element mass and stiffness changes under non-stationary ambient excitations assumed as time modulating white noises. The performances of the proposed algorithm are numerically investigated using the Phase I The American Society of Civil Engineers structural health monitoring benchmark building under stationary or non-stationary ambient excitations, respectively. Moreover, the experimental data with structural element mass and stiffness changes conducted at the Los Alamos National Laboratory are utilized to test the proposed algorithm.

An exact and efficient time-domain method for random vibration analysis of linear structures subjected to uniformly modulated or fully non-stationary excitations

This paper presents a novel time-domain method for non-stationary random vibration analysis of linear structures. Specifically, using the well-known spectral representation method, the stochastic excitation process is first decomposed into a set of trigonometric basis functions (TBFs) modulated by a series of orthogonal basis random variables (BRVs). Next, each TBF is used as a deterministic excitation input for the structural model, and the induced structural responses are derived analytically using the modal superposition method. After superposition of each TBF-induced deterministic responses, an explicit closed-form expression between the structural stochastic responses and the BRVs is derived, based on which the response statistics of structures at any time instant can be evaluated easily. Furthermore, the accuracy of the solution is independent of the time step used. Finally, several numerical examples in civil engineering, including a Euler beam under stochastic moving load and a shear frame structure subjected to uniformly modulated or fully non-stationary seismic motions, are studied to illustrate the performance of the proposed method. The obtained structural response statistics are compared with the solutions obtained by the Monte Carlo and evolutionary spectral methods. The results verified the high accuracy and efficiency of the proposed method.

Data-driven fault diagnosis of bogie suspension components with on- board acoustic sensors

This paper proposes a data-driven approach for fault-detection and isolation of bogie suspension components with on-board acoustic sensors. The fault detection technique is based on the acoustic emissions variation due to structural modal coupling changes in the presence of faulty components. A suspensions component failure introduces an imbalance into the system, resulting in dynamics interferences between the motions. These interferences modify the energy introduced into the system as well as its acoustic emissions. The unknown arbitrary track irregularities generate together with a variable train speed a random nonstationary vehicle excitation. Speech recognition techniques were used to generate features that consider this phenomenon. Frequency spectrums were analysed in different operating conditions to design efficient features. The robustness of the methodology was verified with data from two different test measurement campaigns on a test ring, where the influence of the sensor locations for the fault classification process was studied. The proposed methodology achieved good fault classification performance on the investigated use cases, removed dampers and 50% damper degradation on primary and secondary vertical suspension.

Stochastic sensitivity analysis of coupled acoustic-structural systems under non-stationary random excitations

This paper presents a new sensitivity analysis method for coupled acoustic–structural systems subjected to non-stationary random excitations. The integral of the response power spectrum density (PSD) of the coupled system is taken as the objective function. The thickness of each structural element is used as a design variable. A time-domain algorithm integrating the pseudo excitation method (PEM), direct differentiation method (DDM) and high precision direct (HPD) integration method is proposed for the sensitivity analysis of the objective function with respect to design variables. Firstly, the PEM is adopted to transform the sensitivity analysis under non-stationary random excitations into the sensitivity analysis under pseudo transient excitations. Then, the sensitivity analysis equation of the coupled system under pseudo transient excitations is derived based on the DDM. Moreover, the HPD integration method is used to efficiently solve the sensitivity analysis equation under pseudo transient excitations in a reduced-order modal space. Numerical examples are presented to demonstrate the validity of the proposed method.

Stochastic sensitivity analysis of energy-dissipating structures with nonlinear viscous dampers by efficient equivalent linearization technique based on explicit time-domain method

Nonlinear fluid viscous dampers have been widely used in energy-dissipating structures due to their stable and high dissipation capacity and low maintenance cost. However, the literature on stochastic optimization of nonlinear viscous dampers under nonstationary excitations is limited. This paper is devoted to the stochastic response and sensitivity analysis of large-scale energy-dissipating structures equipped with nonlinear viscous dampers subjected to nonstationary seismic excitations. The analysis procedure is developed in the frame of the equivalent linearization method (ELM) in conjunction with the explicit time-domain method (ETDM). The equivalent linear system and the corresponding statistical moments of responses at a specific time instant are first obtained through a series of stochastic response analyses of the linearized systems. Then the sensitivities of the statistical moments of responses are determined via a series of stochastic sensitivity analyses of the equivalent linear system at the corresponding time instant. The above two iterative procedures are facilitated at high efficiency using ETDM with explicit formulations of the statistical moments of responses and the sensitivities of the statistical moments. This process is repeated for different time instants, and the time histories of the statistical moments and their sensitivities can be obtained. The stochastic response and sensitivity results are further utilized to conduct the stochastic optimal parametric design of the nonlinear viscous dampers. A one-storey building model equipped with a nonlinear viscous damper is analyzed to demonstrate the accuracy of the proposed method, and a suspension bridge with a main span of 1200 m equipped with 4 nonlinear viscous dampers is further investigated to illustrate the feasibility of the proposed method for stochastic optimal design of large-scale structures.

On the Spectral Quality of Time-Resolved CMOS SPAD-Based Raman Spectroscopy With High Fluorescence Backgrounds

The fluorescence background in Raman spectroscopy can be effectively suppressed by using pulsed lasers and time-gated detectors. A recent solution to reduce the high complexity and bulkiness of the time-gated systems is to implement the detector by utilizing time-resolved single-photon avalanche diodes (SPADs) fabricated in complementary-metal-oxide-semiconductor (CMOS) technology. In this study, we investigate the effects of fluorescence-to-Raman ratio, recording time and excitation intensity on the quality of Raman spectra measured by using one of the furthest developed fluorescence-suppressed Raman spectrometers based on a time-resolved CMOS SPAD line sensor. The objectives were to provide information on the significance of the different causes behind the distortion of the measured Raman spectra with various measurement conditions and to provide general information on the possibilities to exploit the high-intensity non-stationary pulsed laser excitation to gain additional improvement on the spectral quality due to laser-induced fluorescence saturation. It was shown that the distortion of the spectra with samples having short fluorescence lifetimes (~2 ns) and high fluorescence-to-Raman ratios, i.e. with challenging samples, is dominated by the timing skew of the sensor instead of the shot noise caused by the detected events. In addition, the actual reason for the observed improvement in the spectral quality as a function of excitation intensity was discovered not to be the conventionally thought increased number of detected photons but rather the laser-induced fluorescence saturation. At best, 26% improvement to the signal-to-noise ratio was observed due to fluorescence saturation.