Non-stationary Response Research Articles

Ride comfort and energy dissipation of vehicle suspension system under non-stationary random road excitation

Response of vehicle suspension system has traditionally been studied under the assumption of stationary road excitation. However, road excitation becomes a non-stationary random input in the time domain when the vehicle travels at a varying speed. In this paper, we investigate the non-stationary response of vehicle suspension under random road excitation. Despite its importance and relevant applications, this problem has not been well addressed in the literature. The non-stationary random road excitations in the time domain are computed using a high precision Gaussian-Legendre integration method. The evolutionary spectral theory is then applied to evaluate the response in the time-frequency domain. Numerical results for a quarter-car model are presented and effects of vehicle acceleration and road roughness on the ride comfort, energy dissipation from the suspension, and time-variant half-power frequency band for the constant speed and various constant acceleration cases are examined. The consideration of energy dissipation provides new insights on the relationships between ride comfort and fuel consumption, which is important for designs of optimized energy-ride comfort suspension systems under non-stationary road excitations.

Survival probability of nonlinear oscillators endowed with fractional derivative element and subjected to evolutionary excitation: A stochastic averaging treatment with path integral concepts

An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.

Nonstationary response statistics of fractional oscillators to evolutionary stochastic excitation

While studying fractional oscillators to the excitation as an evolutionary stochastic process, this paper develops a novel pole-residue method to compute closed-form solutions for the nonstationary response statistics, including evolutionary power spectrum, correlation function and mean square value. The proposed method is not only suitable to the oscillators with an arbitrary fractional order, but also applies to oscillators with multiple fractional derivative terms. A necessary task of the proposed method is to obtain a pole-residue form of the oscillator’s transfer function (TF), which can be achieved through two steps: (1) obtaining a discrete impulse response function (IRF) from taking the inverse Fourier transform of the oscillator’s frequency response function (FRF), which is readily derived from its exact TF; (2) invoking the Prony-SS method to decompose this IRF signal for getting the poles and residues of the TF. Once the pole-residue TF is available, the remaining derivation can follow a similar pole-residue method that has been developed for ordinary linear oscillators. In the numerical studies, three fractional oscillators with one and two fractional derivative terms, respectively, are chosen to demonstrate the proposed method; considering the input to be a uniformly modulated random process modeled with either white noise or Kanai–Tajimi earthquake spectrum, the correctness of the proposed method is verified by Monte Carlo simulations.

Exact benchmark solutions of random vibration responses for thin-walled orthotropic cylindrical shells

The exact solutions of random vibration responses of orthotropic thin cylindrical shells are imperative for verifying numerical approaches. This study proposes the discrete analytical method (DAM) and achieves the exact benchmark responses of thin-walled orthotropic cylindrical shells under stationary and nonstationary random excitations for the first time. Firstly, by revisiting the governing motion equations of elastic orthotropic shells based on the Flügge shell theory, the closed-form modal shape functions of orthotropic cylindrical shell with shear diaphragm boundary condition are introduced into stationary and nonstationary random vibration analyses. The power spectral density functions and root mean squares (RMSs) for stochastic responses of thin shell subjected to various random excitations, including the point, surface, base acceleration and moving excitations, are analytically derived by using the pseudo excitation method. Moreover, DAM is proposed to enhance the computational efficiency by implementing the discretization of the modal coordinates, frequency and temporal domains. Finally, the exact benchmark solutions of stationary and nonstationary stochastic responses are achieved, and the results indicate high accuracy and efficiency of proposed DAM. The excitation bandwidth and spatial distribution have significantly different effects on various types of responses, and maximum response RMS declines with increasing of acceleration of moving load.

IntelligentPooling: Practical Thompson Sampling for mHealth.

In mobile health (mHealth) smart devices deliver behavioral treatments repeatedly over time to a user with the goal of helping the user adopt and maintain healthy behaviors. Reinforcement learning appears ideal for learning how to optimally make these sequential treatment decisions. However, significant challenges must be overcome before reinforcement learning can be effectively deployed in a mobile healthcare setting. In this work we are concerned with the following challenges: 1) individuals who are in the same context can exhibit differential response to treatments 2) only a limited amount of data is available for learning on any one individual, and 3) non-stationary responses to treatment. To address these challenges we generalize Thompson-Sampling bandit algorithms to develop IntelligentPooling. IntelligentPooling learns personalized treatment policies thus addressing challenge one. To address the second challenge, IntelligentPooling updates each user's degree of personalization while making use of available data on other users to speed up learning. Lastly, IntelligentPooling allows responsivity to vary as a function of a user's time since beginning treatment, thus addressing challenge three.

Dynamic reliability analysis for non-stationary non-Gaussian response based on the bivariate vector translation process

Calculation of probability of exceedance for nonstationary non-Gaussian responses remains a great challenge to researchers in the field of structural reliability. In this paper, an analytical solution is proposed for calculating the mean upcrossing rate (MCR) of the non-stationary non-Gaussian responses by approximating the displacement and velocity responses with the bivariate vector translation process, in which the unified Hermite polynomial model (UHPM) is selected as the mapping function. The first four moments (i.e., mean value, standard deviation, skewness, and kurtosis) and cross-correlation function of the displacement and velocity responses needed in UHPM are estimated from some representative samples generated by random function-spectral representation method (RFSRM) and time-domain analysis. Under the Poisson assumption of the upcrossing events, the calculation of extreme value distribution or probability of exceedance for structural response can be determined with the proposed method. The proposed method is applicable to a wide range of structural responses, including asymmetric and hardening or softening responses. Three numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method. It can be concluded that the proposed method provides an accurate and useful tool for dynamic reliability assessment in engineering applications.

Non-stationary response power spectrum determination of linear/non-linear systems endowed with fractional derivative elements via harmonic wavelet

This paper presents a wavelet-based method for determining response evolutionary power spectrum density (EPSD) of linear/non-linear systems endowed with fractional derivative elements. Specifically, first, the generalized harmonic wavelets (GHWs) based Galerkin approximation of the stochastic processes are utilized to transform the fractional-order stochastic linear/non-linear differential equations into a set of linear/non-linear algebraic equations with unknown response wavelet coefficients. Next, the linear algebraic equations are solved in a closed-form, while the non-linear ones are treated by the gradient-based standard numerical methods. Further, an analytical relationship between the EPSD of the excitation and of the response for a linear system is derived by considering the wavelet representation of stochastic processes. For a non-linear system, the response EPSD is estimated by repeated solving of sample algebraic equations. Pertinent numerical examples demonstrate the applicability and accuracy of the proposed method.

Temporal Variability in Climate-Growth Response of Mountain Hemlock at Treeline in Washington and Oregon

Climate-growth relationships in altitudinal treeline forests are particularly informative because they represent the upper limit of a species range where growth is often especially sensitive to climatic variation. Radial growth response to climatic variables typically ranges from energy limited to water limited. In the Pacific Northwest region of North America, treeline forests are typically energy limited. Results from a recent study (Marcinkowski et al. 2015) indicate a change in climate-growth relationships over time in treeline mountain hemlock (Tsuga mertensiana) in northern Washington state. Here, we investigate whether these same relationships hold across 700 km of the range of treeline mountain hemlock in Oregon and Washington. Using cores from trees > 200 years old, we examined temporal and spatial variability of climate-growth relationships since the early 20th century. Results indicate 1) a potential weakening of energy limitation in forest growth at treeline, 2) a nonstationary growth response to climate variables through time, and 3) different patterns of growth response to climate from north to south. Climate-growth relationships were generally similar between different aspects in the same geographic location, with some exceptions. Correlations between climate variables and radial growth differ over time (ranging from significantly positive to significantly negative), indicating the non-stationary nature of climate-growth relationships. These results identify where changes in growth-limiting variables may be occurring at treeline. Additional changes in the effects of climate change on growth at treeline are expected, and may be temporally complex.

Seismic structural health monitoring using the modal assurance distribution

AbstractThanks to emerging techniques in the field of signal processing and due to improvements in smart sensing systems which enable the event‐triggered acquisition of high‐fidelity data at the occurrence of strong ground motion events, seismic structural health monitoring has grown considerably in the last few decades. In this paper, the modal assurance distribution, an alternative time‐frequency representation of the modal features of multivariate and multicomponent signals, is extended for application to short‐term nonstationary vibrational structural responses in which the system may manifest its nonlinear behavior. A general procedure for the extraction of the decoupled normal modes is presented, which allows the identification of instantaneous modal parameters in order to investigate in detail the structural behavior during earthquakes. Valuable information that cannot be recovered by means of traditional criteria can thus be exploited for accurate damage assessment. The results obtained for two case studies consisting of a numerical model with softening nonlinear behavior and a full‐scale experimental reinforced concrete benchmark show the potential and applicability of the method proposed for the integrity assessment of civil structures.

Investigation of the Nonstationary Stochastic Response of Functionally Graded Piezoelectric Material Plates with General Boundary Conditions

The functionally graded piezoelectric material (FGPM) has been widely used in various engineering cases due to its excellent mechanical properties. So far, the investigations on the stochastic dynamic behaviors of FGPM structures are few. This paper develops a unified solution for the nonstationary stochastic responses of the FGPM plate with general boundary conditions, in which the point, distributed and base acceleration random excitations can be considered. The first-order shear deformation theory (FSDT) is adopted to formulate the energy functional of the system and the governing equations are deduced by using the Hamilton's principle. The solutions of the system are obtained based on the modified Ritz method and pseudo excitation method (PEM). The accuracy of the proposed model is validated by comparing the obtained free vibration and random response results with those from the published literature. Finally, the effects of key parameters including boundary conditions, external electric voltage, material constituent and time modulated function on the stochastic response of the FGPM plate are investigated.

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.

Non-stationary response of rain-fed spring wheat yield to future climate change in northern latitudes

The non-stationary response of crop growth to changes in hydro-climatic variables makes yield projection uncertain and the design and implementation of adaptation strategies debatable. This study simulated the time-varying behavior of the underlying cause-and-effect mechanisms affecting spring wheat yield (SWY) under various climate change and nitrogen (N) application scenarios in the Red Deer River basin in agricultural lands of the western Canadian Prairies. A calibrated and validated Soil and Water Assessment Tool and Analysis of Variance decomposition methods were utilized to assess the contribution of crop growth parameters, Global Climate Models, Representative Concentration Pathways, and downscaling techniques to the total SWY variance for the 2040–2064 period. The results showed that the cause-and-effect mechanisms, driving crop yield, shifted from water stress (W-stress) dominated (27 days of W-stress days) during the historical period to nitrogen stress (N-stress) dominated (27 to 35 N-stress days) in the future period. It was shown that while higher precipitation, warmer weather, and early snowmelts, along with elevated CO2 may favor SWY in cold regions in the future (up to 50% more yields in some sub-basins), the yield potentials may be limited by N-stress (only up to 0.7% yield increase in some sub-basins). The N-stress might be partially related to the N deficiency in the soil, which can be compensated by N fertilizer application. However, inadequate N uptake due to limited evapotranspiration under elevated atmospheric CO2 might pose restrictions to SWY potentials even in the least N deficient regions. This study uncovers important information on the understanding of spatiotemporal variability of hydrogeochemical processes driving crop yields and the non-stationary response of yields to changing climate. The results also underscore spatiotemporal variability of N-stress due to N deficiency in the soil or N uptake by crops, both of which may restrain SWY by changes in atmospheric CO2 concentrations in the future.

An integrated approach for structural behavior characterization of the Gravina Bridge (Matera, Southern Italy)

An integrated geophysical approach using non-invasive, non-destructive, and cost-effective seismic and electromagnetic techniques has been implemented to recognize the static and dynamic properties (i.e. eigenfrequencies, equivalent viscous damping factors, and related modal shapes) of the Gravina Bridge and its interaction with foundation soils. The “Gravina” is a bow-string bridge located on outcropping calcarenites in the city of Matera (Southern Italy) and develops for 144 m along a steel-concrete deck. The foundation soil characteristics have been evaluated by means of three high-resolution geo-electrical tomographies, one Vs velocity profile, and two site amplification functions. The main structural characteristics of the bridge have been estimated through permanent and on-demand monitoring using seismic and electromagnetic sensing. The former consisted of accelerometers and velocimeters installed with different geometrical arrangements for permanent earthquake and on-demand ambient vibration test recordings. The electromagnetic sensing was realized by a microwave radar interferometer placed below the deck to measure the displacements of the whole scenario illuminated by the antenna beam providing a continuous mapping of the static and dynamic displacements of the entire target. Acquired data have been analyzed in both frequency and time-frequency domain with the aim to study the stationary and non-stationary response of the monitored bridge. These experimental campaigns allowed us to assess the robustness of the proposed approach and to set up the zero-time reference point of the bridge dynamic parameters.

Operational Modal Analysis for Vibration Control Following Moving Window Locality Preserving Projections for Linear Slow-Time-Varying Structures

Modal parameters can reflect the dynamic characteristics of the structure and can be used to control vibration. To identify the operational modal parameters of linear slow-time-varying structures only from non-stationary vibration response signals, a method based on moving window locality preserving projections (MWLPP) algorithm is proposed. Based on the theory of “time freeze”, the method selects a fixed length window and takes the displacement response signal in each window as a stationary random sequence. The locality preserving projections algorithm is used to identify the transient modal frequency and modal shape of the structure at this window. The low-dimensional embedding of the displacement response data set calculated by locality preserving projections (LPP) corresponds to the modal coordinate response matrix, and the transformation matrix corresponds to the modal shape matrix. The simulation results of the mass slow-time-varying three degree of freedom (DOF) and the density slow-time-varying cantilever beam show that the new method can effectively identify the modal shape and modal natural frequency of the linear slow-time-varying only from the non-stationary vibration response signal, and the performance is better than the moving window principal component analysis (MWPCA).

Modal parameter estimation of civil structures based onimproved variational mode decomposition

This paper proposes an improved variational mode decomposition (IVMD) algorithm for structural modal parameter estimation based on non-stationary responses. In this improved VMD, the mean envelope entropy (MEE) and particle swarm optimization (PSO) are first employed to determine the optimal decomposition parameters for the subsequent VMD analysis. Then the VMD algorithm is used to decompose the non-stationary data into a number of intrinsic mode functions (IMFs). After obtaining the IMFs based on the IVMD, structural modal parameters such as natural frequencies and damping ratios of civil structures can be determined by using Natural Excitation Technique (NExT) and Direct Interpolating approach (DI). The feasibility and accuracy of the proposed procedure are evaluated by both numerical and full-scale examples. The natural frequencies and damping ratios are successfully identified from the vibration responses with high noise and nonstationary characteristics. The results of this study illustrate that the proposed procedure provides a powerful approach to identify the modal parameters of civil structures using non-stationary responses.

Real-time deterministic prediction of wave-induced ship responses based on short-time measurements

This paper studies real-time deterministic prediction of wave-induced ship motions using the autocorrelation functions (ACFs) from short-time measurements, namely the instantaneous ACFs. The Prolate Spheroidal Wave Functions (PSWF) are introduced to correct the large lag time errors in the instantaneous sample ACF, together with a modification of the autocorrelation (AC) matrix for ensuring its positive definiteness. The validity of the PSWF-based ACFs is first examined by using the ship motion measurements from model experiment under stationary wave excitations. It is shown that the use of PSWF-based ACFs leads to better prediction accuracy than direct use of sample ACFs. The validation is then extended to ship motion prediction using in-service data from a container ship, and an improvement of the prediction accuracy by PSWF-based ACFs is again found. Finally, the effectiveness of use of the instantaneous ACFs for non-stationary wave-induced responses is highlighted by comparing with the prediction results based on the ACFs from long-time measurements.

Best linear approximation of nonlinear and nonstationary systems using Operational Modal Analysis

Operational Modal Analysis (OMA) is a widely used class of tools to identify the modal properties of existing civil engineering structures and mechanical systems under ambient vibrations or normal operating conditions by only employing measured responses. The OMA techniques, however, are confined to estimate modal properties of linear and time-invariant systems, which do not always comply with real life applications. Additionally, even though structures are found to be nonlinear and nonstationary (i.e., time-varying), it is often engineering practice to apply linear models due to their straightforward insight into the dynamic behaviour of the considered system and mathematical properties when compared to more time consuming consideration of nonlinearities and nonstationarities. With that point of view, the concept of evaluating a best linear approximation of a nonlinear response is well established within Experimental Modal Analysis, i.e., deterministic modal analysis where both the system excitation and the corresponding response are measured. Along these lines, the paper takes the first step towards extending the concept of a best linear approximation to correlation-driven OMA in terms of estimated natural frequencies, damping ratios, and mode shapes. This paper proves that the correlation function matrix calculated from the measured response of a nonlinear and nonstationary system can be approximated by a set of free decays of the best linear approximation. In other words, the paper derives and shows through appropriately designed numerical simulations that the linear modal model, estimated using the output-only framework of conventional correlation-driven (i.e., covariance-driven) OMA, leads to suboptimal minimisation of the difference between the true response of a nonlinear and/or nonstationary system and the response of the linear modal model in a least squares sense. Moreover, the paper demonstrates, using an additional numerical simulation case study, that the output-only framework causes an underestimation of the error that is inevitably involved when estimating linear modal properties from a nonlinear and nonstationary response.

Moving Window Differential Evolution Independent Component Analysis-Based Operational Modal Analysis for Slow Linear Time-Varying Structures

In order to identify time-varying transient modal parameters only from nonstationary vibration response measurement signals for slow linear time-varying (SLTV) structures which are weakly damped, a moving window differential evolution (DE) independent component analysis- (ICA-) based operational modal analysis (OMA) method is proposed in this paper. Firstly, in order to overcome the problems in traditional ICA-based OMA, such as easy to go into local optima and difficult-to-identify high-order modal parameters, we combine DE with ICA and propose a differential evolution independent component analysis- (DEICA-) based OMA method for linear time invariant (LTI) structures. Secondly, we combine the moving widow technique with DEICA and propose a moving window differential evolution independent component analysis- (MWDEICA-) based OMA method for SLTV structures. The MWDEICA-based OMA method has high global searching ability, robustness, and complexity of time and space. The modal identification results in a three-degree-of-freedom structure with slow time-varying mass show that this MWDEICA-based OMA method can identify transient time-varying modal parameters effectively only from nonstationary vibration response measurement signals and has better performances than moving window traditional ICA-based OMA.

Hierarchical Gaussian Process Models for Improved Metamodeling

Simulations are often used for the design of complex systems as they allow one to explore the design space without the need to build several prototypes. Over the years, the simulation accuracy, as well as the associated computational cost, has increased significantly, limiting the overall number of simulations during the design process. Therefore, metamodeling aims to approximate the simulation response with a cheap to evaluate mathematical approximation, learned from a limited set of simulator evaluations. Kernel-based methods using stationary kernels are nowadays widely used. In many problems, the smoothness of the function varies in space, which we call nonstationary behavior [20]. However, using stationary kernels for nonstationary responses can be inappropriate and result in poor models when combined with sequential design. We present the application of two recent techniques: Deep Gaussian Processes and Gaussian Processes with nonstationary kernel, which are better able to cope with these difficulties. We evaluate the method for nonstationary regression on a series of real-world problems, showing that these recent approaches outperform the standard Gaussian Processes with stationary kernels. Results show that these techniques are suitable for the simulation community, and we outline the variational inference method for the Gaussian Process with nonstationary kernel.

An observation of non-stationary response to non-synoptic wind on the Forth Road Bridge

The GeoSHM project feasibility study for monitoring the Forth Road Bridge is briefly introduced and the instrumentation summarised. The events of January 9th, 2015 are described, when the bridge was struck by storm Elon, which caused widespread damage across Scotland and led to the temporary closure of the bridge when a van was blown over. During this storm an anomalous large amplitude response was observed.The data for January 9th, 2015 are analysed to show that the extreme response and the corresponding wind are non-stationary and non-Gaussian. Further analysis of the rainfall radar data for the same time shows a line of intense rainfall extending for over 100 ​km, which passes the site of the bridge at exactly the time of the peak response. The rainfall intensity was high enough to indicate that this feature was caused by convective activity and this observation was corroborated by records of lightning strikes. It is concluded that non-stationary wind events can give rise to large response of long span bridge structures and that this response can exceed that observed from the stationary wind field. Furthermore, historical data confirm that energetic squall lines are not uncommon in the UK. Therefore, the assumption of stationarity in predicting the wind induced response of long span bridges may be non-conservative and the climatology of large convective systems, such as squall lines, should be considered in assessing the wind hazard for these structures.