Stochastic Response Research Articles

MR Tuned Liquid Colum Damper (MR-TLCD) for Seismic Vibration Control

Seismic vibration control of flexible structure is an important aspect which mostly involves reduction of structure displacement via application of passive damper or mass dampers. Present study focuses on the seismic vibration control of flexible structure against random earthquake, through modified (using MR fluid) tuned liquid column damper (TLCD), known as the MR-TLCD. To this end, stochastic structural response analysis has been performed and results (structure displacement ratio and liquid displacement) are compared with conventional the TLCD. To establish the superior control efficiency of MR-TLCD over conventional TLCD, parametric study has been performed considering wide range of damper, structure, and earthquake loading parameters. Study results shows the existence of optimal value of tuning ratio, head loss coefficient, and yield shear strength of MR fluid of MR-TLCD which minimizes the structure displacement ratio, i.e. maximizes the control efficiency. Another important observation is that, for similar level of control efficiency MR-TLCD requires much less value of mass ratio than that of TLCD, and also at same time it keeps the liquid displacement much less compared to the TLCD, for same value of mass ratio. This aspect increases the applicability of MR-TLCD over TLCD for vibration control of the flexible structure. In comparison to the TLCD, MR-TLCD provides 17 % to 43 % higher structural displacement control efficiency and reduces liquid displacement by 14 % to 45 %.

Dynamic Response and Reliability Analysis of Stochastic Multi-Story Frame Structures under Random Excitation

In earthquake engineering problems, uncertainty exists not only in the seismic excitations but also in the structure's parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of a multi-story moment resisting frame subjected to random ground motion. The North-south component of the Ali Gharbi earthquake in 2012, Iraq, is selected as ground excitation. Using the power spectral density function (PSD), the two-dimensional finite element model of the moment resisting frame's base motion is modified to account for random ground motion. The probabilistic study of the moment resisting frame structure using stochastic finite element utilizing Monte Carlo simulation was presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of the stochastic and deterministic structure based on the first passage failure were checked and evaluated. Results revealed that the probability of failure increased due to randomness in stiffness and mass of the structure. Generally, natural frequencies for the lower modes of vibration and relative displacements for the lower stories were more sensitive to the randomness in system parameters.

Effects of Pavement Roughness and Dynamic Tank Load on the Bridge Response

This study aims to investigate the dynamic Load allowance variation (DLA) of dynamic tank Loads and compare it with Iraq's standard specifications for road bridges. DLA is considered a simple measurement of the dynamic variation magnitude of the tank load for a specific combination of road roughness and speed. In addition to determining the stochastic dynamic response of the bridge. Al-Awsej bridge in Iraq, with a span of 33.2 m and a principal road with four pavement roughness classes (very good, good, average, and poor), was proposed as a case study in this analysis. A spectral closed-form solution was used for evaluating the dynamic tank load due to the passage of a tank type-72A at a constant speed of 40,50,60 and 70 km/hr along a bridge with different types of rough pavement surface. The results show the less value of DLA for very good pavement at 40 km/hr is about 0.032, and the largest value is about 0.293 at 70 km/hr for poor pavement. Also, road surface roughness greatly influences vehicle-bridge interactions and bridge responses. Where at 40 km/hr, the Root mean square of bridge deflection range from 0.31 to 2.75 mm and 0.39 to 3.14 mm at 70 km/hr.

East Asian Monsoonal Climate Sensitivity Changed in the Late Pliocene in Response to Northern Hemisphere Glaciations

AbstractMio‐Pliocene sedimentary archives of the East Asian summer monsoon (EASM) in NE Tibet record a monotonic response to orbital forcing, dominated by eccentricity. By contrast, Pleistocene archives display a more stochastic response that varies regionally and temporally. When and why this response changed is poorly understood. Here, we present a new high‐resolution Rb/Sr ratio data set of EASM intensity from the Sanmenxia Basin, North China, that spans the Plio‐Pleistocene transition. Our results indicate decreased monsoonal rainfall in the late Pliocene, dated at 2.75–2.6 Ma, associated with an intensified response to obliquity and enhanced climate stochasticity. This transition is attributed to the increase of Northern Hemisphere ice volume. Quaternary monsoons display a sensitivity unique to the modern icehouse with large bipolar ice sheets, while pre‐Quaternary monsoons were solely impacted by Antarctic ice sheet dynamics on orbital time‐scales.

Probabilistic Analysis of Composite Materials with Hyper-Elastic Components.

This work is a comprehensive literature overview in the area of probabilistic methods related to composite materials with components exhibiting hyper-elastic constitutive behavior. A practical area of potential applications is seen to be rubber, rubber-like, or even rubber-based heterogeneous media, which have a huge importance in civil, mechanical, environmental, and aerospace engineering. The overview proposed and related discussion starts with some general introductory remarks and a general overview of the theories and methods of hyper-elastic material with a special emphasis on the recent progress. Further, a detailed review of the current trends in probabilistic methods is provided, which is followed by a literature perspective on the theoretical, experimental, and numerical treatments of interphase composites. The most important part of this work is a discussion of the up-to-date methods and works that used the homogenization method and effective medium analysis. There is a specific focus on random composites with and without any interface defects, but the approaches recalled here may also serve as well in sensitivity analysis and optimization studies. This discussion may be especially helpful in all engineering analyses and models related to the reliability of elastomers, whose applicability range, which includes energy absorbers, automotive details, sportswear, and the elements of water supply networks, is still increasing, as well as areas where a stochastic response is the basis of some limit functions that are fundamental for such composites in structural health monitoring.

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

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

Stochastic characterization of multiscale material uncertainties on the fibre-matrix interface stress state of composite variable stiffness plates

This work analyses the stochastic response of fibre and matrix scale stresses of Variable Angle Tow (VAT) laminates affected by multiscale uncertainty defects. The aim is to evaluate the influence of the innermost constituents on the overall structural response via an accurate mechanical characterization of both macro- and microscales. The Carrera Unified Formulation (CUF) is employed to obtain two-dimensional (2D) and one-dimensional (1D) models for both scales. Indeed, 2D layer-wise (LW) and 1D component-wise (CW) approaches are adopted for the macroscale and the microscale, respectively. The use of 2D and 1D models proves to be convenient as a superior computational efficiency is reached, this aspect being of great importance as many analyses are necessary for uncertainty quantification. The numerical results demonstrate the validity of the proposed methodology to obtain an accurate description of the 3D stress state at the different scales. A special focus is made on the fibre-scale stresses and how they may vary when affected by multiscale uncertainty.

Offshore tethered platform springing response statistics

This paper demonstrates the validity of the Naess–Gadai method for extrapolating extreme value statistics of second-order Volterra series processes through application on a representative model of a deep water small size tension leg platform (TLP), with specific focus on wave sum frequency effects affecting restrained modes: heave, roll and pitch. The wave loading was estimated from a second order diffraction code WAMIT, and the stochastic TLP structural response in a random sea state was calculated exactly using Volterra series representation of the TLP corner vertical displacement, chosen as a response process. Although the wave loading was assumed to be a second order (non-linear) process, the dynamic system was modelled as a linear damped mass-spring system. Next, the mean up-crossing rate based extrapolation method (Naess–Gaidai method) was applied to calculate response levels at low probability levels. Since exact solution was available via Volterra series representation, both predictions were compared in this study, namely the exact Volterra and the approximate one. The latter gave a consistent way to estimate efficiency and accuracy of Naess–Gaidai extrapolation method. Therefore the main goal of this study was to validate Naess–Gaidai extrapolation method by available analytical-based exact solution. Moreover, this paper highlights limitations of mean up-crossing rate based extrapolation methods for the case of narrow band effects, such as clustering, typically included in the springing type of response.

A novel Bayesian approach for multi-objective stochastic simulation optimization

Multi-objective stochastic simulation optimization plays an important role in designing complex engineering systems. To identify optimal solutions via simulations, Bayesian optimization, which utilizes metamodels and an acquisition function to determine the next design point, has been popular in machine learning. However, studies on Bayesian optimization of multi-objective stochastic simulation are limited and can give undesired performances in terms of convergence and diversity metrics. Moreover, direct extension of Bayesian optimization algorithms of deterministic responses is unrealistic as stochastic responses are observed with heterogeneous noise. To handle these issues, we propose a novel framework for Bayesian optimization of multiple stochastic responses. Stochastic kriging metamodels are constructed independently for multiple stochastic responses considering heterogeneous simulation noise. To remove undesired effects of random simulation noise on comparing solutions, quantiles of stochastic kriging metamodels are employed to define the Pareto dominance. Acquisition functions are proposed by integrating the hypervolume-based and probability-of-improvement (PoI)-based acquisition functions from deterministic multi-objective optimization with typical acquisition functions from univariate stochastic response optimization. This not only enhances the convergence and diversity of Pareto-optimal solutions but also properly handles heterogeneous noise. Exact calculation formulas of the proposed algorithms are further developed. Typical numerical cases and two engineering examples (a rocket injector design example and a real inventory design scenario) demonstrate that the proposed algorithms give better Pareto-optimal solutions of which the proximity to true Pareto fronts and the diversity are improved.

Effect of wave spectral variability on the dynamic response of offshore wind turbine considering soil-pile-structure interaction

The accurate assessment of the dynamic response of the offshore wind turbine is fundamental to its economical and safe design, where the precise characterization of the incident wind and waves is essential. While reliable aerodynamic models can accurately reproduce wind characteristics, the widely-used empirical JONSWAP spectrum in wave field simulation may not capture the multi-modal feature of the mixed wave field near the wind farm. In this study, the stochastic structural responses based on an integrated FE model of the soil-pile-structure system are assessed employing the empirical and actual wave spectra, along with discussions on the impacts of wave theory and wave-structure interaction. The main conclusions are: (1) The empirical wave spectrum can overestimate the dynamic responses compared with the measured swell-dominant one, but the dominant wind load weakens its influence on displacement. (2) When the wave serves as the decisive loading, using the empirical spectrum overestimates the dynamic performance for the swell-dominant wave spectrum while underestimating it for the wind-induced-wave-dominant one. The approximation of the actual wave spectrum to the JONSWAP spectrum is only recommended when the coupling effect of wind and wave is inapparent. (3) The wave theory imposes an evident effect on the dynamic behavior of the offshore wind turbine subject to joint wind and wave loading, where the JONSWAP wave spectrum is more sensitive. (4) The influence of water-monopile interaction is negligible when using the JONSWAP wave spectrum but prominent for the simulation's actual wave spectrum.

Efficient simulation of stochastic seismic response of long-span bridges in river valleys using hybrid BEM-FEM

Determining ground motions is essential for seismic analysis of long-span bridges located at complex sites. However, there is a lack of earthquake records for local site conditions. Thus, this study focused on the effect of local site conditions on the stochastic response of long-span high-pier rigid frame bridges crossing a valley with an overlying water layer. For this purpose, a hybrid boundary element and finite element method (BEM-FEM) is proposed to investigate the stochastic seismic response of long-span bridges at such a complex site. First, a simulation method for 3-D spatially varying ground motions ascribed to wave scattering from irregular surface topography, sedimentary soil, and water layers was established based on the BEM combined with the spectral representation method. Furthermore, the statistical properties of the bridge response were explored based on the FEM and Monte Carlo simulations. The advantage of this work is that it reveals how the seismic wave propagation in this particular valley topography and water layer affects the stochastic response of the bridge considered. Moreover, the results indicate that seismic wave scattering leads to a significant energy redistribution of the input waves in the tri-directions related to the surface location, and a decrease in the coherency loss functions at all frequencies. Owing to this change, the peak ground accelerations (PGAs), bridge response means, and response variability are doubled. The beneficial effect of the water layer on the PGAs is weakened, while the adverse effect of hydrodynamic pressure dominates. The parameter analysis results demonstrate that a 10-m-deep sedimentary soil layer produces a 47% increase in the internal force of the pier bottom. The solid-liquid dynamic coupling effect should be accurately quantified for bridges located at saturated sites. Considering the local site conditions, the coefficient of variation (COV) of the response of the pier top exceeded 0.3, indicating strong randomness.

Stochastic dynamic analysis of nonlinear MDOF systems with chaotic motion under combined additive and multiplicative excitation

Stochastic response and global dynamic analyses of structures with chaotic motion are challenging issues, especially for nonlinear multi-degree-of-freedom (MDOF) system excited by combined additive and multiplicative excitation. In this paper, a novel direct probability integral method (DPIM) is extended to address these challenges. Firstly, the probability density integral equation (PDIE) of MDOF system under combined excitation is established. By using the DPIM to solve the decoupled deterministic dynamic equation and PDIE successively, the stochastic responses are then achieved, and the stochastic bifurcation of system under combined excitation is explored. Moreover, the important equivalent relationship between PDIE and Dostupov–Pugachev differential equation is derived, exhibiting the superiority of PDIE for MDOF system under additive and multiplicative excitation. Due to the lack of effective numerical tool for global dynamic analysis of nonlinear MDOF stochastic system, another aim of this study is to propose a DPIM-based strategy to attack this problem from the view of probability. As the generalized stochastic basin, the ϵ-committor is introduced, and global integrity measure (GIM) is utilized for evaluating the stability of stochastic basin quantitatively. Finally, two examples of nonlinear systems under combined excitation, including nonlinear MDOF coupled ship model with chaotic motion under random oblique wave, demonstrate the effectiveness of DPIM. It is shown that the safety basin of system under combined excitation can be effectively described in a probabilistic way. The dramatic effect of initial disturbance on stochastic ship system is revealed, i.e., the stochastic safety basin is broken up to a series of discretized regions with increasing of intensity of initial disturbance, resulting in the decreasing of system stability.

Response Analysis of Nonlinear Viscoelastic Energy Harvester with Bounded Noise Excitation

Energy harvesting has become a popular topic in recent years. A number of studies have been conducted in the field of vibration energy harvesting system (VEHS). However, few studies have concentrated on viscoelastic energy harvesters driven by bounded noise excitation. In this paper, the stochastic response of a viscoelastic energy harvester subjected to bounded noise is discussed. Approximate solutions of the system were derived by utilizing the method of multiple scales, and the expressions of the mean square voltage (MSV) and mean output power (MOP) were obtained. The relation between the detuning frequency and first-order steady moment was first revealed. The effectiveness of the approach was verified by a good agreement between theoretical results and numerical results. Furthermore, the variations in the detuning frequency can result in the stochastic jump phenomenon, and stochastic bifurcation is induced with the changes in the viscoelastic parameter and detuning frequency. Finally, the impacts of system parameters on the MSV and the MOP were also analyzed.

Novel triple friction pendulum-tuned liquid damper for the wind-induced vibration control of airport control towers

The airport control tower has been widely constructed for rapidly expanding transportation demand, whose anti-wind performance and high-efficiency upgrading technology are attracting increasing attention. To provide an efficient liquid-incorporated solution for wind-induced vibration control of airport control towers, this study proposes a novel triple friction pendulum tuned liquid damper (TFPTLD) and establishes a control demand-oriented design framework to facilitate the design of TFPTLDs. The configuration and theoretical analysis model of the TFPTLD are introduced, based on which the TFPTLD-equipped tower-like structure is established. Stochastic response analysis is performed to obtain the responses under stochastic wind and white noise. Correspondingly, extensive parametric research is conducted to explore the optimal parameters and clarify the vibration control demand-oriented design procedure and easy-to-use fitted design formula for the TFPTLD. Furthermore, a robustness investigation is undertaken to quantify the advantages and sensitivity of the TFPTLD to the stiffness and damping components. Finally, the TFPTLD is applied to an airport tower subject to experimental wind loads to illustrate the effectiveness of the devices and proposed design method. The results show that the proposed TFPTLD yields a compact solution to the high-level performance upgrading of the airport control tower subjected to wind excitations, which significantly reduces the structural acceleration and displacement responses. Following the proposed demand-oriented design method, the TFPTLD exhibits higher effectiveness, larger frequency bandwidth, and more robust wind-induced vibration control than conventional TLDs with the same liquid mass. Benefitting from the triple friction pendulum, a small installation space is required, and the tuning effect of the liquid mass is independent of the change in liquid mass, which can be comprehensively utilized for vibration control to maintain the daily function of liquid tanks.

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

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

Enhanced energy dissipation benefit of negative stiffness amplifying dampers

Negative-stiffness devices are of rapidly increasing interest for high-performance vibration control. With a tuning spring in series, the negative-stiffness device forms an enhancement mechanism for the dashpot, named the negative stiffness amplifying damper (NSAD), which is much more efficient in dissipating energy than the viscous damper alone. Based on the observed enhancement phenomenon, this study reveals the theoretical basis of the energy dissipation enhancement benefit by deriving the analytical enhanced energy dissipation formula. The basic concept and physical realization methods for the negative-stiffness device are introduced, based on which the mechanical models of NSAD and NSAD-equipped structures are established for dimensionless response analysis. Following a stochastic framework, the stochastic responses of the NSAD-equipped structure are solved in closed form. Correspondingly, simple and easy-to-understand equations are derived and denoted as the “enhanced energy dissipation formula” to quantify each component's contribution in the NSAD explicitly. Inspired by the formula, a design framework with two strategies is proposed for NSADs with a synthetic consideration of the energy dissipation enhancement mechanism and target control performance. Following the design principle, detailed design formulae are derived in a simple form. Finally, a design procedure is provided and applied to a series of design cases for illustration. The results show that the energy dissipation enhancement mechanism enables NSADs to offer improved energy dissipation efficiency over conventional dampers without sacrificing the vibration isolation effect or the excessive damping force. The analytical formula clearly explains the theoretical basis by quantifying the NSAD components’ contribution to the vibration control and energy dissipation enhancement effects. Based on this newly explained benefit, the design framework successively provides a high-efficiency energy dissipation device that fulfills its intended control performance. In addition, the easy-to-use design formulae are derived for the new design strategy to allow the direct design of NSADs.

Structural fatigue reliability evaluation based on probability analysis of the number of zero-crossings of stochastic response process

Structural fatigue failure is closely related to the number of stress cycles. Therefore, the number of the intersections between stochastic response process and zero-boundary is a key problem in structural fatigue reliability analysis. In this paper, the relationship between the number of the multiple intersections and the fatigue cumulative damage index is derived according to the Miner rule-based high cycle fatigue random cumulative damage index. Then, use random walk to approximately simulate Wiener process, and get the first intersection probability between Wiener process and the zero-boundary. Next, further utilize the renewal process to solve the distribution law of the number of intersections between the Wiener process and the zero-boundary in a period of time. Finally, based on the fatigue failure criterion considering the randomness of the failure threshold, the calculation formula the structural fatigue reliability is derived. As a practical example, the fatigue reliabilities of the key part of Metro Bogie under different operation periods are analyzed using the proposed method.

Comparison of PDEM and MCS: Accuracy and efficiency

The accuracy and efficiency of two methods for stochastic analysis, the probability density evolution method (PDEM) and the Monte Carlo simulation (MCS) method, are compared in terms of how well they reflect the physical properties of stochastic systems. The basic principle and the numerical implementation details of PDEM and MCS are revisited. The analytical solutions of generalized probability density evolution equation (GDEE) for three typical stochastic systems are given and are to be used as the basis for comparing the two methods. It is verified that, with the rational partition of the probability space, the PDEM provides a continuous and complete reflection of physical properties over the whole probability space. Meanwhile, with the help of the numerical solution of GDEE, PDEM is efficient and accurate to describe the process of the probability density evolution of stochastic systems. In contrast, the random samples in the MCS may not reflect the physical properties of a stochastic system adequately, and the local cluster of sample points may cause redundant calculation, which leads to lower computational efficiency. Through three typical numerical examples, the paper compares the accuracy and efficiency of PDEM and MCS specifically. It is shown that, as the numerical approaches for the stochastic response of a system, the PDEM could get much higher numerical accuracy than MCS with the same number of samples. To achieve the same level of calculation accuracy, MCS needs a much higher number of samples than PDEM.

Stochastic analysis of structures under limited observations using kernel density estimation and arbitrary polynomial chaos expansion

Over the past decades there has been considerable interest among the scientific community in modeling random system inputs from limited observations and propagating the system responses. In this paper, we develop a novel method for the reasonable modeling of system inputs and efficient propagation of response. Our method firstly constructs the random model of system inputs from observations by developing a novel kernel density estimator (KDE) for Karhunen–Loeve (KL) variables. By further implementing the arbitrary polynomial chaos (aPC) formulation on dependent non-Gaussian KL variables, the associated aPC-based response propagation is then developed. In our method, the developed random model can accurately represent the input parameters from limited observations as the developed KDE of KL variables can incorporate the inherent relation between marginals of input parameters and KL variables, while empowering the input model to preserve the second-order correlations. Furthermore, since the statistical moments of KL variables can be analytically evaluated, the aPC formulation for achieving optimal convergence of system responses can be accurately determined. In addition, the system response can be accurately propagated in an efficient way by developing an aPC-based regression method using space-filling design, in which the conditional distributions in Rosenblatt transformation of aPC variables can be efficiently determined without evaluations of multi-dimensional integrations. In this way, the current work provides an effective framework for the reasonable stochastic modeling​ and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented to highlight the effectiveness of the developed method.

Stability of rectangular Kirchhoff plates using the Stochastic Boundary Element Methods

The main objective in this work is to study an application of the Stochastic Boundary Element Methods implemented due to three different probabilistic approaches to analyze stability of the rectangular thin elastic and isotropic plates. This is completed with the use of polynomial approximations applied for the Least Squares Method recovery of critical forces resulting from some material and geometrical random imperfections in the plate, e.g. Young's modulus, Poisson's ratio, thickness. A deterministic core for solving the plate stability problem is the Boundary Element Method in direct approach and modified formulation of the boundary and domain integral equations. Probabilistic approaches employed here include traditional Monte-Carlo simulation, the semi-analytical approach as well as the iterative generalized stochastic perturbation method. Stochastic response in the form of up to the fourth order characteristics are studied numerically in addition to the input uncertainty level.