Stochastic Response Characteristics Research Articles

Stationary Stochastic Response Analysis of the FG-GPLRC Irregular Quadrilateral Plate Based on the Chebyshev–Ritz Method

The stationary random response characteristics of irregular quadrilateral plates under the base acceleration excitation are investigated in this paper. The computational model chosen in this paper is functionally graded graphene platelet reinforced composites (FG-GPLRC). Simultaneously, the material parameters are calculated employing the modified Halpin–Tsai model. A unified dynamic model based on the first-order shear deformation theory (FSDT) is proposed for FG-GPLRC irregular plates, where the displacement variables are considered as the first kind Chebyshev polynomials combined with coordinate transformation. The Chebyshev–Ritz method is utilized to derive the kinetic equations of irregular quadrilateral plates, and then the pseudo-excitation method (PEM) is employed to solve the stationary stochastic response of the system under the base acceleration excitation. The free vibration and stationary stochastic response of the plates under various boundaries are validated by a sufficient number of numerical studies, and the computational results display a favorable match with the published literature and finite element method (FEM), which verifies the correctness and reliability of the present method. Finally, a detailed parametric analysis of the stationary stochastic response characteristics of FG-GPLRC irregular quadrilateral plates is presented. The effects of various mass fractions, number of layers, geometric dimensions of GPL, and boundary constraints on the stationary stochastic vibration behaviors of FG-GPLRC irregular quadrilateral plates are considered.

Design sensitivity analysis of structural systems with damping devices subjected to fully non-stationary stochastic seismic excitations

In the framework of dynamic excitations to be considered during the design phase of structures, the most crucial one is the ground motion acceleration. To increase the structural performances against seismic actions, one of the most effective design criteria is to introduce damping devices.An efficient approach to define the optimal parameters of the damping system is based on the design sensitivity analysis, which provides a quantitative estimate of desirable design change, by relating the available design variables.In this paper a method to evaluate the sensitivities of stochastic response characteristics of structural systems with damping devices subjected to seismic excitations, modelled as fully non-stationary Gaussian stochastic processes, is proposed. The main steps are: i) to define the time–frequency varying response (TFR) function for non-classically damped systems; ii) to evaluate closed form solutions for the first-order derivatives of the TFR function as well as of the one-sided evolutionary power spectral density function of the structural response, with respect to damping parameters of devices; iii) to perform a design sensitivity analysis selecting as performance measure function the non-geometric spectral moments of nodal displacements.A numerical application demonstrates how the proposed approach is suitable to cope with practical problems of engineering interest.

Response of a stochastic multiple attractors wind-induced vibration energy harvesting system with impacts

Focusing on multiple attractors wind-induced vibration energy harvesting system with unilateral impact under the excitation of Gaussian white noise, the stochastic response characteristics, and energy harvesting performance of the systems are presented and discussed. The stochastic averaging of the energy envelope and a conversion mechanism for decoupling the electromechanical equations are introduced to derive the theoretical probability density functions. Numerical simulation results are given to verify the correctness of such theoretical methods. Effects of the noise intensity, restitution coefficient, time constant ratio and coupling coefficient on the mean square voltage are analyzed. It is found that increments of the restitution coefficient and the piezoelectric coupling coefficient of the electrical part have a positive effect on the output voltage of the system, while other factors are opposite. Analysis can help explain the physical mechanisms of the system and provide ideas for optimizing energy harvesting performance by properly tuning the coefficients.

Dynamics of stochastically excited energy harvesting systems with impact

Focusing on rigid impact (RI) and elastic impact (EI) vibratory energy harvesting systems with the excitation of Gaussian white noise, the stochastic response characteristics, and energy harvesting performance of the systems are presented and compared. The stochastic averaging of the energy envelope is introduced with different processing methods in these systems to derive the theoretical probability density functions. Numerical simulation results verify the effectiveness of such theoretical methods. Effects of the coefficients and positions of impact barriers in the systems on the mean square voltage are analyzed. The analysis can help explain the physical mechanisms of the systems and provide ideas for optimizing energy harvesting performance by properly tuning the coefficients. Effects of the existence of three kinds of impact on the output energy are explored and compared. It is found that the existence of EI can help to enhance the energy harvesting performance of the system in most cases, and setting the impact barrier at the origin is more conducive to harvesting energy compared with other impact positions.

Approximate Fokker–Planck–Kolmo-gorov equation analysis for asymmetric multistable energy harvesters excited by white noise

Due to the broadband response characteristics at low levels of excitations, nonlinear multistable systems have garnered a great deal of attention in the area of energy harvesting. Moreover, various performance enhancement strategies of multistable harvesters have been proposed and discussed extensively for systems with perfectly symmetric potentials. However, it is very difficult or even impossible in practice to modulate a nonlinear system with completely symmetric potentials. Therefore, this paper investigates the stochastic response characteristics of asymmetric potential multistable harvesters excited by Gaussian white noise. Approximate Fokker–Planck–Kolmogorov equation for multistable harvester is provided, and solved by the method of detailed balance. Numerical simulations are carried out to characterize the probability distribution and power output of the response of the harvesters. Results indicate that the theoretical method could well predict the response probability distribution of the systems under white noise excitations. To be more specific, the probability distribution of velocity response at a given excitation level is not affected by the asymmetry of potentials, while the probability density function of displacement is very dependent on the shape of potentials. In terms of the power output, the numerical results agree well with the theoretical results. However, the error between numerical and theoretical results increases with the increase in the number of the steady states and the degree of asymmetry of the harvesters.

On the combination of nonlinear load effects of structures

A new method for combination of nonlinear load effects of structures is suggested in the paper. At first, the concepts of load combination and load effect combination are discussed, respectively. A load combination method based on probability space partition is introduced, through which the load effects of structures could be accurately estimated. Afterwards, the principle of nonlinear load effect combination of structures under multiple loads is presented. In order to reflect the nonlinear performances of structures, the physical equations of solid mechanics with a reasonable elastoplastic damage constitutive model are applied to derive structural deterministic nonlinear responses. The probability density evolution equation (PDEE) is used to acquire the stochastic response characteristics of structures under multiple random loads. Hence, the combination of nonlinear load effects of structures is well realized. Furthermore, the reliability of structures under multiple loads is evaluated by using the physical synthesis method. Finally, in order to illustrate the proposed method, the nonlinear load effect combination analyses for a reinforced concrete (RC) plane frame under gravity load and earthquake and a concrete bridge under long-term vehicular loads and earthquakes are carried out. The analysis results demonstrate the applicability and superiority of the proposed method.

A Kriging–NARX Model for Uncertainty Quantification of Nonlinear Stochastic Dynamical Systems in Time Domain

A novel approach, referred to as sparse Kriging–NARX (KNARX), is proposed for the uncertainty quantification of nonlinear stochastic dynamical systems. It combines the nonlinear autoregressive with exogenous (NARX) input model with the high fidelity surrogate model Kriging. The sparsity in the proposed approach is introduced in the NARX model by reducing the number of polynomial bases using the least-angle regression (LARS) algorithm. Sparse KNARX captures the nonlinearity of a problem by the NARX model, whereas the uncertain parameters are propagated using the Kriging surrogate model, and LARS makes the model efficient. The accuracy and the efficiency of the sparse KNARX was measured through uncertainty quantification applied to three nonlinear stochastic dynamical systems. The time-dependent mean and standard deviation were predicted for all the numerical examples. Instantaneous stochastic response characteristics and maximum absolute response were also predicted. All the results were compared with the full scale Monte Carlo simulation (MCS) results and a mean error was calculated for all the numerical problems to measure the accuracy. All the results had excellent agreement with the MCS results at a very limited computational cost. The efficiency of the sparse KNARX also was measured by the CPU time and the required number of surrogate model evaluations. In all instances, sparse KNARX outperformed other state-of-the-art methods, which justifies the applicability of this model for nonlinear stochastic dynamical systems.

Theoretical analysis of piezoelectric energy harvesting system with impact under random excitation

A kind of mono-stable piezoelectric energy harvesting system with unilateral offset barrier under Gaussian white noise excitation is studied theoretically to reveal the physical mechanism in this paper. After utilizing the Zhuravlev transformation and the stochastic averaging method, stationary joint and marginal probability density functions of the displacement and velocity are obtained, as well as the equation of the mean square voltage. Stochastic response characteristics are discussed by analyzing the functions deducted previously. And the effectiveness of the theoretically approximate methods is validated by Monte Carlo simulation results. In addition, effects of the noise intensity, the restitution coefficient, the mechanical damping exponent, and the ratio coefficient between the mechanical and electrical time constant in the system on the mean square voltage are explored.

Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation

SUMMARYNonlinear viscous dampers are supplemental devices widely used for enhancing the performance of structural systems exposed to seismic hazard. A rigorous evaluation of the effect of these damping devices on the seismic performance of a structural system should be based on a probabilistic approach and take into account the evolutionary characteristics of the earthquake input and of the corresponding system response.In this paper, an approximate analytical technique is proposed for studying the nonstationary stochastic response characteristics of hysteretic single degree of freedom systems equipped with viscous dampers subjected to a fully nonstationary random process representing the seismic input. In this regard, a stochastic averaging/linearization technique is utilized to cast the original nonlinear stochastic differential equation of motion into a simple first‐order nonlinear ordinary differential equation for the nonstationary system response variance. In comparison with standard linearization schemes, the herein proposed technique has the significant advantage that it allows to handle realistic seismic excitations with time‐varying frequency content. Further, it allows deriving a formula for determining the nonlinear system response evolutionary power spectrum. By this way, ‘moving resonance’ effects, related to both the evolutionary seismic excitation and the nonlinear system behavior, can be observed and quantified.Several applications involving various system and input properties are included. Furthermore, various response parameters of interest for the seismic performance assessment are considered as well. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.

LUMPING PROCESS OF RUNOFF MODEL BASED ON EVALUATION OF ITS STOCHASTIC RESPONSE -DERIVATION OF HIGHER ORDER MOMENTS OF DISCHARGE-

Using a runoff system composed of a hillslope runoff generation model and a channel network, this paper derives theoretical differential equations to calculate the first- to third-order central moments of discharge. Numerical simulation shows the results of cross-checking the validity of the derived equations. The results are as follows: σqi2 and μqi3 , in a steady state are regarded as constant when the catchment area is smaller than one hundred and several tens kilometers square. The results of this study can be applicable to the evaluation method of lumping process of models based on their stochastic response characteristics.

STOCHASTIC RESPONSE CHARACTERSITICS OF RAINFALL RUNOFF SYSTEM

Rainfall-runoff phenomena generally belong to a random process. Thus, a rainfall runoff system is described by random differential equations. However, the stochastic response characteristics of rainfall runoff systems for a larger basin are not clear. In this paper, the Kinematic Wave model is used as a rainfall runoff system, derivation of theoretical differential equations to calculate the first-and second-order moments of discharge is shown. The validity of derived equations is shown by the results of simulation. By using a similar way, estimation of the higher-order moments of discharge output and its probability density function that change with time is possible. The results of this study can be applicable to the evaluation method of models based on their stochastic response characteristics.

STOCHASTIC RESPONSE CHARACTERSITICS OF KINEMATIC WAVE MODEL

Generally, it is well known that rainfall-runoff phenomena belong to a random process. So, the runoff system is described by random differential equations. In this paper, I adopted Kinematic Wave model for tow slopes and a river channel and derived differential equations whose solutions provided the average and variance of discharge output from the runoff system under the condition of mutually independent rainfall input. The validity of proposed theoretical equations was cross-checked by a simulation. By using similar way, it is possible to estimate the higher order moments of discharge output and its probability density function which change with time.The results of this study can be applied to the evaluation method of models based on its stochastic response characteristics.

Non-linear dynamics of a stochastically excited beam system with impact

The response of non-linear, dynamic systems to stochastic excitation exhibits many interesting characteristics. In this paper, a strongly non-linear beam-impact system under both broad- and small-banded, Gaussian noise excitations is investigated. The response of this system is investigated both numerically, through a multi-degree-of-freedom model, and experimentally focusing on frequency-domain phenomena such as stochastic equivalents of harmonic and subharmonic solutions. An improved understanding of these stochastic response characteristics is obtained by comparing these to non-linear periodic response features of the system. It will be shown that in modelling such a continuous, linear system with a local non-linearity, the linear part can be effectively reduced to a description based on several modes. Combining this reduced, linear part with the local non-linearity in a reduced, non-linear model is shown to result in a non-linear model, which can be used to accurately predict the stochastic response characteristics of the original, continuous, non-linear system. It is shown that including more modes to the model causes its response to differ significantly from that of a single-degree-of-freedom model and show a better correspondence with experimental results, also in the frequency range of the first mode.

Experimental and Numerical Analysis of Nonlinear Phenomena in a Stochastically Excited Beam System with Impact

The response of strongly nonlinear dynamic systems to stochastic excitation exhibits many interesting characteristics. In this paper, a strongly nonlinear beam-impact system under broad and small banded, Gaussian noise excitations is investigated. The response of this system is investigated numerically as well as experimentally. The emphasis lies on frequency domain characteristics. Phenomena like multiple resonance frequencies and stochastic equivalents of harmonic and subharmonic solutions are found. Improved understanding of these stochastic response characteristics is obtained by comparing them to nonlinear periodic response features of the system. It is shown that these stochastic response phenomena can provide information on the periodic response characteristics of the system. The observed stochastic characteristics have more general value and meaning than mere application to this system suggests.

Modal stochastic-response characteristics of multivariable linear systems†

A method is described which greatly facilitates the determination of the stochastic-response characteristics of a class of multivariable linear systems : the method is illustrated by the presentation of a, numerical example.

Synthesis of linear multivariable feedback systems with prescribed stochastic-response characteristics

The letter presents a simple method of synthetising a class of linear multivariable feedback systems with prescribed steady-state stochastic-response characteristics.