Stochastic Linearization Technique Research Articles

Probabilistic analysis of the steady state of weakly perturbed linear oscillators subject to a class of Gaussian inputs

This paper aims to probabilistically study a class of nonlinear oscillator subject to weak perturbations and driven by stationary zero-mean Gaussian stochastic processes. For the sake of generality in the analysis, we assume that the perturbed term is a polynomial of arbitrary degree in the spatial position, that contains, as a particular case, the important case of the Duffing equation. We then take advantage of the so-called stochastic equivalent linearization technique to construct an equivalent linear model so that its behavior consistently approximates, in the mean-square sense, that of the nonlinear oscillator. This approximation allows us to take extensive advantage of the probabilistic properties of the solution of the linear model and its first mean-square derivative to construct reliable approximations of the main statistical moments of the steady state. From this key information, we then apply the principle of maximum entropy to construct approximations of the probability density function of the steady state. We illustrate the superiority of the equivalent linearization technique over the perturbation method through some examples.

Stochastic Optimization of Electromagnetic Inertial Mass Dampers in Nonlinear Hybrid Base Isolation Systems Under Seismic Excitations

Recently developed inerter-based dampers have been installed at the base layer to reduce base displacement and simultaneously improve the superstructure’s performance. However, in the existing research on optimizing damper parameters, the nonlinear property of base isolation is simplified and stochastic linearization technique is introduced to approximate such nonlinearity, which would inevitably introduce errors. In this paper, an improved method is proposed for stochastic optimization of electromagnetic inertial mass dampers (EIMD) in nonlinear hybrid isolation systems subject to seismic excitations. First, a stochastic seismic excitation is represented by one elementary random variable adopting the dimension-reduction spectral representation method. Then, statistical moments of nonlinear base-isolated structural responses are estimated using two-point estimation method (2PEM). Finally, the EIMD optimal parameters are obtained via the modified directional bat algorithm (MDBA) recently proposed by the authors through minimizing base layer response moments. This proposed improved method could fully consider the base isolation nonlinearity compared with the previous stochastic linearization technique. Moreover, the proposed stochastic optimization of the EIMD only requires two dynamic analyses to achieve its efficiency. A seven-story base-isolated frame building equipped with an EIMD is investigated in the numerical simulation, and results indicate that EIMD optimized by this proposed method obviously outperforms the results by the stochastic linearization technique and the conventional viscous damper (VD) in the base layer displacement reduction.

A novel stochastic linearization technique for structures with nonlinear fluid viscous dampers including soil-structure interaction

A novel stochastic linearization technique for structures with nonlinear fluid viscous dampers including soil-structure interaction

The dynamical behavior of a class of stochastic vegetation models

We investigate the influence of random fluctuations in environmental parameters (e.g. rainfall and grazing rate) on the dynamical behavior of a class of ecological models for vegetation biomass in semi-arid regions. Firstly, we show the global existence and uniqueness of positive solutions for the number of vegetation biomass B(t). Then some sufficient conditions are presented for extinction and persistence of B(t) by using the stochastic linearization technique, the Hurwitz criterion and the Lyapunov function method. In particular, a threshold value R0s for the stochastic vegetation model is established in order to illustrate extinction and persistence of B(t). We also find that R0s=1 can be a warning signal for the emergence of desertification. In the case of R0s>1, we show that the stochastic vegetation system has a unique stationary distribution. Finally, we present conditions for persistence of B(t), and prove the pth moment stability and almost sure stability of the null solution in a general decay rate sense (including the exponential, polynomial and logarithmic rates) where the parameters in the stochastic vegetation system can also be dependent on time. These results are illustrated by numerical simulations.

A novel tuned inerter eddy current damper: modeling, optimization, and evaluation

This study proposes a novel tuned inerter eddy current damper using a rack and gear mechanism (TIECD), in which the configuration and mechanical model are introduced in detail. The TIECD design procedure for the seismic control of primary structures is established based on the effective damping ratio enhancement (EDRE) effect. A closed-form solution for TIECD optimal design under white noise excitation is presented using stochastic linearization techniques (SLT) with force-based and energy-based equivalence criteria. The control performance of the TIECD obtained by the closed-form solution, including the EDRE effect and effective damping ratio, is in good agreement with the numerical results obtained from the Monte Carlo simulation when a suitable TIECD critical velocity is selected; a large critical velocity is conducive to improving the accuracy of the closed-form solution. Note that the effect of a low structural damping level on the optimal design of TIECD can be ignored, which is beneficial in practical engineering applications because structural damping cannot be precisely known in most cases. Furthermore, several numerical studies have been conducted to investigate the availability of TIECD under real seismic excitations. The results indicate that the effectiveness of TIECD in mitigating the seismic response of primary structures is approximately the same as that of a tuned viscous mass damper (TVMD) and much greater than that of the viscous and eddy current dampers with the same damping parameters. It can be considered that the maximum damping force and maximum deformation of TIECD and TVMD are the key design factors for short-period and long-period structures, respectively, and the advantages of TIECD over TVMD are owing to the damping force limiting characteristics and small deformation.

Stochastic seismic analysis of structures with nonlinear eddy current dampers

Eddy current damper (ECD), a contactless energy-dissipating device, is applying to control the vibration induced by earthquake and strong wind in civil structures. Combining with motion magnification mechanisms improves the damping effect of the ECD while significantly strengthens its nonlinearity. The response of single degree of freedom and multi-degree of freedom system with ECDs under a stationary stochastic earthquake characterized by the power spectral density function is evaluated using the stochastic linearization technique and expressions of the equivalent linear damping coefficient based on force criterion and energy criterion have been found, respectively. Comparisons with results obtained by Monte Carlo simulations confirm that for the nonlinearity of eddy current dampers the force-based criterion stochastic linearization technique gives accurate estimation.

Multi-Network Coordinated Hydrogen Supply Infrastructure Planning for the Integration of Hydrogen Vehicles and Renewable Energy

The growing penetration of hydrogen vehicles and modern energy conversion technologies strengthen the coupling of transportation and energy networks. This article proposes a hydrogen supply infrastructure planning model for the integration of hydrogen vehicles and renewable energy. To flexibly meet the energy demand of hydrogen vehicles, the proposed model makes investment decisions for various facilities, including hydrogen pipelines, hydrogen refueling stations, power to gas devices, and renewable energy generators. Besides, with the pipeline transportation method applied, the hydrogen network is constructed and coordinated with the electricity and transportation networks. The multinetwork synergistic effect is thus fully utilized, bringing higher operational flexibility and investment economy. Furthermore, a two-stage stochastic planning model is provided to accommodate the variability of renewable energy and traffic loads. To reduce the computational complexity of the proposed stochastic planning model, both linearization techniques and Benders decomposition algorithm are applied. Simulation results of the 8-node system and the 24-node system demonstrate the effectiveness of the proposed model and algorithm. Compared to the uncoordinated model, the proposed model saves by 11% of the total cost for the 24-node test system. Also, the computational performance of the Benders decomposition algorithm surpasses that of the basic algorithm by more than 24.6% in the two test systems.

Optimum design of tuned mass damper with pinched hysteresis under nonstationary stochastic seismic ground motion

The optimum design of a recent tuned mass damper with pinched hysteresis (TMD-PH) for seismic protection of structures is here investigated. The considered device consists of mixed NiTiNOL-steel wire ropes undergoing combined tension-flexure cycles. Previous studies demonstrated that concurrent interwire friction and phase transformation of the shape memory material exhibited by these composite wire ropes generate a peculiar pinched hysteretic behavior, which can be well described by means of a modified Bouc–Wen model. A stochastic-based optimum design methodology is here proposed for such TMD-PH, which is tailored to determine the best design parameters for the seismic protection of general structural systems. The seismic ground motion is modeled as a filtered nonstationary random process whereas the inherent pinched hysteresis of the device is dealt with via the stochastic linearization technique. The expressions of the linearization coefficients are obtained ad hoc and the optimum design of the TMD-PH is formulated as box-constrained single-objective optimization problem. Three objective function formulations are considered to reflect alternative design philosophies, namely, displacement-, acceleration- and energy-based design strategies. Unlike most literature studies addressing the optimum design of vibration absorbers via finite difference-based numerical schemes or non-gradient-based metaheuristic methods, the search for the optimal parameters of the TMD-PH is here performed by means of an iterative semi-analytical gradient-based numerical technique in which the derivatives for the covariance matrix of the system response are calculated analytically. A comprehensive numerical campaign is finally presented. First, a parametric study is carried out to evaluate the effects of the protected system characteristics and those of the seismic ground motion duration on the optimal parameters and performance of the device under consideration. The performance of the TMD-PH optimized according to the proposed methodology is also assessed in the context of a practical case study which makes use of synthetic and natural seismic ground motion records.

Analysis of bilinear hysteretic structures with nonlinear fluid viscous dampers using modified stochastic linearization technique

Viscous dampers are increasingly used to design seismic resistant structures in modern construction due to their ability to reduce structural and non-structural damage by absorbing a large portion of the earthquake input energy. To facilitate the design of such complex nonlinear systems, this study aims to propose new non-Gaussian probability density functions (PDFs) to improve the accuracy in the determination of the seismic response of bilinear hysteretic structures equipped with linearized nonlinear viscous dampers. The employed linearization method is based on a stochastic linearization technique using stationary response velocity PDFs, while the difference between the governing nonlinear differential equation and the equivalent linear equation is minimized. To generate suitable functions, a Genetic Algorithm (GA) optimization method is adopted for both hard and soft soil-based structures equipped with nonlinear viscous dampers using three different power coefficients. The computational error of the employed modified stochastic linearization technique is then evaluated for single-degree-of-freedom (SDOF) systems under a set of five thousand randomly generated earthquake excitations with peak ground accelerations (PGAs) of 0.2 g to 0.6 g. The results indicate that the proposed method can significantly reduce (up to 80%) the computational errors. Subsequently, the lateral roof displacements of 5-story structures equipped with nonlinear viscous dampers are obtained under a set of natural ground motions using different linearization methods. It is shown that the proposed functions lead to the least error for both hard and soft soil-based structures compared to the accurate results.

Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment

Periodic metafoundations have proven to inherit valuable properties from wave propagating in phononic periodic structures in the very low-frequency regime. Therefore, finite locally resonant metafoundations (LRMs) represent a novel type of seismic isolation for ultralow-frequency applications. In this context, it is still unknown the impact that massive resonators with varying frequencies or devices with hysteretic behavior can entail on the whole system performance. For this purpose, we develop and optimize two finite locally resonant multiple degrees of freedom (MDoF) metafoundations in this paper: i) a foundation endowed with resonators, linear springs and linear viscous dampers tuned to multiple frequencies; and ii) a foundation equipped with fully nonlinear hysteretic dampers. Both are optimized considering the stochastic nature of ground motion, modelled with a modified Kanai-Tajimi filter in the stationary frequency domain, and a massive MDoF superstructure, chosen to be a fuel storage tank. In order to take all of the above-mentioned effects into account, we establish a procedure based on nonlinear programming that is able to optimize any number of parameters. More precisely, to optimize the nonlinear behavior of damper devices we employ a Bouc-Wen hysteretic model. Therefore, we reduce the nonlinear differential equations of Bouc-Wen models to a system of linear equations through the stochastic (equivalent) linearization technique. Moreover, we test the optimized systems against natural seismic records both with linear and nonlinear time history analyses. To investigate the role of hysteresis on the nonlinear band structure, we derive linearized and nonlinear dispersion relationships for the uncoupled periodic metafoundation. Finally, we obtain further detailed information on the nonlinear wave propagation by means of a spectro-spatial analysis.

A new modified stochastic linearization technique to analyze structures with nonlinear fluid viscous dampers

Nonlinear viscous dampers can efficiently improve the seismic performance of structures by dissipating large amounts of earthquake-induced energy. In common practice, the spectral analysis of structures with nonlinear viscous dampers is generally conducted based on an estimated equivalent damping ratio. To this end, the stochastic linearization technique can be used as an effective probabilistic approach to take into account the evolutionary characteristics of the input earthquake excitation. This study aims to present optimal non-Gaussian probability density functions to improve the accuracy of the stochastic linearization technique for nonlinear viscous dampers in both firm and soft soil-based structures. It is shown that by using the optimum probability density functions, the computational error of the stochastic linearization technique for a single-degree-of-freedom structure under simulated ground motions, with a range of peak ground accelerations between 0.1 and 0.6 g, is reduced by up to 70%. The efficiency of the proposed probability density functions is then demonstrated for multi-degree-of-freedom structures, by estimating the roof displacements of a six-story steel frame with nonlinear viscous dampers under a set of natural ground motions using different linearization methods. The comparison of the stochastic linearization technique estimated responses with the exact values confirms that using the proposed probability density functions leads to considerably lower errors in both firm and soft soil-based structures compared with the other linearization techniques.

Vibration mitigation of an MDoF system subjected to stochastic loading by means of hysteretic nonlinear locally resonant metamaterials

In this paper, we intend to mitigate absolute accelerations and displacements in the low-frequency regime of multiple-degrees-of-freedom fuel storage tanks subjected to stochastic seismic excitations. Therefore, we propose to optimize a finite locally resonant metafoundation equipped with massive resonators and fully nonlinear hysteretic devices. The optimization process takes into account the stochastic nature of seismic records in the stationary frequency domain; the records are modelled with the power spectral density S0 and modified with a Kanai–Tajimi filter. Moreover, the massive superstructure of a fuel storage tank is also considered in the optimization procedure. To optimize the nonlinear behaviour of dampers, we use a Bouc–Wen hysteretic model; the relevant nonlinear differential equations are reduced to a system of linear equations through the stochastic equivalent linearization technique. The optimized system is successively verified against natural seismic records by means of nonlinear transient time history analyses. Finally, we determine the dispersion relations for the relevant periodic metafoundation.

A Design Method for Viscous Dampers Connecting Adjacent Structures

This paper investigates the seismic design of fluid viscous dampers connecting adjacent structural systems. A simplified dampers design strategy is proposed, which relies on a linearized reduced order model of the coupled system. A stochastic linearization technique is adopted with the aim of extending the design method to nonlinear viscous dampers. The effectiveness of the design method and of the coupling strategy are assessed via numerical analysis of two adjacent buildings with shear-type behaviour connected by linear or nonlinear viscous dampers and subjected to Gaussian stochastic base acceleration. Different dampers locations are analysed. The accuracy of the reduced order model is assessed, by comparing the relevant response statistics to those provided by a refined multi degree of freedoms model. Finally, a parametric study is performed to assess the effectiveness of dissipative connection for different values of seismic intensity and dampers parameters (i.e., viscous coefficients and velocity exponents).

Earthquake protection of structures with nonlinear viscous dampers optimized through an energy-based stochastic approach

The earthquake protection of structures equipped with energy dissipation devices in the form of nonlinear fluid viscous dampers (FVDs) is investigated. Most of the optimal design strategies from the literature either address a simplified linear (Newtonian) idealization of the devices, or identify the characteristics of the nonlinear FVDs in a later stage, by invoking the concept of “energy-equivalent” dampers to compromise between the nonlinear power law force-velocity behavior and a simplified (equivalent, in terms of energy dissipation) linear modeling. In this paper, the nonlinear power law behavior of the devices is incorporated a priori in the optimal design process. The proposed strategy, based upon a numerical approach to a constrained optimization problem, invokes a performance criterion that is derived from the energy balance equation of the system, expressed in stochastic terms. To handle the nonlinear constitutive behavior of the FVDs, a novel equal-energy non-Gaussian stochastic linearization technique is integrated in the optimal design process. For a given power-spectral-density function of the seismic excitation, the most effective set of nonlinear FVDs that maximize the energy dissipation behavior can be identified. By stochastic dynamic analysis and by nonlinear response-history-analysis with an ensemble of ground motions, the proposed energy-based design philosophy is found to be better able to control the overall seismic response of the structure than alternative procedures that are not based on energy concepts and that minimize other performance indices.

Optimal design and seismic performance of tuned mass damper inerter (TMDI) for structures with nonlinear base isolation systems

SummaryThe tuned mass damper inerter (TMDI) couples the classical tuned mass damper (TMD) with an inerter, a mechanical device whose generated force is proportional to the relative acceleration between its terminals, thus providing beneficial mass‐amplification effects. This paper deals with a dynamic layout in which the TMDI is installed below the isolation floor of base‐isolated structures in order to enhance the earthquake resilience and reduce the displacement demand. Unlike most of the literature studies that assumed a linearized behavior of the isolators, the aim of this paper is to investigate the effectiveness of the TMDI while accounting for the nonlinearity of the isolators. Two nonlinear constitutive behaviors are considered, a Coulomb friction model and a Bouc‐Wen hysteretic model, representative of friction pendulum and of lead‐rubber‐bearing isolators, respectively. Optimal design is based on the stochastic dynamic analysis of the system, by modeling the base acceleration as a Kanai‐Tajimi filtered stationary random process and resorting to the stochastic linearization technique to handle the nonlinear terms. Different tuning criteria based on displacement, acceleration, and energy‐based performance indices are defined, and their implications in a design process are discussed. It is proven that the improved robustness of the TMDI reduces its performance sensitivity to the tuning frequency and to the earthquake frequency content, which are well‐known shortcomings of TMD‐like systems. This important feature makes the TMDI particularly suitable for nonlinear base‐isolated structures that are affected by unavoidable uncertainties in the isolators' properties and that may experience changes of isolators effective stiffness depending on the excitation level.

Improved stochastic linearization technique for structures with nonlinear viscous dampers

The stochastic linearization technique (SLT), in its conventional force-based Gaussian version (FB-G-SLT), is widely adopted in the literature to handle the nonlinear power-law constitutive behavior of fluid viscous dampers (FVDs), either for evaluating the system response, or within optimal design strategies. In this note, the authors prove that, for the kind of nonlinearity induced by FVDs, this version of FB-G-SLT is not the best choice, especially if the velocity-related response is of major importance in an optimal design process. As an example, the energy dissipated by the devices depends upon the square of the velocity at the ends of the dampers. Among six different formulations of SLT examined in this note, an equal-energy non-Gaussian SLT (EE-NG-SLT) is found to be superior to the FB-G-SLT since it more satisfactorily captures the nonlinear response of the structure as predicted by Monte Carlo simulations. This alternative EE-NG-SLT, without requiring higher computational effort than the FB-G-SLT, is therefore recommended, especially when energy-based (or, in general, velocity-related) objective functions are used in the optimization problem of nonlinear viscous dampers.

Asymptotic trends in time-varying oscillatory period for a dual-staged torsional system

The primary goal of this article is to propose a new analysis tool that estimates the asymptotic trends in the time-varying oscillatory period of a non-linear mechanical system. The scope is limited to the step-response of a torsional oscillator containing a dry friction element and dual-staged spring. Prior work on the stochastic linearization techniques is extended and modified for application in time domain. Subsequently, an instantaneous expected value operator and the concept of instantaneous effective stiffness are proposed. The non-linear system is approximated at some instant during the step-response by a linear time-invariant mechanical system that utilizes the instantaneous effective stiffness concept. The oscillatory period of the non-linear step-response at that instant is then approximated by the natural period of the corresponding linear system. The proposed method is rigorously illustrated via two computational example cases (a near backlash and near pre-load non-linearities), and the necessary digital signal processing parameters for time domain analysis are investigated. Finally, the feasibility and applicability of the proposed method is demonstrated by estimating the softening and hardening trends in the time-varying oscillatory period of the measured response for two laboratory experiments that contain clearance elements and multi-staged torsional springs.

Correlation between quasi-static and dynamic experiments for a practical torsional device with multiple discontinuous nonlinearities

Vehicle clutch dampers belong to a family of torsional devices or isolators that contain multi-staged torsional springs, pre-load features, clearances, and multi-staged dry friction elements. Estimation of elastic and dissipative parameters is usually carried out under quasi-static loading and then these static parameters are often assumed when predicting dynamic responses. For the purpose of comparison, this article proposes a new time domain parameter estimation method under dynamic, transient loading conditions. The proposed method assumes a priori knowledge of few nonlinear features based on the design and quasi-static characterization. Angular motion measurements from a component-level laboratory experiment under dynamic loading are utilized. Elastic parameters are first estimated through an instantaneous stochastic linearization technique. A model-based approach and energy balance principle are employed to estimate a combination of viscous and Coulomb damping parameters for seven local (stage-dependent) and global damping formulations for a practical device. The proposed method is validated by comparing time domain predictions from nonlinear models to dynamic measurements. Nonlinear models that utilize the proposed damping formulations are found to be superior to those that solely rely on parameters from a quasi-static experiment.

Generalization of Seide’s problem by the regulated stochastic linearization technique

In this paper, the nonlinear random vibration of a uniform beam clamped at both of its ends is investigated by the regulated stochastic linearization technique. A closed system of nonlinear algebraic equations for linearization coefficients is obtained using the frequency–response function matrix. Root mean-square response of maximum deflection of the beam that is obtained from the present technique is subsequently compared with that furnished by the conventional linearization. The numerical results show that the regulated stochastic linearization technique constitutes an excellent alternative to the classical linearization scheme for analyzing responses of the clamped–clamped beam.

Sixty years of stochastic linearization technique

Stochastic linearization technique is a versatile method of solving nonlinear stochastic boundary value problems. It allows obtaining estimates of the response of the system when exact solution is unavailable; in contrast to the perturbation technique, its realization does not demand smallness of the parameter; on the other hand, unlike the Monte Carlo simulation it does not involve extensive computational cost. Although its accuracy may be not very high, this is remedied by the fact that the stochastic excitation itself need not be known quite precisely. Although it was advanced about six decades ago, during which several hundreds of papers were written, its foundations, as exposed in many monographs, appear to be still attracting investigators in stochastic dynamics. This study considers the methodological and pedagogical aspects of its exposition.