Stochastic Linearization Method Research Articles

Pendulum tuned mass damper (PTMD) with geometric nonlinear dampers for seismic response control

In engineering applications, the geometric nonlinear (GN) arrangement of pendulum-tuned mass dampers (PTMDs) may induce the damping force to deviate from its intended state. Therefore, this study extends the scope of the structural control strategy by introducing a pendulum-tuned mass damper considering geometric nonlinear dampers (GND). Specifically, the damper developed in this study, termed PTMD with GND, is an inconspicuous pendulum-type tuned mass damper with different GN settings in the initial state. A nonlinear mathematical model of the PTMD with GND is developed, identifying its significant nonlinear characteristics, particularly the "unavailable" locations. The stochastic linearization method (SLM) is adopted to simplify the nonlinear model using the Kanai-Tajimi modified spectrum. The SLM agrees well with the numerical results obtained using the Monte Carlo simulation method. Furthermore, the control strategy of PTMD with GND is proposed with the consideration of filtered random excitation. Several numerical cases have been conducted to investigate the availability of PTMD with GND and the proposed optimal design strategy under 178 real seismic excitations. The results show that the PTMD with GND is nearly as effective as the ideal transitional TMD in mitigating the seismic response of the primary structure and more effective than the inverted rough linear adjustment. It can be concluded that the proposed optimal design strategy for the PTMDs with GND efficiently suppressed the structural vibration through multiple performance indices and used the nonlinearity of the damping mechanism of the GND to provide a sufficient damping energy dissipation effect adaptively.

Simultaneous Suspension Control and Energy Harvesting Through Novel Design and Control of a New Nonlinear Energy Harvesting Shock Absorber

Simultaneous vibration control and energy harvesting of vehicle suspensions have attracted significant research attention over the past decades. However, existing energy harvesting shock absorbers (EHSAs) are mainly designed based on the principle of linear resonance, thereby compromising suspension performance for high-efficiency energy harvesting and being only responsive to narrow bandwidth vibrations. In this paper, we propose a new EHSA design -- inerter pendulum vibration absorber (IPVA) -- that integrates an electromagnetic rotary EHSA with a nonlinear pendulum vibration absorber. We show that this design simultaneously improves ride comfort and energy harvesting efficiency by exploiting the nonlinear effects of pendulum inertia. To further improve the performance, we develop a novel stochastic linearization model predictive control (SL-MPC) approach in which we employ stochastic linearization to approximate the nonlinear dynamics of EHSA that has superior accuracy compared to standard linearization. In particular, we develop a new stochastic linearization method with guaranteed stabilizability, which is a prerequisite for control designs. This leads to an MPC problem that is much more computationally efficient than the nonlinear MPC counterpart with no major performance degradation. Extensive simulations are performed to show the superiority of the proposed new nonlinear EHSA and to demonstrate the efficacy of the proposed SL-MPC.

Stochastic seismic response of building with shape memory alloy damper

The present study focuses on the estimation of response of building structure supplemented with the superelastic shape memory alloy (SMA) damper under the stochastic seismic excitation. To this end, the stochastic response has been determined using the stochastic linearization method under random earthquakes and the control efficiency of the SMA damper is compared with the yield damper. As the response output, top floor root mean square (rms) acceleration and displacement are presented here in this study. Response analysis results show the presence of optimal values of damper strength which minimizes the responses. As observed, in this optimal condition of damper strength, compared to the yield damper, the SMA damper provided 17 % and 49 % improvements in the values of top floor rms acceleration and displacement. In the end, the parametric study has been performed under the varying strength and stiffness of the damper, time period of the structure, and earthquake excitation to establish the superior control efficiency of SMA damper over yield damper.

The stationary response of piezoelectric cantilever beam model excited by colored noise

This article provides a method to predict the steady-state response of a piezoelectric cantilever under the excitation of Gaussian colored noise which is a better approximation to the ambient random excitation than the white noise. In this paper, a cantilever energy harvester model, which is fully-paved with piezoelectric ceramic and excited by Gaussian colored noise, is built by the Lagrange’s equations and is nondimensionalized. Then the dynamic equation and piezoelectric equation are transformed into a pair of stochastic differential equations about transient equivalent amplitude and transient phase. After that, the stochastic averaging method is applied to simplify the stochastic differential equations into an Itô type equation about the equivalent amplitude, the drift coefficient and the diffusion coefficient are obtained. This method provides another approach which is different from the stochastic linearization method and moment method. On this basis, some crucial outputs such as the steady-state probability density function (PDF) of the equivalent amplitude, the beam’s displacement, the transient output voltage, the joint probability density function of displacement and velocity as well as the mean square value of output voltage are all obtained. With different noise intensity values and delay factors, the Monte Carlo numerical simulation has verified the correctness of the theoretical prediction.

Exponential mean-square stability properties of stochastic linear multistep methods

The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

Time and Frequency Domain Dynamic Analysis of Offshore Mooring

The numerical procedures for dynamic analysis of mooring lines in the time domain and frequency domain were developed in this work. The lumped mass method was used to model the mooring lines. In the time domain dynamic analysis, the modified Euler method was used to solve the motion equation of mooring lines. The dynamic analyses of mooring lines under horizontal, vertical, and combined harmonic excitations were carried out. The cases of single-component and multicomponent mooring lines under these excitations were studied, respectively. The case considering the seabed contact was also included. The program was validated by comparing with the results from commercial software, Orcaflex. For the frequency domain dynamic analysis, an improved frame invariant stochastic linearization method was applied to the nonlinear hydrodynamic drag term. The cases of single-component and multicomponent mooring lines were studied. The comparison of results shows that frequency domain results agree well with nonlinear time domain results.

Seismic response control of adjacent high-rise buildings linked by the Tuned Liquid Column Damper-Inerter (TLCDI)

A novel passive control device termed Tuned Liquid Column Damper-Inerter (TLCDI) is adopted to control the seismic response of adjacent high-rise buildings. The inerter of the proposed TLCDI scheme exploits the large relative acceleration between the two adjacent buildings, producing a significant resisting force. Firstly, the governing nonlinear equations of motion of the TLCDI-linked adjacent high-rise buildings under earthquake excitation are formulated. For design purposes, the inherent nonlinear behavior of the TLCDI is linearized by adopting an equivalent stochastic linearization method. Subsequently, a benchmark structure consisting of two adjacent high-rise buildings linked by a corridor with a roller support and equipped with a TLCDI is examined. Eight characteristic parameters of the TLCDI are optimized through two distinct constrained multi-objective optimization problems by an evolutionary algorithm aimed at minimizing either the peak acceleration or the interstory-drift ratio of the two adjacent benchmark structures, respectively. Mitigation effects of the optimally designed TLCDIs on the responses of the benchmark structure are assessed considering a set of ten natural ground-motion records. It is found that the optimum TLCDI achieves mean reduction ratios of 15.7 and 26.1% on the peak acceleration of both the buildings, respectively. Conversely, the TLCDI fails to significantly reduce the interstory drift ratios. Robustness analysis of the optimally designed TLCDI is investigated under perturbations to the frequency and damping ratios. Some practical considerations are finally discussed concerning the feasibility of employing the proposed TLCDI in the considered benchmark project.

Optimization of SMA based damped outrigger structure under uncertainty

Outriggers have been proven to be an efficient system to reduce the dynamic responses of core-tube type high-rise buildings by utilizing the axial stiffness of the peripheral columns. However, outer columns and outriggers are subjected to excessive lateral force demands. To reduce the demand, the damped outrigger is used where dampers are installed between the perimeter columns and outriggers. To improve the performance and enhance the residual deformation, in this paper, the use of shape memory alloy (SMA) springs is introduced to dissipate energy. The SMA dissipates energy through a hysteretic phase transformation of its microstructure triggered by cyclic loading. In the mathematical model implementation, the nonlinear behavior of superelastic SMA is linearized by the stochastic equivalent linearization method. Stochastic uncertainty in the ground motion is considered, and the performance is assessed by minimizing the failure probability of the structure. To solve the optimization problem, Kriging is used as a surrogate model. For better accuracy of the surrogate model, an efficient global optimization scheme is used. The purpose of the optimization adopted in this study is to find optimal design parameters associated with the initial stiffness of the SMA and the location of the outrigger. The results clearly demonstrated the efficacy of the proposed SMA based outrigger.

Stochastic proximal linear method for structured non-convex problems

In this work, motivated by the challenging task of learning a deep neural network, we consider optimization problems that consist of minimizing a finite-sum of non-convex and non-smooth functions, where the non-smoothness appears as the maximum of non-convex functions with Lipschitz continuous gradient. Due to the large size of the sum, in practice, we focus here on stochastic first-order methods and propose the Stochastic Proximal Linear Method (SPLM) that is based on minimizing an appropriate majorizer at each iteration and is guaranteed to almost surely converge to a critical point of the objective function, where we also proves its convergence rate in finding critical points.

Frequency Domain Modeling of a Halbach PM Linear Generator Based Two-Body Point Absorber for Wave Energy Conversion

Point absorbers utilize the relative motion between the floating body and the submerged body to harvest the wave energy. This paper presents the parametric study of a two-body wave energy point absorber to investigate the influence of key design factors. The targeting point absorber employs a slotless Halbach linear generator as the power take-off system to improve the energy conversion efficiency. In this work, we have linearized our developed nonlinear time domain model and presented it in frequency domain to improve the simulation efficiency, where the results of the time domain model and the frequency domain model have been compared. The stochastic linearization method is employed for the first time to linearize the nonlinear dragging effect of a point absorber. The case study suggests that the stiffness of the power take-off system and viscous floater dragging coefficient should be as small as possible and the optimal mass of the bodies should be adopted to achieve the maximum power output. Under the irregular wave condition, the power matrix against the wind speed and the damping coefficient of the power take-off system shows that the optimal damping coefficient is determined by the local weather and geographic condition. Regarding the determinants of the damping coefficient, the factors that affect the power take-off system is investigated. The Halbach PM array length and the coil width should be specially optimized to obtain the desired damping coefficient of the power take-off system at a given point absorber dimension. Lastly, a test-rig has been developed to validate the magnetic domain model.

Analysis of block random rocking on nonlinear flexible foundation

In this paper the rocking response of a rigid block randomly excited at its foundation is examined. A nonlinear flexible foundation model is considered accounting for the possibility of uplifting in the case of strong excitation. Specifically, based on an appropriate nonlinear impact force model, the foundation is treated as a bed of continuously distributed springs in parallel with nonlinear dampers. The statistics of the rocking response is examined by an analytical procedure which involves a combination of static condensation and stochastic linearization methods. In this manner, repeated numerical integration of the highly nonlinear differential equations of motion is circumvented, and a computationally efficient semi-analytical solution is obtained. Comparisons with pertinent Monte Carlo simulations demonstrate the efficiency and reliability of the proposed approach. Finally, the survival probability related to toppling of the blocks subject to filtered white noise excitation is investigated, for different kinds of block geometries, foundation materials, and filter parameters.

Influence of higher orders of Neumann expansion on accuracy of stochastic linear elastic finite element method with random physical parameters

Influence of higher orders of Neumann expansion on accuracy of stochastic linear elastic finite element method with random physical parameters

Displacement-Dependent Damping Inerter System for Seismic Response Control

Various inerter systems utilizing velocity-dependent damping for vibration control have been developed. However, a velocity-dependent damping element may exhibit relatively poor performance compared to a displacement-dependent damping element (DDE) of equivalent damping ratio, when the structural deformation is small in the early stage of the seismic response. To address this issue, the advantage of DDE in generating a larger control force in the early stage of excitation is promoted here and enhanced by a supplemental inerter-spring-system, thus realizing a proposed novel displacement-dependent damping inerter system (DDIS). First, the influence of various DDIS-parameters is carried out by resorting to the stochastic linearization method to handle non-linear terms. Then, seismic responses of the DDIS-controlled system are evaluated in the time domain taking the non-linearity into account, thus validating the accuracy of the stochastic dynamic analysis. Several design cases are considered, all of which demonstrated damping enhancement and timely control achieved by the DDIS. The results show that the energy dissipation as well as reduction of structural displacement and acceleration accomplished by the proposed system are significant. DDIS suppresses structural responses in a timely manner, as soon as the peak excitation occurs. In addition, it is demonstrated that interactions among the inerter, spring, and DDE, which constitute the damping-enhancement mechanism, lead to a higher energy-dissipation efficiency compared to the DDE alone.

Fleet Size and Rebalancing Analysis of Dockless Bike-Sharing Stations Based on Markov Chain

In order to improve the dynamic optimization of fleet size and standardized management of dockless bike-sharing, this paper focuses on using the Markov stochastic process and linear programming method to solve the problem of bike-sharing fleet size and rebalancing. Based on the analysis of characters of bike-sharing, which are irreducible, aperiodic and positive-recurrence, we prove that the probability limits the state (steady-state) of bike-sharing Markov chain only exists and is independent of the initial probability distribution. Then a new “Markov chain dockless bike-sharing fleet size solution” algorithm is proposed. The process includes three parts. Firstly, the irreducibility of the bike-sharing transition probability matrix is analyzed. Secondly, the rank-one updating method is used to construct the transition probability random prime matrix. Finally, an iterative method for solving the steady-state probability vector is therefore given and the convergence speed of the method is analyzed. Furthermore, we discuss the dynamic solution of the bike-sharing steady-state fleet size according to the time period, so as improving the practicality of the algorithm. To verify the efficiency of this algorithm, we adopt the linear programming method for bicycle rebalancing analysis. Experiment results show that the algorithm could be used to solve the disordered deployment of dockless bike-sharing.

An interval type-2 fuzzy stochastic approach for regional-scale electric power system under parameter uncertainty

ABSTRACTIn this study, an interval type-2 fuzzy stochastic linear programming method (IT2FSLP) is developed to support regional-scale electric power system (REM) planning. The IT2FSLP-REM model is based on an integration of interval type-2 fuzzy sets boundary programming and stochastic linear programming techniques enable it to have robust abilities to the deal with uncertainties expressed as type-2 fuzzy intervals and probabilistic distributions within a general optimization framework. Moreover, it can reflect dynamic decisions for energy supply and energy conversion processes, as well as provide capacity expansion options with multiple periods. The developed model is applied to a case of planning regional-scale energy and environmental systems to demonstrate its applicability. Based on a two-step solution algorithm, reasonable solutions have been obtained, which reflect tradeoffs among economic cost, environmental requirements, and energy-supply security. Thus, the lower and upper solutions of IT2FSLP-REM would then help energy authorities adjust or justify allocation patterns of regional energy resources and services.

A Stochastic Linear Programming Method for the Reliable Oil Products Supply Chain System With Hub Disruption

This paper focuses on a reliable design of oil products supply chain system, which is conductive to promote the sales of oil products. A multi-scenario MILP (Mixed Integer Linear Programming) model coupled with Monte Carlo sampling is employed for optimizing the supply chain system with considering the transportation process of oil products, the stochastic hub disruption and the demand uncertainty. A real-world case is presented to illustrate the accuracy, applicability and efficiency of the proposed model. High-quality solutions are obtained successfully under deterministic conditions as well as uncertain conditions. Then, the effect of uncertainties on the supply chain system design is also analyzed. Finally, the results demonstrate that the stability and flexibility of the designed supply chain system could be substantially improved with less extra costs.

Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations

The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for $\theta \in \left [0,\frac {1}{2}\right ]$ are stronger than those for $\theta \in \left (\frac {1}{2},1\right ]$ . The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability.

Eco-compensation standards for sustaining high flow events below hydropower plants

Abstract High flow events (including both the high flow pulses and floods) exert many critical ecological functions in riverine ecosystems and their elimination will lead to severe ecological degradation. However, common hydropower operations are only required to legally maintain the minimum environmental flows (e-flows). The extra release of high flow events will result in revenue loss of hydropower producers. In this research, we propose an approach to assessing the eco-compensation standard which is necessary to compensate the loss of hydropower producers and promote the release of high flows to downstream rivers. The scenario tree approach is adopted for the generation of possible scenarios of flows and electricity prices to deal with the uncertainties of the two factors in deregulated electricity markets. A stochastic linear programming method is proposed to determine the maximum mean annual revenues under two e-flow provision scenarios, i.e., only sustaining the minimum e-flows (scenario 1) and sustaining both the minimum e-flows and a specified number of high flow events (scenario 2). The revenue difference under the two e-flow scenarios is set as the eco-compensation standard. A case study in the Wangkuai reservoir, China, demonstrates that the eco-compensation standard is obviously influenced by the installed hydropower generation capacity (IHGC) and the number of high flow events planned to be released. Moreover, the greatest standards may be not corresponding to a very large IHGC.

Analysis of Adaptive Synchronization for Stochastic Neutral-Type Memristive Neural Networks with Mixed Time-Varying Delays

Linear feedback control and adaptive feedback control are proposed to achieve the synchronization of stochastic neutral-type memristive neural networks with mixed time-varying delays. By applying the stochastic differential inclusions theory, Lyapunov functional, and linear matrix inequalities method, we obtain some new adaptive synchronization criteria. A numerical example is given to illustrate the effectiveness of our results.

Edge statistics for a class of repulsive particle systems

We study a class of interacting particle systems on $$\mathbb {R}$$ which was recently investigated by Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi: 10.3842/SIGMA.2016.093 ; Schuler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schuler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).