Equivalent Linearization Research Articles

Damping-Ductility Relationships for Flag-Shaped Hysteresis

AbstractDamping-ductility relationships, as applied in direct displacement-based design (DDBD), are products of equivalent linearization techniques used to estimate the nonlinear response of struct...

Simplified Criteria to Select Ground Response Analysis Methods for Seismic Building Design: Equivalent Linear versus Nonlinear Approaches

ABSTRACT The possible amplification of seismic waves in soil deposits is crucial for the seismic design of buildings and geotechnical systems. The most common approaches for the numerical simulation of seismic site response are the equivalent linear (EQL) and the nonlinear (NL). Even though their advantages and limitations have been investigated in several studies, the relative field of applicability is still under debate. This study tested both methods over a wide population of soil models, which were subjected to a set of acceleration time histories recorded from strong earthquakes. A thorough comparison of the results of the EQL and the NL approaches was carried out, to identify the conditions in which the relative differences are significant. This assessment allowed for the definition of simplified criteria to predict when the two schemes are or are not compatible for large expected shaking levels. The proposed criteria are based on simple and intuitive parameters describing the soil deposit and the ground-motion parameters, which can be predicted straightforwardly. Therefore, this study provides a scheme for the choice between the EQL and the NL approaches that can be used even at the preliminary design stages. It appears that the EQL approach provides reliable amplification estimates in soil deposits with thickness up to 30 m, except for very deformable soils, but this depth range may be extended at long vibration periods. This result reveals a good level of reliability of the EQL approach for various soil conditions encountered in common applications, even for high-intensity shaking.

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.

Research and analysis of resonant and stiffness of cylindrical dielectric elastomer actuator

Dielectric elastomer actuators draw great interest in the emerging technology of soft actuations. The drifting of resonance frequency and variable stiffness are commonly existed in soft actuations. This work studied the dynamic performances of a cylindrical dielectric elastomer actuator with coupled loads of tensile forces and variable voltages. The equation of motion for the axial are derived from Euler–Lagrange equation and are reduced into linear ordinary differential equations by the weighted average equivalent linearization method. It is found that the resonant frequency as well as the dynamic behavior can be tuned by variable electrical stimulus on the actuator. An analytical solution of the resonant frequency and equivalent stiffness have been derived in this paper, and have been verified by numerical simulations and experimental measurements. By changing the excitation voltages, the stiffness can be tuned as variable and controllable, which paves the way for future applications of the DE actuators with better performances and resilience.

Experimental dynamic characterization of friction brace dampers and application to the seismic retrofit of RC buildings

Previous earthquakes have caused extensive damage to reinforced concrete (RC) structures with insufficient lateral force resistance or energy dissipation capacity. There is a need to retrofit vulnerable existing RC buildings, particularly those not originally designed for seismic effects or designed to an outdated seismic specification. This study investigates the use of friction dampers as displacement-dependent energy dissipation devices to retrofit RC moment frame structures. First, an experimental program was conducted to characterize the dynamic behavior of friction brace dampers with several different materials, finding that a sintered metal compound provided a stable friction coefficient of 0.4. A strength-based equivalent linearization design procedure was then developed based on the required friction slip force and considering the cracked state of the existing RC structure. A four-story RC school building was then designed using the proposed retrofit design method, validated using nonlinear response history analysis and compared to a previous retrofit scheme that employed buckling-restrained braces (BRBs) and a stiffness-based equivalent linearization design method. The analysis results suggest that the proposed retrofit design method and friction brace dampers are effective in reducing the maximum story drift.

Evaluation of critical separation distance to avoid seismic pounding between buildings: A spectral approach

AbstractDetermining a minimum separation distance required to avoid seismic pounding between adjacent buildings is a capacity‐demand design issue. This study developed a spectral approach to evaluate the expected minimum separation demand (referred to as the critical separation distance, [CSD]) to prevent pounding between buildings during strong earthquake motions. Presented firstly in this paper is an analytical procedure for evaluating the relative displacement process at the potential pounding location between two adjacent buildings during earthquake motions. The procedure is based on random vibration theory and modal analysis with application of the well‐known CQC rule. Subsequently, an approximate nonlinear closed‐form solution for evaluating CSD is derived, by means of a simplified spectral‐based procedure with introducing the concept of equivalent linearization of nonlinear hysteretic behavior and the effective correlation coefficients of the generalized coordinates of equivalent linearized systems to consider inelastic vibration phase between buildings. Finally, the CSDs between two buildings having different heights and the same height for various combinations of buildings with different fundamental periods are separately examined as verification examples of the proposed approach. The accuracy of solution is demonstrated by comparing computed results with corresponding numerical results from nonlinear response history analyses (RHAs). The results show that the proposed approach provides a reliable solution in predicting the separation demand of buildings during earthquake motions.

Structural optimal hybrid control strategies employing dynamic dual units: inerter and spring

AbstractThe concepts of optimal hybrid control strategies in the time and frequency domains are proposed herein. The time‐domain optimal hybrid control (also called GangGang strategy) consists of two dynamic units, an inerter and a spring, which work with different force mechanisms at different time phases of vibrations. The inerter acts on the structure when the vibration phase is accelerated, whereas the spring works when the structure deviates from its initial position. The combination of the two piecewise working units achieves better performance than their individual full‐time operations. The working time phases of the two units are complementary, covering the entire vibration period. The hysteresis loops of them are also complementary in the quadrant coordinate system. Owing to the piecewise properties of the two units, the equivalent linearization system (ELS) of such systems is formulated, and a statistical linearization procedure is developed to obtain the equivalent linearization parameters. Then, a parametric study on the equivalent damping is performed through sensitivity analysis and a comparative analysis of the ELS was performed in terms of the standard deviation and peak displacement levels under stationary stochastic excitations. The frequency‐domain optimal hybrid control conforms with the concept of intelligent isolation. The inerter and spring units can be switched according to the excitation frequency by an on–off control benefitting structures subjected to excitations with definite frequency range.

Prescribed performance model-free adaptive sliding mode control of a shape memory alloy actuated system

This paper concentrates on a simple and robust control method for the discrete time nonlinear systems to fulfill the requirement of predefined accuracy. A sliding mode control method is designed by introducing equivalent dynamic linearization technique according to the input/output (I/O) information merely. A square-root type error transformation method is presented for the tracking error to be restricted within a preassigned zone. The performance of presented control method is demonstrated through experiments on a nonlinear system. Experiment results show that the presented control method has a superior tracking accuracy compared with PID controller and model-free adaptive control (MFAC).

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.

Data-driven sliding mode control of shape memory alloy actuators with prescribed performance

This paper proposes a data-driven sliding mode control (SMC) method with tracking error constraint for a shape memory alloy (SMA) actuated system. More specifically, a new SMC method with prescribed performance is investigated by adopting the equivalent dynamic linearization technique and a novel transformed error algorithm to guarantee that the tracking error can converge into a preassigned region. Meanwhile, the proposed control method can effectively ensure the convergence rate and the steady-state error within a prescribed bound, merely depending on the input/output data without implicit or explicit information of the systems, which is more valuable for the practical application. The presented control method has been implemented on an SMA actuated system successfully and experimental results have shown that the proposed control is easily applicable and highly precise. In addition, the tracking error of the SMA actuated system can be constrained in a preassigned domain by the proposed control method.

Shaking table test and numerical simulation on ultra-large diameter shield tunnel passing through soft-hard stratum

In this study, the seismic behavior of a shield tunnel with an ultra-large diameter of 15 m passing through a soft-hard stratum was investigated, using a series of 1/30 scaled shaking table model tests and numerical simulations. A modified similitude-ratio design method was proposed, with the soil–structure relative stiffness as the main control factor. The model tunnel was made of a plexiglass tube. The soft soil was modeled with silty clay, whereas the hard soil was modeled with fine and angular gravel. The tests were also simulated using the finite element modeling (FEM) software ABAQUS, and equivalent linearization was employed for the soil nonlinearity in the frequency domain. The results show that the modified similitude-ratio design method is workable and effective. Moreover, it may provide additional possibilities regarding the selection of structural model materials, and broaden the applicability of a shaking table with a conventional loading performance. The strain of the tunnel around the soft-hard interface increases significantly, and the maximum strain occurs within the range of 1 D (D is the outer diameter of the tunnel) in the soft soil near the interface along the direction of tunnel's length. In one observation section, the largest strain appears at the crown or bottom, followed by those at the spandrel, haunch, and knee. The region affected by the soft-hard interface is approximately 1 D in the hard soil and 2 D in the soft soil. The accelerations in the hard and soft soil are amplified, and the acceleration amplification factors gradually decrease with an increase in the excitation amplitude. The peak ground acceleration in the soft soil is smaller than that in the hard soil in the case with a larger excitation amplitude. The tunnel is curved in the vertical direction, owing to the soft-hard stratum. This study may provide a reference for shaking table tests for ultra-large diameter shield tunnels at complex sites.

A Double Shape Memory Alloy Damper for Structural Vibration Control

Shape memory alloy (SMA) dampers are widely investigated passive control systems for structural vibration mitigation. However, the damping robustness of conventional austenite SMA dampers may be affected by environmental temperature. In this study, an innovative double SMA damper (DSD) system is presented to improve the temperature robustness of the SMA dampers. In the proposed system, double SMA hysteretic elements with different phase transition temperatures are arranged in parallel, where the SMA element with lower transition temperature behaves as austenite under room temperature, and the other with higher transition temperature behaves as martensite. To study the vibration control effect, both single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) structures with DSD systems are employed. The thermal and mechanical behaviors of the SMA elements and the working principle of DSD are also introduced. Thereon, the equivalent linearization method for SMA’s output force and the motion-governing equations for SDOF structure with DSD are derived. Moreover, parametric studies are conducted to investigate the performance of the proposed DSD system in both frequency and time domains. Also, numerical analysis for the MDOF structure with DSD systems is carried out to illustrate the trend in response reduction with an increasing number of degrees of freedom. The analytical results show that the DSD can mitigate the structural seismic response more effectively than the conventional one with acceptable residual deformation, and is capable of delaying the degradation of SMA’s energy dissipation capacity. Less SMA material is required for the proposed DSD to fulfill the same mitigation requirement, and it is suitable for general applications for temperature robustness.

Theoretical determination of asymmetric rolling amplitude in irregular beam seas

A methodology for predicting the probability density function of roll motion for irregular beam seas is developed in the author’s previous research. In this paper, two methods for the prediction of the probability density function (PDF) of the rolling amplitude are examined. One of these methods is an approach based on a non-Gaussian PDF with the use of the equivalent linearization and the moment method, which has been used by Maki in the field of naval architecture. In the framework of this method, the instantaneous joint PDF of the roll and roll rate can be calculated. Thereby, the transformation from the joint PDF to the PDF of the total energy H is necessary. In this paper, the transformation from the joint PDF of instantaneous roll and rate to the PDF of H was conducted. The other method is the energy-based stochastic averaging method, whereby the PDF of the total energy H is determined from which the roll amplitude is calculated. It is noteworthy that the proposed theory aims to predict the PDF of the amplitude of the vessel due to wind or flooding. For such conditions, the restoring curve (GZ curve) has asymmetricity. To take into account this asymmetricity, a new theory for the transform of the PDF of the total energy H to the PDFof amplitude is proposed. The obtained theoretical results show almost complete agreement, and both of the two theoretical methods i.e., the energy-based stochastic averaging method and the non-Gaussian PDF method with the use of the equivalent linearization and moment method, show equivalent prediction performance even in the asymmetric condition.

REALIZATION OF GROUND MOTION INPUT IN ABAQUS FOR LAYERED FOUNDATION UNDER SV WAVE OF OBLIQUE INCIDENCE OVER CRITICAL ANGLE

In the finite element numerical simulation, it is an urgent challenge to determine the ground motion input for the layered foundation under SV wave of oblique incidence over critical angle. Exact dynamic stiffness matrix in the frequency domain (i.e., frequency domain stiffness matrix method) of layered foundation is used to derive the formula for calculating the equivalent node force of ground motion input under arbitrary angular oblique incidence of SV wave, and the validity and accuracy of the ground motion input applied to ABAQUS numerical simulation are demonstrated by simulating seismic wave field of homogeneous half-space and layered foundation under oblique incidence of SV wave through ABAQUS software. It is shown that the inclined SV wave incidence with an arbitrary angle can be realized through ABAQUS simulation by means of stiffness matrix method, and the method is of high accuracy, especially in the case of SV wave incidence with an inclined angle greater than the critical one. The oval like motion trails of surface particles in uniform half space and the wave motion characteristics of layered half space are simulated commendably by finite element numerical modelling. On the above basis, the frequency domain stiffness matrix method is further combined with equivalent linearization method, which solves the ground motion input problem of two-dimensional layered half space considering soil nonlinearity for the inclined incidence with arbitrary angle.

Stationary Response of Nonlinear Vibration Energy Harvesters by Path Integration

Abstract A path integration procedure based on Gauss–Legendre integration scheme is developed to analyze probabilistic solution of nonlinear vibration energy harvesters (VEHs) in this paper. First, traditional energy harvesters are briefly introduced, and their nondimensional governing and moment equations are given. These moment equations can be solved through the Runge–Kutta and Gaussian closure method. Then, the path integration method is extended to three-dimensional situation, solving the probability density function (PDF) of VEH. Three illustrative examples are considered to evaluate the effectiveness of this method. The effectiveness of nonlinearity of traditional monostable VEH is studied. The bistable VEH is further studied too. At the same time, equivalent linearization method (EQL) and Monte Carlo simulation (MCS) are employed. The results indicate that three-dimensional path integration method can give satisfactory results for the global PDF, especially when solving bistable VEH problems. The results of this method have better consistency with the simulation results than those of EQL. In addition, different degrees of hardening and softening behaviors of PDFs occur when the magnitude of nonlinearity coefficient increases or the bistable VEH is considered.

Seismic Performance Evaluation of Steel Buildings with Oil Dampers Using Capacity Spectrum Method

This study proposes a capacity spectrum Method (CSM)-based procedure to estimate the maximum seismic performance of steel buildings passively controlled with bilinear oil dampers. In the proposed CSM, the maximum seismic response of a building was estimated, in the acceleration-displacement response spectrum, as the intersection between the capacity curve and the damping-adjusted demand curves, using the equivalent linearization method. The building equivalent damping ratio was determined by the sum of the inherent damping, and the square root of sum of squares (SRSS) of the hysteretic damping and the viscous damping of the supplemental oil devices. The calculation steps of the proposed CSM are explained in detail based on the equivalent single degree of freedom (ESDOF) system, and its accuracy was examined by comparison with time history analysis (THA) results. Two model steel buildings of 4 and 10 stories, uniformly equipped with oil dampers along the height, were subjected to six selected earthquake ground motions scaled to be compatible with Level-2 earthquakes, as defined in the Japanese Building Standard Law. The seismic performance of the buildings was estimated by the proposed CSM procedure and compared with the results of nonlinear THA in terms of the maximum story displacements and the shear forces. It was observed that the proposed CSM scheme provided a satisfactory accuracy to assess the maximum nonlinear response of steel buildings passively controlled with oil dampers.

Displacement Demand for Nonlinear Static Analyses of Masonry Structures: Critical Review and Improved Formulations

This paper discusses different formulations for calculating earthquake-induced displacement demands to be associated with nonlinear static analysis procedures for the assessment of masonry structures. Focus is placed on systems with fundamental periods between 0.1 and 0.5 s, for which the inelastic displacement amplification is usually more pronounced. The accuracy of the predictive equations is assessed based on the results from nonlinear time-history analyses, carried out on single-degree-of-freedom oscillators with hysteretic force–displacement relationships representative of masonry structures. First, the study demonstrates some limitations of two established approaches based on the equivalent linearization concept: the capacity spectrum method of the Dutch guidelines NPR 9998-18, and its version outlined in FEMA 440, both of which overpredict maximum displacements. Two codified formulations relying on inelastic displacement spectra are also evaluated, namely the N2 method of Eurocode 8 and the displacement coefficient method of ASCE 41-17: the former proves to be significantly unconservative, while the latter is affected by excessive dispersion. A non-iterative procedure, using an equivalent linear system with calibrated optimal stiffness and equivalent viscous damping, is then proposed to overcome some of the problems identified earlier. A recently developed modified N2 formulation is shown to improve accuracy while limiting the dispersion of the predictions.

Equivalent Linearization of Bladed Disk Assemblies With Friction Nonlinearities Under Random Excitation

AbstractThis article addresses the vibrational behavior of bladed disk assemblies with nonlinear shroud coupling under random excitation. In order to increase the service life and safety of turbine blades, intense calculations are carried out to predict the vibrational behavior. The use of friction dampers for energy dissipation and suppression of large amplitudes makes the mechanical system nonlinear, which complicates the calculations. Depending on the stage, different types of excitation can occur in a turbine, from clearly defined deterministic to random excitation. So far, the latter problem has only been dealt with to a limited extent in the literature on turbomachinery. Nevertheless, there are in general different approaches and methods to address this problem most of which are strongly restricted with regard to the number of degrees-of-freedom (DOF). The focus of this paper is the application of an equivalent linearization method (ELM) to calculate the stochastic response of an academic model of a bladed disk assembly under random excitation. The nonlinear contact is modeled both with an elastic Coulomb-slider and a Bouc–Wen formulation to reproduce the hysteretic character of a friction nonlinearity occurring in the presence of a friction damper. Both the excitation and the response are limited to mean-free, stationary stochastic processes, which means that the stochastic moments do not change over time. Unlike previous papers on this topic, the calculations are performed on a full bladed disk assembly in which each segment is approximated with several degrees-of-freedom.

Data-Driven Iterative Learning Control for Nonlinear Discrete-Time MIMO Systems.

This article considers the tracking control of unknown nonlinear nonaffine repetitive discrete-time multi-input multi-output systems. Two data-driven iterative learning control (ILC) schemes are designed based on two equivalent dynamic linearization data models of an unknown ideal learning controller, which exists theoretically in the iteration domain. The two control schemes provide ways of selecting learning controllers based on the complexity of the controlled nonlinear systems. The learning control gain matrixes of the two learning controllers are optimized through the steepest descent method using only the measured input-output data of the nonlinear systems. The proposed ILC approaches are pure data-driven since no model information of the controlled systems is involved. The stability and convergence of the proposed ILC approaches are rigorously analyzed under reasonable conditions. Numerical simulation and an experiment based on a Gantry-type linear motor drive system are conducted to verify the effectiveness of the proposed data-driven ILC approaches.

A multi-body control approach for flapping wing micro aerial vehicles

Flapping wing micro aerial vehicle (FWMAV) has a multi-body and periodic dynamics, which is influenced by unsteady aerodynamics. These features make it more difficult to control. Ignoring the wing inertia, dynamics averaging, and using a simple aerodynamic model are the simplifying assumptions for conventional model-based control, although they may result in inaccurate control. To overcome these difficulties, a multi-body control is proposed based on a model-free adaptive variable structure control (MFAVSC) approach. MFAVSC takes advantage of input/output data, while not including any explicit model information. At first, the nonlinear FWMAV dynamics is transformed into an equivalent dynamic linearization description with a concept called pseudo-partial derivative (PPD). After estimating the PPD matrix, model-free adaptive control law is designed based on the optimal criteria. Then, it is augmented by a variable structure control term to guarantee the stability, as well as speeding up its convergence. Finally, simulation results demonstrate the effectiveness of the proposed scheme to trajectory control of the FWMAV.