Bilinear Hysteretic Oscillator Research Articles

Survival probability of nonlinear oscillators endowed with fractional derivative element and subjected to evolutionary excitation: A stochastic averaging treatment with path integral concepts

An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.

Using Approximate Bayesian Computation by Subset Simulation for Efficient Posterior Assessment of Dynamic State-Space Model Classes

Approximate Bayesian Computation (ABC) methods have gained in popularity over the last decade because they expand the horizon of Bayesian parameter inference methods to the range of models for which an analytical formula for the likelihood function might be difficult, or even impossible, to establish. The majority of the ABC methods rely on the choice of a set of summary statistics to reduce the dimension of the data. However, as has been noted in the ABC literature, the lack of convergence guarantees induced by the absence of a vector of sufficient summary statistics that assures intermodel sufficiency over the set of competing models hinders the use of the usual ABC methods when applied to Bayesian model selection or assessment. In this paper, we present a novel ABC model selection procedure for dynamical systems based on a recently introduced multilevel Markov chain Monte Carlo method, self-regulating ABC-SubSim, and a hierarchical state-space formulation of dynamic models. We show that this formulation makes it possible to independently approximate the model evidence required for assessing the posterior probability of each of the competing models. We also show that ABC-SubSim not only provides an estimate of the model evidence as a simple by-product but also gives the posterior probability of each model as a function of the tolerance level, which allows the ABC model choices made in previous studies to be understood. We illustrate the performance of the proposed framework for ABC model updating and model class selection by applying it to two problems in Bayesian system identification: a single-degree-of-freedom bilinear hysteretic oscillator and a three-story shear building with Masing hysteresis, both of which are subject to a seismic excitation.

Vibration analysis of nonlinear systems with the bilinear hysteretic oscillator by using incremental harmonic balance method

This paper considers dynamics of bilinear hysteretic systems, which are widely used for vibration control and vibration absorption such as magneto-rheological damper, metal-rubber. The method of incremental harmonic balance (IHB) technique that hysteresis is considered in the corrective term is improved in order to determine periodic solutions of bilinear hysteretic systems. The improved continuation method called two points tracing algorithm which is stable to the turning point makes the calculation more efficient for tracing amplitude-frequency response. Precise Hsu's method for analysing the stability of periodic solutions is introduced. The effects of different parameters of bilinear hysteretic oscillator on the response are discussed numerically. Some numerical simulations of considered bilinear hysteretic systems, including a single DOF and a 2DOF system, are effectively obtained by the modified IHB method and the results compare very well with the 4-oder Runge-Kutta method.

Nonlinear Oscillator Stochastic Response and Survival Probability Determination via the Wiener Path Integral

A Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first, the nonlinear oscillator is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, relying on the concept of the most probable path, a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman–Kolmogorov (C–K) equation is utilized for advancing the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered as numerical examples. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique.

Studies into Computational Intelligence and Evolutionary Approaches for Model‐Free Identification of Hysteretic Systems

AbstractThis article introduces a robust hybrid computational method for the data‐driven model‐free identification of nonlinear systems that exhibit hysteretic behavior. The proposed approach combines Genetic Programming, which incorporates discontinuous basis functions, for discovering the structure of the governing differential equations, Genetic Algorithms for optimizing the parameters of the differential equations, and Computer Algebra for simplifying mathematical expressions symbolically, and consequently, controlling bloat through condensing dependent terms. A similar technique has been previously proposed by the authors for the identification of nonhysteretic Single Degree of Freedom (SDOF) systems. That technique is extended in this article and is utilized to provide parsimonious differential equations that represent the Bouc–Wen model and the bilinear hysteretic oscillator—both exhibit abrupt change in their memory‐dependent response. The representative models are subjected to validation excitations that are substantially different from the probing signals to confirm the generalizability of the models for different dynamical phenomena. The results verify the effectiveness of the approach and the accuracy of the subsequent differential operators that characterize the behavior of the studied hysteretic systems even under new dynamical conditions. Beside the presented application of this approach, the introduced methodology is more general and can be employed across different disciplines, specifically in data analytics for mathematical modeling of various complex systems.

Modelling Earthquake Risk Based on Approximate Nonlinear Reliability Estimates

This study examines the reliability of a bilinear hysteretic oscillator as a means to model the seismic risk of a structure. Many seismic risk analyses have been performed using linear SDOF oscilators. The objective of the present work is to demonstrate that the use of a more complex oscillator model in seismic risk analyses not only alters the structural response, but significantly changes the perceived number of time domain simulations that are thought to be representative of earthquake dangers in the area. The present study compares the results of a spatial seismic risk analysis, based on the probability of a bilinear hysteretic oscillator exceeding a prescribed displacement level, to an analysis using a linear oscillator at the same site. The results of the analyses are presented spatially, that is, as a function of earthquake magnitude and site-to-source distance, and in terms of a Cumulative Earthquake Risk Function (CERF).

Modelling Earthquake Risk Based On Approximate Nonlinear Reliability Estimates

AbstractThis study examines the reliability of a bilinear hysteretic oscillator as a means to model the seismic risk of a structure. Many seismic risk analyses have been performed using linear SDOF oscilators. The objective of the present work is to demonstrate that the use of a more complex oscillator model in seismic risk analyses not only alters the structural response, but significantly changes the perceived number of time domain simulations that are thought to be representative of earthquake dangers in the area. The present study compares the results of a spatial seismic risk analysis, based on the probability of a bilinear hysteretic oscillator exceeding a prescribed displacement level, to an analysis using a linear oscillator at the same site. The results of the analyses are presented spatially, that is, as a function of earthquake magnitude and site-to-source distance, and in terms of a Cumulative Earthquake Risk Function (CERF).

Chaotic behaviors of a bilinear hysteretic oscillator

In the present research, concentration is focused on dynamic response of a bilinear hysteretic oscillator with clearance, and equivalent damping and stiffness are then determined by means of asymptotic and numerical method. A variety of dynamic responses of the nonlinear system are analyzed, and some new phenomena are discussed.

Equivalent Viscous Damping for a Bilinear Hysteretic Oscillator

A bilinear hysteretic model is commonly used to study elastoplastic structures. In this paper, a damped, bilinear hysteretic oscillator is studied under harmonic loading. We show the existence of an equivalent viscous damping for small values of a loading parameter such that the associated linear structure and the hysteretic structure have the same frequency response curves. We use the Kryloff-Bogoliuboff method of averaging to find the equivalent viscous damping as a function of the steady state amplitude. We present a model of a bilinear elastic oscillator which captures the steady-state dynamics of the hysteretic oscillator for low values of the loading parameter. We also study the nature of the dependence of the equivalent viscous damping on the kinematic hardening parameter.

APPROXIMATE ANALYTICAL EXPRESSIONS FOR THE STOCHASTIC RESPONSE OF A BILINEAR HYSTERETIC OSCILLATOR WITH LOW YIELD LEVELS

Differential equations are derived which exactly govern the evolution of the second order response moments of a single-degree-of-freedom (SDOF) bilinear hysteretic oscillator subject to stationary Gaussian white noise excitation. Then considering cases for which response stationarity will be achieved, i.e., excluding the case of an elastic-perfectly-plastic oscillator, algebraic equations for the response moments are found. By the nature of the problem, these moments depend on the probability of the oscillator being in the plastic state. Upon considering oscillators with low yield levels and using analytically available information, physical reasoning, and approximations supported by empirical observation, an equation for the probability of the oscillator being in the plastic state is derived. Upon numerical solution of this equation, analytical approximations to the response moments can be obtained. All analytical, approximate, and numerical results are verified by extensive Monte Carlo simulations.

A damage‐based definition of effective peak acceleration

This paper presents a rational basis for obtaining the Effective Peak Acceleration (EPA) of a given ground motion process. The proposed formulation considers the statistical variability in the ground motion, and is centred on the idea of explicitly linking EPA with expected cumulative damage in the structures due to the inelastic excursions. The structural behaviour has been modelled by a Single-Degree-Of-Freedom (SDOF) bilinear hysteretic oscillator. EPA is considered to be the expected PGA of a scaled ground motion process such that this oscillator undergoes a specified expected damage under the unscaled process if it is linearly designed for the scaled process. For estimation of the damage, the oscillator has been replaced by an equivalent linear oscillator through stochastic averaging. A parametric study has been carried out to investigate the dependence of EPA on several governing parameters, and it has been shown that despite the strong dependence of EPA on oscillator time period, it may be possible to obtain ‘period-averaged’ EPA values for several ground motion processes for engineering applications. © 1998 John Wiley & Sons, Ltd.

A damage‐based definition of effective peak acceleration

This paper presents a rational basis for obtaining the Effective Peak Acceleration (EPA) of a given ground motion process. The proposed formulation considers the statistical variability in the ground motion, and is centred on the idea of explicitly linking EPA with expected cumulative damage in the structures due to the inelastic excursions. The structural behaviour has been modelled by a Single-Degree-Of-Freedom (SDOF) bilinear hysteretic oscillator. EPA is considered to be the expected PGA of a scaled ground motion process such that this oscillator undergoes a specified expected damage under the unscaled process if it is linearly designed for the scaled process. For estimation of the damage, the oscillator has been replaced by an equivalent linear oscillator through stochastic averaging. A parametric study has been carried out to investigate the dependence of EPA on several governing parameters, and it has been shown that despite the strong dependence of EPA on oscillator time period, it may be possible to obtain ‘period-averaged’ EPA values for several ground motion processes for engineering applications. © 1998 John Wiley & Sons, Ltd.

Some aspects of the statistics of a bilinear hysteretic structure under a non-white random excitation

The response of bilinear hysteretic oscillators to non-white stochastic force is considered. A simple method is presented which utilises the dissipation-energy-balancing procedure and the equivalent linearisation technique to produce estimates for the response variance and the probability function of the energy of the oscillator. The predictions of the method compare well with digital simulation results.

Trispectrum for the response of a non-linear oscillator

Simulation is used to obtain information about non-Gaussian aspects of the absolute response acceleration of a bi-linear hysteretic oscillator with an excitation which is Gaussian white noise. Attention is focused on the frequency content of the fourth cumulant of the response. This frequency content is studied by consideration of the trispectrum and also by the simplified technique of looking at the coefficient of excess for the response of a narrowband linear system mounted on the non-linear oscillator. Attempts are also made to model the non-Gaussian response of the non-linear oscillator by a filtered delta correlated (FDC) process, but it is shown that no process of this type can exhibit some of the significant characteristics of the non-linear response. In particular, the trispectrum of the non-linear response appears to be more narrowband than the power spectral density, and also it sometimes does not have the same sign at every point in the three-dimensional frequency space, and this behavior is distinctly different from that of any FDC process. Modifications of the FDC model are suggested in order to obtain improved approximations of the non-linear response.

Equivalent Linear SDF Response to Earthquakes

The objective of this paper is to examine the accuracy of an equivalent linearization method for prediction of maximum response of a single degree-of-freedom bilinear hysteretic oscillator subjected to earthquake input. Simple expressions are developed for equivalent linear frequency and damping in terms of maximum attained response ductility. Comprehensive numerical results are presented in which the effects on accuracy of elastic frequency, yield strength, bilinear hardening slope, initial viscous damping, and earthquake input are examined. The parameters that most strongly influence accuracy are the hardening slope of the bilinear hysteresis loop and the magnitude of the response ductility. Accuracy is poorest for nearly elastoplastic systems and moderate response ductilities. These trends are interpreted by examining in the frequency domain the response of the bilinear oscillator to simple inputs.

Forced vibration of the damped bilinear hysteretic oscillator

An exact solution is presented for the steady−state motion of a sinusoidally excited single−degree−of−freedom system exhibiting bilinear hysteresis and subjected to viscous damping that may be different in the ’’elastic’’ region from the ’’yielded’’ region. Subject Classification: 40.20.

Transient Random Response of Bilinear Oscillators

Results of an empirical study of the transient mean-square response of bilinear hysteretic oscillators to random excitation are presented. The oscillators considered include a moderately nonlinear system, a nearly elasto-plastic system, and the truly elasto-plastic system. The random excitation considered is an approximate simulation of stationary, Gaussian white noise. Generation of the ensemble of time histories of excitation, and integration of the nonlinear equation of motion to obtain the ensemble of time histories of response were performed on a digital computer. It is found that a constant parameter linear system (such as the Krylov-Bogoliubov system) can give acceptable predictions of the transient mean square response of a moderately nonlinear system, but not of a nearly elasto plastic system. A variable coefficient linear system which does give acceptable response predictions for the nearly elastoplastic system is also presented.

Approximate Technique for Treating Random Vibration of Hysteretic Systems

A technique is presented for predicting response statistics of a nonlinear hysteretic system by seeking an approximate equivalence between the hysteretic system and a nonlinear nonhysteretic system for which certain statistics of stationary response can be computed by an existing analytical method. The response statistics that can be computed include stationary displacement and velocity levels, and probability distributions. The technique is applied to the bilinear hysteretic oscillator, and the results are compared with experimental results and with the results of the equivalent linearization approximate technique of Krylov and Bogoliubov.