Development of adaptive modeling techniques for non-linear hysteretic systems

Adaptive estimation procedures have gained significant attention by the research community to perform real-time identification of non-linear hysteretic structural systems under arbitrary dynamic excitations. Such techniques promise to provide real-time, robust tracking of system response as well as the ability to track time variation within the system being modeled. An overview of some of the authors’ previous work in this area is presented, along with a discussion of some of the emerging issues being tackled with regard to this class of problems. The trade-offs between parametric-based modeling and non-parametric modeling of non-linear hysteretic dynamic system behavior are discussed. Particular attention is given to (1) the effects of over- and under-parameterization on parameter convergence and system output tracking performance, (2) identifiability in multi-degree-of-freedom structural systems, (3) trade-offs in setting user-defined parameters for adaptive laws, and (4) the effects of noise on measurement integration. Both simulation and experimental results indicating the performance of the parametric and non-parametric methods are presented and their implications are discussed in the context of adaptive structures and structural health monitoring.     

OGY法实现混沌控制的参数辩识研究

给出了在系统方程未知情况下，采用OGY控制策略，通过混沌的部分测试信息辩识控制参数的方法，并对Duffing方程及强迫Brusselator振子方程的参数辩识及混沌控制进行了数值计算；讨论了使用这一方法实现控制时的一些关键问题，给出了实现控制的一些重要参数；发现了对Duffing方程使用同一参数辩识值，可将混沌控制到不同的一倍周期轨道及高倍周期轨道上的新现象。     

An intelligent parameter varying (IPV) approach for non-linear system identification of base excited structures

Health monitoring and damage detection strategies for base-excited structures typically rely on accurate models of the system dynamics. Restoring forces in these structures can exhibit highly non-linear characteristics, thus accurate non-linear system identification is critical. Parametric system identification approaches are commonly used, but require a priori knowledge of restoring force characteristics. Non-parametric approaches do not require this a priori information, but they typically lack direct associations between the model and the system dynamics, providing limited utility for health monitoring and damage detection. In this paper a novel system identification approach, the intelligent parameter varying (IPV) method, is used to identify constitutive non-linearities in structures subject to seismic excitations. IPV overcomes the limitations of traditional parametric and non-parametric approaches, while preserving the unique benefits of each. It uses embedded radial basis function networks to estimate the constitutive characteristics of inelastic and hysteretic restoring forces in a multi-degree-of-freedom structure. Simulation results are compared to those of a traditional parametric approach, the prediction error method. These results demonstrate the effectiveness of IPV in identifying highly non-linear restoring forces, without a priori information, while preserving a direct association with the structural dynamics.     

On-line identification of non-linear hysteretic structures using an adaptive tracking technique

System identification and damage detection based on vibration data have received considerable attention recently because of their importance to structural health monitoring. Various technical approaches have been proposed in the literature; however, the on-line identification of the changes of parameters for non-linear structures due to damages is still a challenging problem. In this paper, we propose an on-line adaptive tracking technique, based on the least-square estimation, to identify the system parameters and their changes of non-linear hysteretic structures. The method proposed is capable of tracking abrupt or slow changes of the system parameters from which the damage event and the severity of the structural damage can be detected and evaluated. Simulation results for tracking the parametric changes of non-linear hysteretic structures are presented to demonstrate the application and effectiveness of the proposed technique in detecting the structural damages.     

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

The mathematical phenomenological mapping in non-linear dynamics of spur gear pair and radial ball bearing due to the variable stiffness

In this paper the philosophy of mathematical phenomenological mapping has been applied to the non-linear dynamics of spur gears and radial ball bearings. The spur gear pair dynamics and rolling element bearing dynamics are analyzed separately, but with a tendency to reduce the both of the systems to the same mathematical model. The different reasonable assumptions are taken in every of these analyzes, but they do not have significant influence to the accuracy of the results. The systems are reduced to the single degree of freedom dynamics model. The total gear stiffness and ball bearing stiffness are recognized as the main influent factor of vibration behavior of these machine elements. Therefore, the special attention was paid to the new approach and procedure for stiffness solving and related problems. A single spur gear pair dynamics is solved and the results for total gear stiffness and vibration are shown. The conclusions emphasize the importance of described parallel analyzes in order to reduce the calculation time in solving different phenomena with usage of the principle of mathematical phenomenology.     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

Non-linear dynamic analysis of a HSFD mounted gear-bearing system

An investigation is carried out on the systematic analysis of the dynamic behavior of the hybrid squeeze-film damper (HSFD) mounted a gear-bearing system with strongly non-linear oil-film force and gear meshing force in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless unbalance coefficient, damping coefficient and the dimensionless rotating speed ratio as control parameters. The non-dimensional equations of the gear-bearing system are solved using the fourth order Runge-Kutta method. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents and fractal dimension of the gear-bearing system. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotating speed and highly non-linear regimes.     

A robust online parametric identification method for non-deteriorating and deteriorating distributed element models with viscous damping

The online parametric identification of deteriorating and non-deteriorating distributed element models (DEMs) with viscous damping is studied using a generalization of Masing model to provide the proper framework for identification. The approach renders the hysteretic response of the DEM into a time-independent single-valued mapping from equivalent displacement values into equivalent force values, while considering the effect of damping as a parallel element. This approach allows for parametric identification of this non-linear rate-dependent hysteretic behavior to be performed using non-linear optimization techniques. A changing objective function, defined as a norm of force estimation error over a shifting window of recent data, is employed so that classic non-linear optimization techniques can be used for the online identification problem. A variation of the steepest descent method is used with significant modifications. Special measures are taken to guarantee robustness of the results in presence of noise. The results show that the proposed identification method exhibits a very good performance in identifying the correct values of the parameters in real time, and is robust in dealing with noise. The proposed method can be applied to many other types of hysteretic behavior as well.     

Nonlinear parameter identification of models for masonry

This paper investigates the parameter estimation problem for brick masonry models. An identification procedure is proposed in which the uncertainties of known parameters and/or errors of measurements are its elements of distinction. The minimization process of the discrepancies between experimental data and theoretical measurements takes place by means of a first order iterative method. The identification procedure is applied to two different problems: the calibration of an interface model for brick–mortar joint in its functional form through monotonic experimental tests; to evaluate the unknown parameters of a continuum model for brick masonry walls in its non-holonomic form by means of in-plane cyclic shear–compression test of masonry panels. The general framework of the non-linear estimate methodology, the parameter identification problems and the numerical results are presented.     

Stochastic Vibration Model of Gear Transmission Systems Considering Speed-Dependent Random Errors

A dynamic and stochastic simulation model is developed for analyzing the vibration of gear transmission systems with consideration of the influence of the time-variant stiffness, loads, and gear transmission errors. The gear transmission system is viewed as a non-linear, time-correlated and stationary stochastic system. The transmission errors of gears are decomposed into harmonic and random components based on the spectral analysis. To simulate the random component, a second order Markov process with time-variant parameters considering influence of rotational speed is proposed and the method to determine the model parameters based on the random error of measured gear transmission error is developed. A simulation system is developed. The input to the simulation system is a white Gaussian noise process and harmonic errors, and the output is the rotational vibration acceleration of gears. Experiments are carried out to verify the proposed model. The influences of the random error on vibration acceleration are examined using the developed simulation system.     

Dynamic behavior of gear pumps: effect of variations in operational and design parameters

This work presents a wide number of results about the influence that variations in terms of operational and design parameters play on the dynamic behavior of external gear pumps. These results are obtained by using a non-linear lumped-parameter kineto-elastodynamic model developed and experimentally assessed with the aim of including all the important dynamic effects. On the one hand, the effects of variations in the operational parameters—namely output pressure, rotational speed and oil viscosity—are analysed; on the other hand, the effects of modifications of some design parameters are shown: clearances and relief groove dimension. The results in terms of gear eccentricity, pressure evolution, pressure forces, gear accelerations and variable forces exciting the pump casing enlighten the dynamic behavior of gear pumps and give useful indications for design improvements and vibration and noise reduction. As regards specifically gear accelerations as well as forces exciting the casing, they strongly increase with both output pressure and rotational speed, but variations in rotational speed in the operational range give lower effects. Conversely, the modifications of the clearances give negligible effects, while the relief groove dimension is very important: the larger the relief grooves are, the higher the gear accelerations and forces exciting the casing become.     

Structural health monitoring and damage detection using an intelligent parameter varying (IPV) technique

Most structural health monitoring and damage detection strategies utilize dynamic response information to identify the existence, location, and magnitude of damage. Traditional model-based techniques seek to identify parametric changes in a linear dynamic model, while non-model-based techniques focus on changes in the temporal and frequency characteristics of the system response. Because restoring forces in base-excited structures can exhibit highly non-linear characteristics, non-linear model-based approaches may be better suited for reliable health monitoring and damage detection. This paper presents the application of a novel intelligent parameter varying (IPV) modeling and system identification technique, developed by the authors, to detect damage in base-excited structures. This IPV technique overcomes specific limitations of traditional model-based and non-model-based approaches, as demonstrated through comparative simulations with wavelet analysis methods. These simulations confirm the effectiveness of the IPV technique, and show that performance is not compromised by the introduction of realistic structural non-linearities and ground excitation characteristics.     

计及短周期误差的直齿轮副近周期运动及其辨识

齿轮副中的齿距偏差等短周期误差使系统出现复杂的周期运动, 影响齿轮传动的平稳性. 将该类复杂周期运动定义为近周期运动, 采用多时间尺度Poincaré映射截面对其进行辨识. 为研究齿轮副的近周期运动, 引入含齿距偏差的直齿轮副非线性动力学模型, 并计入齿侧间隙与时变重合度等参数. 采用变步长4阶Runge-Kutta法数值求解动力学方程, 由所提出的辨识方法分析不同参数影响下系统的近周期运动. 根据改进胞映射法计算系统的吸引域, 结合多初值分岔图、吸引域图与分岔树状图等研究了系统随扭矩与啮合频率变化的多稳态近周期运动. 研究结果表明, 齿轮副中的短周期误差导致系统的周期运动变复杂, 在微观时间尺度内, 系统的Poincaré映射点数呈现为点簇形式, 系统的点簇数与实际运动周期数为宏观时间尺度的Poincaré映射点数. 短周期误差导致系统在微观时间尺度内的吸引子数量增多, 使系统运动转迁过程变复杂. 合理的参数范围及初值范围可提高齿轮传动的平稳性. 该辨识与分析方法可为非线性系统中的近周期运动研究奠定理论基础.      

Determination of Ductile Damage Parameters by Local Deformation Fields: Measurement and Simulation

This work comprises the development, implementation and application of methods for the parameter identification of damage mechanical constitutive laws. Ductile damage is described on a continuum mechanical basis by extension of the von Mises yield condition with the Gurson–Tvergaard–Needleman as well as with the Rousselier model. The classical Rousselier model is complemented by accelerated void growth and void nucleation. The non-linear boundary and initial value problem is solved by the finite element system SPC–PMHP, which was developed in the frame of the special research program SFB393 for parallel computers. The material parameters are identified by locally measured displacement fields and measured force–displacement curves. For the material parameter identification a non-linear optimization algorithm is used, which renders the objective function to a minimum by means of a gradient based method. A useful strategy to identify the material parameters was found by careful numerical studies. Finally, using the object grating method the local displacement fields as well as the force–displacement curves are measured at notched flat bar tension specimens made of StE 690 and the parameters of the material are identified.     

Iterative estimation methods for Hammerstein controlled autoregressive moving average systems based on the key-term separation principle

This paper considers iterative identification problems for a Hammerstein nonlinear system which consists of a memoryless nonlinear block followed by a linear dynamical block. The difficulty of identification is that the Hammerstein nonlinear system contains the products of the parameters of the nonlinear part and the linear part, which leads to the unidentifiability of the parameters. In order to obtain unique parameter estimates, we express the output of the system as a linear combination of all the system parameters by means of the key-term separation principle and derive a gradient based iterative identification algorithm by replacing the unknown variables in the information vectors with their estimates. The simulation results indicate that the proposed algorithm can work well.     

Exact traveling wave solutions in the coupled plane-base rotator model of DNA

The internal dynamics of the DNA base pairs is studied starting from the generalized coupled plane base-rotator model of DNA, obtained by Yomosa and later improved by Takeno and Homma. We conceive the double-stranded DNA as an anisotropically coupled spin chain simple model. The generalized Hamiltonian expressed in terms of quasi-spin operators is averaged over the generalized coherent states in the Perelomov sense, in order to obtain the classical non-linear evolution equations of this molecular system where the inhomogeneity has not been considered. This approach provides the equations of motion, which could be reduced to a nonlinear Schrödinger equation with a saturable nonlinearity. This non-linear equation, under certain restrictions in the parametric space, supports traveling periodic triangular, bell, bubble and kink like solutions.     

A fractional non-linear creep model for coal considering damage effect and experimental validation

Accurate prediction of coal׳s creep behavior is of great significance to coalbed methane extraction. In this study, taking into account the visco-elastic–plastic characteristics and the damage effect, a fractional non-linear model is proposed to describe the creep behavior of coal. The constitutive and creep equations of the proposed fractional non-linear model are derived via the Boltzmann superposition principle and discrete inverse Laplace transform. Furthermore, uniaxial creep tests under different axial stress conditions were carried out to validate the proposed model. It is found that the present model can describe the experimental data from creep tests with better accuracy than classical models. Particularly, the present model can predict the accelerating creep deformation of coal which classical models fail to reproduce. Finally, the parametric sensitivity analysis is performed to investigate the effects of model parameters on the creep strain. It is verified that the introduction of fractional parameters and damage factor in the present model is essential to accurate prediction of the full creep stage of coal.     

Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium

The classical mass balance equation is usually used to model the transfer of humidity in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hydraulic conductivity are defined as functions of geometry and local characteristics of the porous medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second medium (the mortar), on the global behavior of the system is investigated. Numerical results for the Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of the model in relation to the effective hydrodynamic parameters of the porous structure is presented.