Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

2021 ◽  
Vol 88 (5) ◽  
Author(s):  
Fan Kong ◽  
Pol D. Spanos
Keyword(s):  
Harmonic Balance Method ◽  
Linearization Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Single Degree Of Freedom ◽  
Stochastic Component ◽  
Linear Differential

Abstract A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

scholarly journals Approximate Stochastic Response of Hysteretic System With Fractional Element and Subjected to Combined Stochastic and Periodic Excitation

2021 ◽  
Author(s):  
Fan Kong ◽  
Renjie Han ◽  
Yuanjin Zhang
Keyword(s):  
Fractional Order ◽  
Harmonic Balance Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Hysteretic System ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Linear Differential

Abstract A method based on statistical linearization is proposed, for determining response of the single-degree-of-freedom (SDOF) hysteretic system endowed with fractional derivatives and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two sets of coupled fractional-order non-linear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization for the fractional-order deterministic and stochastic subsystems are used, to obtain the Fourier coefficients of the deterministic component and the variance of the stochastic component, respectively. This yields two sets of coupled non-linear algebraic equations which can be solved by appropriate standared numerical method. Pertinent numerical examples, including both softening and hardening Bouc-Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

2019 ◽  
Vol 86 (10) ◽  
Author(s):  
Pol D. Spanos ◽  
Ying Zhang ◽  
Fan Kong
Keyword(s):  
Harmonic Balance ◽  
Equation Of Motion ◽  
Periodic Component ◽  
Statistical Moments ◽  
System Response ◽  
Statistical Linearization ◽  
Ensemble Averaging ◽  
Stochastic Component ◽  
Stochastic Excitations ◽  
Linear Elements

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

2015 ◽  
Vol 82 (10) ◽  
Author(s):  
Ioannis A. Kougioumtzoglou ◽  
Alberto Di Matteo ◽  
Pol D. Spanos ◽  
Antonina Pirrotta ◽  
Mario Di Paola
Keyword(s):  
Path Integral ◽  
Computational Cost ◽  
High Dimensional ◽  
System Response ◽  
Stochastic Response ◽  
Mdof Systems ◽  
Low Dimensional ◽  
Response Determination ◽  
Efficiency And Reliability

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

Small Vibrations Superimposed on Non-Linear Rigid Body Motion

1997 ◽  
Author(s):  
A. L. Schwab ◽  
J. P. Meijaard
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rigid Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Connecting Rod ◽  
Non Linear ◽  
Linear Differential

Abstract In the case of small elastic deformations in a flexible multi-body system, the periodic motion of the system can be modelled as a superposition of a small linear vibration and a non-linear rigid body motion. For the small deformations this analysis results in a set of linear differential equations with periodic coefficients. These equations give more insight in the vibration phenomena and are computationally more efficient than a direct non-linear analysis by numeric integration. The realization of the method in a program for flexible multibody systems is discussed which requires, besides the determination of the periodic rigid motion, the determination of the linearized equations of motion. The periodic solutions for the linear equations are determined with a harmonic balance method, while transient solutions are obtained by averaging. The stability of the periodic solution is considered. The method is applied to a pendulum with a circular motion of its support point and a slider-crank mechanism with flexible connecting rod. A comparison is made with previous non-linear results.

Statistical Linearization Model for the Response Prediction of Nonlinear Stochastic Systems Through Information Closure Method

2004 ◽  
Vol 126 (3) ◽  
pp. 438-448 ◽  
Author(s):  
R. J. Chang ◽  
S. J. Lin
Keyword(s):  
Maximum Entropy ◽  
Density Function ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Stability Boundary ◽  
Response Prediction ◽  
Linearization Method ◽  
System Response ◽  
Statistical Linearization ◽  
Nonlinear Stochastic System

A new linearization model with density response based on information closure scheme is proposed for the prediction of dynamic response of a stochastic nonlinear system. Firstly, both probability density function and maximum entropy of a nonlinear stochastic system are estimated under the available information about the moment response of the system. With the estimated entropy and property of entropy stability, a robust stability boundary of the nonlinear stochastic system is predicted. Next, for the prediction of response statistics, a statistical linearization model is constructed with the estimated density function through a priori information of moments from statistical data. For the accurate prediction of the system response, the excitation intensity of the linearization model is adjusted such that the response of maximum entropy is invariant in the linearization model. Finally, the performance of the present linearization model is compared and supported by employing two examples with exact solutions, Monte Carlo simulations, and Gaussian linearization method.

scholarly journals On the comparison of the Gaussian heuristic with the statistical linearization methods in random vibrations

1981 ◽  
Vol 3 (3) ◽  
pp. 26-32
Author(s):  
Nguyen Cao Menh
Keyword(s):  
Heuristic Method ◽  
Solution Process ◽  
Random Processes ◽  
Linearization Method ◽  
Degree Of Freedom ◽  
Statistical Linearization ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Linearization Methods ◽  
Linearization Coefficients

In this paper the cumulate description of random processes is used for obtaining general formulae of the Gaussian heuristic method in single-degree-of-freedom and multi-degree-of-freedom systems. These formulae can be easily applied to particular cases. Whereby he Gaussian heuristic method and the statistical linearization method are compared. It can be shown, that if the solution process and the excited process are assumed to be jointly Gaussian, then these two methods are equivalent. Besides, on the basis of the Gaussian heuristic method, the statistical linearization coefficients are presented by more simple expressions than that in the papers [1,7].

MODELING OF NONLINEAR VIBRATION PROBLEMS WITH FLUID FLOWS THROUGH PIPELINES

2018 ◽  
pp. 44-47
Author(s):  
F.J. Тurayev
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Viscoelastic Properties ◽  
Construction Material ◽  
Fluid Flows ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Flowing Liquid ◽  
Vibration Problems

In this paper, mathematical model of nonlinear vibration problems with fluid flows through pipelines have been developed. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integro-differential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integro-differential equations with both constant and variable coefficients as functions of time.A system of algebraic equations is obtained according to numerical method for the unknowns. The influence of the singularity of heredity kernels on the vibrations of structures possessing viscoelastic properties is numerically investigated.It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

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