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.

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.

Amplitude Incremental Method: A Novel Approach to Capture Stable and Unstable Solutions of Harmonically Excited Vibration Response of Functionally Graded Beams under Large Amplitude Motion

2019 ◽  
Vol 20 (5) ◽  
pp. 581-594 ◽  
Author(s):  
B. Panigrahi ◽  
G. Pohit
Keyword(s):  
Large Amplitude ◽  
Harmonic Balance Method ◽  
Functionally Graded ◽  
Vibration Response ◽  
Algebraic Equations ◽  
Unstable Solution ◽  
Linear Algebraic Equations ◽  
Excited Vibration ◽  
Non Linear ◽  
Large Amplitude Motion

AbstractAn interesting phenomenon is observed while conducting numerical simulation of non-linear dynamic response of FGM (functionally graded material) beam having large amplitude motion under harmonic excitation. Instead of providing a frequency sweep (forward or backward), if amplitude is incremented and response frequency is searched for a particular amplitude of vibration, solution domain can be enhanced and stable as well as unstable solution can be obtained. In the present work, first non-linear differential equations of motion for large amplitude vibration of a beam, which are obtained using Timoshenko beam theory, are converted into a set of non-linear algebraic equations using harmonic balance method. Subsequently an amplitude incremental iterative technique is imposed in order to obtain steady-state solution in frequency amplitude plane. It is observed that the method not only shows very good agreement with the available research but the domain of applicability of the method is enhanced up to a considerable extent as the stable and unstable solution can be captured. Subsequently forced vibration response of FGM beams are analysed.

Bending of normally loaded simply supported rectangular plates in the large-deflection range

1969 ◽  
Vol 4 (3) ◽  
pp. 190-198 ◽  
Author(s):  
A Scholes ◽  
E L Bernstein
Keyword(s):  
Linear Equations ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Out Of Plane ◽  
Linear Differential ◽  
Good Agreement

Means of solving the non-linear differential equations of plate bending are revieweed and a method based on minimizing the corresponding energy integral is selected as offering most advantages. The energy intergral can be approximated either by using finite-difference approximatons or by assuming a form of displacement variation. Two sets of non-linear algebraic equations (in the in-plane and out-of-plane deflections) are thus formed and, by substitution alternately in each set, the resulting linear equations are solved. Results for simply supported rectangular plates have been worked out in some detail; these are compared with tests made on plates of various aspect ratios. Good agreement on maximum values of stress and deflection was obtained.

scholarly journals HARMONIC BALANCE THEORY FOR SCHEME TECHNICAL DESIGN

T-Comm ◽  
2020 ◽  
Vol 14 (11) ◽  
pp. 21-32
Author(s):  
Svetlana F. Gorgadze ◽  
◽  
Anton A. Maximov ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Projection Methods ◽  
Algebraic Equations ◽  
Electronic Circuits ◽  
Large Signal ◽  
Linear Algebraic Equations ◽  
Methods Of Synthesis ◽  
Signal Mode

The analysis and generalization of the main publications on the methods of synthesis and analysis of non-linear active microwave circuits based on the use of the harmonic balance method are presented. As a result of some classification of mathematical approaches and techniques used in the context of this method, a selection and review of basic algorithms was made, the sequential application of which makes it possible to obtain the final result for a scheme of any complexity. The principles of drawing up the initial system of differential equations for electronic circuits and reducing it to a system of linear algebraic equations are considered. A detailed and, at the same time, simplified interpretation of the approaches involving the use of projection methods and Krylov subspaces is given in order to make them easier to understand. Both the complete and the restart generalized method of minimal residuals are considered, in which the desired solution is obtained in the course of an iterative process, at each stage of which subspaces of lower dimension are constructed. The possibilities of simulators and application packages intended for circuit design of electronic circuits are considered. The problem of matching a power amplifier in large signal mode using the APLAC simulator, which is NI AWR technology for designing high-frequency circuits, is discussed.

Passivity-based control of islanded microgrids with unknown power loads

2020 ◽  
Vol 37 (4) ◽  
pp. 1548-1573
Author(s):  
Sofía Avila-Becerril ◽  
Gerardo Espinosa-Pérez ◽  
Oscar Danilo Montoya ◽  
Alejandro Garces
Keyword(s):  
Power Flow ◽  
Voltage Regulation ◽  
Algebraic Equations ◽  
Linear Dynamical Systems ◽  
Linear Algebraic Equations ◽  
Control Scheme ◽  
Non Linear ◽  
Local Measurements ◽  
Islanded Mode ◽  
Passivity Based Control

Abstract In this paper, the control problem of microgrids (MGs)operating in islanded mode is approached from a passivity-based control perspective. A control scheme is proposed that, relying only on local measurements for the power converters included in the network representation, achieves both voltage regulation and power balance in the network through the generation of grid-forming and grid-following nodes. From the mathematical perspective, the importance of the contribution lies in the feature that, exploiting a port-controlled Hamiltonian representation of the MG, the closed-loop system’s stability properties are formally proved using arguments from the theory of non-linear dynamical systems. Fundamental for this achievement is the decomposition of the system into subsystems that require a control law and another whose variables can evolve in a free way. From the practical viewpoint, the advantage of the proposed controller lies in the feature that the power demanded by the loads is satisfied without neither computing its specific value nor solving the non-linear algebraic equations given by the power flow, avoiding the computational burden associated with this task. The usefulness of the scheme is illustrated via a numerical simulation that includes practical considerations.

A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Skew Plates

2014 ◽  
Vol 704 ◽  
pp. 118-130
Author(s):  
Hanane Moulay Abdelali ◽  
Mounia El Kadiri ◽  
Rhali Benamar
Keyword(s):  
Mode Shape ◽  
Algebraic Equations ◽  
Skew Angle ◽  
Geometrically Nonlinear Analysis ◽  
Good Convergence ◽  
Linear Mode ◽  
Skew Plate ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Analytical Expressions

The present work concerns the nonlinear dynamic behaviour of fully clamped skew plates at large vibration amplitudes. A model based on Hamilton’s principle and spectral analysis has been used to study the large amplitude free vibration problem, reducing the non linear problem to solution of a set of non-linear algebraic equations. Two methods of solution have been adopted, the first method uses an improved version of the Newton-Raphson method, and the second leads to explicit analytical expressions for the higher mode contribution coefficients to the first non-linear mode shape of the skew plate examined. The amplitude dependent fundamental mode shape and the associated non-linear frequencies have been obtained by the two methods and a good convergence has been found. It was found that the non-linear frequencies increase with increasing the amplitude of vibration, which corresponds to the hardening type effect, expected in similar cases, due to the membrane forces induced by the large vibration amplitudes. The non-linear mode exhibits a higher bending stress near to the clamps at large deflections, compared with that predicted by linear theory. Numerical details are presented and the comparison made between the results obtained and previous ones available in the literature shows a satisfactory agreement. Tables of numerical results are given, corresponding to the linear and nonlinear cases for various values of the skew angle θ and various values of the vibration amplitude. These results, similar to those previously published for other plates, are expected to be useful to designers in the need of accurate estimates of the non-linear frequencies, the non linear strains and stresses induced by large vibration amplitudes of skew plates.

Some the Application of the Taylor Series for the Analysis of Processes in Non-Linear Resistive Circuits

2014 ◽  
Vol 701-702 ◽  
pp. 1173-1176
Author(s):  
Vitaly Viktorovich Pivnev ◽  
Sergey Nikolaevich Basan
Keyword(s):  
Power Series ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
System Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Electrical Circuits ◽  
It Implementation ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Way

The way of calculating the currents and voltages in nonlinear resistive electrical circuits , based on the use of power series (Taylor, Maclaurin) is considered . The advantage of this method lies in the fact that while it implementation it is not necessary to a system of nonlinear equations. To determine the numerical values ​​of the coefficients of the power series corresponding system of linear algebraic equations are solved. Nonlinear operations are limited to the calculation of the numerical values ​​of currents, voltages and their derivatives with respect to the pole equations of nonlinear elements.

scholarly journals A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

2020 ◽  
pp. 146134842092343 ◽  
Author(s):  
M W Ullah ◽  
M S Rahman ◽  
M A Uddin
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Vibration Problems ◽  
Nonlinear Forced Vibration

In this paper, a modified harmonic balance method is presented to solve nonlinear forced vibration problems. A set of nonlinear algebraic equations appears among the unknown coefficients of harmonic terms and the frequency of the forcing term. Usually a numerical method is used to solve them. In this article, a set of linear algebraic equations is solved together with a nonlinear one. The solution obtained by the proposed method has been compared to those obtained by variational and numerical methods. The results show good agreement with the results obtained by both methods mentioned above.

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