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.

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.

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.

ESTIMATION OF NON-STATIONARY PARAMETERS OF NONLINEAR EQUATIONS UNDER UNCERTAINTY

2019 ◽  
Author(s):  
Oleksandr Nakonechnyi ◽  
Petro Zinko ◽  
Iulia Shevchuk
Keyword(s):  
Interpersonal Communication ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Subject Area ◽  
System Of Nonlinear Equations ◽  
Social Community ◽  
Non Linear ◽  
Number Of Individuals ◽  
Communicative Space

This paper presents research of system of nonlinear equations. The algorithms of building a priory estimations, optimal functional estimations and guaranteed estimations of non-stationary parameters of differential equations are offered. These results are spread for discrete-time models. The algorithms of building optimal estimations and guaranteed estimations of non-stationary parameters of difference non-linear equations are offered. The approaches to construct optimal estimations based on Bellman functions and Kalman-Bussi filter. For each algorithms of building optimal functional estimations and guaranteed estimations the error of estimation are offered. We have presented as an example the results of numerical experiments to build guaranteed and optimal estimates for mathematical model of spreading one type of information with external influence. The number of information is taken as key parameter promoting accomplishment of aim. Information is spread in the community along internal (interpersonal communication of the members of social community) and external threads (mass media). Also for simplicity models with a constant number of individuals who are intentionally able to perceive and further spread an informational massage are explored. The model takes the form of non-linear ordinary differential equation with stationary parameters. The peculiarity of such models is that they allow a reasonable level of precision to model the subject area and obtain he results that can be uses in practice. The numerical experiments demonstrated the practical meaning of offered results. The offered approaches except theoretical interest has an important practical meaning. The results can be useful for algorithm development for estimation of dynamic of process in the information-communicative space.

Interacting Fatigue Crack Growth Analysis with Boundary Cracklet Method

2012 ◽  
Vol 445 ◽  
pp. 1017-1022 ◽  
Author(s):  
A. Kadir Yavuz ◽  
A. Deniz Senalp ◽  
Halit S. Türkmen ◽  
S. Leigh Phoenix
Keyword(s):  
Fatigue Crack ◽  
Fatigue Crack Growth ◽  
Power Series ◽  
Crack Growth ◽  
Crack Opening ◽  
Infinite Plate ◽  
Algebraic Equations ◽  
Engineering Structures ◽  
Linear Algebraic Equations ◽  
As Stress

In this study, interacting crack growth in an infinite plate is analyzed with new, fast and accurate Boundary Cracklet Method (BCM) developed by Phoenix and Yavuz. An interior crack is under consideration to watch its propagation because of cyclic loading which is very common for aerospace, naval and civil engineering structures. BCM is very useful to determine the overall stress field as well as stress intensity factors for crack tips and singular wedges at crack kinks. BCM uses integral equations expressed in terms of unknown edge dislocation distributions along crack lines. These distributions derive from an accurate representation of the crack opening displacements using power series basis terms obtained through wedge eigenvalue analysis, which leads to both polynomial and non-polynomial power series. The process is to choose terms of the series and their exponents such that the tractions on the crack faces are virtually zero compared to the far field loading. Applying the method leads to a set of linear algebraic equations to solve for the unknown weighting coefficients for the power series basis terms to make no use of numerical integrations unlike in other methods. Thats why, solution takes just a few seconds on a PC. A simple crack growth emanating from a triangular hole in an infinite plate is analyzed. The fatigue crack growth is assumed to follow Paris-Erdogan Law. The results are compared to those of other numerical methods. A parametric study is performed via graphs and tables to demonstrate the ability of BCM in analysis of fatigue crack growth.

An Analysis of the Large Deflections of Beams using the Rayleigh-Ritz Finite Element Method

1968 ◽  
Vol 19 (4) ◽  
pp. 357-367 ◽  
Author(s):  
A. C. Walker ◽  
D. G. Hall
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Large Deflection ◽  
Energy Functional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Comparison Of The Results ◽  
Element Method

SummaryThe Rayleigh-Ritz finite element method is used to obtain an approximate solution of the exact non-linear energy functional describing the large deflection bending behaviour of a simply-supported inextensible uniform beam subjected to point loads. The solution of the non-linear algebraic equations resulting from the use of this method is effected, using three different techniques, and comparisons are made regarding the accuracy and computing effort involved in each. A description is given of an experimental investigation of the problem and comparison of the results with those of the numerical method, and of the available exact continuum analyses, indicates that the numerical method provides satisfactory predictions for the non-linear beam behaviour for deflections up to one quarter of the beam’s length.

Dynamic Simulation of Compressor Station Operation Including Centrifugal Compressor and Gas Turbine

1990 ◽  
Author(s):  
K. K. Botros ◽  
P. J. Campbell ◽  
D. B. Mah
Keyword(s):  
Dynamic Simulation ◽  
Transfer Matrix ◽  
Numerical Technique ◽  
Field Measurements ◽  
Compressor Station ◽  
Algebraic Equations ◽  
Gas Dynamics Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Simulation Results

Dynamic simulation of the operation of a compressor station requires mathematical modelling of the dynamic behaviour of the compressor unit and various piping elements. Such models consist of large systems of non-linear partial differential equations describing the pipe flow together with non-linear algebraic equations describing the quasi-steady flow through various valves, constrictions and compressors. In addition, the models also include mathematical descriptions of the control system which consists of mixed algebraic and ordinary differential (mad) equations with some inequalities representing controllers’ limits. In this paper a numerical technique for the solution of the gas dynamics equations is described, which is based on the transfer matrix formulation relating the state vector time-difference at one side of an element to that on the other side. This approach facilitates incorporation of all elements transfer matrices into an overall transfer matrix according to the system geometrical connectivity. The paper also presents simulation results and comparison with actual field measurements of three case histories: 1) simulation of a compressor surge protection control process; 2) unit startup; and 3) slow transient of a compressor station responding to changes in the discharge pressure set point. Good agreement between simulation results and field measurements is demonstrated.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

2017 ◽  
Vol 27 (6) ◽  
pp. 1282-1303 ◽  
Author(s):  
M.M. Rahman ◽  
Sourav Saha ◽  
Satyajit Mojumder ◽  
Khan Md. Rabbi ◽  
Hasnah Hasan ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Solar Collector ◽  
Linear Equations ◽  
Bottom Wall ◽  
Algebraic Equations ◽  
Weighted Residual Method ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Induced Flow

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

scholarly journals Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 463-472 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
First Time ◽  
Linear Fractional Differential Equations

AbstractIn this work, we propose a new operational method based on a Genocchi wavelet-like basis to obtain the numerical solutions of non-linear fractional order differential equations (NFDEs). To the best of our knowledge this is the first time a Genocchi wavelet-like basis is presented. The Genocchi wavelet-like operational matrix of a fractional derivative is derived through waveletpolynomial transformation. These operational matrices are used together with the collocation method to turn the NFDEs into a system of non-linear algebraic equations. Error estimates are shown and some illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed technique.

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