Unrestricted Harmonic Balance

1980 ◽  
Vol 35 (10) ◽  
pp. 1054-1061 ◽  
Author(s):  
Friedrich Franz Seelig
Keyword(s):  
Linear Systems ◽  
Harmonic Balance ◽  
Open Systems ◽  
Periodic Structures ◽  
Simulation Method ◽  
Algebraic Equations ◽  
Good Convergence ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Stability

Abstract Periodic structures in chemical kinetic systems can be evaluated by an extension of the well-known method of harmonic balance, which yields very simple expressions in the case of linear systems containing only zero and first order reactions. The far more interesting non-linear systems containing e.g. second order reactions which in case of open systems far from thermodynamic equilibrium give rise to non-classical phenomena like oscillations, chemical waves, excitability, hysteresis, multistability, dissipative structures etc. can be treated in a similar way by introducing new pseudo-linear quantities utilizing certain group properties of harmonic expansions. The resulting complicated implicit non-linear algebraic equations are solved by a method developed by Powell and show good convergence. Since this method - in contrast to the conventional method of simulation - is independent from the stability of the periodic structure to be evaluated it can even be applied to unstable cases where the simulation method necessarily fails. An evaluation of the stability is included in the developed computer program.

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.

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.

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.

Unsteady flow in a supersonic cascade with strong in-passage shocks

1977 ◽  
Vol 83 (3) ◽  
pp. 569-604 ◽  
Author(s):  
M. E. Goldstein ◽  
Willis Braun ◽  
J. J. Adamczyk
Keyword(s):  
Unsteady Flow ◽  
Aircraft Engine ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Trailing Edges ◽  
Linearized Theory ◽  
Closed Form Analytical Solution ◽  
Supersonic Region ◽  
Shock Motion ◽  
The Stability

Linearized theory is used to study the unsteady flow in a supersonic cascade with in-passage shock waves. We use the Wiener–Hopf technique to obtain a closed-form analytical solution for the supersonic region. To obtain a solution for the rotational flow in the subsonic region we must solve an infinite set of linear algebraic equations. The analysis shows that it is possible to correlate quantitatively the oscillatory shock motion with the Kutta condition at the trailing edges of the blades. This feature allows us to account for the effect of shock motion on the stability of the cascade.Unlike the theory for a completely supersonic flow, the present study predicts the occurrence of supersonic bending flutter. It therefore provides a possible explanation for the bending flutter that has recently been detected in aircraft-engine compressors at higher blade loadings.

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.

Sign in / Sign up

Export Citation Format

Share Document