scholarly journals Floquet analysis of linear dynamic RLC circuits

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 264-277
Author(s):  
Mohamed El-Borhamy ◽  
Essam Eddin M. Rashad ◽  
Ismail Sobhy
Keyword(s):  
Dynamic Analysis ◽  
Harmonic Balance ◽  
Necessary Conditions ◽  
Harmonic Balance Method ◽  
Approximate Expression ◽  
Time Varying ◽  
Linear Dynamic ◽  
Rlc Circuits ◽  
Transition Curves ◽  
Theoretical Results

AbstractIn this article, the linear dynamic analysis of AC generators modeled as RLC circuits with periodically time-varying inductances via Floquet’s theory is considered. Necessary conditions for the dynamic stability are derived. The harmonic balance method is employed to predict the transition curves and stability domains. An approximate expression for the Floquet form of solution is constructed using Whittaker’s method in the neighborhood of transition curves. Numerical verifications for the obtained theoretical results are considered. In accordance with the experimental results, a satisfactory agreement is relatively achieved with the closed experimental literature of the problem.

Non-linear dynamic analysis of time varying length flexible body

2020 ◽  
Vol 2020 (0) ◽  
pp. 513
Author(s):  
Masato TAKEUCHI ◽  
Kensuke HARA ◽  
Hiroshi YAMAURA
Keyword(s):  
Dynamic Analysis ◽  
Flexible Body ◽  
Time Varying ◽  
Linear Dynamic ◽  
Non Linear ◽  
Varying Length

scholarly journals Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

2020 ◽  
Vol 26 (13-14) ◽  
pp. 1119-1132 ◽  
Author(s):  
Vinciane Guillot ◽  
Arthur Givois ◽  
Mathieu Colin ◽  
Olivier Thomas ◽  
Alireza Ture Savadkoohi ◽  
...  
Keyword(s):  
Harmonic Balance ◽  
Internal Resonance ◽  
Harmonic Balance Method ◽  
Mode Shapes ◽  
Governing Equations ◽  
Internal Resonances ◽  
Thin Beam ◽  
Beam Focusing ◽  
Theoretical Results ◽  
Theoretical Developments

Experimental and theoretical results on the nonlinear dynamics of a homogeneous thin beam equipped with piezoelectric patches, presenting internal resonances, are provided. Two configurations are considered: a unimorph configuration composed of a beam with a single piezoelectric patch and a bimorph configuration with two collocated piezoelectric patches symmetrically glued on the two faces of the beam. The natural frequencies and mode shapes are measured and compared with those obtained by theoretical developments. Ratios of frequencies highlight the realization of 1:2 and 1:3 internal resonances, for both configurations, depending on the position of the piezoelectric patches on the length of the beam. Focusing on the 1:3 internal resonance, the governing equations are solved via a numerical harmonic balance method to find the periodic solutions of the system under harmonic forcing. A homodyne detection method is used experimentally to extract the harmonics of the measured vibration signals, on both configurations, and exchanges of energy between the modes in the 1:3 internal resonance are observed. A qualitative agreement is obtained with the model.

Harmonic Balance Method for Nonlinear Vibration of Compound Planetary Gear Sets

2015 ◽  
Author(s):  
Weilin Zhu ◽  
Shijing Wu ◽  
Xiaosun Wang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Response Characteristic ◽  
Transmission Error ◽  
Balance Method ◽  
Time Varying ◽  
Algebraic Equations ◽  
Gear Backlash ◽  
Planetary Gear Sets

In this paper, a new nonlinear time-varying dynamic model for compound planetary gear sets, which incorporates the time-varying meshing stiffness, transmission errors and gear backlash, has been presented. The harmonic balance method (HBM), which is an analytical approach widely used for nonlinear oscillators, is employed to investigate the dynamic characteristics of the gear sets. The matrix form iteration algebraic equations has been established and solved by HBM and single rank inverse Broyden method to reveal the effect of transmission error and gear backlash on the frequency response characteristic of the system. Sub-harmonic resonant, super-harmonic resonant and jump phenomenon have been illustrated by several examples.

Dynamic response of Mathieu–Duffing oscillator with Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jianhua Tang ◽  
Chuntao Yin
Keyword(s):  
Dynamic Response ◽  
Caputo Derivative ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Expression ◽  
Balance Method ◽  
Composite Function ◽  
Derivatives Of ◽  
Approximate Expressions

Abstract In this paper, the harmonic balance method and its variants are used to analyze the response of Mathieu–Duffing oscillator with Caputo derivative. First, the exact and approximate expressions of the Caputo derivatives of trigonometric function and composite function are derived. Next, using the approximate expression of the Caputo derivative of the composite function, the resonance of Duffing oscillator with Caputo derivative is analyzed by the harmonic balance method. Finally, Mathieu–Duffing oscillator with Caputo derivative is approximated by three kinds of methods, i.e., the harmonic balance method, the residue harmonic balance method and the improved harmonic balance method. The corresponding numerical simulations are given to illustrate the performance of these methods as well. The results show that the residue harmonic balance method is more precise than the harmonic balance method and the improved harmonic balance method in analyzing the dynamic response of Mathieu–Duffing oscillator with Caputo derivative.

scholarly journals Existence of analytical solution, stability assessment and periodic response of vibrating systems with time varying parameters

2020 ◽  
Vol 14 (2) ◽  
Author(s):  
Jan Dupal ◽  
Martin Zajíček
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Solution Stability ◽  
Time Varying ◽  
Stability Assessment ◽  
Starting Point ◽  
Time Varying Parameters ◽  
Vibrating Systems

The paper is focused on a solution of a vibrating system with one-degree-of-freedom (1 DOF). The goal of this presentation is to deal with the method for periodical response calculation (if exists) reminding Harmonic Balance Method (HBM) of linear systems having time dependent parameters of mass, damping and stiffness under arbitrary periodical excitation. As a starting point of the investigation, a periodic Green’s function (PGF) construction of the stationary part of the original differential equation is used. The PGF then enables a transformation of the differential equation to the integro-differential one whose analytical solution is given in this paper. Such solution exists only in case that the investigated system is stable and can be expressed in exact form. The second goal of the paper is stability and solution existence assessment. For this reason a methodology of (in)stable parametric domain border determination has been accurately developed.

Dynamic analysis of harmonically excited non-linear structure system using harmonic balance method

2001 ◽  
Vol 15 (11) ◽  
pp. 1507-1516 ◽  
Author(s):  
Byung-Young Moon ◽  
Beom-Soo Kang ◽  
Byeong-Soo Kim
Keyword(s):  
Dynamic Analysis ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Linear Structure ◽  
Balance Method ◽  
Structure System ◽  
Non Linear

TRANSITION CURVES AND BIFURCATIONS OF A CLASS OF FRACTIONAL MATHIEU-TYPE EQUATIONS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250275 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
ZHONGJIN GUO ◽  
H. X. YANG
Keyword(s):  
Fractional Order ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Duffing Equation ◽  
Response Amplitude ◽  
Transition Curve ◽  
Homotopy Continuation ◽  
Transition Curves ◽  
Fractional Orders

A general version of the fractional Mathieu equation and the corresponding fractional Mathieu–Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu–Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter.

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