scholarly journals Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method

2017 ◽  
Vol 394 ◽  
pp. 434-450 ◽  
Author(s):  
Louis Guillot ◽  
Pierre Vigué ◽  
Christophe Vergez ◽  
Bruno Cochelin
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Harmonic

Finding Periodic Solutions of Forced Systems With Local Nonlinearities: A Mixed Shooting Harmonic Balance Method

2015 ◽  
Author(s):  
Frederic Schreyer ◽  
Remco Leine
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Mechanical Systems ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Balance Method ◽  
Pros And Cons ◽  
Numerical Approaches

Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.

Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method

2006 ◽  
Vol 1 (3) ◽  
pp. 221-229 ◽  
Author(s):  
J. F. Dunne
Keyword(s):  
Stability Analysis ◽  
High Frequency ◽  
Harmonic Balance ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Equation Error ◽  
Subharmonic Response ◽  
Split Frequency ◽  
Frequency Harmonic

A split-frequency harmonic balance method (SF-HBM) is developed to obtain subharmonic responses of a nonlinear single-degree-of-freedom oscillator driven by periodic excitation. This method is capable of generating highly accurate periodic solutions involving a large number of solution harmonics. Responses at the excitation period, or corresponding multiples (such as period 2 and period 3), can be readily obtained with this method, either in isolation or as combinations. To achieve this, the oscillator equation error is first expressed in terms of two Mickens functions, where the assumed Fourier series solution is split into two groups, nominally associated with low-frequency or high-frequency harmonics. The number of low-frequency harmonics remains small compared to the number of high-frequency harmonics. By exploiting a convergence property of the equation-error functions, accurate low-frequency harmonics can be obtained in a new iterative scheme using a conventional harmonic balance method, in a separate step from obtaining the high-frequency harmonics. The algebraic equations (needed in the HBM part of the method) are generated wholly numerically via a fast Fourier transform, using a discrete-time formulation to include inexpansible nonlinearities. A nonlinear forced-response stability analysis is adapted for use with solutions obtained with this SF-HBM. Period-3 subharmonic responses are obtained for an oscillator with power-law nonlinear stiffness. The paper shows that for this type of oscillator, two qualitatively different period-3 subharmonic response branches can be obtained across a broad frequency range. Stability analysis reveals, however, that for an increasingly stiff model, neither of these subharmonic branches are stable.

Vibration Analysis of a Shape Memory Oscillator by Harmonic Balance Method Verified Numerically

2019 ◽  
Vol 29 (03) ◽  
pp. 1930007 ◽  
Author(s):  
Rafal Rusinek ◽  
Joanna Rekas ◽  
Krzysztof Kecik
Keyword(s):  
Shape Memory ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
System Parameters ◽  
Irregular Motion

This paper focuses on periodic solutions for a one-degree-of-freedom oscillator with a spring made of shape memory alloy (SMA). However, when periodic solutions are unstable, irregular motion is identified numerically. The shape memory spring is described by a polynomial characteristic in this model. The harmonic balance method (HBM) is employed to find periodic solutions near the primary resonance. The solutions are confronted with results obtained by the multiple time scales method and numerical simulations. Finally, the effect of system parameters and temperature on the system dynamics is discussed.

scholarly journals The harmonic balance method for finding approximate periodic solutions of the Lorenz system

2019 ◽  
pp. 187-203 ◽  
Author(s):  
Alexander. N Pchelintsev ◽  
Andrey. A Polunovskiy ◽  
Irina. Y Yukhanova
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Lorenz System ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Trigonometric Polynomials ◽  
Cycle Period ◽  
Power Series Method ◽  
Symbolic Calculations ◽  
The Lorenz System

We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic calculations for obtaining a system of nonlinear algebraic equations with respect to the cyclic frequency, free terms and amplitudes of the harmonics, that make up the desired solution, are shown. To speed up the calculations, this system was obtained in a general form for the first time. The results of the computational experiment are given: the coefficients of trigonometric polynomials approximating the found periodic solution, the initial condition, and the cycle period. The results obtained were verified using a high-precision method of numerical integration based on the power series method and described earlier in the articles of the authors.

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