Harmonic Balance for Large-Amplitude Vibrations in Snap-Through Structures

2014 ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Phase Space ◽  
Large Amplitude ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Large Amplitude Vibrations

A simple nonlinear Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. Using the harmonic balance method we have developed an algorithm to find particular amplitude and frequency relations in two-term and three-term approximations when the solution lies outside of the separatrix on the phase space, i.e. when the oscillator exhibits snap-through behavior.

Harmonic Balance Analysis of Snap-Through Orbits in an Undamped Duffing Oscillator

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Approximate Solution ◽  
Phase Space ◽  
Large Amplitude ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Solution Properties ◽  
Harmonic Balance Analysis ◽  
Balance Analysis ◽  
Large Amplitude Vibrations

A simple nonlinear undamped Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. We present an algorithm that uses the harmonic balance (HB) method to find amplitude and frequency relationships in two- and three-term approximations for solutions that lie outside the separatrix in the phase space. Trends of the approximate solution properties are examined with reference to an analysis of the limit as the trajectory approaches the separatrix.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

ASYMMETRIC PERIODIC MOTIONS WITH CHAOS IN A SOFTENING DUFFING OSCILLATOR

2013 ◽  
Vol 23 (05) ◽  
pp. 1350086 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
JIANZHE HUANG
Keyword(s):  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Periodic Motions ◽  
Complex Motion ◽  
Bifurcation Tree ◽  
Generalized Harmonic Balance Method

In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well. However, asymmetric periodic motions in the softening Duffing oscillators are not investigated analytically yet, and such asymmetric periodic motions possess much richer dynamics than the symmetric motions in the softening Duffing oscillator. For asymmetric motions, the bifurcation tree from asymmetric period-1 motions to chaos is discussed comprehensively. The corresponding, unstable and stable, asymmetric and symmetric, periodic motions in the softening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are completed. This investigation provides a better picture of complex motion in the softening Duffing oscillator.

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Hai-Tao Zhu ◽  
Siu-Siu Guo
Keyword(s):  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Solution Procedure ◽  
Balance Method ◽  
Periodic Response ◽  
Coupled Equations ◽  
The Mean ◽  
Closure Method ◽  
Gaussian Closure

This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

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.

Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance

2011 ◽  
Vol 18 (11) ◽  
pp. 1661-1674 ◽  
Author(s):  
Albert CJ Luo ◽  
Jianzhe Huang
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Balance Method ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Generalized Harmonic Balance Method

In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period- k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters.

A Harmonic Balance Approximation of Dynamic Snap-Through Boundaries in a Single-Degree-of-Freedom Structure

2013 ◽  
Author(s):  
Richard Wiebe ◽  
Lawrence N. Virgin
Keyword(s):  
Large Amplitude ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Nonlinear Phenomena ◽  
Amplitude Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Analytical Approximations

Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. Structures with underlying snap-through static behavior may exhibit highly nonlinear and unpredictable oscillations. Such systems rarely lend themselves to investigation by analytical means. This is not surprising as nonlinear phenomena such as chaos run counter to the predictability of an analytical closed form solution. However, many unexpected analytical approximations of global stability may be obtained for simple systems using the harmonic balance method. In this paper a simple single-degree-of-freedom arch is studied using the harmonic balance method. The equations developed with the harmonic balance approach are then solved using an arc-length method and an approximate snap-through boundary in forcing parameter space is obtained. The method is shown to exhibit excellent agreement with numerical results. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many applications in both civil engineering, where arches are a canonical structural element, and mechanical/aerospace engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.

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