A multiharmonic method for non‐linear vibration analysis

Cardona, Alberto; Coune, Thierry; Lerusse, A.; Géradin, Michel

doi:10.1002/nme.1620370911

Cited by 92 publications

(58 citation statements)

References 14 publications

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“…The jacobian matrix of the system then has to be computed through finite differences, which can be cumbersome in terms of CPU time, or through linearization of the equations [9] .…”

Section: Analytical Expression Of the Nonlinear Terms And Of The Jacomentioning

confidence: 99%

“…Through an approximation of the periodic signals with their Fourier coefficients, which are the new unknowns of the problem, the user can have a direct control on the accuracy of the solutions. First implemented for analyzing linear systems, the HB method was then successfully adapted to nonlinear problems, in electrical (e.g., [8] ) and mechanical engineering (e.g., [9,10] ) for example. The main advantage of the HB method is that it involves algebraic equations with less unknowns than the methods in the time domain, for problems for which low orders of approximation are sufficient to obtain an accurate solution, which is the case if the regime of the system is not strongly nonlinear.…”

Section: Introductionmentioning

confidence: 99%

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The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems

Detroux

Renson

Kerschen

2014

Nonlinear Dynamics, Volume 2

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As a tool for analyzing nonlinear large-scale structures, the harmonic balance (HB) method has recently received increasing attention in the structural dynamics community. However, its use was so far limited to the approximation and study of periodic solutions, and other methods as the shooting and orthogonal collocation techniques were usually preferred to further analyze these solutions and to study their bifurcations. This is why the present paper intends to demonstrate how one can take advantage of the HB method as an efficient alternative to the cited techniques. Two different applications are studied, namely the normal modes of a spacecraft and the optimization of the design of a vibration absorber. The interesting filtering feature of the HB method and the implementation of an efficient bifurcation tracking extension are illustrated.

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“…The jacobian matrix of the system then has to be computed through finite differences, which can be cumbersome in terms of CPU time, or through linearization of the equations [9] .…”

Section: Analytical Expression Of the Nonlinear Terms And Of The Jacomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems

Detroux

Renson

Kerschen

2014

Nonlinear Dynamics, Volume 2

View full text Add to dashboard Cite

show abstract

“…The jacobian matrix of the system can then be computed through finite differences, which is cumbersome in terms of CPU time, or through linearization of the equations [15] .…”

Section: Harmonic Balance Methodsmentioning

confidence: 99%

“…It approximates the periodic signals with their Fourier coefficients, which become the new unknowns of the problem. First implemented for analyzing linear systems, the HB method was then successfully adapted to nonlinear problems, in electrical [14] and mechanical engineering [15,16] for example. The main advantage of the HB method is that it involves algebraic equations with less unknowns than the methods in the time domain, for problems for which low orders of approximation are sufficient to obtain an accurate solution; this is usually the case if the regime of the system is not strongly nonlinear.…”

Section: Introductionmentioning

confidence: 99%

The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems

Detroux

Renson

Masset

et al. 2016

Nonlinear Dynamics, Volume 1

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Because nowadays structural engineers are willing to use or at least understand nonlinearities instead of simply avoiding them, there is a need for numerical tools performing analysis of nonlinear large-scale structures. Among these techniques, the harmonic balance (HB) method is certainly one of the most commonly used to study finite element models with reasonably complex nonlinearities. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. For this reason, the present paper proposes to extend the method to the detection and tracking of codimension-1 bifurcations in the system parameters space. As an application, the frequency response of a spacecraft is studied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonance curves. This example illustrates how bifurcation tracking using the HB method can be employed as a promising design tool for detecting and eliminating such undesired behaviors.

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“…The most popular methods to find periodic steady-state responses of nonlinear differential equations are the Harmonic Balance Method (HBM) [1,2,3,4] and the shooting method [5]. The standard HBM approximates the periodic solution in frequency domain and is very popular as it is well suited for large systems with many states.…”

Section: Introductionmentioning

confidence: 99%

A Mixed Shooting – Harmonic Balance Method for Unilaterally Constrained Mechanical Systems

Schreyer¹,

Leine²

2016

Archive of Mechanical Engineering

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In this paper we present a mixed shooting -harmonic balance method for large linear mechanical systems on which local nonlinearities are imposed. The standard harmonic balance method (HBM), which approximates the periodic solution in frequency domain, is very popular as it is well suited for large systems with many degrees of freedom. However, it suffers from the fact that local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transform, then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the best of both worlds.

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A multiharmonic method for non‐linear vibration analysis

Cited by 92 publications

References 14 publications

The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems

The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems

The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems

A Mixed Shooting – Harmonic Balance Method for Unilaterally Constrained Mechanical Systems

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