Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Peeters, Maxime; Viguié, Régis; Sérandour, Guillaume; Kerschen, Gaëtan; Golinval, Jean‐Claude

doi:10.1016/j.ymssp.2008.04.003

Cited by 446 publications

(383 citation statements)

References 32 publications

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“…In practice, the ratio r = N * /N k is bounded to achieve smooth change of step size [39]. For small-scale equations, N * = 3 is a desirable value.…”

Section: The Arc-length Methodsmentioning

confidence: 99%

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Dou

Jensen

2015

Journal of Sound and Vibration

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We present an optimization procedure for tailoring the nonlinear structural resonant response with time-harmonic loads. A nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis. An optimization procedure based on a gradient-based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities. We consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super-harmonic resonant responses.

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“…In practice, the ratio r = N * /N k is bounded to achieve smooth change of step size [39]. For small-scale equations, N * = 3 is a desirable value.…”

Section: The Arc-length Methodsmentioning

confidence: 99%

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Dou

Jensen

2015

Journal of Sound and Vibration

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“…For conservative systems, different reference algorithms to compute NNMs exist [11,13,32] . For instance, the "exact" manifolds can be computed using the technique developed in [13] , which combines shooting and pseudo-arclength continuation.…”

Section: Algorithm Demonstration On Conservative Examplesmentioning

confidence: 99%

“…For instance, the "exact" manifolds can be computed using the technique developed in [13] , which combines shooting and pseudo-arclength continuation. For each NNM, the graphical depiction in phase space of the periodic orbits at different energy levels provides a reference solution to which we can compare the solution computed using the invariant manifold approach.…”

Section: Algorithm Demonstration On Conservative Examplesmentioning

confidence: 99%

Nonlinear Normal Modes of Nonconservative Systems

Renson¹,

Kerschen²

2013

Topics in Nonlinear Dynamics, Volume 1

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Linear modal analysis is a mature tool enjoying various applications ranging from bridges to satellites. Nevertheless, modal analysis fails in the presence of nonlinear dynamical phenomena and the development of a practical nonlinear analog of modal analysis is a current research topic. Recently, numerical techniques (e.g., harmonic balance, continuation of periodic solutions) were developed for the computation of nonlinear normal modes (NNMs). Because these methods are limited to conservative systems, the present study targets the computation of NNMs for nonconservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, a new finite element technique is proposed to solve the set of partial differential equations governing the manifold geometry.

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“…Yet the first applications of nonlinear simulation to industrial structures are very recent. In particular Renson and Kerschen [3] applied a time-integration method with a shooting algorithm [4] to compute the nonlinear vibrations of the SmallSat spacecraft, and focused on internal resonances between linear modes. Sinou [5] applied a Harmonic Balance Method to a nonlinear model of an industrial rotor and computed a multi-harmonic frequency response.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the Harmonic Balance Method is compared to the time-integration method with a shooting algorithm [4], to demonstrate the efficiency of both methods, and complementary experimental measures are performed to substantiate the model developed.…”

Section: Introductionmentioning

confidence: 99%

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

Claeys

Sinou

Lambelin

et al. 2014

Communications in Nonlinear Science and Numerical Simulation

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-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions. Communications in Nonlinear Science and Numerical Simulation, Elsevier, 2014, 19 (12) AbstractThis study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped-clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped-clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics' frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped-clamped beam and the effect of the non-ideal input signal.

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Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Cited by 446 publications

References 32 publications

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Optimization of nonlinear structural resonance using the incremental harmonic balance method

Nonlinear Normal Modes of Nonconservative Systems

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

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