Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Warmiński, Jerzy

doi:10.1007/s11071-010-9679-5

Cited by 40 publications

(21 citation statements)

References 24 publications

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“…Pioneered in the 1960s by Rosenberg [1], nonlinear normal modes (NNMs) are a rigorous extension of linear normal modes (LNMs) to nonlinear systems. The concept was then further developed [2,3] and enjoyed various applications in nonlinear structural dynamics, see, e.g., [4,5,6,7,8,9]. First defined as families of vibration in unison of the undamped, unforced system [1], NNMs were latter defined as (non-necessarily synchronous) periodic oscillations in order to account for modal interactions [10,11].…”

Section: Introductionmentioning

confidence: 99%

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

2014

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. AbstractThis paper addresses the numerical computation of nonlinear normal modes defined as two-dimensional invariant manifolds in phase space. A novel finite-elementbased algorithm, combining the streamline upwind Petrov-Galerkin method with mesh moving and domain prediction-correction techniques, is proposed to solve the manifold-governing partial differential equations. It is first validated using conservative examples through the comparison with a reference solution given by numerical continuation. The algorithm is then demonstrated on nonconservative examples.

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Section: Introductionmentioning

confidence: 99%

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

2014

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“…In this example we assume that the parametric resonance frequency is 2 ω while the external excitation frequency is ω. This case corresponds to practical engineering applications [29]. The differential equations of motion of the model in Fig.…”

Section: A Loop Phenomenon In the Dynamics Of A Self And Parametricalmentioning

confidence: 99%

“…The modes are determined on the basis of free vibrations of a nonlinear system. Details of that formulation are presented in [29]. The NNMs include geometric nonlinear terms therefore these terms vanish after the coordinate transformation.…”

Section: A Loop Phenomenon In the Dynamics Of A Self And Parametricalmentioning

confidence: 99%

Nonlinear phenomena in mechanical system dynamics

Warmiński

Kęcik

Mitura

et al. 2012

J. Phys.: Conf. Ser.

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Abstract. The goal of the paper is to present selected, untypical, and intuitively unexpected phenomena from nonlinear mechanics. Particular attention is paid to the dynamics of self-, parametric and external excited systems. Interactions between these various vibration types lead mainly to quasi-periodic responses. However, in the selected domains of system parameters, the effect of frequency locking is observed. Furthermore, external harmonic force imposed on such a system produces a specific internal loop inside a resonance zone. As an example of nonlinear autoparametric systems, a structure (oscillator) with an attached pendulum is presented. The nonlinear terms introduced by pendulum motion cause instabilities in the resonance region. This instability transits the pendulum to rotation or chaotic motion. An application of nonlinear couplings for the reduction of unwanted vibrations is also studied. In order to reduce vibrations, the main structure is coupled to an electrical oscillator by a quadratic term. It has been shown that such a coupling leads to the amplitude saturation phenomenon which can then be used to design a nonlinear control strategy.

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“…Indeed, there are many interesting nonlinear phenomena present in this class of dynamical systems. There exists a large body of analytical and numerical studies documenting multistability, symmetry breaking, attractors merging, synchronisation, existence of exotic attractors and various transitions to chaos (Bi, 2004 Sabarathinam et al, 2013;Warmiński, 2010).…”

Section: Introductionmentioning

confidence: 99%

Metamorphoses of resonance curves in systems of coupled oscillators

Kyzioł

Okniński

2016

jtam

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We study dynamics of two coupled periodically driven oscillators in a general case and compare it with two simplified models. Periodic steady-state solutions to these system equations are determined within the Krylov-Bogoliubov-Mitropolsky approach. Amplitude profiles are computed. These two equations, each describing a surface, define a 3D curve -intersection of these surfaces. In the present paper, we analyse metamorphoses of amplitude profiles induced by changes of control parameters in three dynamical systems studied. It is shown that changes of the dynamics occur in the vicinity of singular points of these 3D curves.

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Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Cited by 40 publications

References 24 publications

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

Nonlinear phenomena in mechanical system dynamics

Metamorphoses of resonance curves in systems of coupled oscillators

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