Keywords: vibration, normal mode, nonlinear structure, periodic motion.

Extending the concept of normal modes of vibrating systems to the case where the restoring forces contain nonlinear terms, has been a challenge to many authors, mainly because the principe of linear superposition does not hold for nonlinear systems. However, the existence of "synchronous" periodic oscillations has great potential for applications to nonlinear free and forced vibration problems and it is becoming more important to generate of effective reduced-order models for multi-degree-of-freedom nonlinear systems. Following the pioneer work by Rosenberg [2] on conservative systems, several attempts have been made to develop methods of calculating Nonlinear Normal Modes (NNMs). These include the normal form theory [3,4], the invariant manifold method [5,6] (which led to a new definition of NNMs extending the concept to non conservative systems), the perturbation method [7], the balance harmonic procedure [7], the method of multiple scales [8], and various combinations.

The aim of this paper is to present two approachs for calculating the NNMs of vibrating nonlinear undamped mechanical systems: the time integration periodic orbit method and and the amplitude-phase representation method.

In the time integration periodic orbit method, the NNMs are defined by continuation of periodic orbit. To do this, an energy conserving time stepping for time variable is used combined with the asymptotic-numerical method [10]. This approach allows us to compute the bifurcation diagram of the NNMs. This approach provides the period of the oscillations on NNMs and it leads also to a numerical representation of the invariant manifold associated to the NNM[5].

In the amplitude-phase representation method [12], as in the linear case, an expression is developed for the NNM in terms of the amplitude, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The dynamics of the periodic response is defined by a one dimensional nonlinear differential equation governing the total phase motion. The period of the oscillations, depending only on the amplitude, is easily deduced. It is established that the frequency and the mode shape provide the solution to a -periodic nonlinear eigenvalue problem from which a numerical Galerkin procedure is developed for approximating the NNMs. This formulation allows us to characterize the similar NNMs. It leads also to an analytical (parametric) expression of the invariant manifold and it permits to compute the NNM even in the case of resonance relations between the eigenvalues of the linearized system. The extension of the formulation for the case of a damped autonomous mechanical systems will not be considered (see [9]).

The two approachs will be compared for several simple discrete and continuous vibrating systems.

1

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2

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3

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4

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5

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7

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8

Nayfeh, A.H., Mook, D.T., "Nonlinear oscillations", John Wiley & Sons, New-york, 1984.

9

Bellizzi, S., Bouc, R., " A new formulation for the existence and calculation of nonlinear mode", EUROMECH 457,7-9 june 2004, Fréjus, France, 2004.

10

Cochelin, B, Damil, N., Potier-Ferry, M., "Asymptotic numerical methods and pade approximants for non-linear elastic strutures", International journal of numerical methods in engineering, 37, 1187-1213,1994. doi:10.1002/nme.1620370706

11

Boutyour, E.H., Poitier-Ferry, M., Boudi, M., "Asymptotic-numerical method for buckling analysis of shell structures with marge rotations", Journal of computational and applied mathematics, 2004.

12

Bellizzi S., Bouc, R., "A new formulation for the existence and calculation of nonlinear normal modes" , Journal of Sound and Vibration, submitted, 2004. doi:10.1016/j.jsv.2004.11.014

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