A new formulation is pres ented here for the exis tence and calculation of nonlinear normal modes in undamped nonlinear autonomous mechanical systems. As in the linear case an expression is developed for the mode 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 dynamic of the periodic response is defined by a one-dimens ional nonlinear differential equation governing the total phas e motion. The period of the oscillations, depending only on the amplitude, is easily deduced. It is established that the frequency and the mode s hape provide the s olution to a 2p-periodic nonlinear eigenvalue problem, from which a numerical Galerkin procedure is developed for approximating the nonlinear modes. The procedure is applied to various mechanical s ys tems with two degrees of freedom.

International audienceThe aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped non-linear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms ''periodic orbits'' means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems

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We present an asymptotic analysis -in the "white-noise limitu-of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an It0 stochastic P.D.E. We then establish an expansion in powers of E ( 1 /~ being a measure of the bandwith of the driving noise) of the first moment of the solution.INTRODUCTION. We study here the "white-noise limitn of a PDE,