Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures

Abstract

International audienceIn order to build efficient reduced-order models (ROMs) for geometrically nonlinear vibrations of thin structures, a normal form procedure is computed for a general class of nonlinear oscillators with quadratic and cubic nonlinearities. The linear perturbation brought by considering a modal viscous damping term is especially addressed in the formulation. A special attention is focused on how all the linear modal damping terms are gathered together in order to define a precise decay of energy onto the invariant manifolds, also defined as nonlinear normal modes (NNMs). Then, this time-independent formulation is used to reduce the dynamics governing the oscillations of a structure excited by an external harmonic force. The validity of the proposed ROMs is systematically discussed and compared with other available methods. In particular, it is shown that large values of the modal damping of the slave modes may change the type of nonlinearity (hardening/softening behaviour) of the directly excited (master) mode. Two examples are used to illustrate the main features of the method. A two-degrees-of-freedom (dof) system allows presentation of the main results through a simple example. Then a water-filled circular cylindrical shell with external resonant forcing is considered, in order to show the ability of the method to substantially reduce the dynamics of a continuous structure. © 2006 Elsevier Ltd. All rights reserved

Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration

“…In the presence of weak to moderate viscous damping, as shown in [17,20] and experimentally confirmed in this study, the damped dynamics can be interpreted based on the topological structure of the NNMs of the underlying conservative system. For large damping, it is important to note that the type of nonlinear behavior that is observed (e.g., hardening or softening) may be modified as shown in [24].…”

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

“…Normal forms can then be used in bifurcation theory to classify the generic families of bifurcations in dynamical systems [8,11]. For mechanical vibratory systems, it can also be used to define Nonlinear Normal Modes (NNMs) and build reduced-order models [14,25,26]. However, one recognized drawback of the method is that the validity limit of the change of variables is not given.…”

“…It is a general and powerful method that allows simplification of nonlinear terms of a dynamical system, by distinguishing resonant and nonresonant terms, so that eventually one is able to derive the "skeleton" of the dynamical system containing only the important terms for dynamical behaviors that are responsible for bifurcations and the nature of solutions, in the vicinity of a special solution such as fixed points or periodic orbits [8-10, 16, 17, 21-24]. In the mechanical context, normal form can be used to derive several important analytical results, as well as for showing equivalences with other methods such as NNMs, or appearance of small denominators in perturbative schemes, hence making it a cornerstone of all perturbation techniques [14,15,25,26]. The computation technique for deriving normal forms and coefficients of the associated nonlinear change of coordinates can be efficiently automatized by using symbolic computational toolboxes, as shown, for example, in [18,19].…”

“…The computation technique for deriving normal forms and coefficients of the associated nonlinear change of coordinates can be efficiently automatized by using symbolic computational toolboxes, as shown, for example, in [18,19]. Recently, it has also been applied to the second-order vibration problem [20], in a manner sharing many common points with the real normal form method developed in [25,26].…”

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

“…The most diffused ones are probably the nonlinear normal modes (under this name are classified the techniques based on the centre manifold theorem, the normal form theory and the inertial manifold) [1][2][3][4][5][6][7][8][9], including the most diffused version with asymptotic approach, the discretization of the equations of motion by using global (i.e. defined on the whole structure) admissible functions [10][11][12][13], the proper orthogonal decomposition method [14][15][16][17][18] and the natural mode discretization [19][20][21][22][23].…”