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

Abstract

In 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.