A Direct Approach to Order Reduction of Nonlinear Systems Subjected to External Periodic Excitations

In this work, the basic problem of order reduction of nonlinear systems subjected to an external periodic excitation is considered. This problem deserves special attention because modes that interact (linearly or nonlinearly) with external excitation dominate the response. These dominant modes are identified and chosen as the “master” modes to be retained in the reduction process. The simplest idea could be to use a linear approach such as the Guyan reduction and choose those modes whose natural frequencies are close to that of external excitation as the master modes. However, this technique does not guarantee accurate results when nonlinear interactions are strong and a nonlinear approach must be adopted. Recently, the invariant manifold technique has been extended to forced problems by “augmenting” the state space, i.e., forcing is treated as an additional state and an invariant manifold is constructed. However, this process does not provide a clear picture of possible resonances and conditions under which an order reduction is possible. In a direct innovative approach suggested here, a nonlinear time-dependent relationship between the dominant and nondominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various primary and secondary resonances present in the system. One obtains various reducibility conditions in a closed form, which show interactions among eigenvalues, nonlinearities and the external excitation. One can also recover all “resonance conditions” obtained via perturbation or averaging techniques. The “linear” as well as the “extended invariant manifold” techniques are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.