Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds

Abstract

A methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented. The approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations. Due to the existence of external and parametric periodic excitations, however, the geometry of the manifold varies with time. As a result, the manifold is constructed in terms of temporal and dominant state variables. The governing partial differential equation (PDE) for the manifold is nonlinear and contains time-varying coefficients. An approximate technique to find solution of this PDE using a multivariable Taylor–Fourier series is suggested. It is shown that, in certain cases, it is possible to obtain various reducibility conditions in a closed form. The case of time-periodic systems is handled through the use of Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant; however, the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric excitation frequencies. This warrants some structural changes in the proposed manifold, but the solution procedure remains the same. Two examples; namely, a 2-dof mass–spring–damper system and an inverted pendulum with periodic loads, are used to illustrate applications of the technique for systems with constant and periodic coefficients, respectively. Results show that the dynamics of these 2-dof systems can be accurately approximated by equivalent 1-dof systems using the proposed methodology.