A novel order reduction method for nonlinear dynamical system under external periodic excitations

Abstract

The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper. Using the modal transformation, a large nonlinear dynamical system is split into a 'master' subsystem, a 'slave' subsystem, and a 'negligible' subsystem. Accordingly, a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the 'negligible' subsystem and slaving the 'slave' subsystem into the 'master' subsystem using the extended AIM. As a comparison, Method II accounting for the effects of both 'slave' subsystem and the 'negligible' subsystem is also applied to obtain the reduced order subsystem. Then, a typical 5-degree-of-freedom nonlinear dynamical system is given to compare the accuracy and efficiency of the traditional Galerkin truncation (ignoring the contributions of the slave and negligible subsystems), Method I and Method II. It is shown that Method I gives a considerable increase in accuracy for little computational cost in comparison with the standard Galerkin method, and produces almost the same accuracy as Method II. Finally, a 3-degree-of-freedom nonlinear dynamical system is analyzed by using the analytic method for showing predominance and convenience of Method I to obtain the analytically reduced order system. Galerkin method, nonlinear systems, dimension reduction, post-processed method, model truncation Citation:Cao D Q, Wang J L, Huang W H. A novel order reduction method for nonlinear dynamical system under external periodic excitations.

Monostable Dynamic Analysis of Microbeam-Based Resonators via an Improved One Degree of Freedom Model

“…In fact, high-order vibration items have important influence on dynamic behavior [ 25 ]. Fortunately, the Nonlinear Galerkin method can solve it with introducing higher modes into the single degree of freedom model [ 35 ].…”

“…The Nonlinear Galerkin method and the well-known Galerkin method can be used to obtain the low dimensional manifold by some projection onto a sub manifold [ 35 , 36 ]. However, the well-known Galerkin method restrict the sub-manifold at being a flat sub-manifold; the Nonlinear Galerkin method tries to improve on this by not restricting the sub-manifold to an affine sub-space.…”