Dimensional reduction for nonlinear time-delayed systems composed of stiffand soft substructures

This paper presents a new approach, based on the center manifold theorem, t o reducing the dimension of nonlinear time-delay systems composed of both s tiff and soft substructures. To complete the reduction process, the dynamic equation of a delayed system is first formulated as a set of singular pert urbed equations that exhibit dynamic behavior evolving in two different tim e scales. In terms of the fast time scale, the dynamic equation of system c an be converted into the standard form of a functional differential equatio n in critical cases, namely, to a form that can be treated by means of the center manifold theorem. Then, the approximated center manifold is determin ed by solving a series of boundary-value problems. The center manifold theo rem ensures that the dominant dynamics of the system is described by a set of ordinary differential equations of low order, the dimension of which is identical to that of the phase space of slowly variable states. As an appli cation of the proposed approach, a detailed stability analysis is made for a quarter car model equipped with an active suspension with a time delay ca used by a hydraulic actuator. The analysis shows that the dimensional reduc tion is surprisingly effective within a wide range of the system parameters .