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

This paper presents a new approach, based on the center manifold theorem, to reducing the dimension of nonlinear time-delay systems composed of both stiff and soft substructures. To complete the reduction process, the dynamic equation of a delayed system is first formulated as a set of singular perturbed equations that exhibit dynamic behavior evolving in two different time scales. In terms of the fast time scale, the dynamic equation of system can be converted into the standard form of a functional differential equation in critical cases, namely, to a form that can be treated by means of the center manifold theorem. Then, the approximated center manifold is determined by solving a series of boundary-value problems. The center manifold theorem 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 application of the proposed approach, a detailed stability analysis is made for a quarter car model equipped with an active suspension with a time delay caused by a hydraulic actuator. The analysis shows that the dimensional reduction is surprisingly effective within a wide range of the system parameters.