Identifying parameters of multi-degree-of-freedom nonlinear structural dynamic systems using linear time periodic approximations

Abstract

The authors recently presented a new nonlinear system identification method, here dubbed the NL-LTP method, in which the system of interest is forced harmonically so that it responds in a stable periodic orbit, and then it is perturbed slightly and its response is recorded as it returns to the orbit. Under mild assumptions the response about the periodic orbit can be approximated using a linear time periodic system model, which can be identified from the measurements using techniques that are akin to linear modal analysis. While the prior work focused on simulated measurements from single degree-of-freedom systems, this work presents several tools that are needed in order to use this approach on multi-degree-of-freedom systems and focuses on applying the method to experimental hardware. The proposed system identification methodology is unique in that it identifies both the order of the nonlinear system and a mathematical model for the nonlinear restoring forces without assuming the mathematical form for the nonlinearities a priori. Towards these ends, this work explains how to extract the underlying nonlinear system model, or nonlinear restoring force versus displacement relationships, from the time periodic model that governs deviations of the system from its periodic orbit, and presents various metrics that can be used to determine which terms in the model are meaningful. These new tools are used to apply the identification method to a continuous, multi-degree-of-freedom structure with a discrete geometric nonlinearity, using both simulated and experimental measurements. The experimental hardware consists of a cantilever beam with a nonlinear spring attached to its tip, which is driven in a periodic limit cycle by an electromagnetic shaker.