Numerical continuation of periodic orbits for harmonically forced nonlinear systems with iwan joints

Abstract

A common numerical model for bolted or riveted joints is the Iwan model, which uses a number of discrete Jenkins elements to capture the hysteretic behaviour of the system. Previously, the Iwan model has been primarily implemented within time-domain simulations using the Newmark integrator. However, in many applications it is desirable to predict the nonlinear Frequency Response Functions (FRFs) of a structure that contains joints. In order to do so, the steady-state response must be estimated over a range of frequencies, and it is very time consuming to simply compute the time response until steady-state is reached. Furthermore, the implicit nature of the state variables (i.e. the appearance of “hidden” state variables in the form of slider displacements) makes it non-trivial to use continuation to compute the frequency response using already established techniques such as the shooting method. This paper presents a novel method to numerically compute the non-linear FRFs of a single degree-of-freedom (SDOF) system with an Iwan element. The maximum displacement over the response period is included as a state variable, along with the initial displacement and velocity, and we demonstrate that the position of all sliders can be calculated using these states. The shooting method is modified to account for the added state variable. The method has been tested by computing the FRFs of a SDOF system containing of an Iwan element at multiple force amplitudes. The results show that the proposed method is able to compute the steady-state response even at large force amplitudes, when the system behaves quite non-linearly.