MS15.1: Time-periodic systems

In this work an exact expression is derived for the fold (jump-down) point of the main harmonics of frequency response curve on the model of harmonically excited single degree of freedom viscose damped Duffing oscillator. The derivation is made based on the Poincare expansion method. In its leading term it provides an exact solution for the fold frequency point which then can be used for predicting hysteresis bandwidth depending on the nonlinearity, the static deflection and the viscose damping factor.

We apply control-based continuation (CBC) to a periodically forced active micro-cantilever with integrated sensing and actuation elements. Previous work with an experimental test rig has shown that this micro-electromechanical system (MEMS) is described well by a set of ordinary differential equations. We use this mathematical model to demonstrate that CBC with non-invasive position control is able to follow synchronised periodic solutions through stability changes at fold points.

This study presents a unique nonlinear frequency response NLFR extraction method introduced for smooth nonlinear vibrating systems. The proposed methodology is built on the geometrical representation of the harmonic balance solution for the undamped cubic Duffing oscillator. It is demonstrated that a curvilinear coordinate transform (CCT), defined by the so-called backbone curve (BBC) of the nonlinear normal mode (NNM), allows the comparison of linear and nonlinear systems in its atomic representation. Then, a unique way of extracting unstable periodic orbits is shown based on the CCT method.

The Hyperloop is a potential new mode of transport that will have very large target velocities. One possible design comprises an electromagnetically suspended pod that moves in a tube which is periodically supported by columns. This design has two potential sources of instability: (i) the electromagnetic force and (ii) parametric excitation. A canonical problem has been analysed as a first step to understanding the interplay between these two sources of instability. The analysis shows that the introduction of a parametric excitation expands the region of instability, although this increase is limited compared to the system without parametric excitation.

The current study addresses the stability of a moving mass suspended magnetically from a periodically supported beam. The stability of the free vibration about the periodic steady state has been studied. Preliminary findings employing periodic solutions to find the stability boundary show that parametric resonance can take place in the range of target velocities of the Hyperloop system, although the region of existence is limited. Further analysis is needed to determine the part of the stability boundary that is unrelated to parametric resonance.