MS12.3: Nonlinear Dynamics for Engineering Design

Differently from linear dynamical systems, the analysis of global dynamics is crucial for ensuring the safety of nonlinear systems due to potential phenomena of multi-stability. Trajectories of transient motions in the phase space contain significant information about global dynamics. However, extracting this information is challenging, leading to its frequent oversight. In this study, we demonstrate how transient trajectories can provide valuable information about nearby regions in the parameter and phase space. Specifically, an irregular decay rate may indicate the presence of a fold bifurcation. By measuring this decay rate, we can extract information that can be used to predict the occurrence of fold bifurcations. It is important to note that fold bifurcations are practically significant as they often delineate regions of multi-stability. The proposed method is initially validated numerically on a mass-on-moving-belt model, a van der Pol-Duffing oscillator with an attached tuned mass damper, and a pitch-and-plunge airfoil undergoing flutter instability. Subsequently, the method is experimentally validated on a towed wheel experiencing shimmy vibrations.

In this talk we provide an archetypal double-winged novel nonlinear oscillator composed of torsional springs and rigid rods which behaves smooth and discontinuous dynamics and a kind of special collision pendulum depending on the varying of a geometrical parameter. The system behaves complicated equilibrium bifurcations with different geometrical states, leading to a series of buckling phenomenon and the loss of uniqueness near the limit points. A ‘collision’ parameter is introduced to determine the motions near the limit points which causes a discontinuous jump of velocity and exhibits the standard dynamics as an inverted pendulum, even though the mechanic model is rather different from a pendulum. We also distinguish the coexistence of periodic and chaotic motions and also a smooth period 1 located in a small region under the presence of damping and external force using amplitude-frequency analysis and numerical simulations. This research provides a brand new understanding on geometrical nonlinear oscillators and enriches the complicated dynamical behaviors of nonlinear systems, which also has great potential in isolation vibration engineering.

This research work is centred around coupling nonlinear resonators for bio-inspired acoustic sensing to improve sound processing and thus close the performance gap to human hearing. Critical oscillators tuned near a Hopf bifurcation were shown to model part of the remarkable functionalities of human hearing, i.e. a large dynamic range, a high frequency resolution, and improved signal-to-noise ratios. Sensors built on this concept exhibit high quality factors i.e. high gain and small bandwidth, due to the nonlinear and resonant operation principle. However, to cover the audible frequency range, a high amount of sensors would be required. Thus, we explore how coupling of nonlinear resonators can overcome this issue. Our nonlinear resonators are silicon cantilevers combined with an electronic feedback. Diffusive, output-signal based coupling is implemented using the difference between the sensing signals of two resonators as their driving signal. Combining experiments and simulations of two coupled resonators, we show that depending on the polarity of the coupling two bifurcation points with different sensing properties are obtained. Near one bifurcation point a dampened response of both resonators to sound is observed. Close to the other bifurcation point a strong increase in gain and a nonlinear response is obtained. Finally, Results from extending the system to a chain of four resonators will be discussed.

Long, slender piles may experience stability problems during driving owing to the dynamic load. A simple model with one degree of freedom shows the main parameters involved: the ratio of the driving frequency and the eigenfrequency, the relative damping and the strength of the impact. An analytic formula that gives a first estimation of the danger of instability is derived. More detailed finite-element calculations essentially confirm the applicability of this formula in practice.