Piecewise nonlinear systems are often modeled and studied using numerical integration techniques that cannot deal with the effects of discontinuity boundaries. Freeplay, multi-segmented stiffness, and rigid impacts are present in articulated systems, gears, a mass oscillating between two springs or rigid walls, and in many other real mechanical systems. This leads to abrupt changes in the system’s response, due to grazing/sliding bifurcations. Grazing bifurcations of limit cycles are one of the most commonly found discontinuity-induced bifurcations (DIBs). These types of bifurcations are caused by a limit cycle that becomes tangent to the discontinuity boundary of the available piecewise-smooth function.

In this work, we perform several studies to analyze the effects of different types of discontinuous nonlinearities in a forced system. Our method is inspired by the Henon’s method implemented in Matlab’s^®^ *ode45* time integration solver, using its Event Location feature to accurately capture the effects of discontinuous terms in a system equation of motion. This method effectively performs piecewise time integration, in which integration stops and then restarts with the initial conditions and discontinuous terms updated every time a discontinuity boundary is encountered. We experimentally validate this numerical method against a physical system developed by deLangre in 1996. This system consists of a forced, single degree-of-freedom Duffing oscillator with a freeplay nonlinearity. The freeplay nonlinearity comes from two linear springs that can impact a rigid block during oscillation, and this is capable of causing chaotic responses in the system. We first describe our numerical time integration method and how it works in detail. Next, we describe the base physical system we used in this work. We then carry out various studies with different modifications to the base system. The first study we perform is of the original system, with a low spring constant for both the impact springs in order to represent a small freeplay nonlinearity. In the second study, we modify the impact springs to be cubic instead of linear. The third study features multi-segmented impact springs. In the fourth study, the two impact springs have different stiffnesses. In the fifth and final studies, the spring stiffnesses are set to a very high value to simulate impact with rigid walls. The features we analyze include modal features such as time histories, frequency responses, and power spectra, along with chaos-indicating features including Poincare maps and Lyapunov exponents. The main contribution of this research is to provide accurate results to understand grazing/sliding bifurcations in piecewise systems with strong nonlinearities, and to understand how the strength and type of piecewise combination affects the stability of the system’s response.

 

 

 


Numerical Bifurcation Analysis of a Piecewise-Smooth Nonlinear Oscillator System With Impacts

Division / Sponsors: 
Design Engineering Division and Computers & Information In Engineering Division


Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2020/volume/83907

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