a b s t r a c tIn this paper the backbone curves of a two-degree-of-freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing. The backbone curves describe the loci of dynamic responses of a system when unforced and undamped, and are represented in the frequency-amplitude projection. In this study we provide an analytical method for relating the backbone curves, found using the second-order normal form technique, to the forced responses. This is achieved using an energy-based analysis to predict the resonant crossing points between the forced responses and the backbone curves. This approach is applied to an example system subjected to two different forcing cases: one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes. Additionally, a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced, and these predictions are then compared to responses found using numerical continuation.
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.
This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the frequency-displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response.This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.
Displays have revolutionized the way we work and learn, and thus, the development of display technologies is of paramount importance. The possibility of a free-space display in which 3D graphics can be viewed from 360 without obstructions is an active area of research-holograms or lightfield displays can realize such a display, but they suffer from clipping and a limited field of view. Here, we use a phased array of ultrasonic emitters to realize a volumetric acoustophoretic display in which a millimetric particle is held in midair using acoustic radiation forces and moved rapidly along a 3D path. Synchronously, a light source illuminates the particle with the target color at each 3D position. We show that it is possible to render simple figures in real time (10 frames per second) as well as raster images at a lower frame rate. Additionally, we explore the dynamics of a fast-moving particle inside a phased-array levitator and identify potential sources of degradation in image quality. The dynamics are nonlinear and lead to distortion in the displayed images, and this distortion increases with drawing speed. The created acoustophoretic display shows promise as a future form of display technology.
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