67 research outputs found
Experimental investigation of a single-plane automatic balancing mechanism for a rigid rotor
No full textWe present an experimental investigation of a single-plane automatic balancer that is fitted to a rigid rotor. Two balls, which are free to travel around a circular race, are used to compensate for the mass imbalance in the plane of the device. The experimental rig possesses both cylindrical and conical rigid body modes and the performance of the automatic balancer is assessed for a variety of different levels of imbalance. A non-planar mathematical model that also includes the observed effect of support anisotropy is developed and numerical simulations are compared with the experimental findings. In the highly supercritical frequency range the balls act to balance the rotor and a good quantitative match is found between the model and the experimental data. However, during the rigid body resonances the dynamics of the ball balancer is highly nonlinear and for this speed range the agreement between theory and experiment is mainly qualitative. Nevertheless, the model is able to successfully reproduce many of the solution types that are found experimentally.<br/
Robust heteroclinic cycles in the one-dimensional complex Ginzburg-Landau equation
No full textNumerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one-dimensional complex Ginzburg–Landau equation (CGL) on the unit, spatially periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions as t?±?. A careful analysis of the connections is made using a projection onto five complex Fourier modes. It is shown first that the time-periodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincaré maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in specific parameter regions where the cycles are found to be of Shil’nikov type. This criterion is also applied to a much higher-mode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shil’nikov–Hopf or blow-out bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatio-temporal intermittency in situations modelled by the CGL are discusse
Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities
No full textThis paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking
Travelling solitary waves in the discrete nonlinear Schrodinger equation with saturable nonlinearity: existence, stability and dynamics
No full textThe present work examines in detail the existence, stability and dynamics of travelling solitary waves in dynamical lattices with saturable nonlinearity. After analszing the linear spectrum of the problem in the travelling wave frame, a pseudo-spectral numerical method is used to identify weakly non-local solitary waves. By finding zeros of an appropriately crafted tail condition, we can obtain the genuinely localised pulse-like solutions. Subsequent use of continuation methods allows us to obtain the relevant branches of solutions as a function of the system parameters, such as the frequency and inter-site coupling strength. We examine the stability of the solutions in two ways: both by imposing numerical perturbations and observing the solution dynamics, as well as by considering the solutions as fixed points of an appropriate map and computing the corresponding Floquet matrix and its eigenvalues. Both methods indicate that our solutions are robustly localised. Finally, the interactions of these solutions are examined in collision type phenomena, observing that relevant collisions are near-elastic, although they may, under appropriate conditions, lead to the generation of an additional pulse
Friction-induced reverse chatter in rigid-body mechanisms with impacts
No full textThe focus of this paper is on the possibility of formulating a consistent and unambiguous forward-simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painlev´e paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-sti ness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward-simulation model is a orded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-sti ness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left-accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model
Automatic two-plane balancing for rigid rotors
No full textWe present an analysis of a two-plane automatic balancing device for rigid rotors. Ball bearings, which are free to travel around a race, are used to eliminate imbalance due to shaft eccentricity or misalignment. The rotating frame is used to derive autonomous equations of motion and the symmetry breaking bifurcations of this system are investigated. Stability diagrams in various parameter planes show the coexistence of a stable balanced state with other less desirable dynamic
Homoclinic orbits in reversible systems II : multi-bumps and saddle-centres
No full textThis article extends a review by the author in Physica D, vol.112, pp.158-186 of the theory and application of homoclinic orbits to equilibria in even-order, time-reversible systems of autonomous ordinary differential equations, either Hamiltonian or not. Recent results in two directions are surveyed. First, a heteroclinic connection between a saddle-focus equilibrium and a periodic orbit is shown to arise from a certain codimension-two local bifurcation; a degenerate Hamiltonian-Hopf bifurcation. Under a transversality hypothesis, perturbation from normal form causes this isolated solution to break into a snaking bifurcation curve under which a primary homoclinic becomes a multi-bump with arbitrarily many bumps. Taking as a model a fourth-order equation arising in many contexts, the snaking is terminated by the existence of a heteroclinic connection to an equilibrium. Second, multi-bump homoclinic orbits are considered in the case where the equilibrium is a four-dimensional saddle-centre (having two real and two imaginary eigenvalues). If the system is Hamiltonian, then it is known that a sign condition determines whether or not cascades of multi-bumps accumulate on the parameter values of a primary homoclinic solution. For non-Hamiltonian reversible systems cascades always occur, albeit from one sign of parameter perturbation only. Finally, aided by numerical methods, possible applications are considered to localised cylindrical shell buckling and to a generalised massive Thirring model arising in nonlinear optics
Hidden dynamics in models of discontinuity and switching
No full textSharp switches in behaviodr, like impacts, stick slip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewise-smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity. (C) 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).</p
Bifurcation analysis of surge and rotating stall in the Moore-Greitzer compression system
No full textA simple compression system model, described by a set of three ordinary nonlinear differential equations (the Moore-Greitzer model) is studied using bifurcation analysis to give a qualitative understanding of the presence of surge and rotating stall. Firstly, three parameter values are chosen and a reduced planar system is studied to detect the local bifurcations of pure surge modes. The global bifurcation diagrams are then completed with the help of the continuation software AUTO. A special feature of this 2D system is a set of parameter values where two Takens-Bogdanov points merge. As a next step, the interaction of surge and rotating stall modes is analysed using the same branch tracking technique. Several novel bifurcation scenarios are described. Two-parameter bifurcation maps are computed and a satisfactory agreement with experimental results is found. An explanation is given for the onset of deep surge, rotating stall, classic surge and the hysteresis effects experienced in measurements
Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation
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