A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant

Chimera states in heterogeneous networks

Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in a heterogeneous model for which the natural frequencies of the oscillators are chosen from a distribution. We obtain exact results by reduction to a finite set of differential equations. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.Comment: Revised text. To appear, Chao

Bifurcation analysis of a model for atherosclerotic plaque evolution

We analyze two ordinary differential equation (ODE) models for atherosclerosis. The ODE models describe long time evolution of plaques in arteries. We show how the dynamics of the first atherosclerosis model (model A) can be understood using codimension-two bifurcation analysis. The Low-Density Lipoprotein (LDL) intake parameter (dd) is the first control parameter and the second control parameter is either taken to be the conversion rate of macrophages (bb) or the wall shear stress (σσ). Our analysis reveals that in both cases a Bogdanov-Takens (BT) point acts as an organizing center. The bifurcation diagrams are calculated partly analytically and to a large extent numerically using AUTO07 and MATCONT. The bifurcation curves show that the concentration of LDL in the plaque as well as the monocyte and the macrophage concentration exhibit oscillations for a certain range of values of the control parameters. Moreover, we find that there are threshold values for both the cholesterol intake rate dcritdcrit and the conversion rate of the macrophages bcritbcrit, which depend on the values of other parameters, above which the plaque volume increases with time. It is found that larger conversion rates of macrophages lower the threshold value of cholesterol intake and vice versa. We further argue that the dynamics for model A can still be discerned in the second model (model B) in which the slow evolution of the radius of the artery is coupled self-consistently to changes in the plaque volume. The very slow evolution of the radius of the artery compared to the other processes makes it possible to use a slow manifold approximation to study the dynamics in this case. We find that in this case the model predicts that the concentrations of the plaque constituents may go through a period of oscillations before the radius of the artery will start to decrease. These oscillations hence act as a precursor for the reduction of the artery radius by plaque growth

Interaction of torsion and lateral bending in aircraft nose landing gear shimmy

In this paper we consider the onset of shimmy oscillations of an aircraft nose landing gear. To this end we develop and study a mathematical model with torsional and lateral bending modes that are coupled through a wheel-mounted elastic tyre. The geometric effects of a positive rake angle are fully incorporated into the resulting five-dimensional ordinary differential equation model. A bifurcation analysis in terms of the forward velocity and the vertical force on the gear reveals routes to different types of shimmy oscillations. In particular, we find regions of stable torsional and stable lateral shimmy oscillations, as well as transient quasiperiodic shimmy where both modes are excited

Semi-analytical method to study piecewise linear oscillators

Acknowledgements The authors would like to thank the Balseiro Institute and the National Commission of Atomic Energy for the support. In particular, financial support for the Invited Professor Programme of the Balseiro Institute, which made international collaboration possible.Peer reviewedPublisher PD