Unlocking a nose landing gear in different flight conditions: folds, cusps and a swallowtail

This paper investigates the unlocking of a non-conventional nose landing gear mechanism that uses a single lock to fix the landing gear in both its downlocked and uplocked states (as opposed to having two separate locks as in most present nose landing gears in operation today). More specifically, we present a bifurcation analysis of a parameterized mathematical model for this mechanical system that features elastic constraints and takes into account internal and external forces. This formulation makes it possible to employ numerical continuation techniques to determine the robustness of the proposed unlocking strategy with respect to changing aircraft attitude. In this way, we identify as a function of several parameters the steady-state solutions of the system, as well as their bifurcations: fold bifurcations where two steady states coalesce, cusp points on curves of fold bifurcations, and a swallowtail bifurcation that generates two cusp points. Our results are presented as surfaces of steady states, joined by curves of fold bifurcations, over the plane of retraction actuator force and unlock actuator force, where we consider four scenarios of the aircraft: level flight; steep climb; steep descent; intermediate descent. A crucial cusp point is found to exist irrespective of aircraft attitude: it corresponds to the mechanism being at overcentre, which is a position that creates a mechanical singularity with respect to the effect of forces applied by the actuators. Furthermore, two cusps on a key fold locus are unfolded in a (codimension-three) swallowtail bifurcation as the aircraft attitude is changed: physical factors that create these bifurcations are presented. A practical outcome of this research is the realization that the design of this and other types of landing gear mechanism should be undertaken by considering the effects of forces over considerable ranges, with a special focus on the overcentre position, to ensure a smooth retraction occurs. More generally, continuation methods are shown to be a valuable tool for determining the overall geometric structure of steady states of mechanisms subject to (external) forces.

Generalized r-Lambert Function in the Analysis of Fixed Points and Bifurcations of Homographic 2-Ricker Maps

This paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the [Formula: see text]-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is [Formula: see text]. A generalized [Formula: see text]-Lambert function is defined on the 3D parameters space to determine the existence and variation of the number of nonzero fixed points of the homographic [Formula: see text]-Ricker maps considered. The singularity points of the generalized [Formula: see text]-Lambert function are identified with the cusp points on a fold bifurcation of the homographic [Formula: see text]-Ricker maps. In this approach, the application of the transcendental generalized [Formula: see text]-Lambert function is demonstrated based on the analysis of local and global bifurcation structures of this three-parameter family of homographic maps. Some numerical studies are included to illustrate the theoretical results.

Generalised model-independent characterisation of strong gravitational lenses

We determine the transformation matrix that maps multiple images with identifiable resolved features onto one another and that is based on a Taylor-expanded lensing potential in the vicinity of a point on the critical curve within our model-independent lens characterisation approach. From the transformation matrix, the same information about the properties of the critical curve at fold and cusp points can be derived as we previously found when using the quadrupole moment of the individual images as observables. In addition, we read off the relative parities between the images, so that the parity of all images is determined when one is known. We compare all retrievable ratios of potential derivatives to the actual values and to those obtained by using the quadrupole moment as observable for two- and three-image configurations generated by a galaxy-cluster scale singular isothermal ellipse. We conclude that using the quadrupole moments as observables, the properties of the critical curve are retrieved to a higher accuracy at the cusp points and to a lower accuracy at the fold points; the ratios of second-order potential derivatives are retrieved to comparable accuracy. We also show that the approach using ratios of convergences and reduced shear components is equivalent to ours in the vicinity of the critical curve, but yields more accurate results and is more robust because it does not require a special coordinate system as the approach using potential derivatives does. The transformation matrix is determined by mapping manually assigned reference points in the multiple images onto one another. If the assignment of the reference points is subject to measurement uncertainties under the influence of noise, we find that the confidence intervals of the lens parameters can be as large as the values themselves when the uncertainties are larger than one pixel. In addition, observed multiple images with resolved features are more extended than unresolved ones, so that higher-order moments should be taken into account to improve the reconstruction precision and accuracy.

Workspace and Joint Space Analysis of the 3-RPS Parallel Robot

The Accurate calculation of the workspace and joint space for 3 RPS parallel robotic manipulator is a highly addressed research work across the world. Researchers have proposed a variety of methods to calculate these parameters. In the present context a cylindrical algebraic decomposition based method is proposed to model the workspace and joint space. It is a well know feature that this robot admits two operation modes. We are able to find out the set in the joint space with a constant number of solutions for the direct kinematic problem and the locus of the cusp points for the both operation mode. The characteristic surfaces are also computed to define the uniqueness domains in the workspace. A simple 3-RPS parallel with similar base and mobile platform is used to illustrate this method.

TWO-DIMENSIONAL COUPLED PARAMETRICALLY FORCED MAP

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible nonsingular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of cylindric algebraic decomposition, Gröbner bases, and discriminant varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.

Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem

I. Vekua’s integral representations of holomorphic functions, whosem-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whosem-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue spaceLp(⋅)(Γ;ω)with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certainpandω. This makes it possible to obtain analogues ofI. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight functionω(t)and a functionp(t).

Considering the Ability for Nonsingular Transitions of Assembly Mode for Dimensional Synthesis

An important difficulty in the design of parallel manipulators is their reduced practical workspace, due mainly to the existence of a complex singularity locus within the workspace. The workspace is divided into singularity-free regions according to assembly modes and working modes, and the dimensioning of parallel manipulators aims at the maximization of those regions. It is a common practice to restrict the manipulator’s motion to a specific singularity-free region. However, a suitable motion planning can enlarge the operational workspace by means of transitions of working mode and/or assembly mode. In this paper, the authors present an analytical procedure for obtaining the loci of cusp points of a parallel manipulator as algebraic expressions of its dimensional parameters. The purpose is to find an optimal design for non-singular transitions to be possible.