Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled with Piecewise Nonlinear Energy Sink

The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behaviour of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behaviour of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectrums on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

A Stable and High-Precision Downward Continuation Method of Magnetic Data

Downward continuation is an effective technique that can be used to transform the magnetic data measured on one surface to the data that would be measured on another arbitrary lower surface. However, it suffers from amplitude attenuation and is susceptible to noise, especially when the continuation distance is large. To solve these problems, we present a stable and high-precision downward continuation method combining the ideas of equivalent source technique, Tikhonov regularization, radial logarithmic power spectrum analysis, and constrained strategy. To implement this method, the observed data is used to construct the equivalent source in the study area, and the small amount of measured magnetic data at the lower surface (relative to the original observation surface) is employed to constrain the calculation procedure simultaneously. Then the magnetic data at the target surface can be obtained by using a forward calculation procedure instead of the risky downward continuation procedure. The proposed method is tested on both synthetic model data and real magnetic data collected in the South China sea. Various obtained results demonstrate that the method reported in this study has higher accuracy and better noise resistance than the traditional downward continuation methods.

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.

Revisiting non-convexity in topology optimization of compliance minimization problems

PurposeThis is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. The authors trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods.Design/methodology/approachStarting from the global optimum of the compliance minimization problem, the authors employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum and how the initial guess has some weight in determining the final optimum.FindingsThe non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application.Originality/valueIn this article, the authors present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. The authors put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.

Homotopic approach for turnpike and singularly perturbed optimal control problems

The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

Bifurcation analysis for a novel flight vehicle with pitch-control single moving mass

Moving mass flight vehicle is a strongly nonlinear system under high speed flying conditions. The system attitude dynamics becomes even more complex due to the coupling between the internal moving mass with large mass ratio and the vehicle body. This article investigates the open-loop nonlinear dynamics of a novel flight vehicle with pitch-control single moving mass from the prospective of bifurcation theory and continuation methods. Of particular interest is the influence of moving mass parameters on the number of system equilibrium points, stability of equilibrium curves, bifurcation characteristics, and the longitudinal static stability. Numerical results reveal the bifurcation phenomena existing in the proposed flight vehicle; the generated bifurcation diagrams illustrate that the multiple sets of limit points and Hopf points divide the moving mass parameter space into different regions with different values and types of stability, thus indicating the significant role of the moving mass parameters in the system nonlinear dynamics. Finally, a design strategy for the moving mass parameters is concluded based on the bifurcation analysis results.