Planar Case Research Articles

The problem of "gymnast" motion control in free flight and during the transition in the state of contact with the support

The study from the position of theoretical mechanics discusses some aspects of the jump down of the "gymnast" – athlete (or the robot – "gymnast") from the "crossbar". We discuss all phases of the jump in the planar case. "Gymnast" body is modeled as three-links physical pendulum, however, after departing from the "crossbar" and lowering the "hands" model the dynamic system becomes two-links pendulum. For a two-links model in the regime of "kinematic" control, we deal with the process of lowering hands, free flight, the output of the "legs" on the frame (absolutely inelastic impact) and holding operation in support phase with the aim of vertical stabilize of the pendulum system. Mode "kinematic control" means the ability to instantly change the angle between the links of the body (in some limits). For each of the phases of movement, we found a convenient form of describing dynamical equations. These equations are based on the use as a variable momentum of the system relative to various points of the body or space. The order of this system of equations is lower than for the full order system. The stage of calm "gymnast" that occurs after the foot on the surface of the support, will also be examined on the basis of special system of equations of this type. It was shown, that we can use numerical analysis to build the region of controllability for transition of the two-links model in a state of stabilization, corresponding to the equality of the horizontal coordinate of the support foot and the center of gravity of the two-links pendulum system. The algorithm of stabilizing control is designed. The results of presented of the analysis of the problem allow us to construct a convenient approximate model of the phenomenon as a whole, and to use it to control a robotic counterpart. As an example, it is considered one of the cases of motion that corresponds to the anthropomorphic model.

Combustion of rich hydrogen–air mixture stabilised near a cylindrical porous burner

In this paper, the effect of geometry on the dynamics of the hydrogen–air flame stabilised on the surface of a burner is investigated numerically. Two main regimes of combustion front stabilisation near the cylindrical burner: attached and detached flames are investigated and compared to the planar case. The neutral stability boundary for diffusive-thermal instability is studied in detail. The attached flame regime is characterised by the fact that for rich mixture compositions the flame becomes less stable in comparison to the combustion front stabilised near a planar burner. The detached flame regime is influenced by the divergent outflow, which significantly perturbs the flow field in the product regions of the flame. In contrast to the detached regime in the attached regime curvature stabilises the flame front. The flame properties investigated in this study are extremely sensitive to chemical kinetic models. The results and observations made can be used to support experimental design to set-up experiments for additional validation of detailed mechanisms of chemical reactions.

SMALLNESS OF THE SET OF CRITICAL VALUES OF DISTANCE FUNCTIONS IN TWO‐DIMENSIONAL EUCLIDEAN AND RIEMANNIAN SPACES

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

On strong solutions of fractional nonlinear viscoelastic model of Voigt type

This paper concerns the existence and uniqueness of the strong solutions to the initial‐boundary value problems of fractional analog of Voigt model in the planar case.

Capturing an Asteroid via Triangular Libration Points

This paper develops a new asteroid capture mechanism via triangular libration points based on their stable structure and chaotic motion nearby. Different from collinear libration points, the triangular libration points are linearly stable surrounded by long- and short-period orbit families in the planar case. First, their dynamical structure is explored by the Lie series method and an analytical approximation of long-period orbits is proposed. Then, massive escaping trajectories from the parking orbits are generated by a chaotic-assisted method on the finite time Lyapunov exponent map. The trajectories from the asteroid’s natural orbit to the Earth–moon system are constructed by modified Lambert transfers. Next, the fuel consumption for all potential near-Earth asteroids is evaluated ergodically in both perihelion and aphelion cases based on their initial condition, which guides the selection of the target. Finally, it is applied to 20 asteroids, and the minimum fuel consumption () appears capturing asteroid 2015DU. Furthermore, a postcapture maneuver strategy based on a Poincaré section is proposed using long-period orbits to adjust the parking orbits and to satisfy the mission requirements.

Computer Simulation of Motion of an Inhomogeneous Granular Medium

A model for the motion of a granular medium is proposed, which is based on the elastic interaction of particles each of which is a perfectly rigid body (a disk in the planar case and a sphere in the spatial case) of mass m and radius r, surrounded by a sufficiently thin elastic shell. An algorithm for solving the formulated problem is proposed. At each step, the forces and moments of forces arising from the interaction of a particle with neighboring particles are calculated. The Cauchy problem for the resulting system of differential equations was numerically solved. Based on the mathematical model, a set of programs was developed that simulates the behavior of granular media in moving containers.

Geometrical and electronic properties of unstrained and strained transition metal dichalcogenide nanotubes

We study geometries and electronic properties of unstrained and strained zigzag $\mathrm{Mo}X{}_{2}$ ($X$ = O, S, and Se) nanotubes and $\mathrm{Mo}XY$ ($X,Y$ = O, S, and Se) nanotubes in the framework of density functional theory. Using the local density approximation, we obtain fully relaxed geometries of (10,0) and (14,0) nanotubes for all $\mathrm{Mo}X{}_{2}$, and (7,0) and (28,0) nanotubes for selected $\mathrm{Mo}X{}_{2}$. From the detailed analysis of the optimized geometries it is revealed that, although in the planar case these transition metal dichalcogenides (TMDs) normally take so-called H structures, zigzag nanotubes turn into geometries between H-like and T-like nanotubes having different height for inner and outer chalcogen site. Electronic property investigations reveal that TMD nanotubes without oxygen atoms are found to have narrower gaps than TMD nanotubes with oxygen atoms. These narrow direct gaps of TMD nanotubes should be attractive properties for infrared optoelectronic devices. On the other hand, energy-gap values of the TMD nanotubes with oxygen atoms are found to depend rather strongly on the strain. Therefore, they should be potential future device materials for mechanical sensors. We also find a global strong correlation between the fundamental-gap values and some combined structural parameters in strained $\mathrm{Mo}X{}_{2}$ ($X$ = S, Se) nanotubes and planar $\mathrm{Mo}X{}_{2}$ ($X$ = S, Se), which will be useful to predict gap values of other strained and unstrained ($n,0$) nanotubes.

Variational approximation of functionals defined on 1-dimensional connected sets in ℝ n

Abstract In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝ n {\mathbb{R}^{n}} . Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n ≥ 3 {n\geq 3} .

Runups of Unusual Size: Rogueness and Variability of Swash

AbstractThe occurrence of swash larger than those predicted by the Rayleigh distribution is examined using a phase‐resolved numerical model. Multiple realizations of spectral boundary conditions and beach configurations are considered in the analysis. Planar beaches characterized by surf similarity parameters between 0.3 and 0.6 are found to be more likely to produce unusually large swash. Maximum runup is found to correlate with the maximum infragravity (IG) wave height at the location of swash initiation, the landwardmost continuously wet point. The analysis is extended to beaches with bilinear profiles, where the foreshore slope is steeper than the shoaling slope. It is found that the still water level plays a controlling role in the magnitude of large swash. For these beaches it is found that the still water level with respect to the beach toe influences both the IG and shortwave height at swash initiation. Similar to the planar beach case, swash is found to be maximized when the sum of both IG and shortwave height is maximum; this occurs at toe depths 2–3 times the offshore significant wave height.

Oxidation-enhanced Si self-diffusion in isotopically modulated silicon nanopillars

Silicon (Si) self-diffusivity in a Si nanopillar under dry oxidation was quantitatively evaluated by atom probe tomography of Si isotope heterostructure interfaces. Dry oxidation of a nanopillar with 200 nm diameter at 920 °C for 4 h revealed that the Si self-diffusivity was the same as the one measured for the standard planar oxidation despite the fact that the diffusion region probed in the pillar was surrounded by approximately seven times more oxidation interface areas than the simple planar oxidation case. This finding can be understood by considering the large diffusion length of ∼300 μm of the interstitials for our thermal oxidation condition. The excess interstitials injected by the pillar oxidation as well as those injected by the oxidation of the base (100) plane can easily diffuse through the sample, including interiors of the pillars, making the concentration of the excess interstitials practically equal to those injected for the (100) planar oxidation case.

An iso-parametric $$\pmb {\mathrm {G}^1}$$-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case

A geometrically exact nonlinear iso-parametric $$\mathrm {G}^1$$ -conforming finite element formulation for the analysis of Kirchhoff rods, based on the cubic Bezier curve interpolation, is presented. In this work, the formulation is restricted to the planar 2D case. Introducing the $$\mathrm {G}^1$$ -map, the interpolation preserves the continuity requirement during the deformation process of the rod. In this way, the $$\mathrm {G}^1$$ -conformity is implicitly accounted at the element formulation level.

Optimization of Interplanetary Trajectories Using the Colliding Bodies Optimization Algorithm

In this paper, a recent physics-based metaheuristic algorithm, the Colliding Bodies Optimization (CBO), already employed to solve problems in civil and mechanical engineering, is proposed for the optimization of interplanetary trajectories by using both indirect and direct approaches. The CBO has an extremely simple formulation and does not depend on any initial conditions. To test the performances of the algorithm, missions with remarkably different orbital transfer energies are considered: from the simple planar case, as the Earth-Mars orbital transfer, to more energetic ones, like a rendezvous with the asteroid Pallas.

The Insphere of a Tetrahedron

A contiguous derivation of radius and center of the insphere of a general tetrahedron is given. Therefore a linear system is derived. After a transformation of it the calculation of radius and center can be separated from each other. The remaining linear system for the center of the insphere can be solved after discovering the inverse of the corresponding coefficient matrix. This procedure can also be applied in the planar case to determine radius and center of the incircle of a triangle.

Constrained Manipulability for Humanoid Robots Using Velocity Polytopes

Robot performance measures are important tools for quantifying the ability to carry out manipulation tasks. Generally, these measures examine the system’s kinematic transformations from configuration to task space. This means that environmental constraints are neglected in spite of the significant effects they may have on the robot’s admissible motions. In this paper, we propose a new measure called the constrained manipulability polytope (CMP) that considers the system’s kinematic structure, including closed chains or composite sub-mechanisms, joint limits and the presence of obstacles. For an illustrative planar case, we demonstrate how the CMP can evaluate a robot’s performance in a cluttered scene and how this evaluation can be extrapolated to obtain a workspace visualization. Additionally, we show the advantages and limitations of the CMP compared to the state of the art. Furthermore, the method is demonstrated both in simulation and experimentally for NASA’s Valkyrie robot. We show how the CMP provides a measure for single-arm and dual-arm manipulation tasks, analyze the workspace and be used to optimize the robot’s posture.

On regular reflection to Mach reflection transition in inviscid flow for shock reflection on a convex or straight wedge

The regular reflection to Mach reflection ( ) transition in inviscid perfect air for shock reflection over convex and straight wedges is investigated. Provided that the variation of shock intensity only has a second-order effect on the wave transition, the possible cases for the occurrence of the transition for curved shock reflection over a wedge are discussed. For a planar shock reflecting over a convex wedge, four different flow regions are classified and the mechanism of the disturbance propagation is interpreted. It is found that the flow-induced rarefaction waves exist between disturbances generated from neighbouring positions and isolate them. For a curved shock reflecting over a convex wedge, although the distributions of the flow regions are different from those in planar shock reflection, the analysis for the planar shock case can be extended to curved shock cases as long as the wedge is convex. When a diverging shock reflects off a straight wedge, the flow-induced rarefaction waves are absent. However, the disturbances generated earlier cannot overtake the reflection point before the pseudo-steady criterion is satisfied. In the cases considered, the flow in the vicinity of the reflection point will not be influenced by the unsteady flow caused by the shock intensity and the wedge angle variations. This is clearly a local property of the shock–wall interaction, no matter what the history of the shock trajectory is. For validation, extensive inviscid numerical simulations are performed, and the numerical results show the reliability of the pseudo-steady criterion for predicting the transition on a convex wedge.

On Some Average Properties of Convex Mosaics

In a convex mosaic in we denote the average number of vertices of a cell by and the average number of cells meeting at a node by Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the plane (except for the unphysical strips). Nevertheless, in d = 3 dimensions if we plot the 28 points corresponding to convex uniform honeycombs, the 28 points corresponding to their duals and the 3 points corresponding to Poisson-Voronoi, Poisson-Delaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the plane. To explore this phenomenon we introduce the harmonic degree of a d-dimensional mosaic. We show that the observed narrow strip on the plane corresponds to a narrow range of We prove that for every there exists a convex mosaic with harmonic degree and we conjecture that there exist no d-dimensional mosaic outside this range. We also show that the harmonic degree has deeper geometric interpretations. In particular, in case of Euclidean mosaics it is related to the average of the sum of vertex angles and their polars, and in case of 2 D mosaics, it is related to the average excess angle.

Space Debris Attitude Control During Contactless Transportation in Planar Case

The use of ion flow for the implementation of contactless force action transmission from an active spacecraft to a passive object is a promising technology for the creation of active large space debris removal systems. Keeping a certain orientation of large space debris with respect to the oncoming ion stream in the process of its removal from the orbit can significantly reduce the time for this transport operation. The aim of the work is to develop a control law that stabilizes the space debris in a certain position. A cylindrical spent rocket stage is considered as the space debris. The planar motion of a mechanical system consisting of the active spacecraft and the space debris is considered. The two ways to control the ion flux are considered and compared with each other: by changing the thrust of the ion engine and by turning the ion flow direction. Lyapunov’s theorems on stability and asymptotic stability in the first approximation and Bellman method are used to build the control laws. The results of numerical simulation prove that controlling the ion flow direction is a more efficient way of stabilizing attitude motion in terms of minimizing the time spent.

Morphology of cylindrical cell sheets with embedded contractile ring

The behavior of large deformations of cellular tissues is usually affected by the local properties of cells and their interactions, resulting in folding which acts as an important role in the embryonic development, as well as growing and spreading of a tumor, which can rapidly promote the stereo complexity of the architecture of the tissues. In the present study, a cylindrical vertex model is constructed to explore the morphology of the tubular cell sheets subject to an embedded contractile ring. It is found that an inner region of the contractile ring in equilibrium will protrude from the tube wall, and it will suddenly collapse when the contractile strength exceeds a threshold, indicating the occurrence of a bifurcation. These results on the effect of embedded contraction in the tubular shell are quite different from the planar cases, which can reveal the importance of the interaction between the geometric and material non-linearity in cylindrical geometry. The dependence of the large deformation on the bending modulus parameters and contraction strength is also analyzed for the cylindrical cell shell.

3-D slip analyses of listric faults with ideal geometries

Abstract Studying the kinematics of listric faults is of great importance in petroleum geology and seismicity. The deformation related to listric faults differ from the planar fault plane case. This study involves 3-D co-ordinate geometry with spherical co-ordinates to deduce the effective-slip ( e s ) and the net-slip ( n s ) of listric faults of three ideal geometries: spherical, paraboloid and ellipsoidal. We perform 3D slip analyses on two types of listric faults (Type-1 and 2). Previous authors have considered these shapes of fault planes in their modeling. Using these e s , one can track material points before and after faulting, similar to Mukherjee (2019) applied on translational faults with planar fault planes. The n s can be divided into a strike-slip ( s n s ) and a dip-slip component ( d n s ). This resolution being done on the listric fault plane itself, n s ≠ ( s n s 2 + d n s 2 ) 0.5 . This is unlike the planar fault plane case. The purpose of 3D-geometric analysis of listric faults could be, in long run, the restoration of rock blocks that underwent listric faulting.

Circularly polarized flexible antenna on liquid crystal polymer substrate material with metamaterial loading

AbstractIn this work, a metamaterial loaded circularly polarized (CP) flexible CPW fed antenna is presented. The flexible behavior of the antenna is analyzed and tested at various bending angles. The overall dimension of the liquid crystal polymer (LCP) antenna is 0.44 λ0 × 0.37 λ0, which is having thickness of 0.1 mm. The planar LCP antenna works at WiMAX (3.5 GHz), WLAN (5.8 GHz), and at vehicular communication band (5.9 GHz). Using metamaterial exhibiting split ring resonators and CSRR placement on the either side of the feedline and ground, we obtained circular polarization. Both planar and bending antenna cases are analyzed and results such as axial ratio, reflection coefficient, and radiation patterns with respect to planar and bending angles are validated with measurement results.