Path Equation Research Articles

Rigorous treatment of barycentric stellar motion

High-precision astrometric and radial-velocity observations require accurate modelling of stellar motions in order to extrapolate measurements over long time intervals, and to detect deviations from uniform motion caused for example by unseen companions. We aim to explore the simplest possible kinematic model of stellar motions, namely that of uniform rectilinear motion relative to the Solar System Barycentre, in terms of observable quantities including error propagation. The apparent path equation for uniform rectilinear motion is solved analytically in a classical (special-relativistic) framework, leading to rigorous expressions which relate the (apparent) astrometric parameters and radial velocity to the (true) kinematic parameters of the star in the barycentric reference system. We present rigorous and explicit formulae for the transformation of stellar positions, parallaxes, proper motions, and radial velocities from one epoch to another, assuming uniform rectilinear motion and taking into account light-time effects. The Jacobian matrix of the transformation is also given, allowing accurate and reversible propagation of errors over arbitrary time intervals. The light-time effects are generally very small but exceeds 0.1 mas or 0.1 m/s over 100 yr for at least 33 stars in the Hipparcos Catalogue. For high-velocity stars within a few tens of pc from the Sun light-time effects are generally more important than the effects of the curvature of their orbits in the Galactic potential.

Principle and Design of Noncircular Pin-Cycloid Gear Transmission

Noncircular pin-cycloid transmission is proposed,which can implement variable gear ratio transmission between parallel axes. The meshing coordinate system of the noncircular pincycloid gears is set up,with the forming principle of the conjugate profile described. The tooth profile equation of the noncircular pin gear is established,then that of the cycloidal gear is derived through coordinate system transformation,based on the conjugate tooth profile theory,the circular pin gear cutting tool profile equation,and the noncircular pin gear cutting tool path and mesh equation. The parametric design of the gear pair is obtained by Matlab program. The design example of the noncircular pin-cycloid gear is achieved. And the three dimension model is established by SolidWorks. Based on the kinematics analysis of the noncircular pin-cycloid drive by Adams software,the validity of the principle and design of noncircular pin-cycloid drive is verified through the contrast analysis between the experimental and theoretical arithmetic results.

Unified approach of filament winding applied to complex shape mandrels

The filament winding process faces up limiting fabrication inconveniences when designing complex geometries of composite structures. Even the complete coverage of a cylindrical mandrel requires introducing deviation from geodesic trajectories. As a consequence, models for non-geodesic paths have been developed. The present research aims to establish, to solve and to validate a generic mathematical model that contributes either to wind complex shapes, or to solve common filament winding disadvantages, on the basis of an integrated strategy. This so-called unified approach leads to benefit of composite structures made by filament winding despite the limitations of the manufacturing process. Based on the mathematical description of the mandrel geometry, the theory of surfaces leads to express the local curvatures. Considering the slippage tendency of the fiber tow over the surface, a local stability criterion involving mathematical parameters of the mandrel surface is established, and a general fiber path equation can be formulated. A numerical tool is developed and applied to predict the evolution of the filament winding angle of the fiber tow placed over the surface of two axisymmetric geometries: a convex and a concave one. Experimental validation is carried out by manufacturing these geometries using a four axis filament winding machine.

Quadrant based incremental planning for mobile robots

Path planning of a mobile robot in the presence of multiple moving obstacles is found to be a complicated problem. A planning algorithm capable of negotiating both static and moving obstacles in an unpredictable (on-line) environment is proposed. The proposed incremental algorithm plans the path by considering the quadrants in which the current positions of obstacles as well as target are situated. Also, the governing equations for the shortest path are derived. The proposed mathematical model describes the motion (satisfying constraints of the mobile robot) along a collision-free path. Further, the algorithm is applicable to dynamic environments with fixed or moving targets. Simulation results show the effectiveness of the proposed algorithm. Comparison of results with the improved artificial potential field (iAPF) algorithm shows that the proposed algorithm yields shorter path length with less computation time.

Tailoring Strongly Nonlinear Negative Stiffness

Negative, nonlinear stiffness elements have been recently designed as configurations of pairs or groups of linear springs. We propose a new design of such a system by using a single linear spring with its moving end rolling on a path described by an equation of varying complexity. We examine the effect that the selection of the path has on the size of the deflection regime where negative stiffness is achieved. The stability properties of the equilibrium positions of the system are also investigated, highlighting the influence that the complexity of the path equation brings. The latter naturally affects the characteristics of the forcing functions around these positions. It is demonstrated that the properties of the system can be tailored according to the nature of the equation used and we show how essentially nonlinear negative stiffness elements, (i.e., with no linear parts) can be designed. These results provide a useful standpoint for designers of such systems, who wish to achieve the desired properties in reduced space, which is a common requirement in modern applications.

How to eliminate non-positive circuits in periodic scheduling: a proactive strategy based on shortest path equations

Usual periodic scheduling problems deal with precedence constraints having non-negative latencies. This seems a natural way for modelling scheduling problems, since task delays are generally non-negative quantities. However, in some cases, we need to consider edges latencies that do not only model task latencies, but model other precedence constraints. For instance in register optimisation problems devoted to optimising compilation, a generic machine or processor model can allow considering access delays into/from registers. Edge latencies may be then non-positive leading to a difficult scheduling problem in presence of resources constraints. This research result is related to the problem of periodic scheduling with storage requirement optimisation; its aims is to solve the practical problem of register optimisation in optimising compilation. We show that pre-conditioning a data dependence graph (DDG) to satisfy register constraints before periodic scheduling under resources constraints may create circuits with non-positive distances, resulted from the acceptance of non-positive edge latencies. As a compiler construction strategy, it is forbidden to allow the creation of circuits with non-positive distances during the compilation flow, because such DDG circuits do not guarantee the existence of a valid instruction schedule under resource constraints. We study two solutions to avoid the creation of these problematic circuits. A first solution is reactive, it tolerates the creation of non-positive circuit in a first step, and if detected in a further check step, makes a backtrack to eliminate them. A second solution is proactive, it prevents the creation of non-positive circuits in the DDG during the register optimisation process. It is based on shortest path equations which define a necessary and sufficient condition to free any DDG from these problematic circuits. Then we deduce a linear program accordingly. We have implemented our solutions and we present successful experimental results.

Central force problem in space with SU(2) Poisson structure

The central force problem is considered in a three dimensional space in which the Poisson bracket among the spatial coordinates is the one by the SU(2) Lie algebra. It is shown that among attractive power-law potentials it is only the Kepler one that all of its bound-states make closed orbits. The analytic solution of the path equation under the Kepler potential is presented. It is shown that except the Kepler potential, in contrast to ordinary space, all of the potentials for which all of the almost circular orbits are closed are non-power-law ones. For the non-power-law potentials examples of the numerical solutions of the path equations are presented.

Computational investigation of electron path inside SEM chamber in mirror effect phenomenon

Present work aims at describe the behaviour of an accelerated probing electron that orientated towards a charged insulator sample and hence producing mirror images. The distribution of the trapped charges at the sample surface is approximated as a point charge. Hence, analytical derivation for the path equation of this electron has been presented. The derived model expresses the probing electron path in terms of the scanning potential, incident angle, trapped charges and the sample relative permittivity. Some of experimental data are import from published literatures so as to justify the introduced procedure. The obtained results have shown that important information could be predictable through the use of this procedure. Hence this procedure may be considered as a tool for describing, analysing and providing hints for further investigation of electron mirror images.

Generalised model for heteroazeotropic batch distillation with variable decanter hold-up

A general model of batch heteroazeotropic distillation is proposed. Both liquid phases present in the decanter can be refluxed or withdrawn as distillate, their hold-up can be increased, decreased or kept constant, as well. By assuming maximal separation, that is, that the composition of the condensate always equals to that of the heteroazeotrope, the still path equation was derived. The still path directions are determined for all the 16 possible operational policies. It is possible to steer the still path in a desired direction by changing the operational parameters, which allows the recovery of a pure component in the still. The still path directions are validated by rigorous simulations for three policies not published yet using the mixture water – formaldehyde – propyl formate. From the 16 operational policies, 11 can be considered as useful in practice. To demonstrate the advantage of using a non-traditional policy, the separation of the mixture aniline – ethylene glycol – water was investigated, as well. By using a non-traditional operational policy with hold-up reduction in the decanter a higher purity of ethylene glycol was obtained in the still.

Modeling and predicting for surface topography considering tool wear in milling process

This paper presents a model for the prediction of surface topography considering tool wear during the milling process. First, the cutting edge path equation, which can be transformed into equivalent polynomial equations and solved for discrete positions along the feed direction, is established including the effect of tool wear. Then, cutting edge is divided into a series of cutting points and an algorithm is proposed to determine the range of divided position angle. Finally, surface topography model is established based on the established cutting path equation, the range of position angle, the calculated cutting time, and spiral lag angle. By using this model, surface topography generation is simplified with respect to other models in literature and the modeling method of surface topography does not need to mesh the workpiece and the model can easily be extended to include other factors on surface generation. Based on the established surface topography model, an algorithm is proposed to simulate generation of surface profile in milling operation. Experimental work and validation of the established model is performed on a five-axis milling center by using stainless steel 1Cr18Ni9Ti and cemented carbides milling cutter. Cutting test results about the topography generation of the plane and cylindrical surface show good agreement with model predictions.

First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and -Symmetries

We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.

Computer Simulation Technology of Precision Lapping Process of Polycrystalline Diamond

Polycrystalline diamond, (PCD) is currently used in the industry for cutting tools of difficult-to-machine materials because of its superior characteristics such as hardness, toughness and wear resistance. But the PCD material is difficult to machine due to its above properties. Before making the PCD blank in varies tools, the lapping process is essential and necessary. The lapping quality depends heavily on the parameters. Therefore it is of great significance to study the path and velocity of lapping process for ensuring the lapping quality and uniform wear of abrasive disk. In this paper, on the basis of kinematics and geometry principle, the path equation was established. The complicated lapping process can be converted into two rotation movements, i.e. the movement of part against the center of lapping disk and rotation of fixture around its center. The relative path and relative velocity of random point of the part against abrasive disk were also analyzed and calculated. The computer simulation software was developed for figuring the path curves and instantaneous velocity of the lapping process. The results show that the abrasive disk can wear uniformly and the lapping quality can be raised. The application of computer simulation software provides the possibility to analyze the lapping principles, which is a simple, convenient and effective method.

Analysis on Isothermal Elastohydrodynamic Lubrication of Orthogonal Face Gear

Face-gear drives have been applied in aviation transmissions, in particular, helicopter transmissions, and the lubrication characteristics are an important indicator for estimating the load-carrying capacity of face-gear teeth. In order to analyze the lubrication performance of the face gear under load, equations for the contact path of the face gear loaded tooth contact analysis (LTCA) were established on the basis of the load equivalent error of alignment (LEEA) and the load distribution among the teeth was calculated. Then a method for calculating the contact area and tooth surface velocity of face-gear drives was studied. Face-gear isothermal elastohydrodynamic lubrication (EHL) dimensionless equations are presented. A multigrid algorithm was used to complete the solution of the minimum film thickness and film pressure of face-gear drives. The lubricant film thickness and film pressure variation in the mesh cycle are expressed using example calculations that provide a theoretical basis for face-gear lubrication design.

The Technology Research on Milling Large-Size Spiral Bevel Gears of the Finger Cutter

This paper refers to the finger cutter forming method to process large-size spiral bevel gears by using processing principles, tools, processing technology and path equations. In the actual production process, the method is made a rapid development for the revolutionary innovative process of large spiral bevel gear. The method of research is different from the disc cutter, so the processing of large spiral bevel gears have a new processing method to achievement.

The vacuole model: new terms in the second order deflection of light

The present paper is an extension of a recent work (Bhattacharya et al. 2010) to the Einstein-Strauss vacuole model with a cosmological constant, where we work out the light deflection by considering perturbations up to order M3 and confirm the light bending obtained previously in their vacuole model by Ishak et al. (2008). We also obtain another local coupling term −5πM2Λ/8 related to Λ, in addition to the one obtained by Sereno (2008, 2009). We argue that the vacuole method for light deflection is exclusively suited to cases where the cosmological constant Λ disappears from the path equation. However, the original Rindler-Ishak method (2007) still applies even if a certain parameter γ of Weyl gravity does not disappear. Here, using an alternative prescription, we obtain the knownterm −γR/2, as well as another new local term 3πγM/2 between M and γ. Physical implications are compared, where we argue that the repulsive term −γR/2 can be masked by the Schwarzschild term 2M/R in the halo regime supporting attractive property of the dark matter.

Tool Positioning Algorithm Based on Smooth Tool Paths for 5-axis Machining of Sculptured Surfaces

The current research of the 5-axis tool positioning algorithm mainly focuses on searching the local optimal tool position without gouging and interference at a cutter contact(CC) point, while not considering the smoothness and continuity of a whole tool path. When the surface curvature varies significantly, a local abrupt change of tool paths will happen. The abrupt change has a great influence on surface machining quality. In order to keep generated tool paths smooth and continuous, a five-axis tool positioning algorithm based on smooth tool paths is presented. Firstly, the inclination angle, the tilt angle and offset distance of the tool at a CC point are used as design variables, and the machining strip width is used as an objective function, an optimization model of a local tool positioning algorithm is thus established. Then, a vector equation of tool path is derived by using the above optimization model. By analyzing the equation, the main factors affecting the tool path quality are obtained. Finally, a new tool position optimization model is established, and the detailed process of tool position optimization is also given. An experiment is conducted to machine an aircraft turbine blade by using the proposed algorithm on a 5-axis blade grinding machine, and the machined blade surface is measured with a coordinate measuring machine(CMM). Experimental and measured results show that the proposed algorithm can ensure tool paths are smooth and continuous, improve the tool path quality, avoid the local abrupt change of tool paths, and enhance machining quality and machining efficiency of sculptured surfaces.

Dynamical System Characterization of the Central Path and Its Variants—A Revisit

The notion of central path plays a fundamental role in the development of interior point methods which, in turn, have become important tools for solving various optimization problems. The central path equation is algebraic in nature and is derived from the KKT optimality conditions of a certain logarithmic barrier problem; meanwhile, the primal variable portion of the very same central path can also be cast precisely as the integral curve, known as the affine scaling trajectory, of a certain gradient-type dynamical system. The justification is easy to establish in the context of linear programming. Though expected, the generalization of such a concept to semidefinite programming is not quite as obvious due to the difficulty of addressing noncommutativity in matrix multiplication. This paper revisits the dynamical system characterization of these flows and addresses the needed details for extension to semidefinite programming by means of a simple notion of operators and a specially defined inner product. F...

Study of Structure and Properties of New Rolling-Cut Shear

Based on rolling-cut shear simulation, using a kind of single-shaft and double eccentricity rolling-cut shear, which adopts a new structure of asymmetric feature and negative bias, as the calculating model by establishing motion path equation of spatial shear mechanism, comparing with the steel shear forces, link forces and horizontal link force components with or without asymmetric feature, the asymmetric formulation is deduced. Such asymmetric crank structure can decrease horizontal force component between the linkages during rolling-cut process, increase the effective drive force on links while it comes to the maximum shear force, and decrease the extrusion of blade arc on steel edge as well. Theoretical analysis and steel-shearing quality at site indicate that asymmetric and negative bias is an important and efficient way to prolong the lifetime of blade, decrease blade wear, improve shearing quality, and maintain the constant clearance between blades.

A kinematical approach to gravitational lensing using new formulae for refractive index and acceleration

This paper uses the Schwarzschild metric to derive an effective refractive index and acceleration vector that account for relativistic deflection of light rays, in an otherwise classical kinematic framework. The new refractive index and the known path equation are integrated to give accurate results for travel time and deflection angle, respectively. A new formula for coordinate acceleration is derived which describes the path of a massless test particle in the vicinity of a spherically symmetric mass density distribution. A standard ray-shooting technique is used to compare the deflection angle and time delay predicted by this new formula with the previously calculated values, and with standard first order approximations. Finally, the ray shooting method is used in theoretical examples of strong and weak lensing, reproducing known observer-plane caustic patterns for multiple masses.

Group classification for path equation describing minimum drag work and symmetry reductions

The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered (Pakdemirli, M. The drag work minimization path for a flying object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science223 (5), 1113–1116 (2009)). The Lie group theory is applied to the general equation. The group classification with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates.