Mesh Equations Research Articles

Toward computerized determination of envelope to family of parametric planar curves

This paper introduces an original analytical procedure for the computation of the planar curve in meshing with a general two dimensional curve. Considering a planar gear mechanism with a constant transmission ratio, in this work a general expression of the explicit solution of the equation of meshing is presented. Besides, an original method to evaluate the undercutting conditions, based on the explicit solution of the equation of meshing, is shown. Taking into account a generating curve in homogeneous coordinates and the transformation matrixes which define the relative motion between the profiles, the solution of the equation of meshing and consequently the profile conjugated to the generating curve can be directly found, by a simple coordinate transformation. The explicit solution is expressed with the general conventions of the theory of gearing. Once the equation of the conjugated curve is obtained, a direct method for the evaluation of the undercutting condition can be applied. The proposed method allows to implement effective and affordable computer codes for the computation of conjugate planar profiles.

Dynamic simulation for internally heat-integrated distillation columns (HIDiC) for propylene–propane system

This paper reports a dynamic simulation study of the internally heat-integrated distillation column (HIDiC) using equilibrium-based models. First, three different HIDiC structures, i.e. an ideal HIDiC, a HIDiC with a pre-heater, and a HIDiC with a reboiler, are analyzed by control degrees of freedom (DOF). The reboiler is considered to be a necessary part of the HIDiC from DOF analysis, thermodynamic analysis, and engineering judgment. Then, a heuristic HIDiC control configuration including a bottoms reboiler control is proposed. A modular structured simulator for dynamic distillation columns using MESH equations is developed. The simulator also considers: (1) variable column pressure on each tray of the rectifying section, (2) dynamic vapor holdup, and (3) dynamic energy balance. In addition, the SRK equation of state is employed for estimating thermodynamic properties. A typical medium-pressure HIDiC for separation of propylene and propane explored by Olujic et al. [Olujic, Z., Sun, L., de Rijke, A., & Jansens, P. J. (2006). Conceptual design of an internally heat-integrated propylene–propane splitter. Energy, 31, 3083] is adopted as numerical examples for dynamic simulation studies.

A corrected equation of space curve meshing

The micro-elastic meshing wheel transmission (MEMWT) mechanism is a novel mechanical mechanism based on the principle of space curve meshing, which is quite different from conventional surface meshing. Considering the impact of the diameter of the tines on meshing, a corrected equation of space curve meshing is worked out in this article and compared with a space curve meshing equation established in a previous study. The corrected equation should make the design of MEMWT mechanisms more valuable for small devices or microdevices with two crossed shafts under load.

Pareto front of ideal Petlyuk sequences using a multiobjective genetic algorithm with constraints

Petlyuk sequences are a very promissory option to reduce energy consumption and capital costs in distillation. The optimal design of Petlyuk sequence implies determining 8 integers and 3 continuous variables with the minor heat duty and number of stages possible, but reaching the specified purities. Note that the heat duty and the number of stages are variable in competition, since we cannot decrease indefinitely one without increasing the other. In other words, we have a multiobjective problem with constraints. In contrast with other numerical strategies proposed, we consider the search of optimal designs set, from minimum reflux ratio to minimum number of stages and all designs between them. This set of optimal designs can be achieved by means of Pareto front, which represents a set of optimal solutions not dominated for a multiobjective problem. In this work, we implemented a multiobjective genetic algorithm with constraints to obtain the Pareto front of Petlyuk sequences. This algorithm is coupled to Aspen Plus simulator, so the complete MESH equations and rigorous phase equilibrium calculations are used. Results show clear tendencies in the design of Petlyuk sequence, and can be used to develop a short design method. In addition, the tool developed can be used to optimize not only the distillation columns but also complete chemical and petrochemical plants.

An Optimization Approach to the Displacement Volumes for External Spur Gear Pumps

To increase volume displacement is always concerning in designing external gear pumps (EGPs). Therefore, an approach of optimization analysis intending to enhance their displacement capability is proposed in this study. Through that, design parameters of the spur gears in the pumps are systematically resulted to achieve their optimal volume displacements. Parametrically, a CAD model to visualize the designed gear of the optimal design is also created. Firstly, the study derives tooth profile equations by using the coordinate transformation and equation of meshing for gears on a rack cutter profile. Then, an analytic formula represented the enveloping area by an involute curve is derived. Next, volumes of output and trapped backflow of the EGP are achieved. Therefore the net output volume can be calculated accurately and efficiently. After that, the optimization analysis to maximize the volume displacement is performed. Through that, optimal design parameters for the pumps are achieved under assigned constraints for considerations of design and manufacturing. Additionally, influences of module, pressure angle, and addendum correction factor for the gears in the pumps on their displacement capability are also investigated. Additionally, flowrate fluctuation characteristics under different pressure angles of gears are finally discussed.

Gear geometry of cycloid drives

According to differential geometry and gear geometry, the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion. The correct meshing condition, contact line, contact ratio, calculating method for pin tooth’s maximum contact point are developed. Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1, 2, 3 and −1, respectively. A general method called enveloping method to generate hypocycloid and epicycloid is put forward. The correct meshing condition for cycloid pin wheel gearing is provided, and the contact line and the contact ratio are also discussed.

A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations

A new concept called the dominance of equidistribution is introduced for analyzing moving mesh partial differential equations for numerical simulation of blowup in reaction diffusion equations. Theoretical and numerical results show that a moving mesh method works successfully when the employed moving mesh equation has the dominance of equidistribution. The property can be verified using dimensional analysis. In several aspects the current work generalizes previous work where a moving mesh equation is shown to have this dominance of equidistribution if it preserves the scaling invariance of the underlying physical partial differential equation and uses a small, constant value for τ (a parameter used for adjusting response time of the mesh movement to the change in the physical solution). Also, cases with both constant and variable τ are considered here.

Mesh analysis for toroidal drive with roller teeth

In the current paper, for the toroidal drive with roller teeth, its meshing equation, equations of the contact lines, the limit curves, the induced normal curvature, and the angle of the relative velocity vector to the contact line are determined. Using the equations, contact lines, meshing zones, limit curves, induced normal curvatures, and angles of the relative velocity vector to the contact line are calculated for two kinds of the toroidal drives with roller teeth, respectively. The results are compared with those for the toroidal drive with ball teeth. For the roller tooth surface of the drive, the distribution of the mesh parameters has given. The changes of the mesh parameters along with drive parameters have been investigated. The results are compared with those for the toroidal drive with ball teeth. The results are useful for design and manufacture of the toroidal drive.

GIS-based two-dimensional numerical simulation of rainfall-induced debris flow

Abstract. This paper aims to present a useful numerical method to simulate the propagation and deposition of debris flow across the three dimensional complex terrain. A depth-averaged two-dimensional numerical model is developed, in which the debris and water mixture is assumed to be continuous, incompressible, unsteady flow. The model is based on the continuity equations and Navier-Stokes equations. Raster grid networks of digital elevation model in GIS provide a uniform grid system to describe complex topography. As the raster grid can be used as the finite difference mesh, the continuity and momentum equations are solved numerically using the finite difference method. The numerical model is applied to simulate the rainfall-induced debris flow occurred in 20 July 2003, in Minamata City of southern Kyushu, Japan. The simulation reproduces the propagation and deposition and the results are in good agreement with the field investigation. The synthesis of numerical method and GIS makes possible the solution of debris flow over a realistic terrain, and can be used to estimate the flow range, and to define potentially hazardous areas for homes and road section.

Profile of grinding wheel for grinding TI worm

TI worm drive is a novel kind of hourglass worm, which consists of involute helical gear and enveloping hourglass worm. Accurate grinding for TI worm is the key manufacture technology for TI worm drive being popularized and applied, which means to obtain the profile of grinding wheel corresponding to the current grinding method. With the analysis of the movement relationship and the characteristic of the generating surface, the problem of accurate grinding for TI worm is converted into grinding for involute helical by shaped wheel. According to the theory of gear mesh, mesh equation for grinding involute helicoid by grinding wheel is deduced, the equation of the axial section profile of grinding wheel which can accurately grind TI worm is gained.

Surface Generation and Fabrication of Roller Gear Cam with Spherical Rollers

This paper derived the surface profile equations and the meshing equations of the spherical meshing elements from the homogeneous coordinate transformation matrix and the conjugate surface theory proved by the envelope theory. In the machining aspect, we demonstrated the position equation of cam profile using the coordinate transformation theory and obtained the perfected controlling position point of the tool path under the limitation of the chord errors. Finally, the design and the machining module was developed via Visual Basic that generated the cam profile points and the tool path points. The curves from UNIGRAPHICS to test and verify the cam profile equation. To verify the validation of the tool path, a cutting simulation software, VERICUT, is used to simulate the cutting geometry and kinematics of the roller gear cam, and demonstrate the practical application of the developed method.

Direct analytical solution of a modified form of the meshing equations in two dimensions for non-conjugate gear contact

The current technological state-of-the-art utilises modified gear profiles, which are in part non-conjugate and therefore cannot be analysed using standard conjugate contact theory. Existing non-conjugate mathematical models require the solution of a system of implicit equations, typically with significant computational effort and need for careful monitoring of solution stability, convergence and selection of initial values. This paper derives a modified form for the fundamental gear meshing equations, which are reduced analytically to a single scalar equation, resulting in improved solution speed and stability. The solution is verified in benchmark tests using real gear geometries.

Latency errors in the mathematical modelling and meshing characteristics of the toroidal drive

This paper reveals the latency errors in the manufacturing and assembly of the toroidal drive and integrates these errors in the coordinate systems and the mathematical models of the meshing equations for the conjugate surfaces between the planet worm-gear and the sun-worm and between the planet worm-gear and the stationary internal toroidal gear. The paper further develops the tooth profile model and the contact curves with the latency errors and examines the error effect on the helix model and the helix lead angle model. In addition, two kinds of limit curve model and the induced normal curvature models are further derived. This provides a comprehensive study of error effect on the meshing characteristics of the novel drive.

Generation of planar and helical elliptical gears by application of rack-cutter, hob, and shaper

The developed approaches are directed at obtaining of: (a) the generated surfaces of elliptical gears by enveloping process, and (b) computerization of the process of generation by application of existing equipment and tools. A matrix approach is proposed and developed for generation of equation of meshing. The developed theory is illustrated with examples of generation of planar and helical elliptical gears.

Geometric Design of Cycloidal Drives with Ceramic Ball Meshing Elements

Geometric design procedures for the cycloidal drive with ceramic ball meshing elements were introduced. Using the theory of conjugate surfaces, the equations of meshing of the cycloid drive and the cycloid raceway profiles were derived. Design examples were presented to demonstrate the design procedure and feasibility. The ceramic balls were used as the meshing elements. Prototype of this research was made and tested with satisfying results. The results of this work are suitable for computeraided design and manufacture of industrial applications.

Unstructured surface mesh adaptation using the Laplace–Beltrami target metric approach

This paper develops a set of adaptive surface mesh equations by using a harmonic morphism, which is a special case of a harmonic map. These equations are applicable both to structured and unstructured surface meshes, provided that the underlying surface is given in a parametric form. By representing the target metric of the mesh as a sum of a coarse-grained component and a component quadratic in surface gradients, an improved surface mesh may be obtained. The weak form of the grid equations is solved using the finite element approximation, which reduces the grid equations to a nonlinear, algebraic set. Examples of structured and unstructured meshes are used to illustrate the applicability of the proposed approach.

Manufacture of hypoid gear based on computer numerical control cutting machine

Based on the conceptual model of the five-axis simultaneous CNC bevel gear cutting machine, a new machining theory for the function-oriented active tooth surface of hypoid gear is proposed. The equation of meshing, control equation of the tooth depth and the second order geometric parameters of the gear tooth surface are established. A new algorithm for determining the motion function of machine tool, cutter velocity and point radius of cutter is given. In this process, there is no need for calculating of the moving acceleration and the acceleration rate of the cutter. According to this theory, the second order geometric parameters of the gear tooth surfaces and the designed tooth depth can be generated accurately. At the same time, above-mentioned manufacture method for the hypoid gear is universal and can be used to manufacture many kinds of point mesh gears.

A Rigorous Minimum Energy Calculation Method for a Fully Thermally Coupled Distillation System

Previously, Tanskanen and Pohjola (2000) and Tanskanen and Malinen (2005) have presented a method of rigorous minimum energy calculation for non-ideal multicomponent distillation. The method is based on column simulation with a large number of equilibrium stages to mimic infinitely high columns, and it has been successfully tested with simple distillation column systems. In this paper the rigorous minimum energy calculation method is extended to a fully thermally coupled distillation column system, also known as a Petlyuk column. This paper introduces not only the theoretical background of the applicable solving strategy, but also practical and useful solving tools capable of solving complex distillation column models. In the solving procedure, a modified bounded Newton homotopy method is used to detect the optimum point of operation for the column system, which is described with a full set of MESH equations. The results are promising and indicate that the numerical challenges encountered when rigorously solving complex distillation column problems can be tackled by utilising a modern numerical solving approach.

Geometric design of a pinion with two circularly arrayed conical teeth for roller drives

This article presents a mathematical model and geometric design algorithm for a new type of roller drive. The pinion has conical teeth in two circular arrays instead of one. This work is based on coordinate transformation and envelope theory, from which the equation of meshing of the cycloid drive is derived. The pinion profiles are the equidistant curves of the epicycloid profiles except the contour of the pinion conical tooth holes. Although there are twice as many pinion teeth as conventional rollers, their speed ratios are identical. This approach can design roller drives in which the pinion has two circular arrays of conical and cylindrical rollers. On the basis of these results, the corresponding solid modelling is constructed by CAD. Four examples are presented to demonstrate the feasibility of this approach. These examples can be a useful reference as a design case for other tooth profiles.

Numerical computing method of noncircular gear tooth profiles generated by shaper cutters

A simple and accurate numerical method was proposed for calculating the tooth profile of a noncircular gear. This method is directly based on the real gear shaping process, rather than deducing and solving complicated meshing equations used in the traditional method. The tooth profile is gradually obtained from the boundary produced by continuously plotting the cutter profile on the gear transverse plane. The key point of the method is picking up the graph boundaries. The relative position of the cutter profile on the gear transverse plane is determined by the given pitch line of the noncircular gear, parameters of the shaper cutter, and the shaping process data. In comparison with the traditional method, it is universal and is much more efficient and accurate, especially for noncircular gears, which have nontrivial pitch lines. Special problems in gear design and manufacturing, such as tooth pointing, undercut, and fillet interference, are included in the process. As an application example of the numerical method, a square internal gear is chosen from a new type of hydraulic motor with noncircular planetary gears, and the tooth profile of that gear is computed. The gear is successfully machined by electromagnetic discharge (EMD) using the resulting data.