Mesh Equations Research Articles

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.

Performance enhancement and energy reduction using hybrid cryogenic distillation networks for purification of natural gas with high CO2 content

A novel concept of hybrid cryogenic distillation network has been explored which maximizes the benefits of both desublimation or solid-vapor based separation as well as distillation or vapor-liquid equilibrium based separation during the separation of carbon dioxide from methane or natural gas. Process network synthesis has been performed for four case studies with high carbon dioxide (72 mole%) and medium carbon dioxide (50 mole%) natural gas feed streams. The benefits of optimal locations for cryogenic packed beds were investigated. A conventional cryogenic network consisting of multiple distillation columns with butane as additive for extractive distillation was also studied and presented in this paper. Process modeling of cryogenic distillation network with MESH equations was attempted using an integrated dual loop (C+3) convergence and the results were compared with Aspen Plus simulator for benchmarking. The prediction of solidification region was employed using experimental data from literature to avoid solidification regions in the column. The proposed hybrid cryogenic distillation network showed promising potential for energy and size reduction.

Modeling and Manufacture of a PC-Type Single-Screw Compressor Rotor

This study aimed to effectively design, manufacture and simulate a single-screw compressor. In this paper, a mathematical model of a pc-type single-screw compressor was developed. The pc-type single-screw compressor was assembled using two gate rotors, a screw rotor and a frame. First, the mathematical model of a conical tooth was designed and a gate rotor equation was obtained through coordinate transformation. Then, the coordinate systems between the gate rotor and the screw rotor were set up. Using a homogeneous coordinate transformation matrix, the family of gate rotor surfaces was obtained. Based on gear theory, the equations of meshing between the gate rotor and screw rotor were determined. Substituting the equation of meshing into the family of gate rotor surfaces, the gate rotor and the screw rotor were generated and their geometrical model illustrated using a software package. Based on the developed mathematical models, the contact lines and stress analysis between the gate rotor and screw rotor were displayed. An edge contact between the gate and screw rotors was found, indicating the necessity of a plastic or a composite material to encircle the conical surface. A computer-aided manufacturing software package was used to simulate the manufacture of the screw rotor.

Applications of the Exponential Function of the Euclidean Motion Group to the Theory of Gearing

The present paper provides a first step to a new approach to the theory of gearing, which uses modern differential geometry in order to ensure a strict and coordinate-independent formulation. Here, we are mainly concerned with a basic equation, namely, the equation of meshing, of two rotating surfaces in mesh. Since we are able to solve this equation by the time parameter, we derive parameterizations of the mating pinion from a bevel gear as well as a parameterization for gears produced by special machine tools.

Simulation of the multicomponent distillation of spearmint essential oil by a predictive Soave-Redlich-Kwong equation of state and comparison with experiments

In this study, enrichment in major component of spearmint oil has been studied by continuous multicomponent distillation. A mathematical model was evaluated to predict the enrichment by multicomponent distillation of spearmint oil. The aim of the work was to obtain simulation data to compare with experimental data of spearmint oil enriching. The simulation model is based on bubble point method ,the Wang-Henke algorithm and predictive Soave-Redlich-Kwong (PSRK) equation. In this method, MESH equations are solved and a diagonal matrix method for solving equations is used. The model was validated by experimental data obtained from a pilot-plant system, where a continuous distillation column with trays of one to five numbers has been used. The simulation was able to predict satisfactory the experimental data. The results of simulation and experiments show that the major component of spearmint essential oil is enriched from 31% to 62.04% by the distillation column with five trays. The results show that the addition of a tray to the distillation column with a low number of trays, have more effective relative to add a tray to the distillation column with higher trays. The results indicate that increasing of reflux ratio from 2 to 3 increases separation of the major component from 62.04% to 67.04%.

The Principle of Conjugate Curves for Gear Transmission

The concept of conjugate curves is put forward based on the definition of continuous and tangent contact, and the basic conditions of conjugation contact between a pair of conjugate curves are given. The fundamental principle of conjugate curves for gear transmission is provided: The normal vector to arbitrary direction of contact angle is calculated; the meshing equation is derived and the necessary and sufficient conditions of conjugation meshing for these curves are demonstrated simultaneously; the corresponding conjugated curve can be identified and line of action is also deduced with this method. A general conjugation model is therefore established according to the aforementioned theory and the result is demonstrated with parallel axis gearing as well as cylindrical helix. The geometric properties of conjugation in arbitrary contact angle direction are revealed and the characteristics of meshing motion of conjugate curves are discussed. The versatility of the principle and the outstanding contact features become obvious with treating any gear transmission. The freedom of selecting contact curve suggests the possibility of optimal conjugation design. The results provide a new method for the generation of gear tooth profile and lay a solid foundation for the further study of the gear meshing theory of conjugate curves.

Internal and external HIDiCs (heat-integrated distillation columns) optimization by genetic algorithm

HIDiC (Heat-Integrated Distillation Column) is an effective energy-saving configuration especially for the separation of close boiling point mixtures. In this work, the stochastic methodology has been applied for optimization of both internal and external HIDiCs. The use of GA (Genetic Algorithm) to find the optimal HIDiC configuration is presented while the fitness function is set to be the TAC (Total Annual Cost). HIDiC simulation has been performed based on the modified MESH equations using a rigorous thermodynamic model. Introducing a novel integer variable (the Layout number) leading to a more effective solution for the examined case study. This variable can generate systematically more energy efficient candidates for both internal and external HIDiCs. Benzene-toluene separation has been investigated by the proposed optimization procedure. The multivariable problem can be successfully optimized by GA while a good initial estimation is not essential. Based on the final results, up to 6.60% and 9.75% TAC reduction have been accomplished in external and internal HIDiCs optimization using the proposed method compared to the reported solutions in a previous work for the examined case study. However, VRC (Vapor Recompression Column) optimization at the end of presented work results 7.89% TAC reduction rather than optimal HIDiC.

A modified diffusion coefficient technique for the convection diffusion equation

A new modified diffusion coefficient (MDC) technique for solving convection diffusion equation is proposed. The Galerkin finite-element discretization process is applied on the modified equation rather than the original one. For a class of one-dimensional convection–diffusion equations, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. The application of the derived analytic formula of MDC is extended for other classes of convection–diffusion equations, where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction. The numerical results of the proposed approach for two-dimensional, variable coefficients, with boundary layers, convection-dominated problems show stable and accurate solutions even on coarse grids. Accordingly, multigrid based solvers retain their efficient convergence rates.

Analysis and Manufacture of Skew Bevel Gears with Spherical Involute

Skew bevel gears are fundamental element of transmission system. Processing quality of skew bevel gears has impact on the performance of system. An accurate processing method of tooth surface of skew bevel gears with spherical involute is proposed. According to the space meshing theory, the meshing equation of skew bevel gears is built by the theory of coordinate transform. And the generating principle of generation line of tooth surface about skew bevel gears is studied. In terms of the formation theory of tooth surface, moving process model of milling processing is established. And the generation line is as chord of face circular cutters to cut the tooth surface. Machining of the profile of skew bevel gears is performed using three axes linkage by the method. Finally, the transmission experiment proves that this method can improve dynamics and strength characteristics.

Calculation Equation of Tooth Face of Shaping Gear for Logarithmic Spiral Bevel Gear

According to the processing principle of the spiral bevel gear, analyzed the relative motion of cutter and workpiece in process. Combing the feature of logarithmic spiral bevel gear, this paper took the advantage of the gyration vector expansion to derivate the tooth surface equation of logarithmic spiral bevel gear shaping gear and deduced the meshing equation.

Divided wall distillation column: rationalization of degree of freedom analysis

The present work attempts to rationalize the degrees of freedom analysis of a divided wall distillation column for steady-state simulation using detailed mathematical model, which contains the MESHD equations (conventional MESH equation plus the pressure drop equation). In a divided wall distillation column, pressure difference plays an important role in deciding vapour and liquid flow rates across trays in the divided section. It is observed that for a given divided wall distillation column or Petlyuk column operating under steady state conditions with known feed and condenser pressure, the degrees of freedom is four. This is one higher than that for a conventional distillation column with three product streams (distillate, bottoms and the intermediate product). If this extra degrees of freedom is used to specify liquid split, then no extra degrees of freedom is left to specify vapour split ratio.

Multi-tooth contact behavior of helical gear applying modified meshing equation

In tooth contact analysis, the numerical process of searching for true contact lines or points is always a daunting task. The true contact lines for ideal helical gear transmission can be easily obtained through the helical gear meshing theory; however, in practice, this approach does not work well due to effect of tooth elasticity that shifts the contact lines from their ideal position. To address this problem, an approximate method to seek the true contact positions is proposed applying a modified meshing equation for a pair of helical gear with error free and no shaft deformation. First, the helical gear tooth deformation is calculated with the combination of contact mechanics and finite element method. Second, the deformed tooth flanks are fitted with Chebyshev polynomials. Third, the modified meshing equation is formulated by accounting for the influence of not only gear rigid body movement but also the gear deformation on the meshing process. Finally, the true contact points are determined by solving the correlated tooth flank equations, meshing equations, and force balance equations. The deformations of the gear and pinion teeth within the meshing plane are analyzed, and the contact load distributions along with the Von Mises stresses over a mesh cycle are examined. The study indicates that the proposed numerical method can yield true contact lines quickly while avoiding costly computational time in the traditional true contact line searching process that searches iteratively in the whole meshed tooth flank.

Optimal design of distillation systems with less than N − 1 columns for a class of four component mixtures

The optimal design of complex distillation systems is a highly non-linear and multivariable problem, with several local optimums and subject to different constraints. In addition, some attributes for the design of these separation schemes are often conflicting objectives, and the design problem should be represented from a multiple objective perspective. As a result, solving with traditional optimization methods is not reliable because they generally converge to local optimums, and often fail to capture the full Pareto optimal front. In this paper, a method for the multiobjective optimization of distillation systems, conventional and thermally coupled, with less than N−1 columns is presented. We use a multiobjective genetic algorithm with restrictions coupled to AspenONE Aspen Plus; so, the complete MESH equations and rigorous phase equilibrium calculations are used. Results show some tendencies in the design of intensified sequences, according to the nature of the mixture and feed compositions.

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

We propose and study a new parallel one-shot Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible Navier--Stokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems iteratively solve the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally are not easy to converge as they are basically nonlinear Gauss--Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization problems in which the mesh equations are implicitly coupled with the optimization problems. Second, we introduce an LNKSz framework based on an overlapping domain decomposition method for solving the fully coupled problem. Such an approach does not involve any sequential steps that are necessary for the Gauss--Seidel-type reduced space methods. The main challenges in full space approaches are that the corresponding nonlinear system is much harder to solve because it is two to three times larger and its indefinite Jacobian problems are also much more ill-conditioned. Effective preconditioning becomes the most important component of the method. Numerically, we show that LNKSz deals with these challenges quite well. As an application, we consider the shape optimization of an artery bypass problem in two dimensions. Numerical experiments show that our algorithms perform well on supercomputers with hundreds of processors.

Design of a Novel Cycloid Drive with a Cycloid-arc Gear and Analysis of Its Meshing Characteristic

Cycloid drive is used widely in many industrial areas for both power transmission and precision transmission. In this paper a novel cycloid drive is proposed and its meshing characteristics are analyzed. The conjugated gear pair of this novel cycloid drive is composed by an external cycloid-arc gear and an internal ring gear. Firstly, the idea of this novel cycloid drive is described and the tooth profile equation of the external cycloid-arc gear is presented. Secondly, based on coordinate transformation and gear geometry theory, the meshing equation for this novel cycloid drive is established, the equation of the conjugated tooth profile and the formula of induced normal curvature are also derived. Thirdly, the meshing characteristics are analyzed based on a numerical example: the drawing of the conjugated tooth pair is presented and compared with the conventional cycloid drive and the double-enveloping cycloid drive; the contact lines between one tooth pair at different meshing positions are illustrated. Finally, the induced normal curvature is calculated and compared with that of the conventional cycloid drive and the double-enveloping cycloid drive, which shows good characteristic of this novel cycloid drive.

Generation and investigation of a new cycloid drive with double contact

A new cycloid drive with double contact lines between one tooth pair is presented in this paper. The new conjugated tooth profile is generated by applying double-enveloping gear theory in cycloid drives. Based on coordinate transformation and gear geometry theory, the meshing equation for this new cycloid drive is established at first, then the equation of tooth profile and meshing line and the formula of induced normal curvature are also derived. The meshing characteristics are investigated by theoretical analysis and numerical examples: the double contact characteristic is revealed by that the meshing function can be resolved into two independent factors; the contact lines at different instant and meshing line are illustrated based on a numerical example; the superior characteristics of the new conjugated tooth profile is represented by comparison of induced normal curvature with conventional cycloid drives. The physical prototype is trial-produced and the transmission error of the prototype is also tested, which shows that double-enveloping theory can be applied for cycloid drives.

Computer Simulation of Helical Gears Generated by Rack-Type Cutters

This paper studies the equations of rack cutters for generating involute gears with asymmetric teeth. The asymmetry means that different pressure angles are applied for the driving and coast sides. By applying the equations of the designed profile of the rack cutter, the principle of coordinate transformation, the theory of differential geometry, and the theory of gearing, the mathematical models of involute helical gear are given. Undercutting analysis is also investigated by considering the relative velocity and equation of meshing. Furthermore, a computer simulation program is developed to generate the tooth profile of asymmetric involute gears and to illustrate the effect of tool geometry on the generated surfaces. A CAD/FEM procedure for finite element modeling of asymmetric gear teeth is presented. This simulation should be helpful in the design and manufacture of asymmetric gears.

Design Parameters for Spatial Helix Gearing Mechanism

The formulas of design parameters for spatial helix gearing mechanism are deduced based on the equation of spatial curve meshing, including teeth number, helix angle and screw pitch of driving tines, diameter of tines, corrected length of tines, height of teeth etc.. That could supply theoretical basis for industrial application of the spatial helix gearing mechanism

The Relationship of Symmetric Modification Parameters and Contact Line of Worm Gear

By a reasonable transition of the meshing equation and the contact line equation based on the principle of spatial meshing, using matlab numerical calculation and graphical function to draw a diagram of modification parameters and contact line of worm gear. Proposing a fast and accurate method for distinguishing the type of contact line. For further analysis of worm gear meshing performance provided a useful value.

Uniform convergence analysis of finite difference approximations for advection–reaction–diffusion problem on adaptive grids

A kind of singularly perturbed advection–reaction–diffusion problems with the exponential boundary layer are considered on an adaptive mesh. The existence, uniqueness and stability for the solution of the discrete problem are analysed with the maximum principle. The stability of the continuous problem is also considered. For the equidistribution problem composed by the difference scheme and equidistribution mesh equations, we establish a first-order ϵ-independent convergence rate for the numerical scheme defined on the equidistribution mesh and also an estimation for the accuracy of the solution computed on the final mesh generated by the adaptive algorithm. Numerical results are given to examine the validity of our theoretical analysis and the efficiency of the adaptive algorithm.