Function Of Transmission Errors Research Articles

GENERATION AND TOOTH CONTACT ANALYSIS (TCA) OF HYPOID GEAR DRIVE

The present work covers the Face-Hobbing method for generation and simulation of meshing of Face hobbed hypoid gear drive. In this work the generation process of hobbed hypoid gear has been achieved by determination of the generation function of blade cutter. The teeth surfaces have been drawn depending on the simulation of the cutting process and the head cutter motion. Tooth contact analysis (TCA) of such gear drive is presented to evaluate analytically the transmission error function for concave and convex tooth side due to misalignment errors. TCA results show that the gear is very sensitive to misalignment errors and the increasing of the gear teeth number decrease the transmission error for both concave and convex tooth sides and ensure smooth motion with low vibration.

Computerized design, simulation of meshing, and stress analysis of curvilinear cylindrical gears based on the active design of the meshing line function

The computerized design of curvilinear cylindrical gears based on the active design of the meshing line function is presented. The parametric equations of their meshing line functions are based on a parabola to control the curvilinear shape of the teeth along the face width of the gears. Different numbers of control points and different combinations of curves, including parabolic curves, circular arcs, and involutes, smoothly connected with each other at the control points are chosen to form the active tooth profile. The meshing performance, mechanical behavior, and influence of the parameters of combined curves are studied in terms of the contact patterns, variation of the maximum stresses and loaded functions of transmission errors and compared to two types of curvilinear cylindrical gear drives generated by face-milling cutters as a reference. The investigated curvilinear gear drives with five control points show lower maximum contact and bending stresses as well as lower peak-to-peak level of loaded transmission errors compared to other cases of curvilinear gear drives.

A Reliability-Enhanced Forming Grinding Method of Cylindrical Involute Gears for Electrical Vehicles

A new form-grinding method of cylindrical involute gears for electrical vehicles (EVs) is proposed to obtain stable contact performance of gear tooth surfaces and improve their reliability based on a predesigned controllable second-order transmission error function. The predesigned second-order transmission error function is assigned to a gear drive based on design specifications of EVs. Mathematical models of modified gear tooth surfaces of the gear pair can be obtained by the predesigned second-order trans-mission error function. Moreover, a section profile of a form-grinding wheel for cylindri-cal involute gears can be determined by a coordinate transformation matrix during form-grinding and settings of computer numerical control form-grinding programs for this active design method. This approach is ultimately conducted on three cylindrical involute gear pairs to demonstrate its feasibility and effectiveness.

Analysis of tooth contact for cylindrical gear with high-order topological modification and misalignment

The tooth contact analysis model is introduced for cylindrical gear with high-order topological modification and misalignment. According to the generation principle of gear, the tooth surface equation of double-crowned gear by using worm grinding wheel is derived. Based on the gear meshing theory, a tooth contact model of cylindrical gear considering errors and high-order topological modification is established. Taking helical gears as an example, the effects of errors and four different combinations of modification curves on the contact marks and transmission errors of tooth surface were analyzed. Results show that the paths of contact change dramatically with different combinations of modification curves. With the increasing of the order of modification curve, the contact path transitions smoothly in the middle of the tooth width, while the total length of the major axis of the contact ellipses increase significantly compared with taking the quadratic parabola modification. With proper parameters of the modification, the apex of transmission error function is flatter and its amplitude decreases which further reduces the vibration of gear transmission to installation error. The finite element method is applied to validate the proposed analysis model. The results are basically consistent and the maximum error is within 3.5 %.

Study on the degradation law of harmonic gear drive backlash with wear and assembly errors

Harmonic gear drive mechanism has been widely used in space mechanism, robot and other precision actuators. Based on the theory of probability and statistics, this paper establishes a mathematical model of harmonic gear transmission error considering the assembly and wear among components, and conducts sensitivity analysis and normalization of factors affecting harmonic gear transmission error model. Through Kriging model and Monte Carlo method, reliability degradation law of transmission accuracy is obtained by transmission error function and reliability analysis. Some suggestions are given in the design of harmonic gear drive in the future.

Computerized design, simulation of meshing and stress analysis of pure rolling internal helical gear drives with combined tooth profiles

The computerized design of non-generated pure rolling internal helical gear drives with combined transverse tooth profiles based on the active design of the meshing line function is presented. The application of different combinations of curves, namely circular arcs and involutes, to form the gear active tooth profiles is discussed. The parametric equations for tooth surfaces and the basic design parameters and equations of geometric sizing are proposed. The meshing performance and mechanical behavior are studied in terms of the contact patterns, maximum contact and bending stresses and peak-to-peak level of loaded transmission errors compared with a design of internal helical gear drive with corrected fillet. The combination of a circular arc and an involute connected at the pitch point to form the active tooth profile for both the pinion and the internal gear shows reduced maximum contact and bending stresses and lower peak-to-peak level of the loaded function of transmission errors. The proposed design of combined tooth profiles will contribute to lower the noise and vibration of the proposed gear drives, increase the endurance and life of the gear drive, and lower the heat generation due to pure rolling conditions of the achieved bearing contact.

Design and Analysis for Hypoid Gears with Ease-Off Flank Modification

An approach of ease-off flank modification for hypoid gears was proposed to improve the meshing performance of automobile drive axle. Firstly, a conjugate pinion matching with gear globally was developed based on gear meshing theory. Secondly, a modified pinion was represented by a sum of two vector functions determining the conjugate pinion and the normal ease-off deviations expressed by both predesigned transmission error function and tooth profile modification curves to change the initial contact clearance of the tooth. Thirdly, the best ease-off deviations were determined by optimizing the minimum amplitude of loaded transmission error (ALTE) based on tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA). Finally, the results show that effective contact ratios (εe) are established by clearances both teeth space and of contact elliptical, and greatly affect ALTE. The εe is a variable value with increasing loads for the tooth with modification. ALTE decreases with increasing εe. After εe reaches the maximum, ALTE increases with increasing loads. The mismatch of the best ease-off tooth is minimal, which contributes to effective reduction in ALTE, thus significantly improving drive performance.

Investigation of the effect of contact pattern design on the mechanical and thermal behaviors of plastic‐steel helical gear drives

This article describes an investigation that has been conducted to assess the effects of the contact pattern design on the mechanical and thermal behaviors of plastic-steel helical gear drives. The maximum contact pressure, frictional power loss, transmission error function and operating temperature are determined for different designs of the pinion tooth surfaces. Several numerical examples based on loaded tooth contact analyses and steady-state heat transfer analyses are carried out considering different types of micro-geometry modifications of the involute tooth surfaces. The obtained results show that an appropriate design of the pinion tooth surfaces can help reducing the maximum contact pressure and the maximum operating temperature, increasing the durability of the transmission. The frictional power loss and the peak-to-peak transmission error can be diminished, improving the performance of the transmission and increasing its efficiency. Also, it has been shown that the design of the pinion tooth surfaces can provide some capability to the transmission to absorb angular misalignments.

An elastic model for tooth contact analysis of spur gears using FEM simulation

Gears are a class of machine parts very often used in the most diverse industries due to their many advantages. In addition to the many benefits they provide, gears also have some disadvantages related to the noise and vibrations they can introduce into the system they are part of. The operating drawback which may occur is closely related to the geometry of contact and manufacturing and assembling errors. Investigation of the contact error effect on the operating condition should be first carried out through simulation approach. Gear efficiency can be investigated through the tooth contact analysis (TCA) method and stress analysis. The main target of TCA is to set up the contact path and to determine the transmission error function caused by clearance, gear misalignments or backlash. Generally, TCA is performed considering the elastic behaviour, which frequently implies to get data for different contact positions for full meshing analysis. Nevertheless, TCA outcome delivers noteworthy data about kinematic errors, contact ellipses and the estimation of the instantaneous contact point of the meshing gears. A numerical tool was developed in this work to obtain the contact path in spur gears, which allowed to compute the displacements, stress and strain fields in the contact regions. Furthermore, an analysis of the influence of manufacturing and assembling errors on the contact path and mesh quality was performed. Such an approach may provide an optimal design solution for spur gears.

Application of the Bilateral Filter for the Reconstruction of Spiral Bevel Gear Tooth Surfaces From Point Clouds

Abstract Reconstruction of gear tooth surfaces from point clouds obtained by noncontact metrology machines constitutes a promising step forward not only for a fast gear inspection but also for reverse engineering and virtual testing and analysis of gear drives. In this article, a new methodology to reconstruct spiral bevel gear tooth surfaces from point clouds obtained by noncontact metrology machines is proposed. The need of application of a filtering process to the point clouds before the process of reconstruction of the gear tooth surfaces has been revealed. Hence, the bilateral filter commonly used for 3D object recognition has been applied and integrated in the proposed methodology. The shape of the contact patterns and the level of the unloaded functions of transmission errors are considered as the criteria to select the appropriate settings of the bilateral filter. The results of the tooth contact analysis of the reconstructed gear tooth surfaces show a good agreement with the design ones. However, stress analyses performed with reconstructed gear tooth surfaces reveal that the maximum level of contact pressures is overestimated. A numerical example based on a spiral bevel gear drive is presented.

Computerized design, simulation of meshing and stress analysis of pure rolling cylindrical helical gear drives with variable helix angle

The geometric design, meshing performance, and mechanical behavior of pure rolling cylindrical helical gear drives with variable helix angle are preseted. This new type of pure rolling gear drives are based on the active design of the meshing line function. Seven types including linear, parabolic, and combined meshing line functions with variable helix angle and parametric equations for convex-to-convex meshing type are discussed. The contact patterns, loaded functions of transmission errors, and the evolution of contact and bending stresses along two pitch angles of rotation of the pinion being evaluated are compared to each other and to a pair of non-standard involute helical gears. Results show that one of the cases of design of pure rolling helical gears with variable helix angle reduces bending and contact stresses compared to the case of pure rolling helical gears with constant helix angle. Compared to involute helical gear drive, contact and bending stresses are increased. However, pure rolling cylindrical helical gears show potential for application as plastic gears in which their complex geometry can be reproduced with injection molding and the phenomenon of pure rolling meshing would contribute to a low heat generation and durability of the gears.

Geometric design, meshing simulation, and stress analysis of pure rolling cylindrical helical gear drives

The geometric design, meshing performance, and mechanical behavior of pure rolling helical gear drives are presented. Parametric equations for contact curves on the pinion and gear are determined by coordinate transformation of the active designed pure rolling meshing line for the whole cycle of meshing. Moreover, parametric equations for the tooth surfaces of helical gears with convex-to-convex meshing type are derived according to the motion of generatrices in the transverse section along the calculated contact curves. Then, the basic design parameters are analyzed and formulas for calculation of the geometric size are given. The meshing performance and mechanical behavior, including contact patterns, loaded function of transmission errors, and variation of stresses for two pitch angles of meshing are compared with those of a reference design of micro-geometry modified involute helical gears. Besides, the influence of basic design parameters on tooth contact analysis and stress analysis is studied. The analysis of the results shows that the proposed pure rolling helical gears have the advantage of reducing the relative sliding between tooth surfaces and the possibility of designing pure rolling helical gears with a small number of teeth, though the contact strength of the surfaces is impaired. However, if the appropriate design parameters and Hermite curve parameters for the fillets are properly evaluated as proposed here, the mechanical behavior of the proposed pure rolling helical gear drive, in terms of contact patterns and variation of bending stresses can be superior to that of the micro-geometry modified involute helical gear drives.

Geometric Design, Meshing Simulation, and Stress Analysis of Pure Rolling Rack and Pinion Mechanisms

Abstract The geometric design, meshing simulation, and stress analysis of pure rolling rack and pinion mechanisms are presented. Both the pinion and the rack are based on the active design of the meshing line to provide pure rolling for the whole cycle of meshing. The parametric equations of the contact curves on the rack and pinion tooth surfaces are determined by coordinate transformation of the meshing line equations. Three types of meshing are derived according to the motion of the generatrices along the calculated contact curves: convex-to-concave meshing, convex-to-plane meshing, and convex-to-convex meshing. Then, the basic design parameters are analyzed and formulas for calculation of the geometric size are given. Four different cases of design are considered to compare the meshing performance and mechanical behavior of the proposed gear mechanisms. The results include contact patterns, the unloaded function of transmission errors, and the evaluation of stresses along two cycles of meshing. The analysis of the results shows that the proposed method of design of pure rolling meshing reduces the relative sliding between tooth surfaces, whereas it decreases the contact strength of the tooth surfaces. However, if the design parameters are properly evaluated as a result of simulation and applied as proposed here, the mechanical behavior of the proposed rack and pinion mechanisms can be more favorable than that of the standard geometry of involute rack and pinion sets.

Dynamic characteristic prediction of multistage gear-cylinder transmission system

ABSTRACTDynamic behavior of gear-cylinder directly affecting the printing quality and efficiency is one of the key elements in offset printing. In order to study the problem of ink bar and ghosting caused by excessive vibration, a new model taking into account a variety of key physical factors is developed to predict the dynamic characteristics of a gear-cylinder system using the lumped parameter method. An appropriate mathematical tool is utilized to describe the meshing stiffness and transmission error function, thereby deducing the equations of motion with 16 degrees of freedom. In addition, the partial differential equation of sensitivity analysis in matrix form is obtained after the completion of formula transformation. The simulation results obtained from dynamic response and sensitivity analysis show that the proposed model is reliable enough to be used for dynamic characteristic prediction of a gear-cylinder system.

The mathematical model and a case study of the cosine face gear drive that has a predesigned fourth order function of transmission errors and a localized bearing contact

Traditionally, a face gear drive uses an involute pinion. If the number of teeth of pinion is smaller than the minimum number of teeth without undercutting, the pinion will be undercut. To avoid un...

Computerized generation and meshing simulation of face gear drives manufactured by circular cutters

The procedure of computerized generation and the results of simulation of meshing and stress analysis of face gear drives manufactured by circular cutters without considering the feed motion of the circular cutters is presented. A cutter profile tilt angle is introduced and investigated to approximate the generating surfaces of the circular cutter as much as possible to those of the theoretical reference shaper. Six different cases of design are considered to investigate the influence of design parameters on the contact patterns of the gear tooth surfaces. The results include the distribution of deviations of the face gear tooth surfaces with respect to the reference surface geometry generated by a shaper, the contact patterns, function of transmission errors, and the evaluation of stresses all over two cycles of meshing. The analysis of the results shows that the surface deviations due to the generation of the face gear by circular cutters can result in edge contacts, interference, and high stress concentration. However, if the appropriate design parameters for the circular cutters are properly evaluated as a result of simulation and applied as proposed here, the mechanical behavior of the face gear, by means of contact patterns and evolution of stresses can be as good as for the standard geometry of the face gears.

A method for grinding face gear of double crowned tooth geometry on a multi-axis CNC machine

The stability requirement of bearing contact and a favorable function of transmission errors motivates the investigation of design and manufacture technology of the face-gear drives. Application of a grinding disc wheel with axial profile modification and a longitudinal ease-off topography design of face gear tooth flanks could be proper utilized to obtain a double crowned tooth geometry, which results in the pre-designed bearing contact and desired transmission errors. With the profile crowned tooth geometry of the gear generated by the modified wheel cutter, finding of the execution motions to generate the tooth geometry with the longitudinal ease-off topography design on a multi-axis Computer Numerical Control (CNC) machine with the same tool is a major issue. Therefore, a universal method on the Design Optimization Techniques is proposed to obtain the execution motions for such problem. The proposed optimization methodology is also applicable for the tooth flank error correction by a worm hob or a grinding cutter on the CNC generator. With the term of additional movement of execution motion is defined, a detailed description of additional movements for the ease-offs of the real crowned tooth geometry is also investigated. The feasibility of proposed method is validated by numerical examples.

Analysis of loaded characteristics of helical curve face gear

As a new gear pair, the helical curve face gear pair have not been widely applied as normal face gear. It can be used to substitute crank-slider mechanism to achieve variable transmission ratio and improve the structure and characteristics to some extent, such as grinders, planers and so on. The mathematical model of the helical curve face gear pair and the tooth contact analysis are necessary for the theoretical analysis of this gear pair. Based on the theory of gear meshing and coordinate transmission, the meshing equations for non-circular gear contacting with helical curve face gear without misalignment are obtained. On the basis of Hertz's theory, the principal curvatures at contact point are represented. And the contact area on every tooth and the half cycle of helical curve face gear have been established separately. As well as the contact stress on every tooth and half cycle of helical curve face gear. However, there may be misalignment during the meshing of this gear pair, this paper also covers the contact path and the functions of transmission errors with misalignment, includes the comparison of helical curve face gear and straight curve face gear. The developed theories for complex tooth surface are verified by experiment. These analysis show the contact characteristics of this gear pair during the meshing process and provide the theoretical support for further analysis and application of helical curve face gear pair.

Numerical approach for determination of rough-cutting machine-tool settings for fixed-setting face-milled spiral bevel gears

Spiral bevel gear drives manufactured by the so-called five-cut method are designed to achieve the best conditions of meshing and contact since convex and concave pinion tooth surfaces are generated independently, allowing the achievement of optimized contact patterns and favorable functions of transmission errors for both flanks of gear tooth surfaces, and reducing with it noise and vibration and increasing the endurance of the gear drive. As part of the process of manufacturing, roughing of both flanks of the pinion tooth surfaces is needed. In this paper, a novel numerical approach for determination of machine-tool settings for roughing of the pinion by using a spread-blade face-milling cutter is proposed. The main purpose of the proposed procedure is the reduction of the manufacturing cycle time and maximizing the material cut during the rough-cutting operation, without damaging the to-be-finished surfaces. The procedure is based on the resolution of a constrained optimization problem. A numerical example shows the applicability of the developed numerical approach. The obtained numerical results disclose that acceptable rough-cut geometries are possible to be managed during the first iterations as trade-off solutions.

Design of double-crowned tooth geometry for spiroid gear produced by precision casting process

A novel double-crowned tooth geometry is proposed by the application of ease-off topography for spiroid gear manufactured by precision casting process, with the goals of localizing the bearing contact and obtaining a perfect function of transmission errors. The modified tooth surface is applied as the reference geometry to machine the die cavity geometry that will produce such geometry of the gears. The tooth geometry of crowned gear was achieved first from a pre-designed controllable function of transmission errors along the desired contact path. Then, the desired ease-off topography along the contact line is designed and calculated computationally from the given mathematic model of surface modification. The geometry of double-crowned spiroid gear could be reconstructed by superimposing the ease off of contact line direction on the profile-crowned tooth surface. The article provides numerical examples to validate the feasibility of ease-off modification methodology that was used to produce the double-crowned tooth geometry for the gears, while tooth contact analysis is performed computationally to investigate the stability of bearing contact and function of transmission errors to alignment.