Theory Of Gearing Research Articles

Synthesis of roller screws

The synthesis of screw gears with specified transfer function is considered. The theory, classification, and design of screw gears are discussed.

A generating milling method for a spur face gear using a five-axis computer numerical control milling machine

To improve the machining accuracy and production efficiency of face gears, this article presents a generating milling method for the spur face gear using a five-axis computer numerical control milling machine and proposes a milling principle of the spur face gear using a milling cutter. Based on the milling principle and theory of gearing, a mathematical model of the spur face gear milling cutter is established, and the design parameters of the cutter are determined. Considering the requirements of machine motion during the face gear milling process, a five-axis computer numerical control milling machine is developed to produce the face gear. Based on the structure of the milling machine, a mathematical model of the spur face gear is established, and a milling method is proposed. The spur face gear milling cutter is manufactured, and the face gear milling experiments are conducted on the self-developed milling machine. The measurement results of the tooth surface deviations indicate that the maximum deviation is 22.6 µm, and that the machined spur face gear accuracy can meet the requirements of roughing. The experimental results verify the validity of the proposed generating milling method and demonstrate that the generating milling method is an effective approach to improve the accuracy and efficiency of the spur face gear.

Generation of Noncircular Spiral Bevel Gears by Face-Milling Method

Noncircular bevel gears are applied in variable-speed transmissions with intersecting axes. Since dedicated machines for manufacturing noncircular bevel gears are not available, noncircular bevel gears are normally manufactured using universal computer numerically controlled (CNC) machining centers, resulting in poor productivity. This paper describes a face-milling method for generation of noncircular spiral bevel gears, which is analogous to the generation of spiral bevel and hypoid gears using CNC hypoid gear generators, such as Gleason free-form hypoid generators. As a result, the productivity is significantly improved. Based on the theory of gearing, this paper first describes the basic concept of generation of conjugate noncircular spiral bevel gears. Generation of the tooth surfaces using crown-gear generation concept is analytically discussed with association to the face-milling process of generation of the proposed noncircular spiral bevel gears. The tooth surface geometries are represented by the position vectors and normals. The kinematical model of free-form machines is developed. The machine motion parameters are determined based on the theoretically defined tooth surfaces using the crown-gear generation concept. The developed method is verified by manufacturing a real pair of noncircular spiral bevel gears with satisfactory contact patterns which agree well with those modeled using a commercial cae software program.

Online machining error estimation method of numerical control gear grinding machine tool based on data analysis of internal sensors

This paper presents an online estimation method of cutting error by analyzing of internal sensor readings. The internal sensors of numerical control (NC) machine tool are selected to avoid installation problem. The estimation mathematic model of cutting error was proposed to compute the relative position of cutting point and tool center point (TCP) from internal sensor readings based on cutting theory of gear. In order to verify the effectiveness of the proposed model, it was simulated and experimented in gear generating grinding process. The cutting error of gear was estimated and the factors which induce cutting error were analyzed. The simulation and experiments verify that the proposed approach is an efficient way to estimate the cutting error of work-piece during machining process.

Cutter position calculation of machining equal base circle bevel gears with a pot-shaped milling cutter

Aiming at achieving the application of numerical control (NC) machining equal base circle bevel gears with a pot-shaped cutter, with the equal base circle bevel gear theory, the model of pot-shaped cutter is established and the main parameters deciding the positions and postures of cutter are solved. Furthermore, the cutter positions in the gear blank coordinate system are obtained by calculating the cutter axis vector and centric coordinates. On this basis, combined with the general five-axis NC machine tool, the general expressions of trajectory for five-axis NC machining this kind of gears are presented; the results of cutter positions and states are verified by 3D modeling. Finally, the correctness of the tool trajectory is proved through the machining simulation and actual experiment. The results show that machining equal base circle bevel gears with a pot-shaped cutter can be achieved upon a general NC machine tool.

New methodology for determining basic machine settings of spiral bevel and hypoid gears manufactured by duplex helical method

The duplex helical method is an advanced and primary manufacturing method of face milled spiral bevel and hypoid gears for its characteristics of higher efficiency, lower cost, higher strength and power dry cutting. A new methodology for calculating the basic machine-tool settings of spiral bevel and hypoid gears manufactured by the duplex helical method is proposed in this work. Firstly, the blank design theory of spiral bevel and hypoid gears with the duplex depth taper is described in detail. In order to obtain the optimal machine settings, three reference points are defined. The basic machine settings for spiral bevel and hypoid gears manufactured by the duplex helical method in accordance with the five steps are determined. Finally, a hypoid gear set generated by the duplex helical method is used as an example to validate the developed methodology. The results reveal that the tooth bearings and the transmission error functions on the drive and coast sides are favorable, and the results of the simulation and experiment are consistent.

Russian school of the theory and geometry of gearing: Its origin and golden period (1935-1975)

During the second half of the twentieth century, Soviet scientists were among the world leaders in the theory and geometry of gearing or the theoretical fundamentals of designing gears and gear-cutting tools. The authors of the current paper review works related to the development of the theory of gearing (TG) written in Russian from the mid-1930s to the present. Overall, the review covers more than 400 publications of nearly 200 authors (among more than 1000 studied works). This review involves some works on gear strength and accuracy to attract more experts.

Study on modular modeling and performance evaluation of a conical gear for marine transmission system

Conical involute gears, which are also known as beveloid gears, are the involute gears with cone-shape gear body rather than cylindrical gear body. Generally, conical gears are used in anti-backlash transmission systems and marine transmission systems. In marine transmission system, in order to obtain optimal propulsive force, 7–14 degree inclination angle was needed. In this paper, a 7 degree cone angle conical gear used in 375 kW class marine power transmission system was selected as research objective. And in order to establish the design theory of conical gear, a design system named conical gear modular modeling system was developed. The prototype conical gear was developed by the above design system to be used to simulate in a certain condition. The simulated and optimized prototype conical gear was manufactured which is as a specimen to complete a series of performance tests which contains gear tooth profile test and contact rate test. As the results, the developed conical gear can well meet the performance requirements of the 375 kW class marine power transmission system.

Coupled translation-rotation vibration and dynamic analysis of face geared rotor system

A fourteen degree-of-freedom (DOF) lumped parameter dynamic model considering the coupled translation-rotation vibration is proposed and the coupled multi-body dynamics of the face geared rotor system are studied. The meshing characteristic of the face gear pair is analyzed by applying the tooth surface equation, the mesh theory of face gear and the tooth contact analysis of the face gear pair. Based on the equations of motion, the coupled multi-body dynamic response of the system is investigated using the Runge−Kutta numerical method and the effects of mean load and backlash on the dynamic responses are explored. The jump phenomenon, periodic window, doubling-periodic bifurcation and chaotic behavior of the system are observed and a path to chaotic motion through doubling-period bifurcation motion is also observed.

Fluid analysis of cylindrical and screw type Roots vacuum pumps

This study uses conjugate gear theory to mathematically model the cycloid rotor profile of a Roots pump. After the rotor profile is generated based on the mathematical model of a rotor, a three-dimensional numerical approach can be developed by using the computational fluid dynamics (CFD) solver PumpLinx. The fluid analysis models of cylindrical and screw type Roots pumps are constructed and then a two-lobe rotor is used to illustrate the differences of flow characteristics between cylindrical and screw type Roots pumps. In particular, the models evaluate which of the two pump types achieves a better average outlet flow rate, showing that the cylindrical type pump is better than the screw type but that the outlet flow rate curve of the latter is steadier than that of the former. The findings therefore suggest that the cylindrical type pump should be chosen for efficiency and the screw type pump for steadiness. The results for a simulated pump are similar to those obtained for the real pump, indicating that simulations could serve as a time-saving reference for Roots pump designers.

Mathematical model and geometrical model of double pitch ZN-type worm gear set based on generation mechanism

The current researches on the tooth surface mathematical equations and the theory of gearing mainly pay attention to the ordinary type worm gear set (e.g., ZN, ZA, or ZK). The research of forming mechanism and three-dimensional modeling method for the double pitch worm gear set is not enough. So there are some difficulties in mathematical model deducing and geometry modeling of double pitch ZN-type worm gear set based on generation mechanism. In order to establish the mathematical model and the precise geometric model of double pitch ZN-type worm gear set, the structural characteristics and generation mechanism of the double pitch ZN-type worm gear set are investigated. Mathematical model of the ZN-type worm gear set is derived based on its generation mechanism and the theory of gearing. According to the mathematical model of the worm gear set which has been developed, a geometry modeling method of the double pitch ZN-type worm and worm gear is presented. Furthermore, a geometrical precision calculate method is proposed to evaluate the geometrical quality of the double pitch worm gear set. As a result, the maximum error is less than 6´10–4 mm in magnitude, thus the model of the double pitch ZN-type worm gear set is available to meet the requirements of finite element analysis and engineering application. The derived mathematical model and the proposed geometrical modeling method are helpful to guiding the design, manufacture and contact analysis of the worm gear set.

A precision generating grinding method for face gear using CBN wheel

In order to improve the machining accuracy and production efficiency of face gear, a generating grinding method for face gear using cubic boron nitride (CBN) wheel is studied in this paper. The mathematical model of the CBN wheel profile is derived based on face gear grinding principle and theory of gearing. Taking the demands of machine motion during face gear grinding into account, a five-axis face gear grinding machine is developed. Then, the mathematical model of the ground face gear and the grinding method are proposed based on the structure of the machine. For reducing the face gear tooth profile errors induced by installation errors, an error analysis model of face gear grinding is established; the effects of installation errors on tooth profile errors are analyzed. The CBN wheel was manufactured, and the grinding experiments were performed on the self-developed grinding machine. The processing parameters of face gear grinding were amended according to the tooth flank measurement results obtained from coordinate measuring machine. The ground tooth profile deviation was reduced significantly after processing parameters amendment. Experiment results demonstrate that the generating grinding method is an effective approach to achieve the precision grinding of face gear.

Modelling and bending strength analysis of cylindrical gears with arcuate tooth trace

Based on the analysis of precision moulding mechanism for cylindrical gears with arcuate tooth trace (arc gear), the equation for tooth flank of arc gear was derived preliminarily and the ideal concave and convex tooth flanks were got by MATLAB simulation to achieve graphics rendering. On this basis, with UG NX8.0, the accurate three-dimensional modelling of arc gear was established to be used for the finite element analysis. Then, for the problem that there is no standardised bending strength calculation formula of arc gear, with the aid of finite element method (FEM) this paper studied the relationship between tooth line radius and bending strength. With FEM analysis performed, it was confirmed that under the same conditions, the bending strength of arc gear could be significantly improved compared with that of spur gear and helical gear, and that variation curve of tooth root stress was obtained under the different tooth line radius. Therefore, the analysis method provides engineering applications value under the high-speed and heavy-load working conditions and theoretical basis for the strength theory and design of arc gear.

Traditional versus improved designs for cycloidal speed reducers with a small tooth difference: The effect on dynamics

Whereas traditional cycloid reducers tend to have a one tooth difference design, a two teeth difference design is common when the transmission ratio is low (≦17). The dynamic characteristics of these small teeth difference designs, however, remain unclear. Hence, this study constructs an analytic model of system dynamics in a cycloid reducer with a small tooth difference. Specifically, taking a transmission ratio of 17 as an example, it analyzes both free-pin and fixed-pin designs to show the shortcomings of traditional designs. It then proposes a nonpin design that improves on these shortcomings, one in which the pinwheel is replaced by a cycloid internal gear. Next, adopting the theory of gearing, it constructs a mathematical model of the nonpin design with a multitooth difference and derives undercutting equations to check for the persistence of undercutting on the tooth profile. It also determines the parameter design range of nonundercutting. In an analysis of system dynamics, it then investigates twelve nonpin cases with a small tooth difference, showing the effect of the design parameters on the reducer's system dynamics. The results indicate that appropriate design parameters can indeed improve on the shortcomings of the traditional design.

Mathematical models of hobs for conjugate-curve gears having three contact points

Conjugate-curve gears are the gears which are point contact and the locus curves of the contact points are conjugate curves. The contact pattern of the conjugate-curve gear tooth surfaces are convex to concave, which reduces the contact stress of the tooth surfaces due to the small value of the relative curvature radius at the contact point. The tubular tooth surfaces of the conjugate-curve gears have one pair of conjugate curves. To decrease the running-in time and increase the load capacity, the conjugate-curve gears having three pairs of conjugate curves are designed. The contact stress of the tooth surfaces having three contact points is much smaller than that of the tubular tooth surfaces in the computer contact analysis. For the further study of the performance of the conjugate-curves gears having three contact points, hobs are considered to manufacture the gears. Two mismatched rack cutters having three contact points are applied for the design of hobs. The working edge in the normal section profile of the rack cutter for the hob generating the pinion is a circular arc and the working edges in the normal section of the rack cutter for the hob generating the gear are two parabolic curves that are tangent to the convex circular arc. By applying the designed rack cutter profiles, the principle of coordinate transformation, the differential geometry theory, and the theory of gearing, mathematical models of the hobs are established. To verify the proposed tooth profile and the hobs, the experimental cutting trials and the load capacity test are carried out. The final accuracy of the gear satisfies the design requirements. The results demonstrate the feasible of the proposed design method.

Algorithms for Noncircular Gear Pitch Curves Generation

The paper resumes the traditional hypotheses for the noncircular mating centrodes design, based on the definition of i) the driving centrode geometry ii) the transmission ratio variation and iii) the rotational motion of the driven centrode. Each procedure is transferred within the gear theory and developed considering both specific gear data, such as gear modulus, gear number of teeth and gear number of rotations during a rotational period. Original codes are developed in Php amd MyQSL to illustrate gears conjugate pitch curves geometry and their kinematics for external rolling.

Research on the compensation method for the measurement error of cycloidal gear tooth flank

In the precision measurement of the tooth flank of cycloidal gear, there are many reasons that affect the measurement accuracy. Among the reasons, the alignment angle error of the gear axis play an important role in the measurement results of the profile and tooth trace deviations of the tooth flank. According to the measurement theory of the cycloidal gear, the mathematics model of the tooth flank is established. Based on the measurement process of the profile and tooth trace deviations by the gear measuring machine, a compensation method for the alignment angle error of the gear axis that included in the measurement results is designed. Some simulation experiments were done to verify the proposed method. Experimental results show that after compensation, the alignment angle error of the gear axis reduces significantly, more than 80% of the alignment angle error in profile deviation and more than 74% of the alignment angle error in tooth trace deviation is compensated by the method. The proposed method is helpful to improve the measurement accuracy of the cycloidal gear.

Analysis of the temperature of the rail/wheel contact surface using a half-space model and a moving heat source

The model can be used to analyse the thermal processes due to a frictional block or disc braking or to analyse the temperature conditions close to the ril/wheel contact patch, when macroscopic sliding occurs. It is especially useful in the evaluation of rail chill, head check and other contact tribology-based local heat and stress effects. The contact seizure theory of gear drives developed by Blok is based on a simplified approach to the determination of the temperature due a moving heat source generated by sliding friction. In a simplifying step, the heat flux parallel to the surface of the contacting bodies is disregarded. One-dimensional temperature patterns that vary with time are arranged along the line of motion, based on the displacement of the source. This one-dimensional model is applicable if the velocity (i.e. Péclet number) is ‘high enough’. The influence of the width of the source has not been examined. The literature contains analytical results for various heat flux models, and one can make use of them to overcome the restrictions inherent in the one-dimensional theory. However, their rather complex structure makes them unsuited for further analysis, and only the numerical analysis method appears to be fruitful. One approach is to describe the temperature conditions created in a plate, which extends on a half-plane, by heating a part of its edge. The obtained time-dependent temperature results can be arranged into a sequence in a similar manner as in the approach of Blok. This shows the spatial temperature distribution that can be obtained by enhancing the results obtained using the one-dimensional theory by considering the effect of the crosswise dispersed heat. For heat sources of limited width, the cross-corrected approach better fits the reference values given by a full three-dimensional model. In railway traction, the analysis can be most directly applied to gear drives with a localized load pattern.

Kinematic errors on helical gear of triple circular-arc teeth

Two imaginary skew rack cutter curves with stepped triple circular-arc teeth are presented in this paper. A helical gear pair that includes the gear and pinion surfaces was generated by using two matched imaginary skew rack cutter surfaces. The mathematical models of the helical gear pair with stepped triple circular-arc teeth were developed based on theory of gearing. With the use of these models, the analysis of kinematic errors can be illustrated and evaluated by computer programming and tooth contact analysis (TCA). The assembly errors used had several kinds: ideal assembly errors, vertical misalignment angles, horizontal misalignment angles, manufacturing errors of helical angles, and center distance deviation. TCA revealed that the sensitivity of horizontal misalignment angles was lower than that of manufacturing errors of helix angles but higher than that of vertical misalignment angles.

Study on the NC Machining Theory and Simulation of Hypoid Gear

Based on NC machining principle of hypoid gears and NC machining with high efficiency quality, This paper discusses the feasibility of the hypoid gear processing, establishes the mathematical model of face gear wheel hypoid milling machining adjustment, that will be take the basic data into vertical machining center machine tool, tool, fixture, the installation and adjustment of parameters, and we write a program of the CNC machining and corresponding code, combined with the specific wheel blank parameters to milling simulation test and milling tests, we obtain a new process methods.