Spherical Gear Research Articles

Derivation of design formula for a shifted spherical involute bevel gear pair

The development of shifted spherical involute bevel gear will lead to increased flexibility and convenience in the design of the cross-axis gear system. The shifted spherical involute bevel gear is identified by the axis angle change between the crown rack and work piece. The new design formula for the shifted spherical involute bevel gear pair was derived mathematically from the condition that the arc length sum of the pinion and gear base cones is same as that of the disk of action. Using the formula, it was shown that a bevel gear pair can fit the design requirements with great flexibility. The gear transmission error of the designed and CNC processed gear pair was directly measured to experimentally verify its validity.

Research on the Process of Ring-Involute Tooth Spherical Gear Based on Rapid Prototyping Processing

The ring-involute tooth spherical gear with two degrees of freedom is considered as the research object. Through the combination of theory and experiment, a further study is conducted on it. According to the theory of rapid prototyping, comprehensive consideration of the ring-involute teeth spherical gear tooth profile surface forming principle and transmission features, traditional machining method, transmission error and the actual application, a fast processing method of the ring-involute tooth spherical gear is proposed. Spherical gear model is built in Solidworks surface modeling environment. Through the use of rapid prototyping equipment, the ring-involute tooth spherical gear is worked out. Tooth root thick, addendum thickness and transmission error of spherical gear is further measured on this basis with the method of rapid forming. Basic parameters meet the requirements. The rapid prototyping processing method is proved to be practical, effective and valuable.

Theoretical Investigation of a Planar Rack Cutter with Variable Diametral Pitch

We propose an imaginary planar rack cutter with variable diametral pitch to manufacture the spherical gears of a wrist mechanism. The concept of the imaginary planar rack cutter is derived from the imaginary rack cutter. In general, the diametral pitch of the imaginary rack cutter is constant. In using the imaginary planar rack cutter with constant diametral pitch to cut a spherical blank material, the teeth generated will interfere with each other. Therefore, two matched imaginary planar rack cutters with a variable diametral pitch are proposed. One of the imaginary planar rack cutters consists of convex teeth and is utilized to generate a spherical blank material with concave teeth. The other consists of a concave one and is used to generate a spherical blank material with convex teeth. Through the two-parameter envelope theory and a software package, mathematical models of the spherical gear pairing were determined. Using a developed mathematical model, an investigation on the undercutting analysis of the proposed imaginary planar rack cutter is performed by tooth contact analysis. Finally, a numerical example is used to demonstrate the geometric model of a spherical gear generated by the imaginary planar rack cutter.

Three-dimensional optical measuring method for the tooth surface of high-order oval bevel gears

In view of the complexity of geometry design theory for high-order oval bevel gears’ tooth profile and the difficulties in detecting the tooth surface error, the researches into tooth surface error detection and analysis methods of high-order oval bevel gears were launched. In the view of the feature of three-dimensional (3D) optical measuring method, a 3D optical scanner was used. Meanwhile, the tooth profile parametric equations of a bevel gear cutter with standard spherical involute and non-circular bevel gear were deduced based on the spatial meshing theory and the theory of spherical triangle. Then the 3D optical scanning detection experiment of oval bevel gears was designed and accomplished. After that, the reverse engineering was performed according to the point cloud which was obtained before, and a tooth surface which within a certain tolerance range can be obtained from the reverse engineering. The normal deviation nephogram of theoretical tooth surface and processed tooth surface can be obtained by comparing the tooth surface which is gotten from reverse engineering with theoretical tooth surface. Finally, the measurement and visualisation for processing tooth surface of high-order oval bevel gears were achieved. Meanwhile, the machining accuracy class of five-axis precision milling method was obtained.

전기 모터 구동 대빙급 추진 시스템의 과도 비틀림 진동 분석

A ship’s propulsion shafting system is subjected to varying magnitudes of intermittent loadings that pose great risks such as failure. Consequently, the dynamic characteristic of a propulsion shafting system must be designed to withstand the resonance that occurs during operation. This resonance results from hydrodynamic interaction between the propeller and fluid. For ice-class vessels, this interaction takes place between the propeller and ice. Producing load- and resonance-induced stresses, the propeller-ice interaction is the primary source of excitation, making it a major focus in the design requirements of propulsion shafting systems. This paper examines the transient torsional vibration response of the propulsion shafting system of an ice-class research vessel. The propulsion train is composed of an electric motor, flexible coupling, spherical gears, and a propeller configuration. In this paper, the theoretical analysis of transient torsional vibration and propeller?ice interaction loading is first discussed, followed by an explanation of the actual transient torsional vibration measurements. Measurement data for the analysis were compared with an applied estimation factor for the propulsion shafting design torque limit, and they were evaluated using an existing international standard. Addressing the transient torsional vibration of a propulsion shafting system with an electric motor, this paper also illustrates the influence of flexible coupling stiffness design on resulting resonance. Lastly, the paper concludes with a proposal to further study the existence of negative torque on a gear train and its overall effect on propulsion shafting systems.

Parametric Modeling for Spiral Bevel Gear Based on PRO/E

The author describes tooth profile formatting mechanism of spherical involute spiral bevel gear based on Principles of Gear conjugate, establishes the mathematical model of spiral bevel gears. The precise spherical involute of spiral bevel gears is generated with parametric modeling idea and th secondary development method based on PROGRAM of PRO/E software. Finally, a complete spherical three-dimensional modeling of the involute spiral bevel gear is established, which provides a method of parametric design and manufacturing of spiral bevel gears.

Research on Sliding Ratios of Conjugate Surfaces of Two Degrees of Freedom Meshing Transmission of Spherical Gear Pair

The spherical gear is a gear-driven mechanism with two degrees of freedom (DOF), which can transfer spatial motion. The spherical gear pairs have two types of basic assembly structures including an ideal mechanism and a gimbal mechanism, and whose kinematic characteristics are analyzed. The 2-DOF gearing principle of conjugate tooth surfaces of the spherical gear pair is introduced first. Then, the relative slide between two tooth surfaces in the mesh is analyzed. Finally, the equations of the meshing coned surface and the conjugate surface are established based on the meshing models of the spherical gear pairs. Furthermore, the sliding ratios of the tooth surfaces of the spherical gear pairs are obtained when they mesh in different meshing conditions based on two-parameter movement analysis. The computational results show that the sliding ratios of the spherical gear teeth are related to the angle velocity ratio in the ideal mechanism and they are not only related with angle velocity ratio but also related with the assembly axes in the gimbal mechanism, which are useful in theory for further studying the wear of the spherical gear.

Mathematical Model of Spherical Gears with Helix Traces

Spherical gears with helix tooth traces are proposed to enhance loaded capacity and improve the sensitivity to assembly errors. The hobbing process for spherical gears is illustrated, and then developed the mathematical model of imaginary rack cutter. The tooth surface mode of spherical gears was developed by coordinate translation and gearing theory. Finally, the paper presents a numerical example to validate the proposed new spherical gears drive

The Generation Principle and Mathematical Models of a Novel Spherical Gear Drive

A new type of spherical gear drive, which takes micro-segment involute profile as tooth profile, is presented in this paper. The generation principle of the spherical gear is described. The mathematical models including the equation of micro-segment involute profile, the equation of the conjugate tooth profile and the engagement equation, are established based on the meshing theory. This new type of spherical gear drive has the potential to improve the load capacity and the performance of spherical gear transmission.

CAD simulation and generation machining of discrete ring-involute spherical segment gear pair

A spherical segment gear pair is useful in robotic wrist and similar such applications that require two degrees of freedom. From the first principle, tooth surface of spherical gear can be mathematically derived from a rack-cutter profile using the coordinate transformation and one-parameter family of envelope theory. Alternatively, the spherical gear tooth surface is obtained as an envelope of a series of cutter surfaces formed during the kinematic motion between work blank and cutter. Conventional gear cutting machines cannot be used to generate the spherical gears and hence the spherical gears have remained a conceptual mechanism. In order to establish the feasibility of generation machining, a CAD-based simulation is carried out first where work blank and cutter are taken as solid models and the simulation is performed using the Boolean operation to remove the interfered material in an incremental manner, maintaining the kinematic relationship. The actual machining of the spherical segment gear is then carried out on a five-axis CNC machining centre using a specially developed conical end-mill cutter having a semi-cone angle of 20°. The script file for the CAD simulation has also been used for creating the program for CNC machining.

Contact Characteristics of Spherical Gear and Ring-Rack

The contact characteristics of spherical gear and ring-rack are researched in this paper. First, the tooth profile surface equation of spherical gear is established on the basis of the transmission theory of spherical gears. After that, the profile surface equation of ring-rack is established. Then, kinematics of the transmission of spherical gear and ring-rack is researched, the appreciate coordinates are established, and the profile surfaces of spherical gear and ring-rack are described in the same coordinate. Then, the contact analysis is conducted between spherical gear and ring-rack, and the contact point and ellipse parameters are acquired, as well as the rules of their evolvements. The conclusions above are benefit to further research on the transmission of spherical gear and ring-rack.

Tooth flank, undercutting and tooth pointing of spherical gears

Abstract By the developed mathematical model of spherical gears from literature [16], conditions of tooth undercutting of spherical gears with convex and concave teeth and tooth pointing conditions of the proposed concave spherical gear are derived. Besides, tooth flank profile deviations, limit curves of tooth undercutting and the beginning points of tooth undercutting and tooth pointing of the proposed spherical gear can be obtained by applying the spherical gear tooth mathematical model and the developed computer simulation programs, as illustrated by five numerical examples.

Design and Research on Two-Degree of Freedom Spherical Gear

This paper deals with the analysis and studies in the tooth profile of robot flexible joint spherical gear. A new type of two-degree of freedom conical concave spherical gear is proposed, calculated by analysis of teeth with tapered concave conjugate of the convex teeth shape. The undercut, overlap coefficient are analyzed and calculated to ensure the rational structure of the spherical gear. The conical concave tooth is using in the spherical gear, which makes the process of manufacturing easier and can improve the manufacturing precision and reduce the costs.

Basic Principle and Mathematical Model of Cutting Hypoid Gears by Generating Line Method

Generating line method, which is based on the generating process of spherical involute curve, is a new processing theory of cutting ideal spherical involute gears. This paper proposed the geometry and basic principle of cutting hypoid gears by this method, and defined the planar conjugated relationship between generating lines of the pinion and gear. A mathematical model of tooth surfaces is established based on cutting process. This model can be applied to any shapes and parameters of the gear generating line.

Mathematical Model of Spiral Bevel Gears Manufactured by Generating Line Method

Generating line method for designing and manufacturing spiral bevel gears is proposed in this paper. The tooth surface of spiral bevel gears produced by generating line method is formed by exact spherical involutes, the mathematical model to describe tooth surface has been derived based on gear meshing theory and the cutting motion. This study can provide some fundamentals for manufacturing and contact analysis of spherical involutes spiral bevel gears.

Mathematical models of grinding manufacture and tooth contact analysis of spherical gears

The spherical gear is a gear-driven mechanism with two degrees of freedom (DOF), which can transfer spatial motion and power. Grinding machining of the spherical gear is performed with a plate grinding wheel by using generating method, and based on mathematical model of the plate grinding wheel, the mathematical model of the spherical gear is set up. And then, the meshing model of a spherical gear pair is developed and the tooth contact analysis (TCA) of the spherical gear pairs is performed. After that, the position of a contact point on teeth surfaces of the spherical gear pair can be solved by using a nonlinear solver. Finally, the paths of contact points on teeth surfaces of the spherical gear are studied: Since every tooth of the spherical gear is diverse, the contact paths of the meshing of the spherical gear pair are complex and multiform. These are very useful to further investigations on transmission theory and practical applications of spherical gear pairs.

On grinding manufacture technique and tooth contact and stress analysis of ring-involute spherical gears

The spherical gear is a new gear-driven mechanism with two degrees of freedom (DOF), which can transfer spatial motion and power. A grinding machine is designed for manufacturing spherical gear with a plate grinding wheel by using generating method. Based on the mathematical models of spherical gears, the tooth contact analysis (TCA) of spherical gear pairs is performed, and the contact point paths on tooth surfaces of spherical gears are investigated. Finally, the tooth contact and bending stress analysis are simulated using Finite element method (FEM).

A new kind of spherical gear and its application in a robot's wrist joint

The meshing principles and the manufacturing methods for spherical gears with concave cone surface teeth are presented in this paper. The profile of the convex teeth of this kind of gear is analyzed and calculated. The theoretical feasibility of this kind of gear is proven by the analysis of the profile, contact ratios and undercut. It makes the process of manufacturing easier than before, improving the manufacturing precision and simplifying the manufacturing process. This paper introduces the theory and manufacturing method of concave tooth profile, that is spherical cone gearing. The analysis and calculations on the convex tooth profile engaging with the concave cone tooth is also proved.

Contact characteristics of spherical gears

In this paper, the theory of gearing and the proposed mechanism of spherical gear cutting are applied to develop the mathematical model of spherical gears. Based on the developed mathematical model of spherical gears, computer graph of the gear set is plotted, and tooth contact analysis (TCA) of spherical gear sets is also performed. The TCA results provide useful information about the kinematic errors (KEs), contact ellipses and contact patterns of spherical gear sets.

Research on transmission principle and kinematics analysis for involute spherical gear

A new kind of transmission-type spherical gear called ‘ring involute spherical gear mechanism’ is introduced. Compared to the famous TRALLFA spherical gear, this new spherical gear has an involute tooth profile, and a ring tooth that is distributed continuously on the surface of the sphere. This allows the gear to overcome two disadvantages of the TRALLFA spherical gear: the drive principle error and the manufacturing difficulty. The new transmission and the formation principle of the tooth’s surface of the new spherical gear mechanism are first introduced, then another mechanism, called disk rack, is introduced, which is derived from the spherical gear mechanism when one of the spherical gear’s tooth number reaches infinity. To make the research more convenient, every part of the new spherical gear mechanism is named. In the following sections, some problems are discussed, such as the assembly form, the construction characteristics, the correct meshing condition, the continuum transmission condition and so on. Furthermore, the paper deduces the surface formula of the conjugate teeth profiles, which proves that the conjugate teeth profiles is also one of the ringed involute spherical gear. In order to analyze the relationship between two coordinate systems, which is attached respectively to the diving spherical gear and the driven spherical gear, the orientation cosine matrix method is utilized. By series rotational transformation, the kinematics model and inverse kinematics model are deduced. Using the method for calculating the transmission ratio of planet gear train, the relationship of the two oscillating angle between output axis and the bracket is established. Based on the research, the kinematics graphic simulation of spherical gear mechanism and disk rack are made respectively. The results prove correctness of the kinematics model.