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The variable hyperbolic circular arc tooth trace cylindrical (VH-CATT) gear is a new type of gear that can be efficiently processed with special machine tools. However, geometric errors can affect the machining accuracy of these special machines, leading to a reduction in gear surface quality and potential failure during application. To address this issue, this paper establishes a machine tool coordinate system based on the gear forming principle and derives a motion transfer matrix between components using the Huston lower numbered body array method. An error model for the tooth surface, which includes 40 sources of machine tool error, is established, and the equation for the tooth surface with error terms is derived. A second-order approximation is used to represent the characteristics of the gear error surface, and the impact of individual error sources is analyzed through simulation. Then, comprehensive simulation of all error sources is conducted, and gear machining experiments and data inspection are performed to obtain gear surface error data for comparison. By comparing the measured gear surface error data with the comprehensive error simulation, it is shown that the actual error trend is consistent with the simulation, confirming the correctness of the geometric error model of the special machine tool. This model demonstrates the influence of machine tool error sources on gear tooth surface accuracy, and provides theoretical support for error compensation. [ABSTRACT FROM AUTHOR]

The cutter error is an important factor in the accurate forming of the variable hyperbolic circular arc tooth trace (VH-CATT) cylindrical gears. Also, the study of the relationship between the cutter error and the forming of accurate teeth is beneficial for the gear modification design and the improvement in contact performance. Firstly, based on the principle of forming VH-CATT cylindrical gears, the sources of error in the tooth forming related to accuracy were analysed, including the errors in the cutter position and the cutter geometry, such as the error of rotation around the x-axis g, the error of rotation around the y-axis b, the error of translation along the x-axis Δx, the error of translation along the y-axis Δy, the error of translation along the z-axis Δz, the pressure angle error Δa and the tooth line radius error ΔRT. Next, based on the meshing theory and processing, an ideal tooth surface equation and a tooth surface equation with cutter errors were derived, and the tooth surface reconstruction was done. Then, the gear tooth thickness error was defined to characterize the accuracy of gear forming. Finally, the influences of the cutter error on the tooth thickness error and gear contact were investigated. The study shows that all cutter errors have certain influence on the tooth thickness error, contact area and load distribution; Δx has basically no effect on the gear tooth thickness error; Δx and g make the actual meshing point deviate from the middle section. The study content and the applied methods are helpful in the tooth surface error traceability, the counter-adjustment of the tooth surface processing and the modification design. This study provides also a basis for gear design and load-bearing contact analysis. [ABSTRACT FROM AUTHOR]

The traditional geometry of non-circular cylindrical gears provides a contact line between tooth surfaces. In a misaligned non-circular cylindrical gear drive, causing edge contacts that result in noise and vibration. The point contact of the tooth surface is achieved by modification of the driving gear in the longitudinal direction, in order to reduce the sensitivity of tooth surface contact to installation errors. The proposed longitudinal modification method is based on the application of a grinding worm by controlling the center distance of non-circular cylindrical gear and worm. Based on the gear meshing principle, a mathematical model of the tooth surfaces of non-circular cylindrical gears is established, and a parametric tooth surface equation with a unified expression is obtained. Tooth contact analysis (TCA) of non-circular cylindrical gears shows that the contact path and contact area on the tooth surface are shifted with the change of installation error, but still distributed in the middle of the tooth surface without creating edge contact. The transmission error is increased with the growth of installation error. Tooth modification in a longitudinal direction has an obvious improvement effect on the bad tooth contact condition, but too large a tooth modification parameter will make the tooth surface load distribution concentrated in a smaller area, which results in the decrease of gear load capacity. [ABSTRACT FROM AUTHOR]

Cylindrical gear with variable hyperbolic circular arc tooth trace (VH-CATT) is a new type of gear. Sliding wear is the main mode of the surface failure of multiple mechanical parts. Both the lubrication state and contact temperature considerably influence wear characteristics, which may aggravate the transmission performance of gear pairs. Wear, contact temperature, as well as lubrication states are jointly explored. Therefore, an integrated wear prediction model was proposed through taking into account flash contact temperature and surface roughness of VH-CATT cylindrical gears in mixed elastohydrodynamic lubrication. According to the equivalent ellipse contact model of VH-CATT cylindrical gears and tooth surface equation, normal curvature and velocity relations for VH-CATT cylindrical gears were observed, and the normal meshing force was obtained through the consideration of load sharing coefficients and quality grades. Flash contact temperature was estimated by using the literature. This study proposes analytical solutions for investigating how various surface roughness, operation, and geometric parameters affect asperity contact ratio (ACR), asperity contact pressure (ACP), flash contact temperature (FCT), as well as wear depth (WD) related to driving gears. ACR, ACP, FCT, as well as WD initially decrease and then increase from engaging-in to engaging-out processes. The minimum occurs at the pitch point. The WD declines as module, cutter radius, and rotational velocity increase while augmenting when surface roughness and torques increase. The maximum and minimum wear depths in driving gears occur at the dedendum and pitch point, respectively. Its overall wear is reduced by 23.16 % compared to the wear of spur gears. The results are valuable for the studies of tooth pitting, wear resistance, and fatigue life improvement for VH-CATT cylindrical gear. These studies can provide verification data and references required for engineering designs and VH-CATT cylindrical gear operations. [ABSTRACT FROM AUTHOR]

The 3-DOF spherical gear is proposed and its tooth surface is composed of involute ring tooth surface and semi-spherical bevel tooth surface， the error-free transmission can be realized. Based on the involute forming principle and the coordinate transformation theory， the curved surface equations of annular tooth and arc tooth are derived. According to the meshing principle and motion analysis， the conjugate tooth surface of the arc tooth surface is obtained. The simulation tooth surface and 3D model of 3-DOF spherical gear are obtained by computer， the correctness of tooth surface equation is verified. The basis for the kinematics and dynamics analysis of the 3-dof spherical gear mechanism is provided.

The three-dimension coordinate equation of tooth surface of spiral bevel gear is obtained based on the forming principle of spiral bevel gear. According to the spiral bevel gear computational adjustment cards provided by the Gleason company,the three-dimensional model of spiral bevel gear pair is established by writing the MATLAB program to solve out the 3 D coordinates of discrete points on the tooth surface and using three dimensional modeling software. The contact analysis of bevel gear pair is completed by importing the 3 D model into the finite element software Abaqus to create the finite element model of spiral bevel gear. According to the analysis results,the change of the contact positive pressure of a pair of gear teeth in a complete meshing cycle is obtained,and the accuracy of the simulation results is verified by experiments which provides reference and technical support for the design,manufacture and optimization of spiral bevel gear.

In view of the shortcomings of involute gear drive,the advantages of synthetic face gear drive are summarized. The circular arc profile is applied to the face gear,and the gear drive with circular tooth profile is put forward. By means of theoretical analysis,numerical calculation and physical experiment,the design of the tooth surface of the circular tooth profile is carried out. Based on gear meshing theory and differential geometry theory,the tooth surface equation of the conjugate tooth profile of the basic tooth profile of the circular rack is modeled firstly,and then use the envelope forming theory to derive the equation of gear face machining arc face gear,and the cutter and circular arc tooth profile gear for each envelope forming work on tooth surface and the derivation of the face gear transition surface equation. Finally,the MATLAB software program is used to solve the set of tooth points and imported into the CATIA software. The EXCEL table with macro program runs the macro program and generates the contour curve in the CATIA software. And then the 3D physical tooth surface of the arc gear profile is obtained by surface fitting method. Compared with the simulation model of gear cutting envelope,the accuracy is verified.

According to the gear meshing theorem, the ordinal design of the pitch curve and tooth surface of the gear pair is presented to realize the predicted transmission function for different application. Based on the complex motion and applications of curve face gear, the transmission ratio and axial movement speed are proposed by space coordinate transformation, also the slip velocity equation on the tooth surface curve face gear is established. The sliding characteristics on the tooth surface with different parameters are given. The results showthat the size and direction of sliding velocity vary with the tooth height, and the sliding characteristics of the 5 teeth are different during the half cycle. [ABSTRACT FROM AUTHOR]

The quadruple arc spiral bevel gear is a new type of gear with quadruple arc tooth profile applied to the spiral bevel gear. The three-dimensional model is built in Solid Works based on the tooth surface equation of spiral bevel gear with circular arc tooth profile. The static finite element analysis is carried out by using a pair of teeth based on Abaqus. Compared with the double arc spiral bevel gear,it is concluded that the load capacity of quadruple arc spiral bevel gear is 30% higher than that of double arc spiral bevel gear.

Based on generating method machining pinion of hypoid gear,according to the cutting coordinate system of milling cutter and tooth surface equation,the mathematical model of machining pinion of hypoid gear is established,the theoretical tooth surface equation of pinion. The theoretical tooth surface is dispersed,the coordinates of the discrete points are solved,the tooth surface points are imported into UG and the modeling of tooth surface is carried out. According to the anti-conjugate principle,the tooth surface equation and tooth surface points of tool gear that fully conjugated with pinion are solved and three dimensional modeling is carried out. The motion simulation of three dimensional model is carried out,through the contact area to verify the correctness of modeling. According to the principle of equal volume,the rolling round billet is solved. Introduced the rolling models into DEFORM to carry out numerical simulation. The feasibility of the processing technology is verified by the simulation result,and theoretical guidance for the follow-up experiment is provided.

In order to explicitly solve the tooth surface model of forming face gear,and exactly solve the minimum inner radius of face gear with top-cut. The orthogonal face gear envelope coordinate system is established and the tooth surface equation of generating gear is given. The face gear working tooth surface equation,transition surface equation and top-cut surface equation are derived. According to the tooth surface equation,the simulation analysis of face gear tooth surface is carried out by using the MATLAB 7. 5 and CATIA V5. According to the simulation comparison of tooth surface and the analysis of top-cut boundary line,the top-cut minimum inner radius key point is got,the key point is the intersection of the face gear tip surface and top-cut boundary line. Through the explicit solving the key points,the minimum inner radius of the face gear without top-cut can be got. Comparing with the existing literature results of the implicitly solving inner radius,and the difference on numerical is small between the two. Finally,through the simulation cutting tooth way,it shows the accuracy of this method is superior to the method of literature.

The cylindrical gear with arc tooth trace is a novel cylindrical gear with the features of good meshing performance,large overlap coefficient,smooth transmission,low noise,high efficiency,long service life and so forth. It can make up the deficiency of the existing gear transmission pairs. In order to further study its characteristic,the tooth surface equation is deduced according to the processing principle of rotationary cutter for the cylindrical gear with arc tooth trace and meshing principle. Also,in order to improve the modeling efficiency,the parametric three- dimensional modeling of the rotating disk method is completed based on the secondary development technology. This provides the correct 3D model for the following simulation analysis.

This paper aims to accurately simulate straight bevel gear transmission, and analyse its exact meshing performance. To this end, the equations for tooth surface, tooth root surface and tooth root transition surface of straight bevel gear were derived based on the gear planning principle. Then, the parametric finite-element modelling of straight bevel gear transmission was realized in ANSYS using surface interpolation technique, and the gear pair meshing was simulated by transient analysis. The tooth surface contact stress and the bending stress were obtained throughout the meshing process. Later, the proposed model was contrasted with three traditional models to verify its feasibility. The results show that the modelling accuracy directly bears on the gear strength, and both the contact stress and the bending stress of the gear varied with the meshing position. The proposed method eliminates the huge errors of the traditional equivalent gear design, and pioneers in the analysis of the meshing features of gear pair. Suffice it to say that this research lays the basis for the accurate design and optimization of straight bevel gear transmission. [ABSTRACT FROM AUTHOR]

Aiming at the cycloidal gear used in RV reducer of robots,based on the concept of the single tooth without side clearance,the tooth surface equation of the cycloidal gear after modification is deduced by directly adding the second- order parabolic modification amount to the theoretical conjugate tooth profile along the normal direction. The modification amount of different meshing position on tooth profile curve could be flexibly controlled by controlling the tooth profile modification coefficient. The differences between the parabolic modification and the traditional modification method of the combination of isometric and travel distance,the proposed parabolic modification method is much closer to the complete conjugate tooth profile on the main meshing areas.The TCA program of cycloidal- pinwheel transmission which is written in Matlab formed parabolic transmission errors and the correctness of this modification is proved,which provides a new thought for the modification design of cycloidal gear used in RV reducer.

• Meshing point track and contact line range on tooth surface are rigidly divided. • Modeling approaches of tooth stiffness under line and point contact are proposed. • Models of load distribution ratio and meshing stiffness are established. • Effect of force on load distribution ratio and meshing stiffness is analyzed. • Coupling relationship among force, distribution ratio and stiffness is revealed. Analytical models of load distribution ratio (LDR) and meshing stiffness for orthogonal spur-face gear drive under point contact are established to provide the parameter basis for dynamics studies in this paper. Meshing point tracks and contact line ranges on tooth surfaces are strictly divided according to the point contact ratio. Local contact stiffness at meshing point is calculated based on Hertz contact theory and calculation approaches of global tooth stiffness under line and point contact are proposed and compared. Then, models of LDR and meshing stiffness are constructed and analyzed. The results show that the global tooth stiffness decreases after considering point contact on the basis of line contact. LDR is significantly affected by the error on tooth surface. The meshing stiffness increases with the increase in meshing force. The contact deformation accounts for a high proportion in the total deformation. It expands the existing modeling methods to cover line and point contact, and lays the foundation of studies for orthogonal spur-face gear drive. [Display omitted] [ABSTRACT FROM AUTHOR]

For the main purpose of studying the tooth surface equation and undercutting conditions of the cylindrical gears with curvilinear shaped teeth processed by parallel translating mechanism, the tooth surface equation and meshing condition of the cylindrical gears with curvilinear shaped teeth processed by parallel translating mechanism were deduced by the method of differential geometry and coordinate transformation, based on considering cutting tool shape and installation position error. The undercutting line of the cylindrical gears with curvilinear shaped teeth processed by parallel translating mechanism was also calculated. The manufacturing process of the parallel translating mechanism was simulated in threedimensional software and then the three dimensional solid model of cylindrical gears with curvilinear shaped teeth was obtained. The studies referred to in the present paper have a certain reference value for research, development and design of the cylindrical gears with curvilinear shaped teeth processed by parallel translating mechanism. [ABSTRACT FROM AUTHOR]

• Iterative equations of restrictive conditions are constructed. • The contact line midpoint tracks and meshing point tracks are obtained. • Calculation equations of line contact ratio in shaping are established. • The calculation method of point contact ratio in meshing is elaborated. • The connection between line contact ratio and point contact ratio is disclosed. The contact ratio of face gear drive is hard to calculate precisely because it is point contact adjusted from line contact to avoid jamming. To obtain the calculation methods and improvement measures for the contact ratio of face gear drive, the line contact ratio (LCR) in shaping and the point contact ratio (PCR) in meshing are investigated in this paper. Tooth surface equations of face gear shaping are established based on its generation principle and the contact line midpoint tracks are obtained. Iterative equations to accurately calculate restrictive conditions (undercutting and pointing) for face gear tooth are constructed and verified, their approximate equations are given further. Meshing tracks in face gear drive are obtained by solving the established meshing equations. Calculation methods of the LCR and PCR are elaborated according to their definitions and the fitting equations are given for engineering application. The results show that the PCR is markedly lower than the LCR. The advantage of high contact ratio of face gear drive is significantly affected because it is very sensitive to the realization of point contact. It is acceptable to use the LCR to approximate the PCR when the distribution range of the meshing track on the face gear tooth surface is determined. It provides a theoretical basis for the design and optimization of orthogonal spur-face gear drive. [Display omitted] [ABSTRACT FROM AUTHOR]

Misalignment and gear modification were introduced to gear transmission coordinate system and tooth surface equations of the modified helical gears were established in the movable coordinate systems that were rigidly connected to the gears. The meshing equation was obtained by applying coordinate transformation matrix from the movable coordinate systems to the fixed coordinate system that was rigidly connected to the frame. The equation of meshing was presented to meet the condition of continuous tangency. The minimization of distance between tooth surfaces combined with deviation between unit normal vectors to tooth surfaces was performed to obtain the results of the equation of meshing by application of a constrained nonlinear optimization function in Matlab. The influence of the combination of the various modification style and misalignment on the contact path and transmission error was investigated. The presented results highlight that the modification pattern proposed can improve the tooth meshing condition and the adaptability to the misalignment. [ABSTRACT FROM AUTHOR]

In order to theoretically research the tooth surface maximum contact stress of a Cylindrical Gear with Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT), the computing formula of maximum contact stress of VH-CATT cylindrical gear is investigated according to Hertz formula in this paper. Insufficient contact fatigue strength will lead to pitting corrosion, plastic deformation of tooth surface and other damages. Therefore, the maximum contact stress of tooth surface must be carried out. The contact stress calculation formula is particularly considering the effect of normal force, total carrying length, synthetical curvature radius, and position angle. The present work establishes analytical solutions to research the effect of different parameters for the contact stress of VH-CATT cylindrical gear incorporating elastic deformation on the tooth surface, and which have shown that the different module, transmission ratio, pressure angle, tooth width, and the cutter head radius have a crucial effect on the contact stress and contact ellipse of VH-CATT cylindrical gear along the tooth width direction. Moreover, a finite element analysis is carried out to verify the correctness of the theoretical computing formula of contact stress of VH-CATT cylindrical gear. By contrast with the theoretical calculated value and the stress value of finite element analysis, its error is very small. It is indicated that the derived formula of contact fatigue strength of VH-CATT cylindrical gear has high accuracy and can accurately reflect the real contact stress value of tooth surface, which is beneficial for research on tooth break reduction, pitting, wear resistance and fatigue life improvement of the VH-CATT cylindrical gear. The study results also have a certain reference value for the design and check calculation of the VH-CATT cylindrical gear. [ABSTRACT FROM AUTHOR]