Pinion Tooth Surface Research Articles

Reconstruction of real tooth surfaces of spiral bevel pinions with modified offset

Reverse engineering has been successfully applied in many fields; however, it is not the case for spiral bevel gears. In order to reconstruct the real tooth surfaces of spiral bevel pinions based on the original tooth surface grids (OTSG) provided by coordinate measuring machine, meshing performances were achieved by tooth surface scanning method. Three meshing conditions necessary for local synthesis, i.e. the directional angle, the magnitude of major axis of contact ellipse and the second derivative of transmission errors at the mean contact point, were filtered from the meshing performances obtained by discrete tooth contact analysis. Computational tooth surface grids (CTSG), corresponding to OTSG node to node, were expressed by manufacturing parameters derived by local synthesis. Tooth surface grid difference was formed by making the centre node of CTSG through rotation operation coincided with that of OTSG. Optimization objective was built up by summing up the deviations of all nodes on either OTSG or CTSG. The three meshing conditions, the mean contact point position and the higher order coefficients of blank offset correction were taken as the design variables. The internal function fminsearch of MATLAB was chosen to minimize the tooth surface grid difference. After the manufacturing parameters were achieved, the pinion tooth surfaces were reconstructed based on the reversed CTSG. A zerol spiral bevel pinion in an aviation engine accessory gear box was employed to demonstrate the proposed methodology. The results show that OTSG is unable to give perfect meshing performance. The convergent results of reversing show that many of the field of distance between the reversed CTSG and OTSG is less than 1 μm. This demonstrates the feasibility of the proposed reversing techniques.

Remanufacturing the Pinion: An Application of a New Design Method for Spiral Bevel Gears

Damages of a large spiral bevel gear drive as used in heavy industry typically affect the pinion. Even if the gear still could be used, the complete pair has to be changed. This leads to long off times, high costs, and unnecessary waste. This paper applies a recent design technology for spiral bevel gears to the production of a replacement pinion for the sake of energy saving, reduction of costs and off times, and for the realization of green engineering. The process involves the following steps. First, the real tool surface of the gear is measured by a CMM. Based on the new design method, the tooth surface of the mating pinion is derived from this discrete point cloud. In order to improve the meshing performance, the resulting surface of the pinion is modified in the third step. Finally, the pinion is produced on a CNC machining center. In contrast to other approaches, none of these steps needs the parameters of the special machine tool defining the original gear pair. It is worth noting that our technology can also be profitable to gain more freedom in the design of new gear pairs.

Mathematical Model of Klingelnberg Cyclo-Palloid Hypoid Gear

The geometry of the tooth surface is important for tooth contact analysis, load tooth contact analysis and the ease-off of gear pairs. This paper presents a mathematical model for the determination of the tooth geometry of Klingelnberg face-hobbed hypoid gears. The formulation for the generation of gear and pinion tooth surfaces and the equations for the tooth surface coordinates are provided in the paper. The surface coordinates and normal vectors are calculated and tooth surfaces and 3D tooth geometries of gear and pinion are obtained. This method may also applied to other face-hobbing gears.

Pinion tooth surface generation strategy of spiral bevel gears

Aviation spiral bevel gears are often generated by spiral generated modified (SGM) roll method. In this style, pinion tooth surface modified generation strategy has an important influence on the meshing and contact performances. For the optimal contact pattern and transmission error function, local synthesis is applied to obtain the machine-tool settings of pinion. For digitized machine, four tooth surface generation styles of pinion are proposed. For every style, tooth contact analysis (TCA) is applied to obtain contact pattern and transmission error function. For the difference between TCA transmission error function and design objective curve, the degree of symmetry and agreement are defined and the corresponding sub-objective functions are established. Linear weighted combination method is applied to get an equivalent objective function to evaluate the shape of transmission error function. The computer programs for the process above are developed to analyze the meshing performances of the four pinion tooth surface generation styles for a pair of aviation spiral bevel gears with 38/43 teeth numbers. The four analytical results are compared with each other and show that the incomplete modified roll is optimal for this gear pair. This study is an expansion to generation strategy of spiral bevel gears, and offers new alternatives to computer numerical control (CNC) manufacture of spiral bevel gears.

An Improved Method of Solving Contact Trajectory Boundary for Spiral Bevel Gear

A method of solving contact trajectory boundary is developed for spiral bevel gear. Tooth surfaces of pinion and gear are projected to axial cross section based on rotation transformation, vector operation is used to distinguish whether contact point belongs to tooth surface or not. Distances between contact point and every boundary of tooth surface are calculated, if the minimum distance is less than preset value, this contact point is considered to be contact trajectory boundary. Starting from initial point of TCA, contact point approaches contact trajectory boundary by adaptive step size, when currently step size is greater than preset step size, the value of the last contact point is used as initial value for new contact point, otherwise initial value is calculated by particle swarm optimization with penalty function, this method can improve the solving speed greatly while keeping stable. Finally, the validity and practicability of this method are proved by a numerical example.

Generation of Hypoid Gears on CNC Hypoid Generator

In this study, polynomial functions of orders up to five are applied to induce variations in the cradle radial setting and the velocity ratio in the kinematic scheme of the machine tool for the generation of the pinion tooth surfaces corresponding to reduced transmission error amplitudes of a hypoid gear pair. The new CNC hypoid generators have made it possible to perform this nonlinear correction motions for the cutting of the face-milled hypoid gears. An algorithm is developed for the execution of motions on the CNC hypoid generator for the generation of face-milled hypoid gear tooth surface, based on the machine tool setting variation on the cradle-type hypoid generator induced by the optimal polynomial functions up to fifth-order. By using the corresponding computer program, the motion graphs of the CNC hypoid generator are determined for the generation of hypoid gear tooth surface, based on the optimal variation in the velocity ratio in the kinematic scheme and on the variation in the cradle radial setting on a cradle-type generator. The results presented indicate that the variation of the velocity ratio in the kinematic scheme of the hypoid generator induced by a fifth-order polynomial function resulted in a 62% reduction of the maximum transmission error of the gear pair.

Generation and TCA of Straight Bevel Gear Drive with Modified Geometry

In order to reduce the sensitivity of straight bevel gear drives to misalignment, a new geometry of such gear drives is proposed in longitudinal direction. Point contact instead of line contact of tooth surfaces is achieved by longitudinal crowning of pinion tooth surface. The tooth surface modeling and tooth contact analysis (TCA) of straight bevel gear drives have been established. TCA program of a pair of straight bevel gears was performed in MATLAB and tooth bearing contact and transmission errors were obtained.

Manufacturing Process for Circular-Arc Curvilinear Cylindrical Gears with Predesigned Fourth Order Transmission Error

A fourth order transmission error was employed to improve the stability and tooth strength of circular-arc curvilinear cylindrical gears. The coefficient of fourth order polynomial curve was determined, the imaginary rack cutter which formed by the rotation of a head cutter and the imaginary pinion were introduced to determine the pinion and gear tooth surfaces, respectively. The numerical simulation of meshing shows: 1) the fourth order transmission error can be achieved by the proposed method; 2) the stability transmission can be performed by increasing the angle of the transfer point of the cycle of meshing; 3) the tooth fillet strength can be enhanced.

An Ease-Off Based Method for Loaded Tooth Contact Analysis of Hypoid Gears Having Local and Global Surface Deviations

Actual hypoid gear tooth surfaces do deviate from the theoretical ones either globally due to manufacturing errors or locally due to reasons such as tooth surface wear. A practical methodology based on ease-off topography is proposed here for loaded tooth contact analysis of hypoid gears having both local and global deviations. This methodology defines the theoretical pinion and gear tooth surfaces from the machine settings and cutter parameters, and constructs the surfaces of the theoretical ease-off and roll angle to compute for the unloaded contact analysis. This theoretical ease-off topography is modified based on tooth surface deviations and is used to perform a loaded tooth contact analysis according to a semi-analytical method proposed earlier. At the end, two examples, a face-milled hypoid gear set having local deviations and a face-hobbed one having global deviations, are analyzed to demonstrate the effectiveness of the proposed methodology in quantifying the effect of such deviations on the load distribution and the loaded motion transmission error.

Contact Characteristics of Circular-Arc Curvilinear Tooth Gear Drives

Abstract In this paper, the circular-arc curvilinear tooth gear drive is proposed. The gear and pinion tooth surfaces are generated by two complemented circular-arc rack cutters with curvilinear tooth-traces. According to the theory of gearing, the mathematical model of the proposed gear is developed. The tooth contact analysis technique is utilized to investigate the kinematical errors of circular-arc curvilinear tooth gear drives under different assembly errors. Contact patterns of the circular-arc curvilinear tooth gear drive are simulated by the developed computer-aided tooth contact analysis programs and surface topology method. Numerical examples are presented to show the kinematical errors of the circular-arc curvilinear tooth gear set under different assembly conditions. Relations among the circular-arc tooth profile, curvilinear tooth-trace, contact ratio, contact pattern, and kinematical error are also demonstrated by numerical examples.

Design and Manufacture of Spiral Bevel Gears With Reduced Transmission Errors

A method for the determination of the optimal polynomial functions for the conduction of machine-tool setting variations in pinion teeth finishing in order to reduce the transmission errors in spiral bevel gears is presented. Polynomial functions of order up to 5 are applied to conduct the variation in the cradle radial setting and in the cutting ratio in the process for pinion teeth generation. Two cases were investigated: In the first case, the coefficients of the polynomial functions are constant throughout the whole generation process of one pinion tooth-surface; in the second case, the coefficients are different for the generation of the pinion tooth-surface on the two sides of the initial contact point. The obtained results have shown that by the use of two different fifth-order polynomial functions for the variation in the cradle radial setting for the generation of the pinion tooth-surface on the two sides of the initial contact point, the maximum transmission error can be reduced by 81%. By the use of the optimal modified roll, this reduction is 61%. The obtained results have also shown that by the optimal variation in the cradle radial setting, the influence of misalignments inherent in the spiral bevel gear pair and of the transmitted torque on the increase in transmission errors can be considerably reduced.

Head-cutter for optimal tooth modifications in spiral bevel gears

Abstract A method for the determination of optimal tooth modifications in spiral bevel gears based on improved load distribution and reduced maximum tooth contact pressure and transmission errors is presented. The modifications are introduced into the pinion tooth-surface by using a head-cutter with bicircular profile and with optimal diameter. In the optimization of tool parameters the influence of relative position errors of the mating gears is included. By using the computer program developed for load distribution calculation, the influence of head-cutter parameters and relative position errors of the mating gears on tooth contact and transmission errors is investigated. Based on the results obtained, the radii and the position of the circular tool profile arcs and the cutter diameter for pinion teeth finishing were optimized. By applying the optimal tool parameters, the maximum tooth contact pressure is reduced by 45% and the maximum angular position error of the driven gear by 60%, in regard to the spiral bevel gear pair with a pinion manufactured by a cutter of straight-sided profile and of diameter determined by the commonly used methods.

Computer simulation of tooth contact analysis of mismatched spiral bevel gears

Abstract A new method for computer aided tooth contact analysis in mismatched spiral bevel gears is presented. As the introduced tooth surface modifications in spiral bevel gears are relatively small and the conjugation of the mating surfaces is relatively good, thus it is assumed that the theoretical point contact of mismatched spiral bevel gears under load spreads over a surface along the whole or part of the “potential” contact line made up of the points of the meshing teeth surfaces in which the geometrical separations of these surfaces along the tooth face width are minimal. The method is based on the minimization of the function that determines these separations, defined as the distances of the corresponding surface points that are the intersection-points of the straight line parallel to the common surface normal in the instantaneous theoretical contact point with the pinion and gear tooth surfaces. The points with these minimal separations make up the potential contact lines. The method includes the determination of path of contact and transmission errors, too. A computer program implements the method. By using this program the path of contact, the potential contact lines, the separations of the meshing tooth surfaces along these contact lines, the angular displacements of the gear member and the variation of the angular velocity ratio through a mesh cycle are calculated. The influence of machine tool setting and relative position errors of meshing pinion and gear on tooth contact is investigated and discussed.

Diagonal shaving of an involute pinion: optimization of the geometric and kinematic parameters for the pinion finishing operation

Investigation of diagonal shaving of an involute pinion and optimization of parameters for the pinion finishing operation and of the shaving cutter design are the objectives of the current research. The paper covers the following topics: (1) analysis of the local topology of the pinion tooth surface and shaving cutter; (2) exploration of the parameters of geometry of contact between the involute tooth surfaces of the pinion and the shaving cutter; (3) determination of the optimum parameters of the shaving cutter design and of the kinematics of the diagonal shaving operation. An approach for the computation of the optimum parameters for the diagonal shaving operation of an involute pinion is developed. The approach is based on fundamental results in the theory of gearing obtained by Buckingham (Analytical mechanics of gears, Dover Publications, New York). The developed approach enables the computation of the optimum parameters for the gear finishing operation. Its application can be easily extended to and employed for the optimization of parameters for other methods of rotary shaving operations of involute and non-involute pinions, i.e., axial (transverse or conventional) shaving, tangential (or underpass) shaving, and plunge shaving operations. Computer simulation of the diagonal shaving operation proves that application of the developed approach enables shaving time reduction or reduction of the actual pinion tooth surface deviation from the true involute tooth surface.

Technological methods for noise/vibration reduction in driveline/transmission of trucks and all-wheel-drive vehicles

The paper is targeting solution to the problem of noise/vibration reduction in driveline/transmission of a truck and all-wheel-drive vehicle. It is customary to modify the gear micro-geometry to accommodate misalignment and minimise transmission error in a truck transmission. Various possible kinds of modification of a pinion tooth surface are considered. Topological modification of a pinion tooth surface, if a proper design is accomplished, yields up to two times reduction of transmission errors. Reduction of transmission errors results in corresponding reduction of noise/vibration excitation by a truck driveline/transmission. An advanced technology of finishing of topologically modified pinion tooth surface for low-noise/noiseless vehicle transmission is proposed.

Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis

The main defects of misaligned helical gear drives with parallel axes are: edge contact, noise, and not favorable conditions of bearing contact. One of the greatest concern of manufacturing of helical gear drives (with parallel axes) is the edge contact of tooth surfaces that is caused by misalignment. At present, attempts to avoid edge contact are based on providing chamfers of the tooth surfaces of the gears that may be obtained by modification of the profile of the cutting hobs. The zones of tooth surfaces with chamfers and zones with conventional screw involute surfaces are not connected smoothly, in many cases the magnitude of required chamfers is not determined analytically, and the edge contact is not avoided. The finishing process of helical gears with chamfers is a complicated one. The existing design has to be complemented with TCA (tooth contact analysis). These are the reasons why a new topology of modified helical gear tooth surfaces with involute and crowned zones is proposed. An involute zone is provided in the central area of gear tooth surfaces that will allow line contact if misalignment does not occur. Zones at the top, bottom, front, and back sides of tooth surfaces are crowned and allows localization of the bearing contact when misalignments occur. Crowned zones with smooth connections to the involute zone are obtained as the result of profile and longitudinal crowning. The function of transmission errors is provided at each cycle of meshing as the sum of three branches. Two branches of a parabolic function at the extremes of the cycle of meshing are in tangency with the middle branch of zero transmission errors. The generation of gear tooth surfaces is accomplished by a grinding worm. This has required the solution of two problems, (i) determination of the worm thread surface that is conjugated to the theoretical pinion tooth surface, and (ii) generation by the grinding worm of a crowned pinion tooth surface. Simulation of meshing of misaligned gear drives is accomplished by application of TCA program developed by the authors. The formation of the bearing contact is analyzed considering more than one cycle of meshing. Existence of areas of severe contact stresses is avoided. The developed approach is illustrated with numerical examples.

Mathematical model of face-milling spiral bevel gear with modified radial motion (MRM) correction

Conventional machine setting calculation for the cutting of face-milling spiral bevel and hypoid gears is based on the local synthesis at the mean point. The bias of contact path and motion curve are usually coupled with each other when applying commonly-used tooth flank corrections such as the helical motion, the modify roll method, and the cutter tilt. In this paper, we propose a method to determine the machine settings with modified radial motion (MRM) correction at specified contact point with predetermined motion curve and contact path bias on the pinion tooth surface. Parameters of MRM correction are calculated according to the equations of meshing and relationship between mating curvatures at the specified contact point. As shown by the numerical examples, the bias of contact pattern and the motion curve were controlled separately. The proposed methodology could be used to gain more control on the contact pattern and the motion characteristics.

Fourth-Order Kinematic Synthesis for Face-Milling Spiral Bevel Gears With Modified Radial Motion (MRM) Correction

The use of a fourth-order motion curve is proposed by Stadtfeld and Gaiser to reduce the running noise of a bevel gear set recently. However, the methodology of synthesizing the tooth surfaces was not clearly shown in the literature. In this work, we proposed a methodology to synthesize the mating tooth surfaces of a face-milling spiral bevel gear set transmitting rotations with a predetermined fourth-order motion curve and contact path. A modified radial motion (MRM) correction in the machine plane of a computer numerical control (CNC) hypoid generator is introduced to modify the pinion tooth surface. With MRM correction, an arbitrary predetermined contact path on the pinion tooth surface with predetermined fourth-order motion curve can be achieved. Parameters of MRM correction are calculated according to the predetermined contact path and motion curve. As shown by the numerical examples, the contact path and the motion curve were obtained as expected by applying the MRM correction. The results of this work can be applied to the pinion, which is generated side-by-side (for example, fixed setting method, formate method, and Helixform method) and can be used as a basis for further study on the motion curve optimizations.

Computation of Parameters of a Form Grinding Wheel for Grinding of Shaving Cutter for Plunge Shaving of Topologically Modified Involute Pinion

Abstract The paper is targeting on the finishing of precision gears for low-noise/noiseless transmission for cars and light trucks. Transmission error is the predominant cause of gear noise. The application of a topologically modified pinion results in reduction of transmission error up to two times. The required modification of the pinion tooth surface is provided on a plunge shaving operation with application of a shaving cutter of an appropriate design. A novel approach for computation of parameters of a form grinding wheel for grinding of the shaving cutter for plunge shaving of a precision involute pinion with topologically modified tooth surface is reported in the paper. The developed approach for computation of parameters of the form grinding wheel is focused on application of the shaving cutter grinder with a lack of CNC articulation. The problem under consideration is solved using the DG/K-based approach of part surface machining earlier developed by the author. (The DG/K-approach is based on fundamental results obtained in differential geometry of surfaces, and in kinematics of multi-parametric motion of a rigid body in E3 space (See Radzevich, S.P., Sculptured Surface Machining on Multi-Axis CNC Machine. Monograph, 1991, Vishcha Shkola Publishers, Kiev (in Russian). See also Radzevich, S.P., 2001, Fundamentals of Surface Machining. Monograph, Rastan, Kiev (in Russian).) An analytical solution to the problem is discussed in the paper. The solution has been used for developing software for the Mitsubishi ZA30CNC shaving cutter grinder for the needs of the automotive industry. Computer simulation reveals high accuracy of the ground shaving cutter.

Optimal Tooth Modifications in Hypoid Gears

A method for the determination of optimal tooth modifications in hypoid gears based on improved load distribution and reduced transmission errors is presented. The modifications are introduced into the pinion tooth surface by using a cutter with bicircular profile and optimal diameter. In the optimization of tool parameters the influence of shaft misalignments of the mating members is included. As the result of these modifications a point contact of the meshed teeth surfaces appears instead of line contact; the hypoid gear pair becomes mismatched. By using the method presented in (Simon, V., 2000, “Load Distribution in Hypoid Gears,” ASME J. Mech. Des., 122, pp. 529–535) the influence of tooth modifications introduced on tooth contact and transmission errors is investigated. Based on the results that was obtained the radii and position of circular tool profile arcs and the diameter of the cutter for pinion teeth generation were optimized. By applying the optimal tool parameters, the maximum tooth contact pressure is reduced by 16.22% and the angular position error of the driven gear by 178.72%, in regard to the hypoid gear pair with a pinion manufactured by a cutter of straight-sided profile and of diameter determined by the commonly used methods.