Tooth Surface Deviation Analysis for Continuous Generation Grinding of Internal Gears

Face gear drives exhibit smooth transmission, low noise, and insensitivity to pinion axial installation error. Nonorthogonal face gear pairs enable torque transmission and speed variation under diverse shaft intersection angles, thus expanding their engineering applicability. At present, mathematical modeling of face gears generally considers only the tooth flank surface, while neglecting the tooth bottom surface. This leads to missing data in partial regions of the accurate tooth geometry of face gears. This paper focuses on the precise modeling algorithm and grinding methodology for the full tooth surface of nonorthogonal face gears. The tooth surface generation principle is analyzed, and the meshing kinematics between the grinding wheel (acting as the generating gear) and the nonorthogonal face gear are established. The full tooth surface is divided into five regions, and boundary conditions are defined separately for each region. Mathematical expressions for the generating gear and the full tooth surface of the nonorthogonal face gear are derived. The parameter boundaries for dish-wheel grinding are determined to guarantee full-tooth-surface completion. According to the common gear grinding machine configuration, the kinematic coordinate system for dish-wheel grinding is formulated, and virtual machining and grinding test are implemented. Comparative results demonstrate that the theoretical tooth surface matches the virtual machining outcome without noticeable deviation, validating the accuracy of the full tooth surface modeling and grinding motion. In this study, the proposed grinding method overcomes the limitation that non-orthogonal face gear grinding can only be performed on a few dedicated machine tools, thereby further expanding the application scope of face gear grinding technologies.

A general mathematical model for the generated gear tooth surfaces of spiral bevel and hypoid gears is established based on the three-axis CNC bevel gear machine. According to the principle and the method for the function-oriented design of point-contact tooth surfaces, the locus of spatial tooth contact points on the tooth surface is described on the axial plane of the gear, and then the detailed formulae for the design with the generated gear are derived from the mathematical model. The mathematical model and the formulae can be used in the function-oriented design of point-contact tooth surfaces of spiral bevel and hypoid gears with the gear generated in different types on both the three-axis CNC bevel gear machine and the conventional cradle one.

In the measurement process of the large, complex surface, due to the errors of the installation, the errors of the machine bed itself and other factors, the probe can be over pressed. The situation of measuring interruption can be occurred. In the paper, take the measurement of the spiral bevel gear for example, by using of the gear meshing theory and differential geometry, through a series of coordinate transformations, the coordinates of the tooth surface grid nodes are calculated. Through measuring the tooth surface of the spiral bevel gear by using of Gear Measuring Center(GMC) and 3D scanning probe system, the follow-up control strategy of continuous scanning measurement based on the constant measuring force was proposed. The measurement and evaluation of the tooth surface were achieved through continuous scanning motion along the measurement path planned. In the measurement process, the measurement path was planned according to the theoretical models. The offset values of the probe were real-time detected, the constant deformation of the probe was ensured through controlling the motion of X,Y,Z axes. The mathematical model of the relationship between the velocity of each axis and the deformation vector of the probe was established. The continuous scanning measurement along the path planned based on the constant measuring force can be implemented. The requirement of the steady-state errors and real-time were meted by using of PID control strategy of subsection variable parameters. The method was finally simulated and experimented. The results showed that the occurrence of unexpected interruption in the measuring process can be effectively solved. It plays an important role for the highly-efficient and highly-precise measurements of the tooth surface.

Measurement and Compensation of Deviations of Real Tooth Surface of Spiral Bevel Gear

Study on an internal gear with asymmetric involute teeth

The manufacturing error of spiral bevel gear tooth surface has a great influence on transmission efficiency and gear life. The error of the gear tooth surface needs to be measured accurately and fed back to the machine tool to adjust the parameters. When measuring the spiral bevel gear using a gear measuring machine, combined with the measurement theory of the tooth flank of spiral bevel gear, this paper proposed a method to compensate the inclination error in the measurement result precisely. Based on the iterative search method, a precision matching method for the theoretical and the measured tooth surface of the spiral bevel gear was designed to calculate the compensation results. The experimental results show that the inclination errors included in tooth surface measurement results reduced from more than 3 μm to less than 0.5 μm, and more than 70% of the errors are compensated by the proposed method. The accuracy of the measurement results improved significantly after compensation, and furthermore, it can provide a more accurate basis for the adjustment of machine tool parameters in the manufacturing process.

In this study, we call an internal gear having conical angle as the internal conical gear. In the case of conjugating with spur gear, this gear can transmit rotational motion between intersecting shafts. Also in the case of conjugating with conical gear, there is possibility of adjusting backlash by axial movement. Therefore, as a first step of the study we obtain theoretical tooth surface of the internal conical gear cut by gear shaper. Next we measured tooth surface of the test gears and compared them with theoretical one. Finally the tooth-bearing test confirms that theoretical analyses holds true.

To further develop the performance testing and practical application of internal meshing gear with curve configuration, the grinding manufacturing method is proposed in this paper. Based on basic principle of internal meshing gear with curve configuration, Newton’s iterative method is used to derive the contact line condition equation during the process of form grinding. Then, the spatial coordinate transformation relationship of the pinion is established, and normal profile of the grinding wheel is derived. An improved transversal algorithm is proposed to analyze the tooth profile error after form grinding simulation, and the results verify the correctness of the designed grinding wheel. A meshing model with single tooth is established. The effects of different parameters on the grinding temperature field under dry and wet grinding conditions are simulated respectively, which will be helpful to the optimization and control of temperature in grinding process. The research results lay the essential theoretical foundation and engineering analysis for practical form grinding applications of the new internal gears. Further study on experimental analysis of grinding manufacturing process of the new tooth surfaces will be carried out.

Conical gear is an involute gear having a cone angle. When the conical gears are meshed on the intersecting axis or nonintersecting- nonparallel axis, the contact between tooth surfaces is a point contact. So the tooth surface strength is generally small. It is the Concave conical gear that was developed in order to overcome this weak point. Its tooth surface is a curved surface having the reverse crowning on the tooth surface of the conical gear. Conventionally, this tooth surface has been generated by the conical shaped wheel. By this method, however, it is not necessarily easy to make an ideal tooth surface. Therefore, we devised a new rack for generating the concave conical gear. This is the rack in which the tooth thickness of normal rack is corrected in the direction of the tooth trace. We obtained the theoretical tooth surface generated by the new rack. And, we made the gear having the theoretical tooth surface by the five axis machining center. In this research, first we developed a coordinate calculation method of the theoretical tooth surface, and next a 3D model, and finally an optimal tool path by using CAM. And we measured the curved line of the tooth surface in the plane of principal radii of curvature of the test gear and confirmed that this line agrees well with the ideal curved line. From the above results, the validity of this design and production system is confirmed.

Purpose Crossed helical gears have point contact and their surfaces are subject to high surface stress. Contact stress and root tooth stress are the most common sources of failure in crossed helical gear. This paper aims to study the load-carrying capacity and performance of crossed helical gear teeth with different gear tooth profiles. To overcome defects and reduce the sensitivity to small shaft angle changes. The combined tooth profile is designed to reduce the bending stresses, contact stresses and tooth deflection, and prevent pinion failure in the gearbox. Design/methodology/approach The principle of the method is a line contact is introduced instead of a point contact between two teeth in mesh with each other. The tooth surface of the helical gear is designed and cut by a modified tool. Higher normal pressure angles like 25° and 35° are used. The modified involute is accomplished to eliminate interference between the teeth. Engineering software packages have been applied to generate all crossed helical gears gear profiles. The modification is compounding curves consisting of an epicycloid-involute-hypocycloid gear teeth profile generated by the cutter modified. Findings The stresses in the crossed helical gear teeth profile were reduced by increasing the normal pressure angle values. Using a 35° pressure angle the enhancement percentage in contact stress and teeth fillet region will be about 29.345% and 15.421%, respectively. The best enhancement in a gear’s resistance is the epicycloid-involute-hypocycloid gear teeth profile. The enhancement was 32.610% and 18.588%. The skew in line of action in skewed helical gear will be sensitive when the crossing angles are small. Their teeth surface tends to be easily worn out; however, the wearing process will be reduced by using a proposed gear teeth profile. Practical implications The gear teeth to be modified are cut by a shaper process. The modifying rack cutter of this study can be used as a reference for creating a different helical gears sample. The helical teeth surface is modified to become an envelope of the other. This makes an original point contact change into a line contact. The epicycloid-involute-hypocycloid gear teeth profile is preferred for a higher contact ratio and a large load capacity. This work explicitly introduces a new method of kinematic consideration to improve the load capacity of crossed helical gear. Originality/value This paper showed some novel results by the unique shape of the rack-cutter designed to generate different helical angles and different gear positions. In the future, it will make valuable contributions to the further development of the dynamic performance of a crossed helical gear system through the study in the field of using asymmetric teeth profiles of helical gears with tip relief as the manner to enhance the crossed helical gear performance. Investigation of crossed helical gear by applying a predesigned parabolic function of transmission error enables the absorption of linear discontinuous functions caused by misalignment.

To address the issue of cutting textures formed on gear tooth surfaces during the manufacturing process, a design method is proposed where the cutting textures are distributed according to a sine function. Initially, a simplified mathematical model of the actual gear tooth surface cutting texture was constructed based on a four-axis machining center and cylindrical milling cutter. Ten design cases were created with different sine function phases, aiming to minimize angular acceleration. These cases were compared with normally processed gear pairs to identify the optimal design. Subsequently, using the established actual tooth surface model, the optimal design gear pair was compared with normally processed gear pairs through noise simulation experiments. The correctness of the simulation results was verified through vibration and noise testing experiments. The results indicated that the maximum amplitude of the first-order frequency of vibration and noise sound pressure levels decreased by 14.93% and 11.16%, respectively, validating the effectiveness of the surface cutting texture optimization method. This method provides a novel solution for reducing vibration and noise in gears.

The advantages of the five-axis flank milling of (developable) ruled surfaces include that (1) the machined surfaces could be very accurate and smooth and (2) the machining efficiency is high. Currently, spiral bevel gears are machined on the machine tools specially used for gear manufacturing. The disadvantages are that the cost is high for small batch, prototype, or repair. If a small group of spiral bevel gears are needed, the current methods are not valid. Thus, it is expected to machine the gears on five-axis computer numerical control (CNC) milling centers. Unfortunately, when tooth surfaces are designed based on the conventional gear manufacturing methods, they cannot be accurately machined in five-axis flank milling. This work is to develop the new technique for the five-axis flank milling of spiral bevel gears. First, a new method of designing the tooth surface of spiral bevel gears with ruled surface is proposed. Second, the cutter locations and orientations are calculated for five-axis flank milling the tooth surfaces. Third, the actual tooth surfaces are accurately represented with the cutter envelope surface in five-axis flank milling. It is confirmed that the difference of the actual tooth surface and the designed tooth surface is within the tolerance. Then, a pinion is generated to mesh with the gear, and the tooth contact analysis (TCA) is conducted. The good result demonstrates that the proposed method is valid, thus it can be used in industry.

This paper investigates the geometrical design principal of the spiral bevel gears with two contact paths from spatial conjugate curve theory. Differential geometry and gearing kinematics are introduced to derive this model. In this process, the calculation method of contact paths and tooth surface generating method are presented. According to the arguments in this paper, a process of designing the tooth surface of logarithmic spiral bevel gears with two contact paths is investigated. Then, through this process, the design of a pair of logarithmic spiral bevel gears with two contact paths is completed. Besides, the prototype is manufactured and the performance experiment is completed. Results show the maximum contact stress of spiral bevel gears with two contact paths is reduced compared to those with one contact path. Besides, the transmission efficiency of the spiral bevel gears with two contact paths can reach 98.2%.

In many situations, the polymer gears are the perfect replacement for traditional metal gears. Compared to metal gears, the polymer gears will generate more heat at the gear tooth contact surface and more wear loss. Hence, this study aims to decrease the wear loss and increase the load-carrying capacity of the polymer gears. To achieve this, the reinforcement materials were added to the base polymer gear material. In this study, Nylon-66 was chosen as the polymer matrix material and an isolated glass fibre and steel grid were used as the reinforcement materials. Three different types of polymer spur gears such as Polyamide Gear (Nylon-66), Reinforced Polyamide Gear (Nylon-66+Glass Fibre), and Hybrid Glass Fiber Steel Grid Gear (Nylon-66+Glass Fibre+Steel Grid) named PG, RPG and HGFSGG were fabricated using injection moulding machine and each type of gears were compared for wear loss and specific wear rate under the similar testing conditions. The wear test was performed in the Forschungsstelle fur Zahnrader und Getriebebau (FZG) test rig at 1500 rpm with an applied load of 20 N for 12.5 h continuously. From the test results, it was identified that HGFSGG polymer spur gears have exhibited a lower surface temperature and better wear properties compared to the other two types of spur gears. From the Scanning Electron Microscope (SEM) analysis, it was noticed that the PG spur gear exhibited deep and uniform wear at the gear tooth surface. In contrast, RPG spur gear exhibited glass fibre distortion and abrasive-type wear. The SEM micrograph of the HGFSGG spur gear reveals the presence of very few surface flaws in the gear tooth surface.

As a complex space enveloped surface, the tooth surface of dual-lead helical gear enveloping hourglass worm is characteristic with difference tooth profile, unequal pitch and variable tooth thickness. To grasp the geometry characteristics and precision manufacturing method of it, the analytical method for toroidal helix is presented. The mathematical model of toroidal helix is developed, and the geometry characteristics of helix angle, axial tooth thickness, and tooth surface undercutting are analyzed. The helix angle of toroidal helix presents the convex shape, and the undercutting characteristic is affected by the pressure angle, helix angle, transmission ratio, and normal modulus. Based on the toroidal helix of tooth surface, the precision manufacturing method is proposed, and the measuring method is developed. The hourglass worm sample was manufactured and measured, and the transmission accuracy of hourglass worm drive is tested. The results show that the manufacturing and measuring method of toroidal helix is effective and feasible.