Tooth Surface Error Research Articles

Compensation method of the spiral bevel gear tooth surface error based on eight tooth-surface fitting

Abstract The misalignment error between the gear theoretical coordinate system and actual gear coordinate system is illustrated, as well as its influence on the measurement of tooth surface deviation is discussed. On the analysis of the tooth convex and concave surface error measurement process, the measured points in the theoretical coordinate system are transformed from the actual measured points in the actual coordinate system. Then, an eight tooth-surface fitting method is utilized to calculate the misalignment error and the compensation method is given. Finally, actual measurements of three different spiral bevel gears are analyzed, and the misalignment error are calculated. Using the proposed compensation method, both the eccentricity and tilt are compensated to some extent. And the machining suggestions of each spiral bevel gear are also given according to its compensated tooth surface error distribution.

Optimizing circumferential assembly angle of rotor parts connected by curvic couplings based on acquired tooth surface error data

Due to the excellent self-centering and load-carrying capability, curvic couplings have been widely used in advanced aero-engine rotors. However, curvic tooth surface errors lead to poor assembly precision. Traditional physical-master-gauge-based indirect tooth surface error measurement and circumferential assembly angle optimization methods have the disadvantages of high cost and weak generality. The unknown tooth surface fitting mechanism is a big barrier to assembly precision prediction and improvement. Therefore, this work puts forward a data-driven assembly simulation and optimization approach for aero-engine rotors connected by curvic couplings. The origin of curvic tooth surface error is deeply investigated. Using 5-axis sweep scan method, a large amount of high-precision curvic tooth surface data are acquired efficiently. Based on geometric models of parts, the fitting mechanism of curvic couplings is uncovered for assembly precision simulation and prediction. A circumferential assembly angle optimization model is developed to decrease axial and radial assembly runouts. Experimental results show that the assembly precision can be predicted accurately and improved dramatically. By uncovering the essential principle of the assembly precision formation and proposing circumferential assembly angle optimization model, this work is meaningful for assembly quality, efficiency and economy improvement of multistage aero-engine rotors connected by curvic couplings.

Research on influence mechanism of geometric error sources on tooth surface error of VH-CATT gear special machine tool

Research on influence mechanism of geometric error sources on tooth surface error of VH-CATT gear special machine tool

A Bootstrap-Based Tooth Surface Errors Statistics Methodology for Batch Hypoid Gears after Heat Treatment

In the manufacturing and production of hypoid gears, it is a necessary key problem to improve the tooth surface heat treatment precision and production efficiency of the hypoid gears. How to use advanced statistical theory and methods to evaluate the whole batch machining quality of the tooth surface after heat treatment is particularly urgent. In this connection, for the same batch of hypoid gears with the same gear material, numerical control gear milling method, and heat treatment specifications, a bootstrap-based statistics scheme of tooth surface errors after heat treatment is proposed in this paper. The bootstrap statistics model of the tooth surface errors for the batch hypoid gears is established. The bootstrap probability eigenvalues and confidence intervals of the measurement sequence points on the tooth surface errors are solved, and the optimizing selection of the single sampling numbers and the repeated sampling times is completed. On this basis, by applying the cubic NURBS surface fitting method, the mean value difference surface of the batch tooth surface errors data is constructed, the statistics laws of the whole batch tooth surface errors after heat treatment is determined, and the effective evaluation of the whole batch tooth surface accuracy is realized. Finally, the feasibility and correctness of the bootstrap-based statistics scheme are verified by the tooth surface errors bootstrap statistics application experiment of one kind of hypoid gear. The results show that with the help of the bootstrap statistics method proposed in this paper, it is not necessary to accurately measure the tooth surface errors of the whole batch of hypoid gears one by one. Only by randomly selecting 10 tooth surface samples and repeatedly sampling 2000 times, the original sample characteristic values of the whole batch tooth surface errors can be accurately estimated, and the heat treatment deformation statistics laws of the whole patch tooth surfaces can be also counted.

Transmission performance evaluation of non-circular spur bevel gear based on a novel isochronal measurement method

This paper aims to study the transmission performance evaluation of non-circular spur bevel gear (NCSB-gear) based on a novel isochronal measurement method. Firstly, the mathematical model of NCSB-gear is established. Then a novel isochronal measurement method is proposed, and the actual tooth surfaces are reconstructed by non-uniform rational B-spline interpolation. The advantages of the proposed method are discussed based on the geometric characteristics of tooth surface and measurement fitting error. A pair of forged NCSB-gears is measured through a coordinate measurement machine and the maximum tooth surface fitting error by the proposed method reduces 2μm compared with the traditional method. Moreover, deviation tooth contact analysis (DTCA) and load tooth contact analysis (LTCA) are conducted to evaluate the transmission performance. Finally, the transmission performance of measured NCSB-gear is obtained by a gear rolling tester. By comparing the TCA calculation of different measurement methods, it is found that the relative error of the maximum transmission error based on the isochronal measurement method reduces 4.719% compared with the traditional measurement method, thus verifying the validity of the proposed measurement method.

Modelling method, simulation and experimental verification of hypoid gear involved tooth surface deviation under manufacturing process

Tooth surface errors are inevitable in manufacturing process, and directly affect the meshing performance and NVH properties. In this paper, the tooth profile deviation and midpoint tooth pitch error were considered. Firstly, the theoretical tooth surface model was established by simulating the manufacturing process of face-hobbed hypoid gear. Then, a novel modelling method of tooth surface errors was proposed based on the relationship between discrete tooth surface points and measured error points. A solid model and finite element model by using CAD and CAE technologies were developed to study the effects of tooth surface errors and meshing phase on the contact pattern and transmission error. Results show that the gear tooth surface error was the primary factor while the tooth meshing process contributes the least of total transmission error. The peak-to-peak of the transmission error values of pinion and gear with error was 24.786 times higher than standard. The meshing phase of gear had greater impact on the contact pattern. The range of transmission error fluctuation was 210 μrad within different meshing phases. The tooth surface points measurement and rolling experiment were performed and the fluctuation range was around 233.86 μrad with a 10.2% difference.

Pivotal errors identification of the face gear worm grinding machine tool with a piecewise sensitivity analysis

As one of the main error sources, geometric errors greatly affect the machining accuracy of the face gear worm grinding machine tool. To reveal the mapping rules between the geometric errors and the tooth surface errors (TSEs) of face gear, the tool comprehensive error model (TCEM) and the tooth surface error model (TSEM) were successively established. The TCEM expressed the relationship between 41 geometric errors of the machine tool and 6 tool comprehensive errors (TCEs) of the tool while the TSEM expressed the relationship between 6 TCEs and 3 TSEs. Then, using the improved Sobol method, a piecewise sensitivity analysis was carried out by calculating the sensitivity indexes of the input of the TCEM and TSEM individually. Finally, the joint sensitivity indexes were calculated and the pivotal geometric errors were identified. By modifying the pivotal geometric errors in each TSE direction separately, all the main error reduction rates (ERR) are greater than 70%. Besides, by modifying all the pivotal geometric errors, the maximum values of 3 TSEs are reduced by 73.7%, 74.7% and 86.96%, respectively, and the minimum values are reduced by 57.1%, 71.79% and 76.19%, respectively, confirming the validity of the identified results.

Acquisition and Assessment of Gear Holistic Deviations Based on Laser Measurement

Line laser measurement technology is effective for obtaining the 3D point clouds of complex surfaces. Thus, a 3D gear measurement model is established in this study. All the point clouds of gear surfaces are rapidly obtained using a line laser sensor, and the holistic deviations of gear surfaces are acquired by calculating the normal errors of the gear surface. These informative and complete data include traditional gear error items and considerable valuable information that is not deconstructed. To exploit the acquired gear holistic deviations, the structured 3D model is constructed to express tooth surface errors. In this model, the control method of the statistical process is used to define highly representative assessment indicators, and a new gear accuracy assessment system is developed by selecting reasonable indicators. The measurement practice shows that this method exploits the complete information of the gear surface, including all the current accuracy assessment indicators, which can be used to expand the new assessment indicators. This method can be used to characterize the 3D topological tooth surface completely and comprehensively and realize the deep mining and extended applications of the 3D full information of gear surfaces.

Mist-edge-fog-cloud computing system for geometric and thermal error prediction and compensation of worm gear machine tools based on ONT-GCN spatial–temporal model

The geometric precision of worm gears (WGs) determines the service performance and life of precision machine tools, indexing turntables and other equipment. The machining accuracy of worm gear machine tools (WGMTs) is the core to guarantee the geometric precision of WGs, and is greatly affected by the thermal and geometric errors. To improve the machining accuracy of WGMTs, the thermal and geometric errors should be controlled and compensated. But the control system has a poor real-time performance, and the synchronous control of the geometric and thermal errors cannot be currently achieved, and the thermal error model has a low prediction accuracy and low robustness. To make up for the above gap, a mist-edge-fog-cloud computing system is designed for the error prediction and compensation to relieve the bandwidth pressure of the industrial Internet. Moreover, a sensor network composed of multiple sensors is constructed to obtain the thermal information, and then the ordered neuron temporal-graph convolutional network (ONT-GCN) is proposed based on the ordered neuron-long short term memory network (ON-LSTMN) and graph convolutional network (GCN) for the first time to conduct the spatial and temporal modeling of the thermal error data. The interaction among multiple sensors is explicitly considered, and the dependence of the temporal information of the thermal error data on and spatial information of sensors is taken into account. Besides, to realize the error control, the mapping relationship between the tooth surface error and geometric-thermal errors is established. The error mapping model converts 51 geometric errors and 4 thermal errors into the spatial errors of the hob. Moreover, the sensitivity of errors is analyzed, and then the key error items that affect the geometric precision of the tooth surface are identified and compensated. The results show that the ONT-GCN is superior to traditional time-series modeling methods and that the mist-edge-fog-cloud computing system can effectively shorten the executing time compared with other system frameworks, and can improve the machining accuracy of WGMTs.

Optimization of high-efficiency tooth surface accuracy of spiral bevel gears considering machine-tool motion errors

With the continuous improvement of the requirements of spiral bevel gear transmission for low noise and high precision, the optimization of high-efficiency tooth surface accuracy considering the motion axis error of machining machine tool has become a key point in the process of designing and manufacturing spiral bevel gear. Based on the theory of the multi-body dynamics system and the principle of gear meshing, a new tooth surface error model of spiral bevel gear considering the error of machine-tool moving axis is proposed and optimized, so as to obtain higher tooth surface accuracy and meshing efficiency. Firstly, according to the motion and machining principle of spiral bevel gear NC grinding machine, the corresponding relationship between the motion error of each axis of the NC gear grinding machine and the variation of cutting parameters of spiral bevel gear is analyzed, and the geometric motion error model of spiral bevel gear grinding machine is given; secondly, the tooth surface equation of the spiral bevel gear considering the machine-tool motion axis error is established; then, in view of Powell's dog leg optimization algorithm, the tooth surface considering the motion axis error of the machine tool is optimized, and the optimal cutting parameters of spiral bevel gear tooth surface machining are obtained. Finally, the simulation results show that the difference between the peak and peak values of transmission error and the maximum contact stress of the tooth surface is reduced by 72.6% and 1.62%, respectively, and the meshing efficiency is improved from 0.9798 to 0.9803 after optimization; by comparing the rolling experiment and simulation results, the maximum error between the quantitative parameters of the two groups of tooth surface contact marks is no more than 9%, which verifies the effectiveness and feasibility of this method, and provides a certain theoretical basis for practical production and processing.

Simultaneous geometric and thermal error control of gear profile grinder based on analytical correlation between tooth surface error and position error of grinding wheel/workpiece

The gear profile grinder is important to ensure the geometric accuracy of tooth surfaces. However, its machining accuracy is significantly reduced by the geometric and thermal errors. So the geometric and thermal errors should be controlled simultaneously. The geometric error modeling and decoupling of the gear profile grinder are studied, and the mapping relationship between the geometric accuracy of tooth surfaces and machine errors is constructed. To establish a robust thermal error model, a physically-based model is established to prove the feasibility of using the bidirectional long short term memory network (Bi-LSTMN) to establish a self-learning error model. The high-frequency noises are filtered and singular values are reduced by the Savitzky–Golay (SG) filtering algorithm. An attention-gate-based Bi-LSTMN (AGBi-LSTMN) is proposed to reduce the time complexity. Then the mixed-variants whale optimization algorithm (MVWOA) is proposed to optimize the hyper-parameters, and SG-MVWOA-AGBi-LSTMN is proposed for the thermal error prediction. Finally, the simultaneous geometric and thermal error control is implemented. The results show that the total tooth profile precision is increased by ISO level 3, and the tooth profile tilt precision is increased by ISO level 5 with the simultaneous geometric and thermal error control.

An alignment angle error compensation method of spiral bevel gear tooth surface measurement based on tooth surface matching

In the precision measurement of spiral bevel gears (SBGs) by gear measuring center (GMC), as the manufacturing and assembly error of the GMC exists, there is an alignment angle error between the upper and lower centers. The tooth surface error caused by this alignment angle error is analyzed. A method of accurate registration between the measured tooth surface and the theoretical tooth surface is proposed to calculate the tooth surface deviation variables. And a matching point search method is proposed to solve the corresponding relationship between the measured tooth surface and the theoretical tooth surface of SBG. Then, cubic B-spline interpolation and spiral interpolation are combined to interpolate the points around the theoretical points. Finally, experimental results show that the tilt angles and eccentricities are calculated precisely, which verifies the accuracy and feasibility of the registration algorithm proposed in this paper. After compensation, the alignment angle error of the gear axis that is included in the measurement results of the SBG tooth surface deviation reduces remarkably, and this method can improve the measurement accuracy of the gear measuring machine significantly when measuring the tooth surface deviation.

Tooth surface error correction of hypoid gears machined by duplex helical method

In this work, synchronous cutting of concave and convex surfaces was achieved using the duplex helical method for the hypoid gear, and the problem of tooth surface error correction was studied. First, the mathematical model of the hypoid gears machined by the duplex helical method was established. Second, the coordinates of discrete points on the tooth surface were obtained by measurement center, and the normal errors of the discrete points were calculated. Third, a tooth surface error correction model is established, and the tooth surface error was corrected using the Levenberg-Marquard algorithm with trust region strategy and least square method. Finally, grinding experiments were carried out on the machining parameters obtained by Levenberg-Marquard algorithm with trust region strategy, which had a better effect on tooth surface error correction than the least square method. After the tooth surface error is corrected, the maximum absolute error is reduced from 30.9 µm before correction to 6.8 µm, the root mean square of the concave error is reduced from 15.1 to 2.1 µm, the root mean square of the convex error is reduced from 10.8 to 1.8 µm, and the sum of squared errors of the concave and convex surfaces was reduced from 15471 to 358 µm2. It is verified that the Levenberg-Marquard algorithm with trust region strategy has a good accuracy for the tooth surface error correction of hypoid gear machined by duplex helical method.

Research on Spiral Bevel Gear Detection Based on Environmental Tooth Surface Error Correction Method

This paper studies the influence of the machine tool adjustment parameter error on the tooth surface error of the spiral bevel gear wheel processed by the double helix method. Based on the gear meshing principle, MATLAB software was used to establish the error tooth surface equation, and the analytical formula of the discrete point error of the tooth surface containing the machine tool adjustment parameter errors was derived. The topological map of the concave and convex tooth surface errors corresponding to the machine tool adjustment parameter errors was obtained and analyzed. The mapping relationship between machine tool adjustment parameter errors and tooth surface errors is determined, and the adjustment parameters that have a greater influence on the tooth surface errors are determined. With the help of the approximate expression of the topological diagram of the tooth surface error by the second-order surface, the influence weight of the machine tool adjustment parameter error on the tooth surface error is obtained. Based on the tooth surface error and the two-tooth surface error sensitive matrices, a tooth surface error correction model was established. The least square method of the generalized inverse matrix was used to transcend the equations to obtain the correction amount of the machine tool adjustment parameters. Finally, an example is used to verify the correction effect of the error tooth surface. The results show that the error of the tooth surface on the concave and convex sides after the correction is greatly reduced.

Analysis of the mathematical correlation between post-sharped hob and worm gear tooth surface

The production geometry development of the worm gear drives with circle arched profile in axial section requires the combined management of complex mathematical, geometric and manufacturing knowledge. In this paper has been presented an analysis of the relation between the post-sharpening of the cylindrical hob with circle arched profile in axial section and gear tooth surface error according to the functions determined by previous research work, that fits to the Dudás type ProMAT general mathematical model, which is suitable for the analysis of technological processes in manufacturing. During the research the extension of this method to any type of worm gear drives has been aspired.

A measurement method for tooth surface errors of straight bevel gears based on 3D model

Due to the inconsistency between a theoretically designed tooth surface and the actual measured reference surface, the tooth surface error data obtained by the traditional measurement method is unreliable, and incorrectly reflects the machining accuracy of straight bevel gears. This has become a key problem in the current measurement process, and needs to be resolved. In this paper, a new measurement method is proposed to achieve accurate measurement and evaluation of the machining error of straight bevel gears. The 3-dimensional (3D) model surface designed by the manufacturer is directly extracted as a theoretical reference to complete the measurement, which ensures the correctness of the error results. A measurement strategy of normal rotation of the measured point is adopted to make up for the disadvantages of the one-dimensional (1D) probe and improve the measurement accuracy. The problem of the sensitive direction of the probe deviating from the normal direction is compensated for by increasing the dimension of the 1D probe and reducing the dimension of the 3D normal. According to the constructed error calculation model, the minimum tooth surface error is obtained through reconstruction and optimal matching techniques. Finally, the accuracy and effectiveness of the measurement method are verified through measurement comparison experiments. This method solves the technical problem of the inconsistency of the measured reference surface, and uses a low-cost 1D probe to achieve precise measurement of the 3D surface. It not only improves the disadvantages of current measurement methods, but also provides effective technical support for other gear measurements.

Simulation and Analysis on Tooth Meshing Ease-Off Surface of Gears Based on Polynomial Topology Modification

Based on the principles of common-generating conjugation mapping of pinion-rank-gear meshing, the constructing of ease-off surface has been done by means of the topology modification of the 4-order polynomial surface. The simulation method of tooth surface mesh is purposed based on the ease-off surface. By analyzing topological structure characteristics of ease-off surface such as modification gradient curves, contact line-off and contact path-off, the meshing characteristics such as the contact area, the contact line and transmission error are determined. It is stated on the interaction relations of coefficients of the 4-order polynomial to the topological construction of surface. The methods for reconstruction and analysis of ease-off surface with considering axis misalignments are presented by the direct transformation of coordinate systems. Taking the topological construction of four kinds of surfaces as examples, characteristics parameters such as modification gradient of tooth surface, contact path and transmission errors are simulated. They are expounded on the corresponding parameters of controlling form and adjusting methods for tooth surface. These methods take goals of adjusting-modification and controlling-property for tooth surface mesh. It is improved and developed the meshing analysis method of ease-off surface of tooth. The presented method is liable to provide the basis data for the further loaded contact analysis of complicated tooth surface and the more direct technical for the 3-D topological optimization of tooth surface.

Research on the Effect Law of Machine Tool Errors on Tooth Surface Errors of Curvilinear Gear

Research on the Effect Law of Machine Tool Errors on Tooth Surface Errors of Curvilinear Gear

Crucial geometric error compensation towards gear grinding accuracy enhancement based on simplified actual inverse kinematic model

The geometric error, which will significantly affect grinding accuracy of the gear form grinding machine tool, should be compensated. This paper proposes a crucial geometric error compensation method towards gear grinding accuracy enhancement, and it can directly obtain simplified explicit analytical expressions of compensated motion commands for more efficient and effective compensation. Firstly, an actual forward kinematic model is established by rapid reconstruction of the geometric error modules of motion axes. Next, the conventional actual inverse kinematic model (IKM) directing at tool posture error compensation is developed to calculate analytical expressions of compensated motion commands. Then, the material removal mechanism of gear form grinding process is considered to construct the geometric error- tooth surface posture error model, which directly reveals the mapping rules between the geometric error and the tooth surface errors. On this basis, the crucial error elements regarding tooth profile and helix deviations are identified based on global sensitivity analysis, and the compensated expressions obtained by the actual IKM are further simplified and modified directly towards gear accuracy enhancement. Finally, the crucial error identification and compensation process was verified by both numerical analysis and machining tests. The results show that tooth profile deviations will reduce by more than 80% theoretically but only by 43.27% in fact due to other errors, and it proves the feasibility and effectiveness of the proposed method.

An identification method for crucial geometric errors of gear form grinding machine tools based on tooth surface posture error model

The identification for crucial geometric errors of gear form grinding machine tools is essential for the later error accurate compensation, but complicated and difficult. To reveal the mapping rules between geometric errors and machining errors, the tool posture error model (TPEM) was first constructed but it failed to consider the effects of grinding process. Hence, the tooth surface posture error model (TSPEM) was established based on the discretization of the grinding trajectory and the calculation of the grinding contact lines. Then the improved Sobol method was performed to analyze the error sensitivity and identify the crucial errors and sensitive components by considering the stochastic and intercoupling characteristics of geometric errors. To validate the effectiveness of the proposed method, the identified results were compared with that of the Morris method and the modification experiment of crucial errors was conducted. It shows that the tooth surface errors in the δy-direction are reduced by 68.75% for the TSPEM and 43.36% for the TPEM and that the TSPEM is more effective for the identification of crucial geometric errors.