Tooth Profile Error Research Articles

Influence of manufacturing errors on dynamic floating characteristics for herringbone planetary gears

Owing to the present of manufacturing errors, the dynamic floating characteristics of herringbone planetary gear train (HPGT) can be changed in comparison with the original ideal design. In this research, based on the actual structure of herringbone gears, taking into consideration manufacturing eccentric errors and tooth profile errors, bearing deformation, time-varying meshing stiffness, gyroscopic effect, and so on, a novel and generalized bending–torsional–axial coupled dynamic model of a herringbone planetary gear train is presented to investigate the dynamic floating performances applying the lumped-parameter approach. The model is capable of being employed for the vibration behavior analysis of the HPGT with different types of manufacturing errors and arbitrary number of planets. The variable step Runge–Kutta algorithm is utilized to compute the dynamic responses of the HPGT system. In combination with the proposed computational approach of the component floating displacement amount, the relationship among manufacturing errors, component floating displacements, and different floating forms is obtained, and the effects of manufacturing errors on the HPGT dynamic floating performances are discussed. Meanwhile, sun gear radial floating trajectories in two cases of sun gear float and non-float are compared and analyzed. Results indicate that the manufacturing error and component float prominently affect the dynamic floating characteristics in the HPGT system.

Erratum to: A distributed dynamic mesh model of a helical gear pair with tooth profile errors

The first author’s affiliation was no longer in use in the original version of the article and it should be replaced as follows: School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China.

A distributed dynamic mesh model of a helical gear pair with tooth profile errors

A dynamic model of a helical gear rotor system is proposed. Firstly, a generally distributed dynamic model of a helical gear pair with tooth profile errors is developed. The gear mesh is represented by a pair of cylinders connected by a series of springs and the stiffness of each spring is equal to the effective mesh stiffness. Combining the gear dynamic model with the rotor-bearing system model, the gear-rotor-bearing dynamic model is developed. Then three cases are presented to analyze the dynamic responses of gear systems. The results reveal that the gear dynamic model is effective and advanced for general gear systems, narrow-faced gear, wide-faced gear and gear with tooth profile errors. Finally, the responses of an example helical gear system are also studied to demonstrate the influence of the lead crown reliefs and misalignments. The results show that both of the lead crown relief and misalignment soften the gear mesh stiffness and the responses of the gear system increase with the increasing lead crown reliefs and misalignments.

Experimental and Theoretical Study of Gear Dynamical Transmission Characteristic Considering Measured Manufacturing Errors

In the research of gear transmission, the vibration and noise problem has received many concerns all the times. Scholars use tooth modification technique to improve the meshing state of gearings in order to reduce the vibration and noise. However, few of researchers consider the influence of measured manufacturing errors when they do the study of tooth modification. In order to investigate the efficiency of the tooth modification in the actual project, this paper proposes a dynamic model of a helical gear pair including tooth modification and measured manufacturing errors to do a deterministic analysis on the dynamical transmission performance. In this analysis, based on the measured tooth deviation, a real tooth surface (including modification and measured tooth profile error) is fitted by a bicubic B‐spline. With the tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) on the real tooth surface, the loaded transmission error, tooth surface elastic deformation, and load distribution can be determined. Based on the results, the time‐varying mesh stiffness and gear mesh impact are computed. Taking the loaded transmission error, measured cumulative pitch error, eccentricity error, time‐varying mesh stiffness, and gear mesh impact as the internal excitations, this paper establishes a 12‐degree‐of‐freedom (DOF) dynamic model of a helical gear pair and uses the Fourier series method to solve it. In two situations of low speed and high speed, the gear system dynamic response is analyzed in the time and frequency domains. In addition, an experiment is performed to validate the simulation results. The study shows that the proposed technique is useful and reliable for predicting the dynamic response of a gear system.

Analysis of tooth profile and accumulative pitch errors of end-toothed disc and its tolerance development

Tooth profile errors of the end-toothed disc describe the tooth profile deviations which mainly contain tooth trace error and tooth profile half-angle error while accumulative pitch error describes the tooth position deviation of the end-toothed disc. In traditional analysis, neither the deviation of the benchmark of tooth trace error nor that of tooth profile half-angle error is considered, while the deviations exist in engineering. Hence, tooth width error is proposed in order to describe the deviations of the benchmarks and the indexing accuracy error model is also established according to these four errors in this paper. Meanwhile, Tolerance Proportional Distribution Method is presented to establish the tolerance proportional model of end-toothed disc. Therefore, according to the indexing accuracy error model and tolerance proportional model, the pass rate model of end-toothed disc is established and a new approach for tolerance development is put forward based on this pass rate model. Finally, a simulation program is developed for random sequence generation, which replaces the processing errors measurement, and meshing process simulation, and verifies the correctness of the approach for tolerance development.

Effects of random tooth profile errors on the dynamic behaviors of planetary gears

In this paper, a nonlinear random model is built to describe the dynamics of planetary gear trains (PGTs), in which the time-varying mesh stiffness, tooth profile modification (TPM), tooth contact loss, and random tooth profile error are considered. A stochastic method based on the method of multiple scales (MMS) is extended to analyze the statistical property of the dynamic performance of PGTs. By the proposed multiple-scales based stochastic method, the distributions of the dynamic transmission errors (DTEs) are investigated, and the lower and upper bounds are determined based on the 3σ principle. Monte Carlo method is employed to verify the proposed method. Results indicate that the proposed method can be used to determine the distribution of the DTE of PGTs high efficiently and allow a link between the manufacturing precision and the dynamical response. In addition, the effects of tooth profile modification on the distributions of vibration amplitudes and the probability of tooth contact loss with different manufacturing tooth profile errors are studied. The results show that the manufacturing precision affects the distribution of dynamic transmission errors dramatically and appropriate TPMs are helpful to decrease the nominal value and the deviation of the vibration amplitudes.

A realistic dynamic model for gear fault diagnosis

A reliable dynamic model that predicts the vibrations of a gear transmission containing a fault may allow us to recognize the effects of that fault, to identify its severity, and to develop methods for characterizing its type by using a limited number of experiments. In this research, a new generic model for gear dynamics is proposed. This model is solved numerically, and combines the most advanced approaches for gear modeling, including: modeling the contact between gear pairs as a set of springs, modeling the effect of faults on the mesh stiffness, and integrating the geometric errors. As a result, the model simulates successfully the acceleration signal of a realistic gear pair in a realistic environment. In this paper, we present a different approach to simulate partial tooth face faults, which takes into account the partial contact loss along the tooth contact line due to the fault's geometry. We propose a new analytical formulation for modeling complete tooth face faults that are located across the entire contact area. Additionally, a modified analytical expression for describing the tooth profile errors is proposed. Using a realistic simulation, we were able to experimentally validate the model.

和時計から日本の歯車の源流を探る

The purpose of this study is to understand the nature of one of the oldest gears used in traditional Japanese clock. Today's gear manufacturing technology in Japan came mostly from Europe and America, but we do not know exactly, when and how the gears were manufactured for the first time in Japan. It is interesting to search for this history. It is also exciting to study the tooth profile, precision and accuracy of the gears, and materials of the gears at that time. So far, there have been some studies performed for the mechanism of traditional Japanese clocks/watches, but not for gears. Fortunately we have a chance this time to investigate gears for Japanese watch drive that was made in 1688. Tooth profile and pitch error were measured, and transmission error analysis was also performed. It revealed that the precision of the watch was extremely high without any rust for more than 300 years, even though they were all handmade by Japanese mechanism technician named Sukeza-emon Tsuda the III. In the old days, there was no study on conjugate tooth profile theory available, but mysteriously, tooth profile was nearly made in the form of cycloid. Moreover, the gear material investigation was very interesting: The texture of the gear material was very homogeneous and grain size is far smaller than that of today's comparable steel kind. Impurities in it were very small and scattered well in the matrix. The steel was surely made by Japanese sword smith. The ore of the steel was perhaps sand-iron and it was refined with pine charcoal. The steel was forged and forged by hand very hardly. As the result the quality of the steel of 1688 looks far better than today's industrial steel. This research enabled us to discover how Japanese gear technology was born and developed.

Geometric error detection of curve-face gear pairs

A new type of gear pair, the curve-face gear pair, composed of a curve-face gear and a non-circular gear with mutual engagement, was proposed to achieve variable ratio transmission of motion and power between the intersecting axes. Geometric error detection is required to evaluate the grade of the manufactured gears. Due to the complexity of the curve-face gear, direct detection has been conducted in a very limited way. In this article, a measurement method aimed at curve-face gear pair artifacts is presented based on the computer numerical control gear measuring center. By comparing the measured coordinate data of the surface points with the corresponding theoretical data, various errors such as tooth profile error and pitch deviation can be obtained, and accuracy grades are evaluated with reference to accuracy standards for cylindrical gear and bevel gear. The developed method is simple and robust without requiring a special measuring device; hence, it can be applied for the industrial practice as a means for measuring the tooth profile and pitch deviations which cannot be measured by conventional methods.

The dynamic coupling behaviour of a cylindrical geared rotor system subjected to gear eccentricities

A general dynamic model for the cylindrical geared rotor system with local tooth profile errors and global mounting errors is developed. This model includes a finite element (FE) model for the shaft structure, a lumped-parameter bearing model and a 3 dimensional (3D) gear mesh model based on previous work. Gear geometric eccentricities, unloaded static transmission error, as well as the fluctuation of the mesh stiffness are included in this model to study the dynamic coupling behaviour of the transverse and rotational motions of gears subjected to gear eccentricities. The simulation results are compared against the experimental results provided in the literature, which proves that the proposed modal is suitable to study the dynamic behaviour of a cylindrical geared rotor system. The influence of the helical angle, the number of teeth, gear profile error, and gear eccentricities on the dynamic coupling behaviour were studied. It was found that the dynamic coupling between the transverse and rotational motions of the gears becomes apparent in the low speed range when resonances are excited either by the gear profile error or mesh stiffness fluctuation. The analysis results of this research are useful for the dynamic analysis of gear transmission systems subjected to gear eccentricities.

A model for analyzing stiffness and stress in a helical gear pair with tooth profile errors

A model is introduced for analyzing the influence of tooth shape deviations and assembly errors on the helical gear mesh stiffness, loaded transmission error, tooth contact stress and tooth root stress. The helical gear is approximated as a series of independent spur gear slices along axial direction whose face-width is relatively small. The relative position relationships among those sliced teeth in mesh are developed with tooth profile errors and the stiffness of the sliced tooth is calculated by the potential energy method. From the equilibriums of the forces, gear mesh stiffness, loaded transmission error, tooth contact stress and tooth root stress are calculated. Then two cases are presented for validation of the model. It is demonstrated that the model is effective for calculating the stiffness of helical gear pairs. Finally, the effects of the tooth tip reliefs, lead crown reliefs and misalignments on the gear mesh stiffness, transmission error, tooth contact stress and tooth root stress are analyzed. The results show that mesh stiffness decreases, loaded transmission error, the maximum tooth contact stress and the maximum tooth root stress grow with the increasing tooth tip relief, lead crown relief and misalignment. And tooth edge has concentrated tooth contact stresses with a gear misalignment.

A precision generating hobbing method for face-gear based on worm hob

In order to improve the machining accuracy and production efficiency of face-gear, a method of face-gear generating hobbing by worm is provided in this paper. The principle of face-gear hobbing worm is analyzed, and the mathematical model of the worm is presented based on the principle and the theory of meshing. Taking the hobbing needs into account, the special machine tool is provided, and the movement control method of face-gear hobbing by the worm on the machine tool is proposed. The equation of face-gear tooth surface is calculated, and the 3-D model of face-gear is established based on CATIA software. To reduce the face-gear tooth profile errors induced by ratio errors, an error analysis model of face-gear hobbing is established. The experiment is carried out, and the completed specimen is detected by Coordinate Measuring Machining (CMM). The processing parameter is amended according to the tooth flank detection results, and the maximum normal deviation of the whole tooth surface of the face-gear specimen is improved from 243.2 µm to 61.0 µm. Experiment results demonstrate that the method of face-gear hobbing by worm is an effective approach to achieve the precision face-gear with high dimensional accuracy.

GS1111 3Dプリンタ製平歯車の性能試験

3D Printer is expected as a major tool for additive manufacturing. However, almost all components made by 3D printers are used to investigate shapes of some product, and it still not work as machine elements to drive some machines. In gear manufacturing, 3D printer hardly applied to make gears because of the strength and precision issues. In this study, the authors try to build some test gears with using 3D printer for the purpose of simplifying gear manufacture. The test gears are built in several different laminating angles, and measured there tooth profile errors and pitch errors. Then the authors compared those 3D printer-made gears with hobbed gear in accuracy. In conclusion, when using 3D Printer to build a gear, its layers would be piled up vertically to its gear shaft to reduce tooth profile errors and pitch errors.

歯形誤差・実かみ合いおよび歯剛性変動を考慮した多段歯車系の歯打ち解析

歯形誤差・実かみ合いおよび歯剛性変動を考慮した多段歯車系の歯打ち解析

Error Detection and Classification of Tangential Radial Single Machining of Non-Circular Gear

The studies on non-circular gear cutting and using errors were generally based on theory of cylindrical gear errors, and the similarities and differences between cylindrical gear and non-circular gear were combined to study the change rule of non-circular gear contacting line and obtained corresponding conclusions. In this paper, from a different perspective, by starting from the non-circular gear engagement principle, on the basis of analyzing the concrete geometric error conditions, new type tooth profile algorithm routine is reasonably modified, and needed non-circular gear tooth profile error contrast figure is obtained. Furthermore, theoretical derivation is carried out by incorporating analytical method and contacting line incremental method, validity of computer simulation is verified, a reasonable classification method is proposed based on the characterization of the error characteristic code in the general case, Summed up the error genetic map which possess enormous significance to direct practice generation[1].

Analogue simulation and analysis of non-circular gear reentry error

The studies on non-circular gear cutting and using errors were generally based on theory of cylindrical gear errors, and the similarities and differences between cylindrical gear and non-circular gear were combined to study the change rule of non-circular gear contacting line and obtained corresponding conclusions. In this paper, from a different perspective, by starting from the non-circular gear engagement principle, on the basis of analyzing the concrete geometric error conditions, new type tooth profile algorithm routine is reasonably modified, and needed non-circular gear tooth profile error contrast figure is obtained. Furthermore, theoretical derivation is carried out by incorporating analytical method and contacting line incremental method, validity of computer simulation is verified, and the characterization of the tangential error characteristic code is obtained, Summed up the error sources for error genetic which possess enormous significance to direct practice generation.

Research on the theoretical tooth profile errors of gears machined by skiving

This study focuses on the theoretical tooth profile errors of gears machined by conventional skiving. The influences of skiving cutter rake angle on tooth profile errors of gears are analyzed. It is found that the inherent error of skiving cutter and the longitudinal deviation of cutting edge contributed to the gear profile errors. The influences of skiving cutter resharpening are then analyzed, and it is found that the resharpening amount of skiving cutter has obvious influence on the tooth profile error of workpiece. Therefore, conventional skiving tools are defective in high accuracy gear machining.

Frequency spectrum and vibration analysis of high speed gear-rotor system with tooth root crack considering transmission error excitation

A finite element node dynamic model of gear-rotor-bearing system with different lengths of crack by taking the time-varying mesh stiffness, backlash, transmission error excitation, flexible shaft and supporting bearing into account is proposed. The time-varying mesh stiffness of gear-pair with cracked tooth is obtained by applying the improved potential energy method. Due to the periodic features of the dynamic responses of the system when the tooth is cracked, the short term component (tooth profile error and tooth pitch error) and long term component (accumulative pitch error) transmission error excitations are introduced, the RMS values, kurtosis values, and frequency spectrum diagrams of the dynamic response with respect to input speed considering different forms of transmission error excitation are gained. The influences of transmission error excitation and crack length on the dynamic responses are investigated. The effectiveness of the RMS value, kurtosis value and frequency spectrum in the judgment of the crack length is analyzed.

Application of an Adaptive Neuro Fuzzy Inference System for Low Speed Planetary Gearbox Vibration Control

The gear pair assembly remains one of the major vibration sources in power transmission systems of mechanical rotating machinery. The gear vibration signature is often dominated by several high-level tonal peaks that occur at the fundamental gear mesh and its harmonics. The primary forcing function is produced by the gear transmission error excitation resulting from tooth profile errors, misalignment and elastic deformation. The excessive dynamic response generated can frequently lead to structural fatigue failure. Therefore, it is highly desirable to control the low-speed planetary gearbox rotational vibration acceleration levels. To achieve this goal, the present study demonstrates a method by which these vibrations produced due to the primary forcing function originating from the mesh points can be controlled. Hence, to deal with both the meshing stiffness generation and vibration transmissibility processes more effectively, the control system must be applied close to the gear's connection. This close proximity will allow the application of a single controller. Selection of the proper rule base depending upon the situation can be achieved by the use of an adaptive neuro-fuzzy inference system (ANFIS) controller as an integrated approach for purposes of control to yield excellent results, and this is the highlight of this paper. The results presented in the paper show that the outputs take less time to stabilize. Moreover, due to incorporation of the ANFIS controller with the plant, it is observed that the gearbox reaches the desired vibration very quickly in relatively shorter time.

A precision generating grinding method for face gear using CBN wheel

In order to improve the machining accuracy and production efficiency of face gear, a generating grinding method for face gear using cubic boron nitride (CBN) wheel is studied in this paper. The mathematical model of the CBN wheel profile is derived based on face gear grinding principle and theory of gearing. Taking the demands of machine motion during face gear grinding into account, a five-axis face gear grinding machine is developed. Then, the mathematical model of the ground face gear and the grinding method are proposed based on the structure of the machine. For reducing the face gear tooth profile errors induced by installation errors, an error analysis model of face gear grinding is established; the effects of installation errors on tooth profile errors are analyzed. The CBN wheel was manufactured, and the grinding experiments were performed on the self-developed grinding machine. The processing parameters of face gear grinding were amended according to the tooth flank measurement results obtained from coordinate measuring machine. The ground tooth profile deviation was reduced significantly after processing parameters amendment. Experiment results demonstrate that the generating grinding method is an effective approach to achieve the precision grinding of face gear.