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In order to solve the problem of calculation efficiency and accuracy of time-varying meshing stiffness of spur bevel gears, a slice calculation method of meshing stiffness of spur bevel gears considering the coupling effect of tooth pairs is proposed. Based on the slice discretization method, the bevel gear tooth and the gear body are discretized into several slices with the same width along the tooth width direction and the axis direction respectively, and the slices are expanded based on the back cone equivalence principle. When the slice width is small enough, the bevel gear can be approximated as a series of spur gears, and the meshing stiffness of each slice can be calculated by the energy method. In order to consider the coupling effect of multi-tooth meshing, the correction coefficient of gear stiffness is introduced in the calculation of double tooth meshing stiffness, which reduces the calculation error of double tooth meshing stiffness. The meshing stiffness model of bevel gears under the error condition is established. The meshing stiffness of spur bevel gears, the angular displacement of the center point of each slice and the load distribution of the tooth surface under the ideal and assembly error conditions are calculated and analyzed. Finally, the accuracy and efficiency of the proposed method are verified by comparing with the finite element calculation results.

A face gear is typically used in fishing spinning reel mechanism owing to its easy fabrication and good reel handle rotation. As face gears without tooth flank modification are sensitive to assembly errors, the fishing reel sensation becomes unstable in mass production. Previously, tooth flank modification with a transmission error (TE)-controlled curve was used to solve this problem. The TE-controlled curve stabilized the fishing reel sensation for assembling. However, in mass production, the manufacturing error of a pinion and face gear must be considered along with the assembly error. In a previous study, we analyzed all the eight error factors, including the manufacturing error, individually. Thus, we clarified that pressure angle deviation of the pinion and tooth rotational error of the face gear have a significant effect on TE. However, in mass production, these errors are combined in a complicated manner. In this study, an experiment was conducted using the L18 orthogonal array to elucidate the interaction between error factors. No peaks and troughs were observed in the curve of each factor, thus showing that there is little interaction due to the combination of factors. Although the proposed model was less affected than the conventional model by most errors, the proposed model was affected by the pressure angle deviation of the pinion and the tooth rotational error of the face gear. These results were the same as that of the single-factor experiment in the previous report. By analyzing the path of contact using a three dimensional simulation, we confirmed that uneven TE waveforms were caused by edge contact due to insufficient tooth flank modification. The unevenness in the TE waveform occurred when the factor levels of the assembly and manufacturing errors were large, when these error factors were combined, and when the amount of tooth flank modification of the face gear was small. This phenomenon was regarded as a trochoidal interference, which was caused by the contact of the tooth tip of the pinion, and it was also confirmed using prototype gears. The combination of these errors must be carefully considered in the manufacturing process. Consequently, we have elucidated that avoiding the interference due to combination of errors was significantly important to improve the fishing reel sensation.

Tooth surface equations of a curvilinear cylindrical gear with parabolic modification are derived based on the gear meshing theory. The solving method of the initial contact position of the gear finite element model is studied, a finite element model of the gear pair is established using the parameterized finite element technology, and the loaded transmission error is extracted from the finite element analysis results. With more than 20 sets of calculation examples, the effects of assembly errors, modification amount, curvature radius of the cutter, and offset of modification curve on loaded transmission errors are studied, respectively. The research results show that, assembly errors can reduce the fluctuation range of loaded transmission errors; larger modification amount can reduce the fluctuation range of loaded transmission errors; larger curvature radius of the cutter can reduce the value of loaded transmission errors and its fluctuation range; the offset of the modification curve has slight effect on loaded transmission errors. In summary, the use of a larger curvature radius of the cutter can be used to machine curvilinear cylindrical gears with better transmission performance.

The face gear is a crossed gear similar to a hypoid gear. Owing to its compactness and good assemblability, it is used in geared motors and transmissions for helicopters. Studies on various aspects such as design, processing, and tooth modification methods have been conducted for face gears, which are typically used in fishing reels as spinning reels owing to their easy fabrication and small vibration during reel handle rotation. Because face gears without tooth flank modification are sensitive to assembly errors, the fishing reel sensation becomes unstable during mass production. Therefore, assembling and adjusting face gears to achieve a good fishing reel sensation is time consuming. Previously, tooth flank modification with a transmission error (TE)-controlled curve was used to solve this problem. The TE-controlled curve stabilized the fishing reel sensation for assembly. However, in mass production, the manufacturing error of a pinion and face gear must be considered in addition to the assembly error. In this study, we investigated the assembly and manufacturing errors when the pinion and face gear come in contact. Furthermore, we elucidated the meshing mechanism of face gear pairs under various assembly and manufacturing errors. As a result, it was confirmed that the TE-controlled curve was affected by manufacturing errors only. Moreover, a new interesting finding was obtained regarding the meshing of the face gear with a TE-controlled curve. The TE occurrence ought to be attributed to the pressure angle deviation. Hence, it is probable to replace the main cause of the transmission error with the pressure angle deviation. This consideration is the same for a cylindrical gear pair. The meshing of face gear, which was considered complicated, can be reconsidered as the meshing of a simple cylindrical gear pair. Consequently, we revealed that the TE-controlled curve is significantly useful for considering the meshing and design of a face gear pair, which is an advantageous feature.

When spur gear pair meshing with the assembly error，it will cause the edge contact and stress concentration，and affect the service life. In the meantime，because the parallelism error of vertical plane will produce a larger deviation of tooth contact，therefore，taking the spur gear pair with the parallelism error of vertical plane as the research object，then based on the systematic introduction of the numerical calculation theory of gear contact stress，and the contact stress，contact width and the corresponding load distribution coefficient along the tooth in the case of different assembly error and load conditions are calculated，and the results calculated by numerical calculation method and national standard calculation method in the case of no assembly error are made a comparative analysis. The results show that the results calculated by two methods in the case of no assembly error are close to each other，and the numerical calculation result can reflect the edge effect of the gear pair meshing. Under the same load，with the increase of the installation error，the uneven contact in the tooth width direction becomes worse，and one side disconnects. Under the same assembly error，the uneven contact is improved with the increase of load，the maximum contact stress presents nonlinear increase with the increase of assembly error and load.

In order to solve the problem of measuring the assembly error of gear shafting in engineering， a method of measuring the assembly error of gear shafting based on stereo vision is proposed. In this method， the circular mark points arranged on the feature to be measured are used as the medium to obtain the spatial pose of the corresponding features through 3D reconstruction and mathematical fitting， and then the assembly error is calculated. Taking a bevel gearbox as an example， the complete technical process of the proposed method is given， and the influence of the number of States and the installation angle of the target plate on the fitting accuracy is systematically studied. A complete hardware and software system is developed， and the effectiveness and efficiency of the measurement method are verified by experiments.

Focusing on a planetary gear transmission that uses line gear as core transmission elements， and the influence of assembly errors on its transmission precision is studied. The structural composition of the line gear transmission and the characteristics of different transmission schemes are analyzed， the space meshing coordinate system for the gear train are established， and the quantitative relationship of the parameters at the meshing moment is established among the sun line gear， the outer ring line gear and the planetary line gear. The causes of the angle error and displacement error are analyzed， both between the planetary line gear and the sun line gear and between the outer ring line gear and the planetary line gear. Tooth Contact Analysis （TCA） is subsequently adopted to carry out the numerical analysis of the assembly errors model for the line gear transmission. The results show that， among assembly errors of the planetary gears， the angular errors of its rotation around the horizontal axis， the displacement errors in the longitudinal section of the outer ring line gear， and the displacement errors in the cross section of the planetary line gear， are of more effect on its transmission. That is of significant to design and assembly of the planetary gears.

The tooth root crack fault is a common fault type of the spiral bevel gear pair (SBGP). Affected by the strong bearing capacity, the early crack fault of the SBGP cannot be found in time. In this study, a finite element (FE) model of the SBGP is established and assembled through the tooth contact analysis. The maximum tooth root stress is analyzed considering the variation of assembly errors. Meanwhile, this study simulates the tooth root crack fault of the bevel pinion with different crack degrees. The initial position of the crack is located where the maximum tooth root stress appears. The time-varying mesh stiffness (TVMS) of the SBGP considering different degrees of the pinion tooth root crack fault is obtained. The TVMS and the non-load transmission error are brought into a hybrid FE dynamic model, and steady responses are solved. Based on this, the sensitivities of various statistical indicators for identifying the tooth root crack fault of SBGP under the influence of assembly errors are verified. This paper can provide the necessary theoretical basis for the analysis and diagnosis of tooth root crack faults in the SBGP transmission system.

The aeroengine industry has set strict upper limits for assembly errors in rotor-connecting processes, because assembly errors significantly affect aeroengine stability. Applications of multi-axis mechanisms have the potential to solve the low efficiency of traditional manual connection processes. However, multiple error sources are simultaneously introduced. Thus, an accurate prediction method of rotor assembly error considering multiple error sources is of vital importance, by which the applicability of the new mechanism to rotors can be tested. In this study, a new prediction method for rotor assembly errors is proposed based on the use of a novel multi-axis measuring and connecting mechanism. First, the error propagation among the rotor errors, measurement errors, mechanism errors, and mounting errors is analyzed. Second, reasonable characterization models for these error sources are established using homogeneous transformation matrices. Third, based on the abovementioned error models, a new rotor assembly error prediction algorithm is constructed. It is highly consistent with the actual connection processes. Finally, verification experiments are conducted. The experimental results show that deviation rates of the average values of six types of assembly errors relative to the predictions are all lower than 14%. The proposed prediction method has acceptable accuracy and practical significance.

Assembling plays a significant role in the performance of large-scale CNC gear profile grinding machines. An approach in the deviation evaluation of ground tooth surfaces taking into account assembly errors is proposed in this paper. Based on the error transmission chain of the profile grinding system and the conjugate motion relationship between the grinding wheel and the workpiece, the ground tooth surface model including assembly errors was established using the surface envelope method. Then, the effect of assembly errors on deviations of the profile grinding tooth surface was quantitatively analyzed. The optimized distribution of assembly errors and machining verifications were performed on large-scale CNC gear profile grinding machines. The results show that the proposed method is effective at ensuring ground tooth surface deviations for large-scale CNC gear profile grinding machines.

The speed integration gear is a new type of gear mechanism,which is meshed with cylindrical gears and non-circular face gear and can achieve speed integration transmission. Based on gear mesh theory,the tooth surface equation of non-circular face gear is deduced. Considering the angle error,intersected axis error and axis shift error,the contact model of tooth surface is established. An analysis of distribution characteristics of the contact path is made to obtain the influence of the assembly error on the point contact trace of the teeth. Simulation results indicate that the gradient of the contact path on each tooth of non-circular face gear increases with rate of change of the transmission ratio. The assembly errors cause the shift of the contact path to inner or outer radius,with its movement direction depending on the error direction.

Based on the principle of gear meshing,the precise 3 D solid geometry model of spiral gear is established. Then the finite element model of spiral gear for loaded meshing characteristics analysis is built,and the grid size of model is reasonably determined by the Hertz contact theory. Taking six teeth as research objects,the positions of the maximum contact and bending stresses of the spiral gear are found,and the stress curves are plotted in a meshing period. The analysis of meshing characteristics including contact patterns and paths on the gear surface is carried out by selecting the loading condition and assembly error as the variable respectively. The results provide a remarkable guiding role in the design and practical application of spiral bevel gear.

This paper proposes a method of directly designing the surface of action of a hypoid gear. In the proposed method, the characteristics of the gear meshing are independent of the manufacturing process. The conjugate gear surfaces can then be accurately obtained by coordinate transformation. A plane was selected as the surface of action to achieve hypoid gears with a higher performance than face gears. The proposed hypoid gear may have the same features as cylindrical involute gears that also mesh in the plane of action and are unaffected by translational assembly error. A design example is presented in this paper to verify that tooth surface meshing in the plane of action can be achieved for a high-reduction ratio hypoid gear. The influence of different types of assembly error on the tooth flank error was examined numerically by comparing with face gears.