Tooth Profile Error Research Articles

Accuracy Analysis of Complex Transmission System with Distributed Tooth Profile Errors

Tooth profile errors are the internal excitations that cause gear meshing errors, which are critical error factors affecting gear transmission accuracy. In existing studies, it is usually regarded as a constant or random distribution function. However, the actual machined tooth profile error is not a constant, so this estimation is inconsistent with the actual situation, resulting in an inaccurate evaluation of transmission accuracy. This paper proposes a method for representing tooth profile errors using distribution errors (including systematic and random errors), and a mathematical model of distributed tooth profile errors is presented. The contact stresses of the complex transmission system were compared with those obtained by formulas, proving that tooth profile errors increase contact stress. A method for calculating gear meshing error is proposed to evaluate the actual output accuracy of the complex transmission system. Compared with the test, the output accuracy is reduced by 13.8% under the temperature environment and distributed tooth profile errors. The proposed methods can accurately predict the transmission accuracy of precision transmission systems at the design stage and provide theoretical support for reducing systematic and random errors at the gear machining stage.

Study on Spline Stress of Separator Plates in a Wet Multi-Plate Clutch

The spline teeth fracture of separator plates in wet multi-plate clutches compromises driving safety and the vehicle’s lifespan. Tooth fracture is mainly caused by stress concentration at the tooth root and uneven circumferential load distribution. This paper considers parameters such as torque, teeth count, tooth profile, and misalignment errors, establishing the corresponding finite element (FE) model to analyze the impact of the above-mentioned parameters on the strength of the separator plates. Analysis under even and biased load circumstances demonstrated that an optimum tooth count and profile can significantly increase the strength of the separator plates, offering advice for the optimized design of wet multi-plate clutch separator plates.

The influence of different load distribution considering geometric error on the fatigue life of ball screw

The ball screw pair is a precision drive component that converts rotary motion into linear motion. In practical applications, because the feed system usually has rotational torque and geometric errors, it may increase the contact load of the ball screw pair and reduce the fatigue life. In order to study the influence of different loads and geometric errors on the maximum contact load and fatigue life of ball screw pairs, the following research work was carried out. Firstly, based on the composite load and rotational torque, the ball load distribution model is established, and the accuracy of the model is verified by finite element modeling analysis. Then, the influence of different composite loads, rotation times and geometric errors (including ball size error, lead error and raceway tooth profile error) on the ball load distribution is analyzed. Finally, the influence of compound load, rotating torque and geometric error on the fatigue life of ball screw pairs is studied. The results show that the rotational torque and lead error have a great influence on the load distribution and fatigue life of the ball, and the influence of lead error on load distribution is greater than that of dimension error and tooth profile error. The fatigue life of the ball screw pair with non-uniform load distribution is shorter than that of the ball screw pair with uniform load distribution.

Investigation of Effect of the Tooth Profile Modification on the Engagement Characteristics of the Gear Pair

As a vital component in the electric vehicle, the higher requirements for the gear precision are urgently needed in the service environment with high speed and heavy load. The tooth profile modification can greatly improve the accuracy and strength of the gear transmissions. Therefore, accurately obtaining the optimum modification parameters of the gear system under the action of specific loads is the premise of the research of gear vibration and noise reduction. The time-varying mesh stiffness calculation model is established in this paper considering the coupling effect of the tooth profile errors and structural coupling effect. The minimum variations of the loaded static transmission error are selected as the objective function to obtain the optimum modification parameters of gear pair. In addition, the effect of torque, modification amount and modification length on the engagement characteristics of the gear pair are studied. The research can provide some references for the design and optimization for the gear transmission system in the electric vehicle.

Grinding wheel profile design and temperature field analysis of the pinion for internal meshing gear with curve configuration

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.

Spiral bevel gears: Bifurcation and chaos analyses of pure torsional system

The goal of this paper is to provide a detailed analysis of the complex dynamic scenario of spiral bevel gears by considering the torsional shaft stiffness. The dynamic model takes into account: time-dependent stiffness and non-smooth nonlinearity due to the backlash, i.e., teeth contact loss. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allowed an accurate evaluation of the global and local teeth deformation in both directions: forward and reverse motions. Due to intentional tooth profile errors, e.g., profile modification, manufacturing error, or assembly error, the loaded and unloaded tooth contact analyses were conducted. The dynamic model was validated by comparisons with a verified SDOF model in terms of linear natural frequencies and nonlinear dynamic response. The present study provides amplitude frequency diagrams and bifurcation diagrams; for specific regimes, periodic, quasiperiodic, and chaotic responses were found. The dynamic behavior of the systems was evaluated using various tools such as modal analysis, nonlinear time series analysis, spectra, 3D-phase diagrams, and Poincaré maps. This study proves that decreasing the DOFs of the system can lose some dynamics. Different phenomena have been found such as trapping and boom-and-bust cycle. The dynamic response of different cases is evaluated by estimating the largest Lyapunov exponent and the correlation dimension. It is found that the system undergoes complex dynamic phenomena including nT-periodic, trapping, and aperiodic motions, which are evidenced in 3D-bifurcation diagrams, spectra, Poincaré maps, and 3D-phase diagrams. The chaotic motion found in this study is an undesirable behavior from a practical point of view, which can be avoided by adjusting the designed parameters. The results provide a basis for parameter design and dynamic characteristic control of the spiral bevel gear drive system.

Modeling and compensation of twist error of helical gear with lead modification in continuous generating grinding

A compensating method based on continuously deflecting the dressing wheel is proposed to eliminate the twist error of the lead modification tooth flank. Combining the tooth flank contact trace and the lead modification curve, the mechanism of the twist error is first analyzed. Second, a tooth profile error model is developed at the upper and lower end faces. The flank twist amount is calculated by subtracting the tooth profile error at the upper and lower end faces. Then, the profile slope deviation of the twisted tooth flank is transformed into the pressure angle deviation of the tooth profile, and the method of compensating the pressure angle of the worm wheel is applied to eliminate the pressure angle deviation of the tooth flank caused by the twist error. The grinding partition of the worm wheel is performed according to the diagonal grinding process, and a worm wheel dressing method for continuously deflecting the dressing wheel is provided to compensate for the twist error. Finally, the continuous generating grinding experiment of the lead modification gear verifies the feasibility of the proposed method.

Dynamic Modeling and Analysis of an RV Reducer Considering Tooth Profile Modifications and Errors

Due to their advantages of compact size, high reduction ratio, large stiffness and high load capacity, RV reducers have been widely used in industrial robots. The dynamic characteristics of RV reducers in terms of vibratory response and dynamic transmission error have a significant influence on positioning accuracy and service life. However, the current dynamic studies on RV reducers are not extensive and require deeper study. To bridge this gap, a more effective and realistic lumped parameter dynamic model for RV reducers is developed, considering the tooth profile modification of cycloid gears and system errors. Firstly, for an efficient solution, the equivalent pressure angle and equivalent mesh stiffness of the cycloid–pin gear pair are introduced in the dynamic model based on the loaded tooth contact analysis. Secondly, the differential equations of the system are derived by analyzing the relative displacement relationships between each component, which are solved using the Runge–Kutta method. With this, the effects of errors such as machining errors, assembly errors and bearing clearances on the dynamic behaviors and transmission precision are investigated by comparison to quantify or qualify their influence. This research is helpful in characterizing the multi-tooth mesh and dynamic behavior, and revealing the underlying physics of the RV reducer.

Analytical Investigation on Load Sharing Performance of Planetary Gear Transmission under Loop Maneuver

In order to explore the influence of airframe space motion on the load sharing performance (LSP) of planetary gear transmission (PGT) inside an aircraft, the combined influence of the rotation of the carrier (internal non-inertial system, INIS) and space motion of the fuselage (external non-inertial system, ENIS) is considered. Firstly, the motion equations of PGT members under the airframe loop maneuver are derived, and then the time-varying meshing stiffness, backlash, and tooth profile error are considered; a system-level dynamics model of PGT with casing is established and solved by numerical integration. The results show that the loop maneuver intensifies the fluctuation of the dynamic meshing force in the time domain and does not change its spectral distribution, but it will affect the peak value of the low frequency part. After considering the loop maneuver, there are differences in the load sharing coefficient (LSC) of each branch, and the fluctuation range of LSC increases. The interactive influence of working conditions, bearing stiffness, manufacturing eccentricity errors and loop motion parameters on the LSP is investigated and compared in depth. This research can be utilized to guide the design of aircraft’s drivetrain.

Simple mathematical approach for analyzing gear tooth profile errors of different gears cut using same power-skiving tool

Power-skiving is one of the most efficient methods for cutting internal gears. However, the same power-skiving tool can only be used to cut gears with the same parameters. Accordingly, this paper proposes a simple mathematical method for analyzing the gear tooth profile errors produced when using the same power-skiving tool to cut gears with different parameters. The feasibility of the proposed method is demonstrated by means of an illustrative example. It is shown that a reduction in the number of gear teeth (or gear helix angle) has a greater impact on the tooth profile error than an increase in the number of gear teeth (or gear helix angle). Overall, the proposed error analysis method provides a useful insight into the machining flexibility of a specific power-skiving tool.

Profile design and simulation verification of hard whirling tool for ZC1 worm machining

The profile of the hard whirling tool for ZC1 worm machining is challenging to design precisely. An inaccurate tool profile results in irreversible overcutting or additional machining allowance. This study aimed to develop an accurate tool profile design method and digital verification approach to enhance the quality of ZC1 worms. The precise measurement and axial section data acquisition of the worm are conducted. A tool profile design method is proposed based on the envelope of the equidistant axial section. The motion law of whirling is studied, and a 3D model generation procedure for worm teeth is developed. The effect of the tool profile design is verified, and the impact of cutting parameters on worm tooth profile error is analyzed. The results show that the presented tool design method can effectively avoid overcutting. The employed modeling approach of worm teeth provides an intuitive verification means for tool profile design. Variable cutting parameters lead to different machining errors, and the maximum error is less than 3 μm. Changing the cutting parameters will not induce macro profile error but will impact the micromorphology. These findings should be valuable for designing envelope machining tools with complex motion and realizing the superior performance of hard whirling for ZC1 worm machining.

Research on Processing Error of Special Machine Tool for VH-CATT Cylindrical Gear

The variable hyperbolic circular arc tooth trace (VH-CATT) cylindrical gear is a new gear suitable for heavy loads and high speed. The special structure of the gear provides excellent mechanical properties but also increases the processing difficulty. The special machine tool for VH-CATT gear provides a prerequisite for mass production, but the machining accuracy remains to be improved. Therefore, this paper proposes a Kriging model based on the glowworm swarm optimization algorithm of scene understanding (SGSO) to study the relationship between input parameters and output precision. Then, the SGSO algorithm is used to optimize the parameters of the Gaussian mutation function in the Kriging model to improve its fitting accuracy. When solving four groups of tooth profile and tooth direction errors, the key precision index, R2, of SGSO-Kriging all exceed 0.95. Additionally, the feasibility of the model is verified by the residual diagram and the box diagram. The contour diagram and error results show that reducing the feeding velocity, vf, can improve accuracy most efficiently, and the increase of rotational speed, n, is more conducive to the accuracy of the tooth surface than the acceleration of the coolant, vQ. The above research provides an optimization strategy of gear machining accuracy and a theoretical basis for the promotion of the VH-CATT gear.

Machining principle and cutter design of face gear skiving

Current processing methods of face gear are mainly limited to gear shaping, gear hobbing, and other processes. In order to achieve high-precision and high-efficiency machining of face gear, this paper proposed the cutting technology and tool design of face gear skiving. Solving the tooth surface of the skiving cutter according to the meshing principle, and fitted by NURBS surface. Intersecting the rake and conjugate face to obtain cutting edge data, and import it into the CAD software to build the skiving cutter model, the tooth surface enveloped by the cutting edge of the skiving tool is solved according to the envelope principle. This paper proposed the machining principle and tool design method of face gear skiving and analyzed the influence of tool tooth profile error and installation error on the face gear tooth surface deviation, which provides a reference for improving the machining accuracy and efficiency of face gears and further improving the gear skiving process.

Research on the meshing characteristics of asymmetric gears considering the tooth profile deviation

This paper establishes a time-varying meshing stiffness (TVMS) analytical model for asymmetric gear pair based on the potential energy principle to analyze the TVMS, transmission error, and load share ratio of asymmetric gears. The model considers the adjacent teeth' influence on the body deformation of the gear teeth and is verified the reasonableness by finite element analysis. The effect of tooth tip relief on the characteristics of asymmetric gear pairs is analyzed in this paper. Considering the tooth profile deviation due to tip relief and tooth profile error of the asymmetric gear can obtain results that are closer to the original meshing condition. The results show that the gear contact ratio decreases with the increase of one side of the gear pressure angle. Conversely, the TVMS of spur gear will increase in a certain extent. During modification section of tooth profile, some significant variations will happen on the behaviors of asymmetric gears such as TVMS, transmission error and load share ratio. Slight tooth profile errors on the surface produce conspicuous fluctuations in gear meshing characteristics.

Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts

The circumferential vibration of a gear pair is a parametric excitation caused by nonlinear tooth stiffness, which fluctuates with meshing. In addition, the vibration characteristics of the gear pair become complicated owing to the tooth profile error and backlash. It is considered that the circumferential vibration of the gear pair is affected by the torsional vibration of the shafts. It is important to understand quantitatively the vibration characteristics of the gear system considering the shafts. Therefore, the purpose of this research was to clarify the nonlinear vibration characteristics of a gear pair considering the influence of the shafts using theoretical methods. To achieve this objective, calculations were performed using equations of motion in which the circumferential vibration of the gear pair and the torsional vibration of the shafts were coupled. The nonlinear tooth stiffness was represented by a sine wave. The influence of tooth separation was considered by defining a nonlinear function using backlash and the tooth profile error. For the numerical calculations, both stable and unstable periodic solutions were obtained by using the shooting method. The effect of the shafts on the gear system vibration were clarified by comparing the results in the cases in which the shaft was not considered, one shaft was considered, and both shafts were considered.

Effect of Tooth Surface Roughness on Wear of Spur Gears

Since gears transmit motion by meshing teeth, it is hard to avoid problems which wear and damage to the tooth surface. Because Tooth surface wear is difficult to predict, the approach to the wear problem still often relies on empirical rules and symptomatic measures. If the undermining by wear can be predicted by mathematical formulas, it may allow engineers to solve wear issues in model-based development, or to consider wear resistance from the design state. Therefore, the development of wear prediction formulas seemed like a very valuable goal, so the authors decided to work on it. In this paper, as a step for the final goal, the gear driving test was conducted using multiple gears with different tooth surface roughness. Then to detect changes in tooth profile, the tooth profile error was measured, and wear depth was calculated. Moreover, surface roughness on teeth was measured, and the relationship between surface roughness and wear depth was investigated.

Research on performance test analysis method of mechanical structure based on measured error / virtual simulation

Mechanical errors seriously affect the transmission performance of mechanical structure, especially for high-precision mechanical structure. With the development of virtual simulation technology, the performance test of mechanical structure in virtual environment is gradually applied. Under the background, this paper takes the movable tooth reducer as the research object and puts forward a virtual performance test method by combing test errors with virtual simulation technology. The tooth profile errors of key components of the reducer are obtained by optical scanning, and the simulation test is carried out. Meanwhile, the accuracy of the simulation test is verified by the implementation of vibration test. The results show that both the simulated vibration acceleration and the experimental test vibration acceleration show a double-peak variation trend, and the frequency difference between the two peaks is less than 5%. Therefore, the simulation test method proposed in this paper can effectively evaluate the performance of mechanical structures.

Analytical calculation models for mesh stiffness and backlash of spur gears under temperature effects

The temperature has a great contribution to the mesh stiffness and backlash of the gear pair. Presence of thermal deformation caused by temperature will complicate the gear teeth interaction. In this paper, the thermal time-varying stiffness model and thermal time-varying backlash model are proposed with the consideration of tooth profile error and total thermo-elastic deformation consists of the teeth deformation, teeth contact deformation, and gear body-induced deformation. The key parameters of thermo-elastic coupling deformation affected by temperature are calculated. Based on the proposed models, the influencing mechanism of temperature on the tooth profile error, mesh stiffness, total deformation, and backlash are revealed. The effects of shaft radius and torque load on the thermal stiffness and thermal backlash are studied. The proposed thermal stiffness and backlash calculation model are proven to be more comprehensive and the correctness is validated.

Measurement of gear tooth profiles using incoherent line structured light

Optical measurement methods are being increasingly used in the profile measurement field, but optical high-precision measurements of gear tooth profiles have not been achieved. Therefore, a high-precision profile measurement method that uses incoherent line structured light was proposed and a corresponding measurement system was developed in this paper. To calibrate the accuracy of this system, profile measurement experiments using roughness comparison specimens were conducted. Finally, the tooth profile measurement experiments using an involute master were performed. The results indicated that the incoherent line structured light method was less affected by speckle noise and had higher measurement resolution than the line laser method. The tooth profile error for the involute master measured by the incoherent line structured light method was ±2.2 μm. This study provides a new method for the industrial application of optical high-precision measurement of gear tooth profiles.

Investigation of Dynamic Load Sharing Behavior for Herringbone Planetary Gears considering Multicoupling Manufacturing Errors

In this study, based on the lumped-parameter theory and the Lagrange approach, a novel and generalized bending-torsional-axial coupled dynamic model for analyzing the load sharing behavior in the herringbone planetary gear train (HPGT) is presented by taking into account the actual structure of herringbone gears, manufacturing errors, time-dependent meshing stiffness, bearing deflections, and gyroscopic effects. The model can be applied to the analysis of the vibration of the HPGT with any number of planets and different types of manufacturing errors in different floating forms. The HPGT equivalent meshing error is analyzed and derived for the tooth profile errors and manufacturing eccentric errors of all components in the HPGT system. By employing the variable-step Runge–Kutta approach to calculate the system dynamic response, in conjunction with the presented calculation approach of the HPGT load sharing coefficient, the relationships among manufacturing errors, component floating, and load sharing are numerically obtained. The effects of the combined errors and single error on the load sharing are, respectively, discussed. Meanwhile, the effects of the support stiffness of the main components in the HPGT system on load sharing behavior are analyzed. The results indicate that manufacturing errors, floating components, and system support stiffness largely influence the load sharing behavior of the HPGT system. The research has a vital guiding significance for the design of the HPGT system.