Gear Tooth Profile Research Articles

A Novel Model of an S-Shaped Tooth Profile Gear Pair with a Few Pinion Teeth and Tooth Contact Analysis Considering Shaft Misalignments

Due to the special meshing of convex and concave tooth profiles, an S-shaped tooth profile gear with a few teeth has the advantage of reducing the contact stress and improving the contact ratio. However, the theoretical inseparability of the center distance of an S-shaped tooth profile and shaft misalignments, including center distance errors and axis parallelism errors caused by manufacturing or installation errors, can easily cause tip contact and produce transmission errors. Accordingly, it is necessary to analyze the influence of shaft misalignments on the meshing performance. In this paper, a unified mathematical model of spur and helical gears with an S-shaped tooth profile is established and an analytic method for determination of the contact pattern, meshing path, and transmission error of a gear pair with shaft misalignments is proposed. Then, the validity of the analytic method is proved by practical examples and the influence of shaft misalignments on the contact pattern of an S-shaped tooth profile gear pair is obtained. Through comparison with an involute gear with the same parameters, it is found that shaft misalignments within a certain range result in an acceptable transmission error and contact area migration of the S-shaped tooth profile gear pair similar to that of the involute gear pair.

Design and fluid analysis of the microsegment gear tooth profile for pump operating at high speed and high pressure

In order to solve the problem of oil trapping in involute gear pumps, a new tooth profile based on the microsegment tooth profile and incorporating the circular arc tooth profile theory has been designed in this study. Firstly, the tooth profile equation of the microsegment gear profile for the pump has been derived. Then, the spatial tooth surface equation of the helical microsegment gear for the pump has been derived, and the feasibility of the new tooth profile has been verified using the contact spot experiment. The characteristics of the new gear profile without oil trapping and the closed “8” type mesh line have been analyzed, from which the output flow rate of the microsegment gear pump has been verified to be constant. A conclusion has been drawn on the applicability of the microsegment gear tooth profile for the pumps. Further, the fluid analysis model of the microsegment gear pump has been proposed to analyze the oil pressure distribution and state of fluid motion at high pressures in the microsegment gear pump. A comparison of the performances of the involute and the microsegment gear pumps shows that the output pressure and maximum pressures of the microsegment gear pump are smoother than that of the involute gear pump. Analysis of the effect of the high rotational speed on the output oil pressure and the maximum pressure shows that the former increases, whereas the latter varies very little with the rotational speed.

Gear Profile Measurement Based on Thin Line Structured Light

In order to achieve the high precision 3D measurement of gear tooth profile, a telecentric optical measurement system with large FOV (field of view), WD (working distance) and measuring range was proposed, the system was applied to the gear tooth profile measurement experiment and obtained a good effect. Firstly, the composition and the principle of the system were introduced. The lights of projection and imaging with an angle of 45° from the normal line of the measured surface can reduce the requirement of DOF (depth of field) of the imaging objective lens. Secondly, in combine with the application background of small modulus gear tooth profile measurement, the magnification, NA (numerical aperture) and other optical parameters were calculated, the optical system was constructed and the system performance parameters are listed. Then, the calibration model of the system was demonstrated and the calibrations of depth pixel resolution and horizontal pixel resolution were completed through experiments. Finally, an involute artifact with a modulus of 2.9992mm and a total tooth profile deviation of 1.1μm was measured. The results show that the deviation of the total tooth profile measured is about 2μm, which is consistent with the calibration curve of the involute artifact.

Monitoring gear wear with transmission error

The use of transmission error (TE) in gear diagnostics has recently become more feasible. TE is a parameter that describes the meshing error between two gears, i.e., the deviation in angular position from the theoretically perfect, rigid involute case, and can be measured very accurately using shaft encoders installed on the unloaded part of the input and output shafts. It is hence one of the measurements closest to the source – the internal excitation from the gearmesh. Wear induces geometric deviations on gear tooth profiles, which will manifest as a geometric TE (GTE). To perform effective gear wear monitoring, one requires reliable sensors or measurements that are sensitive to anomalies due to wear, especially when wear is at an early stage and hard to detect. Vibration, which is usually measured on the gearbox casing, tends to be affected by the complex transfer path from the source to the measurement point and thus poses challenges in monitoring wear. Conventional wear analysis techniques, including image acquisition and characterisation of worn gear surfaces, often involve disruption of the operation and offline analysis, causing some delay. TE, as a measurement much closer to the source, has considerable benefits compared to vibration as well as traditional wear analysis.This paper demonstrates a comparative study on TE measured from dry and lubricated wear experiments, offering new insights on the effect of different wear mechanisms on TE signals. The results show that TE is sensitive to the evolution of gear wear and could be used to perform trending effectively. Such a powerful measurement with good trending ability could revolutionise gear prognostics. With the embedded sensor trend associated with Industry 4.0 and Internet of Things, the use of inbuilt encoders will likely become more prevalent in industry, allowing the use of TE measurements in practical gear monitoring.

Modification and Optimization of Cycloidal Gear Tooth Profile Based on Machining Error Compensation

The rotary vector reducer presents high precision and load capacity characteristics. The shape of the cycloidal gear tooth profile in the rotary vector reducer significantly affects its performance. Meanwhile, the effect of the machining error on the error between the designed and theoretical tooth profiles cannot be ignored. Thus, this paper analyzes the machining error items that affect the shape of the cycloidal gear profile from the machining process perspective. Due to the random characteristics and different distribution rules inherent in various machining errors, this paper proposes a cycloidal gear machining error compensation and modification model based on the Monte Carlo simulation method, providing a new theoretical method for compensating cycloidal gear machining errors. While compensating for machining errors, considering the impact of cycloidal gear modification on the carrying capacity, the NSGA-Ⅱoptimization algorithm is utilized to optimize the cycloidal gear modification parameter and finally, to solve the modification parameters with a more comprehensive performance.

Синтез профілю зубчатих коліс малошумного шестеренного насоса з низькою пульсацією

Low-noise gear pumps with non-involute gear profiles, which have undeniable advantages over pumps with involute gear profiles (low noise characteristics, low pressure and volumetric pulsation, inseparability of gear contact), have not gained wide popularity due to difficulties in designing and manufacturing the working profiles of the gears. Companies manufacturing low-noise pumps provide only promotional information, while information about the meshing geometry, profile, and manufacturing technology is practically absent. The aim of the research is to obtain universal formulas for calculating the complex sinusoidal profile of gears for low-noise pumps. This will allow designing sinusoidal meshings with different input parameters (number of teeth, center distance, etc.). Additionally, for the helical sinusoidal meshing, it is necessary to derive a formula for determining the minimum tooth helix angle that ensures inseparability of contact in the meshing. Several stages are involved in obtaining the universal formulas. In the first stage, the construction of a base sinusoidal curve is performed. The obtained profile is not mutually enveloping, meaning that when positioning the gears at the center distance and forming the meshing, there is interference of curves. Hence the need for forming a unified profile where both gears have identical profiles and can be processed by the same tool. In the second stage, the generation of the conjugate gear profile is performed as an enveloping family of circular sine waves, by rolling a dividing circle of the sine wave around the generating circle of the formed gear. Segments of this profile will be further involved in forming the final gear profile. In the third stage, the circular sine wave and its enveloping positions are overlaid to a common center of coordinates. As a result, the conjugate profile points lying on the dividing circle are formed. The unification of the gear profile is achieved by removing certain sections of the curve within the dedendum and addendum, while the remaining sections that have a common tangent at the end points of the conjugation form the final gear profile. As a result of the conducted research, a mathematical apparatus has been developed that allows building gear tooth profiles for low-noise gear pumps with arbitrary geometric parameters and number of teeth. The minimum value of the tooth helix angle required to ensure contact inseparability and absence of clearances has been determined.

Improving the wear resistance of 3D printed spur gears through a free-form tooth flank optimization process

Involute gears have traditionally been the preferred choice for gear transmission systems due to their simplicity and interchangeability. However, there are applications where they do not provide the most durable and efficient solution. While the cost of implementing optimized non-involute gears in most applications often outweighs their comparative advantages, the advent of additive manufacturing has opened up possibilities for designers to explore alternative gear tooth profiles. This is particularly relevant in the realm of plastic gears, where optimized non-involute gears produced through 3D printing can address their primary drawbacks, such as surface durability and wear resistance. In this study, a comprehensive free-form optimization process was conducted to determine the optimal tooth profile that minimizes wear on 3D printed spur gears during operation. The tooth flank geometry was represented using a 4th order B-spline curve, and a genetic algorithm was employed to determine the optimum positions of the control points aiming to minimize wear depth across the tooth flanks. The spur gears were manufactured using Fused Deposition Modeling (FDM) with PLA material. The parameters of the additive manufacturing process were experimentally fine-tuned to achieve the best possible accuracy. To evaluate the performance of the optimized free-form gears, two case studies were implemented, demonstrating that the optimized gears achieved a remarkable reduction of average wear depth by more than 50% and a reduction of maximum wear depth by more than 69% compared to standard involute gears. To further validate the effectiveness of the optimization method, experiments were carried out using an FZG test rig. The profiles of the tooth flanks were measured on a Coordinate Measuring Machine (CMM) before and after the experiments to compare the wear depth against the standard involute gears. The results revealed a significant improvement in the wear resistance of the tooth flanks, with a reduction of wear depth of 44.1%.

The Study of the Profile Errors of Straight Teethed Shaper Cutters in the CAD-Environment

It is well known that straight-teethed shaper cutters present theoretical profile error, which leads to a deviation of the cut gear tooth profile from the involute profile. Taking advantage of the CAD environment, we have visualized and studied shaper cutters at different stages of sharpening. This is advantageous because the effect of any shape or dimension modification of the rake face can be immediately and easily controlled in the design process, thereby saving precious design time. The aim of our research was to create a 3D model in a CAD environment to visualize different sharpening methods and the profile of shaper cutters at different wear stages, and to estimate the inherent profile errors. The aim of using different sharpening methods was to improve the lateral cutting geometry (especially the lateral relief angle) and also to reduce or – at least – keep the profile error within the accepted limits. During the measurements, both the rake angle and the cutter rack profile angle were changed in order to improve the geometry. The edge was generated using numerically computed points and Autodesk Inventor's special commands. Comparative analyses were performed.

ГЕОМЕТРИЧНЕ МОДЕЛЮВАННЯ РОБОЧИХ ПРОФІЛІВ І ОБ’ЄМІВ РОТОРНО-ПЛАНЕТАРНИХ МАШИН

A method for calculating a rotary planetary machine of the Wankel system is presented. A feature of the Wankel machine is that the movement of the rotor in relation to the body is carried out using a planetary mechanism, which is based on an internal gear. The proposed method makes it possible to take into account the connection between the parameters of the geometric shape of two pairs of mutually conjugate curves characteristic of the Wankel machine. The first pair of curves are the working profiles of the casing and the rotor. The second pair is formed by the tooth profiles of the internal gears. Accounting for this connection made it possible to find a description of the function of the change in time of working volumes limited by the profiles of the body and rotor (i.e., to determine the productivity of the machine). As a result of the research, it was found that the performance of the Wankel machine can be found by taking into account three features of this machine. The shaping of the conjugated working profiles of the rotor and stator is carried out using a planetary mechanism, for the calculation of which the functions of complex variables are used. The performance function of the machine is determined by the change in time of the volumes of space limited by the working profiles of the rotor and housing. An approximate description of the functions of changing the working volumes of the machine with time was carried out using specially developed graphic constructions. To match the geometrical parameters of the Wankel with the parameters of the tooth profiles of the gear train of the internal clutch, a method of graphic running has been developed. The results obtained are useful because the scheme of the considered Wankel machine is embedded in the design of the internal combustion engine of the same name, as well as in the design of various hydraulic machines, pumps, compressors, etc. Key words: geometric modeling, rotary planetary mechanism, Wankel machine, functions of a complex variable, internal gearing.  

Design and Analysis of Novel Non-Involute Cylindrical Gears with a Curved Path of Contact

In this paper, novel non-involute cylindrical gears are designed based on a curved path of contact. Firstly, a parabolic curve is predesigned as the contact path of novel gears. Then, the tooth profiles of the novel gears are calculated using differential geometry and spatial meshing theory. Secondly, the three-dimensional tooth models of the novel non-involute gears are established according to a machining simulation. Thirdly, a simulation model (including misalignment error and longitudinal modification) is established to analyze the performance of the novel non-involute gears. Finally, an example is given, and the results show that the presented novel non-involute gears have greater load-carrying capacity compared with the involute gears. Moreover, whether modified or not, with or without misalignment error, the stresses of the presented novel gears are lower than those of involute gears.

Investigation and optimization of hypoid gear micro-geometry parameters and axle deflections to improve the contact stress distribution and fatigue life of hypoid gears

The automotive industry always demands more torque per kilogramme of weight. On the other hand, there is always a growing demand for increased warranty kilometres for customers. This puts increased pressure on the life of the drive axle, which is the major load and torque-carrying member in commercial vehicles. Drive head unit is a crucial contributor to this aim. This paper focuses on the work done towards the above targets wherein advanced tools helped to achieve them in the minimum time. Due attention is given to the hypoid gear system to sustain higher stress because of the heavy torque demanded by the application. It is accomplished by the optimum design of the gear tooth profile and system design. Gear durability is a major concern in heavily loaded axles. To have an optimum design with higher performance torque is a key challenge. The paper talks about the challenges overcome during the design, development, and validation of a single-reduction large-size gear axle to replace the hub reduction rear axle. It demands a better understanding of gear tooth and system behavior, which was analyzed by advanced software tools and measurement machines. The same is validated on the test bench, and the results are discussed.

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.

Design evaluation of disk-type milling cutter with special inserts applying analytical simulation and the experimental verification in screw rotor milling

A milling cutter with inserted blades is designed for each screw rotor since the geometry profile is unique, corresponding to its purposes and function. Correctness in cutter design, including the cutting-edge insert profile, is necessary to achieve a precise rotor profile; therefore, it is essential to introduce an enhanced cutter design. This study proposes a design of a disk-type milling cutter with special inserts. The insert cutting edge was generated on the axial plane considering the cutter curve curvature and then positioned onto the cutter body according to the modified coordinate system with a revolve angle. The analytical cutting result indicates that the simulated profile is relatively even along the gear tooth profile and the normal deviation is significantly tinier than the insert arrangement area. In addition, the cutter design applying special inserts acquires better results than standard inserts. Finally, the experimental result, including the rotor surface topography, confirms that the proposed cutter design is reliable and realistic in CNC screw rotor milling.

Eliptik Dişli Tasarımı, İmalatı, Analizi ve Eliptik Dişlilerin Endüstriyel Uygulamaları

Eliptik dişliler üzerine yapılan çalışmaların diş profillerinin tasarımı üzerine yoğunlaştığı ve prototip imalatlarına yönelik araştırmaların arttığı görülmektedir. Eliptik dişlilerin diş profillerinin silindirik dişliler gibi özdeş olmayacağından hareketle geliştirilen eliptik dişlinin diş profili; kramayer veya silindirik düz dişli formlu kesicinin bölüm elipsi üzerinde yuvarlanma yöntemine uygun olarak evolvent eğriye sahip diş yanaklarının şekillendirilmesiyle elde edilmektedir. Eliptik dişlilerin imalatında CNC tel erozyon tezgahlarından faydalanmanın yaygınlaştığı, bununla birlikte azdırma tezgâhları için elektronik dişli kutuları geliştirilerek eliptik dişlilerin standart modülde imalatının da mümkün olduğu tespit edilmiştir. Eliptik dişli çiftleri farklı makine ve teçhizatta kullanılmakta, özellikle eliptik dişli çiftine sahip debimetre ve pompaların yaygınlaştığı görülmektedir. Enerji verimliliğinin son derece önem kazandığı günümüzde, yüksek akışkan miktarlarının doğru ve hassas ölçülmesi ve işletmelerin akışkan transferindeki kayıplarının giderilmesi de gerekmektedir. Bu genel değerlendirme çalışmasında eliptik dişlilerin tasarımı, geleneksel ve geleneksel olmayan imal usulleri ile imalatı, dayanım ve titreşim analizleri ve literatürde yer alan muhtelif endüstriyel uygulamaları gözen geçirilmiş ve çalışmanın ortak bulguları elde edilmiştir. Eliptik dişlilerin yakın gelecekte bünyesinde mekanik kısım barındıran birçok akıllı ürünün vazgeçilmez bir organı olacağı öngörülmektedir.

Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers

The current loaded tooth contact analysis of cycloid drives based on the assumption of theoretical positions of ring pins ignores the deviations caused by manufacturing errors and elastic deformations, which are not in agreement with reality. To fill this gap, an improved load distribution model of the mismatched cycloid-pin gear pair with ring pin position deviations is presented for a component-level analysis. Firstly, with the cycloid gear tooth profile geometry defined, the unloaded tooth contact analysis is applied as a pre-processor to determine the potential contact points, the gear backlash, and the rotation angle of the cycloid gear. Secondly, due to the statically indeterminate structure of the multi-tooth contact, a varying nonlinear contact stiffness is introduced to establish the relation between force and deformation. Then, the force and moment equilibrium equations with compatibility conditions are solved by using an iterative approach. With this, detailed parametric case studies are presented to verify the correctness of the proposed model by comparing it with those predicted by the current model and to demonstrate the influences of ring pin position deviations on the distributed load, contact stress, loaded transmission error, and instantaneous gear ratio of the mismatched cycloid-pin gear pair. This study provides a deeper investigation into the load distribution characteristics of the cycloid drive and therefore can be employed to assist in gear design.

Mathematical Modeling and Machining of the Internal Double-Arc Spiral Bevel Gear by Finger Milling Cutters for the Nutation Drive Mechanism

A method of machining the internal double-arc spiral bevel gear with a finger milling cutter was presented. The mathematical model of the internal spiral bevel gear tooth profile was established considering the principle of machining a spiral bevel gear by the generating method, and a three-dimensional (3D) tooth profile graph was developed. Subsequently, by applying the gear meshing theory, the 3D model of the tooth alignment curve for the finger milling cutter was established. Based on the tooth surface equation of crown gear, the cutter intercept equation was derived. The cutter was divided into four finger milling cutters considering the design difficulty of the cutter, which is used to manufacture different arc segments of the double-arc tooth profile, respectively. The special machining tool model of the internal spiral bevel gear was further developed by using SolidCam, and the simulation experiment was carried out. The simulated gear model was compared with the theoretical gear model and the error of the simulation experiment was estimated. Actual machining on the machine tool and the internal spiral bevel gear were inspected. The maximum error is 0.035 mm, and the minimum error is 0.005 mm. The machining accuracy meets the requirements. The feasibility of machining the internal double-arc spiral bevel gear with a finger milling cutter was verified.

Towards general performance diagrams to define optimum profile and lead modifications with regard to transmission error in spur and helical gears

Some theoretical results on mesh elasticity and contact patterns in narrow-faced spur and helical gears are presented, which lead to original semi-analytical definitions of optimum combinations of profile relief and lead crown minimising the time-variation amplitudes of transmission error. Extensive comparisons with results from a benchmark code prove that the proposed methodology is sound and can be used to define optimum modifications with minimum effort and computational time. Dimensionless performance diagrams are derived from systematic simulations over a broad range of gear macro- and micro-geometries. Although no closed-form solutions can be found, general trends are clearly identified, making it possible to define optimum tooth modifications with reasonable precision. The importance of face contact ratio is highlighted as it can lead to highly contrasted results with either one or two solutions minimising the time-variations of transmission error. For a given gear geometry and set tooth profile and lead modifications, it is shown that the specific influence of loading can also be derived from the proposed performance diagrams.

Research on the varying-displacement-coefficient modification method for automobile steering noncircular sector gears

The local drum shape tooth profile is adopted for automobile steering noncircular sector gears, which can be obtained according to a piecewise linear function modification on CNC gear shaper. In the practical process, the actual amount of drum shape is always smaller than the theoretical amount, and we conclude that overcut of the tooth profile occurred. Because the modification coefficient is variable, we call it varying-displacement-coefficient modification(VDCM). Currently, researches on tooth profile of noncircular sector gears are primarily concerned with constant modification coefficient to improve gear transmission performance, while modification functions, overcut, and methods of reducing or eliminating the overcut are rarely touched upon.This paper focus on the VDCM method of automobile steering noncircular sector gears. On the basis of noncircular sector gear and rack pair meshing theory, the modifying motion mathematical model is first established, including kinematic relations, pith curves, track of relative instantaneous centre point, calculation formula for the modified tooth profile. And then, the reasons for the overcut are analyzed, and the local overlap and separation of the tooth profile are found. To reduce the overcut amount, a cosine function modification method is proposed. Furthermore, an inverse calculation method for the rack tool tooth profile is proposed to eliminate the overcut entirely. The VDCM modification methods are verified by simulation, trial cutting and parts measurement.

Static and vibration analysis of functionally graded gears

In this study, static and vibration analyses of functionally graded gears (FGGs) have been performed using the FE-based numerical method. The gear body is divided into uniform thickness homogeneous sub-domains conforming to the gear tooth profile. Material properties vary in the radial direction according to power-law gradation with metal at the innermost and ceramic at the outermost surface. Three FG gear cases are studied, assuming the gradation for the entire body, radially outer half, and the outer quarter of the gear. Parameters like root stress, weight reduction, tooth stiffness, and natural frequencies of FG and bi-material gears are studied and presented with steel gear as a reference. For FGGs, root stress increases and the weight of the gear reduces with an increase in gradient index. It has been observed that the maximum difference in the fundamental frequencies of FGGs and steel gears is 12% compared to 25% for bi-metal gears.

A novel design for enhancing the surface durability of the spur gear systems

In this study, a general mathematical model for defining the gear tooth profile and its geometrical aspects that meet a desirable sliding contact during the meshing cycle is built. This model is based on the path of contact shape predefinition. In the contact path, a straight-line segment through the pitch point and a universal transition curve for the other segments are combined to create a free-form tooth profile with desired radius of curvature in the designed gear. The maximum pressure angle during the meshing cycle and the involute curve length parameter are only introduced to characterize this model. The designed gear in this study is given the title “free-form-gear” since it provided a universal manner of establishing the path of contact that provides desirable sliding conditions during the meshing cycle. The proposed tooth form, interference condition, contact ratio, contact stress, sliding coefficient, and meshing efficiency are derived and investigated in this study. The fillet stress and torsional mesh stiffness of the proposed gear pair are obtained using the finite element method (FEM). The performance of the new gear is compared to that of the standard involute and non-involute gear types. The results indicated that the method proposed in this study can be utilized to construct a gear pair with higher surface durability and strength than typical involute gear of the same size.