Gear Profile Research Articles

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.

Synthesis and CAD-Examination of a Cylindrical Gear Pair Derived from a Cosine Profiled Rack

The problematic of gear profiles has again come to the fore during the power demands imposed by technical development. Most of nowadays gears have involute profile due to its numerous ad-vantages. The most important is that that axis distance variation does not affect the transmission ratio. According to [1], a cosine gear drive has a lower slip coefficient and the contact and bending stresses by charge are significantly lower compared to an involute gear. This paper presents a study regarding the generation synthesis of a cosine profiled gear pair, using a single cosine profiled rack, in a CAD environment. The profile of the generating rack and the mathematical computations of the equations of gearing were performed in a Mathcad environment. The tool models were created in Autodesk Inventor. Then it was implemented in Autodesk AutoCAD and using its special programming language (Auto lisp), a simultaneous subtraction-rolling method was used to obtain a realistic simulation of the meshing of the cut gear with the rack. Implementing the generated gears in Autodesk Inventor environment, tests were performed in order to compare the cosine profile of the generated gears with the profile of the involute gears. In the near future, we intend to exploit the results obtained and compare the transmission quality of the involute gears with the cosine gears.

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%.

Grinding force model for gear profile grinding based on material removal mechanism

Grinding force model for gear profile grinding based on material removal mechanism

Material removal analysis during MR polishing of complex gear teeth profiles

The geometric intricacy of tiny gears makes nano-finishing difficult. In the current study, the magnetorheological (MR) polishing process is used for the nano-finishing of intricate surfaces of tiny gear components uniformly. For polishing, the technique employs a dynamic fluid recognized as magnetorheological polishing fluid (MRPF), that has the ability to stiffen in the presence of a magnetic field. Base media, iron and abrasive particles are utilized to synthesize the MRPF. Permanent magnets produce the necessary magnetic field in the finishing zone. Finite element analysis (FEA) is utilized to model the iron and abrasive particles to understand better how they would react in the external magnetic field. FEA is utilized to analyze the magnetic flux density (MFD) distributions and the amount of magnetic force exerting on gear profiles through iron particles (IPs). It has been observed that the IPs present close to the active abrasives are primarily accountable for indenting active abrasives into the workpiece surfaces. In addition, the influence of particle dimension on the stiffness of iron particle chains in MRPF has been investigated. A mathematical model for material removal is developed by utilizing normal finishing force analysis on active abrasives. Lastly, the finishing surface characteristics of gear profiles are examined using an optical profilometer, field emission scanning electron microscope (FESEM) and spectroscopic analysis. Finally, 92.68% improvement in the surface finish is observed.

A comparative study of fatigue behavior between S-shaped and involute POM gears

Abstract Applications of polymer gears have been gradually expanding from motion transmission to power transmission fields. Design of new gear teeth profiles plays a crucial role in meeting the requirement of replacing metallic gears with polymer ones. This work develops a computational framework consisting of tooth profile generation and fatigue life prediction of S-shaped and involute POM gears. The contact and bending fatigue performances of S-shaped and involute POM gears under oil lubrication are comparatively studied. It is found that the S-shaped profile reduces the tooth root stress by 25% and significantly improves the bending fatigue life by 10 times compared with the involute profile, especially under medium or high loading levels. Furthermore, the S-shaped gear with a single circular-filleted root curve benefits a 20% reduction of tooth root stress and a 30% increase in bending fatigue life compared to a double circular-filleted root curve S-shaped gear.

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.

Experimental Evaluation of the Feasibility and Reproducibility of a Novel Multi-step Forming Process for Cu-OFE Micro Gear Manufacturing Compared to Macro-scale Forming

The increasing miniaturization of technical systems leads to a rising demand for micro components. Miniaturized drive systems with geared micro parts or micro gears are applied in various industries. Due to the demand for high production volumes, productive and resource-efficient fabrication is of major importance. Compared to industrially applied manufacturing methods like cutting and metal powder injection, cold forming provides technological, economic and ecological potential for the mass production of micro gears. It enables high productivity and advantageous mechanical properties of the gears. In the macroscopic size range, forming processes are well established in series production of gears. However, in micro-scale, according to the current state of the art, cold forming of modules m<0.2mm is not possible due to size effects, high tool loads and handling problems. Within this contribution, a multi-stage forming process for the fabrication of micro gears with a module of m=0.1mm is introduced and compared with a conventional forming process on macro-scale with a module of m=1.0mm. The use of coil material provides potential for easy handling of the micro parts in a progressive die for the potential application in a high-speed press. In addition, the multi-stage forming achieves a grain alignment, which enhances the formation of the micro gear profile. Based on the comparison between micro- and macro-scale, the performance of the micro process is analyzed with regard to deviations of process force as well as gear quality and possible size effects are identified. The results prove the feasibility of the process chain and the achievement of equivalent part quality and reproducibility compared to macro-scale forming.

Effect on the measurement for gear involute profile caused by the error of probe position

Gear involute artefact (GIA) is a kind of measurement standard used to calibrate involute measuring instruments. To measure the profile deviation of GIA more accurately, the effect on the measurement of a gear involute profile caused by the error of the probe position is considered, the value of profile deviations would be affected and the measurement curve of profile deviations would be compressed or stretched along the roll path length, caused by the error of the probe position. Moreover, about 60% of the measurement errors are concentrated within the involute root of roll path length ρ= 0–5 mm. Based on the above research, an adjustment method for the probe position with a double roller-guide involute measuring instrument is proposed and measurement experiments are carried out using probes with probe radius ro = 0.5 mm, 1.5 mm and 2.5 mm. As the error of probe position e > 0, the curve of profile deviations at the root of the involute is biased towards the body. As the error of probe position e < 0, the profile deviation curve at the root of the involute except for the vicinity of base circle is biased towards the body, while the profile deviation curve near the base circle is biased out of the body caused by the interference of the transition surface, and the measurement curve of profile deviations at the root of the involute is significantly compressed. These errors decrease as the probe radius increases and the error of the probe position decreases. The difference of profile slope deviations and profile form deviations measured by the three probes after adjustment is 0.11 μm and 0.04 μm respectively, which indicates the feasibility of the probe adjustment method described above. The research supports the machining and measurement of a high-precision gear involute artefact.

Analysis of Flow Characteristics of Straight Conjugate Crescent Gear Pump at Variable Working Conditions

The working conditions (i.e., gas content, working pressure, and working temperature) of oil in the actual hydraulic system are always in a dynamic process, which will cause the basic properties of oil (i.e., density, absolute viscosity, and effective bulk modulus) to change and further generate changes of the flow characteristics in the straight conjugate crescent gear pump (SCCGP). However, no clear explanation has been given for the variation of the flow characteristics of the SCCGP with the working conditions in previous studies. This paper first expounds the gear pair profile equations used to draw the tooth profile curves and the equation applicable to the calculation of the geometric displacement at the given geometric parameters. Subsequently, the mathematical models of the oil’s basic properties are analyzed, and it is explained how the changing working conditions affect the changes in oil properties. After that, the paper designs an orthogonal test scheme and applies it to the study of the flow characteristics of the SCCGP for the first time to optimize its working conditions. Then, the paper performs simulations in commercial CFD numerical software, and the change laws of the internal flow field, flow pulsation, pressure pulsation, volumetric efficiency, and total efficiency of the SCCGP at different working conditions are analyzed in detail. The working conditions for decreasing pulse rate and improving total efficiency are summarized. Finally, for model validation purposes, the paper elucidates the construction process of a high-pressure positive displacement pump test system and carried out experimental research. The simulation and actual measurement results are compared under the same working conditions. The results validate the accuracy of the simulation model and the potential for future analytical studies.

Influence of Contact Surfaces’ Impact on the Gear Profile during Hobbing Process

This work is the result of research on the dynamic process that occurs during milling machining, namely, the influence of the contact surfaces’ impact on the gear and the hob and the influence of their displacements on the resulting profile of the tooth. An acquisition system was placed on the final elements of the milling machine chain to determine the torque moments and displacements during gear milling. The experimental analysis proves that the displacements are within admissible limits and have no major influence on the quality of the processing surfaces. A dynamic simulation of the hobbing process with the finite element method (ANSYS) was performed for a limited period of time, and the values of deformation, equivalent strain, and stress have been determined; the time at which the chips come off and the corresponding value of the equivalent stress that occurs at their break were determined based on the maximum distortion energy von Misses theory. It is required to simulate the entire hobbing process, even if it can be time-consuming to differentiate the influence of the dynamic behavior of the machine’s kinematic supplementary chains on the hob wear and the tooth profile. A modal analysis will be able to support the comparative study related to the obtained experimental data.

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.

Analytical modeling and experimental verification of the depth of subsurface heat-affected layer in gear profile grinding

Analytical modeling and experimental verification of the depth of subsurface heat-affected layer in gear profile grinding

Research on the profile modification of power skiving tool for internal gears

Power skiving provides an effective solution and considerable machining efficiency for the machining of internal gears. The tool profile design and the reusability after resharpening is crucial in gear manufacturing. In this paper, a novel method of tool profile correction based on the inverse error-complement of involute profile is proposed. Firstly, the mathematical model of involute cutter with rake angle and relief angle is established. The profile error relative to the target gear is calculated by the mathematical model. Then, the distribution of the gear profile errors is fitted by a fifth-order multinomial. The fitted multinomial function is attached to the cutter profile. The maximum theoretical error of the target gear profile is in 10e-7 mm order of magnitude through the calculation of fewer iterations. Finally, the distribution of the multinomial coefficients along the resharpening direction is obtained by linear programming. The result shows that the cutter designed by the proposed method possess almost negligible error to the gear profile and good repeatability of resharpening.

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.