Computerized Simulation of Manufacturing Errors in Cylindrical Spur Gears and Their Compensation Through Flank Modifications

A modified surface topology of involute helical gears is proposed. The proposed new topology provides an involute zone in the central area of gear tooth surface smoothly connected to crowned zones at the top, bottom, front, and back sides. The involute zone allows line contact in case of aligned gear drives while the crowned zones provide a localized bearing contact and low levels of transmission errors in case of misaligned gear drives. In many cases, non-standard cutting hobs are required to provide chamfers to gear tooth surfaces for avoiding of edge contact when misalignments are presented, but the chamfers are not connected smoothly, the magnitude of required chamfers is not determined analytically, and consequently, severe contact and linear functions of transmission errors may appear. The new topology is based on generation of gear tooth surfaces by an involute grinding worm, Modified roll is applied to the involved motions of generation in order to provide profile and longitudinal crowned zones with smooth connections to the involute zone. Tooth contact analysis and stress analysis by the finite element method has allowed the advantages of the proposed topology to be verified. Finite element models with three pairs of contacting teeth are considered. A function of transmission errors with a low level is obtained as a precondition of reduction of noise and vibration. An uniform transition of load is guaranteed between the adjacent pairs of teeth avoiding areas of severe contact stresses.© 2005 ASME

There are two types of curvilinear cylindrical gear drives: those with point contact and those with line contact. In this paper, a mathematical model, tooth contact analysis and stress analysis are investigated for curvilinear cylindrical gears with line contact. Here, fixed-setting face-milling cutters are used for the generation of the tooth surfaces of the gears and their matching gears. Based on the theory of gearing, a mathematical model of curvilinear gears is derived, and the instantaneous contact lines on the gear tooth surfaces are examined under ideal assembly conditions. Meanwhile, the instantaneous contact points on the gear tooth surfaces and transmission errors under assembly error conditions are investigated by a tooth contact analysis algorithm. Finally, contact and bending stresses for curvilinear cylindrical gears with tip relief and without tip relief are explored by the finite element method. This research result shows that (1) under ideal assembly conditions, the length of the instantaneous contact lines is greater than the tooth face width, but the effect is very weak. (2) In terms of the transmission errors, crossing shaft angle errors are more sensitive than intersecting shaft angle errors; axial displacement errors are more sensitive than centre distance errors. (3) Severe contact stresses caused by edge contact are eliminated by the tip relief. (4) The maximum bending stress at every contact position is always distributed in both ends of the gear tooth surfaces.

The modern theory of gearing is based on the design of gear drives according to the size and location of the contact pattern as well as the maximum level and type of the obtained function of transmission errors. These factors may reduce the life and the endurance of the gear drive for power transmission or even cause its failure. However, there is a factor that affects the quality perception of the performance of the gear drive and deserves to be taken into account in the design stage of the gear drive. It is called gear whine noise and it depends, among other factors, on the type and maximum level of the function of transmission errors. In this paper, an approach for the determination of the amplitude spectrum of different types and levels of functions of transmission errors is proposed in order to predict one of the sources of the radiated gear whine noise. The evaluation of the amplitude spectrum of the function of transmission errors will be made in terms of Fourier series, using the discrete Fourier transform. Therefore, the influence of the type and level of the function of transmission errors on the amplitude spectrum of their Fourier transforms will be investigated in order to find the suitable type of the function of transmission errors for reduction of the radiated gear whine noise and therefore for designing quiet gears.

Spur gears are widely used in the power transmission mechanism of several machines. Due to the transmitted torque, spur gears experience high stresses which could cause gear tooth failure by surface pitting or root fracture. Tip relief and other gear profile modification have been considered for reducing the induced stresses in the gear tooth. In this work, the influence of tip relief on stresses on a pair of identical spur gear was analyzed using commercial FEA software ANSYS, and formulae for estimating contact and bending stresses were derived. Three cases of gear sets were analyzed; a non-modified pair and another two sets with linear and parabolic tip relief profiles. The non-modified gear set frictionless contact stress was validated against the calculated AGMA pitting resistance, Hertzian contact stress and a reported contact stress value in the literature. The four methods agreed well with each other. Similarly, bending stress was also compared with the AGMA bending strength and Lewis bending stress for validation. Then, friction coefficient was varied from 0.0 to 0.3 with increment of 0.1. The gear contact stress increased up to 11% relative to the frictionless case, whereas bending stress decreased by 6%. Linear tip relief modification was carried out for increasing normalised tip relief values of 0.25 to 1.0 with increment of 0.25. The gear frictionless contact and bending stresses decreased by a maximum of 4% and 2%, respectively. Frictional contact stress increased by up to 7.1% and the bending stress is almost identical with the frictionless case. Parabolic tip relief was also carried out with similar normalised tip relief values. Frictionless contact stress decreased by 5% while frictional contact stress increased by up to 11.5% and the bending stress is also almost identical with the frictionless case. Finally, four formulae were introduced for estimating the contact and bending stresses for a tip modified spur gear.

Computerized generation and meshing simulation of face gear drives manufactured by circular cutters

The damage that occurs on the teeth flank of cylindrical gears is a complex phenomenon and depends on many factors. The most common cause for these damages is the high contact stress of the meshing gears. Although this contact pressure, by itself, cannot be a criterion for determining the durability of the gears, a good correlation has been found between the contact or Hertz pressure and the damage that occurs on the tooth flank. This paper analyzes the influence of the pressure angle α on the contact stress. Analytical calculation according to the ISO 6336-2:2006 and finite element method (FEM) was used for the analysis. Four cases were analyzed with a change in the pressure angle, i.e., cylindrical spur gears and pressure angles of 17.5°, 20°, 22.5° and 25°. In the results, it was noted that by increasing the pressure angle, the contact stress decreases. It can also be concluded that by increasing the pressure angle, the difference in the results between the analytical and FEM analysis, also increases.

To control the meshing performance of hypoid gear drive, current local conjugate theory and local synthesis method establish a relationship between the relative motion and local geometrical properties of the mating surfaces at one reference point. Theoretically, both local conjugate theory and local synthesis method cannot ensure the contact performance on the entire tooth surface resulting in uncontrolled contact pattern and undesirable transmission error. A global synthesis approach consisting of a machine-tool settings calculation model, a tooth contact analysis model, a meshing quality assessment function, and a swarm intelligence algorithm are proposed. The direction, area, and shape of the contact pattern and the amplitude of the intersection of two adjacent transmission error curves are taken as the evaluation indexes. Three curvatures of the design pitch cone of the pinion are taken as the control variables. A global selection space is then established within the reasonable range of the curvatures. An improved multiple population genetic algorithm is employed to find the optimal set of the curvatures to achieve the target values of the evaluation indexes. To avoid the edge contact and corner contact condition, a feasible region is defined on tooth surface and the contact patterns on the gear and pinion surfaces are all confined within the area. The optimized contact patterns obtained by the loaded tooth contact analysis method are similar to those obtained by the proposed approach, thus demonstrating the effectiveness of this methodology. The main novelty of this approach is to translate the derivation of the mathematical relation between the curvatures of the mating surfaces and the contact properties to solving a multi-objective optimization problem of the meshing quality indexes by the intelligence algorithm.

In this paper the problem of the failure of gear in speed reduction gearbox developed by Laxmi Hydraulic Pump (LHP) Pvt Ltd, Solapur is resolved by replacing the existing material by a material SAE8620. The Gear and Pinion using AGMA and FEM analysis Method. In the gear design, the bending stress and surface strength of the gear tooth are considered to be one of the main factors to the failure of the gear. In this paper bending and contact stresses are calculated by using analytical method as well as Finite Element Method. To estimate bending stress modified Lewis beam strength method is used as well as AGMA method. Software is used to generate the 3-D solid model of helical gear. Ansys software package is used to analyze the bending stress. Contact stresses are calculated using modified AGMA contact stress method. Ansys software package is used to analyze the contact stress and bending stress. The main objective of this study has to investigate the stresses induced by gear tooth profile. Finally, the results obtained by theoretical analysis and Finite Element Analysis are compared to check the correctness. Estimated stress is used to calculate module of gear which can withstand easily at given loading conditions. A conclusion has arrived on the material which is best suited for helical gear based on the results.

This article focuses on improving the meshing characteristics of a complementary curvilinear gear set generated by complementary rack cutters. Firstly, the mathematical model of a complementary curvilinear gear set was developed based on the theory of gearing. Tooth contact analysis (TCA) was conducted to explore the influences of assembly errors on the transmission error (TE) and contact pattern. A stable point contact was successfully introduced to replace the original line contact in the improved complementary curvilinear gear set, resulting in less sensitivity to assembly errors. Finally, finite element analysis (FEA) was also utilized to calculate contact stress of the original and improved complementary curvilinear gear set.

Loaded tooth contact analysis (LTCA), deformation, and stress calculations of three types of thin-rimmed spur gears with inclined webs at the left side, the center, and the right side of the tooth are conducted in this paper with finite element methods (FEM) combined with a mathematical programming method when these gears are engaged with a solid mating gear. The contact stresses on the tooth surfaces, the bending stresses at the tooth roots, and the joint stresses of the rim and the web are analyzed as the web position and the web angle are changed. It was found that the web position and the web angle of the thin-rimmed gears have a significant effect on the tooth contact stresses, the root bending stresses, and the joint stresses. The maximum contact stress of the right web gear becomes very small when the web is inclined from 0 to 30 deg, and it increases again when the web angles are changed from 45 to 60 deg. This interesting relationship between the web angle and the tooth contact stresses of thin-rimmed right inclined web gears is used to absorb the misalignment errors in the gears and improve the partial tooth contact pattern resulting from the misalignment errors. A calculation example is provided to illustrate how to improve the partial tooth contact patterns in gears with misalignment errors.

Tooth root bending stress and surface contact stress are two major determinants of the load carrying capacity of gear drives. Generally, a high contact ratio has the potential to enhance the gear strength. In this paper, the basic procedures and methods for constructing a high-contact-ratio (HCR) internal spur gear pair with the arc path of the contact are presented. The effects of design and modification parameters such as deflection angle, modification angle, and addendum coefficient on gear drive are analyzed in detail based on 2D finite-element models (FEMs). A comparison of experimental and finite-element analyses (FEA) of the bending stress, contact stress, and contact ratio between HCR and involute internal spur gear drives was conducted to demonstrate the advantages of the HCR gear drive in terms of load capacity. The results show that appropriately increasing the deflection angle and addendum coefficient is beneficial for reducing bending and contact stresses. It was also observed that the bending and contact stresses of a HCR pinion are lower than those of an involute one. Moreover, the contact ratio increases with input torque.

7 - Detailed Gear Stressing

Undercutting and contact characteristics of longitudinal cycloidal spur gears generated by the dual face-hobbing method

Among all the common mechanical transmission elements, gears still playing the most dominant role especially in the heavy duty works offering extraordinary performance under extreme conditions and that the cause behind the extensive researches concentrating on the enhancement of its durability to do its job as well as possible. Contact stress distribution within the teeth domain is considered as one of the most effective parameters characterizing gear life, performance, efficiency, and application so that it has been well sought for formal gear profiles and paid a lot of attention for moderate tooth shapes. The aim of this work is to investigate the effect of pressure angle, speed ratio, and correction factor on the maximum contact and bending stress value and principal stresses distribution for symmetric and asymmetric spur gear. The analytical investigation adopted Hertz equations to find the contact stress value, distribution, and the contact zone width while the numerical part depends on Ansys software version 15, as a FE solver with Lagrange and penalty contact algorithm. The most fruitful points to be noticed are that the increasing of pressure angle and speed ratio trends to minimize all the induced stresses for the classical gears and the altered teeth shape with larger loaded side pressure angle than the unloaded side one behave better than the symmetric teeth concerning the stress reduction.
 

In order to reduce the sensitivity of straight bevel gear drives to misalignment, a new geometry of such gear drives is proposed in longitudinal direction. Point contact instead of line contact of tooth surfaces is achieved by longitudinal crowning of pinion tooth surface. The tooth surface modeling and tooth contact analysis (TCA) of straight bevel gear drives have been established. TCA program of a pair of straight bevel gears was performed in MATLAB and tooth bearing contact and transmission errors were obtained.