Transverse Contact Ratio Research Articles

Improved analytical model for calculating mesh stiffness and transmission error of helical gears considering root curve: Theoretical analysis and experiments

A modified algorithm is proposed in this study to correct time-varying mesh stiffness (TVMS) and transmission error (TE) in helical gears considering tooth root transition curve. The parametric model of helical gear is established by using the meshing principle of helical gear and hob in order to obtain the accurate tooth root transition curve. Moreover, a TE test rig is set up to compare the theoretical and experimental results. The results of the improved algorithm are very similar to the experimental results compared to the traditional model, and the relative error of TE is also small. In this study, the effect of multi-parameters (e.g. pressure angle, meshing position, transverse contact ratio, and overlap ratio) on TE are investigated by comparing theory with experimental analysis. Through the analysis of parameters, it has been established that an elevation in pressure angle leads to a decrease in TE, followed by an increase. In the event that the meshing position is situated closer to the nodal point, the TE slightly decreases. Moreover, as the transverse contact ratio and overlap ratio increase, the TVMS increases, while the TE decreases. When the overlap ratio approaches an odd multiple of 0.5, the TE displays a significant peak.

A new method for the single mesh stiffness calculation of helical gear considering slice coupling effect

In this paper, a new method is proposed, calculating the single mesh stiffness by cobining the silce principle and contact line analysis. Firstly, this method obtains the length of the contact line of the helical gear through contact line analysis. Secondly, along the tooth face width, the teeth of helical gear is separated into a series of equivalent thin teeth of spur gear. Thirdly, in order to consider the slice coupling effect, a weighting factor is introduced to modify the stiffness of the thin tooth itself according to the contribution of the thin tooth to the overall stiffness, which is distributed in parabola. Finally, by integrating the equivalent thin teeth of the spur gear along the contact line, the single mesh stiffness is calculated. The investigation shows that the influence of the slice coupling effect on single meshing stiffness related to the relationship between the value of transverse contact ratio and overlap ratio. When transverse contact ratio is greater, the influence of the slice coupling effect on the single mesh stiffness is very small an can be ignored, and when the overlap ratio is larger, which is significant.

Research on gear mesh stiffness of helical gear based on combining contact line analysis method

A new analytical model of mesh stiffness of helical gears is proposed using a combination of contact line analysis with slice method, which can well predict the dimensionless time-varying mesh stiffness. Based on the derived formula, the influence of gear basic parameters on mesh stiffness fluctuation coefficient is studied. The simulation shows that the influence of the overlap ratio on the mesh stiffness fluctuation coefficient is greater than that of the transverse contact ratio. By designing reasonable helix angle and tooth width, the overlap ratio is close to an integer, and then the mesh stiffness fluctuation coefficient can obtain a local minimum value, which is an effective way to reduce the vibration and noise of gear system.

Влияние параметров исходного контура на коэффициент перекрытия эвольвентных передач

Previously, GOST 13755–81 specified a unified standard initial contour, i.e. the tool rack teeth profile. The rolling method was dominant; a family of surfaces was enveloped by tools in the process of form-generation of involute gears. The effectiveness of such standardization was enhanced by the widespread large-scale and mass production. Recently, changes in the nature and technology of production have led to a significant expansion of possible standard initial contours and their parameters. In many cases, manufacturers reject the enveloping method as well as the involute engagement in general. However, the contour parameters are still the foundation of the standard Cylindrical Involute External Gearing. Calculation of Geometry. Thus, even with the complete rejection of the initial contour, its effect on the gearing quality, namely on the contact ratio, remains unchanged. The transverse contact ratio is the ratio of the length of the engagement line segment to the normal pitch equal to the pitch of the base circumference of the pinion. In this study, dependencies of the transverse contact ratio on the number of pinion teeth, speed ratio, pressure angle and addendum coefficient are obtained. The results obtained can be used to select the initial contour parameters that provide the required traverse contact ratio, making the enveloping method and the initial contour (tool rack) redundant. It is established that the optimum transverse contact ratio equals 2. However, to compensate for manufacturing and assembly errors, the values of the number of pinion teeth, speed ratio, pressure angle and addendum coefficient should be selected so as to slightly exceed the above-mentioned transverse contact ratio.

An approach for design optimization of helical gear pair with balanced specific sliding and modified tooth profile

A unique perspective in design optimization of helical gear pair has been emerged and presented in this article. Specific sliding needs to be balanced for enhancing wear and scuffing resistance of helical gears. Optimized modification in tooth profile has immense benefits in gear operations. Effect of profile shift and specific sliding on design optimization of helical gear pair have been studied and found to be beneficial of great importance. Preventing undercutting, balancing of wear and bending fatigue strength and centre distance adjustment are some of the advantages of profile tooth modifications. Real-coded genetic algorithm (RCGA) has been used to attain minimum volume of helical gear pair. Profile shift coefficients for gear and pinion have been included as design variables along with mostly used generic design variables, module, face width and number of teeth on pinion. Specific sliding, transverse contact ratio and face width constraint along with other strength requirements are the design constraints. The optimal design solutions obtained with and without profile shift are recorded and compared with commercially used software for validation. 3D computer-aided design (CAD) models have been developed by using the optimized results obtained from RCGA and commercially used software. These CAD models are used for performing finite element analysis (FEA) on the helical gear set for analyzing the stress developed in the gear pair. The developed stress in the helical gear pair is found to be well within the allowable stress limits for the gear pair.

Mathematical model of energy efficiency in internal spur gears

The impact of geometric parameters of teeth and lubricating oils to the efficiency of involute internal spur gears, when the transverse contact ratio is 2 < ?? ?3, has been analyzed in this paper. The mathematical model and computer program for determining the current and the effective value of the efficiency have been developed. The influence of the character of load distribution and energy losses due to heating effects during the meshing period is included in the factor of load distribution. The results of computer simulation are given in the form of a diagram of the current values of the efficiency during the meshing period. Also, the values of effective efficiency for the considered cylindrical gear pairs have been calculated.

A New Model for the Single Mesh Stiffness Calculation of Helical Gears Using the Slicing Principle

In this paper, two types of helical gear pairs are defined based on the relationship between the transverse contact ratio and overlap contact ratio. An improved analytical model using the slicing principle is proposed for the calculation of the single mesh stiffness of helical gears. The main improvement in this model against traditional models, that assume no coupling effects between neighbouring sliced tooth pieces, is that a parabola-like weighting factor distribution along the tooth face width is assigned on the sliced tooth pieces for the consideration of the coupling effect. This allows each tooth piece to be associated with a weighting factor and therefore have a different influence on the total mesh stiffness. The calculation results for the single mesh stiffness of two types of helical gear pairs obtained from various methods are compared and discussed. It was found that, compared with the traditional analytical method, the proposed analytical method yields more accurate results in terms of the shape of the single mesh stiffness curve and maximum value of the single mesh stiffness, especially for helical gears with a wide face and large helix angle.

Development of the positive engagement continuously variable transmission design with the application of graph theory

Various designs of CVTs were explored from two points of view: analysis of the design itself and analysis of the methods and techniques, used in the process of creation of the CVT. To solve the problem of finding new designs of mechanisms, the graph transformation method was developed. It includes the following heuristic techniques: a decrease in the number of graph nodes at simplification of a design or to remove non-essential elements; a change in location of graph edges and comparison of a new location of edges and necessary changes in a design; a search for all possible options for location of the parts that correspond to the same graph; introduction to the graph of the nodes, corresponding to the property, for implementation of which a part (a node) can be added to a design or, vice versa, can be removed from a design, and the property it implemented will be implemented by other parts. The original design of the CVT, in which load is transmitted by gear engagement without using friction disks and flexible sections, was developed. A continuously variable of transition ratio is ensured by a gear wheel with a variable pitch. The gear wheel consists of three gear sectors, two of which are involved in meshing. The third unloaded sector performs a turn with angular velocity that is higher than velocity of rotation of the output shaft until it takes a position before coming into engagement. The CVT does not have such drawback as fluctuations of transmission ratio. Based on the analysis of the bond-graphs, efficiency was determined, which depending on velocity of the output shaft is in the range of 95…97 %. The disadvantage of the developed design is an insufficiently wide gear ratio range R=1.5…1.6, which is limited by transverse contact ratio e α

Helix Angle Effect on the Helical Gear Load Carrying Capacity

The aim of this study is to investigate the helix angle effect on the helical gear load carrying capacity, including the bending and contact load carrying capacity. During the simulation, the transverse contact ratio is calculated with respect to the constant pressure angle. By changing the helix angle, both the overlap contact ratio and total contact ratio are calculated and simulated. The bending stress and contact stress of a helical gear are calculated and simulated with respect to the helix angle. Solid (CAD) modelling of a pinion gear was obtained using SOLIDWORKS software. The analytically obtained results and finite elements method results are compared. It is observed that increasing the helix angle causes an increase of the contact ratio of the helical gear. Furthermore, increasing the contact ratio reduces the bending stress and contact stress of the helical gear. However, with a constant transverse contact ratio, it is possible to improve the total contact ratio depending on the helix angle. It is concluded that a higher helix angle increases the helical gear bending and contact load carrying capacity.

Utilización analítica del Contorno de bloqueo restringido por las tolerancias en engranajes cilíndricos exteriores

Los Contornos de Bloqueo muestran al diseñador de engranajes el conjunto de valores de los coeficientes de corrección, que cumple con las limitaciones geométricas de socavado, factor de recubrimiento, interferencia y espesor del diente en el diámetro exterior. En este trabajo se perfecciona este método, considerando para su construcción las tolerancias de la distancia entre centros, el diámetro exterior y la holgura lateral entre los perfiles de los dientes. De esta manera se garantiza el cumplimiento de las restricciones mencionadas, aun con las combinaciones más desfavorables de las tolerancias, lo que mejora la calidad del diseño de estas transmisiones.

Comparative study of stress analysis of gears with different helix angle using the ISO 6336 standard and tooth contact analysis methods

With increasing demand for high speed and high power density gear applications, the need to optimise gears for minimum stress, noise and vibration becomes increasingly important. ISO 6336 contact and bending stress analysis are used to determine the surface load capacity and tooth bending strength but dates back to 1956 and although it is constantly being updated, a review of its performance is sensible. Methods to optimise gear performance include the selection of helix angle and tooth depth to optimise overlap ratio and transverse contact ratio and thus the performance of ISO 6336 and tooth contact analysis methods requires confirmation. This paper reviews the contact and bending stress predicted with four involute gear geometries and proposes recommendations for stress calculations, including a modification to contact ratio factor Zɛ which is used to predict contact stress and revisions to form factor YF and helix angle factor Yβ which are cited to evaluate bending stress. The results suggest that there are some significant deviations in predicted bending and contact stress values between proposal methods and original ISO standard. However, before the ISO standard is changed, the paper recommends that allowable stress numbers published in ISO 6336-5 are reviewed because the mechanisms that initiate bending and contact fatigue have also changed and these require updating.

Simulation Investigations of Influence of Tooth Depth Coefficient on Dynamic Phenomena in Toothed Gear

The increase of transverse contact ratio (εα) value usually allows reducing general level of gear vibroactivity. Article put to the test influence of coefficient εα value on dynamic forces in mesh zone with use of dynamic model of toothed gear. From theoretical point of view, the optimum value of transverse contact ratio is equal 2, what mean, that in mesh are always two pair of teeth. Obtainment such value of coefficient εα requires another construction of toothed wheels – wheels with HCR (High Contact Ratio) profile teeth. On result of occurrence of different deviations in toothed gears, as well as the dynamic phenomena, obtainment of continuous two-pair cooperation of gear pair is impossible and when this necessary is, solutions with near or exceed optimum value of coefficient are applied.

Load sharing based maximum fillet stress analysis of asymmetric helical gears designed through direct design — A parametric study

Direct gear design method is one of the most effective gear design approaches, which provides the high performance gears without concern for any predefined tooling parameters. In the present work, the load sharing ratio (LSR) and the corresponding LSR based maximum fillet stresses in terms of non-dimensional stress number have been estimated for the conventionally designed and direct designed symmetric helical gears. It is also extended for direct designed asymmetric helical gears. The area of existence procedure is developed and using the same, the possible existence of design solutions for a drive of required constant transverse contact ratio and different addendum pressure angles is chosen to develop the rack dimensions. Finally, the influence of a few parameters such as gear ratio, transverse contact ratio, top land thickness coefficient and the pinion teeth number on area of existence, load sharing ratio and the corresponding values of non-dimensional stress number in the asymmetric helical gear drives designed through the direct design method has been carried out and discussed.

10.5937/vojtehg62-5393 = Influence of the cylindrical gear pairs parameters to the transverse contact ratio

In the field of transmission and power transformation from the power engine to the working machine, gear pairs, are mostly used in mechanical engineering due to their compactness of the structure, high reliability and capacity. One way of improving the performance characteristics of gear pairs, and thus the gear transmitters, is to increase the number of simultaneously meshed pairs of teeth, or increasing the transverse contact ratio. To this end, this paper analyzes in detail the partial and simultaneous influence of the number of teeth and tooth profile shapes, moving through shifting coefficient and pressure angle, to the number of simultaneously meshed pairs of teeth. The obtained results allow us to define the optimum parameters of cylindrical gear pairs, in terms of the size of the transverse contact ratio.

Tooth-root stress calculation of high transverse contact ratio spur and helical gears

In this paper, a non-uniform model of load distribution along the line of contact of spur and helical gears, obtained from the minimum elastic potential criterion, has been used, combined with the equations of the linear elasticity, to evaluate the tooth-root stress of high transverse contact ratio gears. The values of both critical stress and load conditions have been obtained and a complete analysis of the tooth bending strength has been carried out. As the load per unit of length at any point of the line of contact and any position of the meshing cycle has been described by a very simple equation, a complete study of the location and the value of the tooth-root stress has been carried out. From this study, a recommendation for the calculation of the bending load capacity of high transverse contact ratio spur and helical gears is proposed.

Contact stress calculation of high transverse contact ratio spur and helical gear teeth

For contact stress calculations of spur and helical gears, the Hertz equation can be used in combination with a model of load distribution along the line of contact. This load distribution is not uniform due to the changing rigidity of the pair of teeth along the path of contact and has decisive influence on the location and the value of the critical contact stress to consider for calculations. Moreover, the load distribution can be highly influenced for non-standard gearing conditions as the presence of undercut, enlarged tooth addendum or reduced center distance, often present in high transverse contact ratio gears. In this paper, a new calculation method of the contact stress of spur and helical gears with transverse contact ratio greater than 2 is developed. It is based on the Hertz equation and an enhanced model of load distribution, obtained from the minimum elastic potential criterion, suitable for non-standard gearing conditions. A complete study on the critical load conditions and the value of the critical contact stress has been carried out. As a result, a recommendation for pitting load capacity calculations is proposed.

Effects of Construction Changes in the Teeth of a Gear Transmission on Acoustic Properties

This paper presents results of experimental research on the acoustic properties of gear wheels with high-profile teeth with differentiated tooth height. Those results showed that gear transmissions with high-profile teeth have the best acoustic properties, with the value of the transverse contact ratio εα ≈ 2.0. They also showed that a reduction in tooth height, and thereby in contact ratio, increased the sound pressure level.

Stress calculation for plastic helical gears under a real transverse contact ratio

Abstract Calculation methods of plastic helical gears with a real transverse contact ratio have never been studied in depth. The methods used in some references assume a uniform load distribution along the line of contact or don’t take into consideration the effect of premature mesh between one or more pair of teeth during engagement, which is not in good agreement with numerical results. This paper is focused on analysis of plastic helical gears. The nonuniform load distribution obtained from the proposed mathematical model is based on the calculation of the real contact ratio representing the real number of pairs of teeth in contact. This model also leads to the distribution of the tooth bending stress and the contact stress along the area of contact.

Calculation of Tooth Bending Strength and Surface Durability of High Transverse Contact Ratio Spur and Helical Gear Drives

Calculation methods of spur and helical gear drives with a transverse contact ratio higher than 2, accurate enough for preliminary designs or standardization purposes, are not available in technical literature. If the load is assumed to be uniformly distributed along the line of contact, simple equations given by the linear theory of elasticity and the Hertzian contact model are not in good agreement with experimental results. However, these simple equations may work quite accurately with a more realistic model of load distribution accounting the changing rigidity of the pair of teeth along the path of contact. In this paper a nonuniform model of load distribution along the line of contact, obtained from the minimum elastic potential criterion, has been considered for stress calculation of spur and helical gear drives with transverse contact ratio between 2 and 2.5. The determinant load conditions have been calculated and the nominal contact stress and the nominal tooth-root stress have been computed.

Simultaneous Optimization of Tooth Flank Form of Involute Helical Gears in Terms of Both Vibration and Load Carrying Capacity. 1st Report. Fundamentals for Optimization.

The transmission gear noise of passenger car becomes usually of problem during cruising condition, i. e. under light loading condition. Three dimensional tooth flank form must therefore be optimized in designing to satisfy not only strength reliability under the maximum loading, but also low vibrational characteristics under the light loading. For designing tooth flank forms to meet those requirements, it should also be taken into consideration that the amount of misalignment of gear shafts under the light loading is quite different from that under the maximum loading, because the stiffness of gear support of automotive transmission is considerably low. In this report, the influence of profile and lead form on vibrational excitation under the light loading and that on tooth root stress, Hertz stress and flash temperature under the maximum loading are clarified: There are local optimum conditions, where vibrational excitation is low, as function of combination of the amount of tooth profile modification, of lead crowning, and of lead inclination. In a definite local optimum condition, the tooth flank form that brings the minimum average rotational displacement between driving and driven gears makes vibrational excitation minimum. The range of the amount of lead inclination that satisfies the requirements for load carrying capacity and vibrational limit becomes smaller with increasing the amount of profile modification, if transverse geometrical contact ratio and overlap ratio are both over 1. The fundamentals for designing the optimum tooth flank form in terms of both vibration and load carrying capacity are shown.