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Improving the mechanical efficiency is not the most important objective in the design of wind turbine gearboxes since the available wind energy is abundant and costless. The most important criteria for dimensioning the gearbox are the fatigue strength—bending and pitting—, noise emission, vibrations, and maintenance requirements. Nevertheless, mechanical losses increase the lubricant temperature and induce thermal stresses, which increases wear and cracking risk. This means that friction losses should be reduced as much as possible, but always regarding the contact and tooth-root stress levels, as well as the other operating parameters which should be kept for ensuring the required operating conditions.In this paper, a study on the variation of the friction losses with the tooth shift coefficients is presented. All the other geometrical parameters—number of teeth, tooth height, pressure angle, helix angle, face width, and center distance—are unalterable, since all of them have been chosen according to more important design requirements. In addition, to keep the contact and tooth-root stress levels, the shift coefficients of the sun, planets and ring are calculated in such a way that the transverse contact ratios are kept, and therefore the critical load points for bending and pitting are also unchanged. The radial clearance is also kept in order to allow the proper evacuation of the lubricant. Finally, all the geometrical constraints (undercut, pointing, root interference, secondary interference, backlash) are also imposed. With all these restrictions, the optimal shift coefficients for all the gears are calculated to minimize the friction losses.

The loaded tooth deflections, manufacturing deviations, and assembly errors on spur gears induce a delay on the driven gear respect to the driving gear, which is called loaded transmission error. This delay causes an earlier start of contact outside the pressure line, resulting in a mesh-in impact, which is an important source of noise and vibrations. To avoid or mitigate this mesh-in impact, profile modifications on the driven tooth tip are frequently used. The additional gap due to the profile relief delays the effective start of contact, and a suitable amount of relief sets the mesh-in point back to the theoretical inner point of contact, which prevents the mesh-in impact and ensures a smooth contact. However, as the loaded transmission error depends on the transmitted load, a specific amount of relief will be suitable only for a specific input torque. For higher transmitted load mesh-in impact will occur again, while for lower transmitted load the effective contact ratio will be reduced due to an excessive delay of the start of contact. Therefore, under variable load conditions it is not possible to adjust the start of contact to the theoretical inner point of contact, so mesh-in impact, smooth contact and reduction of contact ratio may alternatively occur. In this paper, a simple analytical model for the meshing stiffness, load sharing ratio and transmission error is presented for spur gears with profile modification under non nominal -or variable- load conditions.

The meshing stiffness, load sharing ratio, and quasi-static transmission error of spur gear pairs are related to each other and are dependent on tooth geometry. In addition, profile modifications influence the contact conditions, because modified profile points may only contact under load conditions and owing to load-induced elastic deflections. However, this is not the case under no-load conditions, wherein a gap would exist between the teeth owing to the contact of other tooth pairs at unmodified points. In this paper, the contact conditions of modified teeth under load have been studied, and the influence of the amount, length, and shape of the profile modification on the load sharing and transmission error has been described. A simple, analytic formulation of all these relations is provided, based on a previously developed approximate equation for the meshing stiffness, which provides accurate results. The simplicity of the obtained equations facilitates the study of the influence of the profile modification on the load sharing and transmission error, and the required profile modification for predesigned functions of load sharing ratio or quasi-static transmission error was also obtained.

The start of contact between loaded involute gear teeth occurs before reaching the theoretical inner point of contact due to the load-induced deflections of previous tooth pairs in contact. This sooner contact occurs outside the pressure line and produces a shock between the driving tooth root and the driven tooth tip, which induces noise, vibrations and dynamic load. To avoid these undesirable effects profile modifications are often used, which through a suitable tip relief at the driven tooth delay the actual start of contact until locate it at the theoretical inner point of contact. However, the length and shape of profile modification have also influence on the curves of load sharing and quasi-static transmission error. Specifically, long tip relieves, beyond the interval of minimum tooth pair contact, which are unsuitable for standard contact ratio spur gears, may reduce drastically the load at the inner points of the path of contact of high contact ratio gears, though a peak of load arises at the outer interval of two pair tooth contact. Since the determinant contact stresses are usually located at the inner points of the contact interval and the determinant tooth-root stresses at the outer ones, long tip relieves can be used for balancing both determinant stresses and improving the load capacity.

An enhanced model for strength calculations of internal spur gear pairs is presented. The load sharing among couples of teeth in simultaneous contact is determined from the meshing stiffness, whose value has been calculated at any point of the path of contact, considering bending, compressive, shear and Hertzian deflections. This curve of variation of the meshing stiffness has been expressed by means of an approximate accurate equation, as a function of the effective contact ratio exclusively, allowing to express the load at each couple of teeth in a closed form. The values of the bending and pitting stresses along the path of contact have been accordingly computed, as well as their nominal values and their corresponding determinant load conditions. An adapted formulation of the meshing stiffness curve for non-standard tooth dimensions–including reduced tooth height, frequently required to avoid secondary interference–has been also developed. Calculation methods for the nominal stresses for these non-standard internal gears have been presented as well.

In this paper, the meshing stiffness of spur gear pairs, considering both global tooth deflections and local contact deflections, is evaluated at any point of the path of contact and approximated by an analytical, simple function. With this function, the load sharing ratio is calculated and compared with previous results obtained from the hypothesis of minimum elastic potential energy (MEPE model), considering the tooth deflections, but neglecting the hertzian deflections. Critical bending and contact stresses from both models are also compared both for standard and high contact ratio spur gears.

The geometry of the gear teeth, as well as that of the tooth contact, is complex and not easy to simulate. Thus, computer aided techniques, as finite element analyses or complex tooth contact analyses, should be applied for strength calculations or simulations of contact conditions. These techniques provide reliable results, but usually requires long time preparation and high computational cost. Simple models of the linear theory of elasticity or theoretical, conjugate contact conditions may be not sufficiently accurate for modern gears whose power-capacity to size ratio is continuously increasing. However, these simple models, even simpler approximate equations describing them, could provide preliminary information on the influence of the geometrical parameters on the design factors, and therefore optimization criteria from different points of view. In this work, an approximate equation of the meshing stiffness of spur pairs is applied to the analysis of the load sharing ratio, load capacity, and quasi static transmission error of spur gears. The study extends to both standard and high contact ratio spur gears and considers gear teeth with profile modifications.

A simple, analytical model for load sharing and quasi-static transmission error of spur gears with non-standard tooth height is presented. It is based on the minimum elastic potential energy principium and considers the influence of profile modifications and tooth deflections. In addition, is suitable for different tooth height on pinion and wheel, as well as for standard and high-contact-ratio spur pairs.

Profile modifications are often used on gear teeth to ensure a smooth start of contact, avoiding noise, vibrations and dynamic loads induced by a sooner contact outside the pressure line which occurs due to unavoidable tooth deflections under load. A suitable tip relief on the driven gear teeth delays the start of contact, locating it at the appropriate point of the pressure line. However, the amount of relief depends on the deflection, and consequently on the load, which means that it should be calculated for a specific load, obviously coincident with the nominal load, which may be non-suitable for non-nominal, overloaded or underloaded operating conditions. In this paper, the behaviour of gear teeth with profile modifications under non-nominal loading conditions is studied. The load transfer between couples of teeth is simulated and described by analytic, approximate equations. The influence of the non-nominal load on the effective contact ratio is also presented.

In this paper the determinant load conditions for bending and pitting calculations, for involute, internal spur gears have been established. Calculations are based on a new model of load distribution, which takes into account the changing rigidity of the pair of teeth along the path of contact. This model has been obtained from the minimum elastic potential criterion, resulting in non-uniform load sharing among spur tooth pairs in simultaneous contact. As a result, the theoretical values of both bending and pitting nominal stresses are given, which may be suitable for standardization purposes.

A comparison among different approaches of the meshing stiffness of spur gears has been carried out. The influence of each one on the load sharing, and subsequently on the determinant tooth-root and contact stresses, has also been studied. Equations for the evolution of the meshing stiffness along the path of contact according to all these formulations are also provided.

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.

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.

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 fatigue tooth-root stress. The critical value of the stress and the critical 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 analytic equation, a complete study of the location and the value of the tooth-root bending stress has been carried out. From this study, a recommendation for the calculation of the bending load capacity of spur and helical gears is proposed.

The presence of undercut at the tooth root, non-equal addendum on pinion and wheel, non-standard tooth height or non-standard center distance may have decisive influence on the load distribution along the line of contact of spur and helical gear teeth. The curve of variation of the meshing stiffness along the path of contact, quite symmetric respect the midpoint of the interval of contact, loses its symmetry for non-standard geometries and operating conditions. As a consequence, the critical contact points for bending and wear calculations may be shifted from their locations for standard gears. In this paper, a non-uniform model of load distribution along the line of contact of standard spur and helical gears, obtained from the minimum elastic potential criterion, has been enhanced to fit with the meshing conditions of the above mentioned non-standard cylindrical gear pairs. The same analytical formulation of the initial model may be used for the non-standard gears by considering appropriate values of a virtual contact ratio, which are also presented in the paper.

The presence of undercut at the pinion root may affect the load sharing among couples of spur gear teeth in simultaneous contact, as well as the load distribution along the line of contact of helical gears. This occurs if the outside points of the wheel profile do not find, due to undercut, mating profile to mesh with, which is called vacuum gearing. Under these conditions, the effective contact ratio is reduced, and the critical contact points may be shifted from their locations at non-undercut profiles. In this paper, a non-uniform model of load distribution along the line of contact, obtained from the minimum elastic potential criterion, has been used combined with the Hertz equation for the calculation of the contact stress. A complete study on the critical load conditions and the value of the critical contact stress has been carried out, considering a wide range of the values of the geometrical parameters. As a result, a recommendation for pitting load capacity calculations of vacuum-gearing spur and helical gears is proposed.

Calculation methods of spur and helical gear drives for preliminary designs or standardization purposes available in technical literature, use the Hertz equation to evaluate the contact stress, assuming the load to be uniformly distributed along the line of contact. However, this model presents some discrepancies with experimental results because the changing rigidity of the pair of teeth along the path of contact produces a non-uniform load distribution, which implies that some load distribution factors are required to compute the contact stress. In this paper, a non-uniform model of load distribution along the line of contact, recently developed, obtained from the minimum elastic potential criterion, has been used. This model combined with the Hertz equation yields more accurate values of the contact stress. 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 analytic equation, a complete study of the location and value of the critical contact stress has been carried out. From this study, a recommendation for the calculation of the pitting load capacity of spur and helical gears is proposed.

Profile modifications are commonly used to avoid shocks between meshing gear teeth produced by the delay of the driven gear, and the subsequent sooner start of contact, due to the teeth deflections. A suitable tip relief at the driven tooth shifts the start of contact to the proper location at the theoretical inner point of contact. The shape of the relief governs the loading curve of the tooth pair, while the length of relief determines the intervals in which this actual loading curve differs from the theoretical one of unmodified teeth. As at least one tooth pair should be in contact at the unmodified involute profile interval, the length of modification should be smaller than the length of the intervals of two pair tooth contact; otherwise, a shock at the end of contact of the previous pair is unavoidable. However this problem does not occur for high contact ratio spur gears, in which at least two couples of teeth are in contact at any moment. In this work, a study on the load sharing and the quasi-static transmission error for high contact ratio spur gears with long profile modification has been performed, and a model for the tooth contact has been developed.

Calculation methods of spur and helical gears for preliminary designs or standardization purposes available in technical literature use simple equations given by the linear theory of elasticity (Navier’s equation) and the Hertzian contact model to evaluate the bending and contact stresses, assuming the load to be uniformly distributed along the line of contact. However, these models are not in good agreement with experimental results because the changing meshing stiffness of the pair of teeth along the line of action produces a non-uniform load distribution, causing some load distribution factors to be required to compute bending and contact stresses. In this paper, a model of non-uniform load distribution along the line of contact, obtained from the minimum elastic potential energy criterion, is presented. This model combined with the equations of Navier and Hertz yields more realistic values of the bending and contact stresses. An approximate, accurate equation for the inverse unitary potential, allowing analytic calculations of the load per unit of length at any point of the line of contact and any position of the cycle of meshing, is also presented. The same equation, with a slight modification of the coefficients, is also valid for undercut teeth. Results have been validated by comparison with some studies carried out by the Finite Element Method.

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.

This paper presents a model of load sharing for high contact ratio internal spur gears, which takes into account the changing rigidity of the pair of teeth along the path of contact. It has been obtained from the minimum elastic potential energy criterion, resulting in a non-uniform load distribution among several tooth-pairs in simultaneous contact. This model has been used for the calculation of the determinant load conditions for pitting calculations, providing the theoretical values of the pitting nominal stresses. As no proposals of load sharing for this kind of internal spur gear transmissions can be found in literature, the proposed model can be used for strength calculations which may be suitable for standardization purposes.

The friction power losses in spur gears can be computed as the integral, along the entire path of contact, of the normal load, multiplied by the friction coefficient and the elemental sliding. Good agreement with experimental results are obtained by estimating the load from the minimum potential energy load sharing model, and the friction coefficient from the equation provided by Xu in 2005. From this approach, a complete study of the influence of the design parameters (number of teeth, gear ratio, pressure angle, addendum modification coefficients, etc.) on the mechanical efficiency is presented.

Simple, traditional methods for computation of the efficiency of spur gears are based on the hypotheses of constant friction coefficient and uniform load sharing along the path of contact. However, none of them is accurate. The friction coefficient is variable along the path of contact, though average values can be often considered for preliminary calculations. Nevertheless, the nonuniform load sharing produced by the changing rigidity of the pair of teeth has significant influence on the friction losses, due to the different relative sliding at any contact point. In previous works, the authors obtained a nonuniform model of load distribution based on the minimum elastic potential criterion, which was applied to compute the efficiency of standard gears. In this work, this model of load sharing is applied to study the efficiency of both standard and high contact ratio involute spur gears (with contact ratio between 1 and 2 and greater than 2, resp.). Approximate expressions for the friction power losses and for the efficiency are presented assuming the friction coefficient to be constant along the path of contact. A study of the influence of some transmission parameters (as the gear ratio, pressure angle, etc.) on the efficiency is also presented.

The presence of undercut at the tooth root, non-equal addendum on pinion and wheel, non-standard tooth height or non-standard center distance may have decisive influence on the load sharing between pairs of teeth in simultaneous contact of spur gears. The curve of variation of the meshing stiffness along the path of contact, quite symmetric respect the midpoint of the interval of contact, loses its symmetry for non-standard geometries and operating conditions. As a consequence, the critical contact points for bending and wear calculations may be shifted from their locations for standard gears. In this paper, a non-uniform model of load sharing of standard spur gears, obtained from the minimum elastic potential criterion, has been enhanced to fit with the meshing conditions of the above mentioned non-standard spur gear pairs.

An analytic model to compute the efficiency of spur gears has been developed. It is based on the application of a non-uniform model of load distribution obtained from the minimum elastic potential criterion and a simplified non-uniform model of the friction coefficient along the path of contact. Both conventional and high transverse contact ratio spur gears have been considered. Analytical expressions for the power losses due to friction, for the transmitted power and for the efficiency are presented. From this model, a complete study of the influence of some design parameters (as the number of teeth, the gear ratio, the pressure angle, the addendum modification coefficient, etc.) on the efficiency is presented.Copyright © 2011 by ASME

The traditional methods for computation of the efficiency of cylindrical gear transmissions are based on the hypotheses of constant friction coefficient and uniform load distribution along the line of contact. However, the changing rigidity of the pair of teeth along the path of contact produces a non–uniform load distribution, which has significant influence on the friction losses, due to the different relative sliding at any point of the line of contact. In previous works, the authors obtained a non-uniform model of load distribution based on the minimum elastic potential criterion. This load distribution was applied to compute the efficiency of spur and helical gears, resulting in slightly greater values of the efficiency than those obtained if the load distribution along the line of contact is assumed to be uniform. In this work, this non-uniform model of load distribution is applied to study the efficiency of involute spur gears with transverse contact ratio between 1 and 2 (i.e., the load is shared among one or two pairs of teeth), assuming the friction coefficient to be constant along the path of contact. Analytical expressions for the power losses due to friction, for the transmitted power and for the efficiency are presented. A study of the influence of some transmission parameters (as gear ratio, pressure angle, etc.) on the efficiency is also presented.

This paper presents a new model of load distribution for involute, internal gear drives, which takes into account the changing rigidity of the pair of teeth along the path of contact. The model has been obtained from the minimum elastic potential criterion, resulting in a non-uniform load distribution along the line of contact (for helical gears), as well as among several pairs in simultaneous contact. A study of the nominal contact stress, including the calculation of the determinant load conditions, is also presented.Copyright © 2007 by ASME