A E Angelova, T V Ronkova - Computer-aided design of involute cylindrical gear drives for portable electric tools - страница 1

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УДК 621.771.07

Angelova A.E., Ronkova T.V., Tiufektchian M.A.

Bulgaria

COMPUTER-AIDED DESIGN OF INVOLUTE CYLINDRICAL GEAR DRIVES FOR PORTABLE ELECTRIC TOOLS

The designer of portable electric tools usually is faced with the necessity of putting in gears which have much more relative torque transmitting capacity than the corresponding average level. At the same time, the corresponding levels of noise and vibrations are extremely important criteria evaluating the quality of the gear drive. Those considerations determine the necessity of implementation of special calculation methods, known in the theory and practice of gear design, aiming at a significant increase of the contact ratio of the gear pairs in mesh. It is the objective of the present paper to reveal the shortcomings of those methods and to state the necessity for methods of a new nature to be applied.

Keywords: drive, gear, involute, contact ratio, interference, cutting, undercutting

Introduction. The designer of portable electric tools usually has a hard time calculating gear sizes. Loads vary widely depending on feeds, speeds, intensity of work, and material treated. It is anybody's guess what the user would do with the portable electric tool. Since those tools are extremely competitive in price, the oversized tool may be too expensive to sell. As a rule, the designer is faced with the necessity of putting in gears which have much more relative torque transmitting capacity than the corresponding average level, knowing that very often the portable electric tool is apt to be neglected or overloaded. At the same time, the corresponding levels of noise and vibrations are extremely important criteria evaluating the quality of the gear drive. Those considerations determine the necessity of implementation of special calculation methods, known in the theory and practice of gear design, aiming at a significant increase of the contact ratio of the gear pairs in mesh. Actually, the said methods are reduced to the implementation of simple calculation schemes. It is the objective of the present paper to reveal the shortcomings of those methods and to state the necessity for methods of a new nature to be applied in the computer-aided design of involute cylindrical gear drives for portable electric tools.

Nature of the problem. It is known that through increase of the contact ratio of the gear pair its torque transmitting capacity can be increased significantly while at the same time the respective levels of noise and vibration can be decreased. The methods for increasing the contact ratio known from the reference literature refer to the maximum extension of the lengths of the involute gear profiles and the active section of the mesh line. For this purpose, specific calculation schemes are applied for determining the outside diameters of the gear rims, different from the standard ones, where the said outside diameters obtain limiting values in order to meet simultaneously the requirements for:

-lack of interference of the tooth tips of each gear with the respective tooth root fillets of the mating gear ; -lack of cutting of the tooth tips of each gear as a result of an interference with the tooth root fillets belonging to the respective gear-cutting tool.

This approach has been theoretically discussed in great detail in [1]. The authors of the present paper have no intention to revise the idea of this approach. What is revised is the method of its realization; more specifically, this is the fact that the respective calculating schemes known from the theory and practice of gear design are based on the standard methods of checking whether the above presented requirements are fulfilled, which, according to the authors, is unacceptable from standpoint of the contemporary means of computing and design.

The standard methods of checking for lack of interference of the tooth tips of each gear with the respective tooth root fillets of the mating gear and the standard methods of checking for lack of cutting of the tooth tips of each gear as a result of an interference with the tooth root fillets belonging to the respective gear-cutting tool are reduced to the implementation of simple calculation schemes [1, 2, 3]. Since the cutting of the tooth tips (with the exception of the cases when it is related to radial feeding of the cutting tool) is a technologically determined interference of the same nature, this phenomenon will not be separately reviewed in the present paper. Besides the above-mentioned condition, another additionally defined restriction concerns reviewing interference of gears of external mesh only.

Generally, the known prerequisite for lack of interference can be presented with the formula:

Pn<Pp,   (i = 1,2) (1)

where

- pK - radius of curvature at the lowest point of the involute tooth profile;

- ppi - radius of curvature at the lowest point of the active section of the involute tooth profile.

It is assumed in the prior theory and practice of gear design that if condition (1) is fulfilled, there will be no interference. Obviously, this is a logical incompatibility, since the fulfillment of the prerequisite for the lack of interference is assumed to be a sufficient condition for the lack of one. Historically, this was justified by the limitations in the past, ensuing from the relatively simple calculating means of design in this area. The research in the present paper is focused on the realistic meaning of this gap in the theory and practice of design of cylindrical involute gear drives, as well as the respective approach for overcoming it.

Small-size gears often have a small number of teeth (at least for one of the gear wheels). In this case, in some situations this wheel is undercut in the base of the teeth, while at the same time one of the primary designer tasks is the design of gears with a possibly higher transverse contact ratio єа. The mathematical equations known from the reference literature [1,2,3] for determining the value of єа, when there is an undercut would read values different from the real one. The authors consider that it is not correct to just warn about the undercutting while offering unrealistic values of єа.

Outcomes and discussion. The above-mentioned logical incompatibility of the issue of interference does not need any proof by itself. At the same time, for the sake of a realistic evaluation of the respective consequences, it is appropriate to review a specific example.

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Fig. 1. Software package GEOMER. Main geometrical parameters of the specific example

Such an example is a gear set with a small number of teeth of the pinion (z1=10) and outside diameters of the gear rims determined following the recommendations in [1], so that a maximum transverse contact ratio is provided for the respective specific case. For this purpose, the maximum values of the outside diameters of the gear rims are set to guarantee the simultaneous fulfillment of the prerequisites for:

- lack of cutting of the tooth tips in the process of cutting;

- providing sufficient thickness of tooth tips (on the outside diameters of gear rims);

- lack of interference of the tooth tips of each gear with the respective tooth root fillets of the mating gear (according to formula 1).

The two gear rims are hobbed using a standard generating rack type gear cutting tool. If the checking for lack of interference is limited to the well-known method from the reference literature (formula 1), it should be concluded that there is no interference in the tooth root fillets of the driven gear (z2=47) or even that there is a certain reserve against this phenomenon, since, according to fig. 1,

p,2/m = 10.1770, pvJm = 10.2054, p2 <pp2

In reality, the tooth tips of the pinion (z1=10) interfere along the tooth root fillets of the driven gear (z2=47). Figure 2 presents the gear mesh in a transverse section, at a moment of interference. Figure 2 is a screen image of the respective scale visualization in the software package GEOMER. The latter has been developed in the Department of "Machine Science, Machine Elements and Engineering Graphics" of Ruse University by a team led

by

Prof. P. Nenov. The CAD system GEOMER has been implemented in a number of plants, research units and higher schools. A great number of products have been developed through it.

Fig.2. Software package GEOMER. Screen image of gear mesh transverse section with

interference of the tooth tips of the pinion (z1=10) along the tooth root fillets of the driven gear (z2=47)

Each modern type of software package for gear design should provide an opportunity for visualization of a high precision image of the gear mesh, where a number of quality parameters of the gear set could be evaluated, including interference. At the same time, the

Fig. 3. Scaled precise image of the base between the teeth of the driven gear (z2=47) and the corresponding trajectory of one of

the tooth tips of the pinion (z1=10). Normals in the end points of the tooth root fillets interference should be checked independently from the subjective evaluation of the designer as well. The model of interference in the respective software system should be based on a precise imaginary construction of the tooth root fillet of the specific gear wheel and the trajectory of the tooth tip of the mating gear, following a software analysis for determining whether the two curves cross each other. The drawing presented on fig. 3 corresponds to such construction for the example under review. Some of the involute sections of driven gear (z2=47) have been drawn, as well as the corresponding tooth root fillets and a section of the root circle, and at the same time the corresponding trajectory of one of the tooth tips of the pinion (z1=10) in a coordinate system fixed with respect to the driven gear

(z2=47).

When there is undercutting at the base of the teeth, the real transverse contact ratio єа is smaller than the one calculated using the mathematical equations known from the reference literature. Even in well-known company CAD systems the value of єа is not corrected but only

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Fig.4. Software package GEOMER. Main geometrical parameters of a gear mesh of z1=8, z2=23, m=1,75.

a warning is issued that there is "undercutting". This could mislead the designer, especially in the case of insignificant undercutting that the value of the transverse contact ratio is єа>1. In the software system GEOMER the real value of єа is obtained using numerical methods. Figure 4 shows the geometrical parameters of a gear mesh with undercutting of the teeth of the

Fig.5. Software package GEOMER. Screen image of the gear pair.

pinion (z1=8). It can be seen that the real value of єа=0,995. If it is calculated using the classical formulae, it will be єа>1. Figure 5 visualizes the gear mesh in a transverse section.

Conclusion. 1. Limiting the checks for lack of interference and lack of cutting of the tooth tips to the traditional methods may lead to unacceptable errors in some cases. For providing precise conduction of these checks it is necessary to carry out the respective imaginary constructions and software analysis which should be realized independently from the subjective evaluation of the designer.

2. When there is an undercutting, using the traditional mathematical equations to determine the value of the transverse contact ratio єа could lead to a wrong outcome. With the software package GEOMER the precise value of єа is found using numerical methods.

3. Each modern software system for gear design should provide an opportunity for visualization of a high precision image of the gear mesh. In this way, a visual estimate of a number of quality parameters of the gear set can be obtained.

The study was supported by contract BG051PO001-3.3.04/28, "Support for the scientific staff development in the field of engineering research and innovation". The project is funded with support from the Operational Programme "Human Resources Development" 2007-2013, financed by the European Social Fund of the European Union.

Reference

1. Справочник по геометрическому расчету эвольвентных зубчатых и червячных передач, Под ред. И. А. Болотовского, Машиностроение, Москва, 1986.

2. Dudley, Darle W., Handbook of practical gear design, McGraw-Hill, 1984.

3. Ненов П., Е. Ангелова, А. Добрева, В. Добрев, Машинни елементи, Русенски университет, Русе, 2010.

Конструктор переносных электрических инструментов, как правило, сталкивается с необходимостью использования зубчатых передач, характеризующихся намного большей относительной нагрузочной способностью по сравнению с соответствующим средним уровнем. В это же время уровни шума и вибраций являются исключительно важными критериями оценки качества зубчатой передачи. Этими соображениями предопределяется и необходимость применения специальных расчетных методов, известных в теории и практике проектирования зубчатых передач, предназначенных для значительного повышения коэффициента перекрытия зубчатых пар в зацеплении. Цель настоящего доклада - раскрыть недостатки этих методов и обосновать необходимость применения методов нового содержания.

Ключевые слова: передачи зубчатые, передачи эвольвентные, коэффициент перекрития, интерференция, срезание, подрезание

Конструктор переносних електричних інструментів, як правило, зіштовхується з необхідністю використання зубчастих передач, що характеризуються набагато більшою відносною навантажувальною здатністю у порівнянні з відповідним середнім рівнем. У цей же час рівні шуму та вібрацій є винятково важливими критеріями оцінки якості зубчастої передачі. Цими міркуваннями визначається необхідність застосовування спеціальних розрахункових методів, що відомі в теорії і практиці проектування зубчастих передач, призначених для значного підвищення коефіцієнта перекриття зубчастих пар у зачепленні. Мета даної доповіді - розкрити недоліки цих методів і обґрунтувати необхідність застосування методів нового змісту.

Ключові слова: передачі зубчасті,передачі евольвентні, коефіцієнт перекриття, інтерференція, зрізання, підрізання

Angelova E. А. Assoc. Prof. Ph.D., Department - Machine Science, Machine Elements and

Engineering   Graphics,   Faculty   of Transport,   University   of Ruse, 8 Studentska str., 7017 Ruse, Bulgaria.

Ronkova V. T.

Principal assistant Ph.D., Department - Machine Science, Machine Elements

and Engineering Graphics, Faculty of Transport, University of Ruse, 8 Studentska str., 7017 Ruse, Bulgaria.

Tiufektchian A. M. Ph.D. Candidate, Department - Machine Science, Machine Elements and Engineering Graphics, Faculty of Transport, University of Ruse, 8 Studentska str., 7017 Ruse, Bulgaria.

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A E Angelova, T V Ronkova - Computer-aided design of involute cylindrical gear drives for portable electric tools