Geometric design of a planetary gear train with non-circular gears

Abstract

The paper presents a concept of epicyclical gear train able to generate a variable gear ratio law. The basic mechanical configuration consists of three non-circular gears in a typical planetary arrangement.

Such a mechanism couples the advantages of non-circular gears with the typical performances of epicyclical gear trains. Therefore this kind of planetary gear train is useful when the synthesis of a specific torque curve or the design of a function generator involve a highly variable input/output relationship, especially if small weights and sizes are required.

An application is presented, where a planetary gear train with non-circular gears is proposed in order to design a power drive mechanism for high performance bicycles. Such a device maximizes the human output during a typical low-speed way of pedalling.

Introduction

The paper presents a new concept of epicyclical gear train, consisting of an external, an intermediate and a central gear, whose pitch lines are all variable-radius curves.

In spite of their poor diffusion, non-circular gears can be used in a variety of mechanical systems. In fact, since the gear ratio function they generate is variable, a purely mechanical control can be performed on the input/output relationship. For this reason, non-circular gears are useful in those mechanisms whose task is to force an output element to move according to a specific law of motion [1]. Automatic equipment in printing presses, textile industry, packaging machines and quick-return mechanisms represent the most diffuse applications [2], [3].

Dooner [4], Kochev [5] and, recently, Yao and Yan [6] proposed the use of non-circular gears in order to reduce speed fluctuations in rotating shafts or to balance both shaking moments and torque fluctuations in planar linkages, while Emura and Arakawa extended the use of elliptical gears to steering mechanisms [7].

Although several researches focus on the kinematic analysis of variable-radius pitch lines [8], [9], non-circular gears are not yet widely diffuse in industrial applications, due to design and manufacture difficulties.

Initially, the generation of non-circular gears was performed by means of devices where a master non-circular gear and a master-rack were employed. In order to overcome difficulties arising from the necessity of manufacturing the master-gear, Litvin [10], [11] proposed an enveloping method. This approach is based on the idea of using tools (rack and shaper cutters) similar to those usually employed in circular gear generation. According to this approach, conjugate tooth profiles are generated by performing a pure rolling of the cutter centrode along the given pitch curve. A set of necessary relations between cutter and gear motion was then derived for both rack and shaper cutter generation, in order to fulfill the condition of rolling without sliding. Litvin [10] also proposed a method to design combined non-circular gear mechanisms, suggesting their use when the generation of function with highly variable derivative is required, since in that case only one pair of gears could cause undesirable values of the pressure angle.

In 1996, Chang and Tsay [12] developed a mathematical model of non-circular gears, manufactured with rack cutters. Furthermore, they proposed a method in order to determine the complete mathematical model of non-circular gear tooth profiles, manufactured with shaper cutters, based on the use of the inverse mechanism relation and on the equation of motion [13].

Recently, Bair [14] proposed a computerized method to generate elliptical gear tooth profiles by means of a shaper cutter.

In the above-mentioned methods, proposed to design and to manufacture non-circular gears, a proper selection of the cutter parameters is a basic requirement in order to avoid undercutting. Such a condition involves a reduction of tooth thickness in the region of the fillet, thus causing a lower load capacity of the gears. In the above quoted references [12], [13], Chang et al. proposed a method to analyze undercutting in elliptical gears generated by a rack cutter and in non-circular gears manufactured by shaper cutters. The analysis of undercutting condition is based on the relative velocity and on the equation of meshing, as suggested by Litvin [15].

In the work presented in this paper, teeth profiles are generated by means of a numerical procedure, that integrates a differential equation, describing the contact point displacement along the line of action during the meshing process [16], [17], once the constant pressure angle and the pitch lines are defined. A method to investigate on non-interference condition, during the mathematical generation of conjugate profiles, is then proposed, based on an analysis of the situations in which profile’s singularities will occur.

This paper proposes a procedure to design planetary gear trains, in which non-circular gears are used to generate a variable angular-velocity ratio. A clever combination of non-circular gear pairs, in fact, can be used to accomplish severe tasks of function generation.

A planetary arrangement of variable-radius gears is very useful when the required gear ratio mean value is a rational number. In that situation, such a mechanism offers a greater compactness than a pair of multi-lobed non-circular gears [18].

A wide literature focuses on the kinematics of epicyclical gear trains. Some authors, in particular, have studied the problem of assembly conditions. In 1998, Simionescu [19] proposed a unified approach to the assembly condition of epicyclical gear trains. A set of analytical relations, in order to determine the offset angles between the wheels of compound planets, was also provided for those cases in which an equidistant assembly is not possible. The configuration of the planetary gear train, as proposed in this work, is characterized by a single planet gear. Nevertheless, assembly conditions must be considered, since non-circular gears are involved. A numerical procedure is proposed in order to assure that tooth profiles are generated so that a correct assembly of the gears can be achieved.

A method to analyze the kinematic behaviour of the mechanism is showed, by extending the traditional analysis approach, based on the apparent angular-velocity equation, to epicyclical gear trains in which variable-radius gears are involved.

A test-case is presented, where the method is applied to design a planetary non-circular gear train for a torque synthesis problem. By applying the proposed synthesis method, a new gear drive for high performance bicycles is designed. The bicycle market is characterised by a continuous growth, thus determining the industry interest in finding new technological solutions. A great part of bicycles uses multi-speed gear systems, while the design optimization of racing bicycles is continuously looked for, mainly emphasizing the choice of ultra-light materials. Freudenstein and Chen [20] developed elliptical gear drives useful in bicycles and variable motion transmissions involving band or tape drives.

In this paper a new optimization approach is presented, consisting in the design of a variable-ratio drive mechanism, involving a planetary non-circular gear train [21]. This device is claimed to reduce the typical torque fluctuations of a low-speed way of pedalling, in order to maximize the human output.