Theory and Practice of Gearing and Transmissions

The American period in the development of the theory of gearing by Prof. Faydor L. Litvin is described. Professor Litvin has dedicated his life to the development of the theory of gearing, and has made significant contributions to the development of the theory of mechanisms. He has developed two professional careers, one in Russia and another in the United States, having become a respected and renowned authority in kinematics and the theory of gearing, and having trained and supervised more than 85 Ph. D. students and visiting scholars from all over the world, all of whom went on to receive prominent research and teaching positions in the USA, Japan, Australia, Bulgaria, China, Taiwan, Italy, Spain, and Russia.

A brief overview of selected scientific works by Prof. F.L. Litvin performed up to 1980 is presented. His role in the development of the Russian school of the theory of gearing is shown.

Mechanism Science (MS) has been the core of mechanical engineering and industrial engineering since the beginning of engineering practice in modern times. A short survey is presented to outline the main characteristics of MS and its evolution, with the additional aim of identifying challenges and the role of MS in future developments of technology for the benefit of society. The community of IFToMM working in MS is presented, identifying various members and their respective roles. Modern systems with mechatronic features still need careful attention from a mechanism design viewpoint to properly achieve the goals of forwarding technological developments in helping or substituting for human operators in their activities.

The paper presents a short history of the development of the theory of spiroid gears and gearboxes design in the Institute of Mechanics of Kalashnikov ISTU, and aspects of implementation and development of their production in the “Mechanic” Ltd.

It is obvious that the basics of gear technology is the theory of geometry for conjugate action of mating gears for transmission of motion through contacting tooth flanks. Teeth of actual gears though contact not only on tooth flank to tooth flank, but also at edges of tooth tip and of tooth sides. Such edge contact is usually out of the conjugate meshing theory of gearing. Many causes of failure of highly loaded gear teeth are initiated due to the contact at such tooth edges. Contact of tooth edge produces considerable amount of wear debris. Meshing teeth crush intruded wear debris and damage themselves. With run of gear operation, gear failure develops as a phenomenon of such a positive feedback system.

The paper describes methods of tooth generating by two-parametric families of generating lines (cutting tool edges) proposed for spiroid gears. Several general issues of tooth generating process by means of these families are considered.

In many gear drives, the tooth load on one flank is significantly higher and is applied for longer periods of time than for the opposite one. An asymmetric tooth shape reflects this functional difference. Design intent of asymmetric gear teeth is to improve performance of the primary drive profiles at the expense of the performance off the opposite coast profiles. The coast profiles are unloaded or lightly loaded during a relatively short work period. Asymmetric tooth profiles also make it possible simultaneously to increase the contact ratio and operating pressure angle beyond the conventional gear limits. The main advantage of asymmetric gears is contact stress reduction on the drive flanks, resulting in higher torque density (load capacity per gear size). The paper presents an application of the Direct Gear Design^® method to asymmetric tooth gears. This is an alternative approach to traditional design of involute gears, separating gear geometry definition from the generating (tooling) rack to maximize gear drive performance. The paper describes asymmetric tooth and gear mesh characteristics, limits of asymmetric tooth gearing, tooth geometry optimization, analytical and experimental comparison of symmetric and asymmetric tooth gears, and implementation of gears with asymmetric teeth.

The paper describes the meshing space for worm gears with general type worms, including double-enveloping and bevel worms. A pair of conjugate lines is revealed in this space. Position of these lines is defined by position of surface of movement of the cutting edge which generates a worm but does not depend on this edge profile. The revealed lines are pre-assigned to belong to the meshing surface and, in this respect, they are similar to the axes of meshing of gears with a cylindrical worm. Applying the properties of the pointed lines allows, to some extent, for simplification of the analysis of the known and synthesis of new types of worm gears.

The paper describes application of kinematical method of the theory of gearing to determine surfaces and lines produced by generating method related not only to smooth generating surfaces and lines but with the presence of singular points—jogs—in points of adjacent areas. Three types of jogs are considered: jog at the point of intersection of two plane profiles; jog along the line of intersection of two surfaces; jog at the point of intersection of three surfaces. Three types of geometrical images generated by these three types of jogs are proposed: fan, wedge, bunch of normal lines. Efficiency of applying these images when analyzing the process of tooth generation and meshing is shown. Mathematical models are developed and presented for generating processes in the presence of jogs on generating surfaces and lines. Examples of computer modeling with application of these models are given. It is emphasized that the developed method operates much faster than non-differential methods which have been considered to be applicable only when computer modeling the generating processes in the presence of jogs.

Crossed helical gears are used in cars and many household appliances. The trend towards increased comfort in motor vehicles has led to the utilization of more than a hundred servo-drives in luxury class automobiles. Important advantages of crossed helical gears are their easy and inexpensive design, good noise performance and high ratio that can be realized in one step. Sintered steel is a very favorable material for wheels in crossed helical gears. The hardening obtained after the sinter process will affect the microstructure of the sintered steel, so that the wear load capacity can increase to higher values. This report shows results of iron-based sintered material Fe1.5Cr0.2Mo in the case of crossed helical gears concerning wear resistance and other damage types under different speed, torque and lubricants. As material variants, samples with additional treatment, such as pyrohydrolysis, case hardening, shot peening, sinter-hardening, and 2 % copper addition, are used. The calculation method is given for the determination of wear load capacity of the worm with a helical gear made of Fe1.5Cr0.2Mo sintered steel with sinter-hardening treatment. These results provide product developers with the first important clues for indicators for calculation of the worm with a helical gear. The paper also presents an analytical and FEA procedure for determination of coefficient of friction in meshing zone of crossed helical gears. Obtained results from numerical simulation were compared with experimentally obtained results.

Creation of high-performance machines and equipment implies special requirements for the most loaded parts, units and aggregates, their engineering level and quality influencing the operating characteristics of machines in general. Surface-hardened gearwheels, the feature of which is non-uniform distribution of physical and mechanical properties in a surface-hardened layer sustaining high contact and bending loads, become more widely applied in machine drives and transmissions and in the gearbox industry. That is why ensuring the reliability and functional characteristics of the most loaded power gears of drive mechanisms and transmissions is an urgent problem.

Mathematical, especially graph-theoretical, models of gears can be very useful within the conceptual phase of a gear design activity. In the present paper, three graph methods of modeling of gears are discussed. The applied graphs are as follows: mixed, contour and bond graphs. Some general rules of modeling are roughly explained. Exemplary automotive gear box in case of particular drive (gear) is considered. Some engineering tasks are performed (e.g. kinematical analysis) by means of three discussed graph approaches to show usefulness of the proposed methods. The compatible results were obtained in every case. The bond graph method due to its generality and availability of professional software seem to be most useful.

The paper deals with designs and geometry of planetary transmissions with internal gearing with a small difference of gearwheel tooth numbers. Internal planetary gears with non-standard tooth profiles and their strength parameters are considered. Results of the experimental research of these mechanisms are presented.

The effect of errors of alignment on the bearing contact of spiral bevel gear drives has been the subject of research for many years. Generally speaking, gear misalignment causes transmission errors and edge contacts, leading to incremental levels of noise and vibration, and a reduction of the gear drive service life. Apart from assembly and/or manufacturing errors, supporting shafts deflections caused by torque transmission constitute an important but predictable source of misalignments in gear drives. In the paper, a procedure of determination of the relative spatial position of spiral bevel gear supporting shafts during torque transmission will be proposed, in order to predict the relative errors of alignment between spiral bevel gears. The obtained errors of alignment will be employed as initial data in the local synthesis method, for the purpose of compensating them through modification of the pinion surface microgeometry. Finally, a numerical example will illustrate the proposed procedure, as well as the advantages of its consideration in the design of advanced spiral bevel gear drives in order to achieve the best contact pattern and function of transmission errors for nominal torque transmission.

The ease-off concept was introduced to describe the mismatch, a deviation between conjugation and non-conjugation, between two tooth surfaces from a pair of mating bevel gears in contact. However, a complete mathematical description of ease-off and the algorithm of computation were not found. The application of ease-off concept in tooth contact analysis (TCA) allows for a numerical determination of contact patterns and transmission errors of highly conformable contact or almost conjugate contact of tooth surfaces. The paper analytically describes a generalized theory of ease-off and its application in tooth contact analysis of both face-milled and face-hobbed spiral bevel and hypoid gears with complex tooth surface modifications. The implementation of the ease-off algorithm applied to the tooth contact analysis is illustrated with two examples of, respectively, a face-milled gear drive and a face-hobbed hypoid gear drive.

Due to expansion of the micro- and nanocomposites application for manufacturing machine components and friction units, the problem of calculation methods for determination of deformability, strength and wear resistance parameters of gear drives made of essentially inhomogeneous disperse-reinforced materials is addressed. The potentialities of analytical and numerical methods are analyzed. The original three-level (micro, meso- and macro) method for tribomechanical parameters optimization of the gears by controlling material reinforcing is presented. Through specific examples, the potentialities of polymer reinforcement for obtaining functional materials for gears which allow for an increase in damping capability, shape stability and life time of the gear driven by the criterion of wear and bending strength have been studied.

A mathematical model of excitation of oscillations in tooth contact of a helical gear is refined where oscillations are excited by a pitch error. It is shown that choosing the magnitudes of transverse and overlap contact ratios in a helical gear provides minimization of vibration excitation caused by simultaneous action of three forcing factors: variable rigidity of meshing, impulse loading of teeth and pitch error. Examples of calculation of exciting forces, amplitude-frequency characteristics of oscillations and tooth loads are given.

Main calculation principle of gear wheel geometry with asymmetrical tooth profile is presented. The paper presents calculation of bending stresses by means of YFS coefficient which takes into account the tooth shape and stress concentration. Graphic method of YFS coefficients calculation is described. Graphic method of stiffness calculation of teeth with asymmetrical profile and results of vibration testing are given.

The paper presents the main results of investigations carried out by the author on variation of teeth and base surface accuracy (matching bore and rim faces) of spur gears in the main operations of their machining. The results of the influence of the base surface accuracy and cutting conditions on the teeth accuracy in gear-milling by a worm hob and gear-shaping, as well as the methods of choosing gear-machining manufacturing route and the requirements of their quality in the intermediate operations of their machining providing the necessary quality of completed gears, are described. Information about CTБ 1251-2001 (State Standard of the Republic of Belarus) that has necessary techniques and recommendations to solve this problem is also given.

The paper describes classical and newly developed approaches to design of gears and gearboxes. Advantages and drawbacks of these approaches are considered. Examples of applying the advanced mathematical apparatus and techniques of computer-aided design of gears by example of spiroid gears and gearboxes are given. Conclusions of progressive ways of applying the proposed mathematical apparatus and advanced techniques of gear systems design are made.

The development and manufacturing of gear pairs is determined by a system combining the kinematic basis of the relative motions between workpiece and tool with the necessary production technology. This system unites the two subsystems of theory and technology. Via a multitude of parameters, gears can be assigned to different classes according to the range of existence. The paper describes a mathematical model for determining the range of existence for the general kinematic scheme of gear shaping. This model is based on an analysis of the existing kinematic schemes of theoretical and real shaping. Based on a morphological approach, individual kinematic shaping schemes are classified, and their mathematical models are developed. The shaping schemes, as well as the necessary translational and rotational motion matrices, are exemplarily presented for a gear manufacturing machine using the shaping principle. The modern approaches to the principles of designing equipment and instrumentation systems for gear manufacturing are presented. Methodical basics for selecting the optimal machine configuration depending on the technical and economic requirements for the machining and form of a gear’s tooth profile are stated. The system for theoretical and technological optimization synthesis of instrumentation systems for gear manufacturing is presented.