Advances in Gear Theory and Gear Cutting Tool Design

This chapter deals with geometrically-accurate gears and gear pairs of all possible kinds. Parallel-axes gearing (or just P_a-gearing, for simplicity), intersected-axes gearing (or just I_a-gearing, for simplicity), and crossed-axes gearing (or just C_a-gearing, for simplicity), all three of them are covered in this chapter.

The focus is mainly on three fundamental laws of gearing. These laws of gearing are as follows:

It is proven that in geometrically-accurate gearing of all three kinds, namely, in P_a - , I_a - , C_a-gearing, all three fundamental laws of gearing are fulfilled. Only approximate gearing can be designed and manufactured so as to feature one or more fundamental laws of gearing violated.

Gears and worm gears are an important part of many machines and mechanisms. The reliability and durability of gears largely determine the quality and competitiveness of machines in general. The scientific basis for the design of gears and gear cutting tools is the gearing theory. The main research object of this theory is complex surfaces formed by enveloping methods, and the main tool is software-implemented algorithms for modeling the shaping processes. The creation of reliable simulation algorithms is hindered by the imperfection of the applied methods and mathematical models of gearing theory. A precondition for improving such methods and models is fundamental theoretical research in the field of gearing theory, enabling the creation of computational models more adequate to the real processes of shaping and contact of moving bodies than the existing ones.

This chapter presents the results of further development of the gearing and surface shaping theories fulfilled by Prof. D.T. Babichev during more than 50 years of his life and professional activities. His co-authors attempted to compile and prepare the results of the research, almost as originally outlined by Professor Babichev.

This chapter pertains to the scientific theory of gearing and to generating of envelope curves and surfaces in particular. All three kinds of gearing, namely, parallel-axes gearing (or just P_a- gearing, for simplicity), intersected-axes gearing (or just I_a- gearing, for simplicity), and crossed-axes gearing (or just C_a- gearing, for simplicity), are considered in this chapter.

It is a commonly adopted practice to generate tooth flanks of gears for P_a - , I_a - , C_a- gearing as envelopes to an appropriate family of curves and surfaces. Unfortunately, not every envelope curve and surface can be used in design of gears for geometrically-accurate gear pairs. As rolling motion is a vital component of kinematics of enveloping process, an additional requirement must be fulfilled to get two envelope surfaces to be conjugate to one another. Pair of envelope surfaces of this sort is also referred to as reversibly-enveloping curves and surfaces (or just as R_e-curves and surfaces).

It is proven that in geometrically-accurate gearing of all three kinds, namely, in P_a - , I_a - , C_a- gearing, tooth flanks feature R_e-surface geometry.

The new theory in regard to the meshing limit line is systematically established for the Archimedes worm pair. The equations of an Archimedes helicoid, the meshing function, and the meshing limit function for worm pair are all obtained in different forms and by means of different methods from the past. In the light of these more laconic results, it is strictly proved that the meshing limit line for an Archimedes worm drive always exists in general and cannot be removed by the adjustment and/or re-arrangement of the parameters. As far as the most limit points of the second kind are concerned, the curvilinear coordinate parameters are acquired analytically and no iteration is needed. The numerical outcome states clearly that the meshing limit line usually locates roughly at the middle of the worm thread length. This is the theoretical reason why the working length of an Archimedes worm generally cannot exceed the half of its thread length. Generally speaking, the conjugate line of the meshing limit line lies at the middle of the tooth surface of the worm gear and divides the whole conjugate zone into two parts. Moreover, the conjugate line can commonly go through the full tooth height from the top to the root.

Gear cutting with disc-shaped milling cutters is characterized by a loose connection between the cutting tool and the gear tooth profile and provides possibilities to increase productivity and reduce manufacturing costs compared to gear cutting with shaft milling cutters. Various methods for gear cutting with disc-shaped milling cutters were developed and successfully implemented on the conventional machining centers. To be able to select an optimal machining strategy and suitable process parameters for each specific manufacturing task, knowledge about the characteristics of the known methods for gear cutting with disc-shaped milling cutters is necessary.

This chapter is dedicated to theoretical studies of the known methods for gear cutting with disc-shaped milling cutters. Firstly, by analyzing these methods, it was found that they differ in strategies for material removal, tool movement, and tool engagement and, therefore, provide different productivity, flexibility, or machining quality. Then, mathematical models of the form-shaping kinematics (movements of the cutting tool relative to the workpiece) were developed for process simulation. Finally, trajectories, velocities, and accelerations of the machine tool components as well as material removal rates were calculated when gear cutting by using these mathematical models.

The gear wheels geometry peculiarities of the G-rotor pair are considered. The question concerning profiles-curves synthesis of gearing is solved. To ensure the absence of “degenerate areas” on the equidistant, the distance of its removal should be less than the minimum radius of the epicycloid curvature ρ_min. An analytical description of the enveloping family of equidistant to epicycloids in a single continuous form is performed. This is achieved by reducing the function of the epicycloid angular coefficient to a smooth continuous form, which in turn allowed obtaining a rational analytical solution of the relationship equation between the parameters of the construction and the curves family.

Worm tools for highly efficient processing of the satellite profile and an analytical description of their initial shaping contours are presented.

A new tool for highly efficient finishing of spur wheels with large numbers of teeth is presented.

Modern computer technologies provide conditions for the creation of highly efficient systems for gears shaping. This greatly speeds up the gear design process while ensuring quality. A mathematical description of the surface reduced to a universal form on the basis of unified control parameters is proposed. This provides an approximation of the point representation of surfaces by spline surfaces and creates the basis for the use of unified methodological, algorithmic, and software automation of the shaping process. This approach makes it possible to evaluate the shaping process directly during the interactive analysis of gear shaping, including a preliminary assessment of geometry and kinematics of the shaping process. The used unified parameters of the shaping system provide the possibility of integrating the shaping system into automated design, production, analysis, and control systems.

A novel approach to gearing and gear-cutting technology is described. The results of research of sinusoidal-type gears and substantiation of the new method of gears machining are presented. In the first part of the work, involute and sinusoidal gears are analyzed. Comparison of simulation results has confirmed better performance of the sinusoidal transmission; this follows from the operating parameters, in particular, higher bending strength, lower contact stress and strain, reduced contact friction, and stress in teeth meshing. In the second part of the work, a new method, which is called as radial circular generating method of gear machining is suggested. A significant advantage of this method consists in the possibility for makers of transmissions to use simple tools and conventional gear-cutting machine tools for different types of gears. This new method of machining has been compared with hobbin as a main method of the gear cutting; algorithms and programs for description and simulation of these complicated processes have been developed. Results of comprehensive mathematical modeling and computer simulation of these processes based on modeling of non-deformable chips have confirmed a decrease in cutting forces and that of elastic deformation; this improves the accuracy and gear teeth roughness. It is proved that the suggested method of gear machining is highly productive and efficient; it is suitable for practical application.

The result of a technological system designing is to determine its optimal parameters and structure, which provide the specified properties of the product. This chapter proposes principles for building a technological system based on a systems approach. The principles are implemented in the form of a logical design scheme. This model regulates the design process with information, optimization, and algorithmic systems, regardless of the processing type. Using the developed method, several technological systems have been designed for the gears finishing. Particular attention is paid to technological systems for finishing with a rigid kinematic connection between the tool and the gear to be machined and to systems with free rolling. Examples of technological systems for combined machining, systems with multi-tool set-up, selective tracking, and systems for ensuring the tools shaping are considered and designed. The designed technological systems are tested under real conditions in the gear finishing. Thus, the proposed methodology can be used to design technological systems, in particular, to create the systems for gears finishing.

The architectonics of the gear train information model (digital twin), developed on the basis of the lifetime mechanics of machines, is presented, and its main components – concepts, representations, models and methods – are described. These provisions and methods were created within the framework of the scientific school founded by Igor Tsitovich, Corresponding Member of the Academy of Sciences of Belarus. Brief historic data are given. Currently, this direction is developing at the Joint Institute of Mechanical Engineering of the National Academy of Sciences of Belarus. The developed architectonics contains several new components: three method of gear trains synthesis, universal kinematic-quasi-static calculations for the gear transmissions of any configuration, simulation of dynamics based on concept of a regular mechanical systems, strength calculations in the lifetime form, space of operation conditions, reliability calculations according to multilevel “mechanics-dependability” scheme, and a special method for reliability calculation under the general load conditions of the components. Basic representations of the transmission (kinematic diagram, a set of dynamic schemes, life-strength curves, schemes of limiting states, etc.) appear in the development process and go through all stages of life cycle. These components are complemented at the operational stage by transmission diagnostics and evaluation of its lifetime expense as a complex item and serves as accumulators of new knowledge and data about the item and its operation conditions (environment). The methodological provisions developed for the transmissions are applicable to a wide range of engineering objects. They are also presented in the State standards of the USSR and the Republic of Belarus.

Vehicle transmissions are built on the basis of two types of geartrains: with fixed axles of gear wheels (layshaft geartrains) and planetary geartrains. Layshaft transmissions allow to simply increase the number of speeds by adding the appropriate number of clutches and geartrains with the required gear ratio. But the power flow in layshaft transmissions transmits only by single path; therefore, loads acting on internal transmission’s links are maximum. Planetary transmissions have a higher load capacity and allow to divide power flow into several parallel paths. The creation of multispeed planetary transmissions is complicated by a significant increase in the number of planetary gearsets and clutches. This chapter deals with multipower flow transmissions containing combinations of layshaft and planetary geartrains. The main advantages of planetary-layshaft transmissions are the use of simple geartrains, reduced loads on the internal links, a wide ratio range, and the implementation of more speeds at fewer clutches.

Closed planetary-layshaft transmissions have only two possible structures with two power flows and four structures with three power flows. The chapter shows examples of new kinematic diagrams of multispeed multipower flow planetary-layshaft transmissions. A method for calculating the gear ratios of the geartrains that make up the transmission, which satisfies the specified series of speed ratios, has been developed.

Multiparameter gears and gear-type variators are a further development of traditional (one-parameter) gears and existing variable speed drive’s designs. The purpose is to transfer the torque in kinematic or power variable and transformer mechanisms and machines. They differ in that they allow you to adjust not just one controlled parameter – the angle of rotation of the drive wheel – but two or more parameters. Multiparameter gear drives with one pair of engagement allow you to additionally change the relative position of the axes and gear variators – to constantly adjust the gear ratio. These devices are complex of mechanisms. They can have a continuous or continuous-discontinuous operation and reproduce the required gear ratio theoretically accurately or approximately. Due to using of gears, multiparameter gear drives and gear variators allow power to be transmitted by normal components of forces; therefore, they have a high efficiency and low specific material consumption. The spheres of utilization of these devices are general mechanical engineering, automotive industry, tractor and combine building, drives of technological machines, and robotics.

A multidimensional classification of multiparameter gears and gear variators is proposed. The classification considers the speed, load, and functionality of the devices. It is shown that the structure of multiparameter gears includes mechanisms that allow changing the mutual arrangement of the axes while maintaining the meshing conditions. These mechanisms require a control system (unit). Gear variators also have mechanisms that allow adjusting the trajectories of the internal variator links, causing change in the gear ratio. It needs to be connected to the control system.

The teeth of multiparameter gear drives and gear drives that are part of variators form higher kinematic pairs, the contact point of which to be optimized

The shaping of the tooth of multiparameter gears can be carried out using traditional or additive technologies.

The article justifies both the necessity and technical and economic feasibility of the creation of a new technical system for the synthesis of cylindrical gears and its supporting mathematical models and information technologies, taking into account the life cycle of gears. The structure of the technical system, the concept, and the principles of creation and operation are presented. The considered system will allow synthesizing gears with higher qualitative indicators and providing validation of all the stages of its life cycle.

Generalized unified mathematical and logical model of direct and inverse shaping of flat gear systems with an arbitrary profile and use of the theory of affine space mapping has been developed.

Features of approaches to the solution of mathematical models and creation of algorithms in this paper consist of the following:

It is shown that information cross-cutting technologies of product life cycle stages can be used not only to increase efficiency, productivity, and profitability of business processes but also to classify kinematic schemes of formation, create new mechanisms, or radically improve what is already known, using new processing methods, technological processes, expansion of the field of existence of the stages of a life cycle of products, generalizing matmodels, information technologies, and modern technical means.

A paper on a specific example reveals the features of direct and reverse shaping of gears, taking into account the points of fracture, undercutting, displacement of the original forming contour, the required size and shape of the allowance, and the sequence of technological shaping.

This chapter considers the main stages of the gear system development in time, analyzes the current state of the gear wheel theory and technology used in mechanical engineering, gives information on the basic gear materials, and considers trends and sources of gear wheel hardening.

This chapter shows that the most promising methods of increasing the service life of gear wheels in terms of bending and contact endurance, as well as in terms of wear, are the combined methods of surface engineering. These methods involve the combined action of protective coatings and modification of the surface layer.

Prof. Dmitry Tikhonovich Babichev was a famous scientist in the field of theory and practice of gearing, and this digest describes the memorable meetings between the author and him.

Over the years, I met with Prof. Dmitry Babichev at numerous scientific conferences and workshops in Izhevsk, Russia (1996, 1998, 2004, 2008, 2014, and 2017), Sevastopol, Ukraine (2009–2013), St. Petersburg, Russia (2015), and Krakow, Poland (2019). In addition, we worked together on the global publishing project (unfortunately, unfinished)—a multivolume edition—“Gear Trains,” which, on intention, should incorporate the invaluable experience of masters from famous scientific schools, fully formed in Russia, Belarus, and Ukraine. Prof. Dmitry Babichev was very obligatory, had multiple sides to his personality, and fundamentally worked on any new topic. His responsibility and scrupulousness were evident in a variety of situations, among them the example of a long-term and multifaceted Ph.D. and doctoral thesis preparation.