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Introduction To Gear Trains Analysis


  1. Preface
    1.1 Importance of gears in mechanical systems
    1.2 Four different areas to consider when learning gear technology
    1.3 Special features of this introduction
    1.4 Proper order of learning about gears
  2. Basic principle of gearing and types of gears
    2.1 Functions of gears
    2.2 Fundamental requirements of gears in practical applications
    2.3 Mechanical requirements of the gear tooth curve
    2.4 The gear tooth curve
    2.5 Conditions for establishing involute tooth forms
    2.6 Main properties of an involute gear
    2.7 Conditions for continuous meshing of gears
    2.8 Standards for interchangeable gears (standard involute tooth form)
    2.9 Dimensional specifications of standard involute tooth forms
    2.10 Terminology of various parts of involute gears
    2.11 Involute gear tooth cutting and the problem of undercut
    2.12 Types of gears
  3. Gear train
    3.1 Transmission methods of mechanical motion
    3.2 Definitions of gear trains
    3.4 Features of idler gears in gear trains
    3.5 The relationships among center distance, speed ratio and number of teeth
    3.6 Examples of calculations in gear train applications
  4. Change gear mechanisms
    4.1 Variable speed gear train systems
    4.2 Structure of change gear mechanisms
    4.3 Method to determine the number of teeth in change gears
    4.4 Multi-stage variable speed gear mechanisms
    4.5 Common ratio standard of geometric sequence speed group
    4.6 Rotational speed group distributed with six common ratios
    4.7 Gear ratio chart
    4.8 Variable speed gear system examples of geometric sequence
  5. Planetary gear systems
    5.1 Investigation of broad gear motion
    5.2 Differences between planetary and differential gear systems
    5.3 Basic explanation of analysis of rotational speed of planetary motion
    5.4 Problem of number of teeth, number of planet gears and planet gear spacing
  6. Differential gear systems
    6.1 Fundamental explanation of differential gear systems
    6.2 Basic model of differential gear system
    6.3 Calculation for applying differential gear systems
  7. Other related mechanisms (indexing and continuously variable speed mechanisms)
    7.1 Basic explanation of indexing mechanisms
    7.2 Problem of inertia in indexing mechanisms
    7.3 Examples of types and explanation of their structures
    7.4 Continuously variable speed mechanism
  8. Examples of calculations
    8.1 Sample calculation of indexing head used in universal milling machines
    8.2 Comprehensive and dynamic gear mechanism displays for educational purposes


The gear is an important component serving as a key player in power transmission mechanisms. The reason that gearing is so important is because the gear is the convenient way to obtain reliable power transmission without slippage. Until recently, it was designated as the king of mechanical parts and took a rank next to that of the screw family. It is no exaggeration to say that it is one of the most familiar general purpose as well as specialized mechanical components.

Although universal and important, gears are somehow less understood by mechanical engineering students and active mechanical engineers than expected. As a matter of fact, the author is shocked by the low level of their knowledge about gears.

If this situation remains constant, the level of engineers involved in practical applications in the mechanical industry will decrease and will inevitably result in significant inconveniences and difficulties producing grave concerns that should not be lightly ignored. Several causes seem to account for this situation as below :

  1. Gears are characterized by their relatively complex principles and variety, thus being difficult to understand.
  2. Syllabus and learning materials used at schools related to this field often lack the attention this topic deserves. Learning hours are insufficient. Consistent and systematic learning is practically impossible.
  3. Handbooks with easy explanations and consistent introductory texts are scarce. Learners lack appropriate opportunity to gain the knowledge.
  4. Educational contents and directions are biased, lacks practical applications, and not suited for actual designing of machines and mechanisms. The content itself is esoteric, unbalanced, and not likely to attract the interest of general learners.

The main problem is rooted in the deficiency of the educational system. Gear technologies inherently cover such a wide variety of topics that should be chosen appropriately and according to learner’s interests and objectives. Above all, its basic theory and practical applications should be learned separately. A listing of topics that should be included in gear engineering is given below :

  1. Study of tooth form theory and its geometric basics
  2. Analysis of gear mechanisms and related calculations
  3. Analysis of gear loads and strength calculations
  4. Study of gear materials and thermal treatment technology
  5. Manufacturing technologies such as gear cutting processes
  6. Gear precision grades and their test/measurement techniques
  7. Study of performance, noise level, vibration, and lubrication of gears
  8. Study related to damages to and wearing of gears
  9. Study of other relevant matters, etc.

There are too many topics to list them all.

To master all of the topics mentioned above requires much more than a lifetime. Therefore, it is important to select only those subjects that suits learners’ conditions and objectives. Otherwise, learners cannot obtain the proper learning results and are left with keeping a respectful distance from gears, while knowing the importance of gear technologies and feeling discouraged by the difficulties of understanding them.

For these reasons, the author’s many years of experiences in designing and teaching in the gear field leads him to think that mechanical engineers must at least have some gear knowledge and that the gear mechanism is the easiest and the most effective starting point to obtain some practical results.

This leads to the recognition that mastering the gear mechanism is a very important concept to general mechanical engineers and an essential knowledge to have. Unfortunately, however, for those who have recognized this problem and particularly for beginners, the lack of learning materials and handbooks precludes the opportunity for learning and leads to the author’s concern that it will create large inconvenience and empty feelings.

In view of this, this book was designed to include the most basic knowledge as possible, develop systematic discussions consistent with the gear train structural principle by starting from general knowledge of the gear, incorporating as many illustrations and practical examples as possible, so that just reading this book will enable the readers to understand the design and calculation of gear trains and to be able to utilize its knowledge for practical applications.

This book is published as a part of a gear technology series that is in a planning phase. It would be fortunate if this could be utilized as a textbook or handbook for mechanical related subjects used in colleges, high schools, and academic schools, or as a reference material for active mechanical designers and engineers.

In Taipei, Taiwan
Tamkang University
At a mechanical course
August 1986

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