Kinematic Geometry of Gearing

Preface

This second edition is an expansion of the first edition of The Kinematic Geometry of Gearing; A Concurrent Engineering Approach, introducing a generalized integrated methodology for the design and manufacture of different types of toothed bodies. Several expressions are modified from their original presentation along with a reorganization of the material. Included are changes in nomenclature to reduce subscripts, avoid conflicts with symbols, and aid in the implementation of computer software. The kinematic geometry of toothed bodies in mesh builds upon the original presentation and is supplemented with additional figures. The design and manufacturing sections are expanded to provide a more thorough evaluation of the new geometric methodology. Biographical data on historical individuals are provided in footnotes; much of this information is based on the work of J.J. O’Conner and E.F. Robertson.¹ The ensuing presentation more thoroughly develops the single geometric methodology for integrated design and manufacture of gear pairs. A computer simulation for the integrated design and manufacture of generalized gear pairs has been completed (including a GUI or Graphical User Interface), showcasing the concurrent CAD–CAM of gear pairs. Prototype gear pairs have been fabricated and tested to illustrate the geometric methodology developed.

Two bodies in direct contact where the position and orientation of the output element are specified functions of a given input motion comprise gear pairs, threaded fasteners (i.e., bolts and nuts), as well as CAM systems. The overarching goal is a single geometric framework for the generalized design and manufacture of gear pairs, with consideration to fasteners and CAM systems. The importance of gearing continues in the twenty-first century. Gear elements range in size from 4000 mm or 150 in. to 1 μm or 0.5 μin., where the speeds range from less than 1 RPM to over 6 trillion RPM! The automotive industry is currently the biggest user of gear elements commanding over 60% of the world gear market. Such gears encompass spur and helical gears used in transmission, worm and worm wheels used for window regulation, along with spiral bevel and hypoid used in a rear/front axle assembly. Twenty percent of the automotive gear market is targeted to right angle drive gear pairs. It is estimated that there are over 800 million automobiles worldwide with 70 million produced annually. Gears are also essential machine elements in industrial applications, as well as the aerospace and marine industries.

Current gear practice for spatial gearing does not provide for bevel, hypoid, and worm gears to be treated with the same geometric considerations that are applicable to cylindrical gearing (namely, spur and helical gears). These geometric considerations include general formulations for the tooth profile, addendum and dedendum constants, profile modifications, crown, transverse and axial contact ratios, backlash, spiral angle variation, pressure angle variation, inspection techniques, as well as manufacturing technology. The salient theme of this book is to present a single geometric theory for the concurrent CAD–CAM of toothed bodies in direct contact used to transmit power (motion and load) between two axes. The end result is an axode-based theory analogous to that used to design and manufacture planar spur and helical gears. This unified approach is based on formulating a system of pitch, transverse, and axial surfaces, utilizing special curvilinear coordinates to parameterize the kinematic geometry of motion transmission between skew axes. Screw theory or the theory of screws is used as the basis for this geometric foundation. The same results can be obtained using alternatives such as dual numbers, Lie algebras, geometric algebras, or vector algebra. The presented technique builds upon existing known relations and utilizes screw theory to establish

cylindroidal coordinates,

theorem of conjugate pitch surfaces,

kinematic relations between generalized ruled surfaces,

three laws of gearing,

cylindroid of torsure, and

spatial analog of planar Euler-Savary equations.

This analytical foundation is further expanded by introducing a variable diameter cutter or hob cutter for gear manufacture and developing the accompanying kinematic relations necessary for gear fabrication using the variable diameter cutter. A subtle facet of the entire integrated process for gear design and manufacture is the seamless integration of noncircular gear elements. Novel examples of noncircular gear pairs include

2-dof mechanical function generator (variable NC gear pair),

spiral cylindrical and hypoid NC gear pairs,

variable face width NC gear pairs,

coiled NC gear pairs,

coordinated automotive steering with NC gears,

torque and speed balancing of rotating shafts, and

1-dof mechanism for geared robotic manipulators.

AGMA published Gear Industry Vision in September 2004. The goal of this study was to define a vision for the gear community over the next 20 years where gears remain fundamental and the preferred solution in power transmission and control in the 2025 global marketplace. The final section of this study is Key Technological Challenges and Innovations. The top three objectives of this section are the following:

To establish a single system of design and testing standards

To develop improved tribology modeling

To create predictive tools, virtual testing, and simulation tools

These objectives are central to this book. A single geometric system of design is presented by focusing on noncircular hyperboloidal gears for motion transmission between nonorthogonal axes. Although not immediately useful, such a system of design enables spur and helical, worm, and other forms to be readily obtained as a subset of noncircular hyperboloidal gears. About 15% ($5-8B) of the gear market targets rear axle assemblies consisting of spiral-bevel/hypoid gears. The basic differential gear train and spiral-bevel/hypoid gear set inherent in automotive axle assemblies has remained unchanged for over 75 years. Within the past 100 years, spiral-bevel/hypoid gear machine tool manufacturers have focused on a special fabricating process referred to as face milling and face hobbing. Inherent in this ``face’’ cutting process are limitations on the resulting end gear product. It is estimated that there exist over 15,000 spiral-bevel/hypoid gear cutting machines with over 25 years of life where the process presented in this book can eliminate some of these restrictions and lead to a new generation of gear fabrication.

As indicated, the central theme of this book is the presentation of a unified geometric methodology for the parameterization of gear pairs in mesh. An overall evaluation of the kinematic geometry of the newly synthesized gear pair is taken into account by including design rating formulas. These design rating formulas include fillet and inertial stress determination using finite-element analysis, contact stress, dynamics loads, wear, flash temperature, contact and bending fatigue analysis, reliability analysis, minimum lubricant film thickness, and specific film thickness, in addition to mesh and windage losses. An evaluation of the manufacturing process is performed by providing the cutting time, material removal rate, cutting power, surface cutting speed, and relative position and orientation between cutter and gear blank. This concurrent CAD–CAM methodology enables the designer to synthesize gear pairs with increased efficiency, reduced noise, while improving strength and surface durability. This development differs from current gear design and manufacturing practice.

The book is split into three parts and addresses both theory and practice. The first part revisits the concept of toothed bodies in mesh, their various forms for motion transmission, along with some terminology and nomenclature subsequently used to describe the concurrent design and manufacture of toothed bodies presented in Part Two. Part Two establishes the mathematical model used for the integrated design/manufacturing methodology. It is this part where contributions to the kinematic geometry of ruled surfaces in contact, differential geometry of surfaces in direct contact, along with toothed bodies in mesh are developed. Part Three includes design formulas to rate or evaluate gear pairs generated using the developed methodology. Practicing gear engineers can bypass the analytical treatment of Parts One and Two and focus on Part Three. Part Three discusses the design procedure based on the analytical development and gives several examples to illustrate the capabilities of the new approach. A noteworthy feature of the developed methodology is that the design and manufacturing data for the toothed bodies that satisfy the stated requirements and the cutters used to produce them are synthesized concurrently and interactively in a PC environment. The synthesized shapes of the gear and cutter elements along with the surfaces of the teeth separately or in conjugate action are displayed graphically. The designer can view and evaluate trial designs prior to further analysis or manufacturing. Sample displays are included as part of the final chapter to illustrate the process in a variety of nonconventional as well as conventional applications. Included are 10 appendices.

All the relations presented in this text have been coded and tested. Delgear©, a computer software package developed by the author, is included as part of this book. The requirements necessary to run Delgear© are standard with PCs and laptops today. This software enables the reader to specify motion (circular and noncircular gears), tooth type (involute and cycloidal), gear type, cutter, and manufacturing parameters and view the results of the integrated CAD–CAM process for generalized gear pairs. The included software is bundled into an install package that prepares a windows-based environment to use the Delgear© package. Installation instructions are provided in Appendix D. A user's guide is included with the Delgear© software to assist its usage. Included are 22 illustrative examples of gear pairs, both traditional and nontraditional gear pairs, to illustrate features presented in this book.

Much of the mathematics used in this book is presented in existing textbooks and is not summarized. However, the novel information of this book is preceded with a level of basic mathematics. Intermediate graphical displays of gear results and equations developed in Chapters 2–6 are deferred to Chapter . The fundamental theory developed in Chapters 2–6 is presented with figures and equations for individuals interested in the kinematic geometry of gear elements. Gearing is not a field of study analogous to mathematics, vibrations, FEA, or fluid mechanics; and consequently, exercises at the end of each chapter are not included in a traditional textbook manner. The dedicated reader can use the Delgear© software package to check intermediate values at each stage of the presented methodology.

The webpage www.wiley.com/go/dooner_2e provides supplementary material to the Kinematic Geometry of Gearing. This webpage provides a link that enables interested readers to freely download and use software developed by the author. The developed software facilitates the geometric design and rating of various gear types including spur, helical, spiral bevel, straight bevel, spur and spiral non-circular gears, spur and spiral hypoid gears, non-orthogonal worm gears, along with nontraditional gear types.

Acknowledgments are of the order to express the author's appreciation for facilitating the presentation in this work. First, an acknowledgment is due to the late Prof. Ali Seireg for his collaboration and encouragement on the original work and sharing the importance of system design. Behind-the-scenes facilitators include John Wiley & Sons, Ltd for their willingness to continue with this second edition along with IBM APL Product and Services for disentangling a variety of programming woes; especially Nancy Wheeler and David Liebtag, formerly of IBM APL Product and Services. An extended acknowledgement goes to Dr. Michael W. Griffis for his role in the presentation of this new gear approach by fielding many questions and providing insight into the theory of screws. And finally, recognition to IMPO for the fortitude and patience to bear with me as I pieced together this manuscript and software.

David Dooner

Mayagüez, Puerto Rico

1. http://www-groups.dcs.stand.ac.uk/~history/Mathematicians

Part One

Fundamental Principles of Toothed Bodies in Mesh

Before we can understand the future, we must learn about the past

–Anonymous

1

Introduction to the Kinematics Of Gearing

1.1 Introduction

A brief history of gearing and some established gear concepts are presented in this chapter as an introduction to the development of a generalized kinematic theory for the design and manufacture of gears. The primary objective is to familiarize the kinematician with gear terminology in a format that is familiar to them (compatible with established kinematic theory) as well as to introduce the gear specialist to some of the relevant kinematic concepts that are used in developing a generalized methodology for the concurrent design and manufacture of gear pairs. This approach includes the synthesis and analysis of the gear elements concurrently with the design of the corresponding cutter elements used for their fabrication. These introductory concepts will be built upon throughout this book to develop a generalized methodology based on kinematic geometry for the integrated design and manufacturing of appropriate toothed body to transmit a specified speed and load between generally oriented axes and the constraints that may restrict implementation.

1.2 An Overview

An introduction to the complexities involved in the design and manufacture of toothed bodies in mesh can be achieved by first examining the kinematic structure of conjugate motion between parallel axes. One purpose of this chapter is to introduce the concept of toothed wheels and demonstrate the basic kinematic geometry of toothed wheels in mesh as well as their fabrication. This extended introduction is intended to establish a foundation that will be used as a corollary to exemplify the intricacies of spatial gearing (namely, worm and hypoid gearing). A similar introductory treatment on gears is presented in existing textbooks on kinematics and machine design (e.g., Spotts, 1964; Martin, 1969; Shigley and Uicker, 1980; Erdman and Sandor, 1997; Budynas and Nisbett, 2011). The elementary treatment provided in these textbooks on kinematics and machine design is essentially based on the books by Buckingham 1949 and Merritt (1971). Because of its practical importance, the design and manufacture of toothed bodies continues to attract the attention of researchers in a variety of fields (e.g., geometry, lubrication, dynamics, elasticity, material science, and computer science). Dudley (1969) provides a brief account on the history of gears, and additional information regarding the history of gears is provided by Cromwell (1884) and Grant (1899). An overview on the design and manufacture of gears is presented by Dudley (1984) and Drago (1988). Specialists in the gear industry have contributed to the second edition of Dudley's Gear Handbook edited by Townsend (1991). A more extensive and up-to-date analysis for the design and manufacture of gears is provided by the following organizations:

American Gear Manufacturers Association (AGMA)

International Standards Organization (ISO)

Deutsches Institute für Normung (DIN)

Japanese Gear Manufacturers Association (JGMA)

American National Standards Institute (ANSI)

British Gear Association (BGA)

One of the earliest documented geared devices is the South Pointing Chariot. A model of a South Pointing Chariot is depicted in Figure 1.1. The function of this device is to serve as a mechanical compass in crossing the Gobi dessert. The statue atop of the wheeled cart maintains a constant direction of pointing independent of the cart track. Various claims to the date of the device range from 2700 BC to 300 AD. Heron of Alexandria devised many mechanical systems involving mechanisms (some geared). Example systems include special temple gates, mechanized plays, coin-operated water dispensers, and the aeolipile. Leonardo da Vinci is one of the most celebrated designers of all times. Da Vinci is credited with the various sketching of gears in Figure 1.2.

Figure 1.1 South pointing chariot (reproduced by permission of Science Museum London/Science and Society Picture Library)

Figure 1.2 Gear sketches by da Vinci (reproduced by permission of Biblioteca Nacional)

Norton (2001) credits James Watt as the first kinematician for documenting the coupler motion of a four-link mechanism. This documentation was part of his effort to achieve long strokes on his steam engine. More noted is Euler (father of involute gearing) and his analytical treatment of mechanisms. Yet, Reuleaux is considered the father of modern kinematics for his text Theoretical Kinematics. Reuleaux defined six basic mechanical components (namely, a link, wheel, cam, screw, ratchet, and belt). A gear can be considered a manifestation of the wheel, cam, and screw.

Geared devices remain vital components in many machine systems today. As a result, the field of gearing endures an extensive pedigree and can require a devoted apprenticeship to master the subject. Due to the nature of the evolution of gearing, current research and practice, have for the most part, built on concepts charted by nineteenth century geometricians. These contributions include modern concepts in kinematic synthesis and analysis, methods of manufacture, analysis of vibrations and noise, the development and integration of tribological behavior into the field of gearing, and the widespread availability of digital computers. Improvements in the field of gearing can be achieved by directing new energies toward these areas. In order to give the field of gearing a new genesis, gears (special toothed bodies) are classified in general as elements of a mechanism that are used to control an input/output relationship between two axes via surfaces in direct contact. As this manuscript evolves the discrepancies, limitations, inconsistencies, different design philosophies, and the need for new technology within the gear community will become more apparent and the concept of a gear will take on a new identity. The primary goal of this manuscript is to provide the gear designer with new technology and simultaneously provide the gear designer with a practical and unified approach to design and manufacture general toothed bodies. This unified approach provides the analytical foundation to better establish a correlation between theory and practice for generalized gear design and manufacture. It is written with the assumption that the reader has access to the numerous texts which illustrate traditional methods of gear design and manufacture.

1.3 Nomenclature and Terminology

An essential and important aspect of gear design and manufacture is to identify a nomenclature that distinguishes different phenomena with as few symbols as possible. Currently, each of the different gear types (planar, bevel, hypoid,¹ worm, and worm gears) utilize a nomenclature applicable to the particular gear type. The vernacular of a gear specialist can be misleading and confusing for the novice and may require clarification among gear specialists. Also, due to the interdisciplinary nature of gear, design and manufacture some of the established nomenclature within each discipline becomes nebulous. An attempt is made here to adhere to standard gearing nomenclature whenever possible.

Figure 1.3 Two cylindrical wheels (friction wheels) in line contact. An applied force F exists between the two wheels in order to facilitate motion transmission

The purpose of toothed wheels is to transmit uniform motion from one axis to another independent of the coefficient of friction that exists between the teeth in mesh. Grant was one of the first to document a treatise on toothed wheels in mesh (1899). He reveals that at the close of the nineteenth century the design and manufacture of toothed bodies was becoming more analytical, and less of a craft. As the design and manufacture of toothed wheels became more analytical the nomenclature and terminology attained more significance. The following are some of the common terms presently used in the gear community, and additional nomenclature and terminology will be established throughout this book as the analysis of toothed bodies in mesh increases.

Pitch radius: When two cylindrical wheels (input and output wheel) are in line contact as shown in Figure 1.3, the radii of the input and output cylinders are referred to as the pitch radii upi and upo, respectively. Two cylinders are in line contact when the two axes of rotation are parallel. As the two cylinders rotate, there is no slippage at the line of contact. Motion transmission via two cylinders (friction wheels) in contact is limited by the applied radial force F and the coefficient of friction that exists between the two cylinders.

Number of teeth: In order to maintain a desired speed ratio between two axes of rotation, an integer number of teeth N must exist on each wheel. The combination of the number of teeth on each wheel and the size of the two cylinders determine the load-carrying capacity of the toothed wheels in mesh.

Transverse surface: For motion transmission between parallel axes, a transverse surface of any plane is perpendicular to the axis of rotation. The transverse surface is used to parameterize toothed wheels.

Pitch circle: The pitch circle is the intersection between a cylindrical wheel and a transverse surface. The pitch circle is used as a reference for which many calculations are based. The radius of the input pitch circle is upi, and the radius of the output pitch circle is upo.

Diametral pitch: The diametral pitch Pd is a rational expression for the number of teeth N divided by twice the pitch radius u: . The purpose for introducing such an immeasurable quantity is to specify tooth sizes using integer values. It is customary for SI designated standards to use the module m instead of the diametral pitch Pd to specify gear tooth sizes, where Pd = 1/m. The diametral pitch is always the same for two gears in mesh. Accordingly,

, where the center distance E = upi + upo. The possibility of specifying an irrational I/O relationship is alleviated by defining the pitch radii in terms of the diametral pitch. Pd < 20 is considered coarse pitch; afterward fine pitch (Pd ≤ 20).

Transverse pitch: The transverse or circular pitch pt is an irrational expression for the circumferential distance along the pitch circle between adjacent teeth:

.

Addendum circle: The addendum circle is a hypothetical circle in the transverse surface whose radius is the outermost element of any tooth. The addendum is the region between the pitch circle and the addendum circle. The amount by which the radius of the addendum circle exceeds the radius of the pitch circle is expressed in terms of an addendum constant a: ua = up + a/Pd. The active region of the gear tooth that lies in the addendum is referred to as the gear face.

Dedendum circle: The dedendum circle is a hypothetical circle in the transverse surface whose radius is the innermost element between adjacent teeth. The dedendum is the region between the pitch circle and the dedendum circle.

Center line: The two points in the transverse plane where the two axes of rotation for the input and the output wheel intersect, the transverse plane are instant centers. The line connecting these two instant centers is the center line. When the two axes of rotation are skew, the center line is the single line perpendicular to the two axes of rotation.

Center distance: The distance along the center line between the two axes of rotation is the center distance. This length is sometimes referred to as the interaxial distance.

Line of action: The line that passes through the point that is coincident with the two teeth in mesh and also perpendicular to the two teeth is the line of action.

Pitch point: The pitch point is the intersection between the center line and the line of action.

Clearance: The distance along the center line between the dedendum of one gear and the addendum of its mating gear is the clearance. Like the dedendum and addendum, the clearance is defined in terms of the clearance constant c and the diametral pitch Pd.

Tooth width: The distance along the pitch circle between adjacent profiles of a single tooth is the tooth width tt.

Tooth space: The distance along the pitch circle between two adjacent teeth is the tooth space ts. The sum of the tooth width tt plus the tooth space ts must be equal to the transverse pitch pt (i.e., pt = tt + ts).

Backlash: The amount the tooth space of one gear exceeds the tooth width of its mating gear. AGMA recommends that the face width b be proportional to tooth size. This is accomplished via the following AGMA recommendation:

Pressure angle: The included angle between the common tangent between the two pitch circles and the line of action.

IPS: A US customary system of measurements based on length, force, and time whose units are inches, pounds, and seconds, respectively.

CGS: A SI system of measurements based on length, mass, and time whose units are centimeters, grams, and seconds, respectively.

1.4 Reference Systems

Three distinct coordinate systems are used to parameterize the geometry of a gear pair. The three distinct Cartesian coordinate systems are

1. (X, Y, Z) fixed to the ground,

2. input (Xi, Yi, Zi) attached to the driving or input wheel, and

3. output (Xo, Yo, Zo) attached to the driven or output wheel.

Each reference frame is a conventional right-handed Cartesian coordinate system as depicted in Figure 1.4. The zi-axis of the input reference frame (axis of rotation for the input body) is collinear with the Z-axis of the fixed reference frame. The distance E between the two axes of rotation is a fixed distance directed along the positive X-axis of the stationary reference frame. The zo-axis of the output reference frame (axis of rotation for the output body) is perpendicular to the X–Y plane of the fixed reference frame. Associated with each of the two Cartesian coordinate systems (xi, yi, zi) and (xo, yo, zo) are, respectively, two systems of curvilinear coordinates and . The curvilinear coordinates (u, v, w) are introduced to facilitate the parameterization of gear pairs and are indistinguishable from the cylindrical coordinates (r, θ, z) for motion transmission between parallel axes. The special curvilinear coordinates (u, v, w,) will be introduced in Chapter 3 where a single system of curvilinear coordinates can be used to analyze the general case of toothed bodies in mesh.

Figure 1.4 Three Cartesian coordinates systems (X, Y, Z), (xi, yi, zi), and (xo, yo, zo) are used to parameterize toothed wheels in mesh

1.5 The Input/Output Relationship

The relationship between the fixed coordinate system (X, Y, Z) and the input coordinate system (xi, yi, zi) is defined by the net angular position vi about the input Z-axis as measured from the fixed X-axis (see Figure 1.5). Similarly, the coordinate system (xo, yo, zo) is defined by the net angular position vo about a line parallel to the Z-axis and located at a distance E along the X-axis. The I/O relationship between the angular position vi of the input body to that of the position (angular or linear) vo of the output body is defined as the transmission function. The instantaneous gear ratio g is the ratio between the instantaneous angular displacement dvo of the output and the corresponding instantaneous angular displacement dvi of the input; thus,

(1.1)

Here, the differential displacements dvi and dvo refer to an instantaneous change in angular positions vi and vo, respectively. The displacements dvi and dvo are angular displacements about the zi and zo axes, respectively. The angular speeds ωi and ωo are, respectively, the angular displacements dvi and dvo per unit time dt. For uniform motion transmission between fixed axes, the transmission function is linear and its slope is a constant equal to the gear ratio. When this occurs the gear ratio is also defined by the ratio Ni /No of gear teeth. This ratio is defined to accommodate non-circular gears and is the reciprocal of the gear ratio used by AGMA.

Figure 1.5 Basic terminology for toothed wheels in mesh

The zi-axis of the input moving reference frame and the zo-axis of the output moving frame are parallel for two external gears in mesh. The I/O relationship g is negative in this case for two external gears in mesh. Although the majority of gears are external gears, it is convenient to plot the I/O relationship g as positive for both two external gears and internal–external gears in mesh with clarification on the gear type (namely, external–external or external–internal). The elements of a gear pair are usually identified as either the gear or pinion, where the pinion is the smaller of the two gears.² It is possible in special circumstances regarding a hypoid gear pair that the pinion is physically larger than the gear and yet have fewer teeth! The reason for this phenomenon will be presented in Chapter 5. Use of gear and pinion to identify two gears in mesh does not explicitly indicate if the gear pair is used for speed increasing or speed decreasing. As a result, trailing subscripts i and o are added to identify the input and output respectively. Neither subscript is used in certain situations where a notation is applicable to both the input and the output gears.

The simplest scenario of toothed wheels in mesh is motion transmission between parallel axes. Depicted in Figures 1.5 and 1.6 is some terminology used to describe toothed wheels. In general, gear designers parameterize gear teeth in a plane. This same planar parameterization is also applied to analyze bolts and nuts, presses, rotary compressors, and planar four-bar linkages. Since motion transmission between parallel axes can be adequately illustrated in a plane perpendicular to the axis of rotation it is commonly referred to as planar motion. The ease of visualizing planar motion attributes to its usage.

Figure 1.6 Edge radius, end radius, and top round reduce nicks and burrs encountered in shipping and handling prior to assembly

1.6 Rigid Body Assumption

Initially, when analyzing the kinematic geometry of toothed bodies in mesh, it is assumed that the bodies in mesh are rigid although they will inevitably deform depending on the transmitted load. These deformations are accounted for by the compliance of the housing used to support the bearings, the deflections in the bearing supports, the bending and torsional displacements in the gear blanks and shafts, and the deflection of the teeth relative to the gear blank. The assumption of rigid bodies not only simplifies analysis but also necessary in order to initially determine the geometry of the toothed bodies in mesh. The elastic deformations are subsequently calculated and compensated for by profile modifications such as profile relief and crowning of the teeth. Due to errors encountered in manufacturing, assembly, and operation of a gear pair, the amount of profile modification varies for each gear type and is generally based on experience. If the proper modifications are not incorporated then the smooth transmission of motion from one axis to another can no longer be expected to occur, and the gear teeth will be subjected to impulsive loading producing higher stresses and noise.

1.7 Mobility

Earlier in this chapter gears were described as elements of a mechanism. Reuleaux (1876) defines a mechanism as a closed kinematic chain where one of its links is held stationary. The stationary link or ground is usually indicated by feathered marks as shown in Figure 1.7a. The mobility or the degree of freedom (dof) of a mechanism refers to the number of independent parameters that must be specified to uniquely determine the configuration or arrangement of the remaining links within the mechanism. One task of a kinematician is to specify the configuration of a mechanism given the independent parameters. The mechanism shown in Figure 1.7a has no mobility (over constrained), whereas the special mechanism shown in Figure 1.7b has mobility one. The difference between the two mechanisms is that the links 2 and 3 in Figure 1.7a are connected by a pin, whereas the two links 2 and 3 in Figure 1.7b are tangent to one another at point c. A mechanism with zero or less mobility is a structure or truss. The concept of mobility is important and far reaching when considering toothed bodies in mesh. First, a brief discussion regarding planar three link 1-dof mechanisms is discussed; then, later in Chapter 5 the more general case of a five link 1-dof mechanism will be discussed. More insight on mobility is found in many textbooks on mechanisms and kinematics (e.g., Hunt, 1978; Shigley and Uicker, 1980; Edman and Sandor, 1997).

Figure 1.7 Two mechanisms where (a) has mobility one and (b) has zero mobility

The analysis of mechanisms involves identification of the types of motion that may exists between two objects. The displacement or change in position of a point relative to a fixed coordinate system is defined as absolute displacement. The displacement or change in position and orientation of an object relative to a fixed coordinate system is defined as vehicular displacement. The displacement of a point relative to another moving coordinate system is defined as relative displacement.

Figure 1.8 Fixed coordinate moving coordinate systems

Depicted in Figure 1.8 is a movable lamina 2 (planar body) relative to the fixed coordinate system (X, Y, Z). Three independent parameters X2, Y2, and are used to specify the position and orientation of this movable lamina with respect to the fixed coordinate system (X, Y, Z). The mobility m or freedom of the lamina 2 relative to the fixed coordinate system (X, Y, Z) is three:

1. A translation ΔX2 in the X-direction

2. A translation ΔY2 in the Y-direction

3. A rotation Δθz2 about the Z-axis

Planar displacement can be parameterized by a linear combination of the above three displacement and that at any instant the displacement of lamina 2 can be reduced to a rotation about a fixed line parallel to the Z-axis (Theorem of Chasles). The point where this axis of rotation intersects the X–Y plane is the instant center of rotation for the moving lamina 2. By restricting the movable lamina 2 depicted in Figure 1.7b to only rotations about the Z-axis, the mobility of lamina 2 relative to (X, Y, Z) reduces to one (i.e., a rotation without translation). Similarly, restricting the movable lamina 3 depicted in Figure 1.7b to only rotations about a line parallel to the Z-axis (and located a distance E along the positive X-axis) also restricts the mobility of lamina 3 relative to (X, Y, Z) to one. Rosenauer and Willis (1953) define the connection between two bodies according to its mobility . If the mobility between two bodies is one then it is a lower pair, and if the mobility is greater than one then it is a higher pair. Thus, the connection between body 2 and ground as well as the connection between body 3 and ground both comprise lower pairs.

In order to assess the mobility of the three link mechanism shown in Figure 1.7b, it is necessary to determine the freedom or mobility that exists between bodies 2 and 3. There cannot exists any relative motion along the line of action l at the point of contact if contact is maintained between bodies 2 and 3. The mobility between bodies 2 and 3 increases to two (a higher pair) by restricting the relative displacement between bodies 2 and 3 to rotations about a point of the line of action l. The relative mobility between the links of a planar mechanism is given by the planar mobility criterion (Hunt, 1978)³:

(1.2) Numbered Display Equation

where m is the mobility, n is the number of bodies, k is the number of joints, and fj is the freedom at each joint. The above mobility criterion is frequently referred to as Grübler's mobility criterion and is a special form of a more general mobility criterion to be discussed in Chapter 5. Applying the above mobility criterion to the three link mechanism depicted in Figure 1.7, the mobility becomes

In this case, there are three elements or bodies: two gear elements and a fixed housing element. There are also three joints: one between each of the gear elements and the fixed housing thus comprising a total of two joints, and a third one at the point of contact between the two gear elements. The latter joint has 2 dof. Thus, the three link mechanism is a 1-dof mechanism. In other words, as one of the gears rotate, the other gear must rotate according to Equation (1.1). Caution should be exercised when using the above mobility relation. Misleading or wrong results can occur for special geometries and overconstraints. The above relation treats all joints as active and does not consider idle dof or redundant constraints.

1.8 Arhnold-Kennedy Instant Center Theorem

A point that is common to two planar bodies in motion that has the same absolute velocity is referred to as an instant center of rotation. A transverse section of a three link mechanism is shown in Figure 1.9. The intersections between the two axes of rotation zi and zo for the two bodies shown in Figure 1.9 and a transverse surface (the X–Y plane) are referred to as the instant centers of rotation ¢i and ¢o, respectively. Using the special notation ¢i and ¢o to represent axes of rotation by points is valid for motion transmission between parallel axes. Here, the absolute velocities between the fixed coordinate system (X, Y, Z) and the centers of rotation ¢i and ¢o corresponding to the two moving coordinate systems) and (xo, yo, zo), respectively, are zero, thus instant centers. Since the zi-axis of the input Cartesian coordinate system (xi, yi, zi) is coaxial with the Z-axis of the fixed Cartesian coordinate system (X, Y, Z), the point coordinates of the instant center ¢i relative to the fixed coordinate system are determined by (X, Y, Z) = (0, 0, 0). The transverse surface is defined by the X–Y plane of the fixed Cartesian coordinate system (X, Y, Z). The zo-axis of the output Cartesian coordinate system (xo, yo, zo) is perpendicular to the X–Y plane and intersects the positive X-axis at a distance E from the origin; thus, the coordinates of the instant center ¢o relative to the fixed coordinate system are determined by (X, Y, Z) = (E, 0, 0).

Figure 1.9 Two bodies in direct contact for motion transmission between parallel axes

Uniform motion transmission between two parallel axes is possible only if the line of action passes through a fixed point ¢irp known as the pitch point. The subscript irp signifies that ¢irp is the instantaneous rotation pole. The locus of pitch points (relative to the input coordinate system) for each angular position vi of the input determines the input's pitch curve or centrode. Likewise, the locus of pitch points (relative to the output coordinate system) for each angular position vo of the output determines the output's pitch curve or centrode. For uniform motion, transmission the pitch curves become circles whose radii upi and upo depends on the magnitudes of the I/O relationship g and the center distance E.

Illustrated in Figure 1.9 are the input and output bodies, the line of action l, the point of contact c, and the pitch point for the two bodies in contact. In the majority of gearing applications, the position ¢irp of the pitch point p remains invariant for each angular position vi; nonetheless, circumstances can exists (see Appendix C) where the position of the pitch point p varies for the different input positions vi. Two planar curves in direct contact are conjugate if the line of action l passes through the desired pitch point for each angular position vi of the input. In general, the line of action does not have to pass through the desired pitch point ¢irp as will be explained in Chapter 5.

Two toothed wheels in mesh are in effect two links of a three link 1-dof mechanism. Bodies 2 and 3 depicted in Figure 1.9 are in mesh. The input body is one link, the output body is another link, and ground or the hypothetical link connecting the two axes of rotation is the third link. Considering the toothed wheels in mesh as a three link 1-dof kinematic chain, any one of the three links may be held stationary. The process of holding stationary different links of a kinematic chain is known as inversion. Although the absolute motion of the three link 1-dof kinematic chain is different depending on which link is held stationary, the relative motion between the three links remains unaltered. Knowledge of the relative displacements between two elements of a mechanism is necessary for its design (both kinematic and structural). It is important to understand that the relative motion (planar in this case) between the three links necessary to define a gear pair can be obtained from the special vector loop equation:

(1.3) Numbered Display Equation

where dv12 ¢12 is the relative angular displacement of body 2 with respect to body 1 (i.e., an angular displacement dv12 about point ¢12), dv23 ¢23 is the relative angular displacement of body 3 with respect to body 2, and dv31 ¢31 is the relative angular displacement of body 1 with respect to body 3. Body 1 represents ground or the fixed reference system, body 2 is the input body, and body 3 is the output body. By holding stationary link 1 then its displacement is always zero and can also be specified using the vector loop Equation (1.3). That is, the relative displacement of body 2 with respect to body 1 plus the relative displacement of body 3 with respect to body 2 plus the relative displacement of body 1 with respect to body 3 must always sum to zero for the closed three link 1-dof kinematic chain.

The displacement dv12 ¢12 of the input body with respect to ground is denoted dvi ¢i, the displacement dv13 ¢13 (where dv13 ¢13 = − dv31 ¢31) of the output body with respect to ground is denoted dvo ¢o, and the displacement dv23 ¢23, denoted dvirp ¢irp, is the relative displacement of the output body with respect to the input body. The subscript irp is used to indicate ¢irp is the instantaneous rotation pole between bodies 2 and 3. The instantaneous angular speeds ωi and ωo are extracted from the angular displacements dvi and dvo, respectively, by dividing Equation (1.3) through by the incremental change in time dt, where ωi = dvi/dt and ωo = dvo/dt. Hunt (1978), Bottema and Roth (1979), and Phillips (1984) each present a more general treatment on the closure of general kinematic chains.

In order for two pitch circles to rotate without slippage at the point of contact (for two circles in mesh the point of contact and the pitch point are coincident), the absolute velocity of the point of contact on bodies 2 and 3 must be the same relative to the fixed coordinate system (X, Y, Z). An important theorem from planar kinematics is the Arhnold-Kennedy instant center theorem: for three rigid bodies 1, 2, and 3 in mesh, the instant centers ¢12, ¢23, and ¢31 between bodies 1 and 2, bodies 2 and 3, and bodies 3 and 1 all lie on a straight line. Applying the Arhnold-Kennedy instant center theorem to toothed wheels in mesh reveals that the pitch point (instant center ¢irp) must always lie on the line connecting the two wheel's center of rotation. In a mathematical sense, the linear combination