INTRODUCTION

A gear drive composed of external and internal gears is considered. Application of such a train enables us to reduce the loss of power caused by sliding of tooth profiles. This effect is the result of the reduction of the relative angular velocity ω(12) = ω(1) − ω(2), where |ω(12)| = |ω(1) − ω(2)|. (Recall that the gears perform rotation in the same direction). However, a high reduction of |ω(12)| requires a small difference ΔN of the numbers of teeth of the pinion and the internal gear. The possible interference of tooth profiles by the gear assembly, and the undercutting of the internal gear in the process of generation, should be investigated before designation of the desired value of ΔN. Another advantage of application of an internal gear train is the possibility of reducing the dimensions of the train. Trains with internal gears are applied in planetary transmissions, in transmissions of cranes, in excavators, and so on.

The problems of undercutting of internal involute spur gears in the process of generation and interference by their assembly with pinions have been studied by several researchers. The phenomenon of undercutting of internal involute gears was first discussed by Schreier [1961]. Polder [1991] has extended this research and contributed the idea of the envelope to a family of extended hypocycloids.

INTRODUCTION

Involute helical gears are widely applied in the industry for transformation of rotation between parallel and crossed axes. Figure 16.1.1 shows an involute helical gear drive with crossed axes in 3D-space. A gear drive formed by a helical gear and a worm gear is a particular case of a gear drive with crossed axes (Figure 16.1.2). Gear tooth surfaces are in line contact for involute helical gear drives with parallel axes and in point contact for involute helical gear drives with crossed axes.

The theory of involute gears and research in this area have been presented by Litvin [1968], Colbourne [1987], Townsend [1991], and Litvin et al. [1999, 2001a, 2001c, 2001d] and the theory of shaving and honing technological processes are discussed in the works of Townsend [1991] and Litvin et al. [2001a]. Despite the broad investigation that has been accomplished in this area, the quality of misaligned involute helical gear drives is still a concern of manufacturers and designers. The main defects of such misaligned gear drives are (i) appearance of edge contact, (ii) high levels of vibration, and (iii) the shift of the bearing contact far from the central location.

To overcome the defects mentioned above, some corrections of gear geometry have been applied in the past: (i) correction of the lead angle of the pinion (requires regrinding), and (ii) crowning in the areas of the tip of the profile and the edge of the teeth (based on the experience of manufacturers).

INTRODUCTION

There are two types of worm-gear drives: (i) those with cylindrical worms (Fig. 19.1.1) (single-enveloping worm-gear drives), and (ii) those with hourglass worms (see Chapter 20) (double-enveloping worm-gear drives). The terms “single-enveloping” and “double-enveloping” are confusing because in both cases the surface of the worm-gear tooth is the envelope to the one-parameter family of worm thread surfaces that are generated in the coordinate system rigidly connected to the worm-gear. The thread surface of a cylindrical worm is a helicoid. (We recall that a helicoid is the surface that is generated by a given curve while it performs a screw motion.)

This chapter covers (i) the generation and geometry of cylindrical worms, and (ii) the basic design problems (relations between design parameters). Depending on the method for generation, we differentiate henceforth the following types of cylindrical worms (see German Standards DIN 3975):

1. (i) ZA worms, with surface A. The worm surface is a ruled surface that is generated by a straight line while it performs a screw motion with respect to the worm axis. The generating line intersects the worm axis and therefore the axial section of the worm surface is a straight line that is just the generating line. The cross section of the ZA worm is an Archimedes spiral (see Section 19.4).

2. (ii) ZN worms, with surface N. The worm surface is also a ruled surface. However, the generating line lies in a plane that passes through the perpendicular to the worm axis and forms angle λp with the worm axis (see Section 19.5). Here, λp is the lead angle on the pitch cylinder on the worm. The cross section of the worm is an extended involute (see Section 19.5)

3. […]

INTRODUCTION

Spiral bevel gears have found broad application in helicopter and truck transmissions and reducers for transformation of rotation and torque between intersected axes. Design and stress analysis of such gear drives has been a topic of research by many scientists including the authors of this book [Krenzer, 1981; Handschuh & Litvin, 1991; Stadtfeld, 1993, 1995; Zhang et al., 1995; Gosselin et al., 1996; Litvin et al., 1998a, 2002a; Argyris et al., 2002; Fuentes et al., 2002]. Reduction of noise and stabilization of bearing contact of misaligned spiral bevel gear drives are still very challenging topics of research although manufacturing companies [Gleason Works (USA), Klingelnberg–Oerlikon (Germany–Switzerland)] have developed skilled methods and outstanding equipment for manufacture of such gear drives.

The conditions of meshing and contact of spiral bevel gears depend substantially on the machine-tool settings applied. Such settings are not standardized but have to be determined for each case of design, depending on the parameters of the gears and generating tools, to guarantee the required quality of the gear drives. This chapter covers an integrated approach for the design and stress analysis of spiral bevel gears that has been developed by the authors of the book and their associates. The approach provides the solution to the following problems:

1. Determination of machine-tool settings for generation of low-noise stable bearing contact spiral bevel gear drives.

2. Computerized analysis of meshing and contact of gear tooth surfaces.

3. […]

INTRODUCTION

Generation of worms, screws, and helical gears by milling cutters or grinding wheels is considered. Two types of tools for this process are applied: (i) the finger-shaped tool, and (ii) the disk-shaped one. Figures 24.1.1(a) and 24.1.2 show the finger-shaped and disk-shaped milling cutters, respectively. There are two main problems of design when generation of helicoids is considered: (i) the tool surface Σc is given and surface Σp of the workpiece must be determined, and (ii) the inverse problem, when surface Σp is given and Σc is to be determined.

Henceforth, we use coordinate systems Sc, Sp, and Sf, which are rigidly connected to the tool, the workpiece, and the frame of the cutting machine, respectively. Figure 24.1.3 shows the installation of the disk-shaped tool. Here, rp is the pitch radius of the workpiece; rc is the cutter mean radius; Ec and γc are the shortest distance and the crossing angle between the axes of rotation of the tool and the workpiece. In the case of application of the finger-shaped cutter, coordinate systems Sc and Sf coincide, and the tool axis is xc [Fig. 24.1.1(b)].

The relative motion of the workpiece with respect to the tool is the screw one in both cases and is shown in Fig. 24.1.1(c); ψ and p are the angle of rotation and the screw parameter in the screw motion, respectively. Methods used for generation of helicoids, discussed in this chapter, have been developed by Litvin [1968].

INTRODUCTION

Computerized simulation of meshing and bearing contact is a significant achievement that could substantially improve the technology and quality of gears. Computer programs known as tooth contact analysis (TCA) programs are directed at the solution of the following basic problem:

The equations of pinion and gear tooth surfaces, the crossing angle, and the shortest distance between the axes of rotation are given. The pinion and gear tooth surfaces are in point contact. It is necessary to determine (i) the transmission errors, (ii) the paths of contact points on the gear tooth surfaces, and (iii) the bearing contact as the set of instantaneous contact ellipses.

In the case of determination of the bearing contact, it is considered that due to the elasticity of gear tooth surfaces their contact is spread over an elliptic area, and the center of the contact ellipse is the theoretical point of contact. It is also considered that the surface elastic approach is known (for instance, from the experimental data), and the problem of bearing contact can be solved as a geometric problem (see Section 8.7).

The main idea of TCA is based on simulation of tangency of tooth surfaces that are in mesh. The determination of the instantaneous contact ellipse requires knowledge of the principal directions and curvatures of the tooth surfaces that are in tangency.

A rigid body may be viewed as a special case of a system of particles in which the particles are rigidly interconnected with each other. As a consequence, the basic principles which apply to systems of particles also apply to rigid bodies. However, there are additional kinematical and dynamical formulations which are specifically associated with rigid bodies, and which contribute greatly to their analysis. This chapter will be primarily concerned with a discussion of these topics.

Kinematical preliminaries

Degrees of freedom

The configuration of a rigid body can be expressed by giving the location of an arbitrary point in the body and also giving the orientation of the body in space. This can be accomplished by first considering a Cartesian body-axis frame to be fixed in the rigid body with its origin at the arbitrary reference point for the body. The location of the origin of this xyz body-axis frame can be given by the values of three Cartesian coordinates relative to an inertial XYZ frame. The orientation of the rigid body is specified by giving the orientation of the body axes relative to the inertial frame. This relative orientation is frequently expressed in terms of three independent Euler angles which will be defined.

The number of degrees of freedom of a rigid body is equal to the number of independent parameters required to specify its configuration which, in effect, specifies the locations of all the particles comprising the body.

In the previous chapters, we have considered relatively familiar methods of obtaining the differential equations of motion for a mechanical system. In this chapter, we shall introduce a number of other methods, partly in order to give the student a broader view of dynamics, and partly to present some practical and efficient approaches to obtaining the differential equations of motion. The results presented here are applicable to holonomic and nonholonomic systems alike, but the emphasis will be on nonholonomic systems because greater insight is needed in finding applicable theoretical approaches for these systems. As we proceed, we will show the advantages of using quasi-velocities in the analysis of nonholonomic systems.

Quasi-coordinates and quasi-velocities

Transformation equations

It is often possible to simplify the analysis of dynamical systems by using quasi-velocities (us) rather than qs as velocity variables. An example is the use of Euler's equations for the rotational motion of a rigid body. Here the velocity variables are the body-axis components of the angular velocity ω rather than Euler angle rates. The ω components are quasi-velocities, whereas the Euler angle rates are true qs whose time integrals result in generalized coordinates. As we showed in Chapter 3, the Euler equations are simpler than the corresponding Lagrange equations written in terms of Euler angles and Euler angle rates.

This is a dynamics textbook for graduate students, written at a moderately advanced level. Its principal aim is to present the dynamics of particles and rigid bodies in some breadth, with examples illustrating the strengths and weaknesses of the various methods of dynamical analysis. The scope of the dynamical theory includes both vectorial and analytical methods. There is some emphasis on systems of great generality, that is, systems which may have nonholonomic constraints and whose motion may be expressed in terms of quasi-velocities. Geometrical approaches such as the use of surfaces in n-dimensional configuration and velocity spaces are used to illustrate the nature of holonomic and nonholonomic constraints. Impulsive response methods are discussed at some length.

Some of the material presented here was originally included in a graduate course in computational dynamics at the University of Michigan. The ordering of the chapters, with the chapters on dynamical theory presented first followed by the single chapter on numerical methods, is such that the degree of emphasis one chooses to place on the latter is optional. Numerical computation methods may be introduced at any point, or may be omitted entirely.

The first chapter presents in some detail the familiar principles of Newtonian or vectorial dynamics, including discussions of constraints, virtual work, and the use of energy and momentum principles. There is also an introduction to less familiar topics such as differential forms, integrability, and the basic theory of impulsive response.

In the study of dynamics at an advanced level, it is important to consider many approaches and points of view in order that one may attain a broad theoretical perspective of the subject. As we proceed we shall emphasize those methods which are particularly effective in the analysis of relatively difficult problems in dynamics. At this point, however, it is well to review some of the basic principles in the dynamical analysis of systems of particles. In the process, the kinematics of particle motion will be reviewed, and many of the notational conventions will be established.

Particle motion

The laws of motion for a particle

Let us consider Newton's three laws of motion which were published in 1687 in his Principia. They can be stated as follows:

1. I. Every body continues in its state of rest, or of uniform motion in a straight line, unless compelled to change that state by forces acting upon it.

2. II. The time rate of change of linear momentum of a body is proportional to the force acting upon it and occurs in the direction in which the force acts.

3. III. To every action there is an equal and opposite reaction; that is, the mutual forces of two bodies acting upon each other are equal in magnitude and opposite in direction.