Until now, we have been concerned with the kinematics of particles where the objective has been to determine the motion of a particle or rigid body without regard to the cause of the motion. Clearly in any physical system motion cannot occur without the application of some kind of external stimulus. In particular, in order for a particle to accelerate, it is necessary to apply a force to the particle. In order to study the motion that results from the application of a force (or, in general, the application of multiple forces) to a particle, it is necessary to study the kinetics of the particle. The objective of kinetics is threefold: (1) to describe quantitatively the forces that act on a particle; (2) to determine the motion that results from the application of these forces using postulated laws of physics; and (3) to analyze the motion.

The first topic in this chapter is the development of models for forces that are commonly used in dynamics. In particular, models are developed for contact forces, spring forces, and gravitational forces. These models will be used throughout the remainder of this book when solving problems.

The next topic in this chapter covers Newton's laws, which are the fundamental postulates that govern the nonrelativistic motion of particles.

The first topic in the study of dynamics is kinematics. Kinematics is the study of the geometry of motion without regard to the forces the cause that motion. For any system (which may consist of a particle, a rigid body, or a system of particles and/or rigid bodies) the objectives of kinematics are fourfold: to determine (1) a set of reference frames in which to observe the motion of a system; (2) a set of coordinate systems fixed in the chosen reference frames; (3) the angular velocity and angular acceleration of each reference frame (and/or rigid body) resolved in the chosen coordinate systems; and (4) the position, velocity, and acceleration of each particle in the system. In order to develop a comprehensive and systematic approach, the study of kinematics given in this Chapter is divided into two parts: (1) the study of kinematics of particles and (2) the study of kinematics of rigid bodies.

This Chapter is organized as follows. First, both a qualitative and precise definition of a reference frame is given. In particular, it is discussed that a reference frame provides a perspective from which to observe the motion of a system.

In Chapter 3 we discussed the important principles and methods used in the formulation, solution, and analysis of the motion of a single particle. In this chapter we extend the results of particle kinetics to systems consisting of two or more particles.

The first topic covered in this chapter is the center of mass of a system of particles. Using the definition of the center of mass, the linear momentum of a system of particles is defined. Then, using the definition of linear momentum, the velocity and acceleration of the center of mass of the system are defined.

The second topic covered in this chapter is the angular momentum of a system of particles. In particular, expressions for the angular momentum are derived relative to an arbitrary point, an inertially fixed point, and the center of mass of the system. Then, relationships between these three different forms of angular momentum are derived.

The third and fourth topics covered in this chapter are Newton's 2nd law and the rate of change of angular momentum for a system of particles. In particular, it is shown that the center of mass of the system satisfies Newton's 2nd law. Furthermore, the key results relating the rate of change of angular momentum for a system of particles to moment applied to the system are derived.

Until now we have been concerned with the kinetics of particles, i.e., the kinetics of objects that have nonzero finite mass but do not occupy any physical space. Furthermore, in Section 2.15 of Chapter 2 we studied the kinematics of motion of a rigid body. In this chapter we turn our attention to the kinetics of rigid bodies. To this end, the objectives of this chapter are threefold: (1) to describe quantitatively the forces and moments that act on a rigid body; (2) to determine the motion that results from the application of these forces and moments using postulated laws of physics; and (3) to analyze the motion.

The key difference between a particle and rigid body is that a particle can undergo only translational motion whereas a rigid body can undergo both translational and rotational motion. In general, for motion in ℝ3 it is necessary to specify three variables for the translational motion and to specify another three variables for the rotational motion of the rigid body.

Mechanics is the study of the effect that physical forces have on objects. Dynamics is the particular branch of mechanics that deals with the study of the effect that forces have on the motion of objects. Dynamics is itself divided into two branches called Newtonian dynamics and relativistic dynamics. Newtonian dynamics is the study of the motion of objects that travel with speeds significantly less than the speed of light while relativistic dynamics is the study of the motion of objects that travel with speeds at or near the speed of light. This division in the subject of dynamics arises because the physics associated with the motion of objects that travel with speeds much less than the speed of light can be modeled much more simply than the physics associated with the motion of objects that travel with speeds at or near the speed of light. Moreover, nonrelativistic dynamics deals primarily with the motion of objects on a macroscopic scale while relativistic dynamics deals with the study of the motion of objects on a microscopic or submicroscopic scale.

The subject of dynamics has been taught in engineering curricula for decades, traditionally as a second-semester course as part of a year-long sequence in engineering mechanics. This approach to teaching dynamics has led to a wide array of currently available engineering mechanics books, including Beer and Johnston (1997), Bedford and Fowler (2005), Hibbeler (2001), and Merriam and Kraige (1997). From my experience, the reasons these books are adopted for undergraduate courses in engineering mechanics are threefold. First, they include a wide variety of worked examples and have more than 1000 problems for the students to solve at the end of each chapter. The variety of problems provides instructors with the flexibility to assign different problems every semester for several years. Second, these books are generic enough that they can be used to teach undergraduates in virtually any branch of engineering. Third, they cover both statics and dynamics, thereby making it is possible for a student to purchase a single book for a year-long engineering mechanics course. Using these empirical measures, it is hard to dispute that these books cover a tremendous amount of material and enable an instructor to tailor the material to the needs of a particular course. Given the vast array of undergraduate dynamics books already available, an obvious question that arises is, why write yet another book on the subject of undergraduate engineering dynamics?

INTRODUCTION

Coordinate measurements of gear real tooth surfaces enable us to determine surface deviations from the ideal surface. The goal is to minimize the surface deviations by proper correction of the initial machine-tool settings.

The technological aspects of the problem are as follows:

1. (i) The deviations of real tooth surfaces are inevitable due to surface distortion by heat-treatment, errors in initial machine-tool settings, deflection by manufacturing, and so on.

2. (ii) Application of an additional finishing operation for elimination of the deviations would be too expensive in comparison with the approach based on corrections of initially applied machine-tool settings. The advantage of this approach is the possibility of using the same equipment to correct the deviations. The disadvantage is that the approach will be successful only if the deviations are repeatable.

3. (iii) The coordinate measurements must be performed with high precision, which currently prohibits them from being performed simultaneously with the manufacturing. Therefore, the coordinate measurements are performed after manufacturing, but only the first gear of the whole gear set to be manufactured is tested.

4. (iv) In some cases master-gears are used and the coordinate measurements provide the information about the deviations from the master-surface for the surface being tested. The authors consider this approach less effective than computerized determination of surface deviations and corrections of machine-tool settings.

The described technique has been developed in response to the increasing requirements of high quality gear transmissions.

INTRODUCTION

The information on surface curvatures is required for computerized simulation of contact of gear tooth surfaces (see Chapter 9), and grinding of ruled undeveloped surfaces (see Chapter 26). The main ideas of surface curvatures have been developed in differential geometry by many distinguished scientists. This chapter provides condensed information about the basic equations of surface curvatures. For more details, we refer the reader to Nutbourne & Martin [1988], Finikov [1961], Favard [1957], Rashevski [1956], and Vigodsky [1949]. The chapter covers the following basic topics:

1. Representation of a spatial curve in 3D-space and on a surface

2. Geodesic and normal curvatures

3. Curve and surface torsions

4. First and second fundamental forms

5. Principal curvatures and directions and three types of surface points.

SPATIAL CURVE IN 3D-SPACE

Osculating Plane

Figure 7.2.1 shows spatial curve L1ML2. The osculating plane is the limiting position of such a plane that passes through curve points M1, M, and M2 as M1 and M2 approach M. The osculating plane for a curve at its regular point M is formed by the tangent to the curve and the acceleration vector for the same point.

The osculating plane and the curve are in tangency of second order. The osculating plane is an exceptional tangent plane: the deviations of the curve from the osculating plane are of different signs at two sides from the point of tangency, and the curve is above and below the plane (see points L1 and L2 in Fig. 7.2.1).