Angular displacements that occur about the same axis or parallel axes, on the other hand, do follow the commutative law. Also, infinitesimally small angular displacements are commutative. To avoid confusion, we will treat all finite angular displacements as scalar quantities. However, we will have occasion to treat infinitesimal angular displacements as vectors.

In analyzing motion, the first and most basic problem encountered is that of defining and dealing with the concepts of position, posture, and displacement. Since motion can be thought of as a series of displacements between successive positions of a point or postures of a body, it is important to understand exactly the meaning of the terms position and posture. Rules or conventions are established here to make the definitions precise.

The theory of machines and mechanisms is an applied science that allows us to understand the relationships between the geometry and motions of the parts of a machine, or mechanism, and the forces that produce these motions. The subject, and therefore this book, divides itself naturally into three parts. Part I, which includes Chapters 1–5, is concerned with mechanisms and the kinematics of mechanisms, which is the analysis of their motions. Part I lays the groundwork for Part II, comprising Chapters 6–10, in which we study methods of designing mechanisms. Finally, in Part III, which includes Chapters 11–16, we take up the study of kinetics, the time‑varying forces in machines and the resulting dynamic phenomena that must be considered in their design.

In previous chapters we have concentrated on the analysis of mechanisms where the dimensions of the links are known. By kinematic synthesis we mean the design or creation of a new mechanism to yield a desired set of motion characteristics. Because of the very large number of techniques available, this chapter presents only a few of the more useful approaches to show applications of the planar theory.1

A flywheel is an energy storage device. It absorbs mechanical energy by increasing its angular velocity and delivers energy by decreasing its angular velocity. Commonly, a flywheel is used to smooth the flow of energy between a power source and its load. If the load happens to be a punch press, for example, the actual punching operation requires energy for only a small fraction of its motion cycle. As another example, if the power source happens to be a two-cylinder, four-stroke engine, the engine delivers energy during only about half of its motion cycle. Other applications involve using flywheels to absorb braking energy and deliver acceleration energy for automobiles, or to act as energy-smoothing devices for electric utilities as well as solar- and wind-power-generating facilities. Electric railways have long used regenerative braking by absorbing braking energy back into power lines, but newly introduced and stronger materials now make the flywheel feasible for such purposes.

Now that we are acquainted with both mechanical and electrical systems, we would like to connect these two physical domains so that we are able to model, analyze, and design electromechanical systems. Such systems include many important sensors, actuators, and devices including motors, generators, loudspeakers, accelerometers, industrial machinery, vehicle subsystems, and more. Thus, following our earlier discussions, we will begin by reviewing the fundamental physical effects that are leveraged in these devices, and work toward extending our mechanical modeling approach to this class of multi-physics problems.

Following our discussion of translational and rotational mechanical systems, we will now extend our mechanical modeling approach (via the system decomposition technique) to electrical circuits. As in Ch. 2, we begin by considering the fundamental elements from which many electrical systems are built, and we review the corresponding mathematical element laws and electrical circuit diagrams. Then, moving forward, the focus shifts toward the rules that govern the interconnection of these elements as complete circuits and subsystems of series and parallel elements. Through these discussions, we will find that, just as Newton’s second law guided our modeling approach for mechanical systems, our treatment of electrical systems will follow Kirchhoff’s voltage and current laws.

To motivate our interest in the broad field of , let us begin by first dissociating the descriptor “dynamic” and focus solely on the objects of our interest, the “systems.” Throughout this text, we will considerof many kinds. To understand the true scope of potential applications for the tools that we will develop, it is advantageous to establish a clear vision of what qualifies as a system and why. To ensure suitability across many domains, we will prefer the flexibility of a loose conceptual definition, by which the term system will refer to any collection of elements (i.e., physical or mathematical) that have cause-and-effect relationships. Then, reintroducing “dynamic” as a reference to changes that occur over time, the field of dynamic systems encompasses the study of cause-and-effect relationships that propagate changes over time.

In this chapter, we continue our discussion of feedback control systems from the perspective of frequency-domain system properties, which were briefly introduced in Ch. 6. Accordingly, we begin this chapter with a comprehensive review of frequency-domain analysis techniques relating to the response of systems to inputs, or excitations, of the form.

Recall from our discussions in Part II that dynamic systems can be represented through various mathematical modeling strategies, including systems of ordinary differential equations, systems of Laplace-domain equations, transfer functions, functional block diagrams, and physical models. Through our review of the Laplace domain, we found that the zero-state input–output behavior of dynamic systems could be represented, conveniently, as the ratio of the system’s Laplace-domain output and its Laplace-domain input. These ratios, or transfer functions, simplified our analysis of interconnected systems and enabled the design of feedback control systems in Part III. However, while the Laplace-domain and transfer functions are undoubtedly powerful and useful tools for systems with a single input and a single output, known as(SISO) systems, the implementation of transfer functions can become cumbersome and complex for systems with multiple inputs and multiple outputs, known as(MIMO) systems.

In this chapter we will discuss an important graphical tool in feedback system design known as the . As the name suggests, this technique allows us to graph the closed-loop pole locations (i.e., the roots of the characteristic equation) on the complex plane as a function of a selected parameter (e.g., as the value of a resistor is varied or as controller gains are changed). Thus, not only does this technique allow us to extend our concept of stability analysis from the more mathematical Routh–Hurwitz approach in Ch. 8 but it also allows us to visualize the stability of a system and select its parameters to achieve desired system characteristics.