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.

Master the principles of structural dynamics with this comprehensive and self-contained textbook, with key theoretical concepts explained through real-world engineering applications.

• The theory of natural modes of vibration, the finite element method, and the dynamic response of structures is balanced with practical applications to give students a thorough contextual understanding of the subject.

• Enhanced coverage of damping, rotating systems, and parametric excitation provides students with superior understanding of these essential topics.

• Examples and homework problems, closely linked to real-world applications, enrich and deepen student understanding.

• Curated mathematical appendices equip students with all the tools necessary to excel, without disrupting coverage of core topics.

• Containing all the material needed for a one- or two-semester course, and accompanied online by MATLAB/Python code, this authoritative textbook is the ideal introduction for graduate students in aerospace, mechanical, and civil engineering.

Master the principles of structural dynamics with this comprehensive and self-contained textbook, with key theoretical concepts explained through real-world engineering applications.

• The theory of natural modes of vibration, the finite element method, and the dynamic response of structures is balanced with practical applications to give students a thorough contextual understanding of the subject.

• Enhanced coverage of damping, rotating systems, and parametric excitation provides students with superior understanding of these essential topics.

• Examples and homework problems, closely linked to real-world applications, enrich and deepen student understanding.

• Curated mathematical appendices equip students with all the tools necessary to excel, without disrupting coverage of core topics.

• Containing all the material needed for a one- or two-semester course, and accompanied online by MATLAB/Python code, this authoritative textbook is the ideal introduction for graduate students in aerospace, mechanical, and civil engineering.

Master the principles of structural dynamics with this comprehensive and self-contained textbook, with key theoretical concepts explained through real-world engineering applications.

• The theory of natural modes of vibration, the finite element method, and the dynamic response of structures is balanced with practical applications to give students a thorough contextual understanding of the subject.

• Enhanced coverage of damping, rotating systems, and parametric excitation provides students with superior understanding of these essential topics.

• Examples and homework problems, closely linked to real-world applications, enrich and deepen student understanding.

• Curated mathematical appendices equip students with all the tools necessary to excel, without disrupting coverage of core topics.

• Containing all the material needed for a one- or two-semester course, and accompanied online by MATLAB/Python code, this authoritative textbook is the ideal introduction for graduate students in aerospace, mechanical, and civil engineering.

Master the principles of structural dynamics with this comprehensive and self-contained textbook, with key theoretical concepts explained through real-world engineering applications.

• The theory of natural modes of vibration, the finite element method, and the dynamic response of structures is balanced with practical applications to give students a thorough contextual understanding of the subject.

• Enhanced coverage of damping, rotating systems, and parametric excitation provides students with superior understanding of these essential topics.

• Examples and homework problems, closely linked to real-world applications, enrich and deepen student understanding.

• Curated mathematical appendices equip students with all the tools necessary to excel, without disrupting coverage of core topics.

• Containing all the material needed for a one- or two-semester course, and accompanied online by MATLAB/Python code, this authoritative textbook is the ideal introduction for graduate students in aerospace, mechanical, and civil engineering.