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This book presents a concise, clear, and consistent account of the methodology of phase synchronization, an extension of modal analysis to decouple any linear system in real space. It expounds on the novel theory of phase synchronization and presents recent advances, while also providing relevant background on classical decoupling theories that are used in structural analysis. The theory is illustrated with a broad range of examples. The theoretical development is also supplemented by applications to engineering problems. In addition, the methodology is implemented in a MATLAB algorithm which can be used to solve many of the illustrative examples in the book.

This book is suited for researchers, practicing engineers, and graduate students in various fields of engineering, mathematics, and physical science.

### Chapter 1. Linear Systems and Configuration-Space Decoupling Techniques

Abstract
The equation of motion of linear systems is one of the most commonly used equations in science and engineering. It has long been recognized that coordinate coupling in linear systems is a considerable barrier to analysis and design. In this context, it is common for engineers to seek coordinate decoupling techniques. Coordinate decoupling is the process of simultaneously diagonalizing the coefficient matrices of a dynamical system. The main objective of this chapter is to provide an overview of configuration-space decoupling techniques. Three decoupling algorithms are provided and eleven examples are supplemented.
Rubens Gonçalves Salsa Junior, Fai Ma

### Chapter 2. Phase Synchronization and the Physics of Damping

Abstract
A classically damped linear system possesses classical normal modes, which constitute a linear coordinate transformation that decouples the undamped system. This process of decoupling the equation of motion of a dynamical system is the time-honored procedure termed modal analysis. In general, damping is not classical and thus passive linear systems cannot be decoupled by modal analysis. This chapter shows how classical modal analysis can be extended to decouple damped linear systems in configuration space through a procedure called phase synchronization. Similar to modal analysis, decoupling by phase synchronization possesses ample physical insight based upon a consideration of the physics of damping. In this regard, phase angles are shifted in each non-classically damped mode of vibration so as to transform it into a classical mode. Eleven illustrative examples are provided.
Rubens Gonçalves Salsa Junior, Fai Ma

### Chapter 3. Decoupling of Linear Systems by Phase Synchronization

Abstract
This chapter shows how phase synchronization can be applied to general systems, whose coefficient matrices lack the usual properties of symmetry and positive-definiteness. Systems possessing distinct, repeated, or defective eigenvalues are addressed. An algorithm for decoupling via phase synchronization is provided. Seven examples are supplied for illustration.
Rubens Gonçalves Salsa Junior, Fai Ma

### Chapter 4. Selected Applications

Abstract
This chapter outlines applications for which the method of decoupling by phase synchronization can be used as the main theoretical tool. The applications include: computation of invariant of motion, derivation of a canonical form of the equation of motion, characterization of oscillatory behavior in free vibration and modal reduction of a system under base excitation. Several illustrative examples are supplied for theoretical developments.
Rubens Gonçalves Salsa Junior, Fai Ma