Review in Theoretical and Applied Mechanics

Stability theory is one of the most important and rapidly developing field of applied mathematics and mechanics. There are many texts on the subject of stability, such as books by Liapunov (1892), Chetayev (1961), Bolotin (1963), Ziegler (1968),  Huseyin (1986), Kounadis and Kratzig (1995) and Thomsen (1997), to mention just a few. The present book represents an important addition to the existing literature on the subject.

The presentation given in the book is different, in many respects, from the presentation in other works. The main approach in studying stability of linear systems when the parameters of the system are changed is based on the authors own research. The central theoretical result obtained in this book is new detailed analysis of bifurcation of eigenvalues depending on a set of parameters. This result is consequently used throughout the book.

The book is divided into 12 Chapters, Bibliography with more than 170 references to the (mostly) newer works, and Index.

In Chapter 1, the authors present essentials of the Stability theory, stability based on the first approximation, as well as the basis for deriving equations of motion of mechanical systems. In Chapter 2 the analysis of bifurcation of eigenvalues depending on parameters is presented. This may be considered as the main result used in all subsequent analysis. In Chapter 3 the general system of differential equations with coefficients smoothly dependent on parameters is analyzed. Stability boundary and its singularities are examined. In Chapter 4 the bifurcation theory of roots of characteristic polynomials dependent on parameters is presented. Chapter 5 deals with linear conservative systems. The change of frequencies, that depend on parameters, is examined. Here an important comment concerning the problem (well-known in structural optimization theory) of bimodal optimization is made. In Chapter 6 the problem of gyroscopic stabilization is explained in terms of bifurcation of eigenvalues. In Chapter 7 the Hamiltonian systems are studied and the corresponding singularities of the stability boundary are presented. Chapter 8 treats several mechanical systems (airfoil flutter problem, pipe conveying fluid, double pendulum with follower force, and two aeroelastic stability problems) in which stability boundaries and paradoxes associated with bifurcation are explained. In Chapter 9 the multi-parameter stability theory of periodic systems is analyzed. Here, again, the connection with optimization is stressed (the objective function is excitation amplitude). With those results, in Chapter 10 stability boundaries of periodic systems are analyzed. In Chapter 11 the influence of small damping on the stability of systems with small periodic excitation is presented, and finally in Chapter 12 the non-conservative systems with small periodic excitation are analyzed.

The layout of the book is pleasant and the figures are well drawn. The authors are to be commended for their effort in doing presentation clear without sacrificing mathematical rigor. The fact that the authors are active researchers in the field, and that the book brings some of their own results to the wider readership (up to now those results were mostly available in Journals) makes this book especially valuable.

The material covered in the book could be used as a basis for a graduate course in mechanical, aerospace or civil engineering, as well as in applied mathematics courses on stability. Researchers in those fields will also find this book an important addition to the existing literature. To all those the book is warmly recommended. It is my opinion that it will become classic in the field.