Nonlinear dynamics. Between linear and impact limits
The content of this book is built on a new physical idea that the effectiveness of linear and weakly nonlinear dynamic theories is due to the spatio-temporal nature of harmonic motions associated with a subgroup of rigid-body rotations. As follows from the first two chapters of the book, these most elementary macro-level processes invoke conventional tools of elliptic complex numbers, trigonometric expansions, and the related analytical algorithms of averaging and asymptotic integration. Further, it is shown that another subgroup of rigid-body motions, namely translations with reflections, generates hyperbolic complex numbers whenever the inner non-smooth time of the reflection is introduced. This remarkable observation opens an alternative way to building constructive methods of dynamic analyses for strongly nonlinear and essentially anharmonic processes. Mathematical properties of non-smooth temporal substitutions are discussed in Chapters 4 and 5 with necessary proofs; however Section 3 of the Introduction provides necessary information for a quick start of practical applications. Then, Chapters 6, 7 and 8 illustrate applications of the developed methods to linear, weakly nonlinear and strongly nonlinear vibrating systems, respectively. In particular, applying the non-smooth temporal substitutions to linear and weakly nonlinear systems may be very effective in those cases when non-smooth or discontinuous loadings are present and thus harmonic or quasi harmonic approaches require long expansions for capturing essential features of the dynamics. Chapters 9 through 11 deal with the concept of nonlinear normal modes. In particular, formulations of the normal mode problem for impact systems are introduced. Chapter 12 presents a semi-analytical approach combining NSTT with the shooting method that essentially extends the area of applications. Chapter 13 describes a physical basis for non-smooth time substitutions for the case of essentially non-periodic processes. Finally, Chapter 14 illustrates different applications to spatially oscillating structures such as one-dimensional elastic rods with periodic discrete inclusions and two-dimensional media with discontinuous boundary sources of waves. From the methodological standpoint, this book demonstrates that new generating systems can be found by imposing the `worst case scenario' on conventional methods when a failure of one asymptotic point to its complementary counterpart. Although most of the illustrations are based on mechanical oscillators, the developed approaches are obviously well suited for applications to electric, electro-mechanical, electrochemical and other physical models generating strongly anharmonic temporal signals or spatial distributions. In my view, this original book will be of interest to the experts, professors and post-graduate students in various areas of nonlinear physics, fundamental and engineering mechanics, applied mathematics, and other fields of research dealing with nonlinear dynamic models, non-smooth or discontinuous processes.
scientific article; zbMATH DE number 5726191