Nonlinear Oscillations

Abstract

I shall survey the contents of this book before making some general comments. I close with a few specific comments on the organization, clarity, and viewpoint of the book, and on some of the topics omitted. Chapter 1 constitutes an extended abstract of the entire work. The next four chapters cover conservative, nonconservative, externally, and parametrically excited single-degree-of-freedom oscillators. The authors then go on to treat re-degree-of-freedom systems, continuous systems, and traveling waves. Each chapter ends with a set of exercises, many of which represent nontrivial pieces of research, and the 70-page bibliography contains over 1500 references. The book therefore provides a comprehensive sampler of the nonlinear vibration problems which occur in engineering. This text lies firmly within the classical engineering tradition of nonlinear oscillations; the first half of the book is essentially a new version of such texts as those by Minorsky, Hayashi, or Stoker. However, this serves as an introduction to the authors' main aim of covering recent work on multidegree of freedom and continuous systems, which they do in Chapters 5-8. Topics such as the forced oscillations of rc-degree-of-freedom systems with quadratic and cubic nonlinearities and of finite strings, beams, and plates are covered, as well as longitudinal and transverse traveling waves in bars. In connection with traveling waves, a nice presentation of the method of characteristics is given and shock fitting is discussed. The authors' recent work on modal saturation in coupled quadratic systems is also covered.

Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model

“…These terms appear naturally for large cantilever distortions, as characteristics of the geometrical non-linearity [26,27,32]. We confirmed this origin for our experiments: the non-linear signatures are temperatureindependent [35], and for samples having the same aspect ratios (while having different damping, resonance frequency, and metallic coating) the non-linear coefficients follow the same geometrical scalings (see discussion in Section II D).…”

“…We shall not be concerned with the computation of f . Suffice it to say that it is the solution of a continuum dynamics modeling based primarily on the extended Hamilton principle [41], which can be of the type of [30], or of an extension of Euler-Bernoulli's beam equations [26,27]. We present below the energetic writing resulting from this non-linear formulation.…”

“…The most sophisticated analytical modelings derive from basic continuum mechanics a system of coupled nonlinear equations describing the generic dynamics of the mechanical device [27]. This system is usually reduced to a one dimensional problem by a Galérkin-type procedure or a normal mode expansion of the linearized problem [27,31]. For beams, most of the recent work is based on the formulation of Crespo da Silva and Glynn [30].…”

“…Experimentally, the driven regime is easier to handle since it uses the natural amplification of the system through its Q factor. The most sophisticated analytical modelings derive from basic continuum mechanics a system of coupled nonlinear equations describing the generic dynamics of the mechanical device [27]. This system is usually reduced to a one dimensional problem by a Galérkin-type procedure or a normal mode expansion of the linearized problem [27,31].…”

“…This non-linearity is of pure geometrical origin [26,27], and in its most general form it will contain other terms in the dynamics equation in addition to the cubic restoring force, with a straightforward second-order force F non−lin. ∝ x 2 [28,29], and less intuitively inertia non-linear terms [30][31][32].…”