High-energy dynamics are usually accompanied by most interesting physical effects; however, analyzing them by conventional tools introduces challenging mathematical problems due to strong nonlinearities. The main idea of the talk is to show that an adequate basis for understanding essentially nonlinear phenomena must also be itself essentially nonlinear but still simple enough to play the role of a basis. It is shown that such types of 'elementary' nonlinear models can be revealed by tracking the hidden links between analytical tools and subgroups of rigidbody motions, or in other terms, rigid Euclidean transformations. While the subgroup of rotations represents geometrical 'unfoldings' for linear and weakly nonlinear vibrations, the subgroup of translations with reflections can be viewed as a geometrical core of the strongly nonlinear dynamics associated with vibro-impact behaviors. Furthermore, it is found that weakly and strongly nonlinear dynamics are logically linked to elliptic and hyperbolic algebraic structures, respectively. Interestingly enough, during the last one and a half century, hyperbolic/dual numbers were re-introduced multiple times under different names as abstract counterparts to the conventional (elliptic) complex numbers. However, any constructive applications of the corresponding formalism remained quite limited since no bridge to the differential and integral operations was found. Within the present approach, the relation to hyperbolic elements is established by the normalized velocity of the standard impact oscillator, while the corresponding hyperbolic element may describe any periodic process regardless of its class of smoothness. On one hand, such observations provide the abstract theory of hyperbolic numbers with a clear physical interpretation. On the other hand, findings of this theory provide effective additional tools to nonlinear dynamic analysis. The developed methodology, based on non-smooth temporal substitutions, is illustrated with a series of sample problems, and potential applications related to acoustics in granular media are discussed.
About the Speaker
Valery N. Pilipchuk combines a Research Professor position at Mechanical Engineering of Wayne State University with the research and consulting activities in nonlinear dynamics and control at the R&D of General Motors in Michigan, USA. He graduated from the Department of Applied Theory of Elasticity of the National University of Ukraine in Dynamics and Strength of Machines, and received his Doctor of Science degree in Physics and Mathematics from Moscow Institute for Problems in Mechanics of the Russian Academy of Sciences in 1992. The degree was certified also by Kiev Institute of Mathematics of the Ukrainian National Academy of Sciences. He worked as Associate Professor and then Professor and Head of the Applied Mathematics Department of Chemistry& Technology University in Ukraine until 1998. Shortly after his visit to the University of Illinois (1994), he moves the US and works at Wayne State University (1995-1996, 1997-present) on different research projects in the areas of nonlinear dynamics and material science. He is a coauthor of books Method of Normal Vibrations for Essentially Non-Linear System (Moscow, 1989), and Normal Modes and Localization in Non-Linear Systems (Wiley, New York, 1996), and the author of Nonlinear Dynamics (Springer, 2010).
Host: Professors Larry Bergman and Alex Vakakis