**6. References**


Awrejcewicz, J. (1989). *Bifurcation and Chaos in Coupled Oscillators*. World Scientific, Singapore, 245p.

172 Nonlinearity, Bifurcation and Chaos – Theory and Applications

*Institute for Systems Analysis of RAS, Moscow, Russia* 

programs of Russian Academy of Sci. (projects 1.4, 2.5).

*Differential Equation,* Vol.44 , No.12, pp. 1682-1690.

*Stat. Phys*., Vol. 19, pp. 25--52.

Vol.46, No.11, pp.1552-1560.

Singapore, 126p.

*Math. Journal*, Vol. 26, 1, pp. 61-71.

(ether).

**Author details** 

Nikolai A. Magnitskii

**Acknowledgement** 

Singapore, 360p.

416-433.

**6. References** 

Sharkovskii windows of periodicity where recently found also in (Awrejcewicz *et al*., 2012} for initial-boundary value problems in continuous mechanical systems such as flexible plates and shallow shells. As to the processes occurring in a microcosm they, in opinion of the author, also can be successfully described by nonlinear systems of differential equations and their bifurcations. The first results in this direction are received by the author in (Magnitskii, 2010b; Magnitskii, 2011b; Magnitskii, 2012) where the basic equations and formulas of classical electrodynamics, quantum field theory and theory of gravitation are deduced from the nonlinear equations of dynamics of physical vacuum

Paper is supported by Russian Foundation for Basic Research (grants 11-07-00126-а) and

Feigenbaum, M. (1978). Quantitative universality for a class of nonlinear transformations. *J.* 

Sharkovskii, A. (1964). Cycles coexistence of continuous transformation of line in itself. *Ukr.* 

Magnitskii, N. & Sidorov, S. (2006). *New Methods for Chaotic Dynamics*. World Scientific,

Magnitskii, N. (2007). Universal theory of dynamical and spatio-temporal chaos in complex

Magnitskii, N. (2008). Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations. *Commun. Nonlinear Sci. Numer. Simul*., Vol.13, pp.

Magnitskii, N.(2008b). New approach to analysis of Hamiltonian and conservative systems.

Magnitskii, N. (2010). On topological structure of singular attractors. Differential Equations,

Awrejcewicz, J. (1989). *Bifurcation and Chaos in Simple Dynamical Systems*. World Scientific,

Magnitskii, N. (2011). *Theory of dynamical chaos.* URSS , Moscow, 320p. (in Russian)

Gilmore, R. & Lefranc, M. (2002). *The topology of chaos.* Wiley, NY, 495p.

systems. *Dynamics of Complex Systems.*, Vol. 1, 1, pp. 18-39 (in Russian).


Magnitskii, N. (2012). Theory of elementary particles based on Newtonian mechanics. In *"Quantum Mechanics/Book 1*"- InTech, pp. 107-126.

**Chapter 7** 

© 2012 Fuellekrug et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Non-Linearity in Structural Dynamics** 

During the mechanical design and development of technical systems for power plants, in civil engineering, aerospace or mechanical engineering increasing demands are made concerning the performance, weight reduction and utilization of the material. The consequence is that the dynamical behaviour and the occurrence of vibrations of the load carrying parts, the so-called primary structures are becoming more and more important. It has to be avoided that undesired vibrations can disturb or even jeopardize the intended operation. Thus, the analysis of the dynamics and vibrations of structures is an important task. To perform dynamic analyses and to draw conclusions for possibly needed changes of

First, the dynamic analysis requires computational models which may be setup with the Finite Element Method (FEM) or other adequate techniques. Second, the computational models have to be validated because otherwise no reliable theoretical predictions are possible which can be used for optimizing the mechanical design. For the validation of the computational models it is required to perform experiments on components, prototypes or the structures themselves. In many cases the structures can be considered as linear and thus linear structural dynamics methods and approaches can be applied for modelling and validation. However, in some cases non-linear effects are important and have to be taken into account (Awrejcewicz & Krysko, 2008). If this is the case, it is not sufficient to include non-linearities only in the models. Also the experimental validation has to be able to identify, characterize and quantify non-linearities (Awrejcewicz, Krysko, Papkova, & Krysko, 2012), (Krysko, Awrejcewicz, Papkova, & Krysko,

Let us first consider the dynamic equations of structures with non-linearities, then take a look at experimental dynamic identification and modal analysis before we develop basic

and reproduction in any medium, provided the original work is properly cited.

**and Experimental Modal Analysis** 

Additional information is available at the end of the chapter

the mechanical design several steps have to be carried out.

2012), (Awrejcewicz, Krysko, Papkova, & Krysko, 2012).

ideas for identifying non-linearities of structures.

Ulrich Fuellekrug

**1. Introduction** 

http://dx.doi.org/10.5772/48812
