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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 22
TRENDS IN CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping, L.F. Costa Neves, R.C. Barros
Chapter 14
NonLinear AutoParametric Vibrations in Civil Engineering Systems J. Náprstek
Institute of Theoretical and Applied Mechanics ASCR, v.v.i., Prague, Czech Republic J. Náprstek, "NonLinear AutoParametric Vibrations in Civil Engineering Systems", in B.H.V. Topping, L.F. Costa Neves, R.C. Barros, (Editors), "Trends in Civil and Structural Engineering Computing", SaxeCoburg Publications, Stirlingshire, UK, Chapter 14, pp 293317, 2009. doi:10.4203/csets.22.14
Keywords: nonlinear vibration, autoparametric systems, dynamic stability, semitrivial solution, internal resonance.
Summary
Dynamic independency of individual parts of many nonlinear multidegree of freedom systems is often apparent only. Dynamic behavior of individual parts is independent in subcritical states only whether their character is linear or nonlinear. It means in particular that vibrations of one part does not influence a movement of any other part. It holds generally in conditions of low level excitation, high damping, when dealing with systems free of internal resonance or with nonsymmetric system in subcritical state, etc. Getting through a certain bifurcation point in a space of system or excitation parameters, the system can lose dynamic stability. In this moment the relevant terms are put into effect, which brings corresponding parts into a complicated nonlinear interaction.
Many structures encountered in civil, mechanical, naval or aerospace engineering can show properties of this type. These systems are usually called autoparametric. Their basic attributes are related with mathematical equivalents with above physical phenomena. It refers to semitrivial solution when meaningful solution of the adequate differential system consists of nontrivial and trivial parts. Passing through the bifurcation point, the solution becomes nontrivial in all components as a rule representing the autoparametric resonance or postcritical state. It seems that the first theoretical studies dealing with these effects have been published in the period of 19681974, see e.g. [1] or [2]. As the most comprehensive looks the monograph [3], where leading authors summarized a contemporary state of the art. Some motivations, particular solution steps and selected results can be found e.g. in [4], [5], etc. The aim of this study is to present some overview of dynamic problems in the category of the autoparametric systems. There are mentioned several types of damping devices linked with basic engineering structures. Some problems concerning seismic resistance of high slender structures under vertical excitation, ship stability on water waves and the railway car moving on a deformable track are outlined as well. In order to clarify basic theoretical attributes of autoparametric systems, some important types of postcritical response (chaotic, quasiperiodic, limit cycle, etc.) are discussed. A few hints for engineering applications are given. Some open problems are indicated. References
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