Keywords: non-linear vibration, auto-parametric systems, dynamic stability, semi-trivial solution, internal resonance.

Dynamic independency of individual parts of many non-linear multi-degree of freedom systems is often apparent only. Dynamic behavior of individual parts is independent in sub-critical states only whether their character is linear or non-linear. 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 non-symmetric system in sub-critical 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 non-linear interaction.

Many structures encountered in civil, mechanical, naval or aerospace engineering can show properties of this type. These systems are usually called auto-parametric. Their basic attributes are related with mathematical equivalents with above physical phenomena. It refers to semi-trivial solution when meaningful solution of the adequate differential system consists of non-trivial and trivial parts. Passing through the bifurcation point, the solution becomes non-trivial in all components as a rule representing the auto-parametric resonance or post-critical state.

It seems that the first theoretical studies dealing with these effects have been published in the period of 1968-1974, 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 auto-parametric 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 auto-parametric systems, some important types of post-critical response (chaotic, quasi-periodic, limit cycle, etc.) are discussed. A few hints for engineering applications are given. Some open problems are indicated.

[1]

A. van der Burgh, "On the asymptotic solution of the differential equations of the elastic pendulum", Jour. Mécanique, 7(4), 507-520, 1968.

[2]

R.S. Haxton, A.D.S. Barr, "The autoparametric vibration absorber", ASME Jour. Applied Mechanics, 94: 119-125, 1974.

[3]

A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, "Autoparametric Resonance in Mechanical Systems", Cambridge University Press, Cambridge, 2000.

[4]

J. Náprstek, M. Pirner, "Non-linear behaviour and dynamic stability of a vibration spherical absorber", in Proc. 15th ASCE Engineering Mechanics Division Conference, A. Smyth et al., (editors), Columbia Univ., New York, 2002.

[5]

J. Náprstek, C. Fischer, "Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper", Computers & Structures, special issue, to be published, 2009. doi:10.1016/j.compstruc.2008.11.015

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