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Auto-parametric Stability Loss and Post-critical Behaviour of a Three Degrees of Freedom System

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Computational Methods in Stochastic Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 22))

Abstract

Slender structures exposed to a strong vertical component of an earthquake excitation are endangered by auto-parametric resonance effect. This non-linear dynamic process in a post-critical regime caused heavy damages or collapses of many towers, bridges and other structures due to earthquake attack in the epicenter area.

If an amplitude of a vertical harmonic excitation in a structure foundation exceeds a certain limit, vertical response at the top of the structure looses its stability and a dominant horizontal response component arises. In subcritical linear regime only semi-trivial solution being represented by the vertical response component is observed, while horizontal components remain trivial. Therefore, vertical and horizontal response components are independent.

In post-critical regime (auto-parametric resonance) the dynamic non-linear interaction of vertical and horizontal response components occurs. This process results in dominant horizontal response components leading to a failure of the structure. The broadband random non-stationary excitation of the seismic type can be particularly dangerous in this connection.

Qualitatively different post-critical response types have been detected when sweeping throughout the frequency interval beyond the stability limit. Deterministic as well as chaotic response types have been observed. In general they can occur in a steady state, quasi-periodic and completely undetermined regimes including a possible energy transflux among degrees of freedom. Post-critical limit cycles are discussed. The processes of transition from semi-trivial to post-critical state are investigated in case of time limited excitation period.

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Acknowledgements

The support of the Czech Scientific Foundation No. 103/09/0094, Grant Agency of the ASCR No. IAA200710902, IAA200710805 and AV0Z20710524 research plan are gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics AS CR v.v.i., Prosecka 76, Prague 9, Czech Republic

    Jiří Náprstek & Cyril Fischer

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  1. Jiří Náprstek
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  2. Cyril Fischer
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Correspondence to Jiří Náprstek .

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Editors and Affiliations

  1. Inst. Structural Analysis &, Seismic Research, National Technical Univ. Athen, Iroon Polytechniou Str. 9, Athens, 157 80, Greece

    Manolis Papadrakakis

  2. , Intitute of Structural Analysis and Seis, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    George Stefanou

  3. , Institute of Structural Analysis and Se, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    Vissarion Papadopoulos

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Náprstek, J., Fischer, C. (2011). Auto-parametric Stability Loss and Post-critical Behaviour of a Three Degrees of Freedom System. In: Papadrakakis, M., Stefanou, G., Papadopoulos, V. (eds) Computational Methods in Stochastic Dynamics. Computational Methods in Applied Sciences, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9987-7_14

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  • DOI: https://doi.org/10.1007/978-90-481-9987-7_14

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