Abstract
This paper presents a comprehensive study on counter-intuitive or chaotic dynamic response of prototype discrete parameter models of single or multiple degrees of freedom subject to blast or impact loading. Non-linear dynamic behaviour of a typical single degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems are studied by taking into account cubic and quintic non-linearities, elastic perfect-plastic, and elastic-plastic-hardening and softening behaviours. The first part is founded on Duffing’s equation and Ueda’s work on strange attractors which indicates the presence of chaos in deterministic systems by using chaos detection techniques such as Poincare’s mapping and Lyapunov’s exponents and in some cases by fractal dimensions. In these deterministic problems, the system hesitates to settle between two different possible settling regimes despite the fact that the input parameters to the system are deterministic. In a SDOF model with elastic perfectly-plastic or elastic-plastic-hardening resistance function the sign of the permanent plastic deformation may or may not coincide with that of loading direction hence chaotic behaviour can be observed. For an elastic-plastic-softening system, subjected to blast loading, the problem is sensitive to the ratio of post yield stiffness to initial stiffness and for certain ranges of this parameter a small change can replicate into an abrupt change in response. Examples are included of finite elements models with many degrees of freedom of beams and plates. As most intricate engineering structures are composed of these structural elements the existence of component chaos can imply global counter-intuitive behaviour.
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Luke A. Louca is a Reader in structural engineering in the Department of Civil and Environmental Engineering at Imperial College London. He is engaged in teaching at both undergraduate and postgraduate level in both structural steel design and mitigating the effects of explosions on structures. His principal research interests, where he has published some 40 papers in leading journals and conferences, lie in the areas of behaviour and design of structures subjected to explosions and impact loads. Much of this has focussed on steel structures for offshore applications as well as defence applications using both traditional construction materials and fibre reinforced composite structures. Although much of his work is computational, laboratory testing of small scale specimens is also being conducted under dynamic loading. Large scale testing has also been carried out off-site with a number of sponsors where he has been involved in the design of the tests. Much of the work is funded by the EPSRC, Dstl/MoD, Health and Safety Executive and the Office of Naval Research (USA).
Arash S. Fallah is a Research Associate in structural engineering at Imperial College London specialising in computational and analytical modelling of blast and impact loaded monolithic, composite and hybrid structures and systems. His interests include frequency filtering in phononic metamaterials and lattices, extended finite element formulation of plated structures, nonlinear dynamics and chaos, damage and fracture in composites and constitutive visco-elastic, plastic and visco-plastic formulations for metals and composites. Much of his work is funded by EPSRC, Dstl/MoD, Health and Safety Executive and the Office of Naval Research Global and is conducted in collaboration with University of Cape Town and US Naval Academy.
S. K. A. Shah received his B.Sc. degree in Civil Engineering from University of Engineering and Technology Peshawar, Pakistan in 2007 and an M.Sc. in Earthquake Engineering from Rose School of Earthquake Engineering, Pavia, Italy in 2009. He is currently a postgraduate Research Student at Imperial College London in structural engineering. His research interests are analysis of blast loaded plates and shells, constitutive modeling of composite materials, evaluation of damage in impact and blast loaded fibre-metal laminates, structural dynamics and chaos and earthquake engineering.
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Shah, S.K.A., Fallah, A.S. & Louca, L.A. On the chaotic dynamic response of deterministic nonlinear single and multi-degree-of-freedom systems. J Mech Sci Technol 26, 1697–1709 (2012). https://doi.org/10.1007/s12206-012-0418-3
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DOI: https://doi.org/10.1007/s12206-012-0418-3