J. Michael T. Thompson

Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics.

In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load–deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

Nonlinear buckling behaviour of spherical shells: barriers and symmetry-breaking dimples

The nonlinear axisymmetric post-buckling behaviour of perfect, thin, elastic spherical shells subject to external pressure and their asymmetric bifurcations are characterized, providing results for a structure/loading combination with an exceptionally nonlinear buckling response. Immediately after the onset of buckling, the buckling mode localizes into a dimple at the poles. The relations among the pressure, the dimple amplitude and the change in volume of the shell are determined over a large range of pole deflections. These results allow accurate evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same and evaluation of the influences of pressure versus volume-controlled loadings. Non-axisymmetric bifurcation from the axisymmetric state, which occurs deep into the post-buckling regime in the form of multi-lobed dimples, is also established and discussed. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

Under increasing compression, an unbuckled shell is in a metastable state which becomes increasingly precarious as the buckling load is approached. So to induce premature buckling, a lateral disturbance will have to overcome a decreasing energy barrier which reaches zero at buckling. Two archetypal problems that exhibit a severe form of this behavior are the axially-compressed cylindrical shell and the externally pressurized spherical shell. Focusing on the cylinder, a nondestructive technique was recently proposed to estimate the “shock-sensitivity” of a laboratory specimen using a lateral probe to measure the nonlinear load-deflection characteristic. If a symmetry-breaking bifurcation is encountered on the path, computer simulations showed how this can be suppressed by a controlled secondary probe. Here, we extend our understanding by assessing in general terms how a single control can capture remote saddle solutions: in particular, how a symmetric probe could locate an asymmetric solution. Then, more s...

Nonlinear buckling interaction for spherical shells subject to pressure and probing forces

Elastic spherical shells loaded under uniform pressure are subject to equal and opposite compressive probing forces at their poles to trigger and explore buckling. When the shells support external pressure, buckling is usually axisymmetric; the maximum probing force and the energy barrier the probe must overcome are determined. Applications of the probing forces under two different loading conditions, constant pressure or constant volume, are qualitatively different from one another and fully characterized. The effects of probe forces on both perfect shells and shells with axisymmetric dimple imperfections are studied. When the shells are subject to internal pressure, buckling occurs as a non-axisymmetric bifurcation from the axisymmetric state in the shape of a mode with multiple circumferential waves concentrated in the vicinity of the probe. Exciting new experiments by others are briefly described.

Climate predictions: the influence of nonlinearity and randomness

The current threat of global warming and the public demand for confident projections of climate change pose the ultimate challenge to science: predicting the future behaviour of a system of such overwhelming complexity as the Earths climate. This Theme Issue addresses two practical problems that make even prediction of the statistical properties of the climate, when treated as the attractor of a chaotic system (the weather), so challenging. The first is that even for the most detailed models, these statistical properties of the attractor show systematic biases. The second is that the attractor may undergo sudden large-scale changes on a time scale that is fast compared with the gradual change of the forcing (the so-called climate tipping).

Future perspectives in astronomy and the earth sciences

This article is an overview of the contributions to the Triennial Issue of Phil. Trans. R. Soc. A published in December, 2005, and also plays the role of a Preface. Devoted to the work of young scientists, the issue covers the fields of astronomy and earth science.

Probing Shells Against Buckling: A Nondestructive Technique for Laboratory Testing

This paper addresses testing of compressed structures, such as shells, that exhibit catastrophic buckling and notorious imperfection sensitivity. The central concept is the probing of a loaded structural specimen by a controlled lateral displacement to gain quantitative insight into its buckling behavior and to measure the energy barrier against buckling. This can provide design information about a structure’s stiffness and robustness against buckling in terms of energy and force landscapes. Developments in this area are relatively new but have proceeded rapidly with encouraging progress. Recent experimental tests on uniformly compressed spherical shells, and axially loaded cylinders, show excellent agreement with theoretical solutions. The probing technique could be a valuable experimental procedure for testing prototype structures, but before it can be used a range of potential problems must be examined and solved. The probing response is highly nonlinear and a variety of complications can occur. Here, ...

Nonlinear dynamic Interactions between flow-induced galloping and shell-like buckling

Abstract For an elastic system that is non-conservative but autonomous, subjected for example to time-independent loading by a steadily flowing fluid (air or water), a dangerous bifurcation, such as a sub-critical bifurcation, or a cyclic fold, will trigger a dynamic jump to one or more remote stable attractors. When there is more than one candidate attractor, the one onto which the structure settles can then be indeterminate, being sensitive to infinitesimally small variations in starting conditions or parameters. In this paper we develop and study an archetypal model to explore the nonlinear dynamic interactions between galloping at an incipient sub-critical Hopf bifurcation of a structure with shell-like buckling behaviour that is gravity-loaded to approach a sub-critical pitch-fork bifurcation. For the fluid forces, we draw on the aerodynamic coefficients determined experimentally by Novak for the flow around a bluff body of rectangular cross-section. Meanwhile, for the structural component, we consider a variant of the propped-cantilever model that is widely used to illustrate the sub-critical pitch-fork: within this model a symmetry-breaking imperfection makes the behaviour generic. The compound bifurcation corresponding to simultaneous galloping and buckling is the so-called Takens-Bodganov Cusp. We make a full unfolding of this codimension-3 bifurcation for our archetypal model to explore the adjacent phase-space topologies and their indeterminacies.

James Clerk Maxwell 150 years on

This paper is the preface to a special Issue of Phil. Trans. R. Soc. A reporting selected proceedings of the international conference marking the 150th anniversary of James Clerk Maxwells professorial debut at Marischal College, Aberdeen. Following an introduction to Marischal College, a brief historical note summarizes Maxwells life prior to his entering the college as professor of natural philosophy. The preface provides a short summary of the event and overviews the contributed papers devoted to subjects covering a wide range of Maxwells research interests and their modern developments. The mixture of review and research papers reflects both the fundamental importance and the diverse applicability of Maxwells works in electromagnetics, colour science, dynamics and kinetics. Acknowledgements are given to the individuals and bodies who made the conference the success that it was.

Introduction. Progress in Earth science and climate studies

In this introductory paper, I review the ‘visions of the future’ articles prepared by top young scientists for the second of the two Christmas 2008 Triennial Issues of Phil. Trans. R. Soc. A, devoted respectively to astronomy and Earth science. Topics covered in the Earth science issue include: trace gases in the atmosphere; dynamics of the Antarctic circumpolar current; a study of the boundary between the Earths rocky mantle and its iron core; and two studies of volcanoes and their plumes. A final section devoted to ecology and climate covers: the mathematical modelling of plant–soil interactions; the effects of the boreal forests on the Earths climate; the role of the past palaeoclimate in testing and calibrating todays numerical climate models; and the evaluation of these models including the quantification of their uncertainties.