Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design

Chaotic Behavior Resulting in Transient and Steady State Instabilities of Pressure-Loaded Shallow Spherical Shells

“…This allows one to use various mathematical techniques and numerical tools recently developed for the global investigation of low-dimensional systems. In particular, one can examine the global behavior of the system in terms of transient basins of attraction or safe basins in which transient motions, from a given set of starting conditions, do not exceed a given failure criterion within a specified duration of the excitation [19,27,32]. It will be assumed for this study, based on numerical results, that for shallow spherical caps the first mode is dominant and a simplified one-degree-of-freedom (1 d.o.f) model is capable of describing with a reasonable degree of accuracy the non-linear behavior of the cap; issues such as modal coupling which might result in internal resonances will be considered in future research.…”

“…As parameters are varied, such as the forcing frequency for example, basins of attraction can undergo quantitative and qualitative changes; often over a small parameter change there can be rapid erosion and stratification of the basin at a fraction of the forcing level in which the final steady state solution loses its stability; hence a stability analysis which solely considers the steady state and neglects this global transient behavior, may be seriously non-conservative [27][28][29]. In conclusion, the design implications of this theoretical analysis are assessed; dynamic buckling criteria can be established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading operating under either steady state or transient conditions.…”

“…Since the initial conditions, or even the parameters, may vary widely, and indeed are often unknown, attention is given to analyzing all possible transient motions. Hence, the global behavior of the system is examined in terms of transient basins of attraction; safe basins can be defined in which transient motions, from a given set of starting conditions, do not exceed a given failure criterion within a specified duration of the excitation [19,27].…”

On the Design Synthesis of Multistable Equilibrium Systems

“…This dissertation only considers locally linearized dynamics to characterize the systems. The general area of nonlinear dynamics is extensively studied by many [Simiu, 2002;Simiu and Franaszek, 1997;Thompson, 1996]. Instead of only describing MSE systems by their potential energy and effective inertias, expansion of the MSE design methodology could describe phase portraits or shapes of the Hamiltonian adding both potential and kinetic energy.…”

Dynamic characteristics of a digital transducer

“…An approach used by engineers interested in the stability of marine structures is to subject mathematical models of dynamic structures to a range of sinusoidal input stimuli, [4], this is essentially a computer based approach.…”