Nonlinear and chaotic dynamics of thin-walled open-section deployable structures

Abstract:

This dissertation studies the behaviour of straight, thin-walled shell-beams with a curved cross-section, or "ribs". Recent proposals for the design of large deployable structures exploit the structural simplicity and robustness of ribs as deployment actuators on spacecraft. Under dynamic loading, the low torsional rigidity of ribs, however, may cause coupling between flexural and torsional modes of vibration. Modal interactions of this type, which exhibit an energy-transfer from one mode to another, can be many orders greater than the magnitude of input excitation and may lead to catastrophic failure of the structure. An identification and prediction of nonlinear dynamic effects in ribs is critical to ensure that excessive vibrations do not compromise their performance.

This dissertation describes research into the nonlinear dynamic behaviour of cantilevered ribs subjected to sinusoidal base excitation. Regions of nonlinear resonance with single-and multi-mode periodic and aperiodic responses are studied.

A finite-element model is developed to predict natural frequencies and mode shapes, to investigate flexural and torsional buckling, to characterise geometric nonlinearities quantitatively, and to simulate nonlinear dynamic behaviour at primary resonance. A series of ribs were constructed and tested to validate the results from the finite-element model and to establish three different types of energy-transfer between modes. These include subharmonic resonances, combination resonances and non-resonant interactions between widely separated modes. It was found that subregions of some nonlinear resonance zones exhibit aperiodic motion.

Lyapunov exponents and correlation dimensions are used both to characterise the observed aperiodic motion of the ribs and to establish whether chaotic motion exists. Correlation dimension estimates show, in conjunction with dimension theory, that the bending -torsional instability is low-dimensional and that it can , in principle, be modelled with as few as two or at most three degrees of freedom.

An algorithm is developed to calculate the moment-curvature relationships from the axis of a flexible rib, whose deflected shape has been determined from the finite-element model. These relationships are used with analytical expressions to derive the strain energy in the rib. A two-degree-of-freedom analytical model of the dynamic system is developed using Lagrangre's equations and the assumed-modes method. The resulting equations of motion are integrated numerically to investigate the possible complexities in the system response.

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