The nonlinear dynamical and shock mitigation properties of tapered chains

An analytic and numerical study of the problem of mechanical impulse propagation through a horizontal alignment of progressively shrinking (tapered) elastic spheres that are placed between two rigid end walls is investigated. Particular attention is paid to the shock absorption and nonlinear dynamical properties as they pertain to energy partition. The studies are confined to cases where initial loading between the spheres is zero. The spheres are assumed to interact via the purely repulsive and strongly nonlinear Hertz potential. Two systems are studied, each representing a staggering number of possible chain designs. Propagation of energy is analytically studied in the hard-sphere approximation and parameter space diagrams plotting normalized kinetic energy of the smallest grain at the tapered end are developed for various chain lengths and tapering factors. These details are then compared to congruent diagrams obtained via extensive dynamical simulations. Our figures indicate that the ratios of the kinetic energies of the smallest to largest grains possess a gaussian dependence on tapering and an exponential decay when the number of grains increases. The conclusions are independent of system size, thus being applicable to tapered alignments of micron-sized spheres as well as those that are macroscopic and more easily realizable in the laboratory. The results demonstrate the capability of these chains to thermalize propagating impulses and thereby act as potential shock absorbing devices. While inertial mismatches in these granular chains lead to remarkable energy absorption, short chains are found to be limited in that regard. A second granular system is therefore proposed and investigated which greatly improves performance for any size chain. These new systems feature surprisingly complicated dynamics and are inadequately represented by a hard-sphere approximation. Additionally, such systems have shock absorption capacities that vary as a function of position along the chain, enabling customized shock absorbers.;Additional studies investigate energy partitioning and fluctuations are investigated. Approximate power laws are developed which fit the decay of average fluctuations as the size of the system increases. Advanced simulations of tapered chains utilizing the modern hydrocode, ALEGRA is introduced. These simulations incorporate elastic-plastic equation of state and behavior allowing us to probe very large loading of tapered chains. This leads into the discussion of our continuing work beyond this dissertation including the design of a shock absorbing panel. Historical context is provided which has lead researchers to begin looking seriously at these alluring properties of granular or discretized systems.