Experimental evidence of shock mitigation in a Hertzian tapered chain

Abstract

We present an experimental study of the mechanical impulse propagation through a horizontal alignment of elastic spheres of progressively decreasing diameter phi(n): namely, a tapered chain. Experimentally, the diameters of spheres which interact via the Hertz potential are selected to keep as close as possible to an exponential decrease, phi(n+1) = (1-q)phi(n), where the experimental tapering factor is either q(1) approximately equal to 5.60% or q(2) approximately equal to 8.27%. In agreement with recent numerical results, an impulse initiated in a monodisperse chain (a chain of identical beads) propagates without shape changes and progressively transfers its energy and momentum to a propagating tail when it further travels in a tapered chain. As a result, the front pulse of this wave decreases in amplitude and accelerates. Both effects are satisfactorily described by the hard-sphere approximation, and basically, the shock mitigation is due to partial transmissions, from one bead to the next, of momentum and energy of the front pulse. In addition when small dissipation is included, better agreement with experiments is found. A close analysis of the loading part of the experimental pulses demonstrates that the front wave adopts a self-similar solution as it propagates in the tapered chain. Finally, our results corroborate the capability of these chains to thermalize propagating impulses and thereby act as shock absorbing devices.

Frequency filtering in disordered granular chains

“…In the context of the linear chain, we relate the filtering behavior to the localization of eigenmodes in the presence of disorder. One may imagine applications exploiting disorder effects, as could be pursued in the spirit of [5,10,13,14,32,40,47,49] where such systems were engineered to produce a desired output.…”

“…Additionally, there is significant attention placed on the behavior of "designed" and ordered nonlinear chains, often motivated by energy modification and shock-protection applications. Studies have employed smoothly varying mass distributions [40], "decoration" (e.g., deliberate insertion of different sized masses) [10,13,14,27] tapering [5,32,47,49] and controlled variation of the particle material [3,15]. Combinations of both tapering and decoration have also been employed [6,13].…”

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

“…Geometry and/or material anisotropy between particles composing the systems allows for the observation of interesting dynamic phenomena deriving from the interplay of discreteness and nonlinearity of the problem (i.e. anomalous reflections, breathers, energy trapping and impulse fragmentation) [11,12,19,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. These findings open up a large parameter space for new materials design with unique properties sharply departing from classical engineering systems.…”

Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress

“…They provided a detailed relationship between the shape of the solitary wave train and the beads mass ratio. [28][29][30][31][32][33][34] Lately, the Fermi-Pasta-Ulam (FPU)-type lattices such as the one arising also from the Hertzian chain in the presence of precompression have been studied widely. [35][36][37] The KdV equation describes approximately FPU-type lattices for small-amplitude, long-wave and low-energy initial data, which has been used in the mathematical literature to determine the shape and dynamical stability of solitary waves and even their interactions.…”