In the present work, we consider the mass in mass (or mass with mass) system of granular chains, namely a granular chain involving additionally an internal resonator. For these chains, we rigorously establish that under suitable “anti-resonance” conditions connecting the mass of the resonator and the speed of the wave, bell-shaped traveling wave solutions continue to exist in the system, in a way reminiscent of the results proven for the standard granular chain of elastic Hertzian contacts. We also numerically touch upon settings where the conditions do not hold, illustrating, in line also with recent experimental work, that non-monotonic waves bearing non-vanishing tails may exist in the latter case.
Kevrekidis, Panayotis G.; Stefanov, Atanas G.; and Xu, Haitao, "Traveling Waves for the Mass in Mass Model of Granular Chains" (2015). Mathematics and Statistics Department Faculty Publication Series. 1222.
Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/1222