Abstract
Bifurcations of periodic orbits and band zones of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is first considered without any dissipation or external forcing. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in the in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling simultaneously occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution and to detect the bifurcation location. Although the trajectory of the period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced away from the bifurcation point. Pre-compression is found to affect the periodic orbit frequencies, which in turn result in changes in the wave propagation band zones. These changes are illustrated by studying vibration transmission through a granular chain at different frequencies to ascertain the band zone limits.
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References
Johnson, P.A., Jia, X.: Nonlinear dynamics, granular media and dynamic earthquake triggering. Nature 437(7060), 871 (2005)
Dorostkar, O., Guyer, Robert A, Johnson, P.A., Marone, C., Carmeliet, J.: Cohesion-induced stabilization in stick-slip dynamics of weakly wet, sheared granular fault gouge. J. Geophys. Res. Solid Earth 123(3), 2115–2126 (2018)
Wang, G., Riaz, A., Balachandran, B.: Smooth particle hydrodynamics studies of wet granular column collapses. Acta Geotech (accepted) (2019)
Li, C., Umbanhowar, P.B., Komsuoglu, H., Koditschek, D.E., Goldman, D.I.: Sensitive dependence of the motion of a legged robot on granular media. Proc. Nat. Acad. Sci. 106(9), 3029–3034 (2009)
Juarez, G., Lu, K., Sznitman, J., Arratia, P.E.: Motility of small nematodes in wet granular media. EPL (Europhys. Lett.) 92(4), 44002 (2010)
Li, C., Zhang, T., Goldman, D.I.: A terradynamics of legged locomotion on granular media. Science 339(6126), 1408–1412 (2013)
Brzinski III, T.A., Mayor, P., Durian, D.J.: Depth-dependent resistance of granular media to vertical penetration. Phys. Rev. Lett. 111(16), 168002 (2013)
Birch, S.P.D., Manga, M., Delbridge, B., Chamberlain, M.: Penetration of spherical projectiles into wet granular media. Phys. Rev. E 90(3), 032208 (2014)
Li, R., Yang, H., Zheng, G., Sun, Q.C.: Granular avalanches in slumping regime in a 2d rotating drum. Powder Technol. 326, 322–326 (2018)
Guerrero, B.V., Pugnaloni, L.A., Lozano, C., Zuriguel, I., Garcimartín, A.: Slow relaxation dynamics of clogs in a vibrated granular silo. Phys. Rev. E 97(4), 042904 (2018)
Liu, L., Li, J., Wan, C.: Nonlinear dynamics of excited plate immersed in granular matter. Nonlinear Dyn. 91(1), 147–156 (2018)
Nakashima, H., Fujii, H., Oida, A., Momozu, M., Kawase, Y., Kanamori, H., Aoki, S., Yokoyama, T.: Parametric analysis of lugged wheel performance for a lunar microrover by means of DEM. J. Terrramech. 44(2), 153–162 (2007)
Wang, G., Riaz, A., Balachandran, B.: Computational studies on interactions between robot leg and deformable terrain. Procedia Eng. 199, 2439–2444 (2017)
Fan, F., Parteli, E.J.R., Pöschel, T.: Origin of granular capillarity revealed by particle-based simulations. Phys. Rev. Lett. 118(21), 218001 (2017)
Lee, S., Marghitu, D.B.: Analysis of a rigid body obliquely impacting granular matter. Nonlinear Dyn. 57(1–2), 289–301 (2009)
Jayaprakash, K.R., Vakakis, A.F., Starosvetsky, Y.: Nonlinear resonances in a general class of granular dimers with no pre-compression. Granul. Matter 15(3), 327–347 (2013)
Chong, C., Porter, M.A., Kevrekidis, P.G., Daraio, C.: Nonlinear coherent structures in granular crystals. J. Phys.: Condens. Matter 29(41), 413003 (2017)
Starosvetsky, Y., Vakakis, A.F.: Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Phys. Rev. E 82(2), 026603 (2010)
Jayaprakash, K.R., Starosvetsky, Y., Vakakis, A.F., Peeters, M., Kerschen, G.: Nonlinear normal modes and band zones in granular chains with no pre-compression. Nonlinear Dyn. 63(3), 359–385 (2011)
Nesterenko, V.F.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24(5), 733–743 (1983)
James, G.: Periodic travelling waves and compactons in granular chains. J. Nonlinear Sci. 22(5), 813–848 (2012)
Betti, M., Pelinovsky, D.E.: Periodic traveling waves in diatomic granular chains. J. Nonlinear Sci. 23(5), 689–730 (2013)
Nesterenko, V.F., Daraio, C., Herbold, E.B., Jin, S.: Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys. Rev. Lett. 95(15), 158702 (2005)
Boechler, N., Theocharis, G., Daraio, C.: Bifurcation-based acoustic switching and rectification. Nat. Mater. 10(9), 665 (2011)
Lin, W.H., Daraio, C.: Wave propagation in one-dimensional microscopic granular chains. Phys. Rev. E 94(5), 052907 (2016)
Zhang, Y., Moore, K.J., McFarland, D.M., Vakakis, A.F.: Targeted energy transfers and passive acoustic wave redirection in a two-dimensional granular network under periodic excitation. J. Appl. Phys. 118(23), 234901 (2015)
Cui, J.G., Yang, T., Chen, L.Q.: Frequency-preserved non-reciprocal acoustic propagation in a granular chain. Appl. Phys. Lett. 112(18), 181904 (2018)
Wang, J., Liu, C., Wiercigroch, M., Wang, C., Shui, Y.: Stability of periodic modes and bifurcation behaviors in a bouncing-dimer system. Nonlinear Dyn. 86(3), 1477–1492 (2016)
Norris, A.N., Johnson, D.L.: Nonlinear elasticity of granular media. J. Appl. Mech. 64(1), 39–49 (1997)
Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23(1), 195–216 (2009)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, Hoboken (2008)
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The authors are grateful for the support received from the U.S. National Science Foundation, through grant CMMI-1507612.
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Acar, G.D., Balachandran, B. Dynamics of one-dimensional granular arrays with pre-compression. Nonlinear Dyn 99, 707–720 (2020). https://doi.org/10.1007/s11071-019-05407-6
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DOI: https://doi.org/10.1007/s11071-019-05407-6