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Time-Periodic Solutions of Driven-Damped Trimer Granular Crystals

机译：驱动阻尼三聚体颗粒晶体的时间周期解

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摘要

We consider time-periodic structures of granular crystals consisting of alternate chrome steel (S) and tungsten carbide (W) spherical particles where each unit cell follows the pattern of a 2 :1 trimer: S-W-S. The configuration at the left boundary is driven by a harmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall. Similar to the case of a dimer chain, the combination of dissipation, driving of the boundary, and intrinsic nonlinearity leads to complex dynamics. For fixed driving frequencies in each of the spectral gaps, we find that the nonlinear surface modes and the states dictated by the linear drive collide in a saddle-node bifurcation as the driving amplitude is increased, beyond which the dynamics of the system becomes chaotic. While the bifurcation structure is similar for solutions within the first and second gap, those in the first gap appear to be less robust. We also conduct a continuation in driving frequency, where it is apparent that the nonlinearity of the system results in a complex bifurcation diagram, involving an intricate set of loops of branches, especially within the spectral gap. The theoretical findings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures.
机译：我们考虑由交替的铬钢（S）和碳化钨（W）球形颗粒组成的粒状晶体的时间周期结构，其中每个晶胞遵循2：1三聚体：S-W-S的图案。左边界处的配置由具有给定幅度和频率的谐波及时激励驱动，而右边界是固定壁。类似于二聚体链的情况，耗散，边界驱动和固有非线性的组合导致复杂的动力学。对于每个频谱间隙中的固定驱动频率，我们发现随着驱动幅度的增加，非线性表面模式和线性驱动所指示的状态会在鞍形节点分叉中发生碰撞，超过该范围，系统的动力学就会变得混乱。虽然对于第一和第二间隙内的解决方案，分叉结构相似，但第一间隙中的解决方案似乎不那么可靠。我们还进行了驱动频率的连续化，很明显，系统的非线性会导致复杂的分叉图，其中包括错综复杂的分支环集合，尤其是在光谱间隙内。理论上的发现在时间周期结构的实验性全场可视化上得到了定性的证实。

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  • 来源
    《Mathematical Problems in Engineering》 |2015年第12期|830978.1-830978.15|共15页
  • 作者

    Charalampidis E. G.; Li F.; Chong C.; Yang J.; Kevrekidis P. G.;

  • 作者单位

    Aristotle Univ Thessaloniki, Fac Engn, Sch Civil Engn, Thessaloniki 54124, Greece|Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA;

    Univ Washington, Aeronaut & Astronaut, Seattle, WA 98195 USA;

    Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA|ETH, Swiss Fed Inst Technol, Dept Mech & Proc Engn D MAVT, CH-8092 Zurich, Switzerland;

    Univ Washington, Aeronaut & Astronaut, Seattle, WA 98195 USA;

    Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA|Los Alamos Natl Lab, Ctr Nonlinear Studies, Los Alamos, NM 87544 USA|Los Alamos Natl Lab, Div Theoret, Los Alamos, NM 87544 USA;

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