Kovaleva Agnessa, Manevitch Leonid I
Space Research Institute, Russian Academy of Sciences, Moscow 117997, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022904. doi: 10.1103/PhysRevE.88.022904. Epub 2013 Aug 7.
It is well known that complete energy transfer between two weakly coupled linear oscillators occurs only at resonance. If the oscillators are nonlinear, the amplitude dependence of their frequencies may destroy, in general, any eventual resonance. This means that no substantial energy transfer may occur unless, exceptionally, resonance persists during the transfer. In this paper, the self-sustained resonance is considered for an oscillator array consisting of n coupled linear oscillators (a primary system) initially excited by impulse loading and connected to an essentially nonlinear attachment (NLA). Under the condition of resonance, initial energy is transferred to the NLA and then travels back and forth between the linear and nonlinear oscillators. It is shown that the general mechanism of the energy transport is similar to that in the previously studied system of two coupled oscillators but, in contrast to the two degree-of-freedom case, the multidimensional system requires a proper tuning not only for the NLA but for the entire array. In this work, we develop an order-reduction procedure, which allows the separated dynamical analysis for the pair of nonlinearly coupled oscillators and the remaining (n-1) linear oscillators. Using simplifications based on the low-order reduced model, we detect an admissible domain of parameters ensuring resonance interaction and then derive a closed-form approximate solution adequately describing the transient processes in the entire system. We obtain explicit approximate solutions for both conservative (complete energy exchange) and dissipative (irreversible energy transfer) systems and then illustrate the theoretical results by an example of the four degree-of-freedom system. Analytical results are confirmed by numerical simulations.
众所周知，两个弱耦合线性振子之间的完全能量转移仅在共振时发生。如果振子是非线性的，它们频率的幅度依赖性通常可能会破坏任何最终的共振。这意味着，除非在转移过程中异常地持续存在共振，否则可能不会发生大量的能量转移。在本文中，考虑了由n个耦合线性振子组成的振子阵列（一个主系统）的自持共振，该阵列最初由脉冲载荷激发并连接到一个本质上非线性的附件（NLA）。在共振条件下，初始能量转移到NLA，然后在线性和非线性振子之间来回传播。结果表明，能量传输的一般机制与先前研究的两个耦合振子系统中的机制相似，但与两自由度情况不同的是，多维系统不仅需要对NLA进行适当调谐，还需要对整个阵列进行适当调谐。在这项工作中，我们开发了一种降阶程序，该程序允许对非线性耦合振子对和其余（n - 1）个线性振子进行分离的动力学分析。利用基于低阶简化模型的简化方法，我们检测出确保共振相互作用的参数允许域，然后导出一个封闭形式的近似解，该解能充分描述整个系统中的瞬态过程。我们得到了保守（完全能量交换）和耗散（不可逆能量转移）系统的显式近似解，然后通过一个四自由度系统的例子来说明理论结果。数值模拟证实了分析结果。
