Department of Mechanical Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA.
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
Phys Rev E. 2018 Jan;97(1-1):012219. doi: 10.1103/PhysRevE.97.012219.
In linear time-invariant dynamical and acoustical systems, reciprocity holds by the Onsager-Casimir principle of microscopic reversibility, and this can be broken only by odd external biases, nonlinearities, or time-dependent properties. A concept is proposed in this work for breaking dynamic reciprocity based on irreversible nonlinear energy transfers from large to small scales in a system with nonlinear hierarchical internal structure, asymmetry, and intentional strong stiffness nonlinearity. The resulting nonreciprocal large-to-small scale energy transfers mimic analogous nonlinear energy transfer cascades that occur in nature (e.g., in turbulent flows), and are caused by the strong frequency-energy dependence of the essentially nonlinear small-scale components of the system considered. The theoretical part of this work is mainly based on action-angle transformations, followed by direct numerical simulations of the resulting system of nonlinear coupled oscillators. The experimental part considers a system with two scales-a linear large-scale oscillator coupled to a small scale by a nonlinear spring-and validates the theoretical findings demonstrating nonreciprocal large-to-small scale energy transfer. The proposed study promotes a paradigm for designing nonreciprocal acoustic materials harnessing strong nonlinearity, which in a future application will be implemented in designing lattices incorporating nonlinear hierarchical internal structures, asymmetry, and scale mixing.
在线性时不变动力和声学系统中，互易性由微观可逆性的昂萨格-卡西米尔原理保证，只有通过外部奇异性偏置、非线性或时变特性才能打破互易性。本工作提出了一种基于具有非线性层次结构、不对称性和有意强刚度非线性的系统中从大到小尺度的不可逆非线性能量传递来打破动态互易性的概念。由此产生的非互易大到小尺度能量传递模拟了自然界中发生的类似非线性能量传递级联（例如，在湍流中），这是由所考虑的系统的基本非线性小尺度分量的强频率-能量依赖性引起的。本工作的理论部分主要基于作用角变换，然后对所得非线性耦合振荡器系统进行直接数值模拟。实验部分考虑了一个具有两个尺度的系统——一个线性大尺度振荡器通过非线性弹簧与小尺度耦合——验证了理论发现，证明了非互易大到小尺度的能量传递。所提出的研究促进了利用强非线性设计非互易声材料的范例，在未来的应用中，将在设计包含非线性层次结构、不对称性和尺度混合的晶格时实施。
