a ，带有νo的主动弹丸< 0 is vertically incident against a wall. b,c, The angular deformation of each vertex is plotted as a function of time, revealing a unidirectional wave propagating in the opposite direction of Fig. 2a. See also Supplementary Video 2. d, The ratio between incoming and outgoing momenta pout/pin in the x (round markers) and y (square markers) directions as a function of the odd ratio νo for an initial velocity vin of 0.26 m s−1 (blue) and 0.34 m s−1 (orange). The data of Fig. 2c is found by averaging over these two initial conditions. e, Inset top, the chain is modelled as a continuous ring (see Supplementary Information section 1). Inset bottom, we evolve equation (M12) on a 1D periodic ring using the second harmonic as an initial condition. Colour indicates deviation in curvature. Main, the flux with z = Aeiθ (see Supplementary Information section 1B) quantifies the unidirectionality and amplification of the waves in the experiment. Black lines show flux values calculated from solutions to equation (M12), calibrated with experimentally determined parameters for the ring described in Methods. Note that, in the regime |κa|/κ >0.45 ，有较大的误差线 。这很可能是因为该环位于图4A – D中描述的不稳定状态，其中变形很大 。还要注意 ，尽管该模型假定对平面几何形状的小挠度 ，但我们认为与基于方程式（M12）的预测相对较好的一致性。f，g，撞击前后奇数环的角动量 。该环没有角动量发射 ，但奇数响应赋予旋转动量 ，旋转动量以奇数刚度κA缩放 。角动量通过比例RVIN标准化，其中R是环的半径 ，VIN是传入的速度。
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