The self-adaptive behavior of a clamped–clamped beam with an attached slider has been experimentally demonstrated by several research groups. In a wide range of excitation frequencies, the system shows its signature move: The slider first slowly moves away from the beam’s center, at a certain point the vibrations jump to a high level, then the slider slowly moves back toward the center and stops at some point, while the system further increases its high vibration level. In our previous work, we explained the unexpected movement of the slider away from the beam’s vibration antinode at the center by the unilateral and frictional contact interactions permitted via a small clearance between slider and beam. However, this model did not predict the signature move correctly. In simulations, the vibration level did not increase significantly and the slider did not turn around. In the present work, we explain, for the first time, the complete signature move. We show that the timescales of vibration and slider movement along the beam are well separated, such that the adaptive system closely follows the periodic vibration response obtained for axially fixed slider. We demonstrate that the beam’s geometric stiffening nonlinearity, which we neglected in our previous work, is of utmost importance for the vibration levels encountered in the experiments. This stiffening nonlinearity leads to coexisting periodic vibration responses and to a turning point bifurcation with respect to the slider position. We associate the experimentally observed jump phenomenon to this turning point and explain why the slider moves back toward the center and stops at some point.

中文翻译：

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具有滑动质量的谐波受力梁的自适应动力学的解释

几个研究小组已通过实验证明了带有连接的滑块的夹钳式梁的自适应行为。在广泛的激发频率范围内，系统显示其标志性动作：滑块首先缓慢移动离开光束的中心，在某个点振动跳到很高的水平，然后滑块缓慢向中心移动并停在在某些时候，系统进一步提高了其高振动水平。在我们先前的工作中，我们解释了滑块和梁之间的小间隙允许的单边和摩擦接触相互作用，使滑块从中心的梁振动波腹处意外移动。但是，此模型不能预测签名正确移动。在模拟中 振动水平没有显着增加，并且滑块没有转动。在当前的工作中，我们首次解释了完整的签名动作。我们表明，振动和滑块沿梁的运动的时间尺度是很好地分开的，从而使自适应系统紧密跟随轴向固定滑块获得的周期性振动响应。我们证明了在先前的工作中忽略的梁的几何刚度非线性对于实验中遇到的振动水平至关重要。这种加剧的非线性导致并存的周期性振动响应和相对于滑块位置的转折点分叉。