Explanation of the self-adaptive dynamics of a harmonically forced beam with a sliding mass

摘要

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.