Explanation of the self-adaptive dynamics of a harmonically forced beam with a sliding mass
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O R I G I NA L
Florian Müller
· Malte Krack
Explanation of the self-adaptive dynamics of a harmonically forced beam with a sliding mass
Received: 14 November 2019 / Accepted: 19 February 2020 © The Author(s) 2020
Abstract 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. Keywords Self-adaptive · Self-resonant · Non-smooth dynamics · Geometric nonlinearity 1 Introduction Self-adaptive systems are designed to adjust their dynamical characteristics depending on certain operating conditions. In comparison with active systems, passive systems have the advantage that neither a control unit nor an external energy source is needed, which makes them interesting, e.g., for energy harvesting [2,3] and vibration suppression applications [13,17]. In various experiments, carried out independently by different research groups [2,3,11,12,16,19], a clamped–clamped beam under harmonic base excitation with attached slider has shown self-adaptive behavior. After initially small vibrations, the slider moved to a certain position and the vibration level increased substantially. Hereby a signature move was observed [2,11,12], which is shown in Fig. 1, and described in the following. A video of this intriguing behavior is available online [1]. While the system vibrates initially at small level, the slider moves toward the clamping. At a certain point, the vibrations jump to a higher level and the slider turns back toward the beam’s center. This
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