Toward understanding the self-adaptive dynamics of a harmonically forced beam with a sliding mass
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O R I G I NA L
Malte Krack · Noha Aboulfotoh · Jens Twiefel · Jörg Wallaschek · Lawrence A. Bergman · Alexander F. Vakakis
Toward understanding the self-adaptive dynamics of a harmonically forced beam with a sliding mass Received: 27 April 2016 / Accepted: 30 November 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract A mechanical system consisting of an elastic beam under harmonic excitation and an attached sliding body is investigated. Recent experimental observations suggest that the system passively (self-)adapts the axial location of the slider to achieve and maintain a condition of self-resonance, which could be useful in applications such as energy harvesting. The purpose of this work is to provide a theoretical explanation of this phenomenon based on an appropriate model. A key feature of the proposed model is a small clearance between the slider and the beam. This clearance gives rise to backlash and frictional contact interactions, both of which are found to be essential for the self-adaptive behavior. Contact is modeled in terms of the Coulomb and Signorini laws, together with the Newton impact law. The set-valued character of the contact laws is accounted for in a measure differential inclusion formulation. Numerical integration is carried out using Moreau’s time-stepping scheme. The proposed model reproduces qualitatively most experimental observations. However, although the system showed a distinct self-adaptive character, the behavior was found to be non-resonant for the considered set of parameters. Beside estimating the relationship between resonance frequency and slider location, the model permits predicting the operating limits with regard to excitation level and frequency. Finally, some specific dynamical phenomena such as hysteresis effects and transient resonance captures underline the rich dynamical behavior of the system. Keywords Self-tuning · Self-adaptive · Self-arranging · Moving mass · Non-smooth dynamics · Broadband energy harvesting
1 Introduction Self-adaptive systems have the special ability to passively adjust their dynamical characteristics, depending on certain operating conditions. Two of the most important examples are self-resonant and self-damping systems. M. Krack (B) Institute of Aircraft Propulsion Systems, University of Stuttgart, Pfaffenwaldring 6, 70569 Stuttgart, Germany E-mail: [email protected] N. Aboulfotoh · J. Twiefel · J. Wallaschek Institute of Dynamics and Vibration Research, Leibniz Universität Hannover, Appelstr. 11, Hannover, Germany L. A. Bergman Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright Street, Urbana, IL 61801, USA A. F. Vakakis Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green Street, Urbana, IL 61801, USA
M. Krack et al.
A self-resonant system tunes itself onto the frequency of an applied harmonic excitation and thus achieves and maintains resonance over a wide range of excitation frequencies. In contrast, a self-dampi
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