Asynchronous parametric excitation: validation of theoretical results by electronic circuit simulation
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ORIGINAL PAPER
Asynchronous parametric excitation: validation of theoretical results by electronic circuit simulation Artem Karev
· Peter Hagedorn
Received: 27 April 2020 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract A validation of recent theoretical results on the stability effects of asynchronous parametric excitation is presented. In particular, the coexistence of both resonance and anti-resonance at each combination resonance frequency is to be confirmed on a close-to-experiment simulation model. The simulation model reproduces the experimental setup developed by Schmieg in 1976, remaining the only experimental study on asynchronous excitation to this day. The model consists of two oscillating electronic circuits with feedback-free coupling through parametric excitation. In contrast to a mechanical system, the phase relations of the parametric excitation terms in an electronic system can be easily adjusted. The implementation of the simulation model is performed in the electronic circuit simulation software LTspice. The electronic model itself is first validated against the experimental results obtained by Schmieg and is then used to confirm the theoretical findings. The results of the electronic circuit simulation show excellent qualitative and quantitative agreement with analytical approximations confirming A. Karev (B) Institute of Numerical Methods in Mechanical Engineering, Technical University of Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany e-mail: [email protected] P. Hagedorn Dynamics and Vibrations Group, Institute of Numerical Methods in Mechanical Engineering, Technical University of Darmstadt, Dolivostraße 15, 64293 Darmstadt, Germany e-mail: [email protected]
the coexistence of resonance and anti-resonance effects near a combination resonance frequency. Keywords Parametric excitation · Asynchronous excitation · Stability · Normal forms · Validation
1 Introduction Parametric excitation in mechanical systems is well known for its destabilizing resonance effect and, in recent decades, increasingly also for its stabilizing effect, i.e., parametric anti-resonance. While initially the presence of parametric excitation was rather an undesired feature affecting the system’s dynamics in an unfavorable way, nowadays, there is a rich field of applications with deliberately introduced time periodicity. On the one hand, the destabilizing effects are widely used in energy harvesting applications [2,27] as well as in parametric amplifiers [12]; on the other hand, the anti-resonance effect is introduced in order to attenuate vibrations and to enhance dissipative properties [11]. Another quickly growing field of application is the microelectromechanical systems (MEMS). In the field of MEMS, there are several applications employing stabilizing and destabilizing effects, e.g., highly sensitive mass sensing [21] and rapid switching in mechanical resonators [22], respectively. Most of the theoretical studies and practical applications on time-periodic systems deal with synchro
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