Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos
Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos
复制标题
自持机电换能器的行为和混沌路径
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
W. Mathis
中科院分区：
文献类型：
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作者：
J. Chedjou;K. Kyamakya;I. Moussa;Hans;W. Mathis
研究了一种自持式机电换能器的动力学特性。考察了线性响应中不动点的稳定性。研究了它们的局部分叉，发现了可能发生的不同类型的分叉。给出了发生Hopf分叉的条件。在非共振和共振两种情况下都得到了简谐振动解。在共振情况下对它们的稳定性进行了分析。得到了与最大一维(1-D)数值Lyapunov指数相关的各种分岔图，发现混沌可以通过倍周期、加周期或环面破裂而突然出现。文中还给出了机电系统对初始条件和耦合系数微小变化的极端敏感性。对该机电系统进行了实验研究。为研究机电系统的动态行为，提出了一种合适的电子电路(模拟模拟器)。在机电系统模型的系数和电子电路的部件之间建立对应关系。实验得到了简谐振动解和相图。这项工作最重要的贡献之一是提供了一套描述机电系统行为的可靠的解析表达式(公式)。这些公式对于设计工程师来说是非常重要的，因为它们可以用来预测机电系统的状态，并分别避免它们的破坏。数值分析和实验分析的结果吻合得很好，证明了解析公式的可靠性。
This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.