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该文亮点：

1. Buks和Roukes的实验装置被认为是大型静电耦合微/纳振子阵列的一个代表性工作；

2. 考虑随机激励分量的动力学方程的新型建模，表征不同噪声源对系统动力学的影响；

3. 基于维纳路径积分技术进行了准确、高效的随机响应分析。

Highlights:

1. Experimental set-up by Buks and Roukes considered as a representative case of large arrays of electrostatically coupled of micro/nano-resonators.

2. Novel modeling of the equations of motion considering a stochastic excitation component representing the effect of diverse noise sources on the system dynamics.

3. Accurate and computationally efficient stochastic response analysis based on the Wiener path integral technique.

纳米线被视为未来纳米技术中愈发重要的结构构件。近年来，微纳机电器件的发展使快速、可靠、免标记的分子检测成为现实。该技术在当前及未来可被应用于检测与特定疾病有关的生化标志物等，而其检测效率受多种因素影响，如随机性和非线性。这些因素在理解微纳机电器件的检测原理并最终优化纳米机电系统和设备的设计方面起着关键作用。此外，目前的技术能制造出通过电、磁或弹性力耦合的、由数百到数万个微/纳米梁组成的大型阵列。得益于此耦合机制，纳米机电系统的检测灵敏度有望得到增强。

纳米机械系统及设备的优化设计及其检测能力的提升要求：1)建立微/纳振子在随机激励下的非线性多自由度动力学系统模型；2)求解动力学方程并获得系统随机响应需要有效的不确定性传播分析方法。然而，有关非线性微/纳振子的随机建模和分析文献较少，前人研究工作大都与低维（通常为单自由度）系统的解析或数值求解法相关。少数涉及将大型耦合微/纳梁阵列建模为高维多自由度系统的论文采用了大量简化和近似，不可避免地降低了随机响应估计的精确度。

有鉴于此，哥伦比亚大学土木工程与工程力学系Ioannis A. Kougioumtzoglou副教授团队在《国际机械系统动力学学报（英文）》（International Journal of Mechanical System Dynamics, IJMSD）发表题为“耦合微机械振子阵列的非线性随机动力学”的研究论文。相较于文献[2-3]中的建模和求解方法，该文创新点如下：1)避免了通常采用的线性或高阶多项式的非线性静电力的近似；2)采用概率建模方法，通过考虑一个随机激励分量来表示不同噪声源对系统动力学的影响；3)精确并低成本求解表征微梁阵列动力学的高维、非线性耦合随机微分方程。文章采用了Kougioumtzoglou副教授及其合作者最新建立的维纳路径积分（Wiener path integral, WPI）高效变分公式。

文献[1]给出的微机械系统的随机响应由WPI技术获得。相较于使用对应30,000次实现的蒙特卡洛仿真，使用WPI技术具有很高的准确性和计算效率。此外，该文提出的模型可在很大程度上捕捉到系统响应的丰富频率成分的显著方面，并且至少可以定性重现Buks和Roukes实验装置的频域响应。

总之，WPI技术精确度高、计算成本低，该独特优势可促进大型微机械振子阵列的随机响应分析达到前所未有的水平；因此，有望引起此类系统及设备优化和设计的范式转变。

参考文献

[1] Buks E, Roukes L. Electrically tunable collective response in a coupled micromechanical array. Microelectromechanical systems. 2002; 11: 802-807.

[2] Lifshitz R, Cross MC. Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays. Physical Review. 2003; 67: 134302-13.

[3] Zhu J, Ru CQ, Mioduchowski A. High-order subharmonic parametric resonance of multiple nonlinearly coupled microme-chanical nonlinear oscillators. Acta Mechanica. 2010; 212: 69-81.

作 者 简 介

Maria I. Katsidoniotaki    Maria I. Katsidoniotaki received her 5-year Diploma in Civil Engineering from the National Technical University of Athens (NTUA) in Greece (2017), and her M.Phil. (2022) and Ph.D. (2023) degrees in Civil Engineering and Engineering Mechanics from Columbia University, New York, USA. Since 2023, she has been working as a Postdoctoral Research Scientist at Columbia University, focusing on stochastic engineering dynamics. Specifically, she develops accurate and computationally efficient stochastic methodologies for uncertainty quantification of complex dynamic systems,  including structures and civil infrastructure systems,  nanoelectromechanical devices, and physiological mechanisms.

Ioannis Petromichelakis    Ioannis Petromichelakis received his 5-year Diploma in Civil Engineering from the National Technical University of Athens (NTUA) in Greece (2011), his M.Sc. degree from the University of Dresden in Germany (2013), and his Ph.D. degree in Civil Engineering and Engineering Mechanics from Columbia University, New York, USA (2020). His research focuses on path integral techniques and Gröbner basis approaches for stochastic response analysis and optimization of diverse nonlinear dynamic systems, and he has coauthored more than 10 journal papers. He currently works as an Assistant Vice President in Algorithmic Trading at Citibank, USA.

Ioannis A. Kougioumtzoglou   Ioannis A. Kougioumtzoglou received his 5-year Diploma in Civil Engineering from the National Technical University of Athens (NTUA) in Greece (2007), and his M.Sc. (2009) and Ph.D. (2011) degrees in Civil Engineering from Rice University, TX, USA. He joined Columbia University in 2014, where he is currently an Associate Professor in the Department of Civil Engineering and Engineering Mechanics. He is the author of approximately 150 publications, including more than 85 technical papers in archival International Journals. Prof. Kougioumtzoglou is a National Science Foundation (NSF) CAREER awardee and a European Association of Structural Dynamics (EASD) Junior Research Prize recipient “for his innovative influence on the field of nonlinear stochastic dynamics.” Prof. Kougioumtzoglou and his research group develop primarily analytic and numerical methodologies for stochastic response analysis, reliability assessment, and optimization of complex engineering systems and structures subject to uncertainties. These methodologies lead eventually to robust and efficient design of dynamic systems ranging from the nano-scale (e.g., nano-mechanical oscillators) to the macro-scale (e.g., energy harvesters and civil infrastructure systems). Specific theoretical research themes include nonlinear stochastic dynamics and path integrals, fractional calculus modeling, computational stochastic mechanics, data-driven uncertainty quantification methodologies, and signal processing techniques.