DNA-微悬臂梁的Rayleigh-Ritz法求解

刘承; 何小兵; 沈旭栋

doi:10.21656/1000-0887.460067

DNA-微悬臂梁的Rayleigh-Ritz法求解

doi: 10.21656/1000-0887.460067

刘承^{1, 2,},
何小兵^1, ,,
沈旭栋^3, ,

1.
重庆交通大学土木工程学院，重庆 400074
2.
浙江建设职业技术学院建筑工程学院，杭州 311231
3.
浣江实验室软体机器人与智能器件研究中心，浙江诸暨 311816

基金项目:

浙江省教育厅一般科研项目 Y202352719

详细信息

作者简介:
刘承(1985—)，男，高级工程师，博士生(E-mail: liucheng777888@sina.com)

通讯作者:
何小兵(1978—)，男，教授，博士，博士生导师(通信作者. E-mail: david.hxb@gmail.com)

沈旭栋(1982—)，男，助理研究员，博士(通信作者. E-mail: vibration@zju.edu.cn)

中图分类号: O39
计量
- 文章访问数: 45
- HTML全文浏览量: 12
- PDF下载量: 22
- 被引次数: 0
出版历程
- 收稿日期: 2025-04-09
- 修回日期: 2025-06-09
- 刊出日期: 2026-05-01

The Rayleigh-Ritz Solution for DNA-Microcantilevers

LIU Cheng^{1, 2
,},
HE Xiaobing^{1
, ,},
SHEN Xudong^{3
, ,}

1.
School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R. China
2.
Construction Engineering School, Zhejiang College of Construction, Hangzhou 311231, P.R. China
3.
Center for Soft Machines and Smart Devices, Huanjiang Laboratory, Zhuji, Zhejiang 311816, P.R. China

摘要

摘要: DNA-微悬臂梁因其纳米力学特性在生物传感领域极具应用价值，但其均匀曲率和中性轴轴向应变假设缺乏验证. 为此，建立液晶态DNA分子层自由能与组合梁结构变形耦合的泛函势能模型，创新性地引入三角级数模态叠加法表征位移场进行Rayleigh-Ritz法变分求解，并结合分段积分策略实现全域自由能计算. 采用不动点迭代算法求解关于Ritz系数的非线性方程组后，得到悬臂梁变形及DNA链间距分布. 结果表明，变形后曲率呈现显著的均匀特性，为均匀性假设提供了理论依据. 收敛性分析表明：三角级数展开阶数≥10，且梁段划分数量≥2 500，可确保挠度和应力求解精度，有效验证了均匀性假设在耦合模型中的正确性.
- DNA-微悬臂梁 /
- 纳米力学 /
- 自由能 /
- Rayleigh-Ritz
Abstract: DNA-microcantilevers hold significant potential in biosensing due to their nanomechanical properties, yet the assumption regarding uniform curvature and axial strain of the neutral axis remains unverified. To address this, a functional potential energy model coupling the free energy of liquid crystalline DNA molecular layers with the composite beam deformation, was established. An innovative triangular series modal superposition approach was used to characterize displacement fields, and enable the variational solution via the Rayleigh-Ritz method, and a segmented integration strategy was employed to realize the full-domain free energy computation. Then the nonlinear equations governing Ritz coefficients were solved through the fixed-point iteration to yield microcantilever deformations and DNA strand spacing distributions. The results indicate that, the deformed curvature exhibits pronounced uniformity, providing theoretical support for the uniformity assumption. Convergence analysis demonstrates that, the triangular series expansion order ≥10 and the number of beam segments divided ≥2 500 ensure solution accuracy for deflection and stress, effectively validating the correctness of the uniformity assumption within the coupled model.
- DNA-microcantilever /
- nano-mechanics /
- free energy /
- Rayleigh-Ritz

HTML全文

图 1 DNA生化反应下的微悬臂梁的弯曲响应

Figure 1. Bending responses of the microcantilever induced by DNA biochemical reactions

下载: 全尺寸图片幻灯片

图 2 DNA-微悬臂梁示意图

Figure 2. The DNA-microcantilever schematic

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图 3 级数的阶数m对挠度w, 中性轴位移u_Neu和表面应力σ沿x轴的分布影响

Figure 3. Effects of series' order m on the distributions along the x-axis of deflection w, neutral axis displacement u_Neu and surface stress σ

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图 4 梁段划分数量n对挠度w, 中性轴位移u_Neu和表面应力σ沿x轴的分布影响

Figure 4. Effects of the number of divided beam segments, n, on the distributions along the x-axis of deflection w, neutral axis displacement u_Neu and surface stress σ

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图 5 挠度与表面应力随核苷酸数量的变化曲线

Figure 5. Deflections and surface stresses vs. the number of nucleotides

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图 6 表面应力随种植密度的变化曲线

Figure 6. Surface stresses vs. the grafting density

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图 7 在各种植密度下，挠度随核苷酸数量的变化曲线

Figure 7. Deflections vs. the number of nucleotides at different grafting densities

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