光滑节点插值法:计算固有频率下界值的新方法

杜超凡; 章定国

doi:10.6052/0459-1879-15-146

光滑节点插值法:计算固有频率下界值的新方法

杜超凡,
章定国^,

南京理工大学理学院, 南京210094

基金项目: 国家自然科学基金(11272155,11132007),江苏省“333”工程(BRA2011172)和高校基本科研业务专项资金(30920130112009)资助项目.

详细信息

通讯作者:
章定国,教授,主要研究方向:多体系统动力学.E-mail:zhangdg419@mail.njust.edu.cn

中图分类号: O313.7
计量
- 文章访问数: 1062
- HTML全文浏览量: 104
- PDF下载量: 590
出版历程
- 收稿日期: 2015-05-11
- 修回日期: 2015-07-08
- 刊出日期: 2015-09-17

NODE-BASED SMOOTHED POINT INTERPOLATION METHOD: A NEW METHOD FOR COMPUTING LOWER BOUND OF NATURAL FREQUENCY

Du Chaofan,
Zhang Dingguo^,

School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China

Funds: The project was supported by the National Natural Science Foundation of China (11272155, 11132007), the 333 Project of Jiangsu Province, China (BRA2011172), and the Fundamental Research Funds for the Central Universities of China (30920130112009).

摘要

摘要: 将光滑节点插值法用于悬臂梁的静力学,并首次用于旋转柔性梁的频率分析. 采用梯度光滑技术,用线性插值形函数描述梁的位移场,求解4 阶微分方程. 在静力学分析中,将该方法所得梁中各点位移与假设模态法、有限元法及解析解的结果对比,可知该方法虽用简单的线性插值形函数描述梁的位移场,但精度却很高. 进一步研究表明,采用模态高于9 阶的假设模态法会使刚度阵条件数变差,导致结果发散. 在频率分析中,与有限元法、假设模态法和解析解对比,表明该方法一个重要特性:能提供固有频率的下界值,而有限元法和假设模态法只能提供固有频率的上界值,说明该方法结合有限元法在处理无解析解的问题时可以从上下界最大程度的逼近真实解,提高精度. 光滑节点插值法具有形函数结构简单、独立变量少且能提供固有频率下界值的特性,因此,具有较高的推广及应用价值.
- 柔性梁 /
- 光滑节点插值法 /
- 无网格法 /
- 动力学 /
- 固有频率下界
Abstract: A meshfree method called node-based smoothed point interpolation method (NS-PIM) is proposed for static analysis of cantilever beam and dynamic analysis of rotating flexible beam for the first time. Gradient smoothing technique is utilized to perform the numerical integration required in the weakened weak (W2) form formulation. The shape functions are approximated using linear interpolation functions, which can be used to solve the 4th order differential equation. In static problems, the cantilever beams with two loading conditions are analyzed, and the results are compared with the analytic solution, which shows a high accuracy of this method even if using linear shape functions. A further study shows that if more than 9 modes were used in the assumed mode method, the result will be divergent. In dynamic problem, the natural frequencies of a rotating flexible beam are analyzed. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed modes method (AMM). It is found that NS-PIM can provide unique lower bounds of natural frequencies, while FEM and AMM can provide upper bounds of natural frequencies. That means we can get more accurate results for the problems by using FEM and NS-PIM in case that exact solution can't be obtained. The NS-PIM has easier shape functions and less independent variable than FEM, and can provide lower bounds of natural frequencies, with a great value of application and dissemination.
- flexible beam /
- node-based smoothed point interpolation method /
- meshfree method /
- dynamics /
- lower bound of natural frequency

HTML全文

参考文献(31)

洪嘉振,尤超蓝. 刚柔耦合系统动力学研究进展. 动力学与控制学报,2004,2(2):1-6(Hong Jiazhen, You Chaolan. Advances in dynamics of rigid-flexible coupling system. Journal of Dynamics and Control, 2004, 2(2):1-6 (in Chinese))

Dwivedy SK, Eberhard P. Dynamic analysis of flexible manipulators, a literature review. Mechanism and Machine Theory , 2006, 41(7): 749-777

Yoo HH, Ryan RR, Scott RA. Dynamics of flexible beams undergoing overall motions. Journal of Sound and Vibration, 1995, 181(2): 261-278

陈思佳,章定国,洪嘉振. 大变形旋转柔性梁的一种高次刚柔耦合动力学模型. 力学学报,2013, 45(2): 251-256 (Chen Sijia, Zhang Dingguo, Hong Jiazhen. A high-order rigid-flexible coupling model of a rotating flexible beam under large deformation. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 251-256 (in Chinese))

Yoo HH, Shin SH. Vibration analysis of rotating cantilever beams. Journal of Sound and Vibration, 1998, 212(5): 807-828

Chung J, Yoo HH. Dynamic analysis of a rotating cantilever beam by using the finite element method. Journal of Sound and Vibration, 2002, 249(1): 147-164

Du H, Lim MK, Liew KM. A nonlinear finite element model for dynamics of flexible manipulators. Mech Mach Theory, 1996, 31(8): 1109-1119

张雄,刘岩. 无网格法. 北京:清华大学出版社,2004:1-6 (Zhang Xiong, Liu Yan. Meshless Method. Beijing: Tsinghua University Press, 2004: 1-6 (in Chinese))

刘桂荣,顾元通.无网格法理论及程序设计. 济南:山东大学出版社,2007:45-133 (Liu Guirong, Gu Yuantong. An Introduction to Meshfree Methods and Their Programming. Shandong: Shandong University Press, 2007: 45-133 (in Chinese))

Sanborn GG, Shanana AA. On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation. Multibody Syst Dyn , 2009, 22: 181-197

Sanborn GG, Shanana AA. A rational finite element method based on the absolute nodal coordinate formulation. Nonlinear Dyn , 2009, 58: 565-572

Sugiyama H, Gerstmayr J, Shabana AA. Deformation modes in the finite element absolute nodal coordinate formulation. Journal of Sound and Vibration, 2006, 298: 1129-1149

Lan P, Shabana AA. Integration of B-spline geometry and ANCF finite element analysis. Nonlinear Dyn, 2010, 61: 193-206

Liu YN, Sun L, Liu YH, et al. Multi-scale B-spline method for 2-D elastic problems. Applied Mathematical Modelling, 2011, 35: 3685-3697

范纪华,章定国. 旋转柔性悬臂梁动力学的 Bezier 插值离散方法研究. 物理学报, 2014, 63(15): 154501 (Fan Jihua, Zhang Dingguo. Bezier interpolation method for the dynamics of rotating flexible cantilever beam. Acta Phys Sin , 2014, 63(15): 154501 (in Chinese))

Liu GR, Gu YT. Assessment and applications of point interpolation methods for computational mechanics. International Journal for Numerical Methods in Engineering, 2004, 59: 1373-1397

Liu GR, Gu YT. A point interpolation method for two-dimensional solids. Int J Muner Meth Engng, 2001, 50: 937-951 3.0.CO;2-X">

Liu GR, Dai KY, Lim KM, et al. A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures. Computational Mechanics, 2002, 29(6): 510-519

Wang JG, Liu GR. A point interpolation meshless method based on radial basis functions . Int J Numer Meth Eng, 2002, 54(11) : 1623-1648.

Belytschko T, Lu YY, Gu L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering , 1994, 37(2): 229-256

Cheng RJ, Cheng YM, Ge HX. Element-free Galerkin(EFG) method for a kind of two-dimensional linear hyperbolic equation. Chin Phys B, 2009, 18(10): 4059-4064

Atluri SN, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mehcanics, 1998, 22(2): 117-127

Chen L, Liew KM. A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems. Comput Mech, 2011, 47: 455-467

Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. Int J Numer Meth Fluids , 1995, 20: 1081-1106

Cheng RJ, Liew KM. The reproducing kernel particle method for two-dimensional unsteady heat conduction problems. Comput Mech, 2009, 45: 1-10

Liu GR, Zhang GY, Dai KY. A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. International Journal of Computational Methods, 2005, 2(4): 645-665

Liu GR, Zhang GY. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). Int J Numer Meth Engng, 2008, 74: 1128-1161

Cui XY, Liu GR, Li GY, et al. A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique. CMES, 2008, 38(3): 217-229

杜超凡, 章定国. 基于无网格点插值法的旋转悬臂梁的动力学分析. 物理学报, 2015, 64(3): 034501 (Du Chaofan, Zhang Dingguo. A meshfree method based on point interpolation for dynamic analysis of rotating cantilever beams. Acta Phys Sin , 2015, 64(3): 034501 (in Chinese))

杜超凡, 章定国, 洪嘉振. 径向基点插值法在旋转柔性梁动力学中的应用. 力学学报, 2015, 47(2): 279-288 (Du Chaofan, Zhang Dingguo, Hong Jiazhen. A meshfree method based on radial point interpolation method for the dynamic analysis of rotating flexible beams. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(2): 279-288 (in Chinese))

Liu GR. Meshfree Methods: Moving Beyond the Finite Element Method. USA: CRC Press, 2002: 456-458