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 引用本文: 杜超凡, 章定国. 光滑节点插值法:计算固有频率下界值的新方法[J]. 力学学报, 2015, 47(5): 839-847.
Du Chaofan, Zhang Dingguo. NODE-BASED SMOOTHED POINT INTERPOLATION METHOD: A NEW METHOD FOR COMPUTING LOWER BOUND OF NATURAL FREQUENCY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(5): 839-847.
 Citation: Du Chaofan, Zhang Dingguo. NODE-BASED SMOOTHED POINT INTERPOLATION METHOD: A NEW METHOD FOR COMPUTING LOWER BOUND OF NATURAL FREQUENCY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(5): 839-847.

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

• 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.

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.

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