﻿ 模糊变分原理在求解结构固有频率中的应用
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Application of fuzzy variational principle in solution of natural frequency of structures
Zhang Xudong, Qiu Zhiping, Li Qi
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:When dealing with the natural frequency of structures with fuzzy parameters, the previous methods are limited to the case that the fuzzy parameters are transformed into the interval parameters before the calculation, and then the fuzzy results are constructed by the interval results. The calculation cost of the previous methods is relatively expensive, so the fuzzy Ritz method and the fuzzy finite element method which are based on the fuzzy variational principle were adopted. By introducing the fuzzy parameters into Rayleigh quotient variation, the fuzzy variational principle was developed. The fuzzy Ritz method and the fuzzy finite element method were proposed as the application of the fuzzy variational principle. These two methods can obtain the fuzzy result directly. Compared with the traditional interval analysis methods, the proposed methods can increase computational efficiency. The numerical example demonstrates that both methods can achieve high computational accuracy and reduce computational cost significantly.
Key words: fuzzy     variational principle     Ritz method     finite element method     natural frequency

1 线性峰型模糊数

1) 的左右隶属函数LA(x)和RA(x)严格单调且连续,且0≤LA,RA≤1.

2) 集合Ker有且只有一个元素， 则称为峰型模糊数.可记为

 图 1 线性峰型模糊数 Fig. 1 Image motion by rotation of Ys

2 模糊变分原理

=ω~2,则有

3 模糊里兹法

 图 2 等截面弹性简支直梁 Fig. 2 Simply supported elastic beam with uniform cross section

4 模糊有限元法

ai0为λi0对应的归一化后的特征向量.

5 数值算例

 ξ FAM FFEM FFEM误差/% FRM FRM误差/% 1 97.409 97.419 0.010 97.409 0.000 0.9 [91.679,103.373] [91.457,103.381] 0.242,0.008 [91.448,103.371] 0.253,0.002 0.8 [86.170,109.585] [85.261,109.577] 1.054,0.007 [85.252,109.566] 1.064,0.018 0.7 [80.868,116.062] [78.832,116.007] 2.518,0.048 [78.823,115.995] 2.528,0.058 0.6 [75.763,122.820] [72.168,122.670] 4.744,0.122 [72.161,122.658] 4.754,0.132

 ξ FAM FFEM FFEM误差/% FRM FRM误差/% 1 1558.5 1561.1 0.162 1558.5 0.000 0.9 [1466.9,1654.0] [1465.5,1656.6] 0.091,0.160 [1463.2,1653.9] 0.253,0.002 0.8 [1378.7,1753.4] [1366.2,1755.9] 0.904,0.144 [1364.0,1753.1] 1.064,0.018 0.7 [1293.9,1857.0] [1263.2,1858.9] 2.370,0.104 [1261.2,1855.9] 2.528,0.058 0.6 [1212.2,1965.1] [1156.4,1965.7] 4.600,0.029 [1154.6,1962.5] 4.754,0.132

 ξ FAM FFEM FFEM误差/% FRM FRM误差/% 1 7890.1 7952.5 0.791 7890.1 0.000 0.9 [7426.0,8373.2] [7465.8,8439.2] 0.536,0.789 [7407.3,8373.0] 0.253,0.002 0.8 [6979.7,8876.4] [6960.1,8945.0] 0.282,0.773 [6905.4,8874.8] 1.064,0.018 0.7 [6550.3,9401.0] [6435.2,9469.9] 1.757,0.733 [6384.7,9395.6] 2.528,0.058 0.6 [6136.8,9948.4] [5891.0,10014] 4.001,0.658 [5845.0,9935.3] 4.754,0.132

 ξ E ρ 1 E0 ρ0 0.8 [0.92E0,1.08E0] [0.96ρ0,1.04ρ0] 0.6 [0.84E0,1.16E0] [0.92ρ0,1.08ρ0] 0.4 [0.76E0,1.24E0] [0.88ρ0,1.12ρ0] 0.2 [0.68E0,1.32E0] [0.84ρ0,1.16ρ0] 0 [0.60E0,1.40E0] [0.80ρ0,1.20ρ0]

 ξ λ 1 97.419E0I/(ρ0Al4) 0.8 [85.261,109.577]E0I/(ρ0Al4) 0.6 [72.168,122.670]E0I/(ρ0Al4) 0.4 [58.140,136.699]E0I/(ρ0Al4) 0.2 [43.176,151.662]E0I/(ρ0Al4) 0 [27.277,167.561]E0I/(ρ0Al4)

 图 3 由区间解构造的模糊解 Fig. 3 Fuzzy result constructed by interval results

6 结 论

1) 提出了结构固有频率问题的模糊变分原理;

2) 基于模糊变分原理,推导了可以直接得到问题模糊解的模糊里兹法和模糊有限元法;

3) 所提方法与传统的区间解法相比,减少了计算量,提高了计算效率.

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#### 文章信息

Zhang Xudong, Qiu Zhiping, Li Qi

Application of fuzzy variational principle in solution of natural frequency of structures

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(12): 1773-1779.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0749