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Damage identification method for functionally graded Timoshenko beams
DENG Hao , CHENG Wei
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-09-22; Accepted: 2015-12-25; Published online: 2016-04-01 14:12
Corresponding author. CHENG Wei, Tel.:010-82310409, E-mail:cheng_wei@buaa.edu.cn
Abstract: To acquire high precision damage identification method for functionally graded materials, based on the state space variable replacement, the transfer matrix of the functionally graded Timoshenko beam along the axial exponential distribution is obtained. By analyzing the influence of crack on the local flexibility of structure, the contribution of the crack to the local stiffness of the structure is simulated by the torsion spring. The surface crack transfer matrix of functionally graded Timoshenko beams is established. And the theoretical model of multi-span beam under complex boundary conditions is derived. In this paper, the nonlinear equations are transformed into a single objective function optimization problem. The generalized Lagrange algorithm and differential evolution algorithm are combined to identify the damage of the structure. Computational examples show that the proposed algorithm has the characteristics of high precision and fast convergence and is suitable for damage identification of multi-damage model under complex boundary conditions.
Key words: state space transfer matrix method     damage identification     functionally graded materials     exponential gradient     complex boundary condition

1 理论推导 1.1 基本方程

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 图 1 材料沿轴向指数分布的功能梯度Timoshenko梁 Fig. 1 Functionally graded Timoshenko beams along axial exponential distribution

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1.2 状态空间传递矩阵

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1.3 裂纹对结构局部柔度的影响

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 图 2 结构表面裂纹示意图 Fig. 2 Schematic diagram of structural surface crack

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1.4 裂纹区域传递矩阵

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1.5 复杂边界条件多跨功能梯度Timoshenko梁振动模型

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1.6 递推公式

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1.7 裂纹损伤识别

1.7.1 裂纹损伤识别方法

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1.7.2 差分进化算法

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1) 初始化种群。初始种群

2) 变异操作。差分进化算法通过差分策略实现个体变异，这是与遗传算法区别的重要标志。

g代种群{xi(g)|xj, iLxj, i(0)≤xj, iU, i=1, 2, …, NP; j=1, 2, …, k}，通过变异后产生一个中间体：

3) 交叉操作。将变异的中间体{vi(g+1)}和第g代种群{xi(g)}进行个体间的交叉操作：

4) 选择操作。差分进化算法通过采用贪婪算法来选择进入下一代种群的个体：

 图 3 裂纹损伤识别方法 Fig. 3 Crack damage identification method
2 算例 2.1 算例1

 图 4 简支边界条件下存在裂纹的功能梯度Timoshenko梁 Fig. 4 Functionally graded Timoshenko beam with cracks under simple-supported boundary conditions
 图 5 简支边界条件下裂纹损伤识别方法迭代过程 Fig. 5 Iterative process of crack damage identification method under simple-supported boundary conditions

 Hz 实测输入频率 ω1 ω2 ω3 ω4 数值 22.122 5 60.160 5 116.978 8 192.100 1

 裂纹 实际值 初始值 辨识结果 误差/% 裂纹1 0.20 0.25 0.201 45 0.725 裂纹2 0.80 0.25 0.796 33 0.459

 裂纹 实际值 初始值 辨识结果 误差/% 裂纹1 0.5 0.4 0.499 46 0.108 裂纹2 0.2 0.4 0.200 18 0.090

2.2 算例2

 图 6 复杂边界条件下存在裂纹的功能梯度Timoshenko梁 Fig. 6 Functionally graded Timoshenko beam with cracks under complex boundary conditions
 图 7 复杂边界条件下裂纹损伤识别方法迭代过程 Fig. 7 Iterative process of crack damage identification method under complex boundary conditions

 Hz 实测输入频率 ω1 ω2 ω3 ω4 数值 32.785 9 42.335 2 148.491 6 156.130 9

 裂纹 实际值 初始值 辨识结果 误差/% 裂纹1 0.40 0.20 0.401 25 0.312 5 裂纹2 0.90 0.75 0.890 11 1.098 9

 裂纹 实际值 初始值 辨识结果 误差/% 裂纹1 0.3 0.5 0.293 01 2.330 裂纹2 0.5 0.6 0.490 11 1.978

3 结论

1) 所使用的状态空间传递矩阵法物理概念清晰，通过状态空间变量的替换，不但降低了问题的求解难度，而且减少了计算量。

2) 分析了裂纹对结构局部柔度的影响，推导了裂纹区域的传递矩阵，建立了表面存在裂纹的功能梯度Timoshenko梁的解析模型。

3) 建立了复杂边界条件下多跨功能梯度Timoshenko梁的解析模型。

4) 结构裂纹损伤识别方法中，将多目标优化问题转化为单一目标优化问题，通过采用增广拉格朗日算法与差分进化算法的结合对功能梯度Timoshenko梁进行损伤识别，计算结果表明，识别精度较高。

#### 文章信息

DENG Hao, CHENG Wei

Damage identification method for functionally graded Timoshenko beams

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(10): 2214-2221
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0618