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Equivalent modeling method of open thin-walled beam under action of transverse stiffening member
DENG Hao , CHENG Wei
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-07-07; Accepted: 2015-10-10; Published online: 2015-11-19 10:01
Corresponding author. E-mail: cheng_wei@buaa.edu.cn
Abstract: Satellite structures usually have a lot of open thin-walled beams under the action of transverse stiffening member. Transverse member is generally evenly distributed along the axial direction of the beam. Through the study of such structures, it is theoretically proved that the differential equation of vibration of this structure has the same form as the equation of ordinary open thin-walled beam. Therefore, equivalent calculation was carried out using the open thin-walled beam element. Three kinds of mathematical models of open thin-walled beams, finite element model, transfer matrix model and analytical model, were established. The equivalent cross-section parameters were identified using sequential quadratic programming. At the same time, the influence of different objective functions on the identification results was analyzed. And a method for estimating the initial parameters of the cross-section was presented. For the finite element model, the MATLAB and ABAQUS interactive parameters optimization method was proposed. A combination of both full advantages can quickly and efficiently optimize the cross-section parameters and this method has strong versatility. Finally, the correctness and accuracy of the equivalent modeling method are verified by experiments. The proposed equivalent modeling method can reduce more than 90% of the number of elements. By establishing the simplified model, the efficiency of the structural model updating and structure reanalysis can be greatly improved.
Key words: transverse stiffening member     open thin-walled beam     differential equation of vibration     equivalent modeling     parameter optimization identification

Arpaci等[2-3]研究了考虑扭转效应的开口薄壁梁的耦合振动特性。Prokić [4]研究了任意截面形状的开口薄壁梁的动力学特性。Ambrosini等[5-6]运用Vlasov理论计算了任意截面开口薄壁梁的振动频率。Prokić [7]考虑了复杂薄壁梁振动的耦合效应。本文在此基础上建立了有横向加强构件的开口薄壁梁的弯扭耦合振动方程。通过实验测得的固有频率，分别采用有限元法、传递矩阵法和解析法对模型进行参数优化辨识，从而建立了等效梁单元模型，通过与实测频率进行对比，验证了方法的正确性与精确性。

1 开口薄壁杆件弯扭耦合振动方程

 图 1 开口薄壁梁横截面 Fig. 1 Cross-section of open thin-walled beam
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2 横向加强构件作用下的开口薄壁杆件

 图 2 横向隔板构件作用下的开口杆件 Fig. 2 Open bar under action of transverse partition member
 图 3 含有横向隔板的一个微段 Fig. 3 micro-segment containing transverse partition member
 图 4 横向隔板示意图 Fig. 4 Schematic diagram of transverse partition member

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3 复杂薄壁开口杆件等效截面参数优化识别 3.1 有限元模型

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3.2 传递矩阵模型

Ambrosini[13]建立了考虑剪切效应与转动惯量的开口薄壁梁单元的传递矩阵模型，即

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3.3 解析模型

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3.4 截面参数优化识别流程

3.4.1 目标函数建立

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 图 5 MATLAB与ABAQUS交互优化流程 Fig. 5 MATLAB and ABAQUS interactive optimization process
 图 6 截面参数识别流程 Fig. 6 Cross-section parameter identification procedure

3.4.2 MATLAB与ABAQUS交互式优化

MATLAB是一种影响大、流行广的科学计算语言，其中拥有大量优秀的数值算法，而ABAQUS是一款非常强大的有限元分析软件，可以模拟非常复杂的模型。在运用有限元模型进行参数辨识的过程中，本文将结合二者的优点，同时使用MATLAB和ABAQUS进行交互式的优化参数辨识。INP文件是ABAQUS的输入文件，它包含了对整个模型的完整描述，而INP文件本质上是一种文本文件，因此MATLAB可以对其进行读写从而控制ABAQUS的分析过程。ABAQUS通常可以将其计算结果输出至DAT文件，MATLAB可以直接读取DAT文件中的数据，由此可知只需要通过文件之间的交换即可实现MATLAB与ABAQUS联合优化。

3.4.3 截面参数初始值估算

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n=m时有

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4 仿真算例

 截面参数 梁单元模型 传递矩阵模型 解析模型 二范数 无穷范数 一范数 Ix/(10-6m4) 2.26 2.34 2.24 2.16 2.30 Iy/(10-6m4) 3.88 2.50 3.76 3.76 3.71 Jφ/(10-7m4) 1.47 1.44 1.42 1.39 1.44 Jw/(10-9m6) 2.08 2.19 2.06 2.17 2.13 e1/m 0 0 0 0 0 e2/mm 66.4 68.4 65.8 68.2 67.2 Kx 0.837 0.905 0.824 0.788 Ky 0.411 0.443 0.424 0.352

 % 模态阶次 梁单元模型 传递矩阵模型 解析模型 二范数 无穷范数 一范数 1 0.37 2.12 0.29 0.01 0.27 2 1.08 0.98 1.06 0.08 0.80 3 0.09 1.91 0.15 1.40 0.88 4 0.57 2.11 1.05 1.71 1.36 5 1.68 2.02 2.18 1.22 1.77 6 0.07 1.99 0.06 0.18 0.57 7 0.45 0.45 1.01 0.20 0.53 8 0.67 0.92 0.72 0.22 0.63 9 0.02 2.13 0.09 2.35 1.15 10 2.07 2.18 2.06 2.19 2.12 11 2.05 2.08 1.11 2.18 1.85

 模态阶次 模态应变能/(N·m) 模态应变能误差/% 真实模型 等效模型 1 530 070 538 680 1.62 2 595 130 604 227 1.51 3 923 810 922 350 -0.16 4 2 421 200 2 362 411 -2.41 5 5 509 182 5 574 967 1.18 6 6.456×106 6.460×106 0.06 7 7.208×106 7 040 179 -2.33 8 1.419×107 1.396×107 -1.62 9 2.215×107 2.226×107 0.49 10 2.733×107 2.790×107 2.01 11 3.067×107 3.129×107 1.96

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 图 7 等效截面扭转常数与L的关系曲线 Fig. 7 Relationship curves between equivalent cross-section torsion constant and L
5 实验验证

 图 8 实验模态测试示意图 Fig. 8 Schematic diagram of experimental modal testing
 图 9 不同测点的加速度频响函数Bode图 Fig. 9 Bode diagram of acceleration frequency response function at different points

 截面参数 梁单元模型 传递矩阵模型 解析模型 二范数 无穷范数 一范数 Ix/(10-7m4) 3.07 3.07 3.07 3.07 3.07 Iy/(10-6m4) 1.20 1.23 1.19 1.17 1.22 Jφ/(10-8m4) 1.09 1.05 1.07 1.06 1.07 Jw/(10-10m6) 2.11 2.06 2.16 2.15 2.10 e1/m 0 0 0 0 0 e2/m 0.064 5 0.063 1 0.065 6 0.066 3 0.064 2

 模态阶次 实验频率 计算频率 二范数 无穷范数 一范数 传递矩阵模型 解析模型 1 126.8 125.6 128.1 125.9 126.5 125.1 2 231.1 231.8 233.3 227.9 232.5 227.5 3 234.9 233.4 233.9 232.1 236.8 234.9 4 477.4 478.4 480.4 482.0 482.3 486.2 5 585.2 586.9 577.3 588.4 579.7 581.3 6 621.6 625.6 626.7 622.9 626.1 623.7 7 806.4 815.6 791.3 814.8 811.4 797.5

 图 10 模态频率误差 Fig. 10 Modal frequency error
6 结论

1)本文提出用开口薄壁梁来等效近似计算有均匀分布横向加强构件的薄壁杆件，这种等效方法不仅适用于有隔板的构件，对于缀板或缀条加强的薄壁杆件也有同样的适用性，具有一定的通用性。

2)本文建立了3种开口薄壁梁模型，通过对比不同模型辨识结果说明目标函数为二范数与一范数的有限元模型的精度普遍高于其他模型。

3)本文提出了通过实体有限元在静态载荷作用下产生的静态位移来估算等效截面的扭转常数与翘曲常数，并且这种估计方法有较强的通用性。

4)本文所使用的ABAQUS与MATLAB交互式的参数优化辨识方法，可以有效地将MATLAB中各种优化算法与有限元程序相结合，实现了对ABAQUS的二次开发。

5)用梁来等效替代横向构件作用的薄壁结构可以大幅度缩减单元的数量，同时在一定的频率范围内有较高的精度。

6)本文所提出的等效模型物理意义明确，不仅适用于结构模型修正，同时也适用于结构动态载荷识别与损伤识别，模型应用范围广。

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#### 文章信息

DENG Hao, CHENG Wei

Equivalent modeling method of open thin-walled beam under action of transverse stiffening member

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(7): 1469-1478
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0456