提出了一种考虑截面完整变形的截面插值梁模型，并用于机翼结构的建模与分析。首先引入截面插值函数——拉格朗日函数描述截面形状，以位移向量为未知变量描述截面位移，在此基础上依据插值理论构造梁单元位移场。不同于传统梁单元通过假定的中性轴的挠度和转角来确定梁截面各点位移，该梁模型摒弃了中性轴假设与平截面假设，通过截面插值函数得到梁截面面内、面外变形；然后通过有限元理论推导了梁单元刚度矩阵与质量矩阵；最后利用截面插值梁单元对机翼各部件进行有限元建模，并展开典型工况下的静力学分析以及动力学分析，与Nastran实体单元计算结果进行了对比分析，验证了该梁模型的有效性，为机翼的结构设计和强度分析提供了一种新的简化建模方法。
A cross-section interpolation beam model considering the complete deformation of the cross-section is proposed and used to model and analyze the wing structure. First, the Lagrange functions are introduced as the interpolation function to describe the shape of the beam cross-section, and the displacement vectors are used as unknown variables to describe the displacement of the cross-section. On this basis, the displacement field of the beam element is constructed according to the interpolation theory. While the displacements of the beam in the conventional beam element are determined by the deflection and rotation of the assumed neutral axis, the new beam element rejects the neutral axis hypothesis and flat section hypothesis, and the deformation of the beam cross-section is obtained by the interpolation functions. Then based on the finite element theory, the stiffness matrix and the mass matrix of the beam element are derived. Finally, the wing components are modeled by the cross-section interpolation beam element, and the static analysis and dynamic analysis under typical conditions are carried out. The validity of the beam model is verified by comparing the result with the results of Nastran solid model, showing that the model provides a one-dimensional beam simplified modeling method for the structural design and strength analysis of the wing.
[1] 姚卫星, 顾怡. 飞机结构设计[M]. 北京:国防工业出版社, 2016:46-47. YAO W X, GU Y. Aircraft structure design[M]. Beijing:National Defend Industry Press, 2016:46-47(in Chinese).
[2] 蒋余芬, 朱纪洪, 刘世前. 弹性机翼动力学建模与仿真[J]. 系统仿真学报, 2006, 18(z2):5-7. JIANG Y F, ZHU J H, LIU S Q. Elastic wing dynamics modeling and simulation[J]. Journal of System Simulation, 2006, 18(z2):5-7(in Chinese).
[3] 欧阳星, 余雄庆, 王宇. 采用等效刚度有限元模型的复合材料机翼颤振分析[J]. 振动工程学报, 2015, 28(3):404-410. OUYANG X, YU X Q, WANG Y. Flutter analysis of composite wing based on equivalent stiffness finite element model[J]. Journal of Vibration Engineering, 2015, 28(3):404-410(in Chinese).
[4] WRIGHT J R, COOPERJ E. Introduction to aircraft aeroelasticity and loads[M]. Hoboken, NJ:John Wiley & Sons, Ltd., 2007.
[5] ZHAO Y H, HU H Y. Structural modeling and aeroelastic analysis of high-aspect-ratio composite wings[J]. Chinese Journal of Aeronautics, 2005, 18(1):25-30.
[6] 张旭, 吴志刚, 杨超. 基于等效梁模型的长直机翼动力学与颤振分析[J]. 北京航空航天大学学报, 2010, 36(11):1373-1377. ZHANG X, WU Z G, YANG C. Dynamic and flutter analysis of long straight wing based on equivalent beam model[J]. Journal of Beijing University of Aeronautics and Astronautics, 2010, 36(11):1373-1377(in Chinese).
[7] LIVNE E, NAVARRO I. Nonlinear equivalent plate modeling of wing-box structures[J]. Journal of Aircraft, 2012, 36(5):851-865.
[8] GILES G L. Equivalent plate analysis of aircraft wing box structures with general planform geometry[J]. Journal of Aircraft, 1986, 23(11):859-864.
[9] KRISHNAMURTHY T, TSAI F J. Static and dynamic structural response of an aircraft wing with damage using equivalent plate analysis[C]//49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Reston, VA:AIAA, 2008.
[10] 王宇, 欧阳星, 余雄庆. 采用等效有限元模型的复合材料机翼结构优化[J]. 复合材料学报, 2015, 32(5):1487-1495. WANG Y, OUYANG X, YU X Q. Structural optimization of composite wing using equivalent finite element model[J]. Acta Materiae Compositae Sinica, 2015, 32(5):1487-1495(in Chinese).
[11] 周倩南. 基于位移变形理论的空间梁模型分析与研究[D]. 杭州:浙江大学, 2015:1-2. ZHOU Q N. Spatial beam model analysis and research based on displacement deformation theory[D]. Hangzhou:Zhejiang University, 2015:1-2(in Chinese).
[12] TIMOSHENKO S P. LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars[J]. Philosophical Magazine, 1950, 7(245):239-250.
[13] 胡海昌. 弹性力学的变分原理及其应用[M]. 北京:科学出版社, 1981:139-144. HU H C. Variational principle of elastic mechanics and its application[M]. Beijing:Science Press, 1981:139-144(in Chinese).
[14] HUGHES T J R, TAYLOR R L, KANOKNUKULCHAI W. A simple and efficient finite element for plate bending[J]. International Journal for Numerical Methods in Engineering, 2010, 11(10):1529-1543.
[15] ORAL S. Anisoparametric interpolation in hybrid-stress timoshenko beam element[J]. Journal of Structural Engineering, 1991, 117(4):1070-1078.
[16] OWEN D R J, HINTON E. Finite elements in plasticity:Theory and practice[M]. Swansea:Pineridge Press, 1980.
[17] 王勖成. 有限单元法[M]. 北京:清华大学出版社, 2003:111-113. WANG X C. Finite element method[M]. Beijing:Tsinghua University Press, 2003:111-113(in Chinese).
[18] BATHEK J. Finite element procedures in engineering analysis[M]. Englewood Cliffs, NJ:Prentice-Hall, Inc., 1982.
[19] OÑATE E. Structural analysis with the finite element method[M]. Netherlands:Springer, 2009.
[20] 陈桂彬. 气动弹性设计基础[M]. 北京:北京航空航天大学出版社, 2010:114-128. CHEN G B. Aeroelastic design basis[M]. Beijing:Beihang University Press, 2010:114-128(in Chinese).
[21] 赵永辉. 气动弹性力学与控制[M]. 北京:科学出版社, 2007:130-139. ZHAO Y H. Aeroelasticity mechanics and control[M]. Beijing:Science Press, 2007:130-139(in Chinese).
CJA