EI、Scopus 收录
Volume 55 Issue 2
Feb.  2023
Turn off MathJax
Article Contents
Huang Ke, Zhang Jiaying, Wang Qingyun. Natural vibration analysis of two-dimensional flexible wing based on non-uniform beam model. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 487-496 doi: 10.6052/0459-1879-22-551
Citation: Huang Ke, Zhang Jiaying, Wang Qingyun. Natural vibration analysis of two-dimensional flexible wing based on non-uniform beam model. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 487-496 doi: 10.6052/0459-1879-22-551


doi: 10.6052/0459-1879-22-551
  • Received Date: 2022-11-21
  • Accepted Date: 2023-01-07
  • Available Online: 2023-01-09
  • Publish Date: 2023-02-18
  • In order to improve the flight performance of the aircraft, morphing technologies are used to change aerodynamic characteristics through smooth and continuous structural deformation. Since this new concept requires changing the structural shape to obtain the best performance, its inherent dynamic characteristics will be affected and even change its aeroelastic performance. In this paper, an equivalent modelling method of the two-dimensional flexible wing with camber morphing is developed. The dynamic model of the flexible wing is established based on the hypothesis of a non-uniform beam model. The analytical solution and natural frequencies are obtained by the method of Frobenius and verified by comparison with the finite element method solution. The errors of the first four natural frequencies are within 1% and the corresponding modes are consistent. The flexible wing is prepared by 3D printing engineering plastic (ABS) and silicone rubber skin. The Young's modulus of the 3D printing material and silicone rubber are respectively measured by dynamic measurement method and tensile test. The vibration response test platform is built to carry out vibration test of the flexible wing. It is found that the fundamental frequency obtained by vibration test is consistent with the theoretical model results, and the error is less than 3% compared with the finite element method. The equivalent modelling method of a two-dimensional flexible wing is established through theoretical analysis and experimental verification. The research results will provide theoretical support for applying the flexible trailing edge structures.


  • loading
  • [1]
    Barbarino S, Bilgen O, Ajaj RM, et al. A review of morphing aircraft. Journal of Intelligent Material Systems and Structures, 2011, 22(9): 823-877
    Cistone J. Next century aerospace traffic management: the sky is no longer the limit. Journal of Aircraft, 2004, 41(1): 36-42 doi: 10.2514/1.1847
    Ajaj RM, Parancheerivilakkathil MS, Amoozgar M, et al. Recent developments in the aeroelasticity of morphing aircraft. Progress in Aerospace Sciences, 2020, 120: 100682
    孙杨, 昌敏, 白俊强. 变形机翼飞行器发展综述. 无人系统技术, 2021, 4(3): 65-77 (Sun Yang, Chang Min, Bai Junqiang. Review of morphing wing aircraft. Unmanned Systems Technology, 2021, 4(3): 65-77 (in Chinese) doi: 10.19942/j.issn.2096-5915.2021.3.029
    聂瑞. 变体机翼结构关键技术研究. [博士论文]. 江苏: 南京航空航天大学, 2018

    Nie Rui. Research on key technologies of morphing wing structures. [PhD Thesis]. Jiangsu: Nanjing University of Aeronautics and Astronautics, 2018 (in Chinese)
    Wang C. Design and optimisation of morphing aircraft. [PhD Thesis]. Swansea: Swansea University, 2018
    王彬文, 杨宇, 钱战森等. 机翼变弯度技术研究进展. 航空学报, 2022, 43(1): 144-163 (Wang Binwen, Yang Yu, Qian Zhanshen, etal. Technical development of variable camber wing: review. Acta Aeronautica et Astronautica Sinica, 2022, 43(1): 144-163 (in Chinese)
    Woods BKS, Bilgen O, Friswell MI. Wind tunnel testing of the fish bone active camber morphing concept. Journal of Intelligent Material Systems and Structures, 2014, 25(7): 772-785
    Rivero AE, Fournier S, Manolesos M, et al. Experimental aerodynamic comparison of active camber morphing and trailing edge flaps. AIAA Journal, 2021, 59(7): 2627-2640 doi: 10.2514/1.J059606
    Wang C, Khodaparast HH, Friswell MI. Conceptual study of a morphing winglet based on unsymmetrical stiffness. Aerospace Science and Technology, 2016, 58: 546-558
    Zhang J, Wang C, Shaw AD, et al. Passive energy balancing design for a linear actuated morphing wingtip structure. Aerospace Science and Technology, 2020, 107: 106279 doi: 10.1016/j.ast.2020.106279
    王晨, 杨洋, 沈星等. 用于变体飞行器的波纹板等效强度模型及其优化设计. 航空学报, 2022, 43(6): 526146 (Wang Chen, Shen Yang, Shen Xing, et al. An equivalent strength model of courrgated panel and optimization design for morphing aircraft. Acta Aeronautica et Astronautica Sinica, 2022, 43(6): 526146 (in Chinese)
    张盛, 杨宇, 王志刚等. 变弯度机翼后缘偏心梁设计与验证. 航空学报, 2022, 43(6): 525892 (Zhang Sheng, Yang Yu, Wang Zhigang, et al. Design and validation of eccentric beam for variable camber trailing edge. Acta Aeronautica et Astronautica Sinica, 2022, 43(6): 525892 (in Chinese)
    Woods BKS, Friswell MI. Preliminary investigation of a fishbone active camber concept//The ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, 2012
    Rivero AE, Fournier S, Heeb RM, et al. Design, manufacture and wind tunnel test of a modular FishBAC wing with novel 3D printed skins. Applied Sciences, 2022, 12(2): 652
    杨翰雯, 胡德英, 何景武. 大展弦比机翼振动特性的工程算法平台研究. 飞机设计, 2020, 4(3): 65-77 (Yang Hanwen, Hu Deying, He Jingwu. Research on engineering algorithm platform for vibration characteristic of wing with high aspect ratio. Aircraft Design, 2020, 4(3): 65-77 (in Chinese) doi: 10.19555/j.cnki.1673-4599.2020.06.002
    陈桂彬, 邹丛青, 杨超. 气动弹性设计基础. 北京: 北京航空航天大学出版社, 2004

    Chen Guibin, Zou Congqing, Yang Chao. Fundamentals of Aeroelastic Design. Beijing: Beihang University Press, 2004 (in Chinese)
    陈洋, 王正杰, 郭士钧. 柔性翼飞行器刚柔耦合动态特性研究刚柔耦合动态特性研究. 北京理工大学学报, 2017, 37(10): 1061-1066 (Chen Yang, Wang Zhengjie, Guo Shijun. Analysis of rigid-flexible coupling dynamic characteristics of flexible wing aircraft. Transactions of Beijing Institute of Technology, 2017, 37(10): 1061-1066 (in Chinese)
    张亚滨, 高恒烜, 李书. 复合材料机翼结构动力学分析. 飞机设计, 2015, 35(3): 11-15 (Zhang Yabin, Gao Hengxuan, Li Shu. Dynamic analysis of composite materials wing structure. Aircraft Design, 2015, 35(3): 11-15 (in Chinese) doi: 10.19555/j.cnki.1673-4599.2015.03.003
    王松松, 郭翔鹰, 王帅博. 变截面Z型折叠机翼振动特性的有限元与实验分析. 动力学与控制学报, 2020, 18(6): 84-89 (Wang Songsong, Guo Xiangying, Wang Shuaibo. Finite element analysis and experiment on vibration of Z-shaped morphing wing with variable section. Journal of Dynamics and Control, 2020, 18(6): 84-89 (in Chinese)
    田坤黉, 谷良贤, 王洪伟. 基于Hamilton原理的大展弦比直机翼固有特性分析. 机械强度, 2010, 32(5): 854-858 (Tian Kunhong, Gu Liangxian, Wang Hongwei. Inherence characteristic analysis of high apect ration wing based on Hamilton’s principle. Journal of Mechanical Strength, 2010, 32(5): 854-858 (in Chinese) doi: 10.16579/j.issn.1001.9669.2010.05.025
    Woods BKS, Friswell MI. Structual analysis of the fish bone active camber concept//Proceedings of the AIDAA XXII Conference, 2013
    Woods BKS, Friswell MI. Structural characterization of the fish bone active camber morphing airfoil//22nd AIAA/ASME/AHS Adaptive Structures Conference, 2014
    Woods BKS, Dayyani I, Friswell MI. Fluid/structure-interaction analysis of the fish-bone-active-camber morphing concept. Journal of Aircraft, 2015, 52(1): 307-319 doi: 10.2514/1.C032725
    Zhang J, Shaw AD, Wang C, et al. Aeroelastic model and analysis of an active camber morphing wing. Aerospace Science and Technology, 2021, 111: 106534 doi: 10.1016/j.ast.2021.106534
    Hildebrand FB, Advanced calculus for applications. 2nd Edition. Englewood Cliffs: Prentice Hall, 1976
    徐芝纶. 弹性力学 (上册), 第5版. 北京: 高等教育出版社, 2006

    Xu Zhilun. Elasticity Mechanics (Volume 1), 5th edn. Beijing: Higher Education Press, 2006 (in Chinese)
    Naguleswaran S. Transverse vibration of an uniform Euler-Bernoulli beam under linearly varying axial force. Joumal of Sound and Vibration, 2004, 275(1-2): 47-57 doi: 10.1016/S0022-460X(03)00741-7
    徐腾飞, 向天宇, 赵人达. 变截面Euler-Bernoulli梁在轴力作用下固有振动的级数解. 振动与冲击, 2007, 26(11): 99-101 (Xu Tengfei, Xiang Tianyu, Zhao Renda. Series solution of natural vibration of the variable cross-section Euler-Bernoulli beam under axial force. Journal of Vibration and Shock, 2007, 26(11): 99-101 (in Chinese) doi: 10.3969/j.issn.1000-3835.2007.11.023
    邢誉峰, 李敏. 计算固体力学原理与方法. 北京: 北京航空航天大学出版社, 2011

    Xing Yufeng, Li Min. The Theory and Method of Computational Solid Mechanics. Beijing: Beihang University Press, 2011 (in Chinese)
    杜运兴, 程鹏, 周芬. 变截面功能梯度Timoshenko梁的自由振动分析. 湖南大学学报(自然科学版), 2021, 48(5): 55-62 (Du Yunxing, Cheng Peng, Zhou Fen. Free vibration analysis of functionally graded Timoshenko beams with variable section. Journal of Hunan University (Natural Sciences) , 2021, 48(5): 55-62 (in Chinese) doi: 10.16339/j.cnki.hdxbzkb.2021.05.007
    Şimşek M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Engineering and Design, 2010, 240(4): 697-705
    殷雅俊, 范钦珊, 王晶等. 材料力学, 第3版. 北京: 高等教育出版社, 2019

    Yin Yajun, Fan Qinshan, Wang Jing, et al. Mechanics of Materials, 3rd edn. Beijing: Higher Education Press, 2019 (in Chinese)
    邢誉峰, 李敏. 工程振动基础. 北京: 北京航空航天大学出版社, 2011

    Xing Yufeng, Li Min. The Foundation of Engineering Vibration. Beijing: Beihang University Press, 2011 (in Chinese)
  • 加载中


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(15)  / Tables(6)

    Article Metrics

    Article views (514) PDF downloads(80) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint