Deformation Field Reconstruction of Timoshenko Beam and Optimization of Sensor Placement

doi:10.3901/JME.2020.20.001

Abstract

Abstract: For the problem that KO displacement theory can be only applicable to the reconstruction of the one-dimensional deformation field, a new method is presented for reconstructing six degree of freedom displacement field, which is known as “multidimensional integration method”. According to the static equilibrium equation of Timoshenko beam, the mathematical model is established between displacement, rotation and external load. And the corresponding strain field functions and displacement field functions are deduced and the translation relation between section strain and surface strain is established for different external load. In order to improve the tolerance of the presented method, the accuracy and the robustness of the reconstruction displacement are taken as the optimization objective function to established multi-objective particle swarm optimization model for strain sensor placement. The wing frame is taken as the experimental platform to perform the finite element analysis, establish optimization objection model, and give the optimized strain sensor distribution scheme. And on the basis of this scheme, the displacement field is reconstructed by using the results of finite element analysis and the measured surface strain. The experimental results show that the proposed “dimensional integration method” presents a high reconstruction accuracy under the action of two different forms of external loads.

Key words: KO displacement theory, Timoshenko beam, static equilibrium equation, tolerance, multi-objective particle swarm optimization model

CLC Number:

ZHAO Feifei, CAO Kaituo, BAO Hong, GAO Guoming. Deformation Field Reconstruction of Timoshenko Beam and Optimization of Sensor Placement[J]. Journal of Mechanical Engineering, 2020, 56(20): 1-11.

References

[1] 胡乃岗,保宏,连培园.大型相控阵天线结构与调整机构一体化设计[J].机械工程学报,2015,51(1):196-197. HU Naigang,BAO Hong,LIAN Peiyuan. Synthetic design of structure and adjustment mechanism of large phased array antennas[J]. Journal of Mechanical Engineering,2015,51(1):196-197.
[2] 康明魁.有源相控阵天线机电热耦合建模、误差分析与优化设计[D].西安:西安电子科技大学,2015. KANG Mingkui. Coupled structural electromagnetic thermal modeling,error analysis and optimization design of active phased array antennas[D]. Xi'an:Xidian University,2015.
[3] GHERLONE M,CERRACCHIO P,MATTONE M. Shape sensing methods:Review and experimental comparison on a wing-shaped plate[J]. Progress in Aerospace Sciences,2018,99:14-26.
[4] 程志峰,金超,孙伟.数字摄影测量在大型天线主面测量中的应用[J].河北省科学院学报,2019,36(1):47-50. CHENG Zhifeng,JIN Chao,SUN Wei. Application of digital photogrammetry in main surface measurement of large antenna[J]. Journal of the Hebei Academy of Science,2019,36(1):47-50
[5] 朱永国,张文博,邓正平.基于激光跟踪仪和机器视觉的飞机翼身对接装配偏差动态综合修正[J].机械工程学报,2019,55(24):187-196. ZHU Yongguo,ZHANG Wenbo,DENG Zhengping. Dynamic synthesis correction of deviation for aircraft wing-fuselage docking assembly based on laser tracker and machine vision[J]. Journal of Mechanical Engineering,2019,55(24):187-196.
[6] 刘博学,柏宏武,李冬.基于数字摄影的高低温热变形测量研究[J].空间电子技术,2018,15(3):52-55. LIU Boxue,BAI Hongwu,LI Dong. Research on thermal distortion measurement based on digital photogrammetry[J]. Space Electronic Technology,2018,15(3):52-55.
[7] KIM H I,KANG L H,HAN J H. Shape estimation with distributed fiber bragg grating sensors for rotating structures[J]. Smart Materials and Structures,2011,20:1-11.
[8] RAPP S,KANG L H,HAN J H. Displacement field estimation for a two-dimensional structure using fiber bragg grating sensors[J]. Smart Materials and Structures,2009,18(2):025006.
[9] KANG L H,KIM D K,HAN J H. Estimation of dynamic structural displacements using fiber bragg grating strain sensors[J]. Journal of Sound and Vibration,2007,305(3):534-542.
[10] 封思远,保宏,张旭东.基于模糊网络的机翼天线变形测量方法[J].现代雷达,2015,37(11):59-63. FENG Siyuan,BAO Hong,ZHANG Xudong. An airfoil antenna deformation measurement method based on fuzzy network[J]. Modern Radar,2015,37(11):59-63.
[11] MAO Z,TODD M. Comparison of shape reconstruction strategies in a complex flexible structure[C]//Proceedings of the SPIE 6932,Sensors and Smart Structures Technologies for Civil,Mechanical,and Aerospace System,May 1,2008,San Diego,United States. California:SPIE,2008.
[12] ZHAO Feifei,XU Libo,BAO Hong. Shape sensing of variable cross-section beam using the inverse finite element method and isogeometric analysis[J]. Measurement,2020,185:1-11.
[13] TESSLER A,SCIUVA M D,GHERLONE M. A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics[J]. Journal of Mechanics of Materials and Structures,2010,5(2):341-367.
[14] CERRACCHIO P,GHERLONE M,SCIUVA M D. A novel approach for displacement and stress monitoring of sandwich structures based on the inverse finite element method[J]. Composite Structures,2015,127:69-76.
[15] KEFAL A,TESSLER A,OTERKUS E. An enhanced inverse finite element method for displacement and stress monitoring of multilayered composite and sandwich structures[J]. Composite Structures,2015,179:514-540.
[16] KO W L,FLEISCHER V T. Further development of Ko displacement theory for deformed shape predictions of nonuniform aerospace structures[R]. California:NTRS 2009.
[17] KO W L,RICHARD W L,FLEISCHER V T. Displacement theories for in-flight deformed shape predictions of aerospace structures[R]. California:NTRS 2007.
[18] JUTTE C V,KO W L,STEPHENS C A. Deformed shape calculation of a full-scale wing using fiber optic strain data from a ground loads test[R]. California:NTRS 2011.
[19] KO W L,RICHARDS W L,FLEISHER V T. Applications of the ko displacement theory to the deformed shape predictions of the doubly-tapered ikhana wing[R]. California:NTRS 2009.
[20] AKL W,POH S,BAZ A. Wireless and distributed sensing of the shape of morphing structures[J]. Sensors and Actuators A-physical,2007,140(1):94-102.
[21] 蒋纯志,金桂,陈亚琦.等截面铁木辛柯梁的分布传递函数方法[J].湖南科技学院学报,2009,30(8):51-52. JIANG Chunzhi,JIN Gui,CHEN Yaqi. The distributed transfer function method of uniformed timoshenko beam[J]. Journal of Hunan University of Science and Engineering,2009,30(8):51-52.
[22] ZHAO Feifei,BAO Hong,XUE Song. Multi-objective particle swarm optimization of sensor distribution scheme with consideration of the accuracy and the robustness for deformation reconstruction[J]. Sensors,2019,19(6):1306.
[23] JOSHUA H G,SHIN J C,BOUWENSE M A. Minimum condition number by orthogonal projection rows election of artificial boundary conditions for finite element model update and damage detection[J]. Journal of Sound and Vibration,2018,433(27):179-197.