SHAPE RECONSTRUCTION OF PLANE BEAM WITH FINITE DEFORMATION BASED ON ABSOLUTE NODAL COORDINATE FORMULATION
-
摘要: 现有的对有限变形条件下柔性结构变形重构的研究往往单纯基于曲率与应变间的几何关系, 同时忽略了被测体的纵向变形及其与弯曲变形的耦合效应. 为得到一种更加精确且能借助现有的力学工具进行应用方向扩展的变形重构方法, 以平面梁为对象, 借鉴变形重构逆有限元法的思想, 将平面梁的变形重构问题视作一类最优化问题. 首先, 通过引入绝对节点坐标法(absolute nodal coordinate formulation, ANCF)对柔性结构大变形下非线性的平面梁应变−位移关系进行精确描述, 构造了一种逆梯度缩减ANCF平面索梁单元. 然后, 对此逆ANCF单元进行改进, 在简化节点自由度的同时通过引入罚函数确保单元节点处的曲率连续性, 既保证了本问题的适定性, 也提升了最终解的精确性. 最后, 基于该单元利用Newton法构造了平面梁有限变形下变形重构问题的两种求解算法, 即逐单元算法和多单元整体算法, 以实现不同需求下的稳定求解. 数值仿真结果表明, 本方法在大变形条件下的变形重构误差小于1%, 而且在测点较少的情况下依然保持较高的精度, 同时验证了本方法的收敛性与计算效率.Abstract: Most of the existing researches on deformation reconstruction of flexible structures with finite deformation are only based on the geometric relationship between curvature and strain, which ignores the longitudinal deformation and the coupling effect of the longitudinal deformation and the bending deformation. In order to construct a more accurate deformation reconstruction method which can be extended with the help of existing mechanical tools, this paper takes the plane beam as the object, partially inherits inverse finite element method developed by Tessler A, and regards the deformation reconstruction problem of plane beam as a kind of numerical optimization problem. Firstly, by introducing the absolute nodal coordinate formulation (ANCF) into the description of mapping relationship between strain and displacement, an inverse gradient reduced ANCF plane beam element is derived. Secondly, the inverse ANCF element is modified to simplify the degree of freedom of nodes and ensure the C2 continuity at nodes by introducing the penalty function, which not only ensures the problem is well-posed, but also improves the accuracy of the final result. Finally, based on the inverse ANCF element, the Newton method is used to develop two types of algorithms for deformation reconstruction under different working conditions, one is the element-by-element algorithm and the other is the multi-element algorithm. The numerical simulation results show that the reconstruction relative error of this method is less than 1% under the condition of large deformation, and it still maintains high accuracy under the condition of few measuring points. The convergence and computational efficiency of the method are verified by numerical simulation example.
-
图 1 基于应变信息的平面梁变形重构模型
Figure 1. Shape reconstruction model of plane beam based on strain information
图 2 基于线性应变−位移关系的变形重构
Figure 2. Deformation reconstruction based on linear strain displacement relationship
图 3 梯度缩减ANCF梁单元内部的虚假轴向应变
Figure 3. False axial strain in gradient reduction ANCF beam element
图 4 边界约束及曲率连续性约束示意图
Figure 4. Schematic diagram of boundary constraints and curvature continuity constraints
图 9 大变形条件下不同测点数的重构效果
Figure 9. Reconstruction performance of different number of measuring points under large deformation
图 11 不同方法的悬臂梁变形重构仿真结果
Figure 11. Reconstruction performance of cantilever beam in different methods
图 12 轴向大变形条件下的简支梁示意图
Figure 12. Schematic of simply supported beam under large axial deformation
图 13 轴向大变形条件下曲率连续化方法和iANCF方法的对比
Figure 13. Comparison of curvature-based method and iANCF under large axial deformation
表 1 20个测点下的相对误差及计算用时
Table 1. Reconstruction error and calculation time under 20 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ $ {y_{{\rm{err}}}} $ $ {\theta _{{\rm{err}}}} $ t/s 50 6.28×10−4 3.68×10−4 7.42×10−8 0.137 100 6.20×10−4 3.82×10−4 2.88×10−7 0.138 200 7.96×10−4 6.86×10−4 1.10×10−5 0.145 
下载: 导出CSV
表 2 10个测点下的相对误差及计算用时
Table 2. Reconstruction error and calculation time under 10 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ ${y_ {\rm{err} } }$ ${\theta _ {\rm{err} } }$ t/s 50 8.67×10−4 3.75×10−4 9.12×10−7 0.118 100 8.31×10−4 3.69×10−4 4.12×10−6 0.123 200 9.18×10−4 6.04×10−4 2.56×10−4 0.124
下载: 导出CSV
表 3 4个测点下的相对误差及计算用时
Table 3. Reconstruction error and calculation time under 4 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ ${y_ {\rm{err} } }$ ${\theta _ {\rm{err} } }$ t/s 50 2.38×10−3 1.53×10−3 5.34×10−5 0.113 100 2.68×10−3 1.64×10−3 2.60×10−4 0.113 200 5.29×10−2 2.49×10−2 1.88×10−2 0.118
下载: 导出CSV
-
[1] Rapp S, Kang LH, Han JH, et al. Displacement field estimation for a two-dimensional structure using fiber Bragg grating sensors. Smart Material Structures, 2009, 18(2): 025006 doi: 10.1088/0964-1726/18/2/025006 [2] Roesthuis RJ, Janssen S, Misra S. On using an array of fiber Bragg grating sensors for closed-loop control of flexible minimally invasive surgical instruments//IEEE/RSJ International Conference on Intelligent Robots & Systems, 2014 [3] Ryu SC, Dupont PE. FBG-based shape sensing tubes for continuum robots//IEEE International Conference on Robotics and Automation (ICRA), 2014 [4] Froggatt M, Moore J. High-spatial-resolution distributed strain measurement in optical fiber with rayleigh scatter. Applied Optics, 1998, 37(10): 1735-1740 doi: 10.1364/AO.37.001735 [5] Li J, Zhi Z, Ou J. Interface transferring mechanism and error modification of embedded FBG strain sensor//Smart Structures & Materials. International Society for Optics and Photonics, 2004 [6] Jiang H, Bartel V, Daniel K, et al. Real-time estimation of time-varying bending modes using fiber bragg grating sensor arrays. AIAA Journal, 2013, 51(1): 178-185 [7] Davis MA, Kersey AD, Sirkis J, et al. Shape and vibration mode sensing using a fiber optic Bragg grating array. Smart Materials & Structures, 1999, 5(6): 759-765 [8] Foss GC, Haugse ED. Using modal test results to develop strain to displacement transformations. Proceedings of SPIE-The International Society for Optical Engineering, 1995, 2460: 112 [9] Tessler A, Spangler JL. A variational principle for reconstruction of elastic deformations in shear deformable plates and shells. NASA/TM-2003-212445, 2003 [10] Vazquez SL, Tessler A, Quach CC, et al. Structural health monitoring using high-density fiber optic strain sensor and inverse finite element methods. NASA/TM-2005-2137612005, 2005 [11] Gherlone M, Cerracchio P, Mattone M, et al. An inverse finite element method for beam shape sensing: theoretical framework and experimental validation. Smart Materials & Structures, 2014, 23(4): 045027 [12] Groh R, Tessler A. Computationally efficient beam elements for accurate stresses in sandwich laminates and laminated composites with delaminations. Computer Methods in Applied Mechanics & Engineering, 2017, 320: 369-395 [13] Roy R, Tessler A, Surace C, et al. Shape sensing of plate structures using the inverse finite element method: investigation of efficient strain–sensor patterns. Sensors, 2020, 20(24): 7049 doi: 10.3390/s20247049 [14] Kefal A, Tessler A, Oterkus E. An enhanced inverse finite element method for displacement and stress monitoring of multilayered composite and sandwich structures. Composite Structures, 2017, 179: 514-540 [15] Kefal A, Tabrizi IE, Yildiz M, et al. A smoothed iFEM approach for efficient shape-sensing applications: Numerical and experimental validation on composite structures. Mechanical Systems and Signal Processing, 2020, 152: 107486 [16] Oboe D, Colombo L, Sbarufatti C, et al. Shape sensing of a complex aeronautical structure with inverse finite element method. Sensors, 2021, 21(4): 1388 doi: 10.3390/s21041388 [17] Li T, Cao M, Li J, et al. Structural damage identification based on integrated utilization of inverse finite element method and pseudo-excitation approach. Sensors, 2021, 21(2): 606 doi: 10.3390/s21020606 [18] Ko WL, Richards WL, Tran VT. Displacement theories for in-flight deformed shape predictions of aerospace structures. NASA/TP-2007-214612, 2007 [19] Yi JC, Zhu, XJ, Zhang HS, et al. Spatial shape reconstruction using orthogonal fiber Bragg grating sensor array. Mechatronics, 2012, 2012, 22(6): 679-687 [20] Roesthuis RJ, Kemp M, Dobbelsteen JJ, et al. Three-dimensional needle shape reconstruction using an array of fiber bragg grating sensors. IEEE/ASME Transactions on Mechatronics, 2014, 19(4): 1115-1126 doi: 10.1109/TMECH.2013.2269836 [21] Xu L, Ge J, Patel JH, et al. Dual-layer orthogonal fiber Bragg grating mesh based soft sensor for 3-dimensional shape sensing. Optics Express, 2017, 25(20): 24727 doi: 10.1364/OE.25.024727 [22] Shabana AA, Hussien HA, Escalona JL. Application of the absolute nodal coordinate formulation to large rotation and large deformation problems. Asme Journal of Mechanical Design, 1998, 120(2): 188-195 doi: 10.1115/1.2826958 [23] Shabana AA, Yakoub RY. Three dimensional absolute nodal coordinate formulation for beam elements: theory. Journal of Mechanical Design, 2001, 123(4): 606-613 doi: 10.1115/1.1410100 [24] Yakoub RY, Shabana AA. Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. Journal of Mechanical Design, 2001, 123(4): 614-621 doi: 10.1115/1.1410099 [25] Jiang H, Bartel V, Dolk V, et al. Modal estimation by FBG for flexible structures attitude control. IEEE Transactions on Aerospace & Electronic Systems, 2014, 50(4): 2642-2653 [26] 宋雪刚, 刘鹏, 程竹明等. 基于光纤光栅传感器和卡尔曼滤波器的载荷识别算法. 光学学报, 2018, 38(3): 0328012 (Song Xuegang, Liu Peng, Cheng Zhuming, et al. An Algorithm of dynamic load identification based on FBG sensor and kalman filter. Acta Optica Sinica, 2018, 38(3): 0328012 (in Chinese) doi: 10.3788/AOS201838.0328012 [27] 赵飞飞, 曹开拓, 保宏等. Timoshenko梁的变形场重构及传感器位置优化. 机械工程学报, 2020, 56(20): 15-25Zhao Feifei, Cao Kaituo. Bao Hong, et al. Deformation field reconstruction of timoshenko beam and optimization of sensor placement. Journal of mechanical engineering, 2020, 56(20): 15-25 (in Chinese) [28] 张科, 袁慎芳, 任元强等. 基于逆向有限元法的变形机翼鱼骨的变形重构. 航空学报, 2020, 41(8): 250-260 (Zhang Ke, Yuan Shenfang, Ren Yuanqiang, et al. Shape reconstruction of self-adaptive morphing wings’ fish bone based on inverse finite element method. Acta Aeronautica et Astronautica Sinaca, 2020, 41(8): 250-260 (in Chinese) [29] 谭跃刚, 黄兵, 刘虎等. 基于分布应变的薄板变形重构算法研究. 机械工程学报, 2020, 56(13): 242-248 (Tan Yuegang, Huang Bing, Liu Hu, et al. Research on deformation reconstruction algorithm of thin plate based on distributed strain. Journal of Mechanical Engineering, 2020, 56(13): 242-248 (in Chinese) doi: 10.3901/JME.2020.13.242 [30] 赵士元, 崔继文, 陈勐勐. 光纤形状传感技术综述. 光学精密工程, 2020, 28(1): 10 (Zhao Shiyuan, Cui Jiwen, Chen Mengmeng. Review on optical fiber shape sensing technology. Optics and Precision Engineering, 2020, 28(1): 10 (in Chinese) doi: 10.3788/OPE.20202801.0010 [31] 朱晓锦, 季玲晓, 张合生等. 基于空间正交曲率信息的三维曲线重构方法分析. 应用基础与工程科学学报, 2011(2): 305-313 (Zhu Xiaojin, Ji Lingxiao, Zhang Hesheng, et al. Analysis of 3D curve reconstruction method using orthogonal curvatures. Journal of Basic Science and Engineering, 2011(2): 305-313 (in Chinese) doi: 10.3969/j.issn.1005-0930.2011.02.015 [32] Kefal A, Oterkus E, Tessler A, et al. A quadrilateral inverse-shell element with drilling degrees of freedom for shape sensing and structural health monitoring. Engineering Science & Technology An International Journal, 2016, 19(3): 1299-1313 [33] 张越, 赵阳, 谭春林等. ANCF索梁单元应变耦合问题与模型解耦. 力学学报, 2016, 48(6): 1406-1415 (Zhang Yue, Zhao Yang, Tan Chunlin, et al. The strain coupling problem and model decoupling of ANCF cable/beam element. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1406-1415 (in Chinese) doi: 10.6052/0459-1879-16-127 [34] 章孝顺, 章定国, 陈思佳等. 基于绝对节点坐标法的大变形柔性梁几种动力学模型研究. 物理学报, 2016, 65(9): 148-157Zhang Xiaoshun, Zhang Dingguo, Chen Sijia, Hong Jiazhen, et al. Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation. Acta Physica Sinica, 2016, 65(9): 148-157 (in Chinese) [35] Berzeri M, Shabana AA. Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. Journal of Sound & Vibration, 2000, 235(4): 539-565 [36] Tikhonov AN, Arsenin VY. Solutions of Ill-Posed Problems. Washington DC: Wiston, 1977 -
下载:
点击查看大图
计量
- 文章访问数: 413
- HTML全文浏览量: 130
- PDF下载量: 91
- 被引次数: 0
