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Analysis of planar shear deformable beam using rotation field curvature formulation

## Analysis of planar shear deformable beam using rotation field curvature formulation

Zhang Da-Yu, Luo Jian-Jun, Zheng Yin-Huan, Yuan Jian-Ping
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• #### Abstract

In recent years, research on space debris removal technique has received wide attention in aerospace field. Many novel concepts on active flexible debris remover have been proposed, such as flexible flying net, tethered cable manipulator. In view with the high flexibility and large deformation of this kind of structure, the implementation of attitude control is challenging. An accurate dynamic model of highly flexible structure is important and needed. The beam element is the most common element adopted in flexible remover models. So, in this investigation, a rotation field-based curvature shear deformable beam using absolute nodal coordinate formulation (ANCF) (named RB-curvature ANCF beam) is proposed and its geometrically nonlinear characteristic under large deformation motion is studied. Curvature is first derived through planar rotation transformation matrix between the reference frame and current tangent frame of beam centerline, and written as an arc-length derivative of the orientation angle of the tangent vector. Using the geometrically exact beam theory, the strain energy is expressed as an uncoupled form, and the new curvature is adopted to formulate bending energy. Based on the ANCF, the dynamic equation of beam is established, where mass and external force matrices are constant. In order to validate the performance of proposed beam element, other two types of beams are introduced as the comparative models. One is the classical ANCF fully parameterized shear deformable beam derived by continuum mechanics theory, and the other is position field-based curvature ANCF shear deformable beam (named PBcurvature ANCF beam). The PB-curvature model is evaluated by differentiating unit tangent vector of beam centerline with respect to its arc length quoted from differential geometry theory. A series of static analysis, eigenfrequency tests and dynamic analysis are implemented to examine the performance of the proposed element. In static analysis, both small and non-small deformation cases show that the proposed RB-curvature ANCF beam achieves the faster speed, higher precision and good agreement with analytical solution in the case of cantilever beam subjected to a concentrated tip force, which is compared with other two beam models. The eigenfrequency analysis validates RB-curvature ANCF beam in a simply supported beam case that converges to its analytical solution. Meanwhile, the mode shapes of the proposed ANCF beam could be correctly corresponded to vibration state of element with respect to each different eigenfrequency. In the dynamics test, a flexible pendulum case is used and simulation results show that the proposed RB-curvature ANCF beam accords well with ANSYS BEAM3, classical ANCF shear beam and PB-curvature ANCF beam in vertical displacements of tip point and middle point. Since deformation modes are uncoupled in the cross section of proposed beam element, its shear strain is achieved with much better convergence in the case of lower elastic modulus, and shear locking is significantly alleviated, compared with classical ANCF beam. Therefore, RB-curvature ANCF shear deformable beam element proposed in this paper is able to describe accurately geometric nonlinearity in large deformation problem, and can be a potential candidate in the modeling of flexible/rigid-flexible mechanisms.

#### Authors and contacts

###### Corresponding author: Luo Jian-Jun, jjluo@nwpu.edu.cn
• Funds: Project supported by the Major Program of National Natural Science Foundation of China (Grant Nos. 61690210, 61690211), and the National Natural Science Foundation of China (Grant Nos. 61603304, 11472213).

#### References

 [1] Bonnal C, Ruault J M, Desjean M C 2013 Acta Astronaut. 85 51 [2] Nishida S I, Kawamoto S 2011 Acta Astronaut. 68 113 [3] Liu J Y, Lu H 2007 Multibody Syst. Dyn. 18 487 [4] He X S, Song M, Deng F Y 2011 Acta Phys. Sin. 60 044501 (in Chinese) [和兴锁, 宋明, 邓峰岩 2011 物理学报 60 044501] [5] He X S, Deng F Y, Wang R 2010 Acta Phys. Sin. 59 1428 (in Chinese) [和兴锁, 邓峰岩, 王睿 2010 物理学报 59 1428] [6] Chen S J, Zhang D G, Hong J Z 2013 Chin. J. Theor. Appl. Mech. 45 251 (in Chinese) [陈思佳, 章定国, 洪嘉振 2013 力学学报 45 251] [7] Shabana A A 1997 Multibody Syst. Dyn. 1 189 [8] Tian Q, Zhang Y Q, Chen L P, Tan G 2010 Adv. Mech. 40 189 (in Chinese) [田强, 张云清, 陈立平, 覃刚 2010 力学进展 40 189] [9] Omar M A, Shabana A A 2001 J. Sound Vib. 243 565 [10] Hussein B A, Sugiyama H, Shabana A A 2007 J. Comput. Nonlinear Dyn. 2 146 [11] Dmitrochenko O N, Hussein B A, Shabana A A 2009 J. Comput. Nonlinear Dyn. 4 21002 [12] García-Vallejo D, Mikkola A M, Escalona J L 2007 Nonlinear Dyn. 50 249 [13] Tian Q, Zhang Y Q, Chen L P, Yang J Z 2010 Nonlinear Dyn. 60 489 [14] Gerstmayr J, Matikainen M K, Mikkola A M 2008 Multibody Syst. Dyn. 20 359 [15] Nachbagauer K, Pechstein A S, Irschik H, Gerstmayr J 2011 Multibody Syst. Dyn. 26 245 [16] Nachbagauer K, Gruber P, Gerstmayr J 2013 J. Comput. Nonlinear Dyn. 8 021004 [17] Gerstmayr J, Shabana A A 2006 Nonlinear Dyn. 45 109 [18] Dufva K E, Sopanen J T, Mikkola A M 2005 J. Sound Vib. 280 719 [19] Mikkola A M, Dmitrochenko O, Matikainen M 2009 J. Comput. Nonlinear Dyn. 4 011004 [20] Vesa-Ville A, Hurskainen T, Matikainen M K, Wang J, Mikkola A M 2016 J. Comput. Nonlinear Dyn. 12 041007 [21] Zhang X S, Zhang D G, Chen S J, Hong J Z 2016 Acta Phys. Sin. 64 094501 (in Chinese) [章孝顺, 章定国, 陈思佳, 洪嘉振 2016 物理学报 64 094501] [22] Goetz A 1970 Introduction to Differential Geometry (Reading, Massachussetts: Addison Wesley Pub. Co) pp56-58 [23] Timoshenko S 1940 Strength of Materials (Part I Elementary Theory and Problems Second Edition) (New York: D.Van Nostrand Co) pp147-148

#### Cited By

•  [1] Bonnal C, Ruault J M, Desjean M C 2013 Acta Astronaut. 85 51 [2] Nishida S I, Kawamoto S 2011 Acta Astronaut. 68 113 [3] Liu J Y, Lu H 2007 Multibody Syst. Dyn. 18 487 [4] He X S, Song M, Deng F Y 2011 Acta Phys. Sin. 60 044501 (in Chinese) [和兴锁, 宋明, 邓峰岩 2011 物理学报 60 044501] [5] He X S, Deng F Y, Wang R 2010 Acta Phys. Sin. 59 1428 (in Chinese) [和兴锁, 邓峰岩, 王睿 2010 物理学报 59 1428] [6] Chen S J, Zhang D G, Hong J Z 2013 Chin. J. Theor. Appl. Mech. 45 251 (in Chinese) [陈思佳, 章定国, 洪嘉振 2013 力学学报 45 251] [7] Shabana A A 1997 Multibody Syst. Dyn. 1 189 [8] Tian Q, Zhang Y Q, Chen L P, Tan G 2010 Adv. Mech. 40 189 (in Chinese) [田强, 张云清, 陈立平, 覃刚 2010 力学进展 40 189] [9] Omar M A, Shabana A A 2001 J. Sound Vib. 243 565 [10] Hussein B A, Sugiyama H, Shabana A A 2007 J. Comput. Nonlinear Dyn. 2 146 [11] Dmitrochenko O N, Hussein B A, Shabana A A 2009 J. Comput. Nonlinear Dyn. 4 21002 [12] García-Vallejo D, Mikkola A M, Escalona J L 2007 Nonlinear Dyn. 50 249 [13] Tian Q, Zhang Y Q, Chen L P, Yang J Z 2010 Nonlinear Dyn. 60 489 [14] Gerstmayr J, Matikainen M K, Mikkola A M 2008 Multibody Syst. Dyn. 20 359 [15] Nachbagauer K, Pechstein A S, Irschik H, Gerstmayr J 2011 Multibody Syst. Dyn. 26 245 [16] Nachbagauer K, Gruber P, Gerstmayr J 2013 J. Comput. Nonlinear Dyn. 8 021004 [17] Gerstmayr J, Shabana A A 2006 Nonlinear Dyn. 45 109 [18] Dufva K E, Sopanen J T, Mikkola A M 2005 J. Sound Vib. 280 719 [19] Mikkola A M, Dmitrochenko O, Matikainen M 2009 J. Comput. Nonlinear Dyn. 4 011004 [20] Vesa-Ville A, Hurskainen T, Matikainen M K, Wang J, Mikkola A M 2016 J. Comput. Nonlinear Dyn. 12 041007 [21] Zhang X S, Zhang D G, Chen S J, Hong J Z 2016 Acta Phys. Sin. 64 094501 (in Chinese) [章孝顺, 章定国, 陈思佳, 洪嘉振 2016 物理学报 64 094501] [22] Goetz A 1970 Introduction to Differential Geometry (Reading, Massachussetts: Addison Wesley Pub. Co) pp56-58 [23] Timoshenko S 1940 Strength of Materials (Part I Elementary Theory and Problems Second Edition) (New York: D.Van Nostrand Co) pp147-148
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•  Citation:
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• Abstract views:  575
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##### Publishing process
• Received Date:  31 December 2016
• Accepted Date:  06 April 2017
• Published Online:  05 June 2017

## Analysis of planar shear deformable beam using rotation field curvature formulation

###### Corresponding author: Luo Jian-Jun, jjluo@nwpu.edu.cn;
• 1. National Key Laboratory of Aerospace Flight Dynamics, School of Astronautics, Northwestern Polytechnial University, Xi'an 710072, China;
• 2. School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China
Fund Project:  Project supported by the Major Program of National Natural Science Foundation of China (Grant Nos. 61690210, 61690211), and the National Natural Science Foundation of China (Grant Nos. 61603304, 11472213).

Abstract: In recent years, research on space debris removal technique has received wide attention in aerospace field. Many novel concepts on active flexible debris remover have been proposed, such as flexible flying net, tethered cable manipulator. In view with the high flexibility and large deformation of this kind of structure, the implementation of attitude control is challenging. An accurate dynamic model of highly flexible structure is important and needed. The beam element is the most common element adopted in flexible remover models. So, in this investigation, a rotation field-based curvature shear deformable beam using absolute nodal coordinate formulation (ANCF) (named RB-curvature ANCF beam) is proposed and its geometrically nonlinear characteristic under large deformation motion is studied. Curvature is first derived through planar rotation transformation matrix between the reference frame and current tangent frame of beam centerline, and written as an arc-length derivative of the orientation angle of the tangent vector. Using the geometrically exact beam theory, the strain energy is expressed as an uncoupled form, and the new curvature is adopted to formulate bending energy. Based on the ANCF, the dynamic equation of beam is established, where mass and external force matrices are constant. In order to validate the performance of proposed beam element, other two types of beams are introduced as the comparative models. One is the classical ANCF fully parameterized shear deformable beam derived by continuum mechanics theory, and the other is position field-based curvature ANCF shear deformable beam (named PBcurvature ANCF beam). The PB-curvature model is evaluated by differentiating unit tangent vector of beam centerline with respect to its arc length quoted from differential geometry theory. A series of static analysis, eigenfrequency tests and dynamic analysis are implemented to examine the performance of the proposed element. In static analysis, both small and non-small deformation cases show that the proposed RB-curvature ANCF beam achieves the faster speed, higher precision and good agreement with analytical solution in the case of cantilever beam subjected to a concentrated tip force, which is compared with other two beam models. The eigenfrequency analysis validates RB-curvature ANCF beam in a simply supported beam case that converges to its analytical solution. Meanwhile, the mode shapes of the proposed ANCF beam could be correctly corresponded to vibration state of element with respect to each different eigenfrequency. In the dynamics test, a flexible pendulum case is used and simulation results show that the proposed RB-curvature ANCF beam accords well with ANSYS BEAM3, classical ANCF shear beam and PB-curvature ANCF beam in vertical displacements of tip point and middle point. Since deformation modes are uncoupled in the cross section of proposed beam element, its shear strain is achieved with much better convergence in the case of lower elastic modulus, and shear locking is significantly alleviated, compared with classical ANCF beam. Therefore, RB-curvature ANCF shear deformable beam element proposed in this paper is able to describe accurately geometric nonlinearity in large deformation problem, and can be a potential candidate in the modeling of flexible/rigid-flexible mechanisms.

Reference (23)

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