Application of discrete shape function in absolute nodal coordinate formulation

Zhicheng Song; Jinbao Chen; Chuanzhi Chen; Zhicheng Song; Jinbao Chen; Chuanzhi Chen

doi:10.3934/mbe.2021234

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 4603-4627. doi: 10.3934/mbe.2021234

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Research article Special Issues

Application of discrete shape function in absolute nodal coordinate formulation

Key Laboratory of Exploration Mechanism of the Deep Space Planet Surface, Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received: 09 April 2021 Accepted: 25 May 2021 Published: 27 May 2021

To solve integrals in the absolute nodal coordinate method and address the difficulty in applying it to an arbitrary-section beam, this paper focuses on two methods involving single integrals:the invariant matrix method and the Gerstmayr method, with cross-section characteristics by applying the interpolation of a discrete function. Such single integrals demonstrate that the nodal coordinate method can be applied to an arbitrary-section beam. The Euler–Bernoulli beam used in engineering structures is characterised by a symmetrical cross-section, small section size, zero odd integrals and negligible high-order even integrals, which simplifies the single integrals of the two methods. Finally, the Gaussian integration is adopted to improve the solving efficiency of elastic force and force Jacobian.
- absolute nodal coordinate formulation (ANCF),
- tensor of stress,
- invariant matrices,
- discrete shape function
Citation: Zhicheng Song, Jinbao Chen, Chuanzhi Chen. Application of discrete shape function in absolute nodal coordinate formulation[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4603-4627. doi: 10.3934/mbe.2021234

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Abstract

To solve integrals in the absolute nodal coordinate method and address the difficulty in applying it to an arbitrary-section beam, this paper focuses on two methods involving single integrals:the invariant matrix method and the Gerstmayr method, with cross-section characteristics by applying the interpolation of a discrete function. Such single integrals demonstrate that the nodal coordinate method can be applied to an arbitrary-section beam. The Euler–Bernoulli beam used in engineering structures is characterised by a symmetrical cross-section, small section size, zero odd integrals and negligible high-order even integrals, which simplifies the single integrals of the two methods. Finally, the Gaussian integration is adopted to improve the solving efficiency of elastic force and force Jacobian.

References

[1]	B. Y. Duan, The state-of-the-art and development trend of large space-borne deployable antenna, Electro-Mech. Eng., 33 (2017), 1-14.
[2]	J. Wu, S. Yan, L. Xie, G. Peng, Reliability apportionment approach for spacecraft solar array using fuzzy reasoning petri net and fuzzy comprehensive evaluation, Acta Astronaut., 76 (2012), 136-144. doi: 10.1016/j.actaastro.2012.02.023
[3]	X. M. Gao, D. P. Jin, T. Chen, Nonlinear analysis and experimental investigation of a rigid-flexible antenna system, Meccanica, 53 (2018), 33-48. doi: 10.1007/s11012-017-0708-z
[4]	Y. H. Zheng, P. Ruan, S. Cao, Deployable structure design and analysis for space membrane diffractive telescope, Hongwai Yu Jiguang Gongcheng/Infrared Laser Eng., 45 (2016), 133-137.
[5]	P. Li, Q. W. Ma, Y. P. Song, C. Liu, Q. Tian, S. P. Ma, et al., Deployment dynamics simulation and ground test of a large space hoop truss antenna reflector, Sci. Sin. Phys., Mech. Astron., 47 (2017), 104602.
[6]	X. L. Du, J. L. Du, H. Bao, G. H. Sun, Deployment analysis of deployable antennas considering cable net and truss flexibility, Aerosp. Sci. Technol., 82-83 (2018), 557-565.
[7]	K. N. Urata, J. Sumantyo, N. Imura, K. Ito, S. Gao, Development of a circularly polarized L-band SAR deployable mesh reflector antenna for microsatellite earth observation, in 2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, IEEE, (2017).
[8]	A. A. Shabana, Flexible multibody dynamics: Review of past and recent developments, Multibody Syst. Dyn., 1 (1997), 189-222. doi: 10.1023/A:1009773505418
[9]	D. A. Turcic, A. Midha, J. R. Bosnik, Dynamic analysis of elastic mechanism systems. Part Ⅱ: Experimental results, J. Dyn. Syst. Meas. Control, 106 (1984), 255.
[10]	M. Manish, Kineto-elastodynamic analysis of Watt's mechanism using ANSYS, Procedia Technol., 23 (2016), 51-59. doi: 10.1016/j.protcy.2016.03.073
[11]	P. W. Likins, Finite element appendage equations for hybrid coordinate dynamic analysis, Int. J. Solids Struct., 8 (1972), 709-731. doi: 10.1016/0020-7683(72)90038-8
[12]	J. L. Escalona, H. A. Hussien, A. A. Shabana, Application of the absolute nodal co-ordinate formulation to multibody system dynamics, J. Sound Vib., 214 (1998), 833-851. doi: 10.1006/jsvi.1998.1563
[13]	A. A. Shabana, An absolute nodal coordinates formulation for the large rotation and deformation analysis of flexible bodies, No. MBS96-1-UIC, University of Illinois at Chicago, 1996.
[14]	A. A. Shabana, C. J. Desai, E. Grossi, M. Patel, Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements, Acta Mech., 231 (2020), 1365-1376. doi: 10.1007/s00707-019-02558-w
[15]	Y. Zhang, C. Wei, Y. Zhao, C. L. Tan, Y. J. Liu, Adaptive ANCF method and its application in planar flexible cables, Acta Mech. Sin., 34 (2018), 199-213. doi: 10.1007/s10409-017-0721-4
[16]	C. H. Zhao, Dynamic model of flexible beam with arbitrary rigid motion based on floating frame of reference formulation, J. Shanghai Univ. Eng., 27 (2013), 39-42.
[17]	A. A. Shabana, Geometrically accurate infinitesimal-rotation spatial finite elements, Proc. Inst. Mech. Eng., 233 (2019), 182-187. doi: 10.1177/0954405417712550
[18]	J. L. Sun, Q. Tian, H. Y. Hu, Advances in dynamic modeling and optimization of flexible multibody systems, Chin. J. Theor. Appl. Mech., 51 (2019), 1565-1586.
[19]	M. Dibold, J. Gerstmayr, H. Irschik, A detailed comparison of the absolute nodal coordinate and the floating frame of reference formulation in deformable multibody systems, J. Comput. Nonlinear Dyn., 4 (2009).
[20]	R. Y. Yakoub, A New Three-Dimensional Absolute Coordinate Based Beam Element with Application to Wheel/Rail Interaction, University of Illinois at Chicago, 2001.
[21]	P. Li, C. Liu, Q. Tian, H. Y. Hu, Dynamics of a deployable mesh reflector of satellite antenna: Form-finding and modal analysis, J. Comput. Nonlinear Dyn., 11 (2016), 041017.
[22]	Q. Tian, J. Zhao, C. L. Liu, C. Y. Zhou, Dynamics of space deployable structures, in ASME 2015 international design engineering technical conferences and computers and information in engineering conference, 2015.
[23]	P. Lan, Q. L. Tian, Z. Q. Yu, A new absolute nodal coordinate formulation beam element with multilayer circular cross section, Acta Mech. Sin., 36 (2020), 82-96. doi: 10.1007/s10409-019-00897-4
[24]	Z. Wang, Q. Tian, H. Y. Hu, Dynamics of flexible multibody systems with hybrid uncertain parameters, Mech. Mach. Theory, 121 (2018), 128-147. doi: 10.1016/j.mechmachtheory.2017.09.024
[25]	M. Campanelli, M. Berzeri, A. A. Shabana, Performance of the incremental and non-incremental finite element formulations in flexible multibody problems, J. Mech. Design, 122 (2000), 498-507. doi: 10.1115/1.1289636
[26]	M. A. Omar, A. A. Shabana, A two-dimensional shear deformation beam for large rotation and deformation, J. Sound Vib., 243 (2001), 565-576. doi: 10.1006/jsvi.2000.3416
[27]	D. García-Vallejo, A. M. Mikkola, J. L. Escalona, A new locking-free shear deformable finite element based on absolute nodal coordinates, Nonlinear Dyn., 50 (2007), 249-264. doi: 10.1007/s11071-006-9155-4
[28]	A. A. Shabana, R. Y. Yakoub, Three-dimensional absolute nodal coordinate formulation for beam elements: theory, J. Mech. Design, 123 (2001), 614-621. doi: 10.1115/1.1410099
[29]	D. García-Vallejo, J. Mayo, J. L. Escalona, J. Domínguez, Efficient evaluation of the elastic forces and the Jacobian in the absolute nodal coordinate formulation, Nonlinear Dyn., 35 (2004), 313-329. doi: 10.1023/B:NODY.0000027747.41604.20
[30]	J. Gerstmayr, A. A. Shabana, A. Strain, Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formulation, Multibody Dyn., 2005.
[31]	C. Liu, Q. Tian, H. Y. Hu, Efficient computational method for dynamics of flexible multibody systems based on absolute nodal coordinate, Lixue Xuebao/Chin. J. Theor. Appl. Mech., 42 (2010), 1197-1205.
[32]	K. Otsuka, K. Makihara, ANCF-ICE beam element for modeling highly flexible and deployable aerospace structures, in AIAA Scitech 2019 Forum, 2019.
[33]	G. Orzechowski, Analysis of beam elements of circular cross section using the absolute nodal coordinate formulation, Arch. Mech. Eng., 59 (2012), 283-296. doi: 10.2478/v10180-012-0014-1
[34]	J. Gerstmayr, A. A. Shabana, Analysis of thin beams and cables using the absolute nodal co-ordinate formulation, Nonlinear Dyn., 45 (2006), 109-130. doi: 10.1007/s11071-006-1856-1

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