Abstract
Compared with the conventional rigid origami, the flexible origami has larger deformation and more complicated mechanical property and nonlinear problems due to self-contact and friction. In this paper, the nonlinear dynamic formulation for flexible origami-based deployable structures considering self-contact and friction is investigated. Firstly, a symmetric rigid origami model is presented based on the forward recursive formulation without the inclusion of contact, and then, a discretized dynamic model for flexible origami structures is established by using thin plate element of absolute nodal coordinate formulation. To consider the normal contact, the penalty method is adopted to enforce the nonpenetration condition. In order to improve the precision and applicability, a modified mixed contact method considering the friction effect is developed by integrating the advantages of node-to-surface, edge-to-surface and surface-to-surface contact elements. This proposed method can effectively avoid the mutual penetration of different corner nodes, element edges and contact element surfaces. Moreover, the tangential friction model considering the stick–slip transition and large sliding is established by the regularized Coulomb friction law. A series of numerical examples validate the effectiveness of the proposed mixed contact method considering the friction and show the advantage of the flexible model compared with the rigid origami model. Furthermore, the nonlinear performance of the flexible origami-based deployable structures due to the contact and friction is revealed.
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Acknowledgements
This research was supported by General Program (Nos. 11772186, 11772188) of the National Natural Science Foundation of China and the Key Program (No. 11932001) of the National Natural Science Foundation of China, for which the authors are grateful. This research was also supported by the Key Laboratory of Hydrodynamics (Ministry of Education).
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Appendices
Appendix 1
The shape function matrix \({\mathbf{S}}_{A}\) for triangular thin plate element is given by
The shape functions are as follows
where \(P_{i} = \Delta_{i}^{2} \Delta_{j} + {{\Delta_{i} \Delta_{j} \Delta_{k} \left[ {3\left( {1 - \mu_{k} } \right)\Delta_{i} + \left( {1 + 3\mu_{k} } \right)\left( {\Delta_{k} - \Delta_{j} } \right)} \right]} \mathord{\left/ {\vphantom {{\Delta_{i} \Delta_{j} \Delta_{k} \left[ {3\left( {1 - \mu_{k} } \right)\Delta_{i} + \left( {1 + 3\mu_{k} } \right)\left( {\Delta_{k} - \Delta_{j} } \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}\), \(\mu_{i} = {{\left( {l_{k}^{2} - l_{j}^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {l_{k}^{2} - l_{j}^{2} } \right)} {l_{i}^{2} }}} \right. \kern-\nulldelimiterspace} {l_{i}^{2} }}\), \(l_{i}^{2} = c_{i1}^{2} + c_{i2}^{2}\). In addition, \(\Delta_{i} {\kern 1pt} {\kern 1pt} (i = 1,2,3)\) and \(c_{ij} {\kern 1pt} (i,j = 1,2,3)\) are triangular area coordinates and area coefficients, respectively.
Appendix 2
The shape function matrix \({\mathbf{S}}_{B}\) for rectangular thin plate element is written as
The shape functions are as follows
where \(\xi = {x \mathord{\left/ {\vphantom {x {a_{e} }}} \right. \kern-\nulldelimiterspace} {a_{e} }},\;\eta = {y \mathord{\left/ {\vphantom {y {b_{e} }}} \right. \kern-\nulldelimiterspace} {b_{e} }}\;\;(0 \le \xi ,\eta \le 1)\). \(a_{e}\) and \(b_{e}\) are the lengths of the rectangular element in the direction of x and y, respectively.
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Yuan, T., Tang, L., Liu, Z. et al. Nonlinear dynamic formulation for flexible origami-based deployable structures considering self-contact and friction. Nonlinear Dyn 106, 1789–1822 (2021). https://doi.org/10.1007/s11071-021-06860-y
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DOI: https://doi.org/10.1007/s11071-021-06860-y