Skip to main content

Advertisement

SpringerLink
Log in

Dynamic modeling, simulation and design of smart membrane systems driven by soft actuators of multilayer dielectric elastomers

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A multilayer membrane element of absolute nodal coordinate formulation is proposed for dynamic modeling of multilayer dielectric elastomer actuators (DEAs). The coupled dynamics of rigid-body motion and large deformation interacting with electric fields is considered. For the kinematic description, a modified version of the Kirchhoff–Love assumptions is proposed taking the thickness shrinking of the membrane into account. Two material models are introduced based on the Helmholtz free energy in thermodynamics. One is the material model of ideal deformable dielectrics. The other is the large-deformation St. Venant–Kirchhoff model which is a geometrically nonlinear but material linear model. Afterward, the generalized internal forces and their Jacobians are given. The dynamic equations of the systems are solved by the generalized-α algorithm. Finally, three case studies are presented. First, the proposed modeling method is validated by comparing the analytical and simulated solutions. Second, the statics and dynamics of a bending DEA are investigated and the results are compared with the experimental results. Third, an autonomous membrane machine driven by soft DE joints is proposed for space applications and this case is to demonstrate that dynamic modeling and simulation can aid the design of soft machines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Shian, S., Bertoldi, K., Clarke, D.R.: Dielectric elastomer based “grippers” for soft robotics. Adv. Mater. 27(43), 6814–6819 (2015)

    Google Scholar 

  2. Heng, K.-R., Ahmed, A.S., Shrestha, M., Lau, G.-K.: Strong dielectric-elastomer grippers with tension arch flexures. In: Electroactive Polymer Actuators and Devices (EAPAD) 2017, p. 101631Z. International Society for Optics and Photonics

  3. Chuc, N.H., Vuong, N.H.L., Kim, D.S., Moon, H.P., Koo, J.C., Lee, Y.K., Nam, J.-D., Choi, H.R.: Fabrication and control of rectilinear artificial muscle actuator. IEEE/ASME T. Mech. 16(1), 167–176 (2010)

    Google Scholar 

  4. O’Brien, B.M., Calius, E.P., Inamura, T., Xie, S.Q., Anderson, I.A.: Dielectric elastomer switches for smart artificial muscles. Appl. Phys. A-Mater. 100(2), 385–389 (2010)

    Google Scholar 

  5. Xu, C., Stiubianu, G.T., Gorodetsky, A.A.: Adaptive infrared-reflecting systems inspired by cephalopods. Science 359(6383), 1495–1500 (2018)

    Google Scholar 

  6. Araromi, O.A., Gavrilovich, I., Shintake, J., Rosset, S., Richard, M., Gass, V., Shea, H.R.: Rollable multisegment dielectric elastomer minimum energy structures for a deployable microsatellite gripper. IEEE/ASME T. Mech. 20(1), 438–446 (2014)

    Google Scholar 

  7. Richard, M., Kronig, L.G., Belloni, F., Gass, V., Araromi, O.A., Shea, H., Paccolat, C., Thiran, J.-P.: Uncooperative rendezvous and docking for MicroSats. In: 6th International Conference on Recent Advances in Space Technologies, RAST 2013, vol. CONF. IEEE

  8. Chen, Y., Zhao, H., Mao, J., Chirarattananon, P., Helbling, E.F., Hyun, N.-S.P., Clarke, D.R., Wood, R.: Controlled flight of a microrobot powered by soft artificial muscles. Nature 575(7782), 324–329 (2019)

    Google Scholar 

  9. Pelrine, R., Kornbluh, R., Pei, Q., Joseph, J.: High-speed electrically actuated elastomers with strain greater than 100%. Science 287(5454), 836–839 (2000)

    Google Scholar 

  10. Zhao, X., Suo, Z.: Method to analyze electromechanical stability of dielectric elastomers. Appl. Phys. Lett. 91(6), 061921 (2007)

    Google Scholar 

  11. Suo, Z., Zhao, X., Greene, W.H.: A nonlinear field theory of deformable dielectrics. J. Mech. Phys. Solids 56(2), 467–486 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Zhao, X., Suo, Z.: Method to analyze programmable deformation of dielectric elastomer layers. Appl. Phys. Lett. 93(25), 251902 (2008)

    Google Scholar 

  13. Suo, Z.: Theory of dielectric elastomers. Acta Mech. Solida Sin. 23(6), 549–578 (2010)

    Google Scholar 

  14. Keplinger, C., Li, T., Baumgartner, R., Suo, Z., Bauer, S.: Harnessing snap-through instability in soft dielectrics to achieve giant voltage-triggered deformation. Soft Matter 8(2), 285–288 (2012)

    Google Scholar 

  15. Li, T., Keplinger, C., Baumgartner, R., Bauer, S., Yang, W., Suo, Z.: Giant voltage-induced deformation in dielectric elastomers near the verge of snap-through instability. J. Mech. Phys. Solids 61(2), 611–628 (2013)

    Google Scholar 

  16. Lu, T., Cheng, S., Li, T., Wang, T., Suo, Z.: Electromechanical catastrophe. Int. J. Appl. Mech. 8(07), 1640005 (2016)

    Google Scholar 

  17. Qu, S., Suo, Z.: A finite element method for dielectric elastomer transducers. Acta Mech. Solida Sin. 25(5), 459–466 (2012)

    Google Scholar 

  18. Park, H.S., Suo, Z., Zhou, J., Klein, P.A.: A dynamic finite element method for inhomogeneous deformation and electromechanical instability of dielectric elastomer transducers. Int. J. Solids Struct. 49(15–16), 2187–2194 (2012)

    Google Scholar 

  19. Henann, D.L., Chester, S.A., Bertoldi, K.: Modeling of dielectric elastomers: design of actuators and energy harvesting devices. J. Mech. Phys. Solids 61(10), 2047–2066 (2013)

    MathSciNet  Google Scholar 

  20. Kofod, G., Paajanen, M., Bauer, S.: Self-organized minimum-energy structures for dielectric elastomer actuators. Appl. Phys. A-Mater 85(2), 141–143 (2006)

    Google Scholar 

  21. Kofod, G., Wirges, W., Paajanen, M., Bauer, S.: Energy minimization for self-organized structure formation and actuation. Appl. Phys. Lett. 90(8), 081916 (2007)

    Google Scholar 

  22. O’Brien, B., McKay, T., Calius, E., Xie, S., Anderson, I.: Finite element modelling of dielectric elastomer minimum energy structures. Appl. Phys. A-Mater. 94(3), 507–514 (2009)

    Google Scholar 

  23. Rosset, S., Araromi, O.A., Shintake, J., Shea, H.R.: Model and design of dielectric elastomer minimum energy structures. Smart Mater. Struct. 23(8), 085021 (2014)

    Google Scholar 

  24. Zhou, J., Hong, W., Zhao, X., Zhang, Z., Suo, Z.: Propagation of instability in dielectric elastomers. Int. J. Solids Struct. 45(13), 3739–3750 (2008)

    MATH  Google Scholar 

  25. Zhu, J., Cai, S., Suo, Z.: Nonlinear oscillation of a dielectric elastomer balloon. Polym. Int. 59(3), 378–383 (2010)

    Google Scholar 

  26. Chakravarty, U.K.: On the resonance frequencies of a membrane of a dielectric elastomer. Mech. Res. Commun. 55, 72–76 (2014)

    Google Scholar 

  27. Zhu, J., Cai, S., Suo, Z.: Resonant behavior of a membrane of a dielectric elastomer. Int. J. Solids Struct. 47(24), 3254–3262 (2010)

    MATH  Google Scholar 

  28. Garnell, E., Rouby, C., Doaré, O.: Dynamics and sound radiation of a dielectric elastomer membrane. J. Sound Vib. 459, 114836 (2019)

    Google Scholar 

  29. Brochu, P., Pei, Q.: Advances in dielectric elastomers for actuators and artificial muscles. Macromol. Rapid Comm. 31(1), 10–36 (2010)

    Google Scholar 

  30. Gu, G.-Y., Zhu, J., Zhu, L.-M., Zhu, X.: A survey on dielectric elastomer actuators for soft robots. Bioinspir. Biomim. 12(1), 011003 (2017)

    Google Scholar 

  31. Lu, T., Ma, C., Wang, T.: Mechanics of dielectric elastomer structures: a review. Extreme Mech. Lett. 38, 100752 (2020)

    Google Scholar 

  32. Duduta, M., Wood, R.J., Clarke, D.R.: Multilayer dielectric elastomers for fast, programmable actuation without prestretch. Adv. Mater. 28(36), 8058–8063 (2016)

    Google Scholar 

  33. Shabana, A.A.: An absolute nodal coordinate formulation for the large rotation and large deformation analysis of flexible bodies. Report MBS96-1-UIC, Dept. of Mechanical Engineering, University of Illinois at Chicago (1996)

  34. Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Mikkola, A.M., Shabana, A.A.: A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst. Dyn. 9(3), 283–309 (2003)

    MathSciNet  MATH  Google Scholar 

  36. Dufva, K., Shabana, A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. P I Mech. Eng. K-J. Multi-body Dyn. 219(4), 345–355 (2005)

    Google Scholar 

  37. Liu, C., Tian, Q., Hu, H.Y.: New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation. Nonlinear Dyn. 70(3), 1903–1918 (2012)

    MathSciNet  Google Scholar 

  38. Liu, C., Tian, Q., Yan, D., Hu, H.Y.: Dynamic analysis of membrane systems undergoing overall motions, large deformations and wrinkles via thin shell elements of ANCF. Comput. Methods Appl. Mech. Eng. 258, 81–95 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Luo, K., Liu, C., Tian, Q., Hu, H.Y.: Nonlinear static and dynamic analysis of hyper-elastic thin shells via the absolute nodal coordinate formulation. Nonlinear Dyn. 85(2), 949–971 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Pappalardo, C.M., Zhang, Z., Shabana, A.A.: Use of independent volume parameters in the development of new large displacement ANCF triangular plate/shell elements. Nonlinear Dyn. 91(4), 2171–2202 (2018)

    Google Scholar 

  41. Xu, Q., Liu, J., Qu, L.: Dynamic modeling for silicone beams using higher-order ANCF beam elements and experiment investigation. Multibody Syst. Dyn. 46(4), 307–328 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Sun, J., Tian, Q., Hu, H.Y., Pedersen, N.L.: Topology optimization of a flexible multibody system with variable-length bodies described by ALE–ANCF. Nonlinear Dyn. 93(2), 413–441 (2018)

    MATH  Google Scholar 

  43. Wang, T., Tinsley, B., Patel, M.D., Shabana, A.A.: Nonlinear dynamic analysis of parabolic leaf springs using ANCF geometry and data acquisition. Nonlinear Dyn. 93(4), 2487–2515 (2018)

    Google Scholar 

  44. Ghorbani, H., Tarvirdizadeh, B., Alipour, K., Hadi, A.: Near-time-optimal motion control for flexible-link systems using absolute nodal coordinates formulation. Mech. Mach. Theory 140, 686–710 (2019)

    Google Scholar 

  45. Shabana, A.A., Zhang, D.: ANCF curvature continuity: application to soft and fluid materials. Nonlinear Dyn. 100, 1497–1517 (2020)

    Google Scholar 

  46. Shabana, A.A., Eldeeb, A.E.: Relative orientation constraints in the nonlinear large displacement analysis: application to soft materials. Nonlinear Dyn. (2020). https://doi.org/10.1007/s11071-020-05839-5

    Article  Google Scholar 

  47. Sheng, F., Zhong, Z., Wang, K.-H.: Theory and model implementation for analyzing line structures subject to dynamic motions of large deformation and elongation using the absolute nodal coordinate formulation (ANCF) approach. Nonlinear Dyn. 101(1), 333–359 (2020)

    Google Scholar 

  48. Arnold, M., Brüls, O.: Convergence of the generalized-α scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007)

    MathSciNet  MATH  Google Scholar 

  49. Subbaraj, K., Dokainish, M.: A survey of direct time-integration methods in computational structural dynamics—II. Implicit Methods. Comput. Struct. 32(6), 1387–1401 (1989)

    MathSciNet  MATH  Google Scholar 

  50. Webster III, R.J., Jones, B.A.: Design and kinematic modeling of constant curvature continuum robots: a review. Int. J. Robot. Res. 29(13), 1661–1683 (2010)

    Google Scholar 

  51. Marchese, A.D., Rus, D.: Design, kinematics, and control of a soft spatial fluidic elastomer manipulator. Int. J. Robot. Res. 35(7), 840–869 (2016)

    Google Scholar 

Download references

Funding

This research was supported by National Natural Science Foundation of China under Grants 11902028 (Recipient: Dr. Kai Luo), 11722216 (Recipient: Prof. Qiang Tian) and 11832005 (Recipient: Prof. Haiyan Hu). This research was also supported by Research Fund Program for Young Scholars of Beijing Institute of Technology (Recipient: Dr. Kai Luo).

Author information

Authors and Affiliations

  1. Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Kai Luo, Qiang Tian & Haiyan Hu

Authors
  1. Kai Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haiyan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Dr. Kai Luo led the work and finished the deduction of the theory, code programming, computation, data processing, manuscript writing, etc. Prof. Qiang Tian and Prof. Haiyan Hu reviewed the manuscript and provided suggestions for revising the manuscript.

Corresponding author

Correspondence to Kai Luo.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts of interest to this work.

Availability of data and material

All the data related to the current work are available when required.

Code availability

The custom codes cannot be shared for this time as they are related to another ongoing study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix I

Appendix I

For the membrane element in this study, the matrix of shape functions is defined as [36, 37]

$$ {\mathbf{S}} = \left[ {\begin{array}{*{20}c} {S_{1} {\mathbf{I}}} & {S_{2} {\mathbf{I}}} & \ldots & {S_{12} {\mathbf{I}}} \\ \end{array} } \right] \in \Re^{3 \times 36} , $$
(I1)

where \( {\mathbf{I}} \in \Re^{3 \times 3} \) is an identity matrix and the shape functions are formulated as follows:

$$ \begin{aligned} & S_{1} = - (\xi - 1)(\eta - 1)(2\eta^{2} - \eta + 2\xi^{2} - \xi - 1), \, S_{2} = - \xi (\xi - 1)^{2} (\eta - 1), \\ & S_{3} = - \eta (\eta - 1)^{2} (\xi - 1), \, S_{4} = \xi (2\eta^{2} - \eta - 3\xi + 2\xi^{2} )(\eta - 1), \\ & S_{5} = - \xi^{2} (\xi - 1)(\eta - 1), \, S_{6} = \xi \eta (\eta - 1)^{2} , \\ & S_{7} = - \xi \eta (1 - 3\xi - 3\eta + 2\eta^{2} + 2\xi^{2} ), \, S_{8} = \xi^{2} \eta (\xi - 1), \\ & S_{9} = \xi \eta^{2} (\eta - 1), \, S_{10} = \eta (\xi - 1)(2\xi^{2} - \xi - 3\eta + 2\eta^{2} ), \\ & S_{11} = \xi \eta (\xi - 1)^{2} , \, S_{12} = - \eta^{2} (\xi - 1)(\eta - 1), \\ \end{aligned} $$
(I2)

where \( \xi \) and \( \eta \) are the dimensionless local coordinates along two directions of the mid-surface of the element.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, K., Tian, Q. & Hu, H. Dynamic modeling, simulation and design of smart membrane systems driven by soft actuators of multilayer dielectric elastomers. Nonlinear Dyn 102, 1463–1483 (2020). https://doi.org/10.1007/s11071-020-06001-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-06001-x

Keywords

Search

Navigation