Computational Mechanics

, Volume 64, Issue 6, pp 1517–1535 | Cite as

Flexible actuator finite element applied to spatial mechanisms by a finite deformation dynamic formulation

  • Tiago Morkis SiqueiraEmail author
  • Humberto Breves Coda
Original Paper


A flexible actuator finite element is developed and applied for the modelling of spatial mechanisms present in several industrial applications. A total Lagrangian framework is employed for the development of the finite deformation dynamic equilibrium using solid-like shell and 3D frame finite elements. Exploiting the total Lagrangian aspect of the formulation, the actuator motion is imposed by controlling the element reference configuration. It has the advantage of retaining the actuated bar flexibility, an important factor when simulating flexible mechanisms, and not requiring special treatments as constraint enforcement impositions. As the employed elements use alternative nodal parameters such as positions and generalized vectors to describe their kinematics, a treatment on the introduction of rotational connections—spherical, revolute and pinned joints—largely present in actuated mechanisms, is developed. The nonlinear equations of motion are solved by the Newton–Raphson method. Examples are presented to evaluate the proposed flexible actuator finite element regarding its dynamical behaviour in mechanisms where its use is of importance.


Flexible actuator Nonlinear dynamics Solid-like finite element Rotational joint Generalized vectors 



The authors would like to thank the São Paulo Research Foundation (FAPESP-2016/00622-0 and FAPESP-2018/18321-1) for the research grant. This study was also financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.


  1. 1.
    Coda HB, Paccola RR (2011) A FEM procedure based on positions and unconstrained vectors applied to non-linear dynamic of 3D frames. Finite Elem Anal Des 47:319–333CrossRefGoogle Scholar
  2. 2.
    Coda HB, Paccola RR, Sampaio MDSM (2013) Positional description applied to the solution of geometrically non-linear plates and shells. Finite Elem Anal Des 67:66–75MathSciNetCrossRefGoogle Scholar
  3. 3.
    Thomson MW (1999) The AstroMesh deployable reflector. In: IEEE antennas and propagation society international symposium: wireless technologies and information networks, APS 1999—Held in conjunction with USNC/URSI National Radio Science Meeting. IEEE, pp 1516–1519Google Scholar
  4. 4.
    Takano T, Miura K, Natori M et al (2004) Deployable antenna with 10-m maximum diameter for space use. IEEE Trans Antennas Propag 52:2–11. CrossRefGoogle Scholar
  5. 5.
    Meguro A, Shintate K, Usui M, Tsujihata A (2009) In-orbit deployment characteristics of large deployable antenna reflector onboard Engineering Test Satellite VIII. Acta Astronaut 65:1306–1316. CrossRefGoogle Scholar
  6. 6.
    Mitsugi J, Ando K, Senbokuya Y, Meguro A (2000) Deployment analysis of large space antenna using flexible multibody dynamics simulation. Acta Astronaut 47:19–26. CrossRefGoogle Scholar
  7. 7.
    Madeira RH, Coda HB (2016) Kelvin viscoelasticity and lagrange multipliers applied to the simulation of nonlinear structural vibration control. Lat Am J Solids Struct 13:964–991. CrossRefGoogle Scholar
  8. 8.
    Cardona A, Géradin M (1989) Time integration of equations of motion in mechanism analysis. Comput Struct 33:801–820. CrossRefzbMATHGoogle Scholar
  9. 9.
    Jelenic G, Crisfield MA (2001) Dynamic analysis of 3D beams with joints in presence of large rotations. Comput Methods Appl Mech Eng 190:4195–4230. CrossRefzbMATHGoogle Scholar
  10. 10.
    Gebhardt CG, Hofmeister B, Hente C, Rolfes R (2019) Nonlinear dynamics of slender structures: a new object-oriented framework. Comput Mech 63:219–252. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gay Neto A (2017) Simulation of mechanisms modeled by geometrically-exact beams using Rodrigues rotation parameters. Comput Mech 59:459–481. MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ibrahimbegović A, Mamouri S (2000) On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model. Comput Methods Appl Mech Eng 188:805–831. CrossRefzbMATHGoogle Scholar
  13. 13.
    Crisfield MA, Moita GF (1996) A unified co-rotational framework for solids, shells and beams. Int J Solids Struct 33:2969–2992. CrossRefzbMATHGoogle Scholar
  14. 14.
    Teh LH, Clarke MJ (1999) Plastic-zone analysis of 3D steel frames using beam elements. J Struct Eng 125:1328–1337. CrossRefGoogle Scholar
  15. 15.
    Cardona A (2000) Superelements modelling in flexible multibody dynamics. Multibody Syst Dyn 4:245–266MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wasfy TM, Noor AK (1996) Modeling and sensitivity analysis of multibody systems using new solid, shell and beam elements. Comput Methods Appl Mech Eng 138:187–211. CrossRefzbMATHGoogle Scholar
  17. 17.
    Coda HB, Paccola RR (2009) Unconstrained finite element for geometrical nonlinear dynamics of shells. Math Probl Eng 2009:1–32. MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Simo JC, Vu-Quoc L (1986) On the dynamics of flexible beams under large overall motions—the plane case. I. J Appl Mech 53:849–854CrossRefGoogle Scholar
  19. 19.
    Simo JC, Vu-Quoc L (1986) On the dynamics of flexible beams under large overall motions—the plane case. II. J Appl Mech 53:855–863CrossRefGoogle Scholar
  20. 20.
    Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, ChichesterzbMATHGoogle Scholar
  21. 21.
    Lanczos C (1970) The variational principles of mechanics. Dover Publications, New YorkzbMATHGoogle Scholar
  22. 22.
    Siqueira TM, Coda HB (2017) Total Lagrangian FEM formulation for nonlinear dynamics of sliding connections in viscoelastic plane structures and mechanisms. Finite Elem Anal Des 129:63–77. MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ogden RW (1984) Non-linear elastic deformations. Ellis Horwood, ChichesterzbMATHGoogle Scholar
  24. 24.
    Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New YorkCrossRefGoogle Scholar
  25. 25.
    Bonet J, Wood RD, Mahaney J, Heywood P (2000) Finite element analysis of air supported membrane structures. Comput Methods Appl Mech Eng 190:579–595. CrossRefzbMATHGoogle Scholar
  26. 26.
    Coda HB, Paccola RR (2007) An alternative positional FEM formulation for geometrically non-linear analysis of shells: curved triangular isoparametric elements. Comput Mech 40:185–200CrossRefGoogle Scholar
  27. 27.
    Bischoff M, Ramm E (2000) On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation. Int J Solids Struct 37:6933–6960CrossRefGoogle Scholar
  28. 28.
    Coda HB (2009) A solid-like FEM for geometrically non-linear 3D frames. Comput Methods Appl Mech Eng 198:3712–3722. MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Coda HB, Paccola RR (2010) Improved finite element for 3D laminate frame analysis including warping for any cross-section. Appl Math Model 34:1107–1137MathSciNetCrossRefGoogle Scholar
  30. 30.
    Géradin M, Cardona A (2001) Flexible multibody dynamics: a finite element approach. Wiley, ChichesterGoogle Scholar
  31. 31.
    Ota NSN, Wilson L, Gay Neto A et al (2016) Nonlinear dynamic analysis of creased shells. Finite Elem Anal Des 121:64–74. MathSciNetCrossRefGoogle Scholar
  32. 32.
    Simo JC (1993) On a stress resultant geometrically exact shell model. Part VII: shell intersections with 5/6-DOF finite element formulations. Comput Methods Appl Mech Eng 108:319–339. CrossRefzbMATHGoogle Scholar
  33. 33.
    Betsch P, Sänger N (2009) On the use of geometrically exact shells in a conserving framework for flexible multibody dynamics. Comput Methods Appl Mech Eng 198:1609–1630. MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Warburton GB (1976) The dynamical behaviour of structures, 2nd edn. Pergamon Press, New YorkGoogle Scholar
  35. 35.
    Sokolov A, Xirouchakis P (2007) Dynamics analysis of a 3-DOF parallel manipulator with R-P-S joint structure. Mech Mach Theory 42:541–557. MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Dasgupta B, Mruthyunjaya TS (2000) Stewart platform manipulator: a review. Mech Mach Theory 35:15–40. MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Camarillo DB, Milne CF, Carlson CR et al (2008) Mechanics modeling of tendon-driven continuum manipulators. IEEE Trans Robot 24:1262–1273. CrossRefGoogle Scholar
  38. 38.
    NASA (2015) Reference guide to the International Space Station. National Aeronautics and Space Administration, Johnson Space Center, HoustonGoogle Scholar
  39. 39.
    NASA Integrated Truss Structure. In: Natl. Aeronaut. Sp. Adm. Accessed 27 Sept 2018
  40. 40.
    Wikipedia Contributors (2018) Integrated truss structure. In: Wikipedia, Free Encycl. Accessed 27 Sept 2018
  41. 41.
    NASA (2006) Space ISS Spacewalk 2. AP Archive. In: Natl. Aeronaut. Sp. Adm. TV. Accessed 27 Sept 2018
  42. 42.
    Williams S (2007) ISS STS-120 Radiator Deployment. Youtube. Accessed 27 Sept 2018
  43. 43.
    Siqueira TM, Coda HB (2016) Development of sliding connections for structural analysis by a total Lagrangian FEM formulation. Lat Am J Solids Struct. CrossRefGoogle Scholar
  44. 44.
    Laursen TA, Puso MA, Sanders J (2012) Mortar contact formulations for deformable-deformable contact: past contributions and new extensions for enriched and embedded interface formulations. Comput Methods Appl Mech Eng 205–208:3–15. MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Gay Neto A, Wriggers P (2019) Computing pointwise contact between bodies: a class of formulations based on master–master approach. Comput Mech. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Structural Engineering Department, São Carlos School of EngineeringUniversity of São PauloSão CarlosBrazil

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