Skip to main content
SpringerLink
Log in

Analysis of 2D flexible mechanisms using linear finite elements and incremental techniques

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this paper, a fully linear method for the kinetoelastodynamic analysis of mechanisms of engineering praxis, characterized by large displacements and rotations (rigid body motion) but small elastic displacements and strains, based on the standard Euler–Bernoulli finite elements is proposed and investigated. The method is a co-rotational approach and relies on the principle to decompose the motion of a mechanism in a series of successive time steps, so small that the linear finite element method can be applied within each step, adding a correction procedure in order to compensate errors resulting from not incorporating the exact (non-linear) beam theory. After presentation of the method, we apply it to test cases well known in the literature and discuss its characteristics.

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

Similar content being viewed by others

References

  1. Aarts RGKM (2002) SPACAR User Manual

  2. Alvarez J and Rojo J (2002). An improved class of generalized Runge–Kutta methods for stiff problems. Part I: the scalar case. Appl Math Comput 130: 537–560

    Article  MATH  MathSciNet  Google Scholar 

  3. Alvarez J and Rojo J (2004). An improved class of generalized Runge–Kutta methods for stiff problems. Part II: The separated system case. Appl Math Comput 159: 717–758

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosio JAC and Ravn P (1997). Elastodynamics of multibody systems using generalized inertial coordinates and structural damping. Mech Struct Mach 25: 201–219

    Article  Google Scholar 

  5. Armero F and Romero I (2001). On the formulation of high-frequency dissipative time-stepping algorithms for non-linear dynamics. Part I: low order methods for two model problems and non-linear elastodynamics. Comput Methods Appl Mech Eng 190: 2603–2649

    Article  MATH  MathSciNet  Google Scholar 

  6. Armero F and Romero I (2001). On the formulation of high-frequency dissipative time-stepping algorithms for non-linear dynamics. Part II: second order methods. Comput Methods Appl Mech Eng 190: 6783–6824

    Article  MATH  MathSciNet  Google Scholar 

  7. Berzeri M, Campanelli M and Shabana A (2001). Definition of the elastic forces in the finite-element absolute nodal coordinate formulation and the floating frame of reference formulation. Multibody Syst Dyn 5: 21–54

    Article  MATH  Google Scholar 

  8. Betsch P and Steinmann P (2002). Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int J Numer Methods Eng 54: 1775–1788

    Article  MATH  MathSciNet  Google Scholar 

  9. Caraciolo R and Trevisani A (2001). Simultaneous rigid-body motion and vibration control of a flexible four-bar linkage. Mech Mach Theory 36: 221–243

    Article  Google Scholar 

  10. Chen S and Tortorelli D (2003). An energy conserving and filtering method for stiff non-linear multibody dynamics. Multibody Syst Dyn 10: 341–362

    Article  MATH  MathSciNet  Google Scholar 

  11. Concalves JPC and Ambrosio JAC (2003). Optimization of vehicle suspension systems for improved comfort of road vehicles using flexible multibody dynamics. Nonlinear Dyn 34: 113–131

    Article  Google Scholar 

  12. Connelly JD and Huston RL (1994). The dynamics of flexible multibody systems: A finite segment approach – I Theoretical aspects. Comput Struct 50: 255–258

    Article  Google Scholar 

  13. Connelly JD and Huston RL (1994). The dynamics of flexible multibody systems: a finite segment approach – II example problems. Comput Struct 50: 259–262

    Article  Google Scholar 

  14. Crisfield M (1996). A unified co-rotational framework for solids, shells and beams. Int J Solids Struct 33: 2969–2992

    Article  MATH  Google Scholar 

  15. Crisfield MA and Shi J (1996). An energy conserving co-rotational procedure for non-linear dynamics with finite elements. Nonlinear Dyn 9: 37–52

    Article  Google Scholar 

  16. Cuadrado J, Gutierrez R, Naya M and Gonzalez M (2004). Experimental validation of a flexible MBS dynamic formulation through comparison between measured and calculated stresses on a prototype car. Multibody Syst Dyn 11: 147–166

    Article  MATH  Google Scholar 

  17. Devloo P, Geradin M and Fleury R (2000). A corotational formulation for the simulation of flexible mechanisms. Multibody Syst Dyn 4: 267–295

    Article  MATH  MathSciNet  Google Scholar 

  18. Elkaranshawy H and Dokainish M (1995). Corotational finite element analysis of planar flexible multibody systems. Comput Struct 54: 881–890

    Article  MATH  Google Scholar 

  19. Felippa CA, Haugen B (2005) A unified formulation of small-strain corotational finite elements: I. Theory Center for Aerospace Structures CU-CAS-05-02 report

  20. Gasparetto A (2001). Accurate modeling of a flexible-link planar mechanism by means of a linearized model in the state-space form for design of a vibration controller. J Sound Vibration 240: 241–262

    Article  Google Scholar 

  21. Geradin M and Cardona A (1989). Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra. Comput Mech 4: 115–135

    Article  Google Scholar 

  22. Iura M (1994). Effects of coordinate system on the accuracy of corotational formulation for Bernoulli–Euler’s beam. Int J Solids Struct 31: 2793–2806

    Article  MATH  Google Scholar 

  23. Jonker JB and Aarts RGKM (2001). A perturbation method for dynamic analysis and simulation of flexible manipulators. Multibody Syst Dyn 6: 245–266

    Article  MATH  Google Scholar 

  24. Kanarachos S, Spentzas C (2002) Investigation of the performance of the incremental finite element method of analysis of 2D mechanisms with respect to the stiffness of their arms. In: Proceedings of 4th GRACM Congress on Computational Mechanics 27–29 June Patras, Greece

  25. Kanarachos S, Spentzas C (2004) Analysis of flexible mechanisms using the conventional FEM. In: Proceedings of 1st International Conference From Scientific Computing to Computational Engineering 8–10 September Athens, Greece

  26. Kanarachos S (2004) Analysis of mechanisms using the finite element method. Dissertation, National Technical University of Athens

  27. Kanarachos S (2006) An approximate method for computing kinetoelastic problems. First South-East European Conference on Computational Mechanics 28–30 June Kragujevac, Serbia

  28. Mayo J, Garcia-Vallejo D and Dominguez J (2004). Study of the geometric stiffening effect: Comparison of different formulations. Multibody Syst Dyn 11: 321–341

    Article  MATH  Google Scholar 

  29. Meijaard JP (2003). Application of Runge-Kutta-Rosenbrock methods to the analysis of flexible multibody systems. Multibody Syst Dyn 10: 263–288

    Article  MATH  MathSciNet  Google Scholar 

  30. Negrut D, Haug E and German H (2003). An implicit Runge-Kutta method for integration of differential algebraic equations of multibody dynamics. Multibody Syst Dyn 9: 121–142

    Article  MATH  MathSciNet  Google Scholar 

  31. Olivetto G and Greco A (2002). Some observations on the characterization of structural damping. J Sound Vibration 256: 391–415

    Article  Google Scholar 

  32. Pascal M (2001). Some open problems in dynamic analysis of flexible multibody systems. Multibody Syst Dyn 5: 315–334

    Article  MATH  Google Scholar 

  33. Schiehlen W (1997). Multibody System Dynamics: Roots and Perspectives. Multibody Syst Dyn 1: 149–188

    Article  MATH  MathSciNet  Google Scholar 

  34. Schwertassek R, Dombrowski S and Wallrapp O (1999). Modal representation of stress in flexible multibody dynamics. Nonlinear Dyn 20: 381–399

    Article  MATH  MathSciNet  Google Scholar 

  35. Schwertassek R, Wallrapp O and Shabana A (1999). Flexible multibody simulation and choice of shape functions. Nonlinear Dyn 20: 361–380

    Article  MATH  MathSciNet  Google Scholar 

  36. Shabana A (1998). Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics. Nonlinear Dyn 16: 293–306

    Article  MATH  MathSciNet  Google Scholar 

  37. Shabana A and Schwertassek R (1998). Equivalence of the floating frame of reference approach and finite element formulations. Int J Nonlinear Mech 33: 417–432

    Article  MATH  Google Scholar 

  38. Shabana AA (1996). Finite element incremental approach and exact rigid body inertia. ASME J Mech Design 118: 171–178

    Article  Google Scholar 

  39. Shabana AA (1997). Flexible Multibody Dynamics: Review of Past and Recent Developments. Multibody Syst Dyn 1: 189–222

    Article  MATH  MathSciNet  Google Scholar 

  40. Spentzas KN and Kanarachos SA (2002). An incremental finite element analysis of mechanisms and robots. Forschung im Ingenieurwesen 67: 209–219

    Article  Google Scholar 

  41. Trevisani A (2003). Feedback control of flexible four-bar linkages: a numerical and experimental investigation. J Sound Vibration 268: 947–970

    Article  Google Scholar 

  42. Valembois RE, Fisette P and Samin JC (1997). Comparison of various techniques for modeling flexible beams in multibody dynamics. Nonlinear Dyn 12: 367–397

    Article  MATH  MathSciNet  Google Scholar 

  43. Vetyukov Yu, Gerstmayr J and Irschik H (2004). The comparative analysis of the fully non-linear, the linear elastic and the consistenly linearized equations of motion of the 2D elastic pendulum. Comput Struct 82: 863–870

    Article  Google Scholar 

  44. Wallrapp O and Wiedemann S (2003). Comparison of results in flexible multibody dynamics using various approaches. Nonlinear Dyn 34: 189–206

    Article  MATH  MathSciNet  Google Scholar 

  45. Yakoub R and Shabana A (1999). Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems. Nonlinear Dyn 20: 267–280

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Frederick Institute of Technology, 7 Y. Frederickou St., Palouriotisa, Nicosia, Cyprus

    Stratis Kanarachos

Authors
  1. Stratis Kanarachos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stratis Kanarachos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kanarachos, S. Analysis of 2D flexible mechanisms using linear finite elements and incremental techniques. Comput Mech 42, 107–117 (2008). https://doi.org/10.1007/s00466-008-0240-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0240-z

Keywords

Search

Navigation