Multibody System Dynamics

, Volume 21, Issue 4, pp 325–345 | Cite as

An improved dynamic modeling of a multibody system with spherical joints

  • Ali Rahmani Hanzaki
  • Subir Kumar Saha
  • P. V. M. Rao
Article

Abstract

A dynamic modeling of multibody systems having spherical joints is reported in this work. In general, three intersecting orthogonal revolute joints are substituted for a spherical joint with vanishing lengths of intermediate links between the revolute joints. This procedure increases sizes of associated matrices in the equations of motion, thus increasing computational burden of an algorithm used for dynamic simulation and control. In the proposed methodology, Euler parameters, which are typically used for representation of a rigid-body orientation in three-dimensional Cartesian space, are employed to represent the orientation of a spherical joint that connects a link to its previous one providing three-degree-of-freedom motion capability. For the dynamic modeling, the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices is utilized. It is shown in this work that the representation of spherical joints motion using Euler parameters avoids the unnecessary introduction of the intermediate links, thereby no increase in the sizes of the associated matrices with the dynamic equations of motion. To confirm the efficiency of the proposed representation, it is illustrated with the dynamic modeling of a spatial four-bar Revolute-Spherical–Spherical-Revolute (RSSR) mechanism, where the CPU time of the dynamic modeling based on proposed methodology is compared with that based on the revolute joints substitution. Finally, it is explained how a complex suspension and steering linkage can be modeled using the proposed concept of Euler parameters to represent a spherical joint.

Keywords

Spherical joint Dynamic modeling Euler parameters Closed loop 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs (1988) Google Scholar
  2. 2.
    Garcia de Jalon, J.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, Berlin (1994) Google Scholar
  3. 3.
    Saha, S.K.: A decomposition of manipulator inertia matrix. IEEE Trans. Robotics Autom. 13(2), 301–304 (1997) CrossRefGoogle Scholar
  4. 4.
    Saha, S.K., Schiehlen, W.O.: Recursive kinematics and dynamics for closed loop multibody systems. Mech. Struct. Mach. 2(29), 143–175 (2001) Google Scholar
  5. 5.
    Huston, R.L., Passerello, C.E.: On constraint equation-A new approach. ASME J. Appl. Mech. 41, 1130–1131 (1974) Google Scholar
  6. 6.
    Angeles, J., Lee, S.K.: The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal compliment. ASME J. Appl. Mech. 55, 243–244 (1988) MATHGoogle Scholar
  7. 7.
    Chaudhary, H., Saha, S.K.: Constraint wrench formulation for closed-loop systems using two-level recursions. Mech. Des. 129(12), 1234–1242 (2007) CrossRefGoogle Scholar
  8. 8.
    Angeles, J.: On twist and wrench generators and annihilators. In: Proceedings of the NATO-Advanced Study Institution on Computer Aided Analysis of Rigid and Flexible Systems 1, Troia, Portugal, 27 June–9 July 1993 Google Scholar
  9. 9.
    Duffy, J.: Displacement analysis of the generalized RSSR mechanism. Mech. Mach. Theory 13, 533–541 (1978) CrossRefGoogle Scholar
  10. 10.
    Rahmani Hanzaki, A., Saha, S.K., Rao, P.V.M.: Dynamics modeling of multibody systems with spherical joints using Euler parameters. In: Proceedings of Multibody Dynamics’2007, ECCOMAS Thematic Conference, Milan, Italy, 25–28 June 2007 Google Scholar
  11. 11.
    Robertson, A.P., Slocum, A.H.: Measurement and characterization of precision spherical joints. Precis. Eng. 30, 1–12 (2006) CrossRefGoogle Scholar
  12. 12.
    Attia, H.A.: Dynamic modeling of the double Wishbone motor-vehicle suspension system. Eur. J. Mech. A Solids 21, 167–174 (2002) MATHCrossRefGoogle Scholar
  13. 13.
    Attia, H.A.: Dynamic simulation of constrained mechanical systems using recursive projection algorithm. J. Braz. Soc. Mech. Sci. Eng. XXVIII(1), 37–44 (2006) Google Scholar
  14. 14.
    Norton, R.L.: Design of Machinery—An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edn. McGraw-Hill, New Delhi (2002) Google Scholar
  15. 15.
    Deo, N.: Graph Theory with Application in Engineering and Computer Science. Prentice-Hall, Englewood Cliffs (1974) Google Scholar
  16. 16.
    McPhee, J.J.: On the use of linear graph theory in multibody system dynamics. Nonlinear Dyn. 9, 73–90 (1996) CrossRefGoogle Scholar
  17. 17.
    Smith, D.A.: Reaction force analysis in generalized machine systems. ASME J. Eng. Ind. 95(2), 617–623 (1973) Google Scholar
  18. 18.
    Milner, J.R., Smith, D.A.: Topological reaction force analysis. ASME J. Mech. Des. 101(2), 192–198 (1979) Google Scholar
  19. 19.
    Chaudhary, H.: Analysis and optimization of mechanisms with handmade carpets. Ph.D. thesis, Indian Institute of Technology (IIT), Delhi (2007) Google Scholar
  20. 20.
    Saha, S.K.: Dynamic modeling of serial multibody systems using the decoupled natural orthogonal complement matrices. ASME J. Appl. Mech. 66, 986–996 (1999) CrossRefGoogle Scholar
  21. 21.
    Chaudhary, H., Saha, S.K.: Matrix formulation of constraint wrenches for serial manipulators. In: International Conference on Robotics and Automation (ICRA 2005), pp. 4647–4652, Barcelona, Spain, 18–22 April 2005 Google Scholar
  22. 22.
    Rahmani Hanzaki, A., Saha, S.K., Rao, P.V.M.: Modeling of a rack and pinion steering linkage using multibody dynamics, In: Proceedings of the 12th IFToMM World Congress, Besançon, France, 18–21 June 2007 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Ali Rahmani Hanzaki
    • 1
  • Subir Kumar Saha
    • 2
  • P. V. M. Rao
    • 2
  1. 1.Mechanical Engineering Dept.Shahid Rajaee UniversityLavizanIran
  2. 2.Mechanical Engineering Dept.IIT DelhiNew DelhiIndia

Personalised recommendations