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A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints

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Abstract

In the present study, for the first time, flexible multibody dynamics for a three-link serial robot with two flexible links having active prismatic joints is presented using an approximate analytical method. Transverse vibrations of flexible links/beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the links. In this study, first, vibration analysis of a flexible link sliding through an active prismatic joint having translational motion is considered. A rigid-body coordinate system is used, which aids in obtaining a new and rather simple form of the kinematic differential equation without the loss of generality. Next, the analysis is extended to include dynamic forces for a three-link planar serial robot called PPP (Prismatic, Prismatic, Prismatic), in which all joints are prismatic and active. The robot has a rigid first link but flexible second and third links. To model the prismatic joint, time-variant constraints are written, and a motion equation in a form of virtual displacement and virtual work of forces/moments is obtained. Finally, an approximate analytical method called the “constrained assumed modes method” is presented for solving the motion equations. For a numerical case study, approximate analytical results are compared with finite element results, which show that the two solutions closely follow each other.

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References

  1. Ma, O.: Mechanical analysis of parallel manipulators with simulation, design and control applications. Ph.D. thesis, Department of Mechanical Engineering, McGill University (1991)

  2. Tsai, L.W.: Robot Analysis: The Mechanics of Serial and Parallel Manipulators. Wiley, New York (1999)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Dwivedy, S.K., Eberhard, P.: Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41, 749–777 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Winfrey, R.C.: Elastic link mechanism dynamics. J. Eng. Ind. 93, 268–272 (1971)

    Article  Google Scholar 

  6. Erdman, A.G., Sandor, G.N., Oakberg, R.G.: A general method for kineto-elastodynamic analysis and synthesis of mechanisms. J. Eng. Ind. 94, 1193–1205 (1972)

    Article  Google Scholar 

  7. Cleghorn, W.L., Fenton, R.G., Tabarrok, B.: Finite element analysis of high-speed flexible mechanisms. Mech. Mach. Theory 16, 407–424 (1981)

    Article  Google Scholar 

  8. Turic, D.A., Midha, A.: Generalized equations of motion for the dynamic analysis of elastic mechanism systems. J. Dyn. Syst. Meas. Control 106, 243–248 (1984)

    Article  MATH  Google Scholar 

  9. Yang, Z., Sadler, J.P.: Large-displacement finite element analysis of flexible linkages. J. Mech. Des. 112, 175–182 (1990)

    Article  Google Scholar 

  10. Kitis, L.: Natural frequencies and mode shapes of flexible mechanisms by a transfer matrix method. Finite Elem. Anal. Des. 6, 267–285 (1990)

    Article  MATH  Google Scholar 

  11. Wu, H.T., Mani, N.K., Ashrafiuon, H.: Selection of modal basis for flexible bodies of mechanical systems. Mech. Mach. Theory 30(3), 471–489 (1995)

    Article  Google Scholar 

  12. Shabana, A.A.: Resonance conditions and deformable body co-ordinate systems. J. Sound Vib. 192, 389–398 (1996)

    Article  Google Scholar 

  13. Madani, M., Moallem, M.: Hybrid position/force control of a flexible parallel manipulator. J. Franklin Inst. 348, 999–1012 (2011)

    Article  MATH  Google Scholar 

  14. Junfeng, H., Xiangfu, C., Pei, L.: Vibration suppression of flexible parallel manipulator based on sliding mode control with reaching law. Appl. Mech. Mater. 160, 30–34 (2012)

    Article  Google Scholar 

  15. Ubertini, F.: A contribution to the analysis of flexible link systems. Int. J. Solids Struct. 37, 969–990 (2000)

    Article  MATH  Google Scholar 

  16. Faris, W.F., Ata, A.A., Sa’adeh, M.Y.: Energy minimization approach for a two-link flexible manipulator. J. Vib. Control 15(4), 497–526 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Abedi, E., Nadooshan, A.A., Salehi, S.: Dynamic modeling of tow flexible link manipulators. World Acad. Sci., Eng. Technol. 46, 461–467 (2008)

    Google Scholar 

  18. Ibrahimbegovic, A., Mamouri, S.: 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 (2000)

    Article  MATH  Google Scholar 

  19. Bauchau, O.A.: On the modeling of prismatic joints in flexible multi-body systems. Comput. Methods Appl. Mech. Eng. 181, 87–105 (2000)

    Article  MATH  Google Scholar 

  20. Fattah, A., Angeles, J., Misra, A.K.: Dynamics of a 3-DOF spatial parallel manipulator with flexible links. In: Proceedings of IEEE International Conference on Robotics and Automation, Nagoya, Japan, May 21–27, vol. 1, pp. 627–632 (1995)

    Google Scholar 

  21. Wang, X., Mills, J.K.: Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach. Mech. Mach. Theory 41, 671–687 (2006)

    Article  MATH  Google Scholar 

  22. Munoz, J.J., Jelenic, J.: Sliding contact conditions using the master–slave approach with application on geometrically non-linear beams. Int. J. Solids Struct. 41, 6963–6992 (2004)

    Article  MATH  Google Scholar 

  23. Munoz, J.J., Jelenic, J.: Sliding joints in 3D beams: conserving algorithms using the master–slave approach. Multibody Syst. Dyn. 16, 237–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kang, B., Mills, J.K.: Dynamic modeling of structurally-flexible planar parallel manipulator. Robotica 20, 329–339 (2002)

    Google Scholar 

  25. Zhang, Q., Mills, J.K., Cleghorn, W.L., Jin, J., Zhao, C.: Trajectory tracking and vibration suppression of a 3-PRR parallel manipulator with flexible links. Multibody Syst. Dyn. 33(1), 27–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sharifnia, M., Akbarzadeh, A.: An analytical model for vibration and control of a PR-PRP parallel robot with flexible platform and prismatic joint. J. Vib. Control 22(3), 632–648 (2016)

    Article  MathSciNet  Google Scholar 

  27. Sharifnia, M., Akbarzadeh, A.: Approximate analytical solution for vibration of a 3-PRP planar parallel robot with flexible moving platform. Robotica 34(1), 71–97 (2016)

    Article  Google Scholar 

  28. Sharifnia, M., Akbarzadeh, A.: Dynamics and vibration of a 3-PSP parallel robot with flexible moving platform. J. Vib. Control 22(4), 1095–1116 (2016)

    Article  MathSciNet  Google Scholar 

  29. Tabarrok, B., Leech, C.M., Kim, Y.I.: On the dynamic of an axially moving beam. J. Franklin Inst. 297(3), 201–220 (1974)

    Article  MATH  Google Scholar 

  30. Wang, P.K.C., Wei, J.D.: Vibrations in a moving flexible robot arm. J. Sound Vib. 116(1), 149–160 (1987)

    Article  Google Scholar 

  31. Krishnamurrthy, K.: Dynamic modeling of a flexible cylindrical manipulator. J. Sound Vib. 132(1), 143–154 (1989)

    Article  Google Scholar 

  32. Lee, U., Jang, I.: On the boundary conditions for axially moving beams. J. Sound Vib. 306, 675–690 (2007)

    Article  Google Scholar 

  33. Theodore, R.J., Ghosal, A.: Comparison of the assumed modes and finite element models for flexible multilink manipulators. Int. J. Robot. Res. 14(2), 91–111 (1995)

    Article  Google Scholar 

  34. Theodore, R.J., Ghosal, A.: Modeling of flexible-link manipulators with prismatic joints. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 27(2), 296–305 (1997)

    Article  Google Scholar 

  35. Hwang, R.S., Haug, E.J.: Translational joints in flexible multibody dynamics. Mech. Struct. Mach. 18(4), 543–564 (1990)

    Article  Google Scholar 

  36. Sugiyama, H., Escalona, J.L., Shabana, A.A.: Formulation of three-dimensional joint constraints using the absolute nodal coordinates. J. Nonlinear Dyn. 31, 167–195 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lee, S.H., Park, T.W., Seo, J.H., Yoon, J.W., Jun, K.J.: The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation. Multibody Syst. Dyn. 20, 223–237 (2008)

    Article  MATH  Google Scholar 

  38. Shabana, A.A.: Dynamics of Multibody Systems, 2nd edn. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  39. Meirovitch, L.: Analytical Methods in Vibrations. Macmillan, New York (1967)

    MATH  Google Scholar 

  40. Meirovitch, L.: Elements of Vibration Analysis, 2nd edn. McGraw-Hill, New York (1986)

    MATH  Google Scholar 

  41. Meirovitch, L.: Fundamentals of Vibrations. McGraw-Hill, New York (2002)

    Google Scholar 

  42. Hoffman, J.D.: In: Numerical Methods for Engineers and Scientists, 1st edn., pp. 401–406. McGraw-Hill, New York (1992)

    Google Scholar 

  43. Przemieniecki, J.S.: Theory of Matrix Structure Analysis, pp. 328–339. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  44. ABAQUS: ABAQUS documentation, Dassault Systemes, Providence (2011)

    Google Scholar 

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Authors and Affiliations

  1. Mechanical Engineering Department, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

    Mahdi Sharifnia

  2. Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP), Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran

    Alireza Akbarzadeh

Authors
  1. Mahdi Sharifnia
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  2. Alireza Akbarzadeh
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Correspondence to Mahdi Sharifnia.

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Sharifnia, M., Akbarzadeh, A. A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints. Multibody Syst Dyn 40, 261–285 (2017). https://doi.org/10.1007/s11044-016-9525-8

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  • DOI: https://doi.org/10.1007/s11044-016-9525-8

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