Advertisement

Acta Mechanica Sinica

, Volume 34, Issue 2, pp 409–420 | Cite as

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

  • C. F. Du
  • D. G. Zhang
  • L. Li
  • G. R. Liu
Research Paper
  • 112 Downloads

Abstract

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

Keywords

Meshfree method NS-PIM Rigid-flexible coupled system dynamics Rotating beams Dynamic response 

Notes

Acknowledgements

The authors are grateful for the support from the National Natural Science Foundation of China (Grants 11272155, 11132007, and 11502113), the Fundamental Research Funds for Central Universities (Grant 30917011103), and the China Scholarship Council for one year study at the University of Cincinnati.

References

  1. 1.
    Yoo, H.H., Ryan, R.R., Scott, R.A.: Dynamics of flexible beams undergoing overall motions. J. Sound Vib. 181, 261–278 (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Yoo, H.H., Shin, S.H.: Vibration analysis of rotating cantilever beams. J. Sound Vib. 212, 807–828 (1998)CrossRefGoogle Scholar
  3. 3.
    Li, L., Zhang, D.G., Zhu, W.D.: Free vibration analysis of a rotating hub-functionally graded material beam system with the dynamic stiffening effect. J. Sound Vib. 333, 1526–1541 (2014)CrossRefGoogle Scholar
  4. 4.
    Chung, J., Yoo, H.H.: Dynamic analysis of a rotating cantilever beam by using the finite element method. J. Sound Vib. 249, 147–164 (2002)CrossRefGoogle Scholar
  5. 5.
    Du, H., Lira, M.K., Liew, K.M.: A nonlinear finite element model for dynamics of flexible manipulators. Mech. Mach. Theory 31, 1109–1119 (1996)CrossRefGoogle Scholar
  6. 6.
    Sanborn, G.G., Shabana, A.A.: A rational finite element method based on the absolute nodal coordinate formulation. Nonlinear Dyn. 58, 565–572 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, G.R., Quek, S.S.: Finite Element Method: A Practical Course, 2nd edn. Butterworth-Heinemann, Burlington (2013)zbMATHGoogle Scholar
  8. 8.
    Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer, Dordrecht (2005)Google Scholar
  9. 9.
    Sanborn, G.G., Shabana, A.A.: On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 22, 181–197 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Sugiyama, H., Gerstmayr, J., Shabana, A.A.: Deformation modes in the finite element absolute nodal coordinate formulation. J. Sound Vib. 298, 1129–1149 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lan, P., Shabana, A.A.: Integration of B-spline geometry and ANCF finite element analysis. Nonlinear Dyn. 61, 193–206 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Y.N., Sun, L., Liu, Y.H., et al.: Multi-scale B-spline method for 2-D elastic problems. Appl. Math. Model. 35, 3685–3697 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lucy, L.B.: A numerical approach to testing of the fission hypothesis. Astron. J. 8, 1013–1024 (1977)CrossRefGoogle Scholar
  14. 14.
    Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics: A Meshfree Practical Method. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Monaghan, J.J.: An introduction to SPH. Comput. Phys. Commun. 48, 89–96 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Belytschko, Y., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, W.K., Jun, S., Zhang, Y.E.: Reproducing kernel particle methods. Int. J. Numer. Methods Eng. 20, 1081–1106 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Atluri, S.N., Zhu, T.: A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)Google Scholar
  19. 19.
    Liu, G.R., Gu, Y.T.: A point interpolation method for two-dimensional solids. Int. J. Numer. Methods Eng. 50, 937–951 (2001)Google Scholar
  20. 20.
    Liu, G.R., Dai, K.Y., Lim, K.M., et al.: A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures. Comput. Mech. 29, 510–519 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, G.R., Zhang, G.Y., Gu, Y.T., et al.: A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput. Mech. 36, 421–430 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, J.G., Liu, G.R.: A point interpolation meshless method based on radial basis functions. Int. J. Numer. Methods Eng. 54, 1623–1648 (2002)CrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, G.R., Zhang, G.Y., Dai, K.Y., et al.: A linearly conforming point interpolation method (LC-PIM) for 2-D solid mechanics problems. Int. J. Comput. Methods 2, 645–665 (2005)CrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, G.R., Zhang, G.Y.: Upper bound solutions to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM). Int. J. Numer. Methods Eng. 74, 1128–1161 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, G.Y., Liu, G.R., Wang, Y.Y., et al.: A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. Int. J. Numer. Methods Eng. 72, 1524–1543 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wu, S.C., Liu, G.R., Zhang, H.O., et al.: A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds. Int. J. Heat Mass Transf. 52, 1464–1471 (2009)CrossRefzbMATHGoogle Scholar
  27. 27.
    Liu, G.R.: Meshfree Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2002)CrossRefGoogle Scholar
  28. 28.
    Cui, X.Y., Liu, G.R., Li, G.Y., et al.: A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique. CMES 28, 217–229 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Liu, G.R., Han, X.: Computational Inverse Techniques in Nondestructive Evaluation. CRC Press, Boca Raton (2003)CrossRefzbMATHGoogle Scholar
  30. 30.
    Liu, G.R.: Meshfree Methods: Moving Beyond the Finite Element Method, 2nd edn. CRC Press, Boca Raton (2010)zbMATHGoogle Scholar
  31. 31.
    Wu, S.C., Haug, E.J.: Geometric non-linear substructuring for dynamics of flexible mechanical system. Int. J. Numer. Methods Eng. 26, 2211–2226 (1988)CrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.College of Civil Science and EngineeringYangzhou UniversityYangzhouChina
  2. 2.School of SciencesNanjing University of Science and TechnologyNanjingChina
  3. 3.School of Aerospace SystemsUniversity of CincinnatiCincinnatiUSA

Personalised recommendations