Skip to main content
SpringerLink
Log in

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kinematics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their increments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C 1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motiondeformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.

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. Schiehlen, W. Multibody system dynamics: roots and perspectives. Multibody System Dynamics, 1(2), 149–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Shabana, A. A. Flexible multibody dynamics: review of past and recent developments. Multibody System Dynamics, 1(2), 189–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wasfy, T. M. and Noor, A. K. Computational strategies for flexible multibody systems. Appl. Mech. Rev., 56(6), 553–613 (2003)

    Article  Google Scholar 

  4. Schiehlen, W. Computational dynamics: theory and applications of multibody systems. European Journal of Mechanics A/Solids, 25(4), 566–594 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kane, T. R., Ryan, R. R., and Banerjee, A. K. Dynamics of a cantilever beam attached to a moving base. AIAA Journal of Guidance, Control, and Dynamics, 10(2), 139–151 (1987)

    Article  Google Scholar 

  6. Simo, J. C. and Vu-Quoc, L. The role of non-linear theories in transient dynamic analysis of flexible structures. Journal of Sound and Vibration, 119(3), 487–508 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, S. C. and Haug, E. J. Geometric non-linear substructuring for dynamics of flexible mechanical systems. International Journal for Numerical Methods in Engineering, 26(10), 2211–2226 (1988)

    Article  MATH  Google Scholar 

  8. Wallrapp, O. and Schwertassek, R. Representation of geometric stiffening in multibody system simulation. International Journal for Numerical Methods in Engineering, 32(8), 1833–1850 (1991)

    Article  MATH  Google Scholar 

  9. Berzeri, M. and Shabana, A. A. Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation. Multibody System Dynamics, 7(4), 357–387 (2002)

    Article  MATH  Google Scholar 

  10. Belytschko, T. and Hsieh, B. J. Non-linear transient finite element analysis with convected coordinates. International Journal for Numerical Methods in Engineering, 7(3), 255–271 (1973)

    Article  MATH  Google Scholar 

  11. Banerjee, A. K. Block-diagonal equations for multibody elastodynamics with geometric stiffness and constraints. Journal of Guidance, Control, and Dynamics, 16(6), 1092–1100 (1994)

    Article  Google Scholar 

  12. Mayo, J. and Dominguez, J. Geometrically nonlinear formulation of flexible multibody systems in terms of beam elements: geometric stiffness. Computers and Structures, 59(6), 1039–1050 (1996)

    Article  MATH  Google Scholar 

  13. Cosserat, E. and Cosserat, F. Théorie des Corps Déformables, Hermann et Fils, Paris (1909)

    Google Scholar 

  14. Toupin, R. A. Elastic materials with couple-stresses. Arch. Rational Mech. Anal., 11(1), 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mindlin, R. D. and Tiersten, H. F. Effects of couple-stresses in linear elasticity. Arch. Rational Mech. Anal., 11(1), 415–488 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eringen, A. C. Linear theory of micropolar elasticity. J. Math. Mech., 15(6), 909–923 (1966)

    MathSciNet  MATH  Google Scholar 

  17. Ottosen, N. S., Ristinmaa, M., and Ljung, C. Rayleigh waves obtained by the indeterminate couple-stress theory. European Journal of Mechanics A/Solids, 19(6), 929–947 (2000)

    Article  MATH  Google Scholar 

  18. Fleck, N. A. and Hutchinson, J. W. A reformulation of strain gradient plasticity. Journal of Mechanics and Physics of Solids, 49(10), 2245–2271 (2001)

    Article  MATH  Google Scholar 

  19. Chong, A. C., Yang, F., Lam, D. C., and Tong, P. Torsion and bending of micron-scaled structures. J. Mater. Res., 16(4), 1052–1058 (2001)

    Article  Google Scholar 

  20. Stolken, J. S. and Evans, A. G. A microbend test method for measuring plasticity length scale. Acta. Mater., 46(14), 5109–5115 (1998)

    Article  Google Scholar 

  21. Soh, A. K. and Chen, W. Finite element formulations of strain gradient theory for microstructures and the C 0-1 patch test. International Journal for Numerical Methods in Engineering, 61(3), 433–454 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wood, R. D. Finite element analysis of plane couple-stress problems using first order stress functions. Int. J. Num. Mech. Engng., 26(2), 489–509 (1988)

    Article  MATH  Google Scholar 

  23. Hermann, L. R. Mixed Finite Elements of Couple-Stresses Analysis, John Wiley and Sons, New York, 1–17 (1983)

    Google Scholar 

  24. Fillmore, J. P. A note on rotation matrices. Journal of Computer Graphics and Applications, 4(2), 30–33 (1984)

    Article  Google Scholar 

  25. Palais, B. and Palais, R. Euler’s fixed point theorem: the axis of a rotation. Journal of Fixed Point Theory and Applications, 2(2), 215–220 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, F., Chong, A. C. M., and Lam, D. C. C. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structure, 39(10), 2731–2743 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, College of Resource and Environmental Sciences, Chongqing University, Chongqing, 400030, P. R. China

    Zhan-fang Liu  (刘占芳), Shi-jun Yan  (颜世军) & Zhi Fu  (符 志)

Authors
  1. Zhan-fang Liu  (刘占芳)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shi-jun Yan  (颜世军)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi Fu  (符 志)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhan-fang Liu  (刘占芳).

Additional information

Project supported by the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (No. 11176035), the National Natural Science Foundation of China (No. 11072276), and the National Basic Research Program of China (No. 2011CB612211)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Zf., Yan, Sj. & Fu, Z. Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations. Appl. Math. Mech.-Engl. Ed. 34, 1001–1016 (2013). https://doi.org/10.1007/s10483-013-1723-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-013-1723-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation