Skip to main content
SpringerLink
Log in

Stability and vibration of a helical rod constrained by a cylinder

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler’s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov’s and Euler’s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form.

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. Love A.E.H. (1944). A Treatise on Mathematical Theory of Elasticity. Dover, New York

    MATH  Google Scholar 

  2. Liu, Y.Z.: Nonlinear Mechanics of Thin Elastic Rod. (in Chinese) Tsinghua University & Springer, Beijing (2006)

  3. Liu Y.Z. and Zu J.W. (2004). Stability and bifurcation of helical equilibrium of a thin elastic rod. Acta Mech. 167(1–2): 29–39

    Article  MATH  Google Scholar 

  4. Seemann W. (1996). Deformation of an elastic helix in contact with a rigid cylinder. Arch. Appl. Mech. 67: 117–139

    Article  MATH  Google Scholar 

  5. van der Heijden G.H.M. (2001). The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc. R. Soc. Lond. A 457: 695–715

    Article  MATH  Google Scholar 

  6. van der Heijden G.H.M., Champneys A.R. and Thomson J.M.T. (2002). Spatially complex localization in twisted elastic rods constrained to a cylinder. Int. J. Solids Struct. 39: 1863–1883

    Article  MATH  Google Scholar 

  7. Xue Y., Chen L.Q. and Liu Y.Z. (2004). Problems on equilibrium of a thin elastic rod constrained on a surface. (in Chinese) Acta Phys. Sin. 53(7): 2040–2045

    Google Scholar 

  8. Liu Y.Z. and Xue Y. (2005). Equilibrium and stability of an elastic rod constrained to a cylinder. (in Chinese) J. Appl. Mech. 22(3): 392–394

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai, 200030, China

    Yanzhu Liu & Liwei Sheng

Authors
  1. Yanzhu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liwei Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanzhu Liu.

Additional information

The project supported by the National Natural Science Foundation of China (10472067). The English text was polished by Yunming Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Sheng, L. Stability and vibration of a helical rod constrained by a cylinder. Acta Mech Sin 23, 215–219 (2007). https://doi.org/10.1007/s10409-007-0058-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-007-0058-5

Keywords

Search

Navigation