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A tube dynamics perspective governing stability transitions: An example based on snap-through buckling

Publication ,  Journal Article
Zhong, J; Virgin, LN; Ross, SD
Published in: International Journal of Mechanical Sciences
December 1, 2018

The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum. That is, given a sufficient disturbance, the structure might buckle, perhaps suddenly, to another shape. This paper considers the dynamic mechanisms under which such transitions (typically via saddle points) occur. A two-mode Hamiltonian is developed for a shallow arch/buckled beam. The resulting form of the potential energy—two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations. Then, from Hamilton's equations, the equilibria are determined and linearization of the equations of motion about the saddle is obtained. After computing the eigenvalues and eigenvectors of the coefficient matrix associated with the linearization, a symplectic transformation is given which puts the Hamiltonian into normal form and simplifies the equations, allowing us to use the conceptual framework known as tube dynamics. The flow in the equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we account for the addition of damping in the tube dynamics framework, which leads to a richer set of behaviors in transition dynamics than previously explored.

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Published In

International Journal of Mechanical Sciences

DOI

ISSN

0020-7403

Publication Date

December 1, 2018

Volume

149

Start / End Page

413 / 428

Related Subject Headings

  • Mechanical Engineering & Transports
  • 40 Engineering
  • 0913 Mechanical Engineering
  • 0910 Manufacturing Engineering
  • 0905 Civil Engineering
 

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Zhong, J., Virgin, L. N., & Ross, S. D. (2018). A tube dynamics perspective governing stability transitions: An example based on snap-through buckling. International Journal of Mechanical Sciences, 149, 413–428. https://doi.org/10.1016/j.ijmecsci.2017.10.040
Zhong, J., L. N. Virgin, and S. D. Ross. “A tube dynamics perspective governing stability transitions: An example based on snap-through buckling.” International Journal of Mechanical Sciences 149 (December 1, 2018): 413–28. https://doi.org/10.1016/j.ijmecsci.2017.10.040.
Zhong J, Virgin LN, Ross SD. A tube dynamics perspective governing stability transitions: An example based on snap-through buckling. International Journal of Mechanical Sciences. 2018 Dec 1;149:413–28.
Zhong, J., et al. “A tube dynamics perspective governing stability transitions: An example based on snap-through buckling.” International Journal of Mechanical Sciences, vol. 149, Dec. 2018, pp. 413–28. Scopus, doi:10.1016/j.ijmecsci.2017.10.040.
Zhong J, Virgin LN, Ross SD. A tube dynamics perspective governing stability transitions: An example based on snap-through buckling. International Journal of Mechanical Sciences. 2018 Dec 1;149:413–428.
Journal cover image

Published In

International Journal of Mechanical Sciences

DOI

ISSN

0020-7403

Publication Date

December 1, 2018

Volume

149

Start / End Page

413 / 428

Related Subject Headings

  • Mechanical Engineering & Transports
  • 40 Engineering
  • 0913 Mechanical Engineering
  • 0910 Manufacturing Engineering
  • 0905 Civil Engineering