Skip to main content

Ever wonder what the modes of vibration of a rod contacting a surface look like?

Submitted by oliver oreilly on
research

Dear Colleague,

Our latest work on the dynamics of rods contacting rigid surfaces has just appeared:

N. N. Goldberg and O. M. O'Reilly, On contact point motion in the vibration of elastic rods, Journal of Sound and Vibration, 2020. 

In this work, a systematic method for analyzing the vibrations of elastic rods whose effective length is variable, with particular emphasis on rods in unilateral contact with rigid surfaces. Problems of this type abound in engineering applications at all length scales, from the laying of submarine pipelines to the stiction of cantilevers in microelectromechanical systems (MEMS).

 

By a careful treatment of boundary conditions, we are able to clearly state the circumstances under which a rod of variable length can be treated as one of fixed length for the sake of analyzing small-amplitude vibrations. We then apply our analysis to the free vibrations of a rod in contact with a surface and, for the first time, are able to establish the mode shapes for the linear vibrations of the rod about a nonlinear equilibrium configuration. Animations of the modes can be seen in the movie below:

https://twitter.com/UCBDynamicsLab/status/1291165160107843585?s=20

 

In our analysis of this simple vibrations problem, we encounter an unusual singular limit and observe a close connection between vibration, stability, and existence. We also relate our work to the nonlinear stability criterion developed by Majidi et al. [1,2]. Our results can be considered as an extension to the works of Roy and Chatterjee [3].

 

  1. C. Majidi, O. M. O’Reilly, and J. A. Williams. On the stability of a rod adhering to a rigid surface: Shear-induced stable adhesion and the instability of peeling. Journal of the Mechanics and Physics of Solids, 60(5):827–843, 2012.
  2. C. Majidi, O. M. O’Reilly, and J. A. Williams. Bifurcations and instability in the adhesion of intrinsically curved rods. Mechanics Research Communications, 49(0):13–16, 2013.
  3. A. Roy and A. Chatterjee. Vibrations of a Beam in Variable Contact With a Flat Surface. Journal of Vibration and Acoustics, 131(4):041010, August 2009.