The Vibration and Stability of a Friction-guided, Translating String

Eyelets, capstans and cylindrical surfaces are often used in thread, fiber and paper handling machinery to guide the axially moving element. In addition to providing positional control, these “guides” introduce dry friction forces that alter the vibration and stability characteristics of the system. This paper examines the lateral response of a string that slides through an elastically supported, dry friction guide. Exact expressions are derived for the linear response under free and forced conditions. Solutions for the eigenvalue spectrum exhibit unusual features including multiple divergence instabilities, regions of flutter instability, and regions of curve veering associated with mode localization. A second order perturbation solution is derived to examine the behavior of the eigenvalue spectrum in regions of flutter instability and curve veering. The analysis highlights the common features of these two phenomena and suggests ways to minimize vibration by adjusting various design variables. The analysis also demonstrates that the eigenvalue loci in regions of flutter instability and curve veering are naturally described by a local hyperbolic approximation.