Free vibrations of hanging strings in contact with contact point motion

We study the free vibrations of two hanging strings that remain in mutual contact over a time-varying span and separated elsewhere. The contact is enforced by reversible Johnson–Kendall–Roberts (JKR) type adhesion, so the contact length is an unknown, time-dependent quantity. Using the variational approach, we derive the governing equations, boundary conditions, and a transversality condition that determines the contact length. As exact solutions are unavailable, we apply asymptotic analysis to obtain displacement fields and the motion of the contact point. Static equilibria appear above a critical adhesive strength and undergo a saddle-node bifurcation; one branch is unstable due to divergence instability, and the bifurcation point is independent of material, geometric, and adhesive properties. Two families of normal modes (NMs) emerge: one with fixed contact length (single string behavior) and one with contact-point motion. The associated eigenvalue problems (EVPs) are self-adjoint and the computed NMs satisfy orthonormality properties. Impulsive responses are examined based on the modal expansion. We have explored approximate solutions based on Ritz method which incorporates the moving contact, without invoking the transversality condition. This numerical framework validates the asymptotic predictions for equilibria, frequency spectrum, and contact point dynamics. We identify a counterintuitive threshold of the contact length for sustained oscillations and show that beyond these limits the strings separate irreversibly as t ∞ 8.