On contact point motion in the vibration analysis of elastic rods

We present 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 elucidate 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. In applying our method to a simple free vibration problem, we encounter an unusual singular limit and observe a close connection between vibration, stability, and existence