Vibration of Tensioned Beams with Intermediate Damper. I: Formulation, Influence of Damper Location

Abstract

Exact analytical solutions are formulated for free vibrations of tensioned beams with an intermediate viscous damper. The dynamic stiffness method is used in the problem formulation, and characteristic equations are obtained for both clamped and pinned supports. The complex eigenfrequencies form loci in the complex plane that originate at the undamped eigenfrequencies and terminate at the eigenfrequencies of the fully locked system, in which the damper acts as an intermediate pin support. The fully locked eigenfrequencies exhibit “curve veering,” in which adjacent eigenfrequencies approach and then veer apart as the damper passes a node of an undamped mode shape. Consideration of the evolution of the eigenfrequency loci with varying damper location reveals three distinct regimes of behavior, which prevail from the taut-string limit to the case of a beam without tension. The second regime corresponds to damper locations near the first antinode of a given undamped mode shape; in this regime, the loci bend backwards to intersect the imaginary axis, and two distinct nonoscillatory decaying solutions emerge when the damper coefficient exceeds a critical value.