Stability Analysis of a Tensioned String With Periodic Supports

摘要

This report analyzes the zero-pole locations of an infinite length of tensioned string that has attached periodic supports. The dynamic response of the system is derived for distributed wave number forcing and discrete point forcing acting on the string. These wave number-frequency transfer functions are then written in zero-pole format by a mathematical transformation of their infinite series. Once this is accomplished, the locations of the system's poles and zeros become apparent, and they can be plotted in the wave number frequency plane. It is shown that there are specific regions where an infinite number of poles can exist and specific regions where poles cannot exist. For the system with wave number forcing, the system zeros correspond very closely to the system poles except in the area of the fundamental unsupported string resonance. For the system with point forcing, the zeros can exist in the entire wave number frequency plane except at the fundamental resonance. A numerical example is included, and the different zones of the system are demonstrated.