The Algebraic Frequency Equation of a Taut String Damped at a Fraction Location: Solution Structures and Properties

The frequency equation for a cable damped at a fraction location, generally being a transcendental equation, was reformulated into an algebraic form. Based on the fundamental theorem of algebra and the characteristics of logarithmic function in the complex domain, the solution structures were revealed and several examples with given damper positions were presented to study the variation of the solutions with the damping coefficient. The results show that: 1) All solutions of the frequency equation of the system could be classified into a finite number of solution branches. 2) The solutions of different orders in the same solution branch share an identical real part and any two adjacent solutions share an identical difference. 3) According to different variations in decay rate and frequency with damping coefficient, all the solution branches could be classified into four categories, each of which showed different trend of variation.