Closed-Form Eigen-Solutions for Damped Strings: A Parametric Study Across 1/2-, 1/3-, and 1/4-Span Damper Positions

This paper presents a comparative investigation of the dependence of eigen-solutions on damper positioning, using a dimensionless taut-string model with a point viscous damper. Building on the known closed-form eigen-solutions for dampers at the 1/2- and 1/3-span positions, this study newly derives the novel closed-form solution for the 1/4-span configuration. With this newly derived quarter-span solution, the mid-span, one-third-span, and quarter-span cases form a minimal closed-form analytical sequence for examining how damper position organizes the controlled eigenspectrum. For these three canonical positions, the transcendental frequency equation is transformed into an equivalent algebraic polynomial equation through hyperbolic-function identities and a change of variables, with the order of the resulting polynomial determined by the selected position. As the damper position changes from 1/2 to 1/4 span, the characteristic polynomial rises from first to third order, yielding one, two, and three eigenvalue branches, respectively. These branches exhibit different evolutionary features: the mid-span system shows a global critical spectral transition, the one-third-span system undergoes position-specific local branch coalescence and separation, whereas the quarter-span system contains a smoothly evolving conjugate pair of controlled branches among its three branches. The quarter-span solution therefore identifies, within this minimal canonical sequence, a smooth controlled-branch behavior that is structurally related to the critical spectral transitions revealed by the mid-span and one-third-span cases. Through this systematic comparison, the closed-form solutions clarify the position-dependent organization of algebraic order, controlled-branch structure, and spectral-transition behavior. They also provide analytical benchmarks for numerical eigen-solution, branch tracking, optimal damping identification, and subsequent damper-arrangement studies under practical installation constraints.