Geodesic
Point spectra and normal modes of the Rayleigh loaded string with damping
Theoretical and applied mechanics, Tome 52 (2025) no. 1, p. 17

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We describe the point spectra of some dissipative version of the celebrated ``Rayleigh loaded string'', an elastic string of finite length carrying a number $n\ge1$ of equally spaced, equal point masses, which is a basic model that exhibits a band structure and appears in many applied areas. We consider the case in which the dissipation is due to a viscous damping due to the interaction string-environment, a standard model for internal visco-elastic dissipation (the Kelvin--Voigt model), and their combined presence. We show that the point spectrum of each of these damped versions of the Rayleigh loaded string is a continuous deformation of the point spectrum of the unloaded elastic strings with that damping and that presents a band structure similar to that of the undamped case. We also provide explicit analytical expressions for the eigenfunctions, for any value of $n$.
DOI : 10.2298/TAM241213010F
Classification : 35Q70, 74H45, 47J10
Keywords: Rayleigh loaded string, damped elastic string, viscous elastic string, Kelvin--Voigt viscoelastic string, band spectrum 
Francesco Fassò; Sara Galasso; Antonio Ponno. Point spectra and normal modes of the Rayleigh loaded string with damping. Theoretical and applied mechanics, Tome 52 (2025) no. 1, p. 17 . doi: 10.2298/TAM241213010F
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     title = {Point spectra and normal modes of the {Rayleigh} loaded string with damping},
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