An approximate approach to study the free vibration of a string with an arbitrary variation in mass density and tension, with attached concentrated masses, and its application to hanging chain and rotating cord

A multi-stepped approximation has been developed to study the free vibration of a string with arbitrary variation in the tension and the mass density. To achieve this objective, a novel exact approach is developed to study free vibration of a string with N uniform sections of different mass densities under N different tensions with N+1 attached concentrated masses and springs at the ends. The two advantages of this approach are that it leads to a single frequency equation for any number of sections and the same computer program can be used to solve different problems using pertinent data. This method is applied to cases where tension and/or mass density changes continuously such as hanging chain, rotating chain, or inhomogeneous strings vibrating in either horizontal or vertical position. In all cases, the results obtained using this approach agreed with previous results obtained using exact and approximate methods. It was found that natural frequencies of a fixed string under constant tension T   with linear variation in the mass density, ρ(x)=ρ0(1+αx/L)ρ(x)=ρ0(1+αx/L), can be approximated using a very simple equation, ωn=(nπ/L)[T/{ρ0(1+0.5α)}]1/2ωn=(nπ/L)[T/{ρ0(1+0.5α)}]1/2 and results reasonably agree with previous results for a wide range of αα. New results for many other cases are also presented in what follows.