On Modal Localization in a String on an Elastic Base

Modal localization in a string with coordinate-dependent density is studied. Methods for describing mechanical systems with a heterogeneous structure are developed. The tension in the string is considered to be constant and weak compared to the force in the substrate. The very phenomenon of modal localization consists in a significant change in the eigenmodes with a slight shift in the frequency spectrum due to small deviations in the value of some parameter, which in this study is the distribution of mass along the string. The Sturm–Liouville boundary value problem with a variable coefficient and a small parameter at the highest derivative is formulated to determine its influence on the spectrum and the first mode of string oscillations. The problem is reduced to the integral equation with an unknown spectral parameter by applying a generalized Green’s function. Its solution is constructed using the method of successive approximations based on an iterative sequence that allows determination of the value of the spectral parameter and the oscillation mode. Numerical integration of the equations is used to find the critical value of this parameter and the corresponding eigenmode is constructed. The effect of mode localization in a simple weakly coupled mechanical system with two degrees of freedom is also examined. The consistency of the results obtained when comparing the continuous model with its discrete analogue is noted.