Evolution of a trapped mode of oscillation in a string on the Winkler foundation with point inhomogeneity

We consider a mechanical system with mixed spectrum of natural oscillations. This is an infinite string on the Winkler elastic foundation with point inhomogeneity (the spring with negative stiffness). The discrete part of the spectrum contains a unique positive eigenvalue that corresponds to a trapped mode of oscillation. The aim of the paper is to describe evolution of the amplitude of the trapped mode of oscillation for two perturbed problems with slowly varying parameters. The first problem corresponds to the case of the concentrated spring with slowly varying stiffness, the second one corresponds to the case of the string with slowly varying tension. For both problems, analytic solutions are obtained by means of the asymptotic procedure suggested by Gavrilov & Indeitsev (2002), based on the method of multiple scales. In the first case, namely, for the concentrated spring with slowly varying stiffness, the governing equation is a PDE with constant coefficients. This allows one to easily verify the constructed solution by numerical calculations. The comparison demonstrates a good agreement.