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

Non-stationary localized oscillations of an infinite Bernoulli-Euler beam lying on the Winkler foundation with a point elastic inhomogeneity of time-varying stiffness

“…(1) with respect to x over the infinitesimal vicinity of x = 0 taking into account (2), one gets (4), wherein p = 0. Now we substitute ansatz (19)- (22) and representations (23) into Eq. (4) and equate coefficients of like powers ǫ.…”

“…Combining the solution in the form of Eqs. (19)- (22) with its complex conjugate, we get the non-stationary solution as the following ansatz:…”

“…It is known [2,7,16,21,22,27,28] that applying non-stationary external excitation to a system possessing trapped modes leads to the emergence of undamped oscillations localized near the inhomogeneity. The large time asymptotics for such oscillation can be found [7,21,22,27] by means of the method of stationary phase [29,30].…”

“…It is known [2,7,16,21,22,27,28] that applying non-stationary external excitation to a system possessing trapped modes leads to the emergence of undamped oscillations localized near the inhomogeneity. The large time asymptotics for such oscillation can be found [7,21,22,27] by means of the method of stationary phase [29,30]. Gavrilov in [6,7] suggested an asymptotic procedure based on successive application of two asymptotic methods, namely, the method of stationary phase [29,30] (which we use to describe the steady-state motion in a system with constant properties) and the method of multiple scales [30,31] (which we use to describe a motion in a system with time-varying properties).…”

“…In studies [6,7] the problem concerning non-uniform motion of a point mass along a taut string on the Winkler foundation was considered and solved. The asymptotic procedure suggested in [6,7] was successfully applied to describe the evolution of the amplitude of the trapped mode of oscillations in a taut string on the Winkler foundation with a point inertial inclusion with time-varying mass [21] and in a taut string on the Winkler foundation with a concentrated spring with negative time-varying stiffness [22]. In preprint [28] non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity are considered.…”