The effect of small internal and dashpot damping on a trapped mode of a 1D-waveguide, that is, a semi-infinite string on a Winkler elastic foundation, has been investigated. At the edge of the string a mass–spring–damper system is attached. The string is assumed to have an internal damping. Four models for the internal damping are considered: air damping, Kelvin–Voigt damping, local Kelvin–Voigt damping, and damping related to time hysteresis. Depending on the internal damping and the parameters in the formulated problem, it will be shown that the amplitude of a trapped mode of the string can decrease or increase with time.

In this paper, the dynamics of a compressed Euler-Bernoulli beam on a Winkler elastic foundation under the action of an external nonlinear force, which models a wind force, is studied. The beam is assumed to be long, and the lower part of its spectrum is prescribed. An asymptotic method is proposed to find the parameters of the beam, in order to have this prescribed lower part of the spectrum. All these parameters are necessary to guarantee the stability of the beam and to avoid resonances between the low frequency modes. These modes have special spatial supports that exclude a direct interaction between them. It is shown that the Galerkin system describing the time evolution can be decomposed into a system of almost independent equations which describes n independent nonlinear oscillators. Each oscillator has its own phase and frequency. It is shown that interaction between oscillators can exist only through high frequency modes.

In this paper the dynamics of a weakly nonlinear elastic string on a Winkler elastic foundation is studied. The foundation may be spatially heterogeneous. At one end of the string a mass-spring system is attached, and the other end of the string is fixed. The string is assumed to be long, and the lower part of the spectrum of the string is prescribed. It is shown that localized modes exist and that the dynamics of the string for large times is determined by these localized modes. The frequencies of these localized modes can be controlled by special choices for the spatial heterogeneities in the elastic foundation. Analytical and numerical results are presented to illustrate the findings.

This paper presents a mathematical analysis of an extended model describing a sea ice-induced frequency lock-in for vertically sided offshore structures. A simple Euler–Bernoulli beam as model for the offshore structure is used, and a moving boundary between an ice floe and the structure itself is introduced. A nonlinear equation for the beam dynamics is found by using an asymptotical approach and a Galerkin procedure. It is shown analytically that a frequency lock-in regime occurs during ice-induced vibrations (IIV), when the dominant ice force frequency is closed to a natural frequency of the structure. For beams, perturbed by small nonlinearities and a small damping, the concept of quasi-modes is introduced. A quasi-mode is a linear combination of the usual eigenmodes. The large time behaviour of solutions at the instability onset is determined by a single quasi-mode, which is maximally linearly unstable.The beam model analysis leads to the conclusion that an interaction between a moving ice floe and a structure can lead to a “negative friction” for particular values of the ice floe parameters. From the analysis presented in the paper it follows that the lock-in regime occurs when simultaneously two phenomena are present: a forcing resonance and a “negative friction”.

In this paper, the dynamics and the buckling loads for an Euler–Bernoulli beam resting on an inhomogeneous elastic, Winkler foundation are studied. An analytical, asymptotic method is proposed to determine the stability of the Euler–Bernoulli beam for various types of inhomogeneities in the elastic foundation taking into account different types of damping models. Based on the Rayleigh variation principle, beam buckling loads are computed for cases of harmonically perturbed types of inhomogeneities in the elastic foundation, for cases of point inhomogeneities in the form of concentrated springs in the elastic foundation, and for cases with rectangular inclusions in the elastic foundation. The investigation of the beam dynamics shows the possibility of internal resonances for particular values of the beam rigidity and longitudinal force. Such types of resonances, which are usually typical for nonlinear systems, are only possible for the beam due to its inhomogeneous foundation. The occurrence of so-called added mass effects near buckling instabilities under the influence of damping have been found. The analytical expressions for this “added mass” effect have been obtained for different damping models including space hysteresis types. This effect arises as a result of an interaction between the main mode, which is close to instability, and all the other stable modes of vibration.