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 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.

In this paper, a new simple oscillator model is considered describing ice-induced vibrations of upstanding, water-surrounded, and bottom-founded offshore structures. Existing models are extended by taking into account deformations of an ice floe and a moving contact interaction between an ice rod, which is cut out from the floe, and the oscillator which represents the offshore structure. Special attention is paid to a type of ice-induced vibrations of structures, known as frequency lock-in, and characterized by having the dominant frequency of the ice forces near a natural frequency of the structure. A new asymptotical approach is proposed that allows one to include ice floe deformations and to obtain a nonlinear equation for the simple oscillator vibrations. The instability onset, induced by resonance effects for the oscillator and generated by the ice rod structure interaction, is studied in detail.

The dynamics of a string on an elastic foundation with time- and coordinate-dependent coefficients have been studied. Asymptotic solutions have been constructed for the following cases: for an arbitrary value of the elastic foundation coefficient at small and large time values, and for small and large coefficients of the elastic foundation at arbitrary times. Also a special case originated from an ageing process has been studied. The ageing process is described by an expression approximating some well-known experimental data. The existence of localized modes along the x coordinate is shown. The existence of these localized modes can lead to a spatial resonance phenomenon under certain conditions. For the case of an arbitrary elastic foundation coefficient value at small and at large times, the spatial resonance phenomenon is observed at small, special frequencies. This effect depends also on a special phase and mode number. For large mode numbers, this special resonance seems to be not possible.