The IPMash RAS mathematicians created a model that makes it possible to predict vibration conditions for structures operating in Arctic ice conditions, such as oil platforms, and to reduce the risk of damage to them. The results of the study were published in the scientific journal Chaos, Solitons and Fractals.

Operation of the engineering structures in the icy conditions of the Arctic and subarctic — oil platforms, lighthouses, wind turbine poles — is associated with a serious problem: the impact of moving ice fields, causing dangerous vibrations of the structures.

This phenomenon, known as vibrations caused by interaction with ice, has been studied for more than half a century, but its physical nature has not been fully revealed. The safety and durability of the facilities directly depends on how accurately the models can predict the behavior of a structure in such conditions.

Traditionally the engineers distinguish three reaction modes of the structure: intermittent crushing of ice with moderate vibrations, resonant frequency capture (the most dangerous mode with a critical increase in amplitude) and continuous brittle crushing characterized by complex, irregular behavior.

The existing models often simplified the picture, considering ice as a monolithic medium, and water only as an additional mass attached to the structure. This approach, which works for low frequencies, did not take into account the important fact that water is mixed with ice in the contact area.

The initial idea of accounting for a two-phase medium in this zone was proposed by Dmitry Anatolyevich Indeytsev, Corresponding Member of the Russian Academy of Sciences, former Director of IPMash RAS. However, his proposed model described only the first two vibration modes.

The IPMash RAS scientists proposed to introduce into the model a more detailed description of the destruction zone — the space between the moving ice floe and the structure.

In this zone, as field observations show, the ice does not just press on the barrier, but crumbles, forming a complex two-phase mixture of water and ice fragments of different sizes.

“Taking into account the properties of this mixture, its ability to accumulate or leave the gap, as well as the inclusion in the model of the randomness of the processes of breaking off and the size of pieces of ice, allowed us to describe all three modes. It was this innovation that made it possible to explain how one and the same system can move from a periodic reaction to resonance and then to chaotic behavior in response to the speed of the ice”, — said Andrey Abramian, the chief researcher at the Wave Processes Mathematical Modeling Laboratory at IPMash RAS.

The researchers managed to apply perturbation theory and the Poincare mapping method to this task. Instead of solving continuous equations of motion, they described the result of each ice collision as a discrete step.

This allowed for creation of a so-called «transfer mapping» — a rule according to which the state of the system (position and speed of the structure) changes to a new state after an impact. The developed model made it possible to describe all three classical modes of ice vibrations. The analysis showed that the mode-to-mode transition is governed by the competition of two factors: the time between ice impacts and the rate of energy dissipation. At low ice speeds, the fragments manage to leave the gap, and we observe steady periodic fluctuations of small amplitude.
As the speed increases to a critical one, the conditions in the gap change so that the system enters into resonance: the amplitude begins to rise sharply.

Finally, at even higher speeds, the interval between impacts becomes so short that the energy does not have time to dissipate. The system accumulates it, that leads to the failure of oscillations and the change to a chaotic, broadband mode.

Thus, the proposed model provides a theoretical basis for a more accurate assessment of the forces acting on the structure, depending on the speed of the ice, its strength and the properties of the mixture of water and ice in the destruction zone. This allows us to predict the boundaries of dangerous modes more accurately at the design stage. Knowing the parameters of their structure and the typical ice conditions in the installation area, the engineers can use the model to assess whether the structure will fall into a zone of resonance or chaos and, if necessary, to improve the design to shift the dangerous zones beyond the limits of the realistically possible ice drift speeds.