Stability Analysis of a Rotating Cantilever Beam inside a Geotechnical Centrifuge

Abstract

A geotechnical centrifuge is used to conduct tests on small-scale models to obtain data on the response of real-life structures. In the field of earthquake engineering, centrifuges can be used to model soil-structure interaction and to analyse structural response to earthquakes through scaled models. By rotating the centrifuge at high angular velocities, the model inside the centrifuge can experience the stresses corresponding to a full-scale prototype. Through simplified scaling laws, the dimensions of the small-scale model are estimated, and the results from these tests can be extrapolated to study the behaviour of the prototype.

When the small-scale model of a dynamically sensitive structure, such as an offshore wind turbine, is subjected to high angular velocities inside a centrifuge, it is essential that the model remains stable to obtain meaningful results from the test. Instabilities, such as divergence or flutter, may arise in the small-scale model depending on the system’s parameters due to the action of pseudo-forces and coupling between the displacement fields, which may lead to the failure of the model. Moreover, it is unclear whether the simplified scaling laws can still be applied for such slender small-scale models due to the action of these pseudo-forces and the coupling terms. Therefore, understanding the behaviour of these dynamically sensitive small-scale models under centrifuge conditions is important to ensure their stability and to obtain conclusive results about the prototype.

This research represents the small-scale model of a monopile-founded offshore wind turbine inside a centrifuge as a homogeneous Rayleigh cantilever beam rotating about a vertical axis, with each point along the beam consisting of three translational displacements. A mathematical model is formulated using Lagrangian formalism that incorporates all relevant pseudo-forces and coupling of displacement fields that may affect the stability of the small-scale model. Euler-Lagrange equations are then applied to formulate the governing equations of motion, and a dimensionless form of the equations is presented to draw general conclusions. Moreover, the effect of soil-structure interaction is considered using a lumped spring model.

Subsequently, an eigenvalue analysis is performed using analytical and numerical approaches. The numerical approach is more robust and is applied to calculate the eigenproperties of the system, although it requires a good initial guess. The technique of representing the response as a summation of assumed modes is explored in the analytical approach, and a convergence study is conducted. Afterwards, a parametric study is performed to identify the factors influencing stability. The angular velocity at which the onset of instability happens is also determined. Finally, the findings are applied to a case study of a small-scale model of an offshore wind turbine tested inside the centrifuge facility at ETH Zürich as part of the research project, DONISIS, led by TU Delft.

The results of this study demonstrate that, for the small-scale beam model of a monopile-founded offshore wind turbine tested inside a centrifuge, the dominant instability mode is the divergence of chordwise lateral bending. This instability arises from the axial compression induced by the centrifugal forces acting along the beam. The coupling of displacement fields through Coriolis forces does not affect the onset of instability, although its influence on the eigenproperties becomes apparent only at angular velocities far exceeding the practical operating range of a centrifuge. Since the Coriolis terms have no significant impact on system behaviour, their effect on scaling laws can be regarded as negligible. Consequently, the existing scaling laws remain applicable to small-scale models of rotating beams within the practically operating range of angular velocities of the centrifuge.