## Solution of coupled differential equations arising from imbalance problems

Jenny Niebsch, Ronny Ramlau, and Kirk M. Soodhalter

### Abstract

We investigate the efficient solution of a set of coupled ordinary differential equations arising from a model describing vibrations of a wind turbine induced by imbalances of its spinning blades. The Forward Problem (computing vibrations from imbalances) admits a coupled integral equation formulation. Each integral equation is solved over the same underlying Hilbert space $\mathbb{H}$. We observe that these coupled integral equations can be represented as one compact operator acting on the tensor product space $\mathbb{R}^{N}\otimes \mathbb{H}$, where $N$ is the number of coupled equations. A Galerkin discretization leads to a linear system of dimension $nN$ with corresponding Kronecker product structure, where $n$ is the number of basis elements used to discretize $\mathbb{H}$. Such systems can be solved using a variety of techniques which exploit the Kronecker structure. We demonstrate the effectiveness of exploiting the tensor structure with numerical experiments and show that our results agree with data recorded from actual wind turbines.

Full Text (PDF) [437 KB]

### Key words

mathematical modeling, couple differential equations, integral equation, Kronecker product, tensor product, Hilbert space

### AMS subject classifications

15A69, 34A55, 45D05, 65R20, 65R30

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