Abstract
The article presents a symbolic framework that is used to obtain the linear and non-linear equations of motion of a multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitate the creation and manipulation of models with various levels of fidelity. The symbolic treatment provides analytical gradients and linearized equations of motion. The linear and non-linear equations can be exported to Python code or dedicated software. The application are multiple such as: time-domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, digital twins, etc. In this paper, we describe the method we used to systematically generate the equations of motion of multibody systems. We apply the framework to generate illustrative onshore and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. A Python implementation is provided as an opensource project.
Highlights
The generation of wind turbine digital technologies requires versatile aero-servo-hydro-elastic models, with various levels of fidelity, suitable for a wide range of applications
Tools with linearization capabilities, such as hawc[20] stab[2] (Sønderby and Hansen, 2014) or OpenFAST (OpenFAST, 2021) are dedicated to horizontal axis wind turbines and the 21 evaluation of the gradients are limited to hard-coded analytical expressions or numerical finite-differences
VP = vi + ωi × sP + (u P )i aP = ai + ωi × + ωi × sP + 2ωi × (u P )i +i where: ri, vi and ai are the position, velocity and acceleration of the origin of the body; sP0 is the initial position vector of point P with respect to the body origin; uP is the elastic displacement of the point (0 for rigid bodies); ωi is the rotational velocity of the body with respect to the inertial frame; ( ̇) and ( ̇)i refer to time derivatives in the inertial and body frame respectively
Summary
The generation of wind turbine digital technologies requires versatile aero-servo-hydro-elastic models, with various levels of fidelity, suitable for a wide range of applications. Current mod[17] els are implemented for a specific purpose and usually based on an heuristic structure Aeroelastic tools, such as Flex (Øye, 1983; Branlard, 2019) or ElastoDyn (OpenFAST, 2021), rely on: an assumed chain of connections between bodies, a given set of degrees of freedom, and predefined orientations of shape functions. Small implemen[22] tation changes often require an extensive redevelopment, and the range of applications of the tools remain limited (Simani, 23 2015) To address this issue, we propose a framework for the automatic derivation, processing and parametrization of models with granularity in the level of fidelity.
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