Flexible multibody dynamics using joint coordinates and the Rayleigh‐Ritz approximation: The general framework behind and beyond Flex

Abstract

This article presents a general framework to set up the equations of motion for flexible multibody system using joint coordinates and Rayleigh-Ritz approximation. The framework covers in particular the underlying formulation behind the aeroelastic code Flex developed by Øye. This formulation is strongly related to the one used in the module ElastoDyn of the code OpenFAST.The approach results in models with few degrees of freedom thanks to carefully selected shape functions and the direct account of the constraints. The specificities of the method for open-loop systems and thin beams are presented. The inclusion of torsion and couplings with finite element methods or superelements are discussed. Simple validations cases are used. A four degrees of freedom model of a two-bladed wind turbine is developed to illustrate the method. INTRODUCTIONThe conventional approach for the simulation of flexible multibody systems consists in discretizing the bodies in finite elements and solving the constraints equation together with the equations of motion. The discretization and the account of the constraints usually result in a large number of degrees of freedom (DOF). This approach is for instance used by the aeroelastic code Hawc2 1 for the simulation of wind turbines. Flex 2 is another widespread tool in the wind energy community, which uses a different formulation that results in a very limited number of DOF. The general framework leading to this formulation is the topic of this article. The ElastoDyn module of the aeroelastic code FAST (now OpenFAST) 3,4 uses a similar formulation, in particular, the early versions of both codes used the same DOFs to represent a wind turbine. The minor differences between the two will be highlighted in this article: ElastoDyn relies on Kane's method instead of the velocity transformation matrix, and ElastoDyn accounts directly for the radial shortening of beams.Flex was originally written by Øye and has been further developed by different wind turbine industries. The equations of motion are set using two main methods : the joint coordinate approach and the Rayleigh-Ritz (RR) approximation. The joint coordinate method was described for rigid bodies by Nikravesh 5 and flexible bodies by Book. 6 In this method, a minimal set of DOF is obtained by choosing as coordinates the ones necessary to describe the joint motions. The constraints equation are then automatically satisfied. The RR approximation, or assumed shape function method, consists in projecting the infinite number of DOF necessary to describe a flexible body onto a reduced and converging set of shape functions. The method is well documented in the book of Shabana. 7The current article relies on the mentioned references and the available documentations of Flex. 2,8 The general framework of the method is described in section 2 to ease the understanding of the specialized implementation of Flex. The recursive formulations that are obtained for open-loop systems are presented next. The possibility to couple the equations with finite ele...

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