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An Aeroelastic Implementation for Yacht Sails and Rigs

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An Aeroelastic Implementation for Yacht Sails and Rigs An Aeroelastic Implementation for Yacht Sails and Rigs ARON HELMSTAD, TOMAS LARSSON Degree project in Naval Architecture Stockholm, Sweden 2013 ABSTRACT An aeroelastic model has been developed to determine the elastic response of sailing yachts’ rigs and sails due to the aerodynamic pressure induced on the sails by the wind. A vortex-lattice method (VLM) is used to determine the aerodynamic pressure. The structural response is calculated with a non-linear finite element method (FEM). In the aeroelastic model, pressures are calculated using the VLM module and applied to the finite element representation of the sail. Displacements are calculated and the new shape is used to determine a new pressure distribution. The procedure is iterated until the pressure distribution has converged. A comparison of a fixed and elastic rig is presented which shows the importance of using a fully aeroelastic model when studying the behavior of yacht sails. The VLM-implementation uses vortex ring elements with two different wake models: a fixed wake aligned with the free-stream and a force free wake. The VLM-implementation shows good correlation to commercial VLM-codes as well as with model tests. Results show minor differences between the fixed- and the free wake model. Three types of finite elements have been implemented to model the various parts of the rig: nonlinear bar elements for stays and boom, nonlinear triangular membrane elements for sails and linear beam elements for mast and spreaders. The Newton-Raphson method is used to solve the non-linear structural problem for displacements. The structural implementation show good correlation to other FEM-implementations and analytical solutions. SAMMANFATTNING En aeroelastisk modell har tagits fram för bestämning av strukturell respons hos segelbåtars segel och rigg till följd av det aerodynamiska tryck som verkar på seglen. En s.k. vortex-lattice method (VLM) används för bestämning av det aerodynamiska trycket. Den strukturella responsen beräknas med en ickelinjär finita element metod (FEM). Tryck beräknas, i den aeroelastiska modellen, och överförs till den finita elementa beskrivningen av seglen. Nodförskjutningarna beräknas och den nya formen på seglen används för att bestämma en ny tryckfördelning. Proceduren itereras till tryckfördelningen konvergerat. En jämförelse mellan stel och elastisk rigg visar på betydelsen av att använda en fullt aeroelastisk modell vid analys av segel. VLM har implementerats med ringvirvelelement och två olika vakmodeller: en fix vak i friströmmens riktning och en kraftlös vak. Impelmentationen av VLM visar god överensstämmelse med såväl kommersiella VLM-program som med modelltester. Jämförelser visar på små skillnader mellan den fixa och den fria vakmodellen. Tre typer av finita elements har implementerats för att beskriva riggens delar: ickelinjära stångelement för stag samt bom, ickelinjära triangulära membranelement för segel och linjära balkelement för mast samt spridare. Newton-Raphsons metod används för att lösa det ickelinjära strukturproblemet med avseende på nodförskjutningar. Den strukturella implementationen visar på god överensstämmelse med andra FEM-implementeringar och med analytiska lösningar. NOMENCLATURE FLUID DYNAMIC PART ∆t time step A area c chord length D drag force Fx thrust force Fy heeling force L lift force M total number of panels N total number of vortex elements n normal vector Nz number of panels and vortex elements in span direction p pressure q velocity vector Q∞ free-stream velocity vector r location vector rb bound vortex vector Rcore fictive vortex core size α angle of attack ϕ potential function Г circulation or vortex strength Counters i panel counter j influencing vortex element counter STRUCTURAL PART A area C Cauchy-Green deformation tensor E Young’s modulus Green-Lagrange deformation tensor EGL f external node force vector internal node force vector fint external node force fx,, f y f z unit matrix of size 2-by2- respective 3-by-3 II23, Jacobi transformation matrix JJ, 0 K global tangent stiffness matrix linear stiffness matrix K 0 initial displacement stiffness matrix Ku geometric stiffness matrix K g l length unstrained length l0 n normal vector residual between external and internal forces Ri S second Piola-Kirchhoff stress tensor prescribed stress S0 T axial force t thickness u node displacement vector u nodal displacement in x-direction x vector of node coordinates v nodal displacement in y-direction w nodal displacement in z-direction strain in general engineering strain E Green’s strain G angle Superscripts e element e1, e 2, e 3... element number Subscripts 0 initial configuration, initial guess 1,2, 3... node number 21,32, 43... node difference f free degrees of freedom x,, y z direction CONTENT 1 Introduction ........................................................................................................................................ 1 2 Fluid dynamic model .......................................................................................................................... 2 2.1 Potential flow theory ............................................................................................................... 3 2.2 Implementation ........................................................................................................................ 6 2.3 Verification and validation .................................................................................................... 16 3 Structural model ............................................................................................................................... 20 3.1 Simple example of nonlinear problem .................................................................................. 21 3.2 Bar element formulation ........................................................................................................ 22 3.3 Membrane formulation .......................................................................................................... 25 3.4 The Newton-Raphson method ............................................................................................... 28 4 Aeroelastic model ............................................................................................................................. 30 4.1 Aeroelastic implementation ................................................................................................... 30 4.2 Comparison of stiff and elastic rig ........................................................................................ 32 5 Discussion ........................................................................................................................................ 35 References ............................................................................................................................................. 36 Appendices .......................................................................................................................................... A-1 A. Positioning of vortex ring elements ..................................................................................... A-1 B. VLM validation 1: Bertin & Smith wing ............................................................................ B-1 C. VLM validation 2: Rectangular flat plate ............................................................................ C-1 D. VLM validation 3: 35° swept wing ..................................................................................... D-1 E. VLM validation 4: Warren 12 wing ..................................................................................... E-1 F. Nonlinear problem with one degree of freedom ................................................................... F-1 G. FEM validation 1:Three hanging bars ................................................................................. G-1 H. FEM validation 2:The inextensible catenary ....................................................................... H-1 I. FEM validation 3: Quadratic membrane subjected to point load .......................................... I-1 J. FEM validation 4: Rectangular membrane subjected to pressure ........................................ J-1 K. FEM validation 5: Half-cylindrical membrane ................................................................... K-1 1 INTRODUCTION Performance prediction is an important part in the design process of sailing yachts. Velocity prediction programs (VPP) are often used for this purpose in which the hydro- and aerodynamic forces are calculated and balanced. The aerodynamic forces are often calculated by semi-empirical methods, based on experimental data. A more general and favorable approach would be to evaluate the forces for the actual sail shape. Furthermore, sails and rigs are elastic and will therefore deform when subjected to the aerodynamic loads induced by the wind. Consequently the sail will alter its shape when loaded, to obtain the so called flying shape. This knowledge is essential for the sail maker, as well as for predicting the performance. To determine the flying shape is far from trivial. The reason for this is that the aerodynamic forces on the sail are a function of the shape of the sails which means that the fluid and structure problem must be solved simultaneously. This is generally referred to as fluid-structure interaction (FSI) or aeroelasticity. This thesis describes the implementation of a FSI-tool that is used to model the aeroelastic behavior of yacht sails and rigs
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