HYDROELASTIC TAILORING AND OPTIMIZATION OF A COMPOSITE MARINE PROPELLER

Transcription

1 HYDROELASTIC TAILORING AND OPTIMIZATION OF A COMPOSITE MARINE PROPELLER José P. Blasques, Christian Berggreen and Poul Andersen Department o Mechanical Engineering, Technical University o Denmark Nils Koppels Allé, Building 403, DK-2800 Kgs. Lyngby, Denmark jpb@mek.dtu.dk ABSTRACT The ollowing paper deals with the design and optimization o a lexible composite marine propeller. The blade shape is obtained rom an existing high skew metal propeller. The aim is to tailor the laminate to control the elastic couplings and thereore the deormed shape o the blade. The development o a hydroelastic code is described irst where the inite element method and boundary element method are used or the structural and hydrodynamic sub-models, respectively. The equilibrium between the elastic and hydrodynamic orces is obtained by direct substitution. NOMADm, a mesh adaptive direct search ilter algorithm, is used together with DACE, a surace itting algorithm, to determine the optimal laminate lay-up and blade pitch angle. The optimal conigurations which reduce the uel consumption or the combination o two load cases are ound. The strength requirements are then analyzed using the Tsai- Wu ailure criteria. The results show that it is possible to design a lexible composite marine propeller that will enable a reduction o the uel consumption while withstanding the imposed loads. 1. INTRODUCTION Anisotropic composite materials present dierent levels o elastic couplings which depend on the laminate lay-up. Among these, bending- and extension-twist couplings have been identiied as a possible mean or the passive control o composite structures [1-,2-]. The aim here is to investigate the possibility o tailoring these elastic couplings to control the perormance o a lexible composite marine propeller. The shape o the propeller blade will morph under load thus passively adjusting itsel to the pressure loads o the dierent operational conditions. The optimal composite propeller is the one which most reduces the uel consumption. The use o elastic couplings to control the passive properties o dierent structures like aircrat wings, helicopter rotor blades and wind turbine blades has been extensively reported. The literature concerning the application o composite materials in the design o marine propellers is however limited. Lee et al. [3-] has optimized the laminate layup o a ixed pitch m diameter composite marine propeller using genetic algorithms. The inal design is shown to reduce the torque at one o the load conditions. An evaluation o the strength properties o a composite propeller blade with 1.4 m diameter has been presented by Lin et al. [4-]. The strength is analysed using the Hashin ailure criterion and the results show the inluence o the laminate type on the stress ield. Marsh [5-] has presented an overview o the developments in the composite propeller industry. A ew companies have realized the potential and claim to have designed composite marine propellers with passive properties. Nonetheless, no details o this work have yet been published. The work described in this paper tries to extend the knowledge available in the ield o design and optimization o composite marine propellers, and especially with the ocus on larger marine propellers or merchant or naval vessels. The propeller blade considered throughout this paper is approximately our times larger than those described earlier in literature. The result will hopeully shed some light on the

2 possibility and potential o using composite propeller blades in large merchant or naval vessels. The hydrodynamic model is based on the boundary element method instead o the liting line method applied in earlier papers. Moreover, the optimization o the ibre orientations is combined with the strength analysis in one design process. The two analyses have earlier been treated separately. 2. BLADE GEOMETRY AND MATERIAL PROPERTIES The original propeller blade is part o a controllable pitch propeller installed on a military vessel. The propeller has 4 blades, a diameter o 4.4 meter and an expanded area ratio o Its main geometrical properties are presented in Table 1. r/r θ skew [o] Rake/R P/D c/d t/c /c Table 1 - Geometrical properties o the propeller blade. These types o ships typically operate at two speed conditions cruising and maximum speed which correspond to very distinct operational conditions (Table 2). The nondimensional thrust and torque coeicients K T and K Q are deined as, (1) where T, Q, ρ, n and D are the thrust, torque, water density, rotational speed and propeller diameter, respectively. No. Eng. Power n V s J P/D 07 K T K Q η 0 FC - [kw] [r.p.s.] [m/s] [-] [-] [-] [-] [-] [Kg/h] Cruising Speed Maximum Speed Table 2 Data or the cruising and maximum speed operation conditions. The composite marine propeller was designed using carbon/epoxy uni-directional reinorcements. The mechanical properties are presented in Table METHODOLOGY The initial step o the analysis o the composite propeller consisted o the development o the structural and hydrodynamic sub-models. These models were then combined in the hydroelastic model which, or a steady incoming low, determines the equilibrium between the structural and hydrodynamic orces. Once the hydroelastic model is set up it is possible to analyze the inluence o the laminate lay-up on the propeller perormance. An optimization model was then used to determine the laminate lay-up which will improve the propeller perormance. The strength o the most optimal conigurations was then studied using the Tsai-Wu strength index.

3 Symbol Units Std CF Young s Modulus E 1 GPa UD Stiness Modulus E 1 GPa 135 E 2 GPa 15 E 3 GPa 15 Shear Modulus G 12 GPa 5 G 13 GPa 5.3 G 23 GPa 2.9 Poisson s Ratio ν ν ν Ult. Tensile Strength σ 1,t MPa 1500 σ 2,t MPa 50 σ 3,t MPa 50 Ult. Comp. Strength σ 1,c MPa 1200 σ 2,c MPa 250 σ 3,c MPa 70 Ult. Shear Strength σ 12 MPa 70 σ 13 MPa 64 Density σ 23 ρ MPa kg/m Thickness t mm 0.3 Table 3 Mechanical properties o Carbon/Epoxy uni-directional reinorcement [6-] Hydroelastic model The displacement ield is determined using the inite element method in the structural sub-model, and the hydrodynamic pressure ield is likewise determined using the boundary element method in the hydrodynamic sub-model. The equilibrium between the hydrodynamic and structural parts is presented below. Thus, the combined analysis consists o determining the displacement vector {d} which satisies [K]{d}={}. Since {} is a unction o {d} the problem is nonlinear. Furthermore, {} is only dependent o the state {d} which makes the problem suited or iterative solution. The direct substitution method has been chosen and its implementation is described next. w i-1 w i 3 w i u i-1 1 u i u i+1 Figure 1 - Algorithm or the computation o the hydroelastic steady state. In Figure 1 the subscript i indicates the iteration number, u is the displacements ield determined by the structural model and w is the pressure ield given by the hydrodynamic model. Each o the numbered steps is described below: 1) The displacement ield is determined by the structural model or a given pressure distribution. At the irst iteration the pressure distribution is taken rom the original blade; 2) The hydrodynamic model mesh is updated based on the displacement ield determined in 1).

4 3) A new pressure ield is determined by the hydrodynamic model based on the new mesh. 4) The new pressure ield given in the hydrodynamic model mesh is mapped into the structural model. The process is repeated until convergence is achieved, i.e., equilibrium is ound. The thrust and torque are calculated at each iteration and the method converges when the relative dierence between iterations o the thrust and torque is less than 1%. It can be shown that the method is stable (see [7-]) and thus suited or the problem at hand. The displacement and pressure ields have been mapped based on the closest point and by linear interpolation, respectively Structural sub-model The structural sub-model was developed in the commercial inite element package ANSYS (version 11) and used to determine the displacement ield as a result o a given pressure distribution. In the irst approach, the blade geometry was approximated by the mid-thickness lines o each section. A surace was subsequently deined and meshed with 8 node parabolic layered shell elements (SHELL99). The shape o the blade was obtained by setting the thickness o the elements at each node equal to that o the blade at the corresponding position. Since it was not possible to validate the results against experimental data, a second inite element model was developed ollowing a dierent approach. The shape o the blade was deined by its volume and meshed using 20 node parabolic layered solid elements (SOLID191). A mesh convergence test on the displacements and stresses was conducted or each o the models. Figure 2 (a) Mesh used or displacement analysis (let). (b) Mesh used or stress analysis (right). The two models were compared in terms o their ability to predict the inluence o the laminate lay-up on the thrust and torque orces (see [7-]). The results were ound to be in good agreement. As shall be seen later, the displacements are the only necessary result rom the structural model used in the optimization model. Hence, the irst model using shell elements was chosen because it was less computationally expensive. When comparing the stress ields, there were some discrepancies between the predicted magnitudes o the stress. Generally, the stress ield calculated by the second model should be more accurate than the shell model assuming mesh convergence or the stresses has been attained. However, or practical reasons concerning the element-wise orientation o the laminates, the shell model was used.

5 A total o 1440 and 5760 elements are required or the analyses o the displacement and stress ields, respectively (Figure 2). The model is constrained at the blade root to simulate the attachment to the blade hub. The centriugal orces are applied as accelerations depending on the rotational speed. The number o layers is assumed constant and ixed at 32 throughout the blade and all layers are assumed to have the same thickness. The total thickness o the laminate is equal to that o its metal counterpart blade at the corresponding position. The layer angles may be deined element-wise Hydrodynamic sub-model The hydrodynamic sub-model was used to determine the pressure ield or a given blade shape. The inlow to the propeller is assumed to be uniorm and steady, and viscous orces have not been accounted or. A total o 480 section points deine the 450 elements which are arranged in a 1515 three dimensional structured mesh (Figure 3) laid over the surace o the blade. The pressure is determined at 450 nodes placed at the center o each element whose corners are deined by the section points (which deine the section lines). The pressure is integrated over the hydrodynamic mesh to determine the values o thrust and torque. Figure 3 - Mesh o hydrodynamic sub-model (pressure side) Optimization Model In designing a marine propeller the speed at which the ship must sail is typically given and hence the thrust the propeller must generate. Thus, the optimal propeller will produce the desired thrust and minimize the torque, reduce the risk o cavitation, withstand the pressure orces and satisy noise and vibration constraints. In this investigation the cavitation and vibration constraints have been neglected. The strength constraints have not been included in the optimization model, but have instead been studied separately. The optimization problem was thereore ormally stated as: min,, : Δ..2..., Δ..2..., (2),, 10,, ,,,,

6 where the subscript C, M and O indicate cruising, maximum and original (or required), respectively. The torque and thrust correspond to Q C, Q M, T C and T M, respectively. The design variables are the ply angles, θ, and the blade pitch angles ϕ C and ϕ M. Finally, Δt is the percentage o time the ship operates at each o the orward speed conditions, SFOC is the speciic uel oil consumption and n is the propeller rotational speed. Two operational conditions were considered, namely, cruising and maximum speed. The objective unction is the combined uel consumption. It was assumed that the ship operates 90% o the time at cruising speed and 10% at maximum speed, i.e., Δt C =0.9 and Δt M =0.1. The speciic uel consumptions or each o the operation conditions were estimated to be SFOC C =0.190 Kg/KWh and SFOC M =0.540 Kg/KWh. The design variables are assumed to vary continuously. Since the propeller is assumed to have controllable pitch, the pitch angles can be adjusted or each operational condition independently. The two thrust values are constrained rom below, i.e., it is assumed that they may not be lower than the required thrust values. Note that inequality constraints were used instead o equality constraints on the thrust. This relaxation is possible due to act that the thrust and torque are coupled in their variations. That is, an increase (or decrease) in the thrust will be accompanied by an increase (or decrease) in the torque Optimization algorithm The optimization problem was solved using NOMADm; a derivative ree, patternsearch algorithm [8-]. This algorithm is specially suited or the solution o nonlinear optimization problems where the gradients o the objective unction and constraints C are not available. The decision to opt or a gradient ree method was motivated by the low accuracy o the presented hydroelastic model. With the lack o analytical sensitivities, the use o inite dierences or the evaluation o the gradients was compromised by the low accuracy o and C. The eect o the perturbations in the design variables was impossible to measure as these could not be distinguished rom the variations due to the equilibrium iterations. The NOMADm algorithm is essentially composed o a search and poll step. There are several dierent strategies or the search step. In this case, surrogate models o and C have been used based on the DACE package. Thus, the initial step o the algorithm consisted on the evaluation o the computationally expensive unctions and C using a Latin hypercube sampling scheme. The aim is to generate a relatively large number o data sites that are equally scattered throughout the easible region. A DACE model is subsequently itted to the data to orm an inexpensive surrogate. The MATLAB unction mincon is then used to generate a local minimizer o the surrogate where the unctions and C are consequently evaluated. Any new data site generated throughout is used to calibrate the surrogate. The poll step is invoked i the search step ails to ind an improved point in the mesh. This step consists o evaluations o and C in the mesh points around the incumbent solution Strength analysis The strength o the blade is analyzed using the Tsai-Wu strength index (TWSI) as given in Equation (3). An index above one indicates material ailure whereas an index below one indicates no ailure has occurred.

7 (3) RESULTS Beore proceeding with the optimization o the laminate lay-up several simple tests were conducted. The aim was to establish the inluence o the iber angle on the blade perormance and strength. The study was conducted assuming the blade to be laminated with one single laminate using one single angle or all plies, i.e., [θ] 32 over the entire blade. The ply reerence axis is presented in Figure 4. Figure 4 - Ply angle orientation reerence axis The angle was then rotated between 0 o and 180 o and the variation o the uel consumption, thrust and torque were measured or both load cases (Figure 5). The results are shown as a ratio between the measured values and that o the original blade. The main changes in geometrical properties were measured in the radial distributions o pitch, camber and mid-chord positions. The pitch o the sections tends to be equal or lower than the original, while the camber tends to be larger in sections close to the hub and lower close to the tip. As expected, or both cases the blade has deormed in the upstream direction. A ply angle, θ, in the range between 0 o and 40 o will result in a lexible tip and sti lower part while a ply angle between 60 o and 120 o will reinorce the tip while making the lower part o the blade closer to hub more lexible. Figure 5 - Inluence o the lamina angle on the thrust, torque and combined uel consumption, assuming original pitch angles.

8 The curves in Figure 5 were aected by the blade pitch angle which can be used to compensate or the increase or decrease in the thrust orces. Note that these have to remain equal to those o the original propeller blade. Regarding the strength requirements, the critical areas were identiied as the trailing edge and hub areas. These areas were thereore locally reinorced (Figure 7 (b)) where area 1, 2, 3 and 4 were laminated using [-45 o ] 32, [0 o ] 32, [0 o ] 32 and [90 o ] 32. In the optimization model the angles in these areas is not varied but kept constant at the values speciied Optimization o laminate lay-up In the irst approach it was assumed that the design variables were the orientation o the dierent layers through the thickness o the laminate. Thus, the ollowing laminate conigurations were tested: [θ 1 ] 32, [(θ 1 ) 8, (θ 2 ) 8, (θ 3 ) 8, (θ 4 ) 8 ] and [(θ 1 ) 8, (θ 2 ) 8 ] s. All conigurations converged to approximately the same results, i.e., to the laminate [40 ο ] 32. The resulting values o uel consumption reduction and blade pitch angles are presented in Table 4. At maximum speed the thrust and torque are exactly equal to that o the original blade, however, at cruising speed a 4.2% reduction in the uel consumption (corresponding to 1.12% in the combined case) was achieved. These results indicate that the variation o the ply angles through the thickness has little inluence on the blade response. Thus, the second approach consisted in deining the ply angles at each element as a unction o its position in space. The approach is inspired by the results rom Parnas et al. [9-] where cubic iber paths and Bezier curves are used to optimize the iber angles and thickness o a composite structure. The determination o the angle is as illustrated in Figure 6 (a). The ply angle at an element in position A is equal to the angle θ e. This angle is determined by the tangent to a second degree polynomial deined by the points c 0, c 1 and c 3. The design variables in this case are the measures y 1 and y 2. The point c 0 and the coordinates z 1 and z 2 are kept ixed. Figure 6 (a) Deinition o the second degree curved iber path (let). (b) Resulting iber angles obtained using curved iber paths (right). The ollowing laminates were tested [θ 1 (y 1,y 2 )] 32, [(θ 1 (y 1 1,y 2 1 )) 16, (θ 2 (y 1 2,y 2 2 )) 16 ], [(θ 1 (y 1 1,y 2 1 )) 8, (θ 2 (y 1 2,y 2 2 )) 8, (θ 3 (y 1 3,y 2 3 )) 8, (θ 4 (y 1 4,y 2 4 )) 8 ]. As beore the results were not sensitive to variations o the angles through the thickness. The most optimal laminate was [θ 1 (0.756,0.677)] 32. The resulting iber angles are presented schematically in Figure 6 (b). This approach resulted in a urther decrease o the uel consumption as presented in Table 4.

9 It is interesting to note that the two design conigurations presented above have the act that they promote a lexible tip in common. In Figure 7 (a) the displacement sum or the curved iber path coniguration is plotted, and it is clear the eect o a lexible tip. Table 4 - Results or the two optimal laminate conigurations. Figure 7 (a) Displacement plot or the optimal composite blade with curved iber paths (let). (b) Maximum TWSI over all layers or the optimal propeller blade with curved iber path (right) Strength analysis The last step in the design process consisted o the analysis o the stress ield in terms o the TWSI (Figure 7 (b)). The local reinorcements proved eective. The maximum TWSI measured or both laminate conigurations was around 0.7 indicating that no ailure has occurred. Although this value might correspond to an unacceptably low saety actor, it indicates that with iner tailoring o the laminates it is possible to build a lexible composite marine propeller blade which withstands the subjected loads. 5. CONCLUSIONS A ramework or the design and optimization o composite marine propellers has been presented. The laminate lay-up o a composite propeller blade with high skew was optimized or two dierent load conditions cruising and maximum speed. Two dierent approaches have been explored, where the irst assumes constant ibre angles throughout the blade and the second uses curved ibre paths. The optimal lay-ups have enabled a reduction o the combined uel consumption around 1.2% corresponding to a decrease o 4.5% or the cruising condition and no reduction or the maximum speed condition. The largest variations in the blade shape were observed in the radial distributions o pitch, camber and mid-chord positions. The optimal conigurations promoted a lexible tip and a sti body close to the hub. The strength o the blade was analysed in terms o the Tsai-Wu strength index and the leading edge and hub regions were identiied as the critical areas. The blade was thereore reinorced in these regions prior to the optimization procedure. It is shown that the maximum ailure index is around 0.7 or both optimal conigurations. The results indicate that it is possible to

10 build a lexible composite marine propeller which outperorms its original metal counterpart while resisting the imposed loads. ACKNOWLEDGEMENTS This work has been perormed within the Network o Excellence on Marine Structures (MARSTRUCT) and has been partially unded by the European Union through the Growth Programme under contract TNE3-CT The invaluable support and advice given by Associate Proessor Mathias Stolpe, Department o Mathematics, Technical University o Denmark, is grateully acknowledged. REFERENCES 1- Berring P., Branner K., Berggreen C., and Knudsen H. W., Torsional Perormance o Wind Turbine Blades Part I: Experimental Investigation, 16 th International Conerence on Composite Materials (ICCM-16), Kyoto, Japan, July 8-13, Branner K., Berring P., Berggreen C., and Knudsen H. W., Torsional Perormance o Wind Turbine Blades Part II: Numerical Veriication, 16 th International Conerence on Composite Materials (ICCM-16), Kyoto, Japan, July 8-13, Lee Y.J. and Lin C.C., Optimized design o composite propellers, Mechanics o advanced materials and structures, 2004;11: Lin H.J., Lin J.J., Chuang, T.J., Strength evaluation o a composite marine propeller blade, Journal o Reinorced Plastics and Composites, Vol.24, 2004;17: Marsh G., A new start or marine propellers?, Reinorced Plastics, December, Peters S.T., Handbook o composites, Chapman and Hall, 2 nd Edition, Blasques J.P., Composite materials marine propeller, Master thesis, Department o Mechanical Engineering, Technical University o Denmark, Booker A.J., Dennis J.E., Frank P.D., Seraini D.B., Torczon V., Trosset M.W., A rigorous ramework or the optimization o expensive unctions by surrogates, Technical Report CRPC-TR98739-S, Center or Research on Parallel Computation, Rice University, Parnas L., Oral S. and Ceyhan Ü., Optimum design o composite structures with curved iber paths, Composites Science and Technology, 63, 2003,