Multidisciplinary Design Optimization of a Heavy-Lift Helicopter Rotor in Hover

Transcription

1 Multidisciplinary Design Optimization of a Heavy-Lift Helicopter Rotor in Hover by Smith Thepvongs (Structural Optimization) Robert Wohlgemuth (Aerodynamic Optimization) Professor Panos Y. Papalambros ME 555: Design Optimization University of Michigan Abstract The multidisciplinary design of a heavy-lift helicopter rotor in hover is optimized from aerodynamic and structural perspectives. Each optimization discipline is considered separately and then combined as a unified optimization problem. The aerodynamic optimization finds values of blade length rotor rotation speed root chord chord taper ratio collective pitch and linear twist rate to minimize the power required to hover for a prescribed thrust. The structural optimization considers the blade length root chord chord taper ratio twist rate and wall thickness of the blade structure idealized by a thin-walled box-beam. The configuration for minimum rotor weight is determined while subecting the structure to subect to strength and static/dynamic requirements. The combined optimization minimizes the power required to hover with the rotor weight and blade loading distributions passed between the individual structural and aerodynamic optimizations. The optimum heavy-lift rotor blade design is presented.

2 Table of Contents 1.0 Introduction Nomenclature Aerodynamic Subsystem Mathematical Model Model Analysis Optimization Study Parametric Study Discussion Structure Subsystem Mathematical Model Optimization Study Parametric Study Subsystem Integration Optimization Discussion Conclusion Acknowledgements References 53 Appendix 54 1

3 1.0 Introduction A heavy-lift helicopter has far-reaching applications that benefit society in a number of ways military medical and construction. The effectiveness of such helicopters is directly determined from the flight duration which is a function of the fuel efficiency. Fuel efficiency is related to power that the engine(s) must provide to keep the helicopter in the air. The lower the power required to overcome the induced and profile drag the more fuel efficient the helicopter becomes and the more cost effective and beneficial the helicopter is to the customer. Heavy-lift helicopters spend a very large percentage of their flight time in the hover regime so it is therefore desired by the designer to minimize the power required to hover. Many variables and parameters of the rotor and rotor blades determine the aerodynamic performance of the helicopter. In addition to aerodynamics the rotor blades are flexible structures that add weight to the helicopter and have static and dynamic requirements for safe flight. In any helicopter minimizing weight is also a primary consideration for improving performance so it is also desired to minimize the weight of the rotor in its design. Unfortunately the optimal structural design for minimum weight will most likely not correlate to the optimal aerodynamic design for minimum power consumption. The optimization of a heavy-lift helicopter rotor will therefore consist of an aerodynamic part to design the rotor and blades to minimize the power required to hover subect to a prescribed thrust as well as a structural part to design the blades for minimum weight subect to structural integrity and frequency/deflection requirements. After analyzing the optimal rotor design from the aerodynamic and structural viewpoints the optimization will be combined with the final obective to again minimize the power required to hover. With the combined 2

4 optimization the rotor weight will become a variable in the required thrust such that the aerodynamic and structural optimizations become closely linked. In addition the load distribution rotor speed and pitch angle along the blade as determined from the aerodynamic code will be directly used by the structural analysis code. 2.0 Nomenclature a c c g c i c tip c root h i h tip h root l m g m i m θi n r t t i w i w tip speed of sound chord of blade transformation parameter for Gaussian Quadrature average chord of element i tip/root chord beam width average height of element i tip root beam height blade length or disk radius transformation parameter for Gaussian Quadrature average mass per unit length of element i average rotational inertia per unit length of element i about x number of finite-element nodes dimensionless radial position (x/l) wall thickness ith quad point for Gaussian Quadrature ith weight for Gaussian Quadrature tip vertical deflection 3

5 x x i x 1 x 2 A i C do C l C lα C P C po C T E F x G I xi I yi J i L distance from hub through blade cross section coordinate of element centers along blade length limits of integration in Gaussian Quadrature cross-section area at element i profile drag coefficient sectional lift coefficient lift curve slope power coefficient profile power coefficient thrust coefficient Young s moduls internal axial force at node Shear modulus cross-sectional area moment of inertias at element i load at node M M tip Mach number Mach number at tip of blade M 2 M 3 M x M y N b P T TR W internal flap and lag moments at node aligned with inertial frame internal moments at node aligned with element principal axes number of blades power thrust chord taper ratio (tip chord/root chord) blade weight 4

6 α λ ρ ρ a σ σ + x σ x σ y θ o θ tip θ tw ω x ω y ω z Ω blade angle of attack rotor inflow ratio material density air density rotor solidity ratio maximum stress in cross section in elements adacent to node yield stress collective pitch angle (at blade root) geometric twist at tip of blade blade twist over length torsional flap and lag natural frequency rotor rotation speed 3.0 Aerodynamic Subsystem The goal of the aerodynamic optimization is to minimize the power required to hover for a heavy-lift helicopter for a prescribed thrust. Helicopters spend a significant percentage of their flight time in the hover regime especially for heavy-lift helicopters which are made for purposes such as search and rescue construction and military transport. It is desired to make the helicopter as efficient as possible in this regime to extend the mission capabilities of helicopter and improve fuel efficiency. The factors of the rotor that effect its power consumption are the prescribed thrust the number of blades the rotor blade rotation speed the rotor disk radius the blade airfoil properties the collective pitch angle the geometric twist of the rotor blade the root chord of the blades and the blade taper. It is chosen for analysis and manufacturing simplicity to 5

7 assume a linear twist and chord distribution. For any helicopter rotor the rotation speed and disk radius are closely linked through many different perspectives such as the necessity to produce enough airspeed over the blade sections to lift the helicopter and consideration of the power required to overcome the drag related to the generation of lift which is related to the disk loading. The rotor rotation speed and radius must also be carefully chosen to avoid the drag rise associated with transonic flow. The geometric twist and chord taper effect the distribution of lift and drag over the rotor blade which also determines the power induced by the rotor during the creation of lift. Care must be taken is selecting a twist rate such that the rotor blades have a margin to stall at the prescribed maximum thrust for emergency maneuvers. The optimization variables must also be chosen such that the thrust requirement is satisfied. It can be seen that the aerodynamic optimization of a helicopter rotor involves many variables that must be determined in unison to maximize the performance of the helicopter. 3.1 Mathematical Model The aerodynamic analysis presented is taken from blade element theory (BET) and involves the summation of the sectional airloads along the blade to determine the overall thrust and power on the rotor (Leishman 2000). Obective Function The obective of the aerodynamic optimization is to minimize the power required by the heavy-lift helicopter to hover. This can be expressed mathematically as the following equation (Leishman 2000). min P = ρ πr Ω (1) 5 3 a C p 6

8 It should be noted that the power coefficient depends on all of the variables and it will subsequently be defined. Constraints The aerodynamic optimization of the heavy lift helicopter is subect to the following constraints relating to both physical and practical design requirements. First it should be noted that although not specifically part of the optimization model it was assumed that both the chord distribution along the blade and the twist distribution along the blade are linear. This requirement arises for practical simplicity of both analysis and manufacturing. The thrust produced by the helicopter must greater than or equal to a prescribed maximum take-off thrust in order to satisfy its design mission. g 1 = Tmax T 0 (2) The Mach number at the tip of the rotor blade must be less than a prescribed Mach number to prevent the effects of transonic flow in hover and forward flight. It should be noted that compressibility effects are also included in the mathematical model. g 2 M tip M 0 (3) = crit The chord taper ratio defined as the ratio of the tip chord to the root chord clearly can not exceed a value of 1.0 in order to maintain a favorable distribution of aerodynamic forces. g 3 = TR (4) The angle of attack at ¾ of blade span must not exceed a critical value of the angle of attack in order to provide a margin on blade stall for the design thrust condition. It is desired to maintain at least a small margin for emergency maneuvering. g 4 = ( 3 / 4) l crit α α 0 (5) 7

9 The blade radius is chosen to be limited in size in order for the helicopter to be able to provide its service in places with limited space such as between power lines or on a Navy vessel. g 5 = l lmax 0 (6) The blade twist rate is restricted to be less than or equal to zero from the definition of the geometric twist angle used in this optimization study. Design Variables and Parameters g6 = θ tw 0 (7) The design variables of the heavy-lift helicopter aerodynamic optimization are the rotor rotation speed the length of each blade the chord of the blade at its root the pitch of the blade at its root (collective pitch) the ratio of the chord at the tip of the blade to the chord at the root of the blade (taper ratio) and the geometric linear twist rate of the blade. All design variables with the exception of the geometric twist rate of the blade are positive. Parameters in this study include the number of blades on the rotor which is chosen to be four. The desired take-off thrust is to be chosen as lbs to be roughly equivalent to existing single rotor heavy-lift helicopters. The airfoil is prescribed to be a NACA 0012 which is common for helicopter rotors because it has well documented properties and is symmetric so no aerodynamic moment exists (Leishman 2000). The airfoil choice determines the lift curve slope at low Mach numbers (M=0.1) which is 2π for the airfoil chosen (Leishman 2000). In addition the airfoil choice determines the angle of attack for stall which occurs around 15. The profile drag coefficient is also a parameter based on the airfoil and has a value of The critical Mach number at the tip is chosen to avoid the drag rise due to transonic flow (Leishman 2000). In addition the aerodynamic laws break down in transonic flow so the maximum Mach number at the tip of the blade is 0.8. The speed of sound and air density are atmospheric properties and will be set to 8

10 their sea level values of 1116 ft/sec and slugs/ft 3 respectively such that sea level is the design altitude. The maximum blade radius will be chosen to be approximately 45 ft as the upper end of a typical heavy-lift helicopter (Leishman 2000). The number of degrees of freedom of the aerodynamic optimization is simply the number of design variables six since there are no direct equalities for these quantities to reduce this number. Before looking at the optimization problem it is desired to verify that the design space of the aerodynamic model is feasible. That is it must be shown that at least one set of values of the variables satisfies all of the constraints. One feasible design point found is the following with the associated values of each of the constraints given as well. Feasible Point: l = 40 ft croot = 2. 2 ft TR = θ 0 = 10 θ tw = 0. 1 rad Ω = 200 Constraint Values: g 1 = Tmax T = 25000lb 25690lb = 690lb 0 g 2 M tip M = = = crit g 3 = TR 1.0 = = g 4 ) = α ( 3 / 4 l α crit = = g 5 = l lmax = 40 ft 45 ft = 5 ft 0 g6 θ = = tw Function Value at Feasible Point: P = 2478HP Model Summary RPM The aerodynamic optimization may be summarized in negative null form as the following non-linear optimization problem. Note selected relationships between variables and parameters are given below for completeness of the mathematical model. Minimize P = ρ πr a 5 Ω 3 C p 9

11 Subect to: g 1 = Tmax T 0 g 2 M tip M 0 = crit g 3 = TR g 4 = α ( 3/ 4) l α crit g 5 = l lmax 0 g6 = θ tw 0 Variables: Ω c TR θ 0 0 l root 0 Parameters: N b = 4 Tmax = lbs ρ a = slugs/ft 3 a = 1116 ft/sec M = 0.8 α = 15 l 45 ft C π (1/rad) C = crit crit max = lα 2 = M =0.1 Additional relations needed for the analysis are the following (Leishman 2000). C (8) = 1 p dc p 0 do dc = λ dc + dc (9) p T p 0 C T 1 T = dc = R T ρ aπ Ω (10) dc 1 2 T ) = σcl α ( θr λr dr (11) 2 dc p 1 = σ 3 C r dr (12) 0 2 d θr λ = σc lα σclα (13) N b c σ = 2 πl (14) 10

12 c = c c ( 1 TR r (15) root root ) θ = θ 0 + (16) rθ tw Clα M = 0.1 Clα = (17) 2 1 M r 2 tip Ωl M tip = (18) a λ α = θ (19) r x r = (20) l Because of the integration in equations (8) and (10) of the differential thrust and power to obtain the total thrust and power respectively of the entire rotor and the relative complexity of these terms practical implementation of these formulas into the optimization code requires numerical integration. It is chosen to use ten-point Gaussian Quadrature in this study for its relative accuracy and efficiency (Dyer 2002). The Gaussian Quadrature procedure can be found in Appendix A Model Analysis Before undertaking the aerodynamic optimization of the heavy-lift helicopter in hover it is useful to attempt reduce the order of the model using monotonicity analysis. A monotonicity table can be found in Table 1 below. It should be noted that constraint g7 was incorporated into the model during the optimization process and will be defined subsequently in equation (21). 11

13 Blade Radius Root Chord Taper Ratio Collective Pitch Blade Twist Rotational Speed f g g2 + + g3 + g g5 + g6 + g7 + + Table 1. Monotonicity of Aerodynamic Model It can be seen from the table that by the first monotonicity principle (MP1) g1 is guaranteed to be active because it will provide the only upper bound for the blade radius collective pitch and blade twist. This is an expected result since the thrust produced by the rotor to minimize power should be the prescribed thrust exactly without any excess thrust produced. It can also be seen that the constraint g4 may be active; however it can not be determined before numerical implementation of the code whether or not g1 dominates g4 for the root chord taper ratio and rotational speed. Unfortunately knowledge of g1 being active does not allow reduction of the aerodynamic model because the thrust is solved for implicitly in the analysis due the requirement of numerical integration of the aerodynamic information over the blade span. In addition no additional knowledge of constraint activity can be determined before the numerical optimization analysis. 3.3 Aerodynamic Optimization Study The aerodynamic optimization was first attempted in Matlab with the final results completed in the Optimus software. First attempts at finding an optimal solution in Matlab were completed with the sequential quadratic programming (SQP) algorithm fmincon. A description of the first unsuccessful optimization trials in Matlab can be found in Appendix A2 along with the Matlab script files used for the analysis. 12

14 The results of the first unsuccessful optimization attempts led to the need for refinement of the aerodynamic model in order for the results to be physically meaningful. It can be seen that the original model has no provision to handle the fact that a blade element may have a local angle of attack in the stall regime. The original model simply predicts that the blade element will continue to produce increasing lift with increasing angle of attack. In reality the lift produced by the airfoil becomes unpredictable and significantly reduced when the angle of attack exceeds a critical angle. For the airfoil used in this study α crit = 15. In order to incorporate this into the aerodynamic model it is chosen to check for the value of the local angle of attack before the incremental thrust calculation. If the angle of attack is above 15 the incremental thrust for that section is set to zero. It is also clear that the model does not take into account the fact that the drag increases significantly on a stalled airfoil. The drag is calculated in two parts the induced drag due to the creation of lift and the skin friction drag. In the model with the correction of stall through a penalty of no incremental lift there is also no incremental induced drag such that the only drag component remaining is independent of the angle of attack. This of course does not match the drag-rise found in the stall regime. It is suggested that the profile drag is actually a quadratic function of angle of attack (Leishman 2000); however the particular coefficients are not given and no source is given to obtain the data. It is therefore chosen to make the parasite drag coefficient a piecewise function of the angle of attack such that the stall regime has ten times higher profile drag than the attached flow regime. The refined model therefore defines the parasite drag coefficient as C = forα 15 and C = 0. 1 forα > 15. do Additional problems encountered utilizing the Matlab fmincon SQP algorithm include results that depend heavily on the upper and lower bounds selected for the variables as well as the starting point. In order to enforce that all variables are positive except for the blade twist a do 13

15 lower bound vector must be defined for the variables. The chosen lower bound of the blade twist was varied as a test of the optimization code and the fmincon results depended heavily on the value chosen although it was never an active constraint at the optimum. In addition choice of the starting point significantly varied the optimized solution meaning that the optimum found most likely was local and not global. Because of the numerical difficulties encountered in the early trials it was decided to convert the analysis routine to a Fortran code coupled with LMS Optimus to perform the optimization. Optimus allows for improved insight of the design space though an analysis tool design of experiments. Early optimization trials with LMS Optimus using its SQP algorithm again had trouble finding a global optimum. It was found through the graphical display in Optimus that certain feasible combinations of the variables caused the power to be retuned as NaN and hence terminated the optimization run. Whenever the geometric twist at any station of the blade was negative as determined by the collective pitch and the blade twist variables then the inflow calculation would produce an error because of an attempted square root of a negative number. In order for the optimization routine to continue its runs a seventh constraint had to be added to the aerodynamic model. This constraint requires that the geometric angle at the tip of the rotor blade be greater than or equal to zero. Since the blade twist is negative and the collective pitch is positive if the geometric twist at the tip of the blade is non-negative it is guaranteed to be nonnegative over the entire blade. In negative null form the additional constraint for the geometric angle at the blade tip is as follows. g7 θ 0 (21) = tip where θ = θ 0 + θ (22) tip tw 14

16 In addition to incorporating g7 into the aerodynamic model it was chosen to investigate scaling as a means to improve the numerical behavior of the aerodynamic optimization. The obective function the power required by the helicopter to hover is scaled to be in units of horsepower instead of lb-ft/sec which represents reducing the minimized obective by a factor of 550. In addition it is chosen to scale g1 by the required thrust lbs. With these new modifications to the aerodynamic model reliable optimization results were found. A sequence of optimization algorithms was incorporated to ensure that the optimum found was indeed a global one. This sequence was comprised of first using the sequential quadratic programming algorithm in Optimus and then using a simulated annealing algorithm to ensure that the optimum was global. The refined aerodynamic analysis Fortran code that was coupled with Optimus can be found in Appendix A3. The values of the variables to minimize the power required to hover are l = 45 ft c = ft TR = θ = θ = and Ω = RPM. This o o tw optimal point produces a minimized power of P = HP. In addition the thrust constraint g1 is active at the optimum as predicted by monotonicity analysis. The thrust produced at the optimal solution is T = lbs which is within the precision of the allowed constraint violation. The only other active constraint at the optimum is g5 where the blade radius is l = ft. For this optimal solution M = α 14.5 TR = θ tw = andθ tip = tip ( 3 / 4) l = ; therefore g2 g3 g4 g6 and g7 are not active at the optimal solution. As with all previous trials of the aerodynamic optimization the constraint on the length of the blade is active. Although monotonicity analysis did not predict this behavior it can be seen that the power required to hover depends more strongly on the rotor rotation speed than on the rotor radius so that the radius will be the largest value allowed in this study for the optimal 15

17 aerodynamic solution. In fact during all aerodynamic optimization trials the blade radius was driven to its upper bound in order to minimize power. It should be noted that the optimal values of the variables found from the optimization sequence described above are within 1% of the values found after ust the SQP optimization and the power required to hover decreased by only 0.58% from the simulated annealing optimization after the SQP optimization. Therefore the SQP algorithm found the global optimum within a fairly close tolerance. 3.4 Parametric Study After studying the baseline aerodynamic subsystem of this helicopter rotor optimization it is desired to get a better feel for the sensitivity of the optimal rotor variables to changes in the parameters in the problem. For instance the effect of the number of blades of the rotor on the minimized power required to hover and the associated blade geometry is studied. In addition in case the helicopter is not used at sea-level it is chosen to determine the power required and the values of the rotor variables needed in order to hover at feet above sea-level. Since the maximum blade length constraint is active at the optimal solution it is chosen to see the effect of relaxing the constraint to determine the mathematical optimum solution even if the results are not physically relevant. Also since most of the aerodynamically optimized results determined a value of the rotation speed far lower than those on existing helicopters an optimization study was performed with the rotational speed fixed at 200 RPM. The optimization routine for each parametric study is identical to that of the baseline optimization. That is first the sequential quadratic programming algorithm in Optimus was run with the results verified using the simulated annealing algorithm in Optimus. 16

18 Results from the parametric optimization study on the number of blades on the rotor can be found in Table 2 along with the previously described optimum with four blades. In this study all other parameters remained at their baseline values. It is interesting to note that the power required to hover is nearly identical for 4 and 5 bladed rotors while the 3 bladed rotor required approximately 4.3% more power. Again the blade radius constraint was active at the optimum solution for each rotor. Clearly the thrust requirement constraint was active at the optimum solution for each rotor as well. No other constraints imposed on the optimization were active for any of the parametric studies concerning the number of blades. Baseline (4 Blades) 3 Blades 5 Blades Radius (ft) Root Chord (ft) Taper Ratio Collective Pitch (deg.) Blade Twist (deg.) Rotation Speed (RPM) Total Thrust (lb) Total Power (HP) Table 2. Parametric Study on Number of Blades It is interesting to note that the optimized root chord does not follow any pattern with respect to the number of blades. Looking at the baseline results one may predict that the root chord of the 5 bladed rotor would be smaller to create approximately the same solidity as the 4 bladed rotor and this in fact was true. Interestingly the three bladed rotor does not make up for its fewer number of blades by increasing the blade chord; its root chord is less than half of that for the 4 bladed rotor. The 3 bladed rotor however must spin faster to provide the necessary thrust to lift the helicopter and also has a higher taper ratio (less blade taper) and a lower collective pitch than the baseline case. These results are not intuitive and may be attributed to the complexity of the design space of the problem. 17

19 The optimization results of the parametric study on the altitude chosen can be found in Table 3. The table shows the baseline 4 bladed rotor at sea-level as well as a 4 bladed rotor at feet of altitude. Changing the operational altitude from sea-level to feet reduces the air density to ρ = slugs/ft 3 and the speed of sound to a = ft/sec. All other a parameters in the parametric altitude study remained at their baseline values. Baseline (Sea-Level) feet Radius (ft) Root Chord (ft) Taper Ratio Collective Pitch (deg.) Blade Twist (deg.) Rotation Speed (RPM) Total Thrust (lb) Total Power (HP) Table 3. Parametric Study on Altitude In the foot altitude case the blade radius and thrust constraints were active similar to the baseline case. Interestingly the only two variables that are significantly changed between the optimum solutions at sea-level and feet are the blade twist and the rotor rotational speed while the other variables remain very similar in value. It could have been predicted that the rotor would need to rotate faster at feet than at sea-level because the air is less dense at altitude meaning that increased blade velocity would be needed to produce the same amount of lift. The blade twist being significantly higher for the foot trial than the sea-level trial is not an intuitive result. It should also be noted that the power required to hover is approximately 17.3% higher at feet than at sea-level. This result implies that the production of lift is penalized by induced power more at feet than at sea-level. This confirms the fact that the flight envelope of a helicopter is limited in altitude by the power available by the engine(s). 18

20 The results of a parametric study on the blade radius upper bound constraint are presented in Table 4 along with the baseline results considered. No other parameters were changed for this study. It can be seen that the blade radius constraint remained active at the optimum solution found verifying that the power required by the rotor is a stronger function of the rotation speed than the radius. The thrust constraint was the only other active constraint at the optimum solution found. It can be seen that the power required to hover was reduced by 53% with the 122% larger blade radius. Additionally the rotor rotation speed was reduced by 68% due to the larger blade radius. Clearly the blade is far too long and the rotation speed far too low for a practical helicopter although the results show interesting mathematical results and encourage the combination of the aerodynamic optimization with the structural optimization to determine if rotors in current use are determined by mathematical or practical constraints. Baseline (45 ft Radius) 100 ft Radius Radius (ft) Root Chord (ft) Taper Ratio Collective Pitch (deg.) Blade Twist (deg.) Rotation Speed (RPM) Total Thrust (lb) Total Power (HP) Table 4. Parametric Study on Blade Radius Constraint The results of the parametric study with a fixed rotor rotation speed of 200 RPM can be found in Table 5. It can be seen from the table that the power required to hover with a 200 RPM rotational speed is approximately 10.3% higher than when the rotation speed was allowed to be an optimization variable. It is interesting to note that the blade radius constraint is no longer active at the optimum solution because the blade radius is now limited by the tip Mach number constraint which was active for this parametric study. Also it can be seen that the root chord and taper ratio are much smaller for the high rotational speed than for the baseline case. The 19

21 collective pitch is also smaller than for the baseline case but this is counteracted by a less negative blade twist meaning that the blade is still operating at an efficient angle of attack ust below stall. It should be noted that the values of the variables in this parametric study with the rotor rotational speed fixed represent values that are similar to those of current helicopters (see Table 9). 3.5 Discussion Baseline (Omega varied) Omega = 200 RPM Radius (ft) Root Chord (ft) Taper Ratio Collective Pitch (deg.) Blade Twist (deg.) Rotation Speed (RPM) Total Thrust (lb) Total Power (HP) Table 5. Parametric Study with Rotation Speed Fixed Looking at the results of the aerodynamic optimization and subsequent parametric studies much has been learned about the behavior of the aerodynamic system of a helicopter rotor in hover. The maor design rule learned is that from an aerodynamic point of view reductions in power can be achieved by minimizing the rotor rotational speed at the expense of increasing the blade radius. In all optimization studies considered the blade radius umped to its upper bound constraint while the rotor rotational speed found the lowest value possible that still ensured that the rotor provided enough lift to meet the thrust requirement. Additionally the root chord of the blade seemed to follow no particular trend in the parametric studies so it is assumed that it is more weakly correlated with the power and thrust than the other variables in the study. Again the blade taper ratio did not follow any particular trend in the studies considered and like the root chord was not an important variable in the aerodynamic optimization. The collective pitch angle of the blades and the blade twist rate seemed to be set such that the blade was operating at an 20

22 efficient angle of attack over the outer portions of the blade while ensuring that not much of the blade was in the stall regime. During most of the aerodynamic optimization trials considered the angle of attack at the three-quarters blade span station was found to be approximately 14 which is slightly under the critical stall angle of 15. This implies that an aerodynamically efficient rotor operates such that its rotor operates with the blade at an angle for maximum lift to drag ratio which occurs ust below the stall angle for the aerodynamic model in this study. It is clear that the blade radius and rotor rotational speed are the big factors in determining an operationally efficient rotor considering a blade operating at an efficient angle of attack. The practical constraint on the blade radius is always active at the aerodynamically optimal solution while the rotor rotational speed drops to well below the values of current helicopter configurations (see Table 9). Aerodynamic optimization then is not necessarily the factor that leads to the selection of the blade radius and rotor rotational speed. Factors such as static and dynamic structural requirements aeroelastic and aeromechanical stability and handling qualities of the aircraft may all be maor factors in the choice of the rotor rotational speed and blade radius. A complete and practical optimization for the design of a helicopter rotor would therefore consider such constraints on the design space. It is also worth noting the difficulty of the sequential quadratic programming algorithm to find a global aerodynamic optimum solution for the helicopter rotor. Although the SQP algorithm worked very well for the baseline parameters considered the stability of the algorithm was highly dependent on the values of the parameters and the starting point provided. In addition the fact that SQP allows the variables to be chosen in the infeasible region required that the aerodynamic model be made as robust as possible so that the code could continue to run although the results in those regions could not be made accurate. For example although 21

23 infeasible and far from an optimum solution SQP attempted to choose variable values such that the tip Mach number exceeded the speed of sound. This caused the code to attempt a square root of a negative number in equation (17). Therefore a piecewise formula had to be introduced into the infeasible space on the value of the lift curve slope which affected the SQP algorithm because it attempts to calculate gradients in the infeasible space where the model is not valid. A similar non-linearity existed for the calculation of the inflow ratio in equation (13) due to the possibility that the tip twist would be chosen negative producing another square root of a negative number. In addition the corrections to the aerodynamic model to include the effects of stall are piecewise functions such that gradient calculations are inaccurate. In order to improve the results of SQP for the combined aerodynamic and structural analysis a better model for the lift curve slope was developed. The new model is again a piecewise function but is now piecewise continuous and developed such that no extreme gradients are encountered. The new model on the lift curve slope is defined as a function of the angle of attack at a given section. C α 2π (1/radians) for α 15 l = M =0.1 C = 2π cos(18( 15 )) (1/radians) for 15 < α 20 (23) lα M α = 0.1 C α 0 (1/radians) for α > 20 l = M =0.1 A plot of the new sectional lift coefficient determined by equation (23) at M=0 versus angle of attack can be found in Figure 1 below. It can be seen that there are steep gradients involved in the stall regime but is in fact a better representation of the physics of stall. Using the revised stall model with SQP the optimal values of the variables are l = 45 ft c = ft TR = θ = θ = and Ω = RPM. This optimal point o tw produces a minimized power of P = HP. Again the thrust constraint and the maximum o 22

24 blade length constraint were the only active constraints at the optimum. It should also be noted that use of the simulated annealing algorithm only improved the obective by 0.3 HP which corresponds to a 0.02% decrease in power. The SQP algorithm found a global optimum with the revised aerodynamic model for the lift curve slope which was the goal of the model revision. Cl vs Alpha Cl Alpha (degrees) Figure 1. Sectional Lift Coefficient versus Angle of Attack for Revised Model Comparison of the optimal rotor for hover found in this aerodynamic subsystem study with the expected optimal inflow and lift distributions along the blade for a hovering rotor provides insight into the aerodynamic model and parameter values used for this study. The ideal rotor for hover should have a uniform inflow over the rotor disk to minimize the induced power (Leishman 2000). Figure 2 shows the inflow distribution of the optimal rotors presented in this study along the rotor disk radius. The ideal uniform inflow value calculated using momentum 23

25 theory for the baseline parameters is also included in the plot. It can be seen that the actual inflow for the baseline case is not uniform and is significantly greater than ideal inflow over the outer portion of the rotor disk. This is due to the fact that momentum theory considers the rotor as an actuator disk and does not consider the profile drag and hence power on the blade. In order to obtain uniform inflow over the rotor disk the blade must be twisted such that the geometric pitch angle satisfiesθ ( r) = θtip / r. This is not a physically reasonable pitch distribution since the blade will have infinite pitch at the root. The linear geometric twist assumed in this study provided an inflow ratio that is nearly uniform over the outer portion of the rotor disk where most of the load is distributed. It is interesting to note that the rotor with a fixed rotation speed at 200 RPM had a significantly lower inflow than the other cases considered due to the fact that the inflow ratio is defined as the inflow velocity across the rotor disk divided by the tip speed of the blade. Fixing the rotation speed to a realistic value much higher than those found for the aerodynamically optimized rotors produces a much larger tip speed and hence a much lower value of the inflow ratio. Figure 3 below presents the sectional lift distribution along the blade span for the optimal studies presented. The ideal hovering rotor produces a triangular lift distribution with no lift being produced at the root (Leishman 2000). The ideal lift distribution is shown in the plot for the baseline parameters. The baseline optimal rotor produces a lift distribution that is very close to the ideal distribution for a hovering rotor. All of the lift distributions calculated for the different optimal parametric studies have similar shape with the maor differences coming in the required magnitude to produce the total thrust. 24

26 Baseline 3 Bladed Rotor 5 Bladed Rotor 4 10kft 4 Bladed with Fixed 200 RPM Momentum Theory Uniform Inflow - Baseline Inflow Ratio Blade Station (ft) Figure 2. Optimal Inflow Distribution over Rotor Disk 450 Baseline Sectional Lift (lbs/ft) Bladed Rotor 5 Bladed Rotor 4 10 kft 4 Bladed Fixed Omega Ideal Load Distribution (Baseline) Blade Station (ft) Figure 3. Optimal Lift Distribution over the Rotor 25

27 4.0 Structural Optimization Problem Statement The obective of this subsystem study is to minimize the structural weight of a cantilevered helicopter rotor blade designed for heavy lift. The optimization study focuses on the design of the maor load-carrying portion of the blade structure represented by a thin-walled box-beam (Figure 4). Trade-offs are anticipated as the structure must be sufficiently robust to meet strength requirements while at the same time satisfying minimum natural frequency constraints in three degrees of freedom for positive stability and control characteristics. In addition the blade design is constrained such that stiffness is adequate to prevent excessive static droop. Figure 4. Idealized Blade Structure 4.1 Mathematical Model Obective function The obective of this subsystem design optimization is to minimize the weight of the structure. The weight is expressed as the product of the material density and the structural volume accounting for the effect of linear taper. The volume of the upper and lower walls of the box beam is expressed as: 1 lt ctip + ( croot c ) 2 (24) 2 tip Similarly the volume of the side walls is given by: 1 lt htip + ( hroot h ) 2 (25) 2 tip 26

28 It should be noted that the contribution of geometric twist to the increase in structural volume over a given length is neglected since the total twist rate is constrained to small angles (described below). In addition the above expressions include the corner material in both the horizontal and vertical wall calculations which is not exact but considered to be an acceptable estimate because of the thin-walled assumption (which is posed as a constraint in the following section). This thin-walled assumption allows simplifications in calculating cross-section properties. The obective function of minimizing the total structural weight expressed in terms of the root dimensions and taper ratio is : Minimize W = ρ lt 1+ TR) c + (1 + TR) h ) (26) (( root root There are five physical constraints and six practical constraints in this optimization problem. The physical constraints are first discussed. The airfoil type is assumed to be constant along the length of the blade thus the ratio of the box-beam width to height must be fixed. The ratio was determined by assuming that the box beam was centered at the quarter chord of the NACA 0012 airfoil then using the available space and visually estimating. The resulting constraint is: h = (27) root c root Three additional constraints are derived from standard practice. In order to have both predictable control characteristics and benign stability the torsion flap and lag frequencies are required to be sufficiently greater than the rotating frequency. The exact requirements are derived from typical design practices (Wilbur 2004) and are given as: 4Ω ω x 0 2Ω ω 0 y 3Ω ω 0 z (28) 27

29 The maximum stress due to the combined bending along the two principal axes is constrained to be less than the yield stress for the material (which is fixed). Note that the particular finiteelement code used to analyze the blade structure solves for internal forces/moments at the nodes which separate two elements with different cross-section geometries. Using this set of forces the axial stress due to bending in the element to the left and right of each node are calculated. For each of the elements the maximum of the two stresses is used to compare against the yield stress resulting in a set of constraints corresponding to one for each element: + { σ } σ 0 max σ (29) x x y The four practical constraints will be discussed. First it is necessary to enforce a maximum wall thickness such that the thin-walled assumptions used in the model are valid. The maximum was set at the usual value of 10% of the smallest dimension of the cross section. Since the crosssection is smallest at the tip the constraint is enforced here expressed in terms of the taper ratio. t 0.1TRh 0 (30) root The blade twist rate affects the distribution of lift coefficient along the blade. Negative twist rates improve the uniformity of inflow distribution along the blade and perfectly uniform inflow represents the case of minimum induced power in hover (Leishman 2000). For a twist rate exceeding twice the stall angle of a symmetrical airfoil some part of the blade would be stalling during operation. Also a larger twist rate also reduces the effective range of collective pitch and thus the range of thrust that can be produced. With these considerations the maximum rate of negative twist is chosen to be -20 deg. Additionally in order for all parts of the to produce positive lift the geometric pitch angle must be greater than zero at all span locations which can be enforced by considering the twist rate and collective pitch angles. θ 0 (31) tw 28

30 θ 20 0 (32) tw θ tw θ 0 0 (33) The blade length is important for aerodynamic efficiency in hover and also contributes to performance in autorotation (Leishman 2000). A historical survey of helicopters designed for heavy lift shows a minimum blade radius of 6 m. (Leishman 2000). This value was set as a constraint for the current design study. 6m l 0 (34) Lastly a constraint is gained from imposing a limit on the static non-rotating blade deflection due to self-weight. Static droop may cause problems when starting and stopping the rotor when blade-sailing in windy conditions can cause the blade to flex and impact the airframe (Leishman 2000). Although maximum allowable static droop depends on the helicopter configuration a maximum upper limit of 2.13 m. was chosen by considering typical values (Three Rivers Free-Net 2004). It should be noted that unlike the dynamic analysis the collective pitch at 75% of the blade radius is set to zero to simulate a spin-up or spin-down zerolift situation. 2.13m w 0 (35) Design Variables and Parameters tip The variables for which an optimum is sought are the blade length twist taper ratio root chord root height wall thickness rotation speed and collective pitch angle. The collective pitch is the geometric rotation of the blade root from the zero-lift plane about its maor axis. The rotation speed is the angular velocity of the blade and causes tensile centrifugal loads in the blade and is typically limited by aerodynamic constraints. With the consideration of the first equality constraint the list can be reduced by removing the root height resulting in seven degrees of 29

31 freedom. It should be noted that in fact the structure geometry is sought down the entire length and the above list of variables are sufficient to give a full description under the assumption of linear twist linear taper and constant wall thickness. The thrust at maximum loading material density yield strength and modulus are set as parameters. The thrust per blade at maximum loading is defined as the thrust expected at the most severe operating condition with a safety factor divided by the number of blades. In this study this value was set at 1.5 of the maximum take-off weight divided by 4. The material is chosen to be aluminum which gives a fixed value for the density yield strength Young s Modulus and shear modulus. The parameter values that will be used in the study are given below: Parameter Thrust/blade at maximum loading Value 9375 lb. Material Density lb/in 3 Yield strength lb/in 2 Young s Modulus 10000e3 lb/in 2 Shear Modulus 3770 ksi Using these values it is difficult to obtain example values that satisfy the constraints without running the actual model since the centrifugal loading due to the rotor speed results in a non-linear expression for the displacement using von Karman beam theory from which the stresses are derived. This expression has no closed form solution and computer approximations are required. Similarly it is necessary to perform computer experiments such as Design of Experiments (DOE) to explore monotonicity in the optimization problem. 30

32 Summary Model Collecting the expressions derived above constructs the formal statement of the optimization problem. Minimize W = ρ lt 1+ TR) c + (1 + TR) h ) Subect to: h g g g g g g g g g g = h = t 0.1TRh = 6 l 0 = 2.13 w (( root root = 0.25c = 4Ω ω 0 = 2Ω ω 0 = 3Ω ω 0 = θ θ = θ root tw 0 = θ 0 tw 10: no. of elems x z y tw 20 0 tip root 0 0 = max root 0 + { σ σ } σ 0 x x In order to determine the optimal quantities and test the constraints a number of values are evaluated using the non-linear finite-element analysis code AEROSCOR (Shang 1995). The assumptions used in this model are a fixed number of nodes and neglect of shear deformation. The process of determining the natural frequencies tip deflection and maximum stress is outlined below. First the cross-section bending stiffness in the principal axis of each of the elements is determined from the geometry: y EI EI 1 i = E 12 1 i = E 12 y z c h 3 i i c h 3 i i ( c 2t)( h 3 ( c 2t) ( h i i i 2t) i 3 2t) (36 37) 31

33 GJ i 2 4G( cihi ) t = (38) 2( c + h ) i i where the average chord and height of the box beam at each element is determined by considering the linear taper and the division of the blade length into elements: x c h i i i 1 = + i 2 = c = h root root c h l n root root 1 xi (1 TR) l xi (1 TR) l ( ) Additionally the blade mass and moment of inertia are given by the cross section geometry: 2ρtl( ci + hi ) mi = n 1 1 m [( c 2t) + ( h 2t ] 2 2 θ i = ρcihil ( ci + hi ) ρ( ci 2t)( hi 2t) l i i ) ( ) Assuming a constant distribution of blade loading gives the applied nodal loads for the finite element model: T L = n 1 for.ne. 1 or n (45) 1 T L = for.eq. 1 or n (46) 2 n 1 Using these quantities as inputs to the finite-element code along with the parameters defined earlier the internal axial force internal bending moments in the inertial frame and the natural frequencies in all three degrees of freedom are obtained along with the static (non-rotating) tip deflection. These are expressed as unknown functions with known dependencies: 32

34 33 ) ( ) ( ) ( ) ( ) ( ) ( ) ( l L m EI EI f w l L m EI EI f l L m EI EI f l L m GJ f l L m EI EI f M l L m EI EI f M l L m f F tw i i z i y tip tw i i z i y z tw i i z i y y tw i i x tw i i z i y tw i i z i y tw i x θ θ θ θ ω θ θ ω θ θ ω θ θ θ θ θ θ θ = Ω = Ω = Ω = Ω = Ω = Ω = (47) To obtain the stresses the internal bending moments are transformed to the frame aligned with the principle axes of the elements. ) / sin( ) / cos( ) / sin( ) / cos( l x M l x M M l x M l x M M tw tw y tw tw x θ θ θ θ θ θ θ θ + + = = (48 49) Subsequently the maximum stress caused by the combined axial and bending loads is obtained at the point farthest from the neutral axis and utilize the usual load-stress relationship. The maximum stress in the entire beam is simply the maximum of the set of the element stresses. + + = + + = + z y y x x x z y y x x x I M c I M h A F I M c I M h A F σ σ (50 51) Note that in the combined optimization problem the nodal loads are determined using the blade element theory analysis provided by the full aerodynamic model accounting for the non-uniform lift distribution and also the changes blade planform. In the isolated structural optimization the blade is subected to a triangular load distribution with value zero at the root and increasing linearly to the tip such that the total load applied to the blade is the total thrust: the maximum gross weight multiplied by a 1.5 maneuvering load factor divided by the number of blades. This