NASA-CR-20_285 ,,2 j _'/-T Optimum Design of High-Speed Prop-Rotors Final Report Grant Number: NAG2-771 Principal Investigator Aditi Chattopadhyay Graduate Research Associate - Thomas Robert McCarthy Department of Mechanical and Aerospace Engineering Arizona State University Tempe, Arizona APR 1 41997 Abstract An integrated multidisciplinary optimization procedure is developed for application to rotary wing aircraft design. The necessary disciplines such as dynamics, aerodynamics, aeroelasticity, and structures are coupled within a closed-loop optimization process. The procedure developed is applied to address two different problems. The first problem considers the optimization of a helicopter rotor blade and the second problem addresses the optimum design of a high-speed tilting proprotor. In the helicopter blade problem, the objective is to reduce the critical vibratory shear forces and moments at the blade root, without degrading rotor aerodynamic performance and aeroelastic stability. In the case of the high-speed proprotor, the goal is to maximize the propulsive efficiency in high-speed cruise without deteriorating the aeroelastic stability in cruise and the aerodynamic performance in hover. The problems studied involve multiple design objectives; therefore, the optimization problems are formulated using multiobjective design procedures. A comprehensive helicopter analysis code is used for the rotary wing aerodynamic, dynamic and aeroelastic stability analyses and an algorithm developed specifically for these purposes is used for the structural analysis. A nonlinear programming technique coupled with an approximate analysis procedure is used to perform the optimization. The optimum blade designs obtained in each case are compared to corresponding reference designs. ii Table of Contents Page List of Tables ....................................................................................... viii List of Figures ....................................................................................... ix Nomenclature ........................................................................................ xi I. Introduction ...................................................................................... 1 Helicopter Design Considerations ......................................................... 1 Helicopter Rotor Blade Optimization ...................................................... 3 Sequential Optimization ............................................................ 3 Multidisciplinary Optimization .................................................... 4 High-Speed Rotorcraft Design Considerations .......................................... 5 High-Speed Rotorcraft Optimization ...................................................... 6 Multiple Design Objectives ................................................................. 7 II. Objectives ........................................................................................ 9 III. Multiobjective Optimization ................................................................... 10 Modified Global Criteria Approach ...................................................... 10 Minimum Sum Beta (Min El3) Approach ............................................... 11 Kreisselmeier-Steinhauser (K-S) Function Approach ................................. 12 Problem Statement ......................................................................... 13 Blade model ................................................................................. 14 Optimization ................................................................................. 17 Analysis ..................................................................................... 19 Dynamic, Aerodynamic and Aeroelastic Analyses ............................ 19 Structural Analysis ................................................................ 19 Optimization Implementation .............................................................. 20 in Page Results and Discussions ................................................................... 21 Conclusions ................................................................................. 30 IV. Integrated Helicopter Rotor Blade Optimization ............................................ 33 Problem Definition ......................................................................... 33 Blade Model ................................................................................. 34 Optimization ................................................................................. 38 Analysis ..................................................................................... 40 Dynamic, Aerodynamic and Aeroelastic Analyses ............................ 40 Structural Analysis ................................................................ 41 Optimization Implementation .............................................................. 41 Results and Discussions ................................................................... 41 Conclusions ................................................................................. 53 V. Integrated High-Speed Proprotor Optimization ............................................ 55 Problem Definition ......................................................................... 55 Blade Model ................................................................................. 55 Optimization ................................................................................. 57 Analysis ..................................................................................... 58 Dynamic, Aerodynamic and Aeroelastic Analyses ............................ 58 Structural Analysis ................................................................ 59 Optimization Implementation .............................................................. 59 Results and Discussions ................................................................... 61 Conclusions ................................................................................. 76 VI. References ...................................................................................... 78 iv List of Tables Table Page 1 Summary of Multiobjective Optimization Constraints ................................... 23 2 Summary of Multiobjective Design Variables ............................................ 23 3 Summary of Integrated Helicopter Rotor Blade Optimization Results ................ 44 4 Summary of Integrated Helicopter Rotor Blade Optimization Design Variables ...................................................................................... 45 5 Summary of Integrated High-Speed Proprotor Optimization Results ................. 63 6 Summary of Integrated High-Speed Proprotor Optimization Design Variables ...................................................................................... 65 List of Figures Figure Page 1 Simplifiedrotorblademodelwithlineartaper.......................................... 14 2 Centrifugal stress distribution ............................................................. 24 3 Lead-lagbendingstiffnessdistribution.................................................. 25 4 Nonstructurawl eightdistribution........................................................ 26 5 Comparisonofindividualobjectivefunctions.......................................... 27 6 4/revverticalsheariterationhistory...................................................... 28 7 3/revinplanesheariterationhistory...................................................... 29 8 Objectivefunctioniterationhistory....................................................... 30 9 Double-celled box beam configuration...................................................33 10 Variation of tip shapewith tip shapeparameter,p.....................................3.5 11 Variationoftip shapewithtipshapeparametero,t..................................... 36 12 Variationofbladetwistwithchangesintipshapeparameter_,5...................... 37 13 Comparisonsofnormalizedvibratoryloadsandtotalpower......................... 46 14 Torsionalbendingstiffnessdistribution................................................. 47 15 Bladenonstructurawl eightdistributions................................................ 48 16 Chorddistribution.......................................................................... 49 17 Bladecenterofgravityoffsetdistribution............................................... 50 18 4/revverticalsheariterationhistory...................................................... 51 19 4/rev lagging moment iteration history...................................................52 20 Objective function convergence history..................................................53 21 Swept blade planform ...................................................................... 57 22 Comparisonofindividualobjectivefunctions.......................................... 64 23 Chorddistribution.......................................................................... 66 vi Figure Page 24 Flapwise bending stiffness distribution.................................................. 67 25 Lead-lagbendingstiffnessdistribution.................................................. 68 26 Torsionalbendingstiffnessdistribution................................................. 69 27 Nonstructurawl eightdistribution........................................................ 70 28 Twist distribution ........................................................................... 71 29 Sweepdistribution......................................................................... 73 30 Lifting linedistribution.................................................................... 74 31 Individual objective function iteration history...........................................75 32 Objectivefunctioniterationhistory....................................................... 76 vii Nomenclature base of the mth box beam member, ft bm c chord, ft Co - c3 chord distribution parameters fi natural frequency of the ith mode, per rev (/rev) fr 3/rev radial shear, lb fx 3/rev inplane shear, lb fz 4/rev vertical shear, lb g J, g2, g3 vector of constraint functions A g approximated constraint vector hm height of the mthbox beam member, ft k principle radius of gyration, ft2 mc 3/rev torsional moment, lb-ft 3/rev flapping moment, lb-ft mx 4/rev lagging moment, lb-ft mz P chord distribution shape parameter Pn two point approximation exponent for the nthdesign variable tm box beam wall thickness of the m th member, ft wtj leading edge nonstructural weight at the jth node, lb/fi central nonstructural weight at the jth node, lb/ft Wcj blade skin weight at the jth node, lb/fi Wskj honeycomb filler weight at the jth node, lb/ft Whcj Xe nondimensional center of gravity offset forward of shear center x, y, z reference axes nondimensional radial location A yj distance from the blade root to the center of the jth segment, ft Aj average area of the jth segment, ft2 AI autorotational inertia, lb-fi 2 CT thrust coefficient Cp power coefficient Elxx, Elzz bending stiffnesses, lb-ft 2 Fk kthobjective function Fko values of the objective function Fk at the beginning of an iteration _1, i_2, I_3 multiobjective formulation of the individual objective functions, Fk °°. Vln Fk prescribed target of the objective function Fk A Fk approximated value of the objective function Fk FOM hover figure of merit GJ torsional stiffness, lb-ft 2 Lj length of the jth segment, ft M total number of constraints and objective functions NCON number of constraints NDV number of design variables NOBJ number of objective functions NSEG number of blade segments NMEM number of box beam structural members R blade radius, ft T thrust, lb Wnsi nonstructural weight of the jth segment, lb Wsi structural weight of the jth segment, lb wj total weight of the jth segment, lb W total blade weight, lb c_ chord distribution shape parameter (Xk kth aeroelastic stability root 131,132 pseudo design variables El, _2 lifting line curvature parameters 8 twist shape parameter ktheigenvalue of the Floquet transition matrix inverse taper ratio _t advance ratio design variable vector ithdesign variable 9 K-S function multiplier 0 blade twist, degrees 0o- 03 blade twist distribution parameters cy area solidity cyi stress in ith segment, lb/ft 2 (Ycent centrifugal stress, lb/ft 2 O'vib vibratory stress, lb/ft 2 ix twist ratio 13k minimum allowable aeroelastic damping for the kthmode prescribed bounds for blade distribution parameters rotor angular velocity, revolutions per minute (RPM) 0) rotor angular velocity, rad/sec Subscripts max maximum value r value at the blade root ref reference blade value t value at the blade tip L lower bound U upper bound