hinglesss rotor eom model

8/12/2019 Hinglesss Rotor EOM Model

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Defence Science Journal Vol. 54 No 4 October 2004 pp. 419-4272004 DESIDOC

Validation of Comprehensive Helicopter eroelastic nalysiswith Experimental Data

Shrinivas R Bhat and Ranjan GanguliIndian Institute of Science, Bangalore-560 012

BSTR CT

Th e experimental data for a Cbla ded soft-inplane hingeless main rotor is used to validatea comprehensive aero elastic analysis. A finite element model has been dev eloped for the rotorblade which predicts rotating frequencies quite well acro ss a range of rotation speeds. Thehelicopter is trimmed and the predicted trim-control angles are found to be In the range ofmeasured values for a variety of flight speeds. Power predictions over a range of forwardspeeds also compare w ell. Finally the aeroelastlc analysis is used to s tudy the Importance ofaerodynamic models on the vibration predic t~o n. nsteady aerodynamics and free-wak e modelshave been investigated.Keywords Helicopter vibration validation ae roelastic analysis finite element mode l r otor bladeaerodynam ic models free-wake models unsteady aerodynam ics quasi-steadyaerodynamics

NOMENCLATUREc B l a d e c h o r dC T h r u s t c o e f f i c i e n tC F i n it e e l e m e n t d a m p i n g m a t r ix

F r ~ V e r ti c al r o o t s h e a r f o r c eF F i n i t e e l e m e n t f o r c e v e c to rH S h a p e f u n c t i o n m a t r ixH s) T i m e s h a p e f u n c ti o n

~ M a i n r o t o r p o w e r c o e f f i c i e n t K F i n i t e e l e m e n t s t i f f n e s s m a t r i xd L o n g i tu d i n al h u b s h e a r f o r c e x ~ R o l l in g h u b m o m e n t

F V ~ L a t e r a l h u b s h e a r f o r c e M . y ~ p i tc h i n g h u b m o m e n tF I ~ V e r ti ca l h u b s h e a r f o r c e M I Y a w in g h u b m o m e n tF s ~ L o n g i tu d i n al r o o t s h e a r f o r c e M R o l l i n g r o o t m o m e n tF~~ L a t e r a l r o o t s h e a r f o r c e M~~ P i tc h i n g r o o t m o m e n tReceived l December 2003

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EFSCI VOL 54 NO , OCTOBER 2004

Yawing root momentFinite element mass matrixNumber of spatial finite elementsNumber of bladesNumber of time finite elementsModal displacement vectorGlobal displacement vectorGeneralised load vector for response analysisTemporal nodal displacement vector fortime elementRotor radiusLocal time coordinateKinetic energyStrain energyDisplacement vectorForward speedVirtual workBlade preconeVariationHelicopter trim-control anglesNormal mode transformation matrixFrequencyV I S Z (advance ratio)Density of airSolidity ratioAzimuth angle (time)Rotation speed (scalar quantity)Reference rotor speed (scalar quantity)

1 INTRODUCTIONA key tool in accelerating the helicopter design

process is a comprehensive aeroelastic analysis code,which is able to simulate the behaviour of thehelicopter in forward flight. These codes combinestructural dynamic and aerodynamic modelling anduse a physics-based approach to predict helicopterbehaviour. Over the past two decades, a few suchcodes have been developed1- .

The object ive of this study is to validate theUniversity of Maryland Advanced Rotocraft Code(UMARC) 1.4and understand the basic performanceand vibratory effects on hingeless main rotor inforward flight. The experimental data for a4-bladed soft-inplane hingeless main rotor hasbeen used.2 HELICOPTER AEROELASTIC ANALYSIS2 1 Governing Equation of Motion

The helicopter is represented by a nonlinearmodel of the rotating elastic rotor blades dynamicallycoupled to a six-DOFs rigid fuselage. Each bladeundergoes f lap bending, lag bending, elastic twist,and axial displacement. Governing equation of motionderived using a generalised Hamilton s principleapplicable to non-conservative systems is writtenas

where 6 U 6 T and W are the virtual strainenergy, the kinetic energy, and the virtual work,respectively. The 6 U and 6 T include energycontributions from components attached to theblade, eg, pitch link, lag damper, etc. These equationsare based on the work of Hodges and Dowell4and include second-order geometric nonlinear termsaccounting for moderate deflections in the flapbending, lag bending, and axial and torsion equations.External aerodynamic forces on the rotor bladecontribute to the virtual work variation, 6 W


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BHAT GANGULI: VALIDATION OF HELI OPTER AEROELASTJC ANALYSIS W I TH EXPERIMENTAL DATA

2 .1 .1 Fini te E lement Discre t i sa t ionFinite element method i s used to discretise the

governing equation of motion, and it allows foraccurate representation of complex hub kinematicsand non-uniform blade properties. After the finiteelement discretisation, Hamilton s principle is writtenas

Each of the N beam finite element has 15DOFs. These DOFs correspond to cubic variationsin axial elastic and flap, lag-bending deflections,and quadratic variation in elast ic torsion. Betweenthe elements, there is cbntinuity of displacementsand slope for flap and lag-bending deflections, andcontinuity of displacements for elastic twist andaxial deflections. This element ensures physicallyconsistent linear variations of bending momentsand torsional moments and quadratic variations ofaxial force within the elements. The shape functionshere are Hermite polynomials for lag and flap bendingsand Lagrange polynomials for axial and torsionaldeflections. Substituting = H (where H is theshape function matrix) in the Hamilton s principle(Eqn 2 . one obtains:

The displacements, q are functions of timeand all nonlinear terms have been moved into theforce vector. Spatial functionality has been removedusing finite element discretisation, and partial differentialequations have been converted to ordinary differentialequations.

The finite element equations, representingeach rotor blade, are transformed t o normal modespace for efficient solution of blade using themodal expansion. The displacements are expressedin terms of normal modes as = @ p . Substitutingthis equation in to Eqn 3 ) leads to normal mode

equation having the form:

where the normal mode mass, stiffness, dampingmatrix, and force vector are defined as =Q T ~

= a r c @ , = @ T K O and = Q T F respectively.Integrating Eqn (4) by parts, one obtains:

The RHS of the Eqn. is zero because of theperiodicity condition of the response. Hence, Eqn 5)yields the following system of first-order differentialequations:

where

2 . 1 . 2 Finite Element-Temporal Discre tisationThe above equation is nonlinear because P

contains noniinear terms. The nonlinear, periodic,ordinary differential equations are then solved forblade steady response using the finite element ontime in conjunction with the Newton-Raphson method.Discretising Eqn 6 )over N, time elements aroundthe circumference (where ty = 0 W N, I = 2R ) andtaking a first-order Taylor s series expansion aboutthe steady state value

yields algebraic equations :


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EF SCI 1 VOL. 54, NO. 4, OCTOBER 2004

Here, K i s the t angen t ia l s t i f fness mat r ix fo rt ime e lement i , a n d Q, i s the load vec tor . A lso , themoda l d i sp lacement vec to r can be wr i t t en as

Substituting Eqn 9) and its derivativ e in Eqn 7)a n d s o l v i n g i t e r a t i v e l y , y i e l d s t h e b l a d e s t e a d yr e s p o n s e .2.2 A e r o d y n a m i c M o d e l l i n g2 .2 .1 Quas i - s teady Aerodynamic Mode l l ing

Q u a s i -s t e ad y a e r o d y n a m i c a n a l y s i s a s s u m e st h a t t h e b l a d e a i r l o a d s a r e s o l e l y a f u n c t i o n o ft h e i n s t a n t a n e o u s b l a d e s e c t i o n a n g l e o f a t t a c k .T h e q u a s i - s t e a d y a e r o d y n a m i c a n a l y s i s r e s u l t s i nt h e f i n i te e le m e n t m a s s , d a m p i n g a n d s t i f f n e s sm a t r i c e s , a n d t h e lo a d v e c t o r a s s o c i a t e d w i t h t h ea e r o dy n a m i c l o a d i n g o n t h e b l a d e a n d f u s e l a g e .Th e calculation of this m atrix requires the calculationo f t h e i n c i d e n t a i r v e l o c i t y o n t h e b l a d e i n t h ed e f o r m e d f r a m e , w h i c h i s g i v e n a s

w h e r e w i s the wind ve loc i ty wi th con t r ibu t ionsf r o m t h e v e h i c le f o r w a r d s p e e d a n d r o t o r i n f l o w ,s t h e b l a d e v e l o c i ty r e l a t i v e t o t h e h u b - f i x edf r a m e r e s u lt i n g f r o m b l a d e r o t a t i o n a n d b l a d emotion, and 17 i s the blade veloci ty du e to fuselagem o t i o n . O n c e a i r v e l o c i t y i s c a l c u l a t e d i n t h edeformed plane, 2-D strip theory is used to determinet h e a e r o d y n a m i c l o ad s o n t h e b l a d e . T h e s e a r ewr i t t en as

where ba r iden t i f i e s fo rces and moments in thedeform ed p lane , the sub sc r ip t e iden t i f i e s fo rce o fc i rcu la to ry o r ig in , V i s the incident veloci ty , andC C nd m re the sect ion l i f t , drag, p i tchingmoment coeff ic ient , respect ively .2 .2 .2 Uns teady Aerodynamic Mode l l ing

T he aerodynam ic environment is highly complexa n d u n s t e a d y , w i t h r e g i o n s o f t r a n s o n i c f l o w ,sep ara te d f low, and dyn amic s ta l l . An e f f ic ien t andcomp at ib le ae rodynam ic mod e l was p resen ted byLe ishman5 . Th i s ae rody nam ic mode l cons i s t s o f anat tached potent ia l f low fo r l inear unsteady air loads ,a separated f low form ulat ion for nonl inear unsteadyair loads , and a dynam ic stall formulation fo r vortex-induced air loads .

T h e a t t a c h e d f l o w f o r m u l a t i o n is based ont h e i n d i c ia 1 r e s p o n s e m e t h o d , i n w h i c h r e s p o n s eis computed f rom a f in i t e d i f fe rence approximat iont o D u h a m e l s i n t e g r a l . N o n l i n e a r a e r o d y n a m i cform ulation for sep arate d flow is based on Kirchhofftheory, which re la tes the a i r loads to angle of a t tackand the t r a il ing ed ge separa t ion po in t loca tion . Thedynam ic s ta ll fo rmula t ion accoun ts fo r the vor tex-induced aerodynamic loads. T he formulat ion modelsthe separa t ion o f the concen t ra ted l ead ing edgevortex.2 2 3 Free-wake Aerodynamic Mode l l ing

The main fea tu re o f th i s s tudy i s the use o fpseudo-implicit free -wake m odel, which is modifiedto include elastic fl ap, lag, and torsional deformationaround th e az imuth6. Th e e f fec t o f the ro to r cyc l i ccon t ro l s on the b lad e o r ien ta t ion wr t the hub p laneis a lso considered . Th e wake geom etry is calculatedin an inner loop ins ide the t r im procedure . Theb lade s teady response i s passed on to the wakesubrout ines to obtain the posi t ion of the blade-control points and vortex-release points accurate ly .Th e blade flap, lag, and elastic torsional displacements


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BHAT G A N G U L I V A L I D A T I O N OF HELICOPTER AEROELASTIC ANALYSIS WITH EXPERIMENTAL DATA

as well as rotor-control input are passed on to thewake subroutines. The wake model, however, utilisesa second-order accurate accelerated-convergencepseudo-implicit iteration method that is more stablethan an explicit method.

3 HUB LOADS COU PLED TRIMSteady and vibratory components of the non-

rotating frame hub loads are calculated by theindividual contributions of individual blades. Forthis, the motion-induced aerodynamic and inertialloads are integrated along the blade span to obtainblade loads at the roots and then summed over theblade to obtain the rotor hub loads.

~, , y) = g ~ ; ; osy,,, 1i-y ~inv, p ~ gOSIY,,),=I

Calculation of steady hub loads is needed totrim the helicopter. The harmonics of hub loadsare responsible for vibration and dynamic stresses.

Once the hub loads are obtained, the helicopterneeds to be trimmed. This is defined as the conditionwhere the steady forces and moments acting onthe helicopter sum to zero and simulates the condition

for steady-level flight. The trim solution for thehelicopter involves finding the pilot control anglesQ at which the six steady forces and momentsacting on the helicopter are zero:

The trim equations are solved iteratively usinga Newton-Raphson procedure. A coupled trimprocedure is carried out to solve the blade response,pilot input trim controls, and vehicle orientation,simultaneously. This procedure is called trim sincethe blade response [Eqns 7) and 9 ) ] and trim[Eqn 13)l are simultaneously solved, therebyaccounting for the influence of elastic blade deflectionson the rotor steady forces.

The coupled trim is solved iteratively untilconvergence. The coupled trim procedure is essentialfor elastically-coupled blades since elastic deflectionsplay an important role in the steady net forces andmoments generated by the rotor.4. HELICOPTER BASELINE MODEL

The helicopter is modelled in the Universityof Maryland Advanced Rotocraft Code UMARC)as an aircraft with a single main rotor and tailrotor. The main rotor is modelled as a hingelessrotor system. Each blade is identical and is definedby undergoing flap, lead-lag, torsion, and axial degreesof motion. The blade is divided into 10 finite elements.To validate the structural model, the blade naturalfrequencies are calculated. Figure 1shows the bladefrequencies versus normalised rotor speed. In general,good correlation of calculated frequencies is observedwith the experimental results.

To validate overall aerodynamics, the basicperformance predictions at steady forward flightare compared with flight test data Figs and 3).The main rotor power predictions are plotted inFig. 2 for nondimensional thrust coefficients C,/aof 0.05848 and 0.0708. Satisfactory results areobtained at all flight speeds. However, at low-flight


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D E P S C I J V O L 54 N O 4 O C T O B E R 2004

R E D I C T E D. E X P E R I M E N T A L

igure 1 Hingeless main rotor blade frequencies

I2C a= 0 05848

EROELASTIC ANALYSISlo E X P E R I M E N T A L R E S U L T SH O V E R T H E O R E T I C A L

igure 2 Nondimensional main rotor power coefficients wrt advance ratio


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BHAT GANGULI : VALIDATION O F HELICOPTER AEROELASTIC ANALYSIS WITH EXPERIMENTAL DATA

O N G I T U D I N A L S H A I TT I L T ANGL E

AI N ROT OR COL L E CT I VEkONGITUDINAL CYLtC

AIL ROTOR COLLECTIVE

5 ik


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D EP S I I, VOL. 54, NO. 4, OCTOBER 2004

. REE WAKE --, UNSTEADY UASI STEADY

0.100 275 37 0.100 235V R R V R R

Figure 4. Amplitude of hub forces and momentspredictions with the flight test data demonstratesthe validity of overall aerodynamics. The hub loadcalculations are performed for quasi-steadyaerodynamics and unsteady inflows with linearinflows and free-wake inflow aerodynamic models.REFERENCES1. G. Bir, et al. University of Maryland advanced

rotorcraft code UMARC) theory manual. UM-AERO Report 92-02, 1992.

2. Johnson, W. A comprehensive analytical modelof rotorcraft aerodynamics and dynamics, Part 1:Analysis development. National Aeronautics andSpace Administration NASA), USA. NASATechnical Memorandum No. NASA-TM-8112,June 1980.

3. Lim, J.W. Analytical investigation of UH-60Aflight blade air loads and loads data. In 51thAnnual Forum of the AHS. Fort Worth, Texas,May 1995.

4. Hodges, D. Dowell, E. Nonlinear equationsof motion for the elastic bending and torsionof twisted nonuniform rotor blades. NationalAeronautics and Space Administration NASA),USA. NASA Technical Note No. NASA-TN-D-7818, 1974.

5. Leishrnan, J.G. Indicia1 lift approximations fortwo-dimensional subsonic flow as obtained fromoscillatory measurements. Journal of Aircraft1993, 30 3), 340-5 1.

6. Ganguli,R.; Chopra, I. Weller, W.H. Comparisonof calculated vibratory rotor hub loads withexperimental data. J Amer. Helicopter Soc.1998, 43 4), 312-18.

7. Drees, J.M. A theory of airflow through rotorsand its application to some helicopter problems.J Helicopter Asso. Great Britain 1949, 3.

8. Leishman, J.G. Principles of helicopteraerodynamics. Cambridge University Press, 2000.


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BHAT GANGU LI VALIDATION O HELICOPTER AEROELASTIC ANALYSIS WITH EXPERIMENTAL DATA

C o n t r i b u t o r s

M r S h r in i va s R. B h a t c o m p l e t e d h i s B E M e c h E n g g ) f ro m t h e M . S . R a m ai ahInst i tu te of Technology, Bangalore Universi ty . He i s working as Research A ssis tantin the Dept of Aerospace Engineering, Indian Inst i tu te of Scien ce I ISc), Bangalore .

D r R a n j a n G a n g u l i c om p l e te d h i s B T e c h H o n s ) fr o m t h e I n d ia n I n s ti tu t e o fT echn o logy I IT ) . Kharagp ur and h i s S and P hD bo th f rom the Univer s ity o fMary lan d , Co l l ege Park , USA. H e i s work ing as Ass i s t an t P ro fesso r in the Dep tof Aeros pace Engineering, IISc, Bangalore. H is area s of research include: Helicopterdynamics , hea l th moni to r ing , smar t s t ruc fu res and op t imisa t ion . He has 37papers publ ished in nat ional l in ternat ional journals and 2 papers publ ished1presen ted in confe rences .

hinglesss rotor eom model

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