Dynamics of Helicopter Rotor Blades

8/22/2019 Dynamics of Helicopter Rotor Blades

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DYNAMICS OF HELIC~~ER ROTOR BLADESP. J. MAGARI, L. A. SHULTZ and V. R. MURTHY

Department of Mechanical & Aerospace Engineering, Syracuse University, Syracuse, NY 13244, U.S.A.(Receiv ed 14 Sept ember 1987)

Abstract-A rotating blade finite element with coupled bending, torsion and axial stretching degrees-of-freedom is developed. APL is used for symbolic manipulations required for the development of theelement. The element is implemented in MSC/NASTRAN to generate the numerical results. The resultsare compared with the experimental and other published numerical results and are found to be accurate.There are several immediate potential applications. The implementation of the element in the existing finiteelement software greatly enhances their utility for helicopter applications.

NOTATIONarea of cross section, in2Youngs modulus, lb/in2mass centroid offset from the elastic axis, positivein front of the elastic axis, in.shear modulus, lb/in2coordinates of load paths on clevis, in.blade cross section moment of inertia about y-axis,in4blade across section moment of inertia aboutz-axis, in4St Venants torsional stiffness constant, in4element of stiffness matrix, ith row, jth columnmass radius of gyration (kf,=k& +k$)element length, in.element of mass matrix, ith row, jth columnmass/length, lb-sec2/in2radius of rotor, in.tension, lbstrain energyaxial deflection, in.elastic displacements in the y (chordwise) and z(flapwise) directions, respectively, in.kinetic energyglobal x coordinate to root of element measuredfrom center of rotation, in.undeformed coordinate system rotating with rotorpretwist angle, radrotation about y-axis, radrotation about z-axis, radelastic twist about x-axis, radrotor rotational speed, rad/secfrequency of vibration, radfsec

Subscripts and superscriptsderivative with respect to xderivative with respect to timei ith load path

lNTRODUCIIONThe finite element method is a universally applicableapproach for obtaining numerical solutions to a widevariety of problems in the structural, fluid flowand heat transfer areas. The method is extensivelyused in the aerospace industry for structural analysesand several computer programs such as MSC/NASTRAN, ANSYS, MARC etc. exist for this pur-

pose [l]. A survey of these programs clearly showsthat they are incapable of analyzing rotating helicop-ter blades other than simulating a simple centrifugalforce field. The finite element method is currentlybeing applied to isolated blade dynamic prob-lems [2-131 and shows a great potential for isolatedblade as well as coupled rotor-body analyses. In arecent reference 1141 the finite element method wasapplied to determine the free vibration characteristicsof pretwisted blades with coupled bending degrees-of-freedom. The main limitations of this paper forapplication to helicopter rotor blades are (i) no bladerotation, and (ii) no torsional or axial degrees-of-freedom.The main objective of this paper is to developa rotating blade finite element which can be im-plemented easily in the existing finite element soft-ware such as MSC/NASTRAN. The element is tohave the following features:

i. No assumptions regarding the coincidence ofneutral, elastic and mass axes are to be made.ii. Significant coupling-type terms associated withthe centrifugal forces are to be included in the model.The equations derived by Houbolt and Brooks [I 51provide an example of the coupling that can existbetween the various degrees-of-freedom due to bladerotation.

The derivation of the stiffness and the coupledmass matrices is analytical and involves very tediousalgebraic operations, integrations and differenti-ations, and computer programs are developed in APLfor these symbolic manipulations. AFL is an inter-mediate level computer language, which is very con-venient for scientific calculations involving arrayprocessing. APL was introduced 20 years ago and itsgrowth and development is still going on. BecauseAPL treats arrays uniformly, there is new interest inthe use of APL for describing problems for parallel,vector, or array computing. The computer programsdeveloped in APL to carry out the symbolic oper-ations are given in Appendix I.

763


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76 4 P. J. MAGARIet alIn order to validate the present rotating blade finite

element, it is first utilized to perform a coupledvibration analysis on a conventional propeller. Sol-utions in the form of natural frequencies are gener-ated via the present formulation and compared to theexperimental data [16] for the vibrating propeller.Excellent accuracy is obtained with the use of only afew elements, thereby showing high computationalefficiency. Then the new element is used to performa coupled vibration analysis on a specific advancedconfiguration bearingless multiple-load-path rotorblade[17], similar to those in NASAs ITR/FRRproject [18]. The bearingless rotor technology fromthe ITR/FRR program may well be incorporatedinto the Armys next generation helicopter program.A fast converging method is developed to takeinto account the nonlinearities associated with themultiple load paths and the rotation.The rotating blade finite element developed in thispaper can be implemented in MSC/NASTRAN inthe following ways.

1. Through GENEL, Bulk Data Cards.2. Through DMI Bulk Data Cards.3. Through the INPUTT4 DMAP Module.4. By incorporation as a dummy element.5. By incorporation into the library as a distinctly

recognizable element.The first three of these approaches are used togenerate the results presented in this paper.

HAMILTONS PRINCIPLE

The equations of motion can be derived using theextended Hamiltons principle [191which states: of allthe admissible displacements between specified initialand final conditions at times t, and t2 which satisfythe prescribed boundary conditions of the system, thedisplacements which also satisfy equilibrium andstress boundary conditions are selected by the ex-tremum principle given by the following eqn.

(1)where

V = kinetic energyU = potential energy

6W = virtual work done by the body and surfaceforces

6 E variation.COORDINATE SYSTEMS

The coordinate system (x, y, z) rotates with theblade such that the x-axis falls along the undeformedelastic axis of the blade as shown in Fig. 1. (n, I) arecross-sectional axes located in the cross-sectional

Elahic axisFig. 1. Blade undeformed coordinate system.

plane which is normal to the undeformed elastic axisof the blade. The cross section of the blade is assumedto be symmetric about the chord and the n-axisis defined as the major principal axis along thischord. The C-axis is perpendicular to the chord andpasses through the shear center as shown in Fig. 2.The angle between the global y-axis and the cross-sectional major principal axis n is defined to be thepretwist of the blade and is denoted by /l (see Fig. 2).The locations of a point P before and after defor-mation in the cross-sectional plane are shown inFigs 3 and 4, respectively. The position vector of thispoint in deformed coordinates (x, , y, and z]) can be

Fig. 2. Blade cross-sectional axes.

\ Elastic axisFig. 3. Location of P before deformation.


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Dynamics of helicopter rotor blades 76521

- Y, ------4

i _____& iPi_____)1

I 6I I YlFig. 4, Location of P after deformation.

written asr=ix,+jy,+kz,, (2)

where i, j, k are unit vectors along the undefo~edrotating axes x, y and z, respectively. The relationsbetween the undeformed global (y, z)-axes and cross-sectional principal (Q [)-axes are given by (see Fig. 3)

y=qcosfl--[sin/?z =q sin/3 -4 cosfi. (3)

The relations between the undeformed and deformedglobal coordinates are given by (see Fig. 4)

x,=x+u-yu-zwy,=y+u--z#z,=z+w+y&. (4)

STRAIN ENERGYThe strain energy of the rotor blade undergoing

coupled flapwise bending, chordwise bending, tor-sion, and axial stretching deformation is given by

U=i s ~~~~~~~~sin~ -wWcos&20+EZ~(ucos~ + wsinfl)*+GJ(~)*+ EA(u)* + T(v)2 + T(w)2]dx, (5)

where

sT(x) = Q*mx dx. (6)xThe derivation of the above equation is based on theprinciples of en~nee~ng beam theory and follows thetreatment of Houbolt and Brooks [15].

KINETIC ENERGYThe kinetic energy of the blade can be written as

The velocity vector of a cross-sectional point is givenby ~fferentiating the position vector r given by eqn(2) as shown below.

dr .&=f+wXr,where w is the angular velocity vector of the triad(i, j, k) and is given by

w=kQ. (9)Substitution of eqns (2)-(4), (8) and (9) into eqn (7)yields

V=k oR{m~~ti-*~)+(i+Ctx+*u)Z+ti2]s+2mecos/?[-(ti -Ciu)(ti+Ci)-(tj+Rx+SZu)~u+~~-Rx#(~+nw)]+2me sin/31 -(ti -C&)(6-Q4)-(d+nx+nu)(b+nw)+52*,~,]+ m(k& cos* !I k& sin2 fl)x [(d+sQ*+ (Qu)* + fj-nz@ f2Qrgw+ 2W@] + mfk& sin fl + k& cos2 i)x [(I+ a$)* f (q4 + nw )*]+m sin28(ki,-ki,)[(d+R)(3-04)+SW($ +nw)-n(~o~+~lj~)))dx, (10)

where

ss pydrtd5 =mecos/3A

ss pzdqd[=mesinflA

IJ py2 d? di = m(k& COG f k& sin2 8)A

ss pz* dtl dC = m(kj,, sin* /I + k$ cos2 8)Apy z dq d[ = m cos /? sin /?(k& - k& ).


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166 P. J. MAGARI t al.For the case of conventional helicopter rotor bladeapplications, the above kinetic energy expression canbe simplified to the following eqn:

If=; R{m[li2+dZ+~2+R2U2+R2y2+*2X2s 0- 2nvli + 2R3xu + 2Rxd + 2nuti1+ 2me cos b [R2u + biuj - Q2x+v - R2xv]+2me sin/l [-tifj +~2x~v'-R2v~-~2~~']+m(k&cos2j? + k& sin2j!J)[fj2+ n2$*]+ m(kf,,, sin/l + k& cos2 jI)[d2 + Ci2d2]- 2m cos fl sin fl(k& - k$)Ci2$)} dx. (11)

SHAPE FUNCTIONS

In order to evaluate the integrals of eqns (5) and(11) the displacement fields within the element areassumed as shown to be

Axial stretching u = a, f a,xTorsion 4 = b, + 6,x

Chordwise bending u = c,, + crx + c2x2 + c3x3Flapwise bending w = d9 + d,x + d2x2 + 4x3.

The following shape functions are derived from theabove displacement functions to express the displace-ments in terms of the nodal displacements.

Axial stretchings,= l-x/ls2 = x/l.

Torsions,=l-x/lsq = x/l.

Chordwise bendings5 = 1 - 3x2/12 + 2x3113s - x - 2x211 + x3/12-s, = 3x2/13 - 2x3/13sg = -x2/1 +x3/P.

Flapwise bendings9 = 1 - 3x3/12 + 2x3113

SIO - x + 2x211 -x3/ss,, = 3x3/12 - 2x3/13s,2 = x2/1 - x3/12.

The displacements in terms of nodal displacementsand shape functions are given byu(x) = siui + s3u2 (12)

4(x)=s34,+s442 (13)

~(x)=s,u,+s,e,,+s,v,+s,e,, (14)

w(x) = S9Wl+ sloe,, + si, w2 + s,,e,,. (15)

STIFFNESS, MASSAND DAMPING MATRICESBy taking the variation of eqns (5) and (1 l), the

following eqns can be obtained:

6U= R{[EIY(t(sinj?-wcos/?)]I 0x [sin /I& - cos ~SW]+[El,(v"cos~+w"sin~)]

+ [cos /kSv" + sin /18wn]

+ GJq5 J + EAuGu + TV 6~+ TwGw} dx (16)

6V= oR{m(i22r4R2x +*d)6u +m(ti -Rv)&iI+ [m (fi*v - f&i) + me cos /I R2- me sin bn24]&+ [m(ti + Rx + Qu) -me sin /I$]%+ (mti + me cos fi&)Sti + [ me cos /?R'xw+me sin/?(-n2xv-Q2v)-m cos 2fl(k& - ki,)Ci24- m cos /I sin fl k& - ki,)R2]64+ (me cos /hip - me sin pi, + m ki t$ )SC$+ (- me cos fif12x + me sin j3f12x~)6v'+(-me cosj?R2x~

-me sin jIQ*x)6w} dx. (17)By substituting eqns (16) and (17) into eqn (1) andintegrating by parts with respect to time, one canshow that Hamiltons principle becomes

SH= -6U+6V=O, (18)where 6U is given by eqn (16) and 6 V is given by the


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Dynamics of helicopter rotor blades 161following equation.

6V= a(I-jmii - C(x + 24)- 2fI?rId]6u

+[-m(ii-R2u)+me sin/%$+ R2me (cos B - 4 sin /I) - 2Rmti]6v+ [- mti -me cos /?fJ]Sw+[-Q2mevsin/?+me(iisin/J-ticos/?)- R2m (ki, - k$)(cos j3 sin B + 4 cos 28)- mki$ - Rzmex(-v sin /l + w cos /I)]&$- timex (cos /I - 4 sin /I)&-R2mex(sin/?++cosfl)6w}dx. (19)

If a finite element discretization is employed forHamiltons principle, then eqn (18) becomes

6H= i 6Hi = i (-SUi +6Vi)=0, (20)i =l i =lwhere

6Hi= -SUi+SVi . (21)By substituting eqns (12)-(15) into eqns (16), (19) and(21), and by recognizing that the limits of integrationof eqns (16) and (19) are xi and xi+, the followingequation can be obtained for the variation of theHamiltonian:

6Hi = {hi}T([W{4iii) + [Cl{4,1+ [W(cliI+ {fI)v (22)

where[M] = mass matrix[C] = damping matrix[K] = stiffness matrix

{f } = motion independent force vector.The components of the above quantities are given

byMjk = a2(m)ja~a(sqk)cjk = a2@Hi)/a+%%k)

& = a2@Hi)/%?$(k)f, = WHWqj,

where dH l i is a linear product of the form

(23)(24)(25)(26)

The expressions for the resulting stiffness and massmatrices are too lengthy to present in this paper andtherefore only the stiffness terms that arise due toblade rotation are presented in Appendix II. Thesecan be added to the nonrotating blade stiffness matrixcomputed from the existing finite element codes,thereby allowing fully coupled modeling of therotating systems.

TENSION COEFFICIENTSThe stiffness matrix of the element depends on the

tension, and in the case of a conventional rotor bladethe tension can be calculated in advance from eqn (6).The resulting equation in the element coordinatesystem is given by

1Xi+ 1,T(x) = T2 + R2mx dx, (27)X, X

where

sR

T , = n2mx dx (28)Xi 4ii = length of the ith element

i, i + 1 = nodes of the ith elementxi = distance of the ith node from the axis of

rotation.The fully articulated rotor (Fig. 5) is in use on most

production helicopters in the world today. This rotorsystem contains three hinges: the flapping hinge, thelead-lag hinge and the pitch hinge. The flappinghinge was initially introduced to solve the bodyrolling problem due to the differential lift generationof the advancing and retreating blades in forwardflight. The lead-lag hinge was introduced to relievehigh stresses in the blades resulting from the period-ically varying in-plane forces due to blade flappingmotion. The pitch hinge was provided to facilitatethe blade collective and cyclic pitch controls. Thepresence of these hinges complicates the rotor systemand recently development of bearingless rotors wasinitiated to eliminate this complexity [18]. The

Pitchhinge.

a Hub rotationFig. 5. Articulated rotor system.


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768 P. J. MAGARI et 01.

IIIk1 Load Dath IIII

R 3 I

Load path 3

Fig. 6. Model for a triple-load-path rotor blade.

bearingless rotorcraft offers reduced weight, lesscomplexity and superior flying qualities, and it ismost likely that the next generation rotorcraft will beequiped with those rotors. The recent advances inmaterials and understanding of rotor dynamics areresponsible for the development of the bearinglessrotors. All practical designs of bearingless rotorsinclude multiple load paths, and the one that wasflight tested by Boeing Vertol had three loadpaths [191.The model of such a triple-load-path rotorblade is shown in Fig. 6.

While the tension on the blade is calculated usingeqn (27) as before, the tension in the n load pathsare calculated as follows. Assuming that the n loadpaths are coincident with the blade at the clevis, it canbe shown [17] that the tension in the ith load path atthe clevis is given by

where

fi a,=a,a,...aJaij=l

r' dxa=J .Ei

I = length of the load path.Now, the tension in the ith load path correspondingto the coincident load path is given by

sI

T(x) = Q2mx dx + T :. (30)XOnce the approximate tensions in the load paths andthe blade are calculated as indicated above, the staticstiffness matrices of the system can be calculated andnew tensions in the load paths at the clevis (T,) canbe calculated. These new values are used to updatethe T i in eqn (30). At most, two or three iterations

are needed to achieve the convergence. The finalconverged values are the desired tensions in the loadpaths and the blade.

NUMERICAL RESULTS AND DISCUSSIONIn order to validate the present formulation, a

coupled vibration analysis is performed on a rotatingtwisted propeller blade having a nonuniform, un-symmetrical cross section and cantilever boundaryconditions at the root. The computed results arecompared with experimental data obtained in theform of natural frequencies of the system.The propeller blade analyzed is the WADC S-5scale model [16]. The propeller radius given is 24 in.(measured from the center of rotation), with the bladeroot effectively cantilevered 6 in. from the center ofrotation. Except for rotational speed, all the datanecessary to compute the natural frequencies is givenin Table 1 for a propeller pitch setting of 0 at 0.75R.For different pitch settings the only change inthe input data is a change in the values of twistcorresponding to the desired pitch setting.

The first, second and third mode naturalfrequencies are computed for cases corresponding tothe rotational speeds and pitch settings of the experi-mental data[16,20]. Table 2 shows the very closecorrelation between the natural frequencies generatedTable 1. Data for the WADC S-5 scale model propellerGrid pointlocation, x m(in.) (lb-&/in*)

6 1.026 x 1O-38 0.69610 0.66012 0.60814 0.56416 0.53518 0.52020 0.50622 0.49824(tip) 0.498

Y(lb-n?) Et2 6(lb-in ) (deg)0.200 x 106 63 x lo6 30.50.110 49 25.2

0.083 46 20.10.058 44 14.80.042 43 9.60.03 I 43 4.70.027 44 00.026 47 -4.20.025 51 -7.50.024 56 - 10.0x,=6in.; I =2in.; R=24in.


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Dynamics of helicopter rotor blades 769Table 2. Comparison of natural frequencies (Hz)

Natural frequenciesMode R @pm) FEM EXFT TMM1 1567 40.79 40.08 40.962 109.32 - 109.223 279.95 - 279.791 1589 41.31 41.352 109.97 107.53 109.773 280.62 - 280.471 2609 60.16 58.73 60.072 139.65 - 139.523 319.38 - 319.401 2614 60.44 - 60.162 139.47 137.02 139.683 319.59 319.621 3583 78.27 76.52 78.34

by the present formulation (FEM), experimentally ob-tained values (EXIT), and natural frequencies of thesystem computed by the transfer matrix (TMM) [20].Single- and triple-load-path systems are analyzed.Cantilever boundary conditions are assumed at theinboard ends of all the load path member(s) for thesingle- as well as the triple-load-path system. Theload path members are connected to the bladethrough a rigid clevis as shown in Fig. 6. Noting thatthe clevis is completely rigid, the multipoint con-straints across the clevis are as given below:

U = u + h;e; + /r;e;0 =#-h;4w = w+hjcj4=4ie,=e;6, = 9;. (31)

The subscript i denotes load path quantities and canvary in value (l-n) dictated by the system underanalysis. The coordinates of the location of the loadpaths on the clevis relative to the blade root locationon the clevis (point 0) are given by hi and hi.

Table 4. Comparison of natural frequencies (rad/sec),R=OrpmSingle load path Triple load pathMode FEM TMM FEM TMM

1 7.3184 7.3243 11.4351 11.42982 42.4490 42.4708 60.8580 65.83683 45.9095 45.9001 70.4045 70.36644 128.5689 128.5038 160.3119 160.25495 133.6289 133.6382 181.7632 181.71206 251.9575 251.8232 258.2888 258.72647 266.1199 266.1357 490.8899 400.29518 401.4146 400.5921 430.9312 430.50559 416.5278 416.4330 475.4824 475.018210 622.2776 621.0142 666.7434 665.9004

Table 5. Comparison of natural frequencies (rad/sec),R= 360rpmSingle load path Triple load path

Mode FEM TMM FEM TMM1 36.7365 36.7738 45.0691 45.09982 48.0705 48.1092 72.3661 72.33763 105.1274 104.9309 125.2526 124.98404 138.0839 138.2931 164.1234 164.37215 202.6147 202.4001 232.3592 232.10996 280.5532 280.5927 318.5969 318.30117 336.7368 336.3352 426.8616 426.13328 402.8845 402.5505 477.1384 476.46669 508.3706 507.5868 499.9177 499.388510 671.6609 669.1642 742.3208 740.2850

The physical constants of the single- and triple-load-path systems analyzed are given in Table 3. Thenatural frequencies obtained from the present for-mulation are presented in Table 4 for single- andtriple-load-path blades for the nonrotating case(a = 0 rpm). The natural frequencies for the rotatingcase (Q = 360 rpm) are presented in Table 5. With theexception of the nonrotating single-load-path fre-quencies (which are simply a degenerate case of thesingle-load-path rotating formulation), all the resultspresented in Tables 4 and 5 are compared with thetransfer matrix solutions for the same rotorsystem [17]. This provides further validation of thepresent element and formulation. Both Tables 4 and5 show the excellent agreement between the results

Table 3. Data used for the numerical calculationsRadius of the rotor = 260 in.Distance of the clevis from the root = 52 in.Rotational speed (Q) = 360 rpmFlapwise bending stiffness @I,,) = 0.2977 x 10s lb-in*Chordwise bending stiffness (EI.) = 10 x lo8 lb-in*Torsional stiffness (GJ) = 0.2 x IO8 b-in2Axial stiffness (EA) = 10 lbMass per unit length = 0.0015 lb-sec2/in2Built-in twist = 0.0 degCollective pitch = 15.026 degDistance between mass and elastic axes = - 0.6 in.#,, = 0.8945 x lo- lb-se2mkk = 0.04 lb-sec2

For triple-load-path rotorh,, = 1.0 in.; h,, = 3.0 in.; hy2= - 1.0 in.; h, = - 1.0 in.;h,, = 2.0 in.; b, = - 2.0 in.


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770 P. J. MAGARI et al.Table 6. Comparative convergence of the element. Natural frequencies (rad/sec) for triple-load-path rotor,R=36Orpm

Mode Case 1 Case 2 Case 3Lumned Coupled Lumped Coupled Lumped Coupled TMM1 44.7709 45.0829 44.9860 45.0521 45.0320 45.0691 45.09982 71.7704 72.40753 120.7130 125.28274 163.7905 164.39125 224.20916 306.27597 412.1135! 465.9194469.847910 677.3142

232.4769318.7578427.0327484.0181501.1094753.9558

72.1452 72.2845 72.2875124.2684 125.2544 124.6992164.0283 164.1602 164.0494230.5968 232.3641 231.3711315.7322 318.6079 316.9766423.6169 426.8689 425.0334474.7075 477.8970 475.3533492.0505 499.9707 495.4612732.0105 744.2107 736.2747

72.3661 72.3376125.2526 124.9840164.1234 164.0571232.3592 232.3490318.5969 318.3011426.8616 426.1332477.1384 476.4666499.9177 499.3885742.3208 740.2850Case 1. Three elements for each load path, 7 for the blade.Case 2. Five elements for each load path, 15 for the blade.Case 3. Seven elements for each load path, 20 for the blade.

from the present finite element formulation and thetransfer matrix method. Finally, the rapid con-vergence of the element utilizing both lumped andcoupled mass matrices may be seen from Table 6. Allthe natural frequencies in Table 6 are for a triple-load-path rotating case (Q = 360 rpm) correspondingto the triple-load-path case presented in Table 3.Again, transfer matrix solutions are provided forcomparison.

All the above calculations are done in MSC/NASTRAN by feeding the element through thefollowing means.

1. GENEL Bulk Data Card.2. DMI Bulk Data Card.3, INPUTT4 Module.

The present formulation greatly enhances the capa-bility of MSJNASTRAN for helicopter dynamicsanalyses and implementation can be further im-proved by employing the following means.

1. Incorporation as a dummy element.2. Incorporation into the element library witha distinctly recognizable name.

CONCLUSIONS

In the present formulation, a new universal finiteelement is derived for nonuniform, pretwisted rotat-ing blades with flapwise bending, chordwise bending,torsion and axial stretching degrees-of-freedom.Also, for dynamic applications, a coupled massmatrix consistent with the stiffness matrix is derived.New APL symbolic routines developed here allow thedevelopment of stiffness and coupled mass matricesin an analytical form. Since no numerical integrationsare required to evaluate the matrices for any givenrotating system, the element has a high compu-tational efficiency and yields excellent accuracy.

The element is thoroughly validated by analyzing

a nonuniform, highly twisted propeller blade oper-ating at various high rotational speeds. The resultsgenerated agree very well with the experimentallyobtained results. The element is applied to analyzesingle- as well as multiple-load-path blades. Formultiple-load-path blades, a fast converging iterationscheme for computing the distribution of blade ten-sion in each load path is developed. The resultsfrom the present finite element formulation closelycorrelate with calculated results from other formu-lations.

There are several immediate potential applications.The element can be implemented in MSC/NASTRAN for coupled rotor-body analyses withinNASTRAN. Also, the new element could be usedfor the design and analysis of multiple-load-pathbearingless rotors, without making any equivalentsingle-load-path approximation.

1

234.5.6.7.8.

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1424-1430 (1985).15. J. C. Houbolt and G. W. Brooks, Differential equationsof motion for combined flapwise bending, Chordwisebending, and torsion of twisted nonuniform rotorblades. NACA Report 1346 (1958).16. W. F. Hunter, The integrating matrix method for deter-mining the natiural vibration characteristics of propellerblades. NASA TN D-6064 (1970).17. V. R. Murthy and A. M. Joshi, Free vibration charac-teristics of multiple load path blades by the transfermatrix method. J. Am. Helicopter Sot. 31,43-50 (1986).

18. W. G. Bousman, R. A. Ormiston and P. H. Mirick,

Design considerations for bearingless main rotor hubs.Presented at the 39th Annual Forum Proceedings of theAmerican Helicopter Society (1983).19. P. G. C. Dixon, Design, development and flight demon-stration of the loads and stability characteristics ofbearingless main rotor. USA AVRADCOM-TR-80-D3(1980).20. V. R. Murthy, Dynamic characteristics of rotor blades.J. Sound Vibr. 49, 483-500 (1976).

APPENDIX I: APL SUBROUTINES USED TO PERFORMALGEBRAIC MANIPULATIONS

POL YMULTThe left and right hand arguments of this function areexecutable APL statements of multivariate polynomials ofthe following form:+ ((2 x co1 x co2 + 14) x (X*7))WY constant variable

+((-5xcO6xco7+l)x(x*1))+~~~ww constant variableThe returned value is also an executable APL statementwhich is the multivariate polynomial multiplication of theright and left arguments.

Cl] ~kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkC23 kknkkkkkkkkkkkkkkkkk fiLGEBRb&IC SOFTWnRE BT kkkknkkkkkkkkkkkkkkkkkkkkkkC33 n~t~tr*~~**t~~tf*Lt*** PfiTRICK J XfiGaRI kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkC43 Skkkkkkkkkkkkkkkkkkkkkkkkkkk &ND kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkC53 Skkkkkkkkkkkkkkkkkkkkkk LOUIS fi SHULTR kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt6] kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkC73 n THIS VERSION OF POLYMULT ACCEPTS CIND RETURNS ILGEBR&IC STaTEMENTSC83 R WHICH LRE EXECUTfiBLE @PL STATEMENTS[9] n FIND THE INDEX POSITIONS OF THE PLUS SIGNS BETWEEN TERMSCl01 INDEXSPl+(((iQ(=Pl)h+ =Pl)/\FPl)r(F~l)+lCl11 xNnEXSP2+(((1@( =P2)A+=P2)/\FP2),(FF2)+1Cl21 orRl+l 0 PlP2c[13] a LOOP1 SELECTS CI TERM IN F2 TO OPERfiTE ON Pl WITH[I43 LOOPl:CTR2+1[15] n LOOP2 LOOPS ON GILL TERMS IN Pl FOR ONE TERM IN P2Cl61 t-oop2:Cl73 a FIND ERICH TERM IN Pl fiblO F2Cl81 TERMT~cP~CINDEXSP~[CTR~] TO(INDEXSP1CCTR1+1]-I)]Cl91 TERMT2cP2[INDEXSP2[CTR2] TO(INDEXSF2[CTR2+1]-I)][20] R LOCATE CONST POSITIONS OF TPRHTl &ND TERMT2c211 R==Pl+L/() =TERMTl)/\FTERHTlc221 RP=P2+L/() =TERMT2)/\FTERMT2[23] 1 PULL OUT CONSTANT PaRT OF EaCH TERM


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712 P. J. MAGARIet al.

C241 TERMlt3S(RPCP1-~)~TERMT1C253 TERtQt3+(RPCP2-i)+tERHf2[26-j K FIND THE SIGN OF THE NEW MULTIPLIEQ CONSTRNTC273 CONSTSIGt~t~((~(l~TERwl),'l')X~(lfTEKM~),'1')~'~-'C283 R FIND MULT SIGNS aND DIV SIGNS IN TERM1 TErPM2t291 LTSlt(('x =TERMl)+(+ =TERHl))/\pTERMlc301 LTSZt((x =TERMz)+( +=TERH2))/\fTEHM2[31-j n FIND NUMBER CONSTINT I t4 FRONT OF NEW CONST TERMC321 OUTFTt,(+TERMlC2 TO(l~LTSl)-13)XzTE~~2C2 T'J(l+LT~2)-11C333 A FIND DENOYINRTOR OF CONSTaNTc343 BnCKTt+(+TERnlC((-l~LTsl)+l)+l) TopTERMl])XiTERM2[((-lbLTS2)+1) TOFTERHZ][35] n FINQ THE NEW TOTaL CONST TERML363 NCONSTt+,(( ,CONSTSIGN,OUTFT,(TERMl[((l+LTSl)) TO(l+LTSl)-I]),

TERY2[((lfLTS2)) TO(-lfLTS2)-l],+,BaCKT,)[37] AMULTIPLY OUT THE VaRIaBLES[38-J (I IND VaR TERMS aND DROP OFF TRaILING )Sc391 VTERMlt-l+RPCPlfTERMTlc401 VTERM2+-1+RPCp2+TERMT2c411 CTR3t3 0 NVTERMC ~421 A BRUNCH It= THERE ARENT at31 VCIRI~~LES IN VTERM~c431 +LOOP30UT IF (pVTERH1) =()[44] K TaKE EaCH TERM IN VTERMl aND TRY TO FIND IT IN VTERMZc451C461c471C481c491c501c511E521c531c543c551C563c571t581c591C601Lb11C621C6311643C651C663

A IF FOUND aDD EXP IF NOT JUST CaTENaTE VaR AND EXP TO NEW VTERMLooP3:K BRaNCH IF VIIR IS IN BOTH TERMS

+Ll IF+/VaRWHEREtVTERMl[CTR3]zVTERM~K CaTENaTE VaR TERM FROM VTERHl TO NEW VaR TERM WHICH WaSNT IN VTERM2

NVTERMeNVTERM,VTERM1C(CTR3-2) TO(CTR3+3)-JCTR3tCTR3+6 0 SLOOP3 IF CTR3(FVTERMl 0 -tLOOP-jOUT

Ll:EXPt+(+VTERMl[CTR3+2])+~VTERM2[(V~RWHERE~l)+2]NVTERMtNVTERM,VTERMlC(CTR3-2) TO(CTR3+1)],EXP,)

K DROP OFF VaRS nt4Q EXPS THaT aRE USED IN VTERM2VaRWHERE[(VaRWHERE\l)+ -2 -1 0 ,\3]tlVTERM2+(.wVaRWHERE)/VTERM2CTR3+CTR3+6 0 +LOOP3 IF CTR3(pVTERMl

LOOP30UT : NVTERMtNVTERM, VTERM2, ) ,q COMPILE VAR aND CONST TERMS

PlP2tPlP2,NCOnST,NVTERMR GO BaCK LNV GET a NEW TERM FROM p2 aNn KEEP SaME TERM FROM PlR IF aLL TERMS IN ~2 aRE NOT USED UP

CTR2tCTR2+1 0 +LOOP2 IF CTR~~(FINDEXSP~)-~K GO BaCK aND GET 6 NEW TERM FROM pl aND REPEaT SWEEP aCROsS P21 STOP IF aLL TERMS IN pi HavE BEEN USED UP

CTRltCTRl+l 0 +LOOPf IF CTRli(fINDEXSPl)-1


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Dynamics of helicopter rotor blades 173POL YINT +((-5xCO6xC07tl)x(X*l))+...

The right argument of this function is an executable APL -+statement of a multivariate polynomial of the form constant variableThe left argument is the variable with which the user+ ((2 x co1 x co2 t 14) x (X* 7)) would like to integrate with respect to. The returned valueWW is an executable APL statement which is the result ofconstant variable integration with respect to the desired variable.

C63c71C81c91Cl01Cl11Cl23El33Cl41El5111161

INDEXSPC(((~G(=P)A+=P),~+~CTRlel 0 IPc"

APICK OUT fi TERM IN P 1130 INTEGKfiTE ITLi :TERH+P[ INDEXSPCCTRl] TO INDEXSPECTRl+l]-11F DKC)NCH IF Vf3RICIDLE TO DE INTEGRIITED IS NOT IN TERM

+L2 IF(t/V=TERH)=()A LOCtJTE THE V&RICIBLE aND THE + SIGN IN THE TERM

Cl71 LOCV+(V=TEKH)/~FTERHCl81 LOCDIVt(+=TERM)/\rTEKMCl93 LOCRP+L/(((~G~X~=TEKM)A~)~~TERM)/~~~ERM)[20] A &DQ 1 TO EXPONENT OF VIhRICIDLE AND ClIV COMST BY HEW EXPONENTc211 TERM[LOCVt2]h+(+TERMCLOCV+23)+1c221 TERM+TERM[l TO LOCDIV],(~(~TERM~(LOCDIV~~) TO LOCRP-l])x+TERM[LOCV+23)

,TERH[LOCRP TOpTERM]C231 St-4[247 a TRIKE CARE OF TERMS THhT DONT COMTLIIN THE VC)KICIBLE OF INTEGRATIONC251 L~:TEKMc(-~~.TERM),x(~V~~~))C2.53 Ld):IP+IP,TERM 0 CTRlcCTRltlC271 -tLf IF INDEXSPCCTRl](pP

POLYPARTIAL +((-5xCO6xCO7+l)x(X*l))+~~~The right argument of this function is an executable APL -cyJstatement of a multivariate polynomial of the form constant variableThe left argument is the variable with which the user+ ((2 x co1 x co2 + 14) x (X*7))WCYJ would like to differentiate with respect to. The returnedvalue is an executable APL statement which is the result ofconstant variable differentiation with respect to the desired variable.


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114 P. . MAGARIt al.

t43 kkkkkkkkkkkkkkkkkkkkkkkkkkKk &ND kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkES3C63c73E83c91Cl03Cl13Cl23Cl33Cl41El51Cl63

SkkkkkkkkRkRkkkkkkkkkkk LOUIS fi SHULTf kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkSkkkkkkkkkkkkkkkkkkkkkkk~&kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkK INPUT THE POLYNQMICIL P f&ND THE VFIRICLBLE V WITH WHICH YOUk WOULD LIKE TO DIFFERhNYIIXtE WRTK LOCATE THE TERMS IN P

INDEXSP4(((1)(=F)~+irP)/tpY)to+1CtRl41 0 PP4

K PICK OUT A TERM IN P &ND DIFFERENTIflTE XTL~:TERXcP[IHaEXSF[CTRl] TO INDEXSP[CtR1+1]-l]k BRANCH IF VI)RICiDLE TO I?IFF WRT IS NOT It.8 TERM

+L2 IF(+/VofERM)=(-)K LOCLITE THE VCIRTI\RLE, i-HZ SECOND ( t&ND THE FIRST K

Cl73 LOCV4(V=TERM)/\FTERXEl83 LOCTIMES4~/(~/(w~TERM)/~~TERK),~f(~~TERM)/~~l~RXEl91 LOCOPCC(((~TERX)~~~T~~X)C~][20] k SUBTRACT 1 FROM THE EXFO1.~EWT OF V Gt4D MULT CONST DY OLD EXPCItMEiTc213 TERMfTERMCl TO LOCOPC],(~(~TERM[(LOCOFC+~) TO LaCTIMES-i])K~TERMCLOCV+

2]),TERH[LOCTIMES TOFTERY]L223 LOCV4(V=TERU)/\fTERMC231 TERXCLOCV+234t(+T~RXCLocv+23t-lr.243 SL4f253 fi LKE C&RR OF TERMS THCIT DONT CORTd#XN THE VPlRIClDLE OF VIFFERENTIb3TION[26] L2:TERMt[273 L4:PFcFP,TERX 0 CTRl4CTRftlC28] .$Ll IF INDEX?iP[CTRl]c$P1291 INDEX()4(PFC((VaPP)/~pPF)+~]~())/(V~PP)/\FFFt301 kRnSEt,(8(~sW)fIN~EXO)+(~H~(~IN~E:


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Dynamics of helicopter rotor bladesK53 * -Cl*L2 + (ZnClW3 - C2)*L3K53 - -(K53 + (Cl%4 + 2.%2%3)*14 + C2*c4*15)*cllKS4 = -L2 + (2.%3 - C9)013 + (2.%3%9 + C4)9L4 + c4*c9*uTEH - -I + (2.*c3 - c9)*12 + (2.ac3W9 + c4)*L3TEH - XJ*(TEn + C4*C9*L4)K54 = OMEGAS * H * E * CBETA * (TEH + K54)

K62 = 2.*C5*L2+(3.*C6 B.*C5%7))"13 6.*(2.*C6"C7 C5*C8)*L4K62 - (K62 + S*Cb'W*L5)*(-Cll)TEH = I2 + 2.*C7*L3 + (ca -(2 *c6ac7 + C5*CB)aL5)*L4 - (C6 + 2.*c5*C7)*L5- C6*Ca*L7K62 - K62 : TEnaOnEGAB*nK64 - 2,*C7*L3 + 2.*(C7*C9 + C8)*L4 + 2.*C8*C9*L5;g - (L + (c9+4.*c7)*L2 (4.*c7ac9*3.wa)*L3 + 3.*ca*cg*L4)*xJ

= -OWGAS*WE*SBETA*(TEn + K64)K66-L+a.nC7*L2+(16.aC7*C7+6.*Ca)*L3+24.*C7~Ca*L4+9.*Ca*Ca*L5TEH - L3+4.*c7*L4+(4.*c7w7+2.*ca)*L5+4.~7*Ca*L6+ca*ca*L7K66 - Cl1 * K66 - 0HEGAS"nWEnK71 = -OHEGAS*WClO*(L2 + C9*L3)K77 = -0HEGAS * H * Cl0 * Cl0 * L3K82 = -C11*(4.*C5*CS*L3 12.*CS"C6*L4 g.*C6*C6*L5)TEH = C5"L3 + C6*L4 - C5*C5*L5 - 2.'%5*C6*L6 - Cb*Cb*L7'K82 K82 - TEn'WnEGAS*nK84 =-OtfEGAS*n*E*SBETA*(C5*13 (2.*C6+C5*C9)*L4 2.%6*C9*L5)TEH - XJ*(2.*C5*12 (3.*C6+2.*CJ*C9)*13 3.*c6'W9*14)K84 - K84 - OliEGAS*n*E*SBETAWEHK86 -2.'W.C5*L2 (8.%59c7 + 3.*Cb)*L3 + 6.*(C5*C8 + 2.%6%7)*14+ 9.%6WI*L5TEH - C5*L4 + (2.+c5*c7 C6)*L5+(C5'%8 2.'%6%7)*L6 + CbaC8*L7K86 = cll*K86 - OnEGAS*HWEHK88 = 4.*cS*cS*W + 12.*CS*Cb*lk 9.*C6*6*UTEH - CS*cS*L5 +2.*cSW6*16 + Cb*c6*17K88 = Cl1 * K88 - OMEGAS *n *TEnK93 = -Cl1 * (cml*L3 + 2.aC1sC2*L4 c2%2*L5)K94 = Cl"L3 + (Cl%9 + c2)*L4 + c2*c9*L5TEH - XJ*(Cl*L? + (Cl'W9 + C2)*L3 + CZaC9*L4)K94 - OUEGAS*H*EWBETA*(K94 + TEH)K95 - -Cl*12 + (2*cl*c3 - C2)*L3 + (ClW4 + 2.*C2*C3)*14K95 - (K95 + C2%4*L5)aCllK99 - Cll*(Cl'W*L3 + 2.sCl*C2*L4 C2'%2*L5)K102 = L2 + C5*L4 + 2.*Cb*LSTEH = 2.*C5*L3 + 3.'W6*L4KlO2 - OHEGAS*n*E*SBETA*ClO*(K102 XJ * TEH)K103 - -OnEGAS*n*E*CBETA*C10*(Cl*L4+C2*L5+XJ*(C1*L3C2*L4))K104 - OHEGAS*(Kll2S-KHlS)*ClO*CTBETA*(L2C9*L3)KlO5 = 2.*C3*L4-L3+C4*L5 XJ*(2.%3*L3 - L2 + C4*L4)K105 = OHEGAS*n*E*CBETA*ClO*KlO5


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776 P. J. MAGARIet al .K106 = 2.*(C7*L4 + C8*L5)TEH = L2 + 4.*C7*L3 + 3.'%8*14K106 = -OMEGAS*H*E*SBETA*ClO*(K106 XPCEH)K108 = CS*L4 + 2.*C6*L5TEH = 2.*CS*L3 + 3.*C6*L4K108 = -OtlKGAS*H*E*SBETA*ClO*(K108XJWEH)K109 = OMEGA8*ti*E*CBETA*C1O*(Cl*L4C2*L5 + XJ*(Cl*L3 + C2*L4))KlOlO = OMEGAS * (KH2S - KHlS) * Cl0 * Cl0 * CTBETA * L3K113 = -Cll*(Cl*C3*L3'+ Cl*C4 + C2*C3)*L4 + C2*C4*L5)K114 = C3*L3 + (C4 + C3%9)*L4 + C4*C9*L5TEH = C3*L2 + (C4 + C3*C9)*L3 + C4*C9*L4K114 = OHEGAS*H*E*CBETA*(Kll4 + XWTEH)Kll5 = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~K119 = Cll*(Cl*C3*L3 + (Cl*C4 + C2*C3)*L4 + C2C4*L5)KlllO = C3*L4 + C4*L!i XJ*(C3*L3 + C4*L4)KlllO = OHEGAS*H*E'WBETA*C10*KlllOKllll = Cll*(C3%3*L3 + 2.*cPC4*14 + C4'W4*L5)K122 = -Cll*(4.*cS*c7*L3 6,*(CbaC7+CSaC8)*L4 9.*C6W8*L5)TEH = C7*L3 + C8*L4 - C5%7*L5 - (C6aC7 + C5%8)*L6 - CbaCB*L7K122 = K122 - OklEGAS*VTEHK124 = C7*L3 + (2.*cO + C7*C9)*L4 + 2.*C8*C9*L5TKM = XJ*(2.%7*L2 + (3.*c8 + 2.%7*C9)*13 + 3.%8%9*L4)K124 = -OHEGAS*tPE*SBETA*(Kl24 TEH)K126 -2.*C7*L2+(8.~7*C7+3.*C8)*L3+18.~7aC8*L4+9.*C8*C8*L5TEH = C7*L4 + (2.*c7aC7 C8)*L5 + 3.%7*C8*L6 + C8%8*L7K126 - Cll*K126 - OHEGAS*@TEH.K128 = 4.%5*c7*13 + 6.*(Cb*c7 + C5'%8)*L4 + 9.%6%8*L5TKH = CS*C7%5 + (C6=+C7 CSaC8)*16 + Cb*C8*L7K128 = Cll*K128 - OHEGAS*WVEHK1210 = C7*L4 + 2.'%8*L5 + XJ*(2.*C7*L3 3.*C8*L4)K1210 = -OHEGAS*H*E*SBETA*ClO*Kl210K1212 = 4.%7*C7*L3 + l2.%7%8*L4 + 9.*C8*C8*L5TEH = C7*C7f%5 + 2.%7*C8*L6+ C8=%8*L7K1212 = Cll*Kl212 - OHEGAS*HaTEH

OMEGAS = R*; Cl = 6/12; C2 = -6/lC3=2/1; C4=-3/12; C5=3/12C6 = -2/I; Cl = - l/l; Cg = l /l2C9=-l/l; ClO=l/l; Cll=TL2 = 1212; L3 = 1313; L4 = P/4LS = 15/5; L6 = lb/6; Ll = I /7

I = length of the elementM = mass per unit lengthXJ = distance of the element from the axis ofrotation

E = distance between mass and elastic axes, eSBETA = sin flCBETA = cos p

KMlS=k;,KM2S = k; ,

CTBETA = cos 21RT= I C&nx dxL1

Dynamics of Helicopter Rotor Blades

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