meirovitch - the lure of the mean axes - asme j.applied mechanics vol 74 may 2007

8/10/2019 Meirovitch - The Lure of the Mean Axes - ASME J.applied Mechanics Vol 74 May 2007

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Leonard Meirovitch 1

Department of Engineering Science andMechanics,

Virginia Polytechnic Institute and StateUniversity,

MS 0219, Blacksburg, VA 24061e-mail: [email protected]

Ilhan TuzcuDepartment of Aerospace Engineering and

Mechanics,The University of Alabama,

Tuscaloosa, AL 35487-0280e-mail: [email protected]

The Lure of the Mean Axes A variety of aerospace structures, such as missiles, spacecraft, aircraft, and helicopters,can be modeled as unrestrained exible bodies. The state equations of motion of suchsystems tend to be quite involved. Because some of these formulations were carried out decades ago when computers were inadequate, the emphasis was on analytical solutions.This, in turn, prompted some investigators to simplify the formulations beyond all rea-sons, a practice continuing to this date. In particular, the concept of mean axes has oftenbeen used without regard to the negative implications. The allure of the mean axes lies inthe fact that in some cases they can help decouple the system inertially. Whereas in thecase of some space structures this may mean complete decoupling, in the case of missiles,aircraft, and helicopters the systems remain coupled through the aerodynamic forces. In fact, in the latter case the use of mean axes only complicates matters. With the develop-ment of powerful computers and software capable of producing numerical solutions tovery complex problems, such as MATLAB and MATHEMATICA , there is no compelling reasonto insist on closed-form solutions, particularly when undue simplications can lead toerroneous results. DOI: 10.1115/1.2338060

1 IntroductionA variety of aerospace structures can be modeled as unre-

strained exible bodies. Examples of these are missiles, space-craft, helicopters, and aircraft. In formulating the equations of motion, the methodology used depended largely on the preferenceof the investigators. Because some of these investigations werecarried out decades ago when computers were inadequate, therehas been a tendency to simplify the formulations to an undueextent. At the other extreme, in relatively recent days the tendencyhas been to rely on large computer codes written for purposesother than for the problem under consideration, resulting in verycumbersome and inefcient solutions.

Surprisingly, the literature on dynamics of unrestrained exiblebodies is not very abundant. Using essentially a Newtonian ap-proach extended to nite bodies, Bisplinghoff and Ashley 1 de-rived scalar equations of motion for exible missiles. Making anumber of assumptions, they arrived at entirely decoupled equa-tions for the rigid-body translations, greatly simplied equations

for the rigid-body rotations and decoupled equations in terms of the natural modes for the elastic deformations. Among the fac-tors contributing to the simplication of the equations of motionwe note the use of principal axes as body axes, the assumptionthat the elastic deformations have a negligible effect on the massmoments of inertia and that the independent equations for theelastic deformation are the same as for an undamped multi-degree-of-freedom system. The scalar equations of motion of Bis-plinghoff and Ashley can be expressed in the compact vector-matrix form

m V + V = F , J + J + * + * = M ,

M + 2 = X 1

where m is the total mass of the missile, V the velocity vector of the origin O of the body axes xyz, the angular velocity vector of xyz, a skew symmetric matrix corresponding to Ref. 2 , J =diag J xx J yy J zz the inertia matrix, * = mr dm, in which r is

a skew symmetric matrix corresponding to the radius vector rfrom O to a typical mass element dm and is a matrix of naturalmodes of vibration referred to as shape functions in this paper, as

explained later , and are vectors of generalized displacementsand velocities, respectively, M =diag M 11 M 22 M nn is a matrixof generalized masses and F , M , and X are a force vector actingon the whole missile, a moment vector about O acting on thewhole missile and a vector of generalized forces acting on thegeneralized coordinates. We note that F , M , and X include all theforces acting on the missile. These include the very importantaerodynamic forces and the thrust force during the powered ight.The effect of the aerodynamic forces is to introduce dissipativeeffects into the system, rendering the concept of natural modesmeaningless, as natural modes exist only in undamped, and henceconservative systems. A comparison of Eqs. 1 with the rigorousequations of motion of an unrestrained exible body given later inthis paper will reveal glaring omissions.

Although the developments of Ref. 1 used a exible missileas a model, the implication was that the developments appliedequally well to exible aircraft. Because the aerodynamic forcesresult from a body moving through air, these forces are mostconveniently expressed in terms of local coordinates moving withthe body, i.e., in terms of the body axes xyz, which explains whybody axes were used in Ref. 1 . Body axes can be used as areference frame in the case of exible spacecraft as well. In thecase of spacecraft, however, the translation of the origin of thereference frame, generally coinciding with the system center of mass, follows a predetermined orbit, so that there are no transla-tional rigid-body degrees of freedom. Considering a spacecraftconsisting of a rigid core with exible appendages simulating an-tennas, Meirovitch and Nelson 3 dened the rotational motionsand elastic deformations by means of a reference frame attached

to the rigid core, and hence to the undeformed spacecraft, in es-sence using body axes. However, the use of body axes is not auniform practice in the case of exible spacecraft. Indeed, con-tending that in general spacecraft do not possess rigid cores,Canavin and Likins 4 , argued in favor of using a oating ref-erence frame, referred to as a Tisserand frame, or mean axesframe. This argument is not very convincing, however, as there isno denying that all exible bodies possess an undeformed state. Atany rate, the mean axes were dened by Canavin and Likins bysetting the internal angular momentum relative to the origin of thereference frame, made to coincide with the system mass center, tozero. In terms of our notation, this amounts to

1Author to whom correspondence should be addressed.Contributed by the Applied Mechanics Division of ASME for publication in the

JOURNAL OF APPLIED MECHANICS . Manuscript received July 26, 2005; nal manuscriptreceived April 18, 2006. Review conducted by N. Sri Namachchivaya. Discussion onthe paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering,University of CaliforniaSanta Barbara, Santa Barbara, CA 93106-5070, and will beaccepted until four months after anl publication of the paper itself in the ASMEJOURNAL OF A PPLIED MECHANICS .

Journal of Applied Mechanics MAY 2007, Vol. 74 / 497Copyright 2007 by ASME

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m

r u dm = 0 2

where r is a skew symmetric matrix corresponding to the radiusvector r and u is the elastic displacement vector. They correctlyidentied Eq. 2 as three scalar constraint equations to be im-posed on u. Then, they discuss at length the nature of the con-straints and ways of satisfying them. The example model consistsof two beams connected at one end to a hinge with a torsionalspring and free at the other end. Because the model is not reallyrepresentative of spacecraft and the discussion of the constraintssatisfaction is rather laborious, we do not pursue the subject at thispoint but return to it later in this paper.

Whereas one can make a case for the use of mean axes forexible spacecraft, provided the constraint equations are handledcorrectly, it should be pointed out that spacecraft operate in amuch more benign environment than aerial vehicles. Indeed, theydo not have to cope with one of the most signicant problems inaerial vehicles, namely, the aerodynamic forces. In the case of aerial vehicles, any advantage that may accrue due to the inertialdecoupling resulting from the use of mean axes is negated by thepersistent coupling caused by the aerodynamic forces.

Worthy of notice due to the use of mean axes is a report byDusto et al. 5 , leading to a system of computer programs, namedFLEXSTAB , for evaluating the stability of arbitrary exible aircraftcongurations in subsonic and supersonic ights. In addition to

the common body axes, they used mean axes for deriving theequations of motion and uid axes for describing the aerody-namic forces. They dened the mean axes by

m

u dm = 0, m

r u dm = 0 3

and recognized that Eqs. 3 represented six constraint equations.To satisfy Eqs. 3 , they used Lagranges multipliers 2 . It shouldbe noted that one of the disadvantages of Lagranges multipliersmethod is that it introduces new unknowns into the equations of motion, in the case at hand six of them. There are two issuescasting doubts on the validity of the results. The rst is the factthat they used two different reference frames, one for the aircraftmotions and one for the aerodynamic forces, without transforming

from one to the other, and the second concerns the wayLagranges multipliers method was applied. Of course, these is-sues are dwarfed by the fact that any decoupling resulting fromthe use of Eqs. 3 is only illusory in the presence of aerodynamicforces.

Using different reasoning, Eqs. 3 can lead to similar results inRef. 1 as those obtained in Refs. 4,5 . Indeed, without anyreference to mean axes, Ref. 1 interprets Eqs. 3 to mean thatthe natural modes of vibration are orthogonal to the translationaland rotational rigid-body modes. In Refs. 4 they imply that theangular momentum due to the elastic deformations is zero, and inRef. 5 that the linear and angular momenta are zero. Of course,they all imply that the system is undamped and that all motions,rigid and elastic, are small. This is hardly the case in general, asthe aerodynamic forces are dissipative in nature and the rigid-

body motions tend to be large.The concept of mean axes was eagerly embraced by a variety of investigators in the belief that its use is likely to result in signi-cant simplication of the equations of motion. In particular, it wasbelieved that the use of mean axes would reduce coupling, thusresulting in independent equations for the rigid body translations,rigid body rotations, and elastic deformations. This belief turnsout to be mistaken, as the disadvantages resulting from the re-quirement to satisfy the constraint equations, Eqs. 3 , far out-weigh any apparent advantages that might accrue from the elimi-nation of some inertial coupling terms. Indeed, as pointed outabove, there is no real decoupling, as the equations remaincoupled through the aerodynamic forces. In fact, as shown later in

this paper, the use of mean axes not only makes matters morecomplicated but in some cases it can render the formulation atodds with the physical reality.

Whatever advantages or disadvantages accrue from the use of mean axes, the procedures used in 4,5 are essentially correct.Unfortunately, the same cannot be said about the subsequent in-vestigations, as at some later time the concept of mean axes wasinvoked but not really used. Indeed, some later investigatorslatched onto the notion of mean axes to drop most inertial cou-pling terms from the equations of motion while failing to enforcethe constraints embodied by Eqs. 3 . In fact, some of the inves-

tigators did not even give the denition of the mean axes.Typical of investigators regarding the mean axes, Eqs. 3 , as avehicle to drop the coupling terms with impunity are Nydick andFriedmann 6 and Friedmann, McNamara, Thuruthimattam andNydick 7 , where the rst is a conference presentation and thesecond is basically a journal version of the rst. Indeed, profess-ing to use the mean axes, when in fact they did not, as well asinvoking a variety of other assumptions, and conning themselvesto the case of steady level cruise, they obtained the greatly sim-plied equations of motion

m V 0 x + qV 0 z = X mg sin ,

m V 0 z qV 0 x = Z + mg cos , j yy0 q = M

M g + C g + K g D

T T dD = D

r T T dD + Q

4

in which m is the total aircraft mass, V 0 x the forward velocity, V 0 zthe plunge velocity, q the pitch velocity, the pitch angle, j yy

0 themass moment of inertia about the pitch axis y, assumed to beconstant, X , Z , and M are associated forces and moment, includingthose due to aerodynamics, M g, C g, and K g are generalizedmass, damping and stiffness matrices, respectively, is a vectorof generalized elastic coordinates, a modal matrix, a skewsymmetric matrix derived from the angular velocity vector = 0 q 0 T and Q a generalized force density vector. Equations 4are even simpler than they may seem, and we note that the rst

three are scalar equations and the fourth is a vector equation, asthey can be solved independently for the rigid-body translations,rigid-body rotion, and elastic displacements. Indeed, rst the thirdof Eqs. 4 can be solved for the pitch velocity q, and hence forthe pitch angle , then the rst and second can be solved for V 0 x and V 0 z and nally the fourth can be solved for . However, thereare several problems with this proposition. In the rst place, al-though the authors do list the denition of the mean axes, Eqs. 3 ,they make no attempt to enforce the constraints imposed by them,which can have disturbing implications. Moreover, there is noindication that the aerodynamic forces were ever expressed interms of mean axes components, nor is there any hint that theaerodynamic forces keep the equations coupled, thus preventingindependent solutions of the equations for the rigid body transla-tions, rigid body rotations and elastic deformations.

It is clear from

6,7 that the objective is an aeroelasticity analy-sis, which calls for very simple equations of motion. Yet, thepapers begin with a formulation capable of describing the dynam-ics and control of maneuvering exible aircraft 8,9 , a very com-plex problem, and proceed to strip away the elaborate formulationreducing it to Eqs. 4 . In the process of reducing the equations of motion in terms of quasi-coordinates, the authors of 6,7 gettripped by the misuse of mean axes. It appears that the authors of 6,7 would have been better served by beginning directly with

aeroelasticity equations rather than with the complex formulationof Ref. 8 . To indicate where 6,7 , went wrong and to highlightsome possible negative implication of the use of mean axes, thecorrect use of the mean axes is discussed later in this paper.

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The use, and sometimes abuse, of the concept of mean axes isnot conned to missiles, spacecraft, and aircraft. Indeed, abuse of the concept can be found in the case of helicopters as well. Froma dynamicists point of view, helicopters can be regarded broadlyas consisting of three parts, the main rotor, the tail rotor and thefuselage, all acting together as a single dynamical system. Of these, the main rotor is by far the most critical part, although ahelicopter could not function properly without a tail rotor. Indeed,in the absence of a tail rotor to counteract the angular momentumof the main rotor, the fuselage would spin in a sense opposite tothat of the main rotor. The equations of motion for helicopters can

be derived by modifying the approach used for aircraft, modica-tions made necessary by the fact that helicopters possess spinningrotors. However, this is not the approach used by Cribbs, Fried-mann and Chiu 10 , who consider the problem of a coupledhelicopter rotor/exible fuselage aeroelasticity model in a veryunorthodox way as far as dynamics is concerned. In particular,they concentrated on the fuselage alone and invoked the use of mean axes to derive decoupled equations for the fuselage rigid-body translations, rigid-body rotations and elastic deformations,as follows:

m f xcm = F , J f + J f = M cm , M q e + K q e = Q 5where m f is the total mass of the fuselage, xcm the displacementvector of the fuselage mass center, F the resultant vector of allexternal forces acting on the fuselage, J f the inertia matrix of thefuselage, the angular velocity vector of the fuselage, M

cm the

resultant vector of all external moments about the fuselage masscenter, M and K are mass and stiffness matrices for the fuselage,q e is a vector of generalized coordinates and Q a vector of gen-eralized forces for the fuselage. Equations 5 are very strange inseveral respects. In the rst place, they are for the fuselage alone,while insisting that the main rotor was coupled to the fuselage. Tothis end, they referred to some equations of equilibrium derivedfor the main rotor in a NASA CR dated 22 years earlier, withoutexplaining how equations of equilibrium derived separately forthe main rotor, equations not even given in Ref. 10 , are coupledto equations of motion derived for the fuselage alone. Moreover,the tail rotor is not even mentioned. Finally, it is clear that Eqs. 5are for a ctitious fuselage at best, as they were derived invokingthe use of mean axes, when in fact the mean axes were not used atall.

Later in this paper, it is shown how the problem of helicopterdynamics is to be approached.

2 Proper Formulation for the Dynamics of AerospaceStructures

As discussed in the preceding section, the mean axes have beenportrayed as a useful tool in the treatment of such diverse aero-space structures as missiles, spacecraft, aircraft, and helicopters. Acloser examination of pertinent investigations, however, paints adifferent picture. Improper formulations of the equations of mo-tion and/or misuse of the concept of mean axes do not inspiremuch condence in the usefulness of the approach. In this section,we propose to contrast the approach based on misuse of meanaxes with a proper formulation of the same problem.

From Meirovitch 8 , the motions of a exible body can bedescribed by means of a reference frame xyz Fig. 1 xed in theundeformed body and known as body . Then, the rigid-body mo-tions are dened as three translations and three rotations of thebody axes relative to the inertial space XY Z and the elastic defor-mations as the displacements of points on the body relative to thebody axes. When expressed in terms of components along thebody axes, the rotational velocities are referred to as quasi-velocities , and the corresponding vector is denoted by . It isshown in Ref. 8 that, when expressed in terms of body-axescomponents, the translational velocities can also be treated asquasi-velocities; they are arranged in the vector V . Note that, un-

like ordinary velocities, quasi-velocities cannot be integrated toobtain displacements 2 .

Denoting by u and v the three-dimensional elastic displacementand elastic velocity vectors, respectively, where u and v are mea-sured relative to the body axes xyz, it is shown by MeirovitchRef. 8 that the hybrid dynamical equations of motion in terms of

quasi-coordinates can be written in the compact vector-matrixform

d

dt

L V

+ dL V

C L R

= F

d dt

L

+ V L

V+

L

E T 1 L

= M

t

T

v

T

u+

F

v+ L u = U 6

where explicit provision was made for damping, in which L is thesystem Lagrangian, V

and are skew symmetric matrices derivedfrom V and , respectively, C is a matrix of direction cosinesbetween xyz and XYZ , R = R X RY R Z T is the radius vector fromthe origin O I of XY Z to the origin O of xyz, E is a matrix relatingthe symbolic angular velocity vector = 1 2 3 T to the angular

quasi-velocity vector , T is the kinetic energy density of thebody, F is Rayleighs dissipation density function 11 and L is a3 3 matrix of stiffness differential operators. Moreover, F, M ,and U are generalized force vectors, which must be expressed interms of the same body axes components used to express the mo-tion variables . They can be obtained from the actual distributedforce vector f r , t and the discrete force vectors F i t acting at thepoints r = r i i = 1 , 2 , . . . , p by means of the virtual work expres-sion. Discrete forces can be treated as distributed by writing themin the form F i t r r i , where r r i are spatial Dirac deltafunctions 11 , so that the virtual work can be written in the form

W = D

f T r , t +i=1

p

F iT t r r i R P

* dD 7

where R P* is a virtual displacement vector whose expression can

be obtained by considering the velocity vector of a typical pointP in the body in terms of components along the body axes asfollows:

vP = V + r + u + v = V + r + u T + v 8where r is the radius vector from O to P and r + u is the skewsymmetric matrix derived from the vector r + u . Using the analogywith Eq. 8 with the term u T ignored as small compared to r T ,we obtain

Fig. 1 Flexible body in space

Journal of Applied Mechanics MAY 2007, Vol. 74 / 499



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R P* = R * + r T + u 9

which is the virtual displacement vector of point P Fig. 1 , inwhich R * is the virtual displacement vector of the origin of xyz,

is the virtual angular displacement vector of xyz and u is thevirtual elastic displacement vector of point P relative to xyz, all interms of body-axes components. Inserting Eq. 9 into Eq. 7 , weobtain

W = D

f T r , t +i=1

p

F iT t r r i R

* + r T + u dD

= F T R * + M T + D

U T u dD 10

where

F = D

f dD +i=1

p

F i, M = D

r f dD +i=1

p

r iF i ,

U = f +i=1

p

F i r r i 11

are the desired generalized forces.

Closed-form solutions of hybrid sets of differential equationsdescribing the dynamics of exible aircraft is not within the stateof the art, so that one must be content with approximate solutions.This implies invariably spatial discretization of the elastic vari-ables 11 . We consider spatial discretization of the system byassuming that the elastic variables can be approximated to a rea-sonable degree of accuracy by series of space-dependent shapefunctions multiplied by time-dependent generalized coordinates,as follows:

u = , v = 12

where = r is a 3 n matrix of shape functions 11 , in whichn is the number of elastic degrees of freedom, is an n-vector of generalized coordinates and is an n-vector of generalized ve-locities. A common misconception is to refer to the entries of as

modes when in fact they are merely shape functions. Of course,one can always try to choose the shape functions as the eigenfunc-tions of a somewhat related system, but that does not make themmodes. It should be noted that the system described by Eqs. 6 isnonlinear and subject to viscous damping. Moreover, in the casein which Eqs. 6 represent the equations of motion for a exibleaircraft, F, M, and U include aerodynamic forces. As a result,even after linearization, any modes are likely to be complex,whereas the entries of are real functions. In the spirit of Rayleigh-Ritz, it is shown by Meirovitch 11 that any set of func-tions capable of describing any possible elastic deformation of thesystem to a given desired degree of accuracy represents an accept-able set of shape functions. However, the use of shape functionsrepresents a mere spatial discretization process resulting in a set of ordinary differential equations.

Before discretizing Eqs. 6 , we consider Eqs. 8 and 12 andwrite the two forms of the kinetic energy

T =1

2 D vP

T vPdD =1

2mVT V + VT S

T + VT D

vdD

+ T

D

r + u vdD + 12

T J

+1

2 D vT vdD

1

2mVT V + VT S

T + VT

+ T

D

r + dD + 12

T J

+1

2 T

13

where

= D

dD , S = D

r + dD ,

J = D

r + r + T dD ,

= D

T dD 14

in which S and J are the matrix of rst moments of inertia and the

inertia matrix, respectively, of Rayleighs dissipation function

F = D

F dD =1

2 DcvT vdD

1

2 T

D

c T dD =1

2 T C

15

where

C = D

c T dD 16

is a damping matrix, in which c is a damping density function,and of the strain energy

V str =1

2 Du T L u dD

1

2 T

D

T L dD =1

2 T K 17

where

K = D

T L dD 18

is a stiffness matrix. Both C and K are symmetric. Then assumingthat the Lagrangian does not depend on R and , which is true forying aircraft, the discrete counterpart of Eqs. 6 are simply

d dt

L V

+ L

V= F,

d dt

L

+ V L

V+

L

= M

d

dt

L

T

+ C + K = X 19

in which, from Eq. 13

T

=

T V T + D

T dD D

T 2 r + dD

20

and

X = D

T UdD 21

is the discretized generalized elastic force vector.Finally, including some obvious kinematical identities and car-

rying out the indicated operations, we obtain the discretized set of state equations




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R = C T V, = E 1 , =

mV + S T +

= 2 D

dD + mV

+ S + F

S V + J +

D

r + dD = D

r + + r

+ dD + V

S + S V

J + D

r

+ T dD + M

T V + D

T r + T dD +

= T V T

D

T 2 r

+ dD 2 D

T T dD C K + X 22

Next, we address the problem of helicopter dynamics. To thisend, we consider the typical helicopter conguration shown inFig. 2 and assume that the main rotor consists of nm equallyspaced blades and the tail rotor consists of n equally spacedblades. Then, by analogy with Eqs. 6 and using the notation of Fig. 2, the helicopter equations of motion can be expressed in thegeneric form

d dt

L VO

+ O L

V O C O

L R O

= F O

d dt

L O

+ VO L

VO+ O

L O

E OT 1

L O

= M O

t

T f v f

T f u f

+ F f v f

+ L f u f = U f

t

T mi vmi

T mi u mi

+ F mi vmi

+ L miu mi = U mi , i = 1,2, ... , nm

t

T j v j

T j u j

+ F j v j

+ L ju j = U j, j = 1,2, ... , n

23

The derivation of explicit equations of motion would require agreat deal of perserverance and patience, and is beyond the scopeof this paper. To develop an appreciation of the nature of theequations, however, we will outline some of the steps involved intheir derivation. The kinetic energy has the general expression

T =1

2 m f V f

T V f dm f +1

2 i=1

nm

mi

VmiT Vmidm i +

1

2 j=1

n

m j

V jT V jdm j

24

where, following an orderly kinematical procedure, the velocityvectors of the individual components are given by

V f r f , t = VO t + r f + u f r f , t T O t + v f r j, t

V mi r mi , t = C miV M + r mi + u mi r mi, t T C mi O + M + vmi r mi , t , i = 1,2, .. . , nm

V j r j, t = C jVT + r j + u j r j, t T C j O + T + v j r j, t , j = 1,2, ... , n 25

in which

V M t = V f r OM , t = VO t + r OM + u f r OM , t T O t + v f r OM , t

VT t = V f r OT , t = VO t + r OT + u f r OT , t T O t + v f r OT , t 26

are the velocity vectors of the main rotor hub M and tail rotor hubT , obtained by evaluating V f at M and T , respectively. Moreover,C mi and C j are matrices of direction cosines between the mainrotor blade body axes x mi ymi zmi and the fuselage body axes x f y f z f and the tail rotor blade body axes x j y j z j and x f y f z f due to thespin of the rotors; both C mi and C j depend explicitly on time. Thekinetic energy is obtained by inserting Eqs. 25 and 26 into Eq.24 and carrying out the indicated operations. A cursory exami-

nation of Eqs. 25 and 26 will reveal that the fuselage, mainrotor and tail rotor are all inertially coupled and so are the equa-tions of motion. Furthermore, it is futile to look for mean axescapable of changing this fact. Clearly, the use of mean axes for thefuselage alone, which is the least critical part of a helicopter,cannot be justied.

Although we expressed the fuselage stiffness in terms of a stiff-ness operator matrix L f , this was merely symbolic because it isnot feasible to generate a differential operator for such a complexstructure as the fuselage. In practice, it is necessary to express thefuselage stiffness, in the context of a spatial discretization process,by means of a stiffness matrix generated by the nite elementmethod. A typical main rotor blade can be modeled as a thin beamundergoing torsion about the longitudinal axis x mi and bendingabout axes ymi and zmi. Denoting the displacement vector for blademi by umi = xmi u ymi u zmi T , the corresponding stiffness operatormatrix can be shown to have the form 11Fig. 2 Flexible helicopter




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L mi =

x miGJ mi x mi

x mi 0 0

0 2

x mi2 EI zmi x mi 2 x mi2 0

0 0 2

x mi2 EI ymi x mi 2 x mi2

x miP mi x mi

x mi

27

where

P mi x mi = x mi

Lmi

mi C mi O + M zmi2 d 28

is the axial force on blade m i due to the centrifugal force causedby the spin of the main rotor hub. The operator matrix L j for thetail rotor can be obtained from Eqs. 27 and 28 in an analogousfashion.

Some of the other quantities, such as Rayleighs dissipationfunction densities F f , F mi , and F j and the generalized forces F 0,M 0, U f , U mi , and U j can be obtained by analogy with those forthe aircraft, where the forces include the aerodynamic forces,which are much more complicated than those for aircraft. In thisregard, it must be pointed out that the degree of complexity in-creases signicantly from hover to forward ight. Indeed, in for-ward ight the blade velocity due to the hub rotation adds to thefuselage velocity during half of the rotation and subtracts from itduring the other half. To compensate for this, the blade is made topitch accordingly.

Generic equations of motion for whole exible helicopters canbe obtained by inserting Eqs. 24 28 into Eqs. 23 and carry-ing out the indicated operations, which would involve a great dealof symbolic manipulations, well in excess of those for exibleaircraft. It is not difcult to see that the resulting equations arelikely to be extremely complex. Contrasting Eqs. 23 28 withthe formulation of Ref. 10 , Eqs. 5 of the present paper, oneconcludes that the formulation of Ref. 10 is badly awed, and nosensible simplication of Eqs. 23 28 would reduce the equa-tions of motion to an extent that would make them resemble Eqs.

5 , not even vaguely. Hence, the contention that Eqs. 5 reect aCoupled Helicopter Rotor/Flexible Fuselage AeroelasticModel , as the title of Ref. 10 implies, is more than question-able.

3 The Use of Mean AxesIn deriving Eqs. 22 , a reference frame embedded in the unde-

formed body was used to dene the rigid-body and elastic mo-tions, as well as the forces, moments and distributed forces. Thischoice seems only natural. As can be expected, the equations forthe rigid-body translations, rigid-body rotations and elastic defor-mations are all coupled. There seems to be a belief that a differentchoice of reference frame is able to reduce the coupling, and

hence the complexity of the formulation. We wish to examine thisproposition.The inertial terms can be simplied to some extent by choosing

the body axes as the principal axes with the origin at the masscenter. In this case, we have

D

r dD = 0, D

r r T dD = J 0 29

where J 0 represents the diagonal matrix of the principal momentsof inertia of the undeformed body. Perhaps more extensive sim-plications can be achieved, at least at rst sight, by using themean axes, dened by

D

u dD D

dD = = 0 ,

D

r u dD = D

r u dD D

r dD = * = 0 30

Inserting Eqs. 29 and 30 into the second half of Eqs. 22 , thedynamical part of the state equations reduces to

mV +

= 2 D

dD + mV + F

J + D

dD = D

r + + r + dD

J + D

r + T dD + M

T V + D

T T dD +

= T V T D

T 2 r + dD 2 D

T T dD

C K + X 31

Contrasting Eqs. 31 with the simplied equations of Refs. 6,7 ,Eqs. 4 of the present paper, we conclude that Eqs. 4 do notdescribe any real aircraft but some ctitious one that does notexist, so that any analysis based on Eqs. 4 is an exercise infutility.

4 Implications of Constraints EnforcementThe mean axes dened by Eqs. 30 represent an abstraction in

the sense that the axes are not as easy to visualize as the body axesembedded in the undeformed body, let alone to determine them.More important, however, is the fact that, from a dynamical pointof view, Eqs. 30 represent constraints imposed on the displace-ment vector function u r , t , or the generalized coordinate vector

. Indeed, Eqs. 30 represent six equations of constraint . Lessappreciated and seldom mentioned, if ever, is the fact that theseconstraints must be enforced . Enforcement of constraints has cer-tain implications well-worth exploring. There is nothing to begained by enforcing the constraints on Eqs. 4 because theseequations are awed. Indeed, we can only gain insight into theproblem by enforcing the constraints on valid equations, namely,Eqs. 31 .

For convenience, we conne ourselves to the discrete version of Eqs. 30 ; they represent constraint equations to be satised by then-dimensional generalized coordinate vector . The implication isthat the components of are not independent but subject to the sixscalar equations




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j=1

n

ij j = 0 , j=1

n

ij* j = 0 , i = 1,2,3 32

The six constraint equations, Eqs. 32 , can be enforced by meansof Lagranges multipliers Refs. 2,5 , a somewhat tedious pro-cess. They can be more conveniently enforced by means of acoordinate transformation reecting the fact that only n 6 of thecomponents 1, 2 , . . . , n are independent. To this end, we rewriteEqs. 32 in the form

A ind + B dep = 0 33

where

34

are 6 n 6 and 6 6 matrices and ind = 1 2 . . . n6 T and dep = n5 n4 . . . n T are n 6 -dimensional and six-dimensional

vectors of independent and dependent components, respectively.Then, regarding the original n-dimensional vector as a con-strained vector , we can write

= c = ind dep =

I

B1 A ind = T ind 35

in which

T = I

B1 A 36

represents an n n 6 transformation matrix. Note that choosingthe bottom six components of c as the dependent component wasmerely to demonstrate the procedure and is a valid choice only if B is nonsingular. Inserting Eq. 35 into Eqs. 31 and multiplyingthe third of the resulting equations on the left by T T , we obtain

mV +

T ind = 2 D

T inddD + mV + F

J + D

T ind dDT ind

= D

T ind r + T ind + r + T ind T ind dD

J + D

r + T ind T dDT ind + M

T T T V + T T

D

T T indT dD +

* ind

= T T T V T T T D

T 2 r + T ind dD

2T T

D

T T indT dD C * ind K

* ind + X* 37

in which

J = D

r + T ind r + T ind T dD

* = T T

T , C * = T T CT , K * = T T KT , X* = T T X 38

At this point, we must discuss two important issues, one con-cerning the number of elastic degrees of freedom of the formula-tion and the other, a more important one, whether mean axesshould be used at all. We recall that the dimension of the indepen-dent coordinate vector ind is n 6, so that using the mean axes asa reference frame, there are only n 6 elastic degrees of freedom.This is in contrast with the widespread notion that there are fullyn elastic degrees of freedom. A paradox arises when mean axesare used and the exibility is modeled by a number of shape func-tions smaller than or equal to six , as in these cases the number of elastic degrees of freedom is either negative or zero , which is a

physical impossibility. Moreover, either explicitly or implicitly,the aerodynamic forces depend on the velocities V, , and ,which, according to the formulation leading to Eqs. 31 , are interms of components along body axes embedded in the unde-formed body. Consistent with this, F , M , and X are in terms of thesame body axes. Clearly, if the velocities V, , and are referredto the mean axes as stated in Nydick and Friedmann 6 andFriedmann et al. 7 , then the aerodynamic forces in Eqs. 4 mustalso be expressed in terms of components along the same meanaxes. Yet, there is no indication in Refs. 6,7 that the aerodynamicforces were ever transformed from the original body axes to themean axes, which raises additional doubts about the validity of theformulation. Hence, if one insists on using the mean axes as areference frame, then one must fully understand the implicationsand honor the constraints consistently throughout. Merely invok-

ing the use of mean axes to eliminate terms without enforcing theconstraints to both motions and forces makes for an erroneousformulation and results. In view of this, one must question the usemean axes in the rst place. Indeed, from Eqs. 37 we concludethat the use of mean axes only results in added complications withno visible benets.

In the case of helicopters, the equations of motion, Eqs. 23 ,are not only signicantly more complex than those for aircraft butthey also depend explicitly on time. Clearly, the use of mean axesfor helicopters is hard to comprehend.

Finally, one must wonder why it was necessary to begin withsuch a complete and rigorous formulation as Eqs. 11 of Ref. 8and strip it down beyond recognition see Eqs. 4 of the presentpaper .

5 ConclusionsThe equations of motion for some exible aerial vehicles, such

as missiles, aircraft, and helicopters tend to be very complex.Decades ago, when computers were inadequate, the emphasis wason analytical solutions, which prompted some investigators to re-sort to undue simplication of the equations of motion, a practicecontinuing to this date. In this regard, the concept of mean axesseemed quite appealing, as in some cases it held the promise of inertial decoupling of the equations of motion. This approachturned out to be a blind alley for aerial vehicles, as the equationsof motion remained coupled through the aerodynamic forces.Ironically, as shown in this paper, proper use of the mean axes canonly result in an increase in the complexity of the formulation.

The undue insistence on simplifying the equations of motionthrough inertial decoupling seems to be a throwback to an agepredating the commonplace use of powerful computers. This isparticularly true in view of the availability of efcient computertools capable of implementing solutions to very complex prob-lems. In this regard, we should mention MATHEMATICA for sym-bolic manipulation and MATLAB for matrix manipulation. Indeed,using MATHEMATICA in conjunction with a formulation in terms of momenta rather than velocities, but entirely equivalent to the for-mulation in this paper, Meirovitch and Tuzcu 1214 not onlycarried out time simulations of the rigid-body and elastic re-sponses of ying aircraft but also implemented feedback controlsensuring the ight stability of maneuvering exible aircraft.




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References1 Bisplinghoff, R. L., and Ashley, H., 1962, Principles of Aeroelasticity , Wiley,

New York, Chap. 9.2 Meirovitch, L., 1970, Methods of Analytical Dynamics , McGraw-Hill, New

York reprinted in 2003 by Dover .3 Meirovitch, L., and Nelson, H. D., 1966, On the High-Spin Motion of a

Satellite Containing Elastic Parts, J. Spacecr. Rockets, 3 11 , pp. 15971602.4 Canavin, J. R., and Likins, P. W., 1977, Floating Reference Frames for Flex-

ible Spacecraft, J. Spacecr. Rockets, 14 12 , pp. 724732.5 Dusto, A. R., Brune, G. W., Dorneld, G. M., Mercer, J. E., Pilet, S. C.

Rubbert, P. E., Schwanz, R. C., Smutney, P., Tinoco, E. N., and Weber, J. A.,1974, A Method for Predicting the Stability Characteristics of an ElasticAirplane, Vol. 1-FLEXSTAB Theoretical Description, Report No. NASA

CR-114712.6 Nydick, I., and Friedmann, P. P., 1999, Aeroelastic Analysis of a GenericHypersonic Vehicle, Proceedings of the CEAS/AIAA/ICASE/NASA Langley International Forum on Aeroelasticity and Structural Dynamics , NASA/CP-1999205/36/PT2, Williamsburg, VA, pp. 777810.

7 Friedmann, P. P., McNamara, J. J., Thuruthimattam, B. J., and Nydick, I.,2004, Aeroelastic Analysis of Hypersonic Vehicles, J. Fluids Struct., 19, pp.681712.

8 Meirovitch, L., 1991, Hybrid State Equations of Motion for Flexible Bodiesin Terms of Quasi-Coordinates, J. Guid. Control Dyn., 14 5 , pp. 10081013.

9 Meirovitch, L., 1997, A Unied Theory for the Flight Dynamics andAeroelasticity of Whole Aircraft, Proceedings of the Eleventh Symposium onStructural Dynamics and Control , Blacksburg, VA, pp. 461468.

10 Cribbs, R. C. Friedmann, P P., and Chiu, T., 2000, Coupled Helicopter Rotor/ Flexible Fuselage Aeroelastic Model for Control of Structural Response,AIAA J., 38 10 , pp. 17771788.

11 Meirovitch, L., 1997, Principles and Techniques of Vibrations , Prentice-Hall,Englewood Cliffs, NJ.

12 Meirovitch, L., and Tuzcu, I., 2004, Unied Theory for the Dynamics and

Control of Maneuvering Flexible Aircraft, AIAA J., 42 4 , pp. 714727.13 Meirovitch, L., and Tuzcu, I., 2004, Time Simulations of the Response of Maneuvering Flexible Aircraft, J. Guid. Control Dyn., 27 5 , pp. 814828.

14 Meirovitch, L., and Tuzcu, I., 2005 Control of Flexible Aircraft ExecutingTime-Dependent Maneuvers, J. Guid. Control Dyn., 28 6 , pp. 12911300.


meirovitch - the lure of the mean axes - asme j.applied mechanics vol 74 may 2007

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