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TECHNICAL REPORT ARCCB-TR-87027

i < EFFECT OF ROTATION ON THE LATERAL STABILITYOF A FREE-FLYING COLUMN SUBJECTED TO AN

AXIAL THRUST WITH DIRECTIONAL CONTROL

0J. D. VASILAKIS

00 .J. J. wU ^E

OCTOBER 1987

US ARMY ARMAMENT RESEARCH, DEVELOPMENTA AND ENGINEERING CENTER

CLOSE COMBAT ARMAMENTS CENTERBEN9T WEAPONS LABORATORYWATERVLIET, N.Y. 12189-4050

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ARCCB-TR-870274. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

EFFECT OF ROTATION ON THE LATERAL STABILITY OF A FinalFREE-FLYING COLUMN SUBJECTED TO AN AXIAL THRUSTWITH DIRECTIONAL CONTROL 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(e) S. CONTRACT OR GRANT NUMBER(.)

J. D. Vasilakis and J. J. Wu (See Reverse)

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERS

US Army ARDEC ACMS No. 6126.23.1BLO.OBenet Laboratories, SMCAR-CCB-TL PRON No. lA-7-7Z76A-NMSCWatervliet, NY 12189-4050

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US Army ARDEC October 1987Close Combat Armaments Center 13. NUMBER OF PAGESPicatinny Arsenal, NJ 07806-5000 15

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Approved for public release; distribution unlimited.

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1. SUPPLEMENTARY NOTES

Presented at the Fourth Army Conference on Applied Mathematics and Computinc,Cornell University, Ithaca, New York, 27-30 May 1986.Published in Proceedings of the Conference.

19. KEY WORDS (Continue on rerere side if neceseary eld Identify by block number)

Free-Flying ColumnStabilityFinite ElementsRotationAxial Thrust

,20. A TRACT rcaIhu m rveae ah& If ,-esewy d Identita y by block rnmber)

This report discusses some aspects of the stability problems of a free-flyinccolumn subjected to axial thrusts. In an age of spacecrafts and missiles, thestability of unsupported flying structures is obviously of great importance.Surprisingly though, there has not been a great deal of work addressing thistype of problem. In this report, first the brief history of the lateralstability of a column is reviewed, and then the basic characteristic featuresof the stability problem of a free-free column are described. The mathematical

(CONT'D ON REVERSE)

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7. AUTHORS (CONT'D)

J. J. WuU.S. Army European Research OfficeLondon, England

20. ABSTRACT (CONT'D)

techniques developed to solve these problems depend on a particular problemconsidered. The most general case requires the solution of a nonself-adjoint differential equation/boundary condition system, which ishomogeneous and with zero eigenvalues. Numerical procedures for such asystem appear to work well, although theoretical proof of convergence isstill lacking. Results of these procedures are discussed.

UNCLASSIFIED

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TABLE OF CONTENTSPage

INTRODUCTION 1

PROBLEM STATEMENT 2

VARIATIONAL STATEMENT 3

FINITE ELEMENT AND NUMERICAL FORMULATION 4

CONCLUSIONS AND DISCUSSION 6

REFERENCES 9

LIST OF ILLUSTRATIONS

1. Geometry of the Problem. 10

2. Critical Load Plot for Kq = 0.00 (Follow~er Force). 11

3. Critical Load Plot for K@0 -0.05. 12

4. Critical Load Plot for Ke = -1.00. 13

5. Critical Load Plot for Ko = 1.00. 14

.L.. ...

INTRODUCTION

In this report, a long free-free slender beam is used as a model for a

flexible missile or rocket. The beam behaves as a Bernoulli-Euler column, and

in this case is assumed to be rotating about its longitudinal axis and subject

to an end thrust (Figure 1). Of prime interest is the effect of the rotation on

the lateral stability of the beam. The motion is assumed to be planar.

Different phases of the problem have been investigated in the past. A sum-

mary of the previous work is given in Reference 1. Silverberg (ref 2) was the

first to include thrust on the flying column. The differential equation for a

free-flying beam was given earlier as shown in Reference 3. Beal (ref 4) and

Feodos'ev (ref 5) obtained results with pulsating thrust. In 1972, Matsumoto

and Mote (ref 6) treated a similar problem with directional thrust. In this

case, however, feedback control was included and a time delay was applied to the

control. The next contribution to understanding the problem was given by Peters

and Wu (ref 1). They concentrated on mode shape solutions at zero frequency for

different thrusts. A comorehensive description is also given in Reference I for

the eigenvalues and mode shape near zero thrust and with a thrust direction

close to that of a follower force. Recently, Park and Mote (ref 7) included a

concentrated mass and feedback control. The feedback control included was

allowed to be from different points along the beam.

As stated above, this report assesses the effect of rotation on the stabil-

ity of a free-free beam. The following section is a description of the problem.

Then the variational statement used for the solution is described. Next we show

how the variational statement is used with finite elements to solve the oroblen,

and lastly, we discuss the results of our investigation.

References are listed at the end of this report.

PROBLEM STATEMENT

The geometry of the problem is shown in Figure 1. The neam has a constant

cross-section of area A, density p, Young's modulus E, and moment of inertia I.

It shows a free-flying column subject to axial thrust with directional control

and rotating about its axis. The differential equation for the beam is given by

Eiuiv + p(xE u')' + pAu + oAQ2u = 0 (1

The first three terms represent the column as treated in Reference 2. The last

term on the left-hand side shows the effect of the rotation. The boundary con-

ditions are given by

u"(O) = 0 u"(2) = 0 u -

at2

u"'(0 = 0 EIu"'(f) - KqPu,12 = 0

In dimensionless form with

u u9 , x = xii T = t/l

T A j oA4 pfz- EI Q - w QT (3

and writing

u(x,r) = u(x)e iT 4

the differential equation then becomes

U-.1 + Q(xu')' + k-u w u 4 .

with the boundary conditions

u"(0) = 0

u,"(0) = 0

u"(1) = 0

u"'(1) - K0Q[u,(1)1 = 0)

Rewriting Eq. (5) as (and dropping hats)

u.... * Q(xu')' (X -,w2)u 0 7)

It appears that the addition of rotation simply shifts the frequency of vibra-

2

tion of the system. The boundary conditions, Eq. (6), become

u"(O) = 0

u" '(0) 0

u"(1) = 0

u"'(1) - K0 Qu'(1) = 0 (8)

The spacial variables are made dimensionless by dividing through by the beam's

length f and time is made dimensionless by dividing through by a constant T

(pAl4, EI) which has the units of time.

The parameter X is a complex number in general

X = AR + iX I

where both XR and XI are real numbers.

VARIATIONAL STATEMENT

To find the form of the variational statement, the differential eauat-on is

multlp7,ed by an arbitrary variation of the adjoint field variable, 6v(x), and

integrated over the beam length. Integration-by-parts indicates the form of tne

variational statement and the natural boundary conditions. The variational

statement is given by

6J = 0 9)

where

3 = f (u"v" - Qxu'v' + (X 2+w )uv]dx + Q(1+Ku)u'(1)v(1) (100

Performing the variation of 3 with respect to u and v, one can arrive at the

orig-nal boundary value problem as well as the aajoint. Eauation (10) is the

basis for a finite element solution to the described Droblem.

3

FINITE ELEMENT AND NUMERICAL FORMULATION

The procedure begins by taking the variation of Eq. (10) and allowing the

variations in the problem variable, 6u(x), to be zero, i.e., varying adjoint

variable v(x) only for now,

f [u"6v" - Qxu'6v' + A 2u6v]dx - Q(1+Ko)u'(1)6v(1) = 0 (11)0

where A 2 = X2 + w2. To discretize, the beam is divided into L elements, letting

i-1= L{x - --- I i = 1,2,3 ... ,L 112)

L

be the running coordinate in each element. Substituting Eq. (12) into Eq. (1l)

L

fl [L3 i(i),5v(i),, Q- + (i-i)}u( i ) I 6v(i) + A 2 U( ) v(i)]ds0 L

Q(I+K@)u(L)'(1)6v(L)(1) = 0 (13

In order that the displacements and their derivatives within an element ce

expressed in terms of their nodal values, the coordinate vectors are introduced.

u )T = {UIl ) U2(1) U3 (1) U4 i

V( i ) T = {Vl(,) v (i) V3 V4 i '

UI(i), U2 (i) represent the displacement and slope at the leFt end of the it')

element, and U3 (i) and U4 (i) represent deflection and slope at the right end. A

similar interpretation is applied to the ad'oint coordinate vector V(i). The

transform is indicated by T.

Hermitian polynomials are used to relate the displacements within an ele-

ment to its nodal values, hence, the following shape function is assumed:

aT( ) = (1 - 3 + 2 2 1 + , 3 - 2 ' _ 1 + 3 i 15

so that

4

u(i)( ) = T )U(i)

v(i)( ) = T ( ) 1

Substituting Eq. (16) into Eq. (13)

L-. -T A2 T6 (i - ~+~L( E6v(L

u3(iYT{L3 - Q[D+(i-l)§] + -- A1=(i Q[I+Ko]U(L)T (L) 0

Li=l

with= 1 Td , = 1 d 1 -

= f f , C fl a,,,,Td0 0 0

= aaTd , E = a'(L)aT(L) !80

Rewriting Eq. (17),

L

\ 6(i)T{ A2P( i ) + S(i ) 6V ' ) = 0 (19)

i=1

wnere

p(i) = A/L = 1,2...

S(i) = L3 C - Q[D + (i-1)B] i 1,2,.., L-1

S(L) = L C - Q[D + (L-1)B] Q(I+K 0 )E (201

using certain continuity conditions between the element nodal values

Mi (i-1) Mi (0-1)

U1 = U3 V 1 = V3

Mi (i-1) Mi (i-l)

U2 = U4 V2 = V4 (21)

One can write

(1) (1) (1) (1) (2) (2) (L) (L)UT = (UI U 2 U3 U 4 U 3 U 4 ..... U 3 U4

(1 (1) (1) (1) (2) (2) (L) (L),

VT = {V 1 V 2 V 3 V 4 V 3 V 4 ..... V3 V 4 (22)

5

Finally, [P] and [S] are NxN matrices with N = 2L+2. Since 6v is arbitrary, tne

eigenvalue problem reduces to

UT{A2[P] + [S]} = 0 (23)

which is solved for the eigenvalues.

CONCLUSIONS AND DISCUSSION

In this report, we have included rotation about the longitudinal axis in

the dynamic stability study of a free-flying missile subjected to axial thrusts.

It is assumed that the motions of bending and the thrust are in the same plane.

in the differential equation, the only difference resulting from the introduc-

tion of rotation is a change in the frequency parameter X2 to

A? = X2 + W 2 (24)

,4here w is the rotation. Consequently, all the stability curves obtained pre-

viously (ref 1) can be used with some simple modifications. It should be noted

that -n Reference ', we have written (with u) = 0)

A = A = AR + iAl (25)

and the stability character of the Problem is indicated by: (1) stable vibra-

t'ons = kT 0, .R = 0; (2) unstable by buckling (divergence) = XR # 0, AI = 0;

(3) unstable by flutter = AR * 0, Al # 0; and (4) marginally stable = X, = XR =

0.

For the present case, the stability behavior is indicated as above, but

with Al and AR replacing XI and XR 'n the previous stability curves

A = AR + iA I (26)

and

A = (AR+iA)V = X,' + w' = (w,+') + 27

6m m mm ~ mm m m m ll m

or

(AR+iA') -' = A? - = (AR+iAI) - W2 28)

Prom Eq. (28), when AR 0, A2 = -AI2 - w 2 , hence AR = 0 and Ai2 = AI; +,# .

Thus, originally stable vibrations will remain stable with higher vibration fre-

quency. On the other hand, when A, = 0, No = AR 2 -2, hence A2 = AR? - w..

Thus, originally divergent motions will become stable vibrations when AR" p w,.

In the case of marginal stability A = 0 will certainly be stabilized since AT'

Wu'.

In the case of flutter instability, Eq. (28) states that A is complex )Z

0, AR s 0) if and only 'f A is complex (Al # 0, AR * 0). Therefore, the flutter

instabi7ity is not affected by the introduction of the rotation, which is an

-nteresting observation.

Several demonstrative stability curves with A? (and A,) versus Qin are

shown in Figures 2 through 5. Only the lowest eigenvalue's branches are shown,

since they are the ones which dictate the stability behavior. Figure 2 shows

the two lowest stabie vibration modes and two rigid body modes on the A C

axis. "his is the case of a free-flying missile with a follower thrust (K,)

and w' t a dimensionless rotation of l = 500. The two fluxural modes coalesce

at load Q,,? = 11.18 beyond whicn flutter instability begins. The rigid body

modes without rotation indicate margina' stab,lity. Due to the rotation w, the

ax's is shifted from A- = 0 to N-' = 0, therefore, these oreviously rigid body

modes are now stable modes of vibrations. The thrust that s controlled with a

small negative tangency (K0 = -0.05) is shown in Figure 3. It is noted in th s

figure that the divergence instability without rotation is stabilized by Lu,' =

500. However, the new critical load is lowered from Qin' = 11.18 to 5.30, not

7

because of w', but due to the negative control parameter K(). F-gure 4 sroiwvt

case of KO = -1 or that the thrust has a fixed direction of the inertia ax-i.

It is clear that the divergence instability of the lowest branch is stabiltzed

so that the critical load has been raised from zero to QCR = 1.50 7T. Fin.--

the case for a small positive tangency control parameter (K0 = 0.05) s sro*r,

:gure 5. In this figure, the original divergence instability at Q T;' = 3.00 ,

stabilized by w2. However, the original critical load of flutter insta']it1i,

Q rT = 9.90 is not changed by the rotation. Hence, the critical load in ths

case 's raised from 3.00 to 9.90 due to the rotation of ' = 500.

REFERENCES

1. Peters, 0. A. and Wu, J. J., "Asymptotic Solutions to a Stability Problem,"

Journal of Sound and Vibration, Vol. 59, No. 4, 1978, pp. 591-610.

2. Silverberg, S., "The Effect of Longitudinal Acceleration Upon the Natural

Modes of Vibration of a Beam," Technical Report TR-59-o0OO-00791, Space

Technology Laboratories, 1959.

3. Timoshenko. S. and Young, D. H., Vibration Problems in Enqineerin, D. jcn

Nostrand Inc., New York, 1955, pp. 297-303.

4. Beal, T. R., "Dynamic Stability of a Flexible Missile Under Constant and

Pulsating Thrusts," American Institute of Aeronautics and Astronautics

Journa', Vol. 3, 1965, pp. 486-494.

5. ceodos'ev, V. I., "On a Stability Problem," Journal of Applied Mathemat'cs

ana Mechanics, Vol. 29(2), 1965, pp. 445-446.

6. Matsumoto, G. Y. and Mote, C. D., Jr., "Time Delay Instabilities in Large

Order Systems With Controlled Follower Forces," Journal of Dnamic 5 istems,

Measurement, and Control, December 1972, pp. 330-334.

7. ParK, Y. P. and Mote, C. D., Jr., "The Maximum Controlled Follower Force on

a Free-Free Beam Carrying a Concentrated Mass," Journal of Sound and

Vibration, 1985, Vol. 98, No. 2, pp. 247-256.

9

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FIGURE 1. GEOMETRY OF THE PROBLEU

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