On missile stability

Journal of Sound.and Vibration (1976) 49(l), 141-147

ON MISSILE STABILITY 1. INTRODUCTION

Concerning the stability problem of a uniform missile under constant thrust with and without directional control, the second lowest branch of eigenvalues turns out to be the critical one in deciding the structural stability as it has been reported by the writer in several previous papers [l-3]. This fact was not mentioned in earlier investigations [4-6]. However, due to a numerical error as pointed out by Sundararamaiah and Johns [7], some of the writers’ conclusions also become invalid. This note attempts to clarify some aspects of this crucial branch of eigenvalues. First, an analysis is presented showing that, other than the first branch of zero eigenvalues (corresponding to the rigid body translation mode), there can be only a set of discrete values of the thrust which admit zero eigenvalues. Stability data are then presented which are revised from previous results with the numerical error eliminated. Finally conclusions concerning the second lowest branch of eigenvalues are drawn from these new evidences.

2.

ON D1SCREl-E VALUES OF THE THRUST WHICH

ADMIT ZERO EIGENVALUES

In the notation of reference [l], the lateral stability of a uniform missile under a constant thrust subjected to directional control can be described by the following boundary value problem : D.E. : u” + Q(xu’)’ -t- I2 24= 0, (la) B.C. :

u”(0) = u”(0) = 0,

u”(1) = 0,

(lb, 14

u”(1) - & Qu’(1) = 0.

(ld, le) As mentioned in reference [l], Q is the non-dimensionalized thrust, A the eigenvalue which dictates the stability behavior and K0 the directional control parameter. A positive & denotes a rotation of the thrust in the same direction as the tangent at the tail-end and a negative K8 one in opposition to the same tangent. First, it can be shown by the following analysis that, other than the rigid body translation mode for which the eigenvalue 1 is zero for all values of Q, there can only be a set of discrete values of Q? designated by Qt, i = 0,1,2, . . ., at which L takes zero values and that the Qt’s are independent of K,. For this purpose, set L = 0 in equations (1) and one has D.E. :

u- + Q(xu’)’ = 0,

(W

B.C. :

u”(0) = u-(o) = 0,

(% 24

u”(1) = 0.

(2d, 2e)

u”(1) - & Qu’(~) = 0.

Let

v = u’.

(3)

Equations (2) become D.E.:

v- + Q(xu)’ = 0,

(da)

B.C. :

v’(0) = v”(0) = 0,

(4b, 4~)

o’(l) = 0,

(M*k)

v”(1) - KB Qu(1) = 0. 141

LETTERS TO THE EDITOR

142

From equation (4a) one has ZJ”+ Qxu = constant = c. The constant c must vanish due to equation (4~). In order to satisfy equation (4e), one can simply replace equations (4) by the following system : D.E. : v”+ Q[x-x,&&xB.C. :

l)]r=O,

(5a)

v’(0) = v’( 1) = 0,

5c)

(S

where ag = 1 + &,

(6)

S(x) is the Dirac delta function and E is an infinitesimal number defined by &8(O)= 1.

(7)

It is an easy matter to verify the equivalence between equations (4) and (5). In particular, substituting x = 1 in equation (Sa), one has v”(l) + Q(1 - aJo’

= 0,

which is equation (4e) by noting equation (6). To solve equations (5), the method of finite element-unconstrained variations [2] is again used here. It is hardly necessary to describe the details since they follow so closely those given in reference [2]. Only the unconstrained variational statement is given here. Through integration-by-parts, it is easily shown that the one associated with equations (5) is 6J= 0,

(8a)

J(u, u*) = j (-u’ u*’+ Qw*) dx + sac Qu(l) u*(l),

(8b)

where

0

being the adjoint variable. Note that u and u* are symmetric in equation (8b), indicating that equations (5) is now a self-adjoint problem. The significance that Eis infinitesimally small (equation (7)) is now apparent : as long as a@is finite, the solutions for Q in equations (8) are independent of ag (or, K,, = cl0- 1). The eigenvalue problem of equations (8) also indicates that, other than the trivial solution u = u’= 0 (which corresponds to Iz= 0 for all values of Q), the discrete eigenvalues Ql, i = 0,1,2, . . ., are the only solutions. The first eight Q,‘s obtained are listed in Table 1.

u*

TABLE

1

Thejirst eight lowest values of the thrust which admit zero eigenualues (rigid body translation mode excluded) i

Qh’

0

1

OGOOO 25977

3

2

4

9.7218 21.348

37486

5

6

58,156 83.333

7

8

113.30 149.02

It is of interest to mention a different way to obtain the Q*‘s. Since the Ql’s are independent of KB, one may take K0 = -1 (a0 = 0) in equations (5). One then has? [8] D.E. : u”+ Qxu = 0, B.C. :

u’(0) = o’(l) = 0.

t Thisis exactly the equations solved by Silverberg as was pointed out earlier by the writer [2].

(9a) Pb, gc)

[8],although his interpretation of the result was incorrect

143

LETTERS TO THE EDITOR

Equation (9a) is reducible to Bessel’s equation (see, for example, reference [9]) and it can be readily shown that solutions to equations (9) are those of

J&Q”‘)

(10)

=0

where J in equation (10) is a Bessel function of the first kind. The first non-zero solution of equation (10) turns out to be 2.5992n2 which agrees quite well with Q1 given in Table 1. 3. STABILITY DATA REVISED In this section, the stability data revised from previous results, some of which are misleading due to a numerical error, are presented. Figure 1 is for the case without control. The deformed

20

IO

cd branch 1st bmwh

0

2

(an alnwst zem branch far the 1 mnge of thrust shown) la zero bmnch, rQid body tmnplatia~)

4

6

8

IO

Figure 1. Four lowest branches of eigenvalues,

I2

K.= 0.

parabola corresponds to the third and fourth lowest branch of eigenvalues (for the range of thrust shown). These two branches now coalesce at Q = 11.126~~with A= 23.10, which agree with the results given in reference [7]. The first branch is zero for all values of Q and corresponds to the rigid body translation mode. The second branch is a troublesome one. From the present numerical results, for Q ranging from zero to 1oOn2,say, the eigenvalues of this branch are too smallt to be distinguishable from the first one which are supposed to be true zeroes. On the other hand, the analysis of the previous section has shown that there is no zero eigenvalue solution other than those of Q,. Thus one concludes that this second branch for & = 0 is an “almost zero” branch. Figures 2 through 5 show the first four eigenvalue branches (including the zero branch) for Ke # 0.From inspection of the figures in the order 2,3,1,4 and 5, it is revealing to observe the changing trend of these curves as K0 varies from -1.0 to -0.05, to 0, to 0.05 and to 1.0. The changing pattern of the “second” branch is of particular interesting. It indicates that, for small I&] (additional datainclude cases in which ]Ke]is as small as OWOl), the second branch has a divergence region in 0 < Q < Q1 so long as K0 is negative and it becomes a stable region so long as KBis positive. Hence the case K. = 0 appears to be bordering between stability and t For larger values of Q (Q = 1000z2, for example), this branch appears to be distinguishable again.

from zero

144

LETTERS TO THE EDITOR

1st branch

(zero

branch1

Figure 2. Four lowestbranchesof eigenvalues,Ke = -1.0. divergence instability. In the case of divergence instability, the eigenvalue L indicates the rate at which the structural disturbance will grow. Since the second branch A is very small for K0 = 0, it may be considered a stable mode in a practical sense until the flutter load Q = 11.126a2 is reached. However, unless Q = 0, this small Iz does not represent a rigid body rotation mode and thus must be a bending mode. Finally, Figure 6 shows the three lowest branches of eigenvalues for various values of Ke and for Q between 0 and 4n2. This can be considered as a revision of Figure 5 in reference [l].

40 C ‘h 5 x’

-

30-

0

1 2

I 4

I 6

I 8

I IO

12

Q/l?

Figure 3. Four lowest branches of eigenvalues, K. = -0.05.

145

LETTERS TO THE EDITOR

1st brolch s ‘I. x 2

IO 0

1 2

I 4

(zero branch)

/ 6

1 8

/ To

12

Figure 4. Four lowest branches of eigenvalues, K. = 0.05.

4. CONCLUSIONS

Concerning the stability problem of a uniform missile with and without directional control, one now may summarize as follows. 1. Barring the rigid body translation mode, 1 can be zero only at a discrete set of Q’s (designated by Ql, i = 0, 1, 2, . . ., the first eight of them are given in Table 1). These Q,‘s are independent of KO. 2. As long as KB is positive, there is a region of oscillatory stability between Q,, and Q,. 3. As long as K. is negative, there is a region of divergence instability between QOand Q,.

z 40I, A? ;;

303rd branch

1st bmnch (zero branch)

20 0

, 2

I 4

I 6

I 8

I IO

I2

o/n2

Figure 5. Four lowest branches of eigenvalucs, KB = 1.0.

146

LETTERS TO THE EDITOR

Figure

6. Three lowest

branches of eigenvalues for various values of &.

K. = 0,the second branch of eigenvalues represents a bordering state of stability and instability. This branch may be considered stable in practical sense due to the smallness of the 2s. It is a bending mode nevertheless.

4. For

It should be pointed out that items 2 and 3 above are concluded from numerical results only and have not been substantiated by analytical proofs. In retrospect, the writer was not justified to have stated in his previous papers that the methods used by Feodos’ev [5] and by Matsumoto and Mote [6] had led to incorrect results. He could only state that they did not discuss the important second branch. However, the Galerkin’s procedure used by Beal [4], which results in a second zero branch for all values of Q, is still inconsistent with the analysis presented here. J. J. WV

Bh?t Weapons Laboratory, Water&et Arsenal, Water&et, New York 12189, U.S.A. (Received 27 July 1976) REFJXENCES

J. J. WV 1975 Journal of Soundand Vibration 43,45-52. On the stability of a free-free beam under axial thrust subjected to directional control. 2. J. J. WV 1976 Journal of Soundand Vibration 46, 51-57. On mode shapes of a stability problem. 3. J. J. WV 1976 American Institute of Aeronautics and Astronautics Journal 14, 313-319. Missile stability using linite elements--an unconstrained variational approach. 4. T. R. BEAL1965American Institute of Aeronautics and Astronautics Journal 3,486-494. Dynamic stability of a flexible missile under constant and pulsating thrust. 1.

LETTERSTO THE EDITOR

147

5. V. I. FEQDO~‘EV1965 Prikladnaia Matematika I Mekhanika 29, 391-392. On a stability problem (translated from Russian). 6. G. Y. MATSUMOT~ and C. D. MOTE 1972 Journal of Dynamic Systems, Measurement and Control, Transaction of the American Society of Mechanical Engineers 94, 330-334. Time delay instability in large order systems with controlled follower force. 7. v. SuNoa RARAMAIAH and D. J. JOHNS 1976 Journal of Sound and Vibration 48, 571-574. Comment on “On the stability of a free-free beam under axial thrust subjected to directional control.” 8. S. SILVERBERG1959 Space Technology Laboratories, Inc. Technical Report TR-59-791. The effect of longitudinal acceleration upon the natural modes of vibration of a beam. 9. C. R. WYLIE 1960 Advanced Engineering Mathematics. McGraw-Hill Book Company, Inc., See p. 422.