f^^Aont 9a6 TECHNICAL

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TECHNICAL REPORT ARLCB-TR-79030

IAD

DYNAMIC STABILITY OF COLUMNS

SUBJECTED TO NONCONSERVATIVE FORCES'

I'! !v

J. J. Wu J. D. Vasilakis

■ f\-'-''-- t

Oqtober 1979

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMA"ND LARGE CALIBER WEAPON SYSTEMS LABORATORY

BEN€T WEAPONS LABORATORY WATEKVLIET, N. Y. ;L2189

AMCMS No. 6111.01.91 AC.0 ■

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ARLCB-TR-79030

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4. TITLE (and Subtllte)

DYNAMIC STABILITY OF COLUMNS SUBJECTED TO NONCONSERVATIVE FORCES

i .

5. TYPE OF REPORT & PERIOD COVERED

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7. AUTHORf»;

J. J. Wu J. D. Vasilakis

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Benet VJeapons Laboratory Watervliet Arsenal, Watervliet, NY 12189 DRDAR-LCB-TL

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AMCMS No. 6111.01.91A0.0 DA PR0.NO.1L161101A91A

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US Army Armament Research and Development Command Large Caliber Weapon Systems Laboratory Dover, New Jersey Q78Q1

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October 1979 13. NUMBER OF PAGES

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18. SUPPLEMENTARY NOTES

Presented at the 25th (Silver Jubilee) Conf of Army Mathematicians, 6-8 Jun 79, Johns Hopkins Univ, Baltimore, MD. To be published in the Journal of Sound and Vibration.

19. KEY WORDS (Continue on reverse midm if necmaaary and identify by block number)

Nonconservative Stability Follower Forces Unconstrained Variational Formulation Finite Element I

20. ABSTRACT pCbntftnje aa rararam at^ ff nacaaamry aatl Identify by b/ocfc number;

The numerical results of a class of problems of linear elastic stability problems subjected to nonconservative forces and under various support con- ditions are presented here. A single solution formulation by which these results have been obtained is described. Accuracy of these results compared with those reported in the literature is discussed.

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TABLE OF CONTENTS

SUMMARY

INTRODUCTION

A CLASS OF PROBLEMS SUBJECTED TO FOLLOWER LOADS

SOLUTION FORMULATIONS |

CASE I. P(x) = P, A Constant

CASE II. P(x) = q(l-x)

CASE III. PCx) = q^/2(l-x)^

NUMERICAL RESULTS AND DISCUSSION

CASE I. P(x) = P = Constant

CASE II. P(x) = q(l-x) I

CASE III. P(x) = qQ/2Cl-x)^

REFERENCES

TABLES

I. NUMERICAL VALUES OF FIRST TWO LOWEST EIGENVALUES OF A CANTILEVERED COLUMN WITH P(x) = P, A CONSTANT.

II. NUMERICAL VALUES OF FIRST FOUR LOWEST EIGENVALUES OF A CANTILEVERED COLUMN WITH P(x) = q(l-x).

III. NUMERICAL VALUES OF FIRST FOUR LOWEST EIGENVALUES OF A CANTILEVERED COLUMN WITH P(x) = q(I-x)^/2

ILLUSTRATIONS

1. Boundary Condition Associated with a Follower Force: Constant of Tangency KQ,

2. Two Lowest Eigenvalues vs. Load Parameters Q/TT^ and q/iT^. A Cantilevered Column of Case I and II.

3. Two Lowest Eignevalues vs. Load Parameter q^f-n^. A Cantilevered Column of Case III.

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SUMMARY. The numerical results of a class of problems of linear elastic stability problems subjected to nonconservative forces and under various support conditions are presented here. A single solu- tion formulation by which these results have been obtained is descri- bed. Accuracy of these results compared with those reported in the literature is discussed. ,

I. INTRODUCTION. Any particular subject of investigation in applied sciences is always motivated by the desire to understand some natural phenomena and hopefully to utilize the results of such an investigation for the benefit of human activities. The study of structural behavior under nonconservative loads is of no exception. Since follower forces are a special class of nonconservative forces [1], one IS surprised to encounter frequently the question as to the relation between such a study and a real engineering problem. Physically, a / follower force is simply one whose direction follows the structural i deformation as in comparison with a dead load which acts in a fixed ■ direction independent of deformation. Some obvious examples of follower forces are: thrust at the tail of a flexible rocket, jet engine thrust of an airplane, thrust on the propeller shaft of a ship, etc. Other examples such as the pressure-and-curvature induced forces included in the gun dynamic studies are less obvious [2].

Since the problems of follower forces are non-self-adjoint their treatment is more difficult than that for the self-adjoint problems. In the classical paper by Beck [3], it was demonstrated that the stability nature of a nonconservative problem can be quite different than that of a conservative one. For these reasons, a systematic approach to this class of problems and an understanding of some of the basic problems involving follower forces are desirable.

The purpose of this paper is to present a single solution approach to a class of problems of follower forces, including several classical examples, to present the numerical results so obtained, and to discuss the accuracy compared with those already published in literature.

In Section II, the class of problems will be defined by a general form of a differential equation and a set of boundary conditions. The solution formulation and its basis are given in Section III, Numerical results of some specific problems are given in Section IV together with a discussion and comparisons with data available in literature.

II. A CLASS OF PROBLEMS SUBJECTED TO FOLLOWER LOADS. The class of problems considered in this paper can be described by the differential

equation

y"" + P(x)y" + X^y =0 (1)

where y(x) denotes the lateral disturbance of a beam, as a function of the abscissa x, P(x) is the axial force always tangent to the deformed axis, and X is the eigenvalue. As usual, a prime denotes differentiation

with respect to x.

Eq. (1) is a non-self-adjoint differential equation (thus noncon- servative problem) except for P(x) = constant. If the axial force P(x) remains fixed in the direction of the undeformed axis, the problem would be of conservative nature and the differential equation a self-adjoint

one.

y"" +.[P(x)y']' + XV = 0 (^'^

Both Eqs. (1) and (1') are well known and the derivations are simple and they follow the procedures given in such textbooks as that by Timoshenko and Gere [4], Boundary conditions considered will be in the following

form:

y"' (0) + P(0)y'(0) + k^(0) = 0 (2a)

-y"(0) + k2y'(0) =0 (2b)

- y'" (1) - (l-k5)P(l)y'(l) + k3y(l) = 0 (2c)

y"(l) + k^y'(l) = 0 (2d)

where k^, k2 are the deflection and rotation spring constants at x = 0 and kj, k4 are the same at x = 1. The constant k^ is related to a "constant of tangency" KQ by equation

Kg = kg - 1 C3)

so that Eq. (2c) becomes

-y"'(l) + KgP(l)y'(l) + kjxCl) = 0 (2c')

where now, if P(l) ^ 0, 9 = Kgy'(1) denotes the angle that P(l) is to be rotated with respect to the tangent of the beam at x = 1 (Figure 1).

Eqs. (2) simply state that the total shear force and moment at X = 0 and x = 1 must be zero. As k^^ approaches infinity, Eq. (2a) requires that y(0) = 0. Thus a zero deflection boundary condition is arrived at. Similar options are provided for by other spring constants k2, k3 and k4.

Three different P(x) will be considered in this paper: (1) P(x) = P, a constant, (2) P(x) = q(l-x), and (3) P(x) = qo/2(l-x)^ where P represents a concentrated force at x = 0, q is a uniformly distributed follower force density and q^ denotes the maximum of a linearly varied follower force density. With the special boundary conditions of a cantilever, case (1), (2), and (3) become the classical problems first solved by Beck [3], Leipholz [5], and Hanger [6], respectively.

III. SOLUTION FORMULATIONSo The solution method used here is the finite element unconstrained variational formulation which has proved to be efficient and simple to use for solutions of non-self-adjoint problems [7,8]. Finite elements are used in the usual sense that the unknown function is approximated by piecewise cubic splines. An uncon- strained variational statement is established and used so that none of the boundary conditions need to be satisfied a priori. An outline of the formulation will be given here.

Introducing an adjoint field variable y*(x), it is a simple matter to see that the following variational statement will lead to the differ- ential equation (1) and boundary conditions (2):

6l(y,y*) =0 (4a) -- - ■■ ■:> ■■ ■:■. 1

I = / (y"y*"-P(x)y'y*'-P'(x)y'y*+^Vy*)dx 0

+ k^y(0)y*(0) + k2y'(0)y*'(0) + k3y(l)y*(l) + k^y'(l)y*'(1)

+ k5P(l)y'(l)y*(l) . (4b)

The fact that Eqs. (4) lead to the given differential equation and boundary conditions for y(x) independent of y*(x) implies that one can take the variation of I at y*(x) E 0 and (6l)y*=Q = 0 still leads to the original problem. Hence our formulation begins with

(61) y*so

= 0 (5a)

or, 1 / [y"6y*" - P(x)y'6y*' - P(x)y'(Sy* + XV<5y*]dx 0

■+ k^y(0)6y*(0) + k2y'(0)6y*'(0) + k3yCl)6y*(l) + k4y'(l)6y*'(1)

• + k y'(l)6y*(l) = 0 (5b)

Finite element discretization enters when the beam is divided into L equal elements and Eq. (5b) is written as .

■ I nLV^'^"6y*^'^"-LP^i^(?)y^i^'6y*^i^' i=l 0 . '

2 - Lp(i)'y(i^'6y*(i) . Al/Dfiy* (i)]d^

L

V' - + kjy'^^^(0)6y**^^^(0) + k2LV*-^'''(0)6y**-^-''(0) / .

+ k5y(L)(l)6y*fL)(l) + k^L^^^^' (l)5y*f'^^ * (1)

+ k5y'^^^'(l)6y*'^^^(l) =0 (6)

In obtaining Eq. (6) from (5b), one has effected a change of coordinates . from X (global) to ^ (local) such that

^ = ^Ci) = Lx-i+1

d? = Ldx

(i) ^'^ y(x) = y^^hQ

y'(X) = ~ y(x) = L -^ yf^^(?) = Ly^^^'(?) dx d?

etc.

Introducing generalized coordinates vector Y*-''"-' and shape function vector a(?) such that

yfi)(?) = a'r(?)Y(i) : . (8a)

with yCi^T ^ ^^^H) Y^(i) Y^(i) Y^(i)^ ^g^^

a(C) = 1 0 -3 2 n \ 0 1 -2 1

^

0 0 3 -2 r 0 0 -1 1 \K\

(8c)

where a superscript T denotes the transpose of a matrix. One observes that

(i) - M)

Y (i) 3

= y^-^(O) ^2^'^ -y^'^'w

(i) = y(i)' (1) (8d)

The counterparts for y*(C) can be similarly defined^

In terms of Y'•■'■-', Y*^-"--*, a, Eq. (6) can be written as:

L . 1 1 ri, I 6Y*(^)'r{L^J a"(C)a"'^(5)d^ - L/ pf^^(C)a'(C)a''^(Qd?

' ■ . i=0 0" ~ 0 ~ ~

- L/ pf'-^'(C)a(?)a''^(Qd? + Ai / a(C)a'^(5)d5}Y^^) 0 r ". L 0" ~

+ 6Y*'--^^'^{k.a(0)a'^(0) + k-L^a'(Oa'^^CO) }Y^^^ "

+ 6Y*f^^'^{k_a(l)a'^(l) + k.L^a'(l)a''^(l) + k^a(l)a''''(l) lY^^^ = 0

It will be convenient to define the following matrices:

(9)

A, = J a(Qa^(Qd? . A2 = / a'COa'V^d^ ~ 0 "0

A^ = / a"(Qa"'^(Qd5 ' 64 = / aCaa''^(?)d? 0

A5 = / ^a'(C)a'T(5)dS - 0

1

Ag = / a(?)a'T(^)d^

A7 ^ / ?'a'(?)a' (5)dC 0

(10)

B = a(0)a'(0) ?2 a'(0)a'(0)

B = a(l)a'r(l) , B = a'(l)a''^(l)

B = a(l)a''^(l)

In terms of the matrices defined in (10), Eq. (9) is written as:

+ 6Y*(^^'^{k^B^ + k2L'B2}Y(^5

+ 6Y*'^^^'^{1^383 + k^L^B^ + k^B^lY'^^^ =0 (11)

where the matrix NL is defined as . 1 . 1 .

M = / p'^^^a'a-'^d? + / p'^^^'aa-'^d^ . (12) ~P Q ~ ~ 0 ~"'

To proceed further, it is necessary to know the specific form of P(x). As we have mentioned earlier, three different forms of P(x) will be considered.

CASE I. P(x) = P, a Constant, In this case, one has

P(x) = p'^^hK) = P

P'(x) = LP'^^^'(5) = 0

Thus 1

'"' CASE li. P(x) = q(l-x),

Thus,

P(x) = P^^^(0 = ^ (L-i+1-0

* 1 T 1 T M = f {[L - (i-1)]/ a'a'^dS - / Sa'a'^d? -PL Q~~ O""

or

(13)

M = Pj a'a'dS = PA . , • (14) ~P 0~ ~

(IS)

M = f {[L - (i-DJA, - A } (16)

CASE IIIo P(x) = qo/2(l-x)'

Thus,

or.

>(i) dm Vt

M. qo -p

{CL-i+l)2j a'a''^d5 - 2(1-1+1]/ 5a'a''^d5

1 2L2 O~ ~ 0

1 T + / C^a'a' d^}

0

5° {(L-i+1)/ a'a''^dC - / 5a'a»"d?}

^

qo 2L2

{(L-i+l)2A2 - 2(L-i+l)A5 + A^}

(17)

- 4 {(L-i+l)A - A } (18)

With NL defined for all three cases in Eqs. (14), (16), and (18) respec- tively, one can now assemble Eq. (11) into a global matrix equation. Introducing the global generalized coordinate vectors Y and Y* as: i

YT _ . Y (1) V (1) Y fl^ Y (1) Y f2) Y 1^2) I ~ ^ h ^^2 ''S 4 3 4

(L) (L) }

Y*T - ry *(1) Y *(1) Y *C1) Y *fl^ Y *f2) Y *f2) Y *fL^ Y *'^^h I ~ ^'^l ^^2 3 4 3 4 3 4 ^

(19)

Eq. (11) now can be written in terms of Y and 6Y* as

:..^: <SY*'^{K - X=^M}Y = 0 ; ' (20)

where the global matrices K and M are formed by properly placing the local matrices defined in EqSo (10) according to the correspondence between the local and global generalized coordinates indicated in Eqs. (19). Now since 6Y* are not subject to any constraint conditions, Eq. (18) reduces to

I ■■ ■ ■.■• ■ ■ ' (K - A^M)Y = 0 (21)

which is solved for the eigenvalue \ and the eigenvector Y.

IV. NUMERICAL RESULTS AND DISCUSSION. It is yvell known that the eigenvalue X dictates the stability of the column: a pure imaginary X is associated with a stable vibration, a real X with instability of divergence, and a complex X with instability of flut^ter [10].

Only cantilevered columns will be considered here. It will be seen that in all three loading cases, the cantilevered columns reaches an instability condition of flutter„

CASE I. P(x) = P = Constant. The characteristic equation in close form was obtained by Beck [3] as

2X^ + Q^ + 2A^coshacos6 + QXsinhasinB =0 (22)

where ,

./ Q 2

a^ =n. , . , ^

(25)

,^ J^TTf. Q For a given Q, the eigenvalue X can be calculated from Eq. (22) and there are an infinite number of X solutions for each Q. Eq. (22) is solved for two lowest branches of X using an iterative procedure. The results are given in Table I. The critical load thus obtained is

Q^j^ = 2.03187r^ = 20.053

which agrees well with the value obtained originally by Beck as QQJ^ = 20.,05. The results for four lowest eigenvalues presently obtained using our finite element-unconstrained variational formulations are also shown in Table I. The first two branches obviously agree well with those from the exact characteristic equation. It should be pointed out that the numerical solutions to the Beck problem given in Reference [4] appear to be inaccurate. A plot of the eigenvalue curve showing the coalescense of the two lowest branches is given in Figure 2. The data from Reference [4] are indicated by small circles^ The fact that these data points do not fall on a smooth curve further add to the doubt about their accuracy.

CASE II. P(x) = q(l-x). The numerical values of the four lowest eigenvalues up to the first critical load are giyen in Table II which the critical load is shown to be

KR 4.0591iT^

compared with data given by Leipholz as 4.1238Tr^ = 40.7 [11] and again as 4.205877^ = 41,51 [12]. The coalescence of the first two branches of eigenvalues is again shown in Figure 2,

CASE III. P(x) = qo/2(l-x)^. Similar data for this case are

presented in Table III and in Figure 3. The critical load of flutter is obtained as

%CR "" 15.2687'ff^

In comparison, the value obtained by Hauger was qgCR ~ [6] and that by Leipholz, q^^^ = 150.80 = 15,279iT^ [12]

158.2 = 16.09?7T'

$

REFERENCES

1. H. Zeigler, Principles of Structural Stability, Blaisdell, 1968, p. 33.

2. J. J. Wu, "Gun Dynamics Analysis by the Use of Unconstrained, .;■ .. Adjoint Variational Formulations," Proceedings of the Second

U.S. Army Symposium on Gun Dynamics, September 1978, pp. II80- II99.

3. M. Beck, "Die Knicklast des einseitig einseitig eingespannten, tangential gedruckten Stabes," ZAMP, 1952, Vol. 52, pp. 225-229.

4. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw-Hill, 1961, p. 3. , ,

5r H. Leipholz, "Anwendung des Galerkinschen Verfahren auf nichtkonser- vative Stabilifatsproblerae des elastischen Stabes," ZAMP, 1962, Vol. 13, pp. 359-372.

6. W. Hauger, "Die Knicklasten elastischer Stabe under gleichmassig verteilten under linear veranderlichen, tangentialen Druckfreften," Ingenieur Archiv, 1966, Vol. 35, pp. 221-229.

7. J. J. Wu, "A Unified Finite Element Approach to Column Stability Problem," Development in Mechanics, Vol. 8, 1975, pp. 279-294.

8. J. J. Wu, "On Missile Stability Journal of Sound and Vibration," 1976, Vol. 49(1), pp. 141-147.

9. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw-Hill, 1961, p. 155.

10. D. A. Peters and J. J. Wu, "Asymptotic Solution to a Stability Problem," Journal of Sound and Vibration, 1978, Vol. 59(4), pp. 591-610.

11. H. Leipholz, Stability Theory, Academic Press, 1972, pp. 239-241.

12. H. Leipholz, "On the Calculation of Buckling Loads by Means of Hybrid Ritz Equations," Archives of Mechanics, Warszawa 1973, Vol. 25(6), pp. 895-901.

13. H. Leipholz, "On the Solution of the Stability Problem of Elastic Rods Subjected to Triangularly Distributed Tangential Follower Forces," Ingenier Archiv, 1977, Vol. 46, pp. 115-124.

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