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A direct variational method for non-conservative system stability
Journal of Sound and Vibration (1982) 80(4), 447-459
A DIRECT
VARIATIONAL
NON-CONSERVATIVE
METHOD
SYSTEM
FOR
STABILITY
J. G. PAPASTAVRIDIS School of Engineering Science and Mechanics, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A. (Received 1 May 1981)
Hamilton’s principle is applied to analyze the problem of the stability of equilibrium of a discrete, holonomic, and scleronomic mechanical system under conservative and non-conservative (position and/or velocity dependent) forces. At the stability limit, the vanishing of the second order terms (in the deviations from equilibrium) of the total action change functional leads to the condition that the matrix of a certain quadratic form be singular; this yields the eigenvalue (frequency-load) curve. The flutter loads follow by setting the frequency derivative of the determinant of this matrix equal to zero; the energetic interpretation of this latter is also given. When the non-conservative forces go to zero it is shown that one recovers the well-known discrete conservative system stability criterion. An application follows, and finally in an Appendix various relevant time-integral equalities are summarized and interpreted.
1. INTRODUCTION
In structural mechanics, the rigorous and systematic formulation of variational/extremum methods for the stability analysis of consetvarive systems was first carried out by Trefftz and his school in the thirties of this century (see references [l-5]). Some application has also been made to non-conseruatiue systems, such as elastic-plastic buckling (see, e.g., references [6-g]). Since then, these “principles” have been occupying a fairly central role on both the theoretical and the numerical fronts. The common characteristic of all these cases, however, is that the treatment has been wholly static; i.e., any actual, or virtual, work or action of inertial forces has been completely neglected! Specifically, in static stability the starting point has been the principle of virtual displacements. With it, and some help from kinetics, one derives further either the “principle” of minimum potential energy (Lagrange + Dirichlet + Koiter-see references [9-l l]), or the variational version of the Eulerian udjucent equilibrium configuration method (Trefftz+ Marguerre-see reference [12] for an exceptionally clear treatment). It should be pointed out here that, whereas the minimum potential theorem holds true for conservative systems only, the adjacent equilibrium approach is free from that restriction, provided of course that the non-conservative system’s stability can be satisfactorily treated by completely neglecting all inertial effects. For conservative systems, of course, both methods lead to the same equations, but, in view of what shall follow in this paper, their different concepts, formulations, and interpretations should be clearly kept in mind. When it comes to kinetic treatments of the stability of equilibrium though, whether by choice (e.g., kinetic treatment of conservative Euler column buckling), or by necessity (e.g., kinetic treatment of circulatory “follower-force” buckling), variational principle applications have been, with few exceptions (see references [13-151 and Chapter 5 of reference [16]), peripheral and sporadic-see, e.g., references [17-241; reference [24] contains a sufficiently general 447 0022-460X/82/040447 + 13 $02.00/O
@ 1982 Academic
Press Inc. (London)
Limited
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J. G. PAPASTAVRIDIS
non-conservative formulation of Hamilton’s principle, but the author is unaware of any truly kinetic stability applications of it. Similar remarks apply to the stability of motion problem: e.g., of steady state oscillation, or of steady motion. “Applications” in the preceding statements means applications in the spirit of the direct methods of the calculus of variations, and not merely those in which the principles are utilized as alternative vehicles for the derivation of the equations of motion, as, e.g., in references [25,26]. Here the starting point should be either Hamilton’s principle in its various forms (see, e.g., reference [27] and Chapter 12 of reference [28]), or some other equivalent timeintegral principle. From the existing approaches, as mentioned earlier, those of Leipholz [13-161 should be singled out for special consideration. The reason is that his treatments, in addition to their generality and direct numerical usefulness (see, e.g., reference [29]), also provide conceptual continuity with the rest of the energetic principles of analytical mechanics! The guiding philosophy of this paper is similar, i.e., to extend the familiar variational principles of dynamics so as to cover results already known from the equations of motion method; the solution of new problems should be attempted at a later stage. Specifically, the concept of adjacent motion, motion through states near a given state of equilibrium (or steady motion), is combined in this paper with Hamilton’s principle to derive conditions that the system’s parameters must satisfy at the stability threshold; this is done in section 2. In section 3 an application of this theory to the well-known discrete version of the follower-force buckling of a column is presented. Finally, in an Appendix, the steps leading from the equations of motion to the various relevant time-integral identities are summarized. In this sense the method presented constitutes the legitimate kinetic counterpart of Trefftz’s variational formulation of the static method of adjacent equilibrium. 2. GENERAL THEORY For analytical simplicity and physical clarity the basic ideas and results are presented first for a one degree of freedom system; the several degrees of freedom (holonomic) case then follows. 2.1. ONE DEGREE OF FREEDOM For a one degree of freedom mechanical system with generalized co-ordinate q = q(t), kinetic energy T(q, 4), potential energy V(q), and additional non-conservative generalized force Q(q, 4) (where (. . .)‘=d(. . .)/dt, t being time), Hamilton’s principle states deviation that for any kinematically admissible, i.e., virtual (and contemporaneous), Sq = Sq(t) from a, say, fundamental state I defined by 4, one has 6A(I)=Ir(6T-6V+6W)dt 11
(SW=Q*Sq)
=-I,;[[(%)‘-(!$I-[-~+Q],~Ot+((~) &I,:: (1) where ~1, t2 are arbitrary times; if &(td = Sq(t2)= 0, or if I is an equilibrium state, then SA(I) = 0. Here SA(I) is not the first variation of some action functional A[q(t)], from
ON NON-CONSERVATIVE
SYSTEM
STABILITY
449
state Z to another virtually attainable one, in the conventional variational calculus sense, but it is simply the homogeneous collection of first-order terms in &I (and its derivatives) that results by applying d’Alembert’s principle to state I. Schematically, equation of motion + d' Alembert’s variational equation 4 Hamilton’s Principle t Lagrange’s “Zentralgleichung”;
(2)
see reference [27], and also the Appendix. In the language of modern functional analysis SA is a “GfUeaux” derivative, rather than a “Frechet” one; here, however, we shall refrain from such physically unmotivating (and intimidating!) formalisms and terminology. In the same spirit for the rotal virtual action change dA between the fundamental state Z and any other state ZZ described by q(ZZ) = q(Z) + Sq, to the second order in the (admissible, weak, and contemporaneous) variations Sq, 84, one sets AA r_1I=
fz(AT-AV+AW~,~~)dt=SA(Z)+(1/2!)S2A(Z), I fl
(3)
where S2A(Z)=62A(q,6q)=f2(b’T-S2V+G2W)dt, (1
(4)
S*T = (a2T/a(i2)(&j)2+ 2(a2T/aqad)8q&.j + (a2T/aq2)(6q)‘, s2 v = (a2v/aq2)(sq)2,
S* W = K? . Sq = [(aQ/aq)sq + (aQ/ad)&j]6q,
(5)
and all partial derivatives are evaluated at state I. In a conservative system Q = 0 (or aQ/aq = 0), and S2A is the second variation of the action functional A(q) from the fundamental state Z in the conventional mathematical sense. The main point can be restated: the virtual displacement (or, function) is only required to be kinematically admissible, i.e., to satisfy certain smoothness requirements in (tl, tz), and special conditions, such as Sq(tI) = Sq(t2) = 0, or Sq(tI) = Sq(t2) # 0, at the end points tl, tz, but, in general it does not represent an actual (i.e., dynamical) neighbouring displacement. Returning now to the study of S’A, by simple integrations by parts of equation (4), with use of equations (5), one can show that a2A = f (sq) =
” J(Sq)Sq dt + EPC, I 11
(6)
(a2T/ad2)sg + [(a2T/a(i2). - (aQ/ad)&j +[(a2T/aq aq)‘-(a2T/aq2)+(a2V/aq2)-(aQ/aq)]sq,
EPC = {s[(aT/a~)sq]}:: = {(a’T/aq aQ)(&)‘+ (a2T/aQ2)sq stj}::.
(7)
Here EPC is the (second-order) end points contribution. In particular if, as is commonly the case, 2T = M2 (m being a positive constant) or, in general terms, if the system is scleronomic and the fundamental state Z is one of equilibrium, i.e., q(Z) = 0, 4(Z) = 0, then expression (7) simplifies to Z(sq) = (a2T/ad2)Sj + (-aQ/ad)stj + (a2V/aq2-aQ/aq)Sq, EPC = {(a2T/a(i2)sq tq}::,
(8) where now all the partial derivatives have been reduced to constants; this also happens when I is a state of steady motion. The conditions that determine the sign of the quadratic form-functional S2A are given in the calculus of variations. In general for an arbitrary admissible, but not actual, neighbouring motion S2A f 0, even when Sq may vanish at
450
J. G.
PAPASTAVRIDIS
the endpoints; see references [30, Chapter 51, and also [31, pp. 103-1061 and [32]. If, however, Sq is chosen to represent the admissible and actual variation &, then -S*A(q, 6q) = &‘A = --I *‘J(&q)* 6q dt + EPC = EPC (= EPC evaluated at &q,&), 11 since in this case J(s;) =I = 0.
(9) (10)
Equation (lo), plus appropriate initial conditions, is the well-known Jacobi-Poincare “equation aux variations” of the adjacent (linearized) motion. In the special case when R reduces to zero, all adjacent small motions are described by ?A=O.
(11)
Equations (9) and (11) essentially express the law of energy conservation, or work/energy theorem for the adjacent motion. In the same way Lagrange’s (first-order) “Zentralgleichung”, or Hamilton’s principle, yield the energy equation for state I; see the Appendix, equations (AlO) to (A16). When S*A is the second variation of some functional A, equation (11) then constitutes a well-known theorem of variational calculus; see, e.g., reference [30, pp. 10%1111. A similar approach, and energetic interpretation, has appeared in the variational treatment of elastostatic stability via the well-known adjacent equilibriumconfiguration method; see references [l], and especially [33, pp. 6-121. In kinetics this action-based approach, as expressed by equations (6), (9) and (ll), seems to have been first advanced, for conservative systems, by Routh [31, pp. lOl-1061; see also references [28, Chapter 121, [34, pp. 416-4391, and [35, pp. 250-2541. Despite these early researches from such mechanics masters it seems to us that so far no systematic attempt at a full exploitation of such direct time-integral-of-energy methods to the study of the adjacent motions about equilibrium or motion, under fairly general non-conservative disturbances, has been made. 2.1.1. The stabilityproblem In a kinetic study of the stability of state I under conservative and non-conservative forces, V and Q depend on, in addition to q and 4, a loading parameter A. Therefore, and since equation (10) is a linear constant-coefficient equation, one sets 8q(t) = C exp (pt)
(12)
in equation (9), or (1 l), and determines the conditions between the load A and frequency p during the adjacent motion; the constant amplitude C depends on the initial conditions at state I. The nature of this adjacent motion is determined by the character of p. Setting for convenience p = io, one distinguishes the following cases: (i) w* = real and positive + steady state oscillation, i.e., stability; (ii) w* =real and negative, or zero+aperiodic diverging motion, i.e., divergence instability; (iii) U* = complex-, increasing amplitude oscillation, i.e., flutter instability; for a more complete characterization, especially in the several degrees of freedom case, see reference [36, Chapter 21. Choosing now tl = 0 and t2 = 27r/w in equation (9) one obtains ~*A=.(~).,(~,A;C)+EPC,
(13)
n(o)=-102./“(e2iui)dt+0,
m&A;
a=[($)(--o*)+(--f?)(iw)+($-$)]Cz_d(u,A)C*.
(14)
ON
NON-CONSERVATIVE
SYSTEM
451
STABILITY
Thus equation (11) translates to d(w, A) = 0.
(15)
Due to equation (lo), with equations (8) and (12), the condition expressed by equation (15) holds true for all three cases described earlier. However it is only for the first case of completely periodic oscillation that equation (15) is equivalent to the direct variational statement pA=O,
since for the second and third cases, due to their non-periodicity, zero; in fact, as equation (8) shows, it reduces to
(16)
the EPC term is not
The study of the sign of S*A, for all admissible and actual neighbouring motions, and its significance to the study of the stability/instability of state I, when this latter is a (fairly general) state of mofion, is currently under investigation by the author; the results shall be reported in the near future. Here the problem of interest is the formulation of conditions at the stability limit: i.e., at the threshold beyond which the character of & changes from that of case (i) to that of (ii) or (iii). Therefore, these “critical” conditions can be equivalently described either by the more familiar equation (15), or by the direct variational statement (16). The additional conditions that determine A,, will be described later. Finally, the connection with conservative system stability is made clear by setting, in equations (14) and (15), KI/&.j = 0 and w = 0 (since a conservative system belongs to the divergence type), and absorbing -aQ/aq into a* V/aq*; then equations (14) and (15) reduce to the well-known A(0, A) = a*V(q, A)/ikj* = 0.
A discussion of the flutter-type treatment.
(17)
of buckling will follow the several degrees of freedom
2.2. SEVERAL DEGREES OF FREEDOM For an N degree of freedom holonomic and scleronomic system the homogeneous quadratic part of the total virtual action change from the fundamental equilibrium configuration I, with use of Einstein’s summation convention and all Latin subscripts running from 1 to N, is 62A(q,,Sql)=~‘z(G2TS2V+62W)dr, 11
and the non-conservative
(18)
forces Qk = Ok (41,(i,,,) satisfy the non-potenfidity condition
Q/c,/# Ql,k.
(21)
This formulation covers all known types of autonomous systems, e.g., conservative (gyroscopic or not), dissipative, and circulatory; see, e.g., references [36,37].
452
J. G. PAPASTAVRIDIS
Reasoning as for the one degree of freedom case one finds S2A=-
‘zJ~(6q,)Sq~dt+EPC, I I1
&(&I) = (Tki)aiir + (-Qk.i)%/ + (vkl EPc
=
{( T,#qk&.jl}$
=
{homogeneous
-
Qk.dh,
bilinear form in Sq, Sg}:,
(22)
5
I
ii2A=~2A(qk,&,)=-
11
Jk
‘Jk(&,),
(.& *& ) d t + EPC,
EPC= EPC(s;I, 54,.
(23)
Again, at the stability limit the EPC term goes to zero due to the 27r/w periodicity of 8q and &, and the critical condition becomes &‘A = 0.
(24)
With &k(t) = ck exp (pt) = ck exp (iwt), tl = 0, t2 = 27r/w, and expressions (22) and (23), equation (24) translates to (25) or, since 0(w) =-l,‘*‘-
(eziw’)dt # 0,
(26)
to D(w,h;Ck)“Dkl(W,A)‘CkC,=O,
~k,~(Tfi)(-W’)+(-Qk,l)(iW)+(Vk,-Qk,,).
(27)
From here on the argument is analytically identical with that employed in the static stability analysis of discrete conservative systems: at the stability limit one is seeking non-trivial dynamic states for which S?A/ D = D becomes indefinite (degenerate). As the theory of quadratic forms shows this occurs when the coefficient matrix D = (Dkl) becomes singular: i.e., when det (Dk,)“&,
I\) =
~(~i~)(-W2)+(-Qk.j)(i~)+(Vkr
Qk.,)l
=
0.
(28)
Since the Dk, elements involve the motion frequency w, as well as the loud A, the equation A = 0 supplies the load-frequency relation or hypersurface. For a general discussion of equation (28), and the dependence of the nature of this hypersurface on the signdefiniteness and symmetry/asymmetry properties of the constant matrices_ T = (Tit), Q=(-Qk,[) and V-Q=(vkl -Ok,,), see, e.g., references [36, 371. Equations (14), (15) and (28) are essentially the same as those derived by Leipholz; see references [l4, equations (27) and (86)], [15, equation (3.7)] and [16, equations (5.170) and (5.171)]. Here further discussion is limited to the following two classes. (i) Conservative non-gyroscopic systems. Here Qk,i = 0, Qk.r = Ql.k (i.e., all the ok.1 terms can be absorbed into v,,), and the critical case occurs for w = 0 (i.e., non-trivial adjacent equilibrium configuration: divergence buckling. Equation (28) then yields the well-known condition for the critical load d(O,A)=]Vkl]=O:
(29)
i.e., for such systems inertial effects do not influence the stability investigation. (ii) Non-conservative, purely circulatory systems. Here Qk,i = 0, and Qk,l = skew-symmetric (its symmetric part, if any, can always be absorbed into vkl). Following
ON NON-CONSERVATIVE
SYSTEM
453
STABILITY
Leipholz [14], at the critical load, in addition to equation (28), one has dw2/dA + 00, or dA/dw2 = 0,
(30)
which, since along the eigenoalue curve A = A(w2, A) dA/do2 = aA/&u2+ (ad/ah).
(dA/dw’) = 0,
(31)
translates to aA(w2 , A yaw2 = 0.
(32)
These results are well-known; for detailed discussions see, e.g., references [14,16, pp. 233-239, and 36, pp. 60-631. From our viewpoint, however, it is instructive to examine the energetic meaning of equations (31) and (32): i.e., to determine their connection with equations (22) through (28), and their w2-derivatives. Along the oscillatory and purely periodic (“lower”) part of the eigenvalue curve A(w2, A) = \(Qi)(-w2)
(33)
+ (VU - C&r)] = 0,
the “energy” equation 8’A = 0 is an identity in 02, and A = A(w2). Therefore
along this
part of the curve
d/dw2[2A(W2,
A(w2); Cj,.)]=
0,
(34)
and at the critical point, due to equations (25)-(27), and (30), d/do2(s2A)
= (d0/dw2)D
+ R(dD/dw2)
= R(dD/do2)
= R(aD/aw2) = 0,
(35)
or finally, aDlaw
= 0:
(36)
i.e., aDlaw is also indefinite at the critical point. The equivalence between equation (36) and equation (32) is most conveniently seen through a reduction of the quadratic form D = D&C’, to a diagonal form, stability coefficients a set of D = dk. c:, (37) I:, ( principul co-ordinates’ by means of a constant non-singular linear co-ordinate
transformation
c = Ac’(det A =
A # 0); when, in addition, A is orthogonal the coefficients dk become the eigenvalues of D. Now if d is the discriminant of D in equation (37), as linear algebra shows (see, e.g.,
reference [38]) A2.A=d,
d =dl*d2..
. :dN.
(38)
Equation (28) then implies that d =0:
(39)
i.e., at least one of the coefficients is zero. On the other hand, the indefiniteness of aD 7
au
= 3.
cl = 0, due to equations (36), (37))
requires that its discriminant be zero, or ad, jaw2 . . . ad,f au2 = 0: i.e., at the critical point at least one of the partial stability coefficients adJaw
(41)
vanishes.
454
J.
G. PAPASTAVRIDIS
Equations (41) and (39) imply that ad/h2
=
(adl/aw2)d2 . . . . - dN + a . . + dl a . . . . dN_l(adN/aw2) = 0,
(42)
and this, thanks to equation (38), finally results in equation (32): i.e., aA/aw2= 0. one can say that the critical (flutter-buckling) point of the eigenvalue curve h = A(w2), at which dh/dw2 = 0 (or dw2/dh + CCJ),is determined either by A(w2, A) = 0 and aA(W2, A)/b2 = 0, or equivalently, by $A(w2, A) = 0, and d/do2[?A(W2, A)] = 0. Finally, the first-order virtual change S(&‘A) of s2A produced by the contemporaneous (and fixed frequency) variations around state IIS = SC, exp (pt) that vanish at the endpoints 0 and 27r/w is, by Hamilton’s principle, applied to the actual periodic trajectory 11, zero: i.e., Summarizing,
S(??A) = 0.
(43)
This variation yields as Euler-Lagrange equations jk = 0; see also the Appendix for derivations obtained by substituting z = a(&) in equation (AlO), or 84 = a(&) in equation (All), or finally by taking the S-variation of equation (A12). Equations (24) and (43) are the kinetic counterparts of Trefftz’s static criteria, as expressed by his equations (26) and (27), respectively, in reference [l]; see his comments surrounding these equations, and also the final lines of the same paragraph. A complete correspondence with the static case, however, would require the formulation of extremum conditions and not simply the stationarity ones formulated here. This is a considerably more difficult problem since the set of adjacent states of motion is infinitely richer than that of adjacent states of equilibrium. Finally, it is expected that the methodology presented should prove powerful in handling large (non-linear) displacements and the effect of geometrical imperfections. It is hoped to answer some of these questions in future publications. 3. APPLICATION:
FOLLOWER-FORCE
BUCKLING
As an application of the foregoing theory one may examine the well-known discrete version of the buckling of a column under the “follower” force P applied to its free end;
Figure 1. Two degree of freedom model of follower-force column buckling.
ON NON-CONSERVATIVE
455
SYSTEM STABILITY
the system with all its kinematical, inertial, and physical parameters is shown in Figure 1. For small motions around the fundamental state of equilibrium I one has (see, e.g., reference [14]) T- $&34:
+ zq&+
(if>,
v = &(2q: -2q1q2 + 4%
Q2=0,
Q1=P.Z(q,-qz),
(Q1,2
#
(44)
Q2.11,
and therefore
T=
3m12
(
@ZWWvNdr =
mlz
Q=
ml2 mlz
1
,
v=(a2V/aql&)t=(~~
-7).
(aC?k/dqt)t = (;
-:),
(45)
Equation (28) then yields =O
det[T(-W2)+V-Q]=A(m2,P)=
’
(46) or A(~2,A)=2’Z’4-(7-2h)!Py2+1=0, ?P2= (m12/c)u2,
A = (l/c)P.
(47) (48)
The critical load results from the solution of the system consisting of equation (47) and aA(!P2,A)/a?P2=4rIrZ-(7-2A)=O.
(49)
One thus obtains the well-known values ACr= (7 -J&/2
= 2.086,
*;=&/2=0.707.
(50)
REFERENCES 1. E. TREFFTZ 1933 Zeitschrift fiir angewandte Mathematik und Mechanik 13, 160-165. Zur Theorie der Stabilitiit des elastischen Gleichgewichts. 2. I$. MARGUERRE 1938 Zeitschrift fiir angewandte Mathematik und Mechanik 18, 57-73. Uber die Behandlung von Stabilitiitsproblemen mit Hilfe der energetischen Methode. 3. I$. MARGUERRE 1938 Jahrbuch der Deutschen Veruchsanstalt fiir Luftfahrt pp. 252-262. Uber die Anwendung der energetischen Methode auf Stabilitiitsprobleme. 4. R. KAPPUS 1939Zeitschriftfiirangewandte Mathematik und Mechanik 19,271-285,344-361. Zur Elastizitiitstheorie endlicher Verschiebungen. 5. B. BUDIANSKY 1974 in Advances in Applied Mechanics 14, l-65. New York: Academic Press. Theory of buckling and postbuckling behavior of elastic structures. 6. J. W. HUTCHINSON 1974 in Advances in Applied Mechanics 14, 67-144. New York: Academic Press. Plastic buckling. 7. M. R. HORNE 1961 in Progress in Solid Mechanics II, 279-322. Amsterdam: North-Holland Publishing Company. The stability of elastic-plastic structures. 8. R. HILL 1978 in Adoances in Applied Mechanics 18, l-75. New York: Academic Press. Aspects of invariance in solid mechanics. 9. W. T. KOITER 1972 Stijfheid en sterke 1-Grondslagen. Haarlem: Scheltema & Holkema. 10. W. T. KOITER 1965Proceedings Koninklijke Nederlandse Akademie van Wetenschappen B68, 178-202. The energy criterion of stability for continuous elastic bodies; Parts I and II. 11. W. T. KOITER 1967 Recent Adoances in Engineering Science 3, 197-218. Purpose and achievements of research in elastic stability. 12. K. MARGUERRE 1950in Neuere Festigkeitsprobleme des Zngenieurs, pp. 189-249. Berlin/GGttingen/Heidelberg: Springer-Verlag. Knick- und Beulvorg%nge.
456
J. G. PAPASTAVRIDIS
13. H. H. E. LEIPHOLZ 1971 Ingenieur-Archiu 40,55-67. Uber die Erweiterung des Hamiltonschen Prinzipsauf lineare nichtkonservativeProbleme. 14. H. H. E. LEIPHOLZ 1977 Actu Mechanica 28, 113-138. On the application of the energy
method to the stability problem of nonconservative autonomous and nonautonomous systems. 15. H. H. E. LEIPHOLZ 1980 in Mechanics Today 5, 193-214. Oxford/New York/Toronto/ Sydney/Paris/Frankfurt: Pergamon Press. Stability analysis of nonconservative systems via energy considerations. 16. H. H. E. LEIPHOLZ 1977 Direct VariationalMethods and Eigenvulue Problems in Engineering. Leyden: Noordhoff International Publishing. (Translated from the German, 1975.) 17. T. B. BENJAMIN 1961 Proceedings of the Royal Society London A261, 457-486. Dynamics of a system of articulated pipes conveying fluid; Part I: Theory. 18. G. HERMANN and S. NEMAT-NASSER 1966 in Dynamic Stability of Structures, pp. 299-308. Oxford/New York: Pergamon Press. Energy considerations in the analysis of stability of non-conservative structural systems. 19. T. ROORDA and S. NEMAT-NASSER 1967 American Institute of Aeronautics and Astronautics Journal 5, 1262-1268. An energy method for stability analysis of nonlinear, non-conservative systems. 20. B. BUDIANSKY 1966 in Dynamic Stability of Structures, pp. 83-106. Oxford/New York: Pergamon Press. Dynamic buckling of elastic structures: criteria and estimates. 21. L. W. REHFIELD 1973 International Journal of Solids and Structures 9, 581-590. Nonlinear free vibrations of elastic structures. 22. J. G. PAPASTAVRIDIS 1981 Journal of Sound and Vibration 74, 499-506. An energy test for the stability of rheolinear vibrations. 23. J. G. PAPASTAVRIDIS 1982 (to appear) Journal of Sound and Vibration. Parametric excitation stability via Hamilton’s action principle. Applica24. M. LEVINSON 1966 Zeitschriftfiir ungewundteMuthemutik und Physik 17,431-442. tion of the Galerkin and Ritz methods to nonconservative problems of elastic stability. 25. E. METTLER 1942 Ingenieur-Archiu 13, 97-103. Uber die Stabilitiit erzwungener schwingungen elastischer K&per. 26. E. METTLER 1947 Ingenieur-Archiv 16, 135-146. Eine Theorie der Stabilitat der elastischen Bewegung. 27. L. NORDHEIM 1927 in (old) Hundbuch der Physik 5,43-90. Die Prinzipe der Dynamik. 28. A. 1. LUR’E 1961 Analytical Mechanics. Moscow: Mathematical-Physical Literature (in Russian). 29. R. BOGACZ, H. IRRETIER and 0. MAHRENHOLTZ 1980 Ingenieur-Archiu 49, 63-71. Optimal design of structuressubjected to follower forces. 30. I. M. GELFAND and S. V. FOMIN 1963 Culculus of Variations. Englewood Cliffs, New Jersey:
Prentice-Hall. 31. E. J. ROUTH 1975 in Stability of Motion (editor A. T. Fuller). London: Taylor & Francis Limited (originally appeared 1877). 32. J. G. PAPASTAVRIDIS 1980 Journal of Applied Mechanics, Transactions of the’Americun Society of Mechanical Engineers 47,955-956. On the extremalpropertiesof Hamilton’saction integral. 33. C. LIBOVE 1962 in Handbook of Engineering Mechanics (editor W. Fliigge). Chapter 44, pp.
6-12. New York: McGraw-Hill Book Co. Elastic stability. 34. W. THOMSON (LORD KELVIN) and P. G. TAIT 1923 Treatise on Natural Philosophy: Purtl. Cambridge University Press (originally published 1867). 35. E. T. WHITTAKER 1937 A Treatise on the Analytical Dynamics of Particles und Rigid Bodies. Cambridge University Press. (Also 1944 Dover.) 36. K. HUSEYIN 1978 Vibrations and Stability of Multiple Parameter-Systems. Alphen aan den Rijn: Noordhoff International Publishing. 37. H. ZIEGLER 1968 Principles of Structuralstability. Waltham, Massachusetts: Blaisdell Publishing Company (Ginn & Co.). 38. W. L. FJZRRAR 1957 Algebra: A Textbook of Determinants, Matrices and Algebraic Forms. Oxford: ClarendonPress. 39. W. F. OSGOOD 1937 Mechanics. New York: The Macmillan Company (reprinted by Dover). 40. B. L. MOISEIWITCH 1966 Variational Principles. London/New York/ Sydney: Interscience
Publishers. 41. B. VUJANOVIC 1975 Zeitschrift ftir ungewundte Muthemutik und Mechunik 55, 321-331. variational principle for non-conservative dynamical systems.
A
ON NON-CONSERVATIVE
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457
APPENDIX: SOME USEFUL TIME-INTEGRAL (ACTION) EQUALITIES OF ANALYTICAL MECHANICS Here, too, the discussion is carried out for a one degree of freedom system; the extension to the several degrees of freedom case, for holonomic systems at least, is straightforward. Consider again the equation of motion for the fundamental state, State 1, m@=Q;
for conciseness here and throughout this whether conservative (better, potential) sufficiently differentiable, but otherwise using the basic identity (md)z =
(Al)
Appendix Q comprises all the applied forces or not. Multiplying equation (Al) by the arbitrary, function z = t(t), (md)z = Qz, and (m@)‘- m@,
642)
one obtains the first order “Zentralgleichung” (m4)i + Qz = (m4.z).
(A3)
Integrating now equation (A3) between the arbitrary following fundamental first-order uuriutionul identity :
time limits r1 and t2 yields the
JI, [(mQ)i + Qzl
dt = {W)z)?.
(A4)
So far only smoothness has been required from z(f). Its physical meaning has been left open, since, as will be seen below, all the energetic/variational principles of mechanics can be obtained from equation (A4) by appropriate special choices of z(f) as actual or uirruul displacement (or velocity, etc.). It is also clear that z(t) equipped with this general physical variational meaning is much broader than the mathematical virtual displacement Sq(t) in the variational calculus sense! On this, usually unclarified matter, see the critical comments by Osgood [39, p. 3791. Returning to equation (A4), one can examine the consequences of particular choices of z(t). (i) The choice I = q yields
‘2 J 11
(mg2 + Qq) dt = {rnQq}::.
(A%
Further manipulations of equation (A5) (with special assumptions for q(r), e.g., periodicity) lead to the famous uiriul theorem of Clausius; for details see, e.g., reference [40, pp. 20-221. (ii) The choice z = 4 yields
I
,:‘[(m4)$+Q4Jdt={m42}::,
or, since m4* = 2T, and (m4)tj = f, (T}+AT=j-‘*
(Qd)dt, 11
which is the work/energy equation for State I.
(A6)
458
J. G. PAPASTAVRIDIS
(iii) The choice z = &, with utilization of the well-known commutation yields
rule (Sq)’= 64,
~“(~T+Q~~)dt={(~~)~~}~=[~~]fz=(0,ifGq(~~)=Sq(r~)=O), I1 11
(A7)
which is Hamilton’s principle (of either varying or stationary action) for State I. Clearly equation (A7) holds for the actual displacement #q(t). For the neighbouring actual path 11, described by 4 + 64, one has 0 + &? = m (q + &)“, or Cw = @Q&)s;l
5Q = m&j,
Similar steps as before, applied to equations fundamental second-order variational identity : ‘2 [(m&Ii I 11
+ (aQ/a#&.
NJ, A9)
(A8) and (A9), lead to the following
+ (sh>z] dt = (m&j)z}::.
(AlO)
(iv) Again, as before (with &j = (&I)‘), the choice z = Sq in equation (AlO) yields -
” J11
(m&d - &Q)Sq dl= 0,
(Al 1)
from which, for Sq = 8q,
6412)
follows. This provides some clarification of the transformations (4) to equation (11). (v) The choice z = &( = (&)‘) yields successively
leading from equation
‘2[(m&j)& + (s’a)&] dt = {m(&)‘}~, I 11
Jt2[1/2m(&)*]* ‘I
dt+
J”(8Q)(&j> dt = {m(&)*}$ 11
or, finally, {62T}+d(62T)=
J’* 2(gQ&j)dt, 11
g2T = m(&)‘.
(A13, A14)
Equation (A13), with equation (A14), is the perturbation energy equation for path I. It is not hard to show that these same equations result by direct utilization of the energy rate equation for path 11, T(11) = W(H), T(U) =[1/2m(4+S;i)2]‘= &(ZZ)=(Q+8Q)(q+&)‘=
(A15)
T(r)+(gT)‘+(1/2S7T)‘, @‘(Z)+(8W)‘+(1/23W)‘.
(Al61
Other special choices of z(t) in equations (A4) and (AlO) will yield additional timeintegral theorems. Still further, useful results may be obtained if, instead of & and s’a, one uses the asymptotic expansions Sq’Eql+e2q2i-.
..,
sh-~eQ~+~20~+.
..,
(A17)
ON NON-CONSERVATIVE
SYSTEM
STABILITY
459
where E is a small parameter, in (AlO) and successively sets to zero the resulting integrals of the various E orders; for applications of this latter approach see references [20-221. Also, for more information concerning the total non-conseruutiue virtual work AWr,n of equations (3)~(5), and the work rates of equations (A13)-(A16), see the masterful exposition of Hill [8, Chapter II.A.11, and also reference [7, Chapter 11.8-111. Finally, the decisive use of the commutation rule da(. . .) = &I(. . .> in most of the above derivations implies that all the variations considered here have to be of the contemporaneous type, i.e., performed at a constant time t, just like those envisioned in d’Alembert’s principle. The variational treatment, however, of more general nonconservative (e,g., non-autonomous) or non-holonomic systems may require wider classes ones; see referenof, generally, non-commutative variations, e.g., non-contemporaneous ces [28, Chapter 12.9, lo] and [41].
A DIRECT
VARIATIONAL
NON-CONSERVATIVE
METHOD
SYSTEM
FOR
STABILITY
J. G. PAPASTAVRIDIS School of Engineering Science and Mechanics, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A. (Received 1 May 1981)
Hamilton’s principle is applied to analyze the problem of the stability of equilibrium of a discrete, holonomic, and scleronomic mechanical system under conservative and non-conservative (position and/or velocity dependent) forces. At the stability limit, the vanishing of the second order terms (in the deviations from equilibrium) of the total action change functional leads to the condition that the matrix of a certain quadratic form be singular; this yields the eigenvalue (frequency-load) curve. The flutter loads follow by setting the frequency derivative of the determinant of this matrix equal to zero; the energetic interpretation of this latter is also given. When the non-conservative forces go to zero it is shown that one recovers the well-known discrete conservative system stability criterion. An application follows, and finally in an Appendix various relevant time-integral equalities are summarized and interpreted.
1. INTRODUCTION
In structural mechanics, the rigorous and systematic formulation of variational/extremum methods for the stability analysis of consetvarive systems was first carried out by Trefftz and his school in the thirties of this century (see references [l-5]). Some application has also been made to non-conseruatiue systems, such as elastic-plastic buckling (see, e.g., references [6-g]). Since then, these “principles” have been occupying a fairly central role on both the theoretical and the numerical fronts. The common characteristic of all these cases, however, is that the treatment has been wholly static; i.e., any actual, or virtual, work or action of inertial forces has been completely neglected! Specifically, in static stability the starting point has been the principle of virtual displacements. With it, and some help from kinetics, one derives further either the “principle” of minimum potential energy (Lagrange + Dirichlet + Koiter-see references [9-l l]), or the variational version of the Eulerian udjucent equilibrium configuration method (Trefftz+ Marguerre-see reference [12] for an exceptionally clear treatment). It should be pointed out here that, whereas the minimum potential theorem holds true for conservative systems only, the adjacent equilibrium approach is free from that restriction, provided of course that the non-conservative system’s stability can be satisfactorily treated by completely neglecting all inertial effects. For conservative systems, of course, both methods lead to the same equations, but, in view of what shall follow in this paper, their different concepts, formulations, and interpretations should be clearly kept in mind. When it comes to kinetic treatments of the stability of equilibrium though, whether by choice (e.g., kinetic treatment of conservative Euler column buckling), or by necessity (e.g., kinetic treatment of circulatory “follower-force” buckling), variational principle applications have been, with few exceptions (see references [13-151 and Chapter 5 of reference [16]), peripheral and sporadic-see, e.g., references [17-241; reference [24] contains a sufficiently general 447 0022-460X/82/040447 + 13 $02.00/O
@ 1982 Academic
Press Inc. (London)
Limited
448
J. G. PAPASTAVRIDIS
non-conservative formulation of Hamilton’s principle, but the author is unaware of any truly kinetic stability applications of it. Similar remarks apply to the stability of motion problem: e.g., of steady state oscillation, or of steady motion. “Applications” in the preceding statements means applications in the spirit of the direct methods of the calculus of variations, and not merely those in which the principles are utilized as alternative vehicles for the derivation of the equations of motion, as, e.g., in references [25,26]. Here the starting point should be either Hamilton’s principle in its various forms (see, e.g., reference [27] and Chapter 12 of reference [28]), or some other equivalent timeintegral principle. From the existing approaches, as mentioned earlier, those of Leipholz [13-161 should be singled out for special consideration. The reason is that his treatments, in addition to their generality and direct numerical usefulness (see, e.g., reference [29]), also provide conceptual continuity with the rest of the energetic principles of analytical mechanics! The guiding philosophy of this paper is similar, i.e., to extend the familiar variational principles of dynamics so as to cover results already known from the equations of motion method; the solution of new problems should be attempted at a later stage. Specifically, the concept of adjacent motion, motion through states near a given state of equilibrium (or steady motion), is combined in this paper with Hamilton’s principle to derive conditions that the system’s parameters must satisfy at the stability threshold; this is done in section 2. In section 3 an application of this theory to the well-known discrete version of the follower-force buckling of a column is presented. Finally, in an Appendix, the steps leading from the equations of motion to the various relevant time-integral identities are summarized. In this sense the method presented constitutes the legitimate kinetic counterpart of Trefftz’s variational formulation of the static method of adjacent equilibrium. 2. GENERAL THEORY For analytical simplicity and physical clarity the basic ideas and results are presented first for a one degree of freedom system; the several degrees of freedom (holonomic) case then follows. 2.1. ONE DEGREE OF FREEDOM For a one degree of freedom mechanical system with generalized co-ordinate q = q(t), kinetic energy T(q, 4), potential energy V(q), and additional non-conservative generalized force Q(q, 4) (where (. . .)‘=d(. . .)/dt, t being time), Hamilton’s principle states deviation that for any kinematically admissible, i.e., virtual (and contemporaneous), Sq = Sq(t) from a, say, fundamental state I defined by 4, one has 6A(I)=Ir(6T-6V+6W)dt 11
(SW=Q*Sq)
=-I,;[[(%)‘-(!$I-[-~+Q],~Ot+((~) &I,:: (1) where ~1, t2 are arbitrary times; if &(td = Sq(t2)= 0, or if I is an equilibrium state, then SA(I) = 0. Here SA(I) is not the first variation of some action functional A[q(t)], from
ON NON-CONSERVATIVE
SYSTEM
STABILITY
449
state Z to another virtually attainable one, in the conventional variational calculus sense, but it is simply the homogeneous collection of first-order terms in &I (and its derivatives) that results by applying d’Alembert’s principle to state I. Schematically, equation of motion + d' Alembert’s variational equation 4 Hamilton’s Principle t Lagrange’s “Zentralgleichung”;
(2)
see reference [27], and also the Appendix. In the language of modern functional analysis SA is a “GfUeaux” derivative, rather than a “Frechet” one; here, however, we shall refrain from such physically unmotivating (and intimidating!) formalisms and terminology. In the same spirit for the rotal virtual action change dA between the fundamental state Z and any other state ZZ described by q(ZZ) = q(Z) + Sq, to the second order in the (admissible, weak, and contemporaneous) variations Sq, 84, one sets AA r_1I=
fz(AT-AV+AW~,~~)dt=SA(Z)+(1/2!)S2A(Z), I fl
(3)
where S2A(Z)=62A(q,6q)=f2(b’T-S2V+G2W)dt, (1
(4)
S*T = (a2T/a(i2)(&j)2+ 2(a2T/aqad)8q&.j + (a2T/aq2)(6q)‘, s2 v = (a2v/aq2)(sq)2,
S* W = K? . Sq = [(aQ/aq)sq + (aQ/ad)&j]6q,
(5)
and all partial derivatives are evaluated at state I. In a conservative system Q = 0 (or aQ/aq = 0), and S2A is the second variation of the action functional A(q) from the fundamental state Z in the conventional mathematical sense. The main point can be restated: the virtual displacement (or, function) is only required to be kinematically admissible, i.e., to satisfy certain smoothness requirements in (tl, tz), and special conditions, such as Sq(tI) = Sq(t2) = 0, or Sq(tI) = Sq(t2) # 0, at the end points tl, tz, but, in general it does not represent an actual (i.e., dynamical) neighbouring displacement. Returning now to the study of S’A, by simple integrations by parts of equation (4), with use of equations (5), one can show that a2A = f (sq) =
” J(Sq)Sq dt + EPC, I 11
(6)
(a2T/ad2)sg + [(a2T/a(i2). - (aQ/ad)&j +[(a2T/aq aq)‘-(a2T/aq2)+(a2V/aq2)-(aQ/aq)]sq,
EPC = {s[(aT/a~)sq]}:: = {(a’T/aq aQ)(&)‘+ (a2T/aQ2)sq stj}::.
(7)
Here EPC is the (second-order) end points contribution. In particular if, as is commonly the case, 2T = M2 (m being a positive constant) or, in general terms, if the system is scleronomic and the fundamental state Z is one of equilibrium, i.e., q(Z) = 0, 4(Z) = 0, then expression (7) simplifies to Z(sq) = (a2T/ad2)Sj + (-aQ/ad)stj + (a2V/aq2-aQ/aq)Sq, EPC = {(a2T/a(i2)sq tq}::,
(8) where now all the partial derivatives have been reduced to constants; this also happens when I is a state of steady motion. The conditions that determine the sign of the quadratic form-functional S2A are given in the calculus of variations. In general for an arbitrary admissible, but not actual, neighbouring motion S2A f 0, even when Sq may vanish at
450
J. G.
PAPASTAVRIDIS
the endpoints; see references [30, Chapter 51, and also [31, pp. 103-1061 and [32]. If, however, Sq is chosen to represent the admissible and actual variation &, then -S*A(q, 6q) = &‘A = --I *‘J(&q)* 6q dt + EPC = EPC (= EPC evaluated at &q,&), 11 since in this case J(s;) =I = 0.
(9) (10)
Equation (lo), plus appropriate initial conditions, is the well-known Jacobi-Poincare “equation aux variations” of the adjacent (linearized) motion. In the special case when R reduces to zero, all adjacent small motions are described by ?A=O.
(11)
Equations (9) and (11) essentially express the law of energy conservation, or work/energy theorem for the adjacent motion. In the same way Lagrange’s (first-order) “Zentralgleichung”, or Hamilton’s principle, yield the energy equation for state I; see the Appendix, equations (AlO) to (A16). When S*A is the second variation of some functional A, equation (11) then constitutes a well-known theorem of variational calculus; see, e.g., reference [30, pp. 10%1111. A similar approach, and energetic interpretation, has appeared in the variational treatment of elastostatic stability via the well-known adjacent equilibriumconfiguration method; see references [l], and especially [33, pp. 6-121. In kinetics this action-based approach, as expressed by equations (6), (9) and (ll), seems to have been first advanced, for conservative systems, by Routh [31, pp. lOl-1061; see also references [28, Chapter 121, [34, pp. 416-4391, and [35, pp. 250-2541. Despite these early researches from such mechanics masters it seems to us that so far no systematic attempt at a full exploitation of such direct time-integral-of-energy methods to the study of the adjacent motions about equilibrium or motion, under fairly general non-conservative disturbances, has been made. 2.1.1. The stabilityproblem In a kinetic study of the stability of state I under conservative and non-conservative forces, V and Q depend on, in addition to q and 4, a loading parameter A. Therefore, and since equation (10) is a linear constant-coefficient equation, one sets 8q(t) = C exp (pt)
(12)
in equation (9), or (1 l), and determines the conditions between the load A and frequency p during the adjacent motion; the constant amplitude C depends on the initial conditions at state I. The nature of this adjacent motion is determined by the character of p. Setting for convenience p = io, one distinguishes the following cases: (i) w* = real and positive + steady state oscillation, i.e., stability; (ii) w* =real and negative, or zero+aperiodic diverging motion, i.e., divergence instability; (iii) U* = complex-, increasing amplitude oscillation, i.e., flutter instability; for a more complete characterization, especially in the several degrees of freedom case, see reference [36, Chapter 21. Choosing now tl = 0 and t2 = 27r/w in equation (9) one obtains ~*A=.(~).,(~,A;C)+EPC,
(13)
n(o)=-102./“(e2iui)dt+0,
m&A;
a=[($)(--o*)+(--f?)(iw)+($-$)]Cz_d(u,A)C*.
(14)
ON
NON-CONSERVATIVE
SYSTEM
451
STABILITY
Thus equation (11) translates to d(w, A) = 0.
(15)
Due to equation (lo), with equations (8) and (12), the condition expressed by equation (15) holds true for all three cases described earlier. However it is only for the first case of completely periodic oscillation that equation (15) is equivalent to the direct variational statement pA=O,
since for the second and third cases, due to their non-periodicity, zero; in fact, as equation (8) shows, it reduces to
(16)
the EPC term is not
The study of the sign of S*A, for all admissible and actual neighbouring motions, and its significance to the study of the stability/instability of state I, when this latter is a (fairly general) state of mofion, is currently under investigation by the author; the results shall be reported in the near future. Here the problem of interest is the formulation of conditions at the stability limit: i.e., at the threshold beyond which the character of & changes from that of case (i) to that of (ii) or (iii). Therefore, these “critical” conditions can be equivalently described either by the more familiar equation (15), or by the direct variational statement (16). The additional conditions that determine A,, will be described later. Finally, the connection with conservative system stability is made clear by setting, in equations (14) and (15), KI/&.j = 0 and w = 0 (since a conservative system belongs to the divergence type), and absorbing -aQ/aq into a* V/aq*; then equations (14) and (15) reduce to the well-known A(0, A) = a*V(q, A)/ikj* = 0.
A discussion of the flutter-type treatment.
(17)
of buckling will follow the several degrees of freedom
2.2. SEVERAL DEGREES OF FREEDOM For an N degree of freedom holonomic and scleronomic system the homogeneous quadratic part of the total virtual action change from the fundamental equilibrium configuration I, with use of Einstein’s summation convention and all Latin subscripts running from 1 to N, is 62A(q,,Sql)=~‘z(G2TS2V+62W)dr, 11
and the non-conservative
(18)
forces Qk = Ok (41,(i,,,) satisfy the non-potenfidity condition
Q/c,/# Ql,k.
(21)
This formulation covers all known types of autonomous systems, e.g., conservative (gyroscopic or not), dissipative, and circulatory; see, e.g., references [36,37].
452
J. G. PAPASTAVRIDIS
Reasoning as for the one degree of freedom case one finds S2A=-
‘zJ~(6q,)Sq~dt+EPC, I I1
&(&I) = (Tki)aiir + (-Qk.i)%/ + (vkl EPc
=
{( T,#qk&.jl}$
=
{homogeneous
-
Qk.dh,
bilinear form in Sq, Sg}:,
(22)
5
I
ii2A=~2A(qk,&,)=-
11
Jk
‘Jk(&,),
(.& *& ) d t + EPC,
EPC= EPC(s;I, 54,.
(23)
Again, at the stability limit the EPC term goes to zero due to the 27r/w periodicity of 8q and &, and the critical condition becomes &‘A = 0.
(24)
With &k(t) = ck exp (pt) = ck exp (iwt), tl = 0, t2 = 27r/w, and expressions (22) and (23), equation (24) translates to (25) or, since 0(w) =-l,‘*‘-
(eziw’)dt # 0,
(26)
to D(w,h;Ck)“Dkl(W,A)‘CkC,=O,
~k,~(Tfi)(-W’)+(-Qk,l)(iW)+(Vk,-Qk,,).
(27)
From here on the argument is analytically identical with that employed in the static stability analysis of discrete conservative systems: at the stability limit one is seeking non-trivial dynamic states for which S?A/ D = D becomes indefinite (degenerate). As the theory of quadratic forms shows this occurs when the coefficient matrix D = (Dkl) becomes singular: i.e., when det (Dk,)“&,
I\) =
~(~i~)(-W2)+(-Qk.j)(i~)+(Vkr
Qk.,)l
=
0.
(28)
Since the Dk, elements involve the motion frequency w, as well as the loud A, the equation A = 0 supplies the load-frequency relation or hypersurface. For a general discussion of equation (28), and the dependence of the nature of this hypersurface on the signdefiniteness and symmetry/asymmetry properties of the constant matrices_ T = (Tit), Q=(-Qk,[) and V-Q=(vkl -Ok,,), see, e.g., references [36, 371. Equations (14), (15) and (28) are essentially the same as those derived by Leipholz; see references [l4, equations (27) and (86)], [15, equation (3.7)] and [16, equations (5.170) and (5.171)]. Here further discussion is limited to the following two classes. (i) Conservative non-gyroscopic systems. Here Qk,i = 0, Qk.r = Ql.k (i.e., all the ok.1 terms can be absorbed into v,,), and the critical case occurs for w = 0 (i.e., non-trivial adjacent equilibrium configuration: divergence buckling. Equation (28) then yields the well-known condition for the critical load d(O,A)=]Vkl]=O:
(29)
i.e., for such systems inertial effects do not influence the stability investigation. (ii) Non-conservative, purely circulatory systems. Here Qk,i = 0, and Qk,l = skew-symmetric (its symmetric part, if any, can always be absorbed into vkl). Following
ON NON-CONSERVATIVE
SYSTEM
453
STABILITY
Leipholz [14], at the critical load, in addition to equation (28), one has dw2/dA + 00, or dA/dw2 = 0,
(30)
which, since along the eigenoalue curve A = A(w2, A) dA/do2 = aA/&u2+ (ad/ah).
(dA/dw’) = 0,
(31)
translates to aA(w2 , A yaw2 = 0.
(32)
These results are well-known; for detailed discussions see, e.g., references [14,16, pp. 233-239, and 36, pp. 60-631. From our viewpoint, however, it is instructive to examine the energetic meaning of equations (31) and (32): i.e., to determine their connection with equations (22) through (28), and their w2-derivatives. Along the oscillatory and purely periodic (“lower”) part of the eigenvalue curve A(w2, A) = \(Qi)(-w2)
(33)
+ (VU - C&r)] = 0,
the “energy” equation 8’A = 0 is an identity in 02, and A = A(w2). Therefore
along this
part of the curve
d/dw2[2A(W2,
A(w2); Cj,.)]=
0,
(34)
and at the critical point, due to equations (25)-(27), and (30), d/do2(s2A)
= (d0/dw2)D
+ R(dD/dw2)
= R(dD/do2)
= R(aD/aw2) = 0,
(35)
or finally, aDlaw
= 0:
(36)
i.e., aDlaw is also indefinite at the critical point. The equivalence between equation (36) and equation (32) is most conveniently seen through a reduction of the quadratic form D = D&C’, to a diagonal form, stability coefficients a set of D = dk. c:, (37) I:, ( principul co-ordinates’ by means of a constant non-singular linear co-ordinate
transformation
c = Ac’(det A =
A # 0); when, in addition, A is orthogonal the coefficients dk become the eigenvalues of D. Now if d is the discriminant of D in equation (37), as linear algebra shows (see, e.g.,
reference [38]) A2.A=d,
d =dl*d2..
. :dN.
(38)
Equation (28) then implies that d =0:
(39)
i.e., at least one of the coefficients is zero. On the other hand, the indefiniteness of aD 7
au
= 3.
cl = 0, due to equations (36), (37))
requires that its discriminant be zero, or ad, jaw2 . . . ad,f au2 = 0: i.e., at the critical point at least one of the partial stability coefficients adJaw
(41)
vanishes.
454
J.
G. PAPASTAVRIDIS
Equations (41) and (39) imply that ad/h2
=
(adl/aw2)d2 . . . . - dN + a . . + dl a . . . . dN_l(adN/aw2) = 0,
(42)
and this, thanks to equation (38), finally results in equation (32): i.e., aA/aw2= 0. one can say that the critical (flutter-buckling) point of the eigenvalue curve h = A(w2), at which dh/dw2 = 0 (or dw2/dh + CCJ),is determined either by A(w2, A) = 0 and aA(W2, A)/b2 = 0, or equivalently, by $A(w2, A) = 0, and d/do2[?A(W2, A)] = 0. Finally, the first-order virtual change S(&‘A) of s2A produced by the contemporaneous (and fixed frequency) variations around state IIS = SC, exp (pt) that vanish at the endpoints 0 and 27r/w is, by Hamilton’s principle, applied to the actual periodic trajectory 11, zero: i.e., Summarizing,
S(??A) = 0.
(43)
This variation yields as Euler-Lagrange equations jk = 0; see also the Appendix for derivations obtained by substituting z = a(&) in equation (AlO), or 84 = a(&) in equation (All), or finally by taking the S-variation of equation (A12). Equations (24) and (43) are the kinetic counterparts of Trefftz’s static criteria, as expressed by his equations (26) and (27), respectively, in reference [l]; see his comments surrounding these equations, and also the final lines of the same paragraph. A complete correspondence with the static case, however, would require the formulation of extremum conditions and not simply the stationarity ones formulated here. This is a considerably more difficult problem since the set of adjacent states of motion is infinitely richer than that of adjacent states of equilibrium. Finally, it is expected that the methodology presented should prove powerful in handling large (non-linear) displacements and the effect of geometrical imperfections. It is hoped to answer some of these questions in future publications. 3. APPLICATION:
FOLLOWER-FORCE
BUCKLING
As an application of the foregoing theory one may examine the well-known discrete version of the buckling of a column under the “follower” force P applied to its free end;
Figure 1. Two degree of freedom model of follower-force column buckling.
ON NON-CONSERVATIVE
455
SYSTEM STABILITY
the system with all its kinematical, inertial, and physical parameters is shown in Figure 1. For small motions around the fundamental state of equilibrium I one has (see, e.g., reference [14]) T- $&34:
+ zq&+
(if>,
v = &(2q: -2q1q2 + 4%
Q2=0,
Q1=P.Z(q,-qz),
(Q1,2
#
(44)
Q2.11,
and therefore
T=
3m12
(
@ZWWvNdr =
mlz
Q=
ml2 mlz
1
,
v=(a2V/aql&)t=(~~
-7).
(aC?k/dqt)t = (;
-:),
(45)
Equation (28) then yields =O
det[T(-W2)+V-Q]=A(m2,P)=
’
(46) or A(~2,A)=2’Z’4-(7-2h)!Py2+1=0, ?P2= (m12/c)u2,
A = (l/c)P.
(47) (48)
The critical load results from the solution of the system consisting of equation (47) and aA(!P2,A)/a?P2=4rIrZ-(7-2A)=O.
(49)
One thus obtains the well-known values ACr= (7 -J&/2
= 2.086,
*;=&/2=0.707.
(50)
REFERENCES 1. E. TREFFTZ 1933 Zeitschrift fiir angewandte Mathematik und Mechanik 13, 160-165. Zur Theorie der Stabilitiit des elastischen Gleichgewichts. 2. I$. MARGUERRE 1938 Zeitschrift fiir angewandte Mathematik und Mechanik 18, 57-73. Uber die Behandlung von Stabilitiitsproblemen mit Hilfe der energetischen Methode. 3. I$. MARGUERRE 1938 Jahrbuch der Deutschen Veruchsanstalt fiir Luftfahrt pp. 252-262. Uber die Anwendung der energetischen Methode auf Stabilitiitsprobleme. 4. R. KAPPUS 1939Zeitschriftfiirangewandte Mathematik und Mechanik 19,271-285,344-361. Zur Elastizitiitstheorie endlicher Verschiebungen. 5. B. BUDIANSKY 1974 in Advances in Applied Mechanics 14, l-65. New York: Academic Press. Theory of buckling and postbuckling behavior of elastic structures. 6. J. W. HUTCHINSON 1974 in Advances in Applied Mechanics 14, 67-144. New York: Academic Press. Plastic buckling. 7. M. R. HORNE 1961 in Progress in Solid Mechanics II, 279-322. Amsterdam: North-Holland Publishing Company. The stability of elastic-plastic structures. 8. R. HILL 1978 in Adoances in Applied Mechanics 18, l-75. New York: Academic Press. Aspects of invariance in solid mechanics. 9. W. T. KOITER 1972 Stijfheid en sterke 1-Grondslagen. Haarlem: Scheltema & Holkema. 10. W. T. KOITER 1965Proceedings Koninklijke Nederlandse Akademie van Wetenschappen B68, 178-202. The energy criterion of stability for continuous elastic bodies; Parts I and II. 11. W. T. KOITER 1967 Recent Adoances in Engineering Science 3, 197-218. Purpose and achievements of research in elastic stability. 12. K. MARGUERRE 1950in Neuere Festigkeitsprobleme des Zngenieurs, pp. 189-249. Berlin/GGttingen/Heidelberg: Springer-Verlag. Knick- und Beulvorg%nge.
456
J. G. PAPASTAVRIDIS
13. H. H. E. LEIPHOLZ 1971 Ingenieur-Archiu 40,55-67. Uber die Erweiterung des Hamiltonschen Prinzipsauf lineare nichtkonservativeProbleme. 14. H. H. E. LEIPHOLZ 1977 Actu Mechanica 28, 113-138. On the application of the energy
method to the stability problem of nonconservative autonomous and nonautonomous systems. 15. H. H. E. LEIPHOLZ 1980 in Mechanics Today 5, 193-214. Oxford/New York/Toronto/ Sydney/Paris/Frankfurt: Pergamon Press. Stability analysis of nonconservative systems via energy considerations. 16. H. H. E. LEIPHOLZ 1977 Direct VariationalMethods and Eigenvulue Problems in Engineering. Leyden: Noordhoff International Publishing. (Translated from the German, 1975.) 17. T. B. BENJAMIN 1961 Proceedings of the Royal Society London A261, 457-486. Dynamics of a system of articulated pipes conveying fluid; Part I: Theory. 18. G. HERMANN and S. NEMAT-NASSER 1966 in Dynamic Stability of Structures, pp. 299-308. Oxford/New York: Pergamon Press. Energy considerations in the analysis of stability of non-conservative structural systems. 19. T. ROORDA and S. NEMAT-NASSER 1967 American Institute of Aeronautics and Astronautics Journal 5, 1262-1268. An energy method for stability analysis of nonlinear, non-conservative systems. 20. B. BUDIANSKY 1966 in Dynamic Stability of Structures, pp. 83-106. Oxford/New York: Pergamon Press. Dynamic buckling of elastic structures: criteria and estimates. 21. L. W. REHFIELD 1973 International Journal of Solids and Structures 9, 581-590. Nonlinear free vibrations of elastic structures. 22. J. G. PAPASTAVRIDIS 1981 Journal of Sound and Vibration 74, 499-506. An energy test for the stability of rheolinear vibrations. 23. J. G. PAPASTAVRIDIS 1982 (to appear) Journal of Sound and Vibration. Parametric excitation stability via Hamilton’s action principle. Applica24. M. LEVINSON 1966 Zeitschriftfiir ungewundteMuthemutik und Physik 17,431-442. tion of the Galerkin and Ritz methods to nonconservative problems of elastic stability. 25. E. METTLER 1942 Ingenieur-Archiu 13, 97-103. Uber die Stabilitiit erzwungener schwingungen elastischer K&per. 26. E. METTLER 1947 Ingenieur-Archiv 16, 135-146. Eine Theorie der Stabilitat der elastischen Bewegung. 27. L. NORDHEIM 1927 in (old) Hundbuch der Physik 5,43-90. Die Prinzipe der Dynamik. 28. A. 1. LUR’E 1961 Analytical Mechanics. Moscow: Mathematical-Physical Literature (in Russian). 29. R. BOGACZ, H. IRRETIER and 0. MAHRENHOLTZ 1980 Ingenieur-Archiu 49, 63-71. Optimal design of structuressubjected to follower forces. 30. I. M. GELFAND and S. V. FOMIN 1963 Culculus of Variations. Englewood Cliffs, New Jersey:
Prentice-Hall. 31. E. J. ROUTH 1975 in Stability of Motion (editor A. T. Fuller). London: Taylor & Francis Limited (originally appeared 1877). 32. J. G. PAPASTAVRIDIS 1980 Journal of Applied Mechanics, Transactions of the’Americun Society of Mechanical Engineers 47,955-956. On the extremalpropertiesof Hamilton’saction integral. 33. C. LIBOVE 1962 in Handbook of Engineering Mechanics (editor W. Fliigge). Chapter 44, pp.
6-12. New York: McGraw-Hill Book Co. Elastic stability. 34. W. THOMSON (LORD KELVIN) and P. G. TAIT 1923 Treatise on Natural Philosophy: Purtl. Cambridge University Press (originally published 1867). 35. E. T. WHITTAKER 1937 A Treatise on the Analytical Dynamics of Particles und Rigid Bodies. Cambridge University Press. (Also 1944 Dover.) 36. K. HUSEYIN 1978 Vibrations and Stability of Multiple Parameter-Systems. Alphen aan den Rijn: Noordhoff International Publishing. 37. H. ZIEGLER 1968 Principles of Structuralstability. Waltham, Massachusetts: Blaisdell Publishing Company (Ginn & Co.). 38. W. L. FJZRRAR 1957 Algebra: A Textbook of Determinants, Matrices and Algebraic Forms. Oxford: ClarendonPress. 39. W. F. OSGOOD 1937 Mechanics. New York: The Macmillan Company (reprinted by Dover). 40. B. L. MOISEIWITCH 1966 Variational Principles. London/New York/ Sydney: Interscience
Publishers. 41. B. VUJANOVIC 1975 Zeitschrift ftir ungewundte Muthemutik und Mechunik 55, 321-331. variational principle for non-conservative dynamical systems.
A
ON NON-CONSERVATIVE
SYSTEM
STABILITY
457
APPENDIX: SOME USEFUL TIME-INTEGRAL (ACTION) EQUALITIES OF ANALYTICAL MECHANICS Here, too, the discussion is carried out for a one degree of freedom system; the extension to the several degrees of freedom case, for holonomic systems at least, is straightforward. Consider again the equation of motion for the fundamental state, State 1, m@=Q;
for conciseness here and throughout this whether conservative (better, potential) sufficiently differentiable, but otherwise using the basic identity (md)z =
(Al)
Appendix Q comprises all the applied forces or not. Multiplying equation (Al) by the arbitrary, function z = t(t), (md)z = Qz, and (m@)‘- m@,
642)
one obtains the first order “Zentralgleichung” (m4)i + Qz = (m4.z).
(A3)
Integrating now equation (A3) between the arbitrary following fundamental first-order uuriutionul identity :
time limits r1 and t2 yields the
JI, [(mQ)i + Qzl
dt = {W)z)?.
(A4)
So far only smoothness has been required from z(f). Its physical meaning has been left open, since, as will be seen below, all the energetic/variational principles of mechanics can be obtained from equation (A4) by appropriate special choices of z(f) as actual or uirruul displacement (or velocity, etc.). It is also clear that z(t) equipped with this general physical variational meaning is much broader than the mathematical virtual displacement Sq(t) in the variational calculus sense! On this, usually unclarified matter, see the critical comments by Osgood [39, p. 3791. Returning to equation (A4), one can examine the consequences of particular choices of z(t). (i) The choice I = q yields
‘2 J 11
(mg2 + Qq) dt = {rnQq}::.
(A%
Further manipulations of equation (A5) (with special assumptions for q(r), e.g., periodicity) lead to the famous uiriul theorem of Clausius; for details see, e.g., reference [40, pp. 20-221. (ii) The choice z = 4 yields
I
,:‘[(m4)$+Q4Jdt={m42}::,
or, since m4* = 2T, and (m4)tj = f, (T}+AT=j-‘*
(Qd)dt, 11
which is the work/energy equation for State I.
(A6)
458
J. G. PAPASTAVRIDIS
(iii) The choice z = &, with utilization of the well-known commutation yields
rule (Sq)’= 64,
~“(~T+Q~~)dt={(~~)~~}~=[~~]fz=(0,ifGq(~~)=Sq(r~)=O), I1 11
(A7)
which is Hamilton’s principle (of either varying or stationary action) for State I. Clearly equation (A7) holds for the actual displacement #q(t). For the neighbouring actual path 11, described by 4 + 64, one has 0 + &? = m (q + &)“, or Cw = @Q&)s;l
5Q = m&j,
Similar steps as before, applied to equations fundamental second-order variational identity : ‘2 [(m&Ii I 11
+ (aQ/a#&.
NJ, A9)
(A8) and (A9), lead to the following
+ (sh>z] dt = (m&j)z}::.
(AlO)
(iv) Again, as before (with &j = (&I)‘), the choice z = Sq in equation (AlO) yields -
” J11
(m&d - &Q)Sq dl= 0,
(Al 1)
from which, for Sq = 8q,
6412)
follows. This provides some clarification of the transformations (4) to equation (11). (v) The choice z = &( = (&)‘) yields successively
leading from equation
‘2[(m&j)& + (s’a)&] dt = {m(&)‘}~, I 11
Jt2[1/2m(&)*]* ‘I
dt+
J”(8Q)(&j> dt = {m(&)*}$ 11
or, finally, {62T}+d(62T)=
J’* 2(gQ&j)dt, 11
g2T = m(&)‘.
(A13, A14)
Equation (A13), with equation (A14), is the perturbation energy equation for path I. It is not hard to show that these same equations result by direct utilization of the energy rate equation for path 11, T(11) = W(H), T(U) =[1/2m(4+S;i)2]‘= &(ZZ)=(Q+8Q)(q+&)‘=
(A15)
T(r)+(gT)‘+(1/2S7T)‘, @‘(Z)+(8W)‘+(1/23W)‘.
(Al61
Other special choices of z(t) in equations (A4) and (AlO) will yield additional timeintegral theorems. Still further, useful results may be obtained if, instead of & and s’a, one uses the asymptotic expansions Sq’Eql+e2q2i-.
..,
sh-~eQ~+~20~+.
..,
(A17)
ON NON-CONSERVATIVE
SYSTEM
STABILITY
459
where E is a small parameter, in (AlO) and successively sets to zero the resulting integrals of the various E orders; for applications of this latter approach see references [20-221. Also, for more information concerning the total non-conseruutiue virtual work AWr,n of equations (3)~(5), and the work rates of equations (A13)-(A16), see the masterful exposition of Hill [8, Chapter II.A.11, and also reference [7, Chapter 11.8-111. Finally, the decisive use of the commutation rule da(. . .) = &I(. . .> in most of the above derivations implies that all the variations considered here have to be of the contemporaneous type, i.e., performed at a constant time t, just like those envisioned in d’Alembert’s principle. The variational treatment, however, of more general nonconservative (e,g., non-autonomous) or non-holonomic systems may require wider classes ones; see referenof, generally, non-commutative variations, e.g., non-contemporaneous ces [28, Chapter 12.9, lo] and [41].