On infinitesimal stability and instability of pendulum type oscillations

lnt J. Non-Linear Mechanics,

Vol. 7, pp. 189-198. PergamonPress 1972,Printed in Great Britain

ON INFINITESIMAL

STABILITY

PENDULUM

TYPE

AND

INSTABILITY

OF

OSCILLATIONS

MORRIS M O R D U C H O W t Polytechnic Institute of Brooklyn, Brooklyn, N.Y. 11201, U.S.A.

A b s t r a c t - - F o r the pendulum type of oscillations governed by the equation 2 + ~b(x) = 0, with ~(x) an odd function, it is shown that according to the linearized disturbance equation, stability is predicted if and only if d T d : t - 0, where T is the period and ~ is the amplitude of the non-linear steady-state oscillations. From this it follows that for a given n o n - l i n e a r function qS(x), infinitesimal stability can at most be predicted only for certain discrete values of ct. It is shown analytically that for a simple pendulum, a power-law spring and a cubic hard or soft spring, the oscillations are infinitesimally unstable for all ~. It is further shown, however, that particular cases of non-linear restoring forces do exist for which infinitesimal stability is predicted for certain ~'s, in contrast to the actual Liapunov instability in these cases. 1. I N T R O D U C T I O N

IN A RECENT interesting paper, Robe and Shippy [1] have shown that the linearized disturbance equations of the oscillations of a simple pendulum will imply instability of the disturbances. This is then in agreement with the known Liapunov instability of the disturbances according to the complete non-linear equations, due to the fact that the period of the non-linear oscillations will vary with the amplitude. The demonstration of Robe and Shippy is based on applying a stability criterion taken from the Floquet theory for Hill-type of equations having multiple characteristic multipliers of value unity (e.g. [2] p. 58). This criterion explicitly involves an unknown solution of the perturbation equation. Robe and Shippy used a digital computer to obtain such solutions, and thereby found numerically that this stability criterion is not satisfied in the case of disturbances of a simple pendulum. The purpose of this paper is to investigate, by entirely analytical means, the stability of a more general class of pendulum type of motions according to the linearized disturbance equations. The analysis will concern steady-state oscillations governed by a non-linear restoring-force function of the form q~(x), with ~p(x) odd. It will be shown that a necessary and sufficient condition for the infinitesimal stability + of the steady-state oscillations is simply dT/do~ = 0, where T is the period and ~ the amplitude of the oscillations. This then affords a ready physical and mathematical insight into the infinitesimal stability prediction of such oscillations. From this criterion it will follow that for given non-linear restoring force functions ~o(x), infinitesimal stability can be predicted at most only for certain discrete values of ~. It will be shown analytically that for the tYequently considered cases of a simple pendulum, a power-law spring, and a cubic hard or soft spring, the steady-state oscillations are infinitesimally unstable for all non-zero c~. It will also be shown, however, that non-linear cases of q~(x) do exist for which oscillations would be infinitesimally stable at certain values of ~, in contrast to the actual Liapunov instability of these oscillations. ";'Professor of Applied Mechanics. ~:i.e. stability according to the linearized disturbance equations. 189

190

MORRIS M O R D U C H O W

For the system investigated here, the non-linear (Liapunov) stability p r o b l e m is actually simpler than the linearized stability problem, since the general solution of the non-linear equation 5/ + ~p(x) = 0 can be obtained and analyzed fairly easily. As indicated in [1], however, it is of interest to determine whether, in general, there are physical cases in which the solutions of the linearized stability p r o b l e m would yield conclusions in disagreement with the L i a p u n o v stability or instability of the lull non-linear disturbances. This is the question investigated herein for the case of non-linear p e n d u l u m type of oscillations under a b r o a d class of restoring force functions.

2. B A S I C E Q U A T I O N S

AND GENERAL

INFINITESIMAL

STABILITY

CRITERION

Let the oscillations of a one-degree-of-freedom system be governed by the equation 5~ + cp(x) = 0,

(1)

where dots denote derivatives with respect to the time t, x is a generalized displacement, and ~p(x) is a restoring-torce function which will be assumed to have the following properties : (a) cp(x) is odd; hence also ~p(0) = 0. (b) F o r all x in a region 0 < x < a m, where am > 0 (and m a y be a finite or infinite), ~p(x) > 0; [if a m is finite, then ~P(am) = 0]. Specific mechanical and electrical systems governed by equation (1) are given, for example, by Stoker ([3] pp. 13-26, 54-56). Quite generally, equation (1) m a y be interpreted physically in at least either of the two following ways: (a) x denotes the displacement of a particle of mass m under the action of a spring of variable stiffness, m~p'(x), depending on x. (b) x denotes the inductance flux in a circuit with a capacitance c and an inductance coil in which the current i varies non-linearily with x in the form i = c~p(x). Since the stability of periodic oscillations a b o u t the origin will be considered, there is no loss in generality in considering the stability of a periodic solution x = f(t) of equation (1) whose initial conditions are: f(0) = 0,

)`(0) = vo > 0.

(2)

Then from equation (1),)` will satisfy the (energy) equation x

v2 _ )-2 = V2o _ 2 .[ ~p(u)du.

(3)

0

F r o m equation (3) and the oddness of q~(x), )-2 will be an even function of x. It will now be supposed that ~tm

vg < 2 .f q~(x)dx. 0

Then there will exist a positive n u m b e r c~ < a m such that v~ = 2.f ~(x)dx.

(4)

0

[Note that, in general, q~(a) > 0.] In this case the solution f(t) will be an odd periodic function with a period T depending on v 0, and hence on a. The value f = a will be first attained at time t = T/4. This type of motion, which m a y be regarded as a generalized p e n d u l u m - t y p e of oscillation, is discussed, tor example, by Cesari ([2] pp. 145-146). F r o m

On infinitesimal stability and instability of pendulum type oscillations

191

equations (3) and (4) it is found that dx IF(e) --~]7(x)]~'

T =

(5)

0

where F ( x ) is the potential energy, defined by x

F ( x ) = .f ¢p(u)du.

(6)

0

Let x = f ( t ) + y, where y denotes a disturbance of the above 'steady-state' non-linear oscillation. Then substitution into equation (1), with use of the fact that y + q~[f(t)] = 0

(7)

yields the tollowing linearized equation lor y(t)" j~ + ¢p'[f(t)]y = 0.

(8)

It is now observed (as is well-known) that y = f is a solution of equation (8). This tollows by differentiating equation (7) with respect to t. Since f(t) is bounded, this solution is stable. Hence the stability of the linearized disturbances will be entirely determined by a second, linearly independent, solution of equation (8). For this purpose, a solution of equation (8) satisfying the following initial conditions )~0) = 0,

~0) = h # 0

(9)

will be considered. This solution will be linearly independent of the solution y = )"(t), since the initial conditions of the latter, by virtue of equations (2), (7) and ¢p(0)= 0, are y(0) = v0 # 0, 4(0) = 0.

The nature of the solution y(t) of equation (8) satisfying (9) can now be inferred as follows. Consider a solution of the non-linear equation (1) with the initial conditions x(0) = 0,

x(0) = vo + h.

(10)

Let I'hl ~ Vo and let x(j, t) denote the solution of equation (1) with the initial conditions x(0) = 0, ~c(0) = j. Then x(v o + h, t) - x(v o, t) -

OX(Vo, t) ~3vo

h + ...

(11)

to first powers of h. The solution y of the linearized disturbance equation (8) with the initial conditions (9) can be interpreted as the left side of equation (11) when the latter is expanded to only first powers of h. Since X(Vo, t) = f , it follows that the solution y(t) will be: y = h ~Vo

(12)

where h is a constant ( # 0). Thus the linearized disturbance solutions can be inferred if the general solution of the original non-linear differential equation is known. In the present case, f will be an odd periodic function in t with a period T, and hence will be expressible

192

MORRIS MORDUCHOW

in the (Fourier sine series) form 0o

E

27ri ci(v o) sin 7~-v0) t.

f =

(13)

i=1

F r o m equation (12) it then follows that

where Pl(t) and P~(t) are periodic functions of period T, given by : 0o

Pl(t) = ~- -

c'i(Vo) sin 2~j t: P2(t) = ~c(t). 7"

(14b)

i=l

Infinitesimal stability is predicted if and only if the solution ot' equation (8) under the initial conditions (9) is bounded for all t. Hence, from equation (14a), a necessary and sufficient condition for infinitesimal stability is

d T / d v o = 0.

(15)

When (15) is satisfied, not only ~c(t), but also the general solution of equation (8) will be periodic, with period T, From equation (4), dT/dv 0 = (dT/dct)[Vo/~p(~)]. Hence, since Vo/~p(~) ¢ O, it follows that equation (15) is equivalent to the condition dT/d~ = 0.

(16)

Betore proceeding, it is noted that condition (16) can also be derived by constructing analytically a closed-tbrm solution to equation (8) under the initial conditions (9). This derivation, which may be of interest in itselt~ but is somewhat more involved than that given above, and is shown in the Appendix. Condition (16), which might perhaps have been conjectured intuitively, can be readily interpreted physically. First it is noted, independently of condition (16), that if T is entirely independent of ~ (as in the case of a linear spring) then it is clear that any disturbance, linearized or not, will lead to an oscillation of unchanged period; hence the disturbed motion can be made arbitrarily close to the undisturbed motion tot all t by sufficiently small initial disturbances (Liapunov stability). On the other hand if a disturbance leads to an oscillation of a different period, then it can be easily seen that the oscillation will be unstable in the Liapunov sense. Thus, lor Liapunov stability, it is necessary and sufficient that T be entirely independent of ~, i.e. d T / d ~ = 0 for all ~. Equation (16) states that this condition is sufficient lbr infinitesimal stability. It also states, however, that even 1or those ~0(x) Ibr which T will vary with ~. the non-linear oscillations can still be infinitesimally stable provided that there exists a value (or values) of~ for which dT'.'d~ = 0. In such cases, the oscillations predicted to be infinitesimally stable if they have those particular amplitudes ~ for which dT/dct = 0: the oscillations, however, will still be unstable in the Liapunov sense. This dift~rence between the Liapunov stability condition and the infinitesimal stability condition here can be readily interpreted physically and mathematically. Suppose vo is disturbed, to lead to a disturbed motion. Then ~ -- ~(Vo) will be modified, lf~ however, dT./d~ = 0 at the given undisturbed ~, then to first powers in the disturbance of v o, T remaiJis LIIIL ..... li_l o...l l.g.~ ( ] .

On infinitesimal stability and instability of pendulum type oscillations

193

and hence the steady-state oscillation will be predicted as stable according to the linearized disturbance equations. It is noted that the infinitesimal stability criterion (16) can be applied tor a given (p(x) and e directly, by using equation (5). No knowledge of either the steady-state solution x(t) of equation (1) or of the perturbation solution y(t) of equation (8) is required. Special cases will now be considered to furnish further insight into the implications of the infinitesimal stability criterion derived here.

3. S P E C I A L C A S E S

First, it is noted that for the linear case q~(x) = kx(k > 0), T will be independent of a, and hence the steady-state oscillations will be both infinitesimally and Liapunov stable. For a non-linear restoring-force function tp(x), however, 7" will no longer be independent of a. Hence, for a non-linear spring, it is concluded that the oscillations will be Liapunov unstable. and that infinitesimal stability can be predicted at most for particular discrete values of the steady-state oscillation amplitude a. A few frequently considered special cases of non-linear restoring-force functions will now be investigated. Consider the simple pendulum, for which q~(x) = K sin x, F(x) = K(1 - cos x), K > 0. Letting sin (x/2) -- sin (a/2) sin 0, equation (5) yields the (well-known) result rt/2

4

( dO ~ [ 1 - s i n Z ( ~ / 2 ) sin 20] ~'

T-x/K

(17)

0

where 0 < ~ < ft. By differentiating inside the integral sign it is readily seen that d T / d ~ v~ 0 (in particular, d T / d ~ > 0) for all ~ in the range considered. Hence, the non-linear oscillations of a simple pendulum are proven to be unstable according to the linearized disturbance equation. As a second case, let q~(x) = Kx", where K > 0 and n is an odd (positive) integer ~> 3. Then by letting x = ~., it is readily found from equation (5) that T ~ ~tl-n)]2; hence, dT/dct ~ 0 for all (finite) ~. Finally, let ~o(x) = a(x + fix3), where a > 0 and fl is either positive (hard spring) or negative (soft spring), with - 1 < fl0{2 < :30. The latter is the range of fl~2 when the solutions of equation (1) are oscillatory of the 'pendulum type'. Letting x = ~ , it is found from equation (5) that in the range considered, with fl ~ O, dT/dct ¢ 0 for all non-zero ct. Thus, for a power-law spring and a cubic hard or soft spring the oscillations are proven to be infinitesimally unstable for all (non-zero and finite) steady-state amplitudes In view of the above three non-linear special cases, it may well be asked whether there can exist any non-linear q~(x) at all for which infinitesimal stability would be predicted for some value (or values) of =. It will now be shown that such cases do exist. Consider the (odd) polynomial case m

tp(X) = alx + ~, a2j+lx 2j+x

(al > O, in >~ 2).

(18)

1

(19)

j=l

Let bj - (a2j+lo~2~)/a,,

]bal ~

U • 1).

Then substituting into equation (5), letting x = ~., and expanding T to first powers of b i,

194

MORRIS MORDUCHOW

one finds: m

£ (cj2j + 2)bj-],

T ~ 4a 1-½[(zr/2) -

(20)

j=l

where 1

Cj =

(1

=,'2

-

~2)~ dd. =

0

f,

(k£o= sin 2k O)dO.

0

It is noted that cj > 0 and is finite for all (finite)j, although cj -~ m a s j ~ or. F r o m equations (20) and (19), condition (16) becomes m

Z (cj/j + 1)bj = 0. j=l

(21)

A function cp(x) of the form (18) and an ~, under the inequalities (19), satisfying equation (21) can be obtained, for example, as tollows. Choose a set of values bl, b2 . . . . . bin_l satisfying inequalities (19). Then determine b m by equation (21). Clearly, b 1. . . . . b,,_ 1 can be chosen so that also Ibinl ~ 1. N o w choose a positive ~. Then from the definition (19), a2i+ 1/a~ = bia-2J (j = 1. . . . . m). F r o m equation (18), one then finds in

Q)(X) = al(X -[- ]~ b j a - 2 J x 2 j + l ) .

(22)

j=l

Note that one can satisfy the requirement q~(~) > 0, since according to equation (22), In

¢p(~) = al~(1 +

Z bfl, j=l

and the [bj[ can hence be chosen sufficiently small so that q~(~) > 0. Thus it is shown that cases of q~(x) and ~ do exist tor which the infinitesimal stability condition (16) holds. For, an example of such cases? is given by equation (18), in which the inequalities (19) hold, and in which equation (21) would be approximately satisfied. [Note that the (odd) polynomial here representing q~(x) must be at least of fifth degree.] To exhibit such an example specifically, let ~p(x) = alx + a3 X3 q.- asx 5. Substitute into equation (5), letting x = ~ sin 0, and set dT/dot = 0. Then letting a3o~Z/a1 = 0"1, it is found (numerically) that aso~'*/a1 = -0.0676. N o w let ~ = 1. Then a3/a 1 = 0.1, asia 1 = -0"0676. Thus the case ~p(x) = a~(x + 0"lx 3 - 0"0676x5), 7 = 1 (where a 1 is arbitrary but > 0) is one for which infinitesimal stability would be predicted, in contrast to the actual L i a p u n o v instability for this case. 4. CONCLUSION The question, raised in [1], of whether physical cases exist in which the infinitesimal stability of a system would be in disagreement with its L i a p u n o v stability has been investigated analytically here for the case of one-degree-of-freedom oscillations under odd "['There can, of course, exist other cas'es in addition to the class furnished here.

On infinitesimal stability and instability of pendulum type oscillations

195

non-linear restoring-force functions q~(x). The derived condition dT/d~ = 0 here for infinitesimal stability furnishes a physical and mathematical insight into this question for this class of cases. For a given non-linear ~o(x), infinitesimal stability can be predicted at most only for certain discrete values of co. It has been shown here that cases in which such an infinitesimal stability would be predicted do exist. These may then be considered as physical cases, or at least as cases with a clear physical interpretation, in which the infinitesimal stability prediction would be in disagreement with the Liapunov stability.t

Acknowledgements-- This research was partially sponsored by the Office of Naval Research. Department of the Navy under Contract No. N00014-67 A~0438-0006.

REFERENCES

[1] T. R. ROBE and D. J. SmPPV, On the Liapunov instability of the oscillating pendulum. Int. J. Non-linear Mech. 5, 109 (1970). [2] L. CESAR1, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Academic Press, New York (1963). [3] J. J. STOKER, Non-linear Vibrations in Mechanical and Electrical Systems. Interseience, New York (1950).

(Received 2 April 1971)

APPENDIX

A closed-form solution of the linearized, Hill-type, perturbation equation (8) under the initial conditions (9) will be derived here. In equation (8), f(t) satisfies equation (7) under the initial conditions (2). It will be necessary here to note the qualitative properties of the periodic functions f(t) and ~(t). In particular f(t) will behave quite similarly to a sine curve of period T and amplitude ct, with f(Q) = ct, f(2Q) = 0 and f(3Q) = - ~ , where Q -= T/4, the quarter-period. Moreover, f(t) will behave quite similarly to a corresponding cosine curve, with f(0) = vo, "f(Q~ ~ O. [(2Q) = - v0, and )'(3Q) = 0. [The reader might draw sketches of f(t), f(t) and 1/f2(t), in accordance with this behavior.] It has already been noted in the text that one solution of equation (8) will be y = f(t). Hence, letting y = wf, and substituting into equation (8), a first-order linear equation for w(t) will be obtained, leading to the following form of general solution for y(t): •

'

y(t) = C~f(t) (. ( f ,

dz

+ C2),

(A.1)

where C1 and C2 are arbitrary constants, with the lower limit c arbitrary. ++In applying equation (A.1) it will be necessary to take account of the fact that f'(z) = 0 at z = nQ tor n an odd integer, and that the integral in equation (A. 1) becomes infinite as t ~ nQ. Consequently, equation (A. 1) must be applied piecewise in the time-intervals (each of duration 2Q) in which ~¢(t) ~ 0, and conditions of continuity of y and of ~ must then'be applied at the boundaries of the intervals.

t A n o t h e r physical example of such cases in the restricted three-body problem. It is well-known that the Lagrange equilateral-triangle-configuration solution in this problem is infinitesimally stable for all primary masses m~ and m, which satisfy (m 1 + m2) 2 > 27mlm z. It has been recently shown [A, P. MARKEE , P.M.M. , 105 (1969)] that this solution is indeed stable for all such m 1 and mE, except for two particular ratios ml/m 2 for which the solution is actually unstable. ~:There are, of course, still only two independent arbitrary constants in equation (A, 1), since c and C2 could be combined as a single constant, As will be seen, however, it is convenient here to express the solution in the form given by equation (A.1).

196

MORRIS MORDUCHOW

F o r the interval 0 ~< t < Q, the solution of form (A.1) satisfying the initial conditions (91 will be: t

dr y(t) =-- Yl(t) = h v o J.( t ) f f,2(r)

0 ~< t < Q.

(A.2a)

0

F o r the next interval Q < t < 3Q, a solution of the following form according to equation (A. 1) may be considered: t

y(t) =- 3'2 = hvo)'(t)

(i

~,2(r ) + C

2Q

?

Q < t < 3Q,

(A.2b)

where C is a constant to be determined. Expressing y~ in the form t

[ht:of dr,, j -(r)]/[ l / f ( t ) ] , 0

and similarly lor Y2- it is found that lira 3'1 = lira )'2 -

hvo

hv o

fiom equation (7). Thus, defining yl(Q) and y2(Q) by the above respective limits, it is seen that y(t) given by equations (A.2a) and (A.2b) will be c o n t i n u o u s at t = Q. It is n o w required to find C so that -9 will be continuous there. Keeping in mind the qualitative cosine-like nature of f ( t ) it is seen that for e. > 0 (though < Q), Q

,:

Q+e

)'(Q - e) . ( d r / j " Z ( r )

: f ( Q + e) . ( d z / . f ' 2 ( r ) .

0

2Q

Consequently, if lira ~'1 = A, t~Q

then lira .f'2 = - A + hvoC[-cp(cO].

(A.2c)

C = - 2A/hvoq~(a).

(A.2d)

t~Q

Hence, for continuity of 9 at t : Q.

F o r the interval 3Q < t < 5Q, or, c may similarly consider the solution y(t) ~ Y3 = h v o f ( t )

[[

d r / f ' 2 ( r ) + C'] :

3Q < t < 5Q.

IA.2el

4Q(T)

It will be seen from the nature of f(t) and ~(t) that hv 0 lim 3'2 ~ lim Y3 . . . . • t~3Q ,~3Q~ q°(:0 Hence y(t) given by A.2b) and (A.2e) will be c o n t i n u o u s at t = 3Q. To find C' so that y will also be continuous there, one may infer n o w that lim f,: =

A + hvoC.'~(3 Q) =

A + hvoCq)O:)

t~3Q

lim -93 = A + ht, oC'q~(c0. t~3Q +

Hence, from equation (A.2d), C' --

4A/hvoq)(cO.

(A.21)

The solution for all t can now be rather easily inferred by proceeding as above, from interval to interval, each of form t = (2m + I)Q to (2m + 3)Q (m = 2, 3. . . . ), and satisfying the conditions of continuity at the end points. The following result will thereby be obtained: For 0 ~< t <~ T ( = 4 Q ) , y(t) is given by equation (A.2a)-(A ~!'1: f)r

On infinitesimal stability and instability o f pendulum type oscillations

197

subsequent times, y will satisfy the relation }'(t o + nr) = }'(to)

4A

~o(~)nf(to);(n = 1,2 . . . . . ),

(A.3)

where t o is any value of t in the interval 0 ~< t o ~< T. Equations (A.2a)-(A.2fl and (A.3) constitute the solution y(t), for all t. of equation (8) ~ith the initial co~d il i,m~ (9), and with y and 3) continuous at all t. An explicit expression for A will be derived below [equation A.7)]. As indicated in the text, infinitesimal stability of the given 'steady-state' non-linear oscillation is predicted if, and only if, y(t) is bounded for all t. From equation (A.3) it then follows that infinitesimal stability is predicted i ~ a n d only i ~ A = 0. (A.4) To obtain A, it is is seen from equations (A.2a) and (A.2c) that i f x = x(t) is the solution of equation (l) under the initial conditions (2) then, letting v -= f and considering v = v(x), x

du ) + i.1 . A = hv o lim ~ [ \ [(v ~dr~I dx]jv~lu

(A.5)

0

But.

(A.6a) 0

and hence d~ T = 4 ~lira - ~ [1v - i ~ vOv(u' r' ~ct)/&t d"u J~.

(A.6b}

0

F r o m equations (3} and (47, vdv/dx = - ~(x), and v~v(x, ct)/t3ct = q~(ct). It then follows from equations (A.5) and (A.6b) that

A = (hvo/4)dT/dct.

(A.7)

The infinitesimal stability condition (A.4) thus becomes equation {16) of the text. It may also be noted that equations (14a) and (14b) of the text imply equation (A.3), derived independently in this Appendix. Finally, it is noted that equations (A.2a)-(A.2f), (A.3) and (A.7) indicate explicitly the role of (dT/d~) in the actual construction of the solution.

R 6 s u m 6 - ~ ) n montre, pour des oscillations du type du pendule r6gies par l'6quation 5i + ~b(x) = 0, oO qStx) est une fonction impaire, que selon l'6quation de perturbation lin6aire, on peut pr6dire la stabilit6 si et seulement si dT/dc~ = 0 oil Test la p6riode et ct l'amplitude des oscillations non lin6aires en r6gime permanent. II s'ensuit que pour une fonction non lin6aire donn6e q~(x), on peut, au plus, pr6dire la stabilit6 infinit6simale seulement pour certaines valeurs discr6tes de ~. O n montre analytiquement que pour un pendule simple, un ressort ~ fonction puissance et un ressort cubique doux ou dur, les oscillations sont infinit6simalement instables pour tout ~. Cependant on montre en outre qu'il existe r6+llement des cas particuliers de forces de rappel non linbaires pour lesquelles on pr6dit la stabilit6 infinit6simale pour certains c~,en contradiction avec l'instabilit6 de fait de Liapunov dans ces cas.

Z u s a m m e n f a s s u n g - - F i i r Pendelschwingungen v o n d e r F o r m 5/ + ~b(x) = 0, mit qS(x) einer ungeraden Funktion, wird gezeigt, dass nach der linearisierten StSrungsgleichung Stabilitat dann und nur dann eintritt wenn dT/d~ = O. T ist die Schwingungsdauer u n d c t die Amplitude der nichtlinearen stationaren Schwingungen. Daraus folgt, dass ftir eine vorgegebene nicht lineare Funktion qS(x) Infinitesimalstabilitiit hSchstens fiJr bestimmte diskrete Werte von ct eintreten kann. Auf analytischem Wege wird gezeigt, dass ftir ein einfaches Pendel, eine Potenzfeder und eine kubisch harte oder weiche Feder die Schwingungen fiir alle Werte yon e infinitesimal unstabil sind. Weiterhin wird jedoch gezeigt, dass bestimmte Falle nicht linear RiickstellkrS_fte existieren, fiir welche Infinitesimalstabilitat ftir bestimmte Werte von ct eintritt, im Gegensatz zur eigentlichen Liapunov Stabilit/it in diesen Fallen.

198

MORRISS MORDUCHOW

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