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Stability of periodic solutions based on higher order variational equations
Int. J. Non-Linear Mechanics. Vol. 6, pp. 1-11. Pergamon Press 1971. Printed in Great Britain
STABILITY HIGHER
OF
PERIODIC
ORDER
SOLUTIONS
VARIATIONAL
BASED
ON
EQUATIONS
RICARDO CHICUREL Virginia Polytechnic Institute, Blacksburg, Virginia Abstract--In a non-linear system, a disturbance away from a known periodic solution is expressed as #x, where /Lis a small parameter. Perturbation of the governing equation in powers of# leads to a sequence of equations, the first of which is the conventional variational equation for infinitesimal stability, amenable to Floquet theory. Solutions of the succeeding equations contain terms which are products of periodic functions and exponentials, as well as secular terms. The latter are inconsistent with a simple criterion of stability, but may be removed, as illustrated in an example related to the parametric excitation of a pendulum, by a procedure similar to that of Lindstedt. A comparison with the conventional analysis employing only one variational equation determines the magnitude of the disturbances within which infinitesimal stability may be assumed valid.
INTRODUCTION
THE stability of a periodic solution of a non-linear system is most simply examined by formulation of a variational equation, linear in the variation(s) away from the periodic solution of the dependent variable(s). This method gives rise to the concept of infinitesimal stability [1], which has a limited usefulness, since it furnishes no information on the magnitude of the disturbances that may be tolerated before a stable condition is destroyed. On the other hand, methods that yield stability characteristics in the large, such as Liapunov's second method, are as a rule, considerably more difficult to apply. In the present study, an approach is presented whereby an estimate may be made of the limits on the initial values of disturbances within which infinitesimal stability may be considered valid. The idea of the approach is to recognize that the usual variational equation represents a first variation in the dependent variable(s), and to formulate higher order variational equations, which may be done via a perturbation scheme in which a small parameter is attached as a coefficient to the deviations of the dependent variable(s) away from the periodic solution. Thus, the generating solution becomes the conventional variational equation. The zero order equation in the present perturbation scheme will contain peribdic time dependent coefficients, in contrast to the usual perturbation solutions which are characteristically contrived to produce constant coefficient forms.
METHOD OF APPROACH
Consider a system governed by equations of the form 3), = fi(Yl, Y2. . . . . Yn, t),
i = 1, 2 . . . . . n,
(1)
where the f / a r e periodic in t, with a period P. Also, the f~ are analytic in Yl, Y2 Yn" We assume the existence of a periodic solution Yi = Oi, of period P; and seek another solution near 0;, expressed as yi(Oi +/~txi)./~ is a quantity which may be described as an . . . . .
1
2
RICARDO CHICUREL
amplitude parameter. If initial values of the xi are specified, then the nearness of the solution (0~ +/~x~) to 0~ depends on the value assigned to ~. Treating/~ as a perturbation parameter, we put i = 1, 2 , . . . ,
Xi = Xio + ]lXil -~- ~/12Xi2 "t- . . . .
(2)
n.
Since the initial values of x~ are specified independently of/t, we require that X~o(O) = xi(O),
(3a)
xo.(O) = O,
j = 1, 2, 3, ..
(3b)
Substitution of y~ = (0~ + #xi) = [0i + ~tXio + It2Xi -t- . . . ] pansion in powers of/~, produces the following equation :
into equation (1), and ex-
P:~io + P25qx + . . . . . . .
t) + It
/
ao(t) Xjo j=l
n
+lt2(~
j=l
(4)
n
ao~t) xja +
j,
~= biat(t) XjoXko) + . . . i = 1
1,2 . . . . . n
where a~, blj k ..... are periodic functions of time, derivable from the f~. The first term on the left-hand side and the first on the right-hand side of equation (4) cancel out on account of the fact that 0~ is a solution of (1). Thus, the power of # may be dropped by one in each remaining term of equation (4). This leads to a set of differential equations associated with ascending powers of # :
(It°)"
Xio = ~ aqxjo
(5a)
j=l
(,//1) "
Xil
a o x j l d-
= j=l
j,
2
bijkXjoXko.
(5b)
1
Equation (5a) is the conventional variational equation for the first variation, xi0, whereas (5b) is an equation for the second variation, xil, and so on. The solution of (5a) may be expressed through the use of Floquet theory [2], in the following form : Xil =
~ Aj
j=l
Cij e hit
Fj(t),
(6)
where the h~ are characteristic exponents*; the Fj are periodic functions of period P: % constitute the elements of a non-singular matrix : and the Aj are arbitrary constants. For the solution of (5b), a variation of constants technique will be used. Thus, the xn will be expressed in the same form as the xi0, except that variables zj(t) will now replace the constants Aj:
x . = ~ zj cij e h~t Fj~t). j=l * It will be a s s u m e d t h a t the hj are all distinct.
(7)
Stability o f periodic solutions based on higher order variational equations
3
Substitution of (7) into (5b) produces the following result: j)" + j
hkt Fk)
~,j (cij e h~ F j) = j=l
j,
+
~ j, k,t,m= l
bijk At cj t Am Ckm e(h,+h.OtFtFm.
(8)
The cancellation shown in equations (8) occurs by virtue of the generating equation (5a). Equations (8) may be solved simultaneously for the Fs, so that if we let c=O ~ represent the typical element of the inverse of the matrix of the c~j, we obtain: 11
1 X
C~ii 1 b u k Cfl Ckm A t A m e (h*+h"-hr)t F I F m
z" - F,
i,j, k,l, ra= l
=
~
gtr,,re (h'+h'-h~)'
(9)
t,m = 1
where g(t)tm, =
~
c,~ 1 bok cjl CkmA~Am(Ff m/F,). The g's are periodic functions of time,
i,j,k= l
which may be expressed as Fourier Series: +ao
gzm, =
flZm,~exp
t.
s= -or)
Consequently, Equation (9) becomes fl,.,~ expL~--}-~- h t + h m - h , - t
2, =
(10)
Imm= l s= - oo
All terms in the expression for ~, for which the quantity [(2srci/P) + ht + hm - hr] :/: 0, retain their exponential form when integrated, however, if the bracketed expression is zero, then integration will result in a term of the form (constant) × t. Thus, xil may be expected to contain terms of the form H(t) exp[(2sT~i/P) + hi + hm]t and secular terms of the form tH(t), where H(t) represents a periodic function. The first type are bounded for all time if the real parts of all the characteristic exponents are nonpositive. When secular terms are found, we have a situation which suggests the use of a procedure to suppress those terms in the manner of a classical method such as that of Lindstedt [3]. In this way, the form of the Floquet Solution, from which stability is readily ascertained, is imposed on xzl as well as xt0. It will be seen in the example presented below that this operation produces a connection between the initial values of xt and the characteristic exponents, which reflect the stability of the system. On the other hand, if no secular terms arise in the second variational equation, it is then necessary to proceed to successively higher order variational equations which can be dealt with in a manner similar to that described in connection with equation (5b) until, and il, secular terms are indeed encountered Rather than attempt a formulation of the procedure to eliminate the secular terms based on the general system of the form given by equation (1), the approach will be described through a specific example which has received considerable attention in the literature:
4
RICARDOCHICUREL
the parametric excitation of a pendulum due to an imposed vertical oscillation of the support point. M o t i o n o f a p e n d u l u m with o s c i l l a t i n 9 s u p p o r t
The physical system to be considered has been examined by various authors. Skalak and Yarymovych [4] analyzed the subharmonic oscillations associated with a small amplitude sinusoidal vertical motion of the support of a simple pendulum and applied the method of Andronow and Witt to investigate stability. Struble [5 7] presented a solution based on an asymptotic expansion in powers of a parameter representing the amplitude of the vertical oscillation of the support, and incorporating a principal part term containing slowly varying amplitude and phase parameters. Chhaptar and Dugundji [8], and Dugundji [9] determined theoretically, as well as experimentally transient and steady state responses of the parametrically excited pendulum. The equation of motion appropriate to the motion of a simple pendulum whose support point oscillates harmonically in a vertical direction may be written as 0 +
(-/~ ~092 t) L coso) sinO
= 0.
(11)
Introducing a dimensionless time, z = cot~2, and letting a = (49/o92L), q = 2(ct/L), equation (11) becomes 0" + (a - 2q cos 2z) sin 0 = O.
(12)
We investigate the stability of the solution 0 = O, by putting 0 = p x , and then expressing x as a power series in #: x = Xo + I t x l + p a x 2 + . . . .
(13)
In order to suppress the anticipated secular terms in the higher order variational equations, the parameters a and q are also expressed as power series in p: a = a o + lla I + p e a 2 -t- . . .
(14a)
q = qo + /Llql q- /12q2 -t- ...
(14b)
Thus, a generating solution corresponding to a neighboring point in the a~/space is implied, that point being (ao, qo)Substitution of the series given by (13), (14a), (14b) into equation (12) leads to variational equations corresponding to ascending powers of p: (po) x~ + (ao - 2qo cos 2z) Xo = 0
(15a)
(pl) x'l' + (ao - 2qo cos 2z)xl = - (al - 2ql cos 2z)Xo
(15b)
(It z) x~ + (a o - 2qo cos 2z) x2 = - (a2 - 2qz cos 2z) Xo - (al - 2ql cos 2z) xl + ~ (ao - 2qo cos 2z) x 3.
(15c)
Stability o f periodic solutions based on higher order variational equations
5
Equations (15) are in the form of the Mathieu equation. The first of the group is the conventional homogeneous variational equation with a general solution which may be given as x o = A o e ip' F x + B o e -ip' F z
(16)
where Ao, Bo are arbitrary constants, ip and - i p are the characteristic exponents, pure imaginary in this case, and F1 and F2 are periodic functions of z. Letting G1 = e ip" F1, G2 = e- ip' F2, we have: x0 = Ao GI + Bo G2.
(17)
For the solution of (15b), we write: x 1 = ul(z ) G 1 -k- pl(Z) G2
(18)
in accordance with the method of variation of constants, which also requires that the derivative of xl be of the same form as for the solution of the homogeneous equation: X'x = Ul G'x + 1)1 G[.
(19)
G1 u'l + G2 V'l = 0.
(20)
From (18) and (19), we obtain:
Substituting equation (18) into (15b), with due observance of equation (19) and using the fact that G 1 and G2 are solutions of the homogeneous Mathieu equation, results in the relation G'I u't + G~ v'l = - (al - 2ql cos 2z) Xo.
(21)
The simultaneous solution of (20) and (21) for ut, vI yields: u'~ =
G2(a I -
2ql
G1G'2 -
cos 2z) Xo
,
v'~ =
- Gl(al
G'IG2
-
2ql
GAG'2 -
cos 2z) Xo G'IG2
If the expressions for Xo, G1, G2 are examined, it is apparent that u] and 1)'1will contain the periodic terms: AoF1F2(a I - 2ql G 1 G '2 -
cos 2z)
G'IG 2
and
- BoFxF2(a I -
'
GIG, 2 -
2q1 cos z)
(22)
G'IG 2
respectively. To avoid secular terms in x 1, we then require that the average values of the expressions (22) from z = 0 to z = n vanish. This leads to the choice a 1 = ql = 0. The initial values of x, as represented by Ao and B o, have not, up to this point, had an effect on the solution, and it becomes necessary to consider the next higher order variational equation (15c). Repeating the procedure for the solution of (15b), where, instead of (19) we now have t
P
(23)
X2 = U2G'1 + 1)2G2 leads to the following periodic term in the expression for u~ : A__o0[ 2 ( 0 2 2
2q2cos2z)FiF2
-
AoBoF2F2(ao-
GiG' 2 -
G'IG 2
2qo cos 2z!]
(24) "
The expression for v~ contains the same periodic term, except that the factor Bo replaces
6
RICARDOCHICUREL
A o. Thus, only one condition for the suppression of secular terms exists. This permits us to choose arbitrarily the value of either a2 or q2. For simplicity, set q2 = 0. a2 may be expressed according to the second order approximation: ]A2a2 = a - a o
-/ta
1.
The value of p at this point may be set equal to unity, since in the solution of equation (12), written as px, the factor p can be absorbed by adjusting the function x by application of a multiplicative factor. Additionally, al = 0. Therefore, a 2 = a - a o. From these observations and the requirement that the periodic term (24) have a zero average value, we arrive at the following result:
AoB o =
[
,i
2(a - ao) ! G -~2 ~ G'IGz.J/ d
G,G'2 -- ~-1G2
oz.
(25)
F r o m equation (16) it can be seen that, since x o is real, the functions AoF 1 and BoF 2 are complex conjugates of one another, and because G1, G2 are known only within an arbitrary constant factor, we can assume G1 = G2, F , = F 2, and Ao = Bo. Thus we may put
G2=qb-i~,~ Bo ~ 3
G~ = ~b + i~k,
A o = o r + ifl,
(26)
where the functions q~, ¢ and the constants a, fl are all real. From these definitions and equation (16), we find that xo(O) = x(0) = 2 I ~ ¢ - 3¢)~=o, x~)(o) = x'(o) = 2 ( ~ ¢ ' - 3~"),=o,
and in inverted form: \
/~=\
2w
J,=o
2w
/,=0
where W = (~b~b' - ~b'~k). Thus, A°B°=~z+fl2
= { ~ [1( @ , 2
+ ~,2) x2 _ 2(~b~' + ~b~b')xx' + (~2 + ~b2)x,2] } ,=o.
In the appendix it is shown that ~k(0) = ~b'(0) = 0. Therefore, =
~b
2
The manner in which equations (25) and (27) are used is this : a value of a o, close to a, is assumed. The functions F 1, F2, GI, G2, ~b, ~ are then determined by solution of equation
Stability of periodic solutions based on higher order variational equations
7
15(a). This yields numerical values for the right-hand side of equation (25), and for the coefficients of x2(0) and x'2(0) in equation (27). Equating the right-hand sides of equations (25) and (27~, results in a relationship between x(0) and x'(0). This relationship corresponds to an ellipse in an x(O~-x'(O) phase plane, and its significance is that, for points on the ellipse, the characteristic exponents of the infinitesimal stability analysis govern the character of the solution provided a is replaced by ao. As ao is allowed to differ more from a, i
I
r
0.9 0"8 0-7 0.6 0'5 0"4 0"3 0,2 0.1
-x o -0.1 -0"2 -0'3 -0"4 -0"5 -0'6 -0'7 -0"8 -0.9 I I -0-6 -0-5
I
-0.4 - 0 ' 3
-0-2
-0.1
0
O'l
0-2
0'3
0"4
0-5
0,6
x FIG. 1. Phase p l a n e plot for z = 0, n, 2n . . . . for p a r a m e t r i c a l l y excited p e n d u l u m w i t h a = 3.42, q = 2.00.
the ellipse becomes larger. The solid-line ellipses of Fig. 1 are a family obtained in this way for an example in which a = 3.42, q = 2.00. Consider a point on an ellipse corresponding to a particular value of ao. Such a point represents a set of initial values of x and x' which generate a solution x(t). The values x(rO, x'(n) from that solution can be regarded again as initial values by translation of the
8
RICARDO CH1CUREL
time origin. This new set of initial values should belong to the same ellipse as the original set since the long time character of the solution is determined by the value of ao. In order to determine how well this condition was being satisfied in the example chosen, numerical solutions of the non-linear equation (12) were obtained by computer. Values of x and x' at intervals of n of the independent variable for each such solution are plotted as points in Fig. 1. As can be seen, there is excellent agreement with the approximate solution for small x(0), x'(0), and, as expected, a deterioration of this agreement as x(0), x'(0) increase. Now suppose that a o approaches a in the approximate solution. Then from equations (25), (27) it is apparent that the ellipse in the x(O)--x'(O) plane associated with a0 shrinks to a point and that the limiting ratio of the lengths of its axes is I{[-Ip'(0)J/l-t~(0)-l}a=aol. It is shown in the appendix that the values of x and x' at "c = 0, n, 2n . . . . from a solution of the first variational equation (15a), with a = ao, as in a conventional stability analysis, plot as points on an ellipse of those same proportions, but in this case, the proportions are independent of the size of the ellipse. For small values of x(0), x'(0), no appreciable correction on the solution for x based on the variational equation (15a), is introduced by the present method. The correction will become noticeable when the ratio of axes of the phaseplane ellipse ceases to remain essentially constant as the ellipse size is increased. This can be detected by means of a simple graphical construction in the phase-plane consisting of drawing a curve through a number of points, one for every ellipse, with coordinates equal to the ellipse semi-axes. Beyond the point on this curve where a straight line approximation is no longer acceptable, the conventional infinitesimal analysis is not reliable. From Fig. 1, it is seen, for example, that if x'(0) = 0, then infinitesimal stability may be assumed to remain valid for Ix(0)l ~< 0.15 approximately. In the same figure are also shown, by means of a brcken line representation, the fixed proportion ellipses of the infinitesimal stability analysis. In the problem examined, the solution of the first variational equation was corrected by altering the value of the parameter a in a manner dictated by the second and third variational equations, using the criterion that secular terms not be generated by the latter. The solution can thus be described as a third order approximation. Its deviation from the solution of the non-linear equation (12) is therefore due to fourth and higher order corrections. Although no bounds on these corrections are available from the third order approximation, it can be said that, as long as the latter is in good agreement with the first order approximation, the fourth and higher order corrections are expected to be negligible.
CONCLUDING REMARKS
The procedure of the present paper constitutes a refinement of the method to determine the stability characteristics of an oscillation by examination of the variational equation. By introducing higher order variational equations, but requiring that they have solutions of the same form as for the first equation, the notion of characteristic exponents is retained. However, the latter are now linked to the initial values of the solution representing a disturbance of the oscillation. As shown in the example analyzed, a comparison with the results of a conventional infinitesimal stability study reveals the range of initial values within which those results may be considered valid. The procedure does not depend on a small parameter in the governing equation(s), as is the case, for example, in the method of Kryloff-Bogoliuboff [10], which when applied
Stability of periodic solutions based on higher order variational equations
9
to the problem studied in this paper, would require that the parameter q be sufficiently small. Application of the procedure to study the stability of rotational locks of satellites is being attempted by Abbitt [11]. Publication of his results is expected in the near future.
REFERENCES [1] N. MINORSKY,Nonlinear Oscillations, D. Van Nostrand Co., Inc., Princeton, N. J. (1962). [2] G. FLOQUET, Sur les equations differentielles linears a coefficients periodiques, Ann. Ecole Norm. Sup., Paris 12, 47 (1883). [3] A. LINDSTEDT,Mere. de rAc. Imper. de St. Petersburg 31 (1883). [4] R. SKALAKand M. I. YARYMOVYCH,Subharmonic oscillations of a pendulum, Trans. ASME, J. appl. Mech. 27, 159-164 (1960). [5] R. A. STRUBLE,Oscillations of a pendulum under parametric excitation, Q. appl. Math. 21, 121 (1963). [6] R. A. STRUBLE,On the subharmonic oscillations of a pendulum, Trans. ASME. J. appl. Mech. 39, 301-302 (1963). [7] R. A. STRUBLE,Oscillations of a pendulum under parametric excitation, Q. appl. Math. 22, 157-159 (1964). [8] C. K. CHHAPTARand J. DUGUNmI, Dynamic stability of a pendulum under coexistence of parametric and forced excitation, AFOSR Report 68-0001, M.LT. ASRL-TR-134-5, (1967). [9] J. DUGUNDn, Dynamic stability of a pendulum under parametric excitation, AFOSR Report 69-0019, M.LT. ASRL-TR-134-4 (1968). [10] N. KRYLO~ and N. Boc_,OLiUaOrr,Introduction to Non-Linear Mechanics, Princeton (1943). [11] M. ABBITX,Unpublished Study. (Received 11 July 1969)
APPENDIX
Initial values of ~b, ~p' The two linearly independent solutions of equation (15a) used to express Xo are:
Gl(Z )
G2(z) =
---- e ip~ f , ( z ) ,
GI('L') = e-ip,: FI(T).
G l ( - z ) is also a solution, as may be seen by replacing z by - z in equation (15a). But Gl(-Z) = e-ip~Fl(-z). The only solution of the form e -ip~ × (periodic function) is 62, and hence we must have F l ( - z ) = Fl(z ). Therefore, G2(z ) = G l ( - z ). From equation (26), ~5 = ~ (G1 + 62) --i
~, = - ~ - ( 6 ~
:
[GI(T) + G 1 ( - z l], i
- 62) = - ~ [G~(r) - 6 1 ( - ~ j 3 .
Thus, ~b is an even function, and ~b is an odd function, which implies tk'(0) = ~b(0) = 0.
10
RICARDO CHICUREL
R e l a t i o n between x o, x'o when z is a multiple o f n If t is r e s t r i c t e d t o t h e v a l u e s 0, n, 2 n . . . . . G 1 = e ip~ GI(0), G', = e ' ~ G',(0), B u t G~(0) = G2(0 ) = ¢(0), G'~(0) =
t h e n , G ~, G 2 a n d t h e i r d e r i v a t i v e s a r e g i v e n b y : G 2 = e - i P ' G2(0), G i = e - ' p ' Gi(0).
- G ~ ( 0 ) = i~b'(0), s i n c e ~b'(0) = ~b(0) = 0. T h e r e f o r e ,
X0(Z) = A o G 1 "b B o G 2 = (A0 eip' + B0e-ip~)~b(0),
]
(A.11 t = 0, rc, 2rc . . . .
X'o(Z)
A o G ' ~ + B o G '2
(A o e ~p" - B o e-~P') i~'(O).
(A.2)
D i v i d i n g e q u a t i o n (A. 1) b y 4>(0) a n d (A.2) b y ~'(0), s q u a r i n g e a c h o f t h e r e s u l t i n g e q u a t i o n s , and adding, produces the following result: [~b(0)]2 + [~b,(0)] 2 -
4 A o B o, z = 0, rt, 2 n . . . .
(a.3)
E q u a t i o n (A.3) r e p r e s e n t s a n e l l i p s e w i t h r a t i o o f a x e s e q u a l t o I ff'(0)/~b(0)[. T h i s r a t i o is i n d e p e n d e n t o f t h e size o f t h e e l l i p s e if ao in e q u a t i o n (15a) is set e q u a l t o a, as in a c o n v e n tional stability analysis.
R6sum~-Dans un syst6me non lin6aire on exprime un 6cart par rapport ~ une solution p6riodique connue par px. oh t~est un param6tre petit. Le d6veloppement en puissances de p des 6quations r6gissant le syst6me conduit une s6rie d'6quations, la premi6re 6tant l'6quation variationelle classique de la stabilit6 infinit6simale, justiciable de la th6orie de Floquet. Les solutions des 6quations suivantes contiennent des termes qui sont des produits de fonctions p&iodiques et d'exponentielles, ainsi que des termes s6culaires. Ces derniers ne sont pas compatibles avec un crit~re simple de stabilit6, mais par une m6thode semblable h celle de Lindstedt on peut les 61iminer, comme le montre un exemple 1i6 A l'excitation param6trique d'un pendule. En comparant avec une analyse classique utilisant seulement une 6quation variationelle on d6termine l'ordre de grandeur des 6carts pour lesquels on peut supposer valable la stabilit6 infinit6simale.
Zusammenfassnn~In einem nichtlinearen System wird eine st6rungsbedingte Abweichung von einer bekannten periodischen L6sung als laX ausgedrfickt, wobei Ix einen kleinen Parameter darstellt. Eine Stfrungsentwicklung der Hauptgleichung in Potenzen von tt ffihrt zu einer Folge von Gleichungen, von denen die erste die konventionelle Variationsgleichung ffir unendliche Stabilit/it darstellt und die mit der Floquet-Theorie behandelt werden kann. L6sungen der nachfolgenden Gleichungen enthalten sowohl Produkte von periodischen Funktionen und Exponentialgr6ssen als auch s/ikulare Ausdrficke. Die letzteren sind mit einem einfachen Stabilitatskriterion unvereinbar, kfnnen jedoch mit einem dem von Lindstedt ~ihnlichen Verfahren entfernt werden, wie in einem Beispiel, das mit der parametrischen Anregung eines Pendels in Beziehung steht, gezeigt wird. Dutch Vergleieh mit der normalen Analyse, bei d e r n u r eine Variationsgleichung benfitzt wird, kann die Gr6ssenordnung der St6rungen bestimmt werden, innerhalb derer unendliche Stabilit/it als giiltig angenommen werden kann.
A H H O T a l [ H J t - - ]~ lleJlllli0~illOJi CHCTeMt~ CMelL[elllle OT II:H~eCTIIOI'O ll(+])HOiI,HqeC HOI'O p(!lll(~lllUI BbI[)aH{cIIO qODP;3 /~X, F;[e /z HB~qHeTC}I MaJIBIM napaMeTpoM. Bo:~Myttienne y t l p a B ; l n l O L K e r 0 ~'[)aBftt}ltll;I B CTe[IeHflX /x B e ~ T 1{ IlO.qOqetlllH HOC2Ie~OBaTezqsIIOCTII y[)aBHl
Stability of periodic solutions based on higher order variational equations
11
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STABILITY HIGHER
OF
PERIODIC
ORDER
SOLUTIONS
VARIATIONAL
BASED
ON
EQUATIONS
RICARDO CHICUREL Virginia Polytechnic Institute, Blacksburg, Virginia Abstract--In a non-linear system, a disturbance away from a known periodic solution is expressed as #x, where /Lis a small parameter. Perturbation of the governing equation in powers of# leads to a sequence of equations, the first of which is the conventional variational equation for infinitesimal stability, amenable to Floquet theory. Solutions of the succeeding equations contain terms which are products of periodic functions and exponentials, as well as secular terms. The latter are inconsistent with a simple criterion of stability, but may be removed, as illustrated in an example related to the parametric excitation of a pendulum, by a procedure similar to that of Lindstedt. A comparison with the conventional analysis employing only one variational equation determines the magnitude of the disturbances within which infinitesimal stability may be assumed valid.
INTRODUCTION
THE stability of a periodic solution of a non-linear system is most simply examined by formulation of a variational equation, linear in the variation(s) away from the periodic solution of the dependent variable(s). This method gives rise to the concept of infinitesimal stability [1], which has a limited usefulness, since it furnishes no information on the magnitude of the disturbances that may be tolerated before a stable condition is destroyed. On the other hand, methods that yield stability characteristics in the large, such as Liapunov's second method, are as a rule, considerably more difficult to apply. In the present study, an approach is presented whereby an estimate may be made of the limits on the initial values of disturbances within which infinitesimal stability may be considered valid. The idea of the approach is to recognize that the usual variational equation represents a first variation in the dependent variable(s), and to formulate higher order variational equations, which may be done via a perturbation scheme in which a small parameter is attached as a coefficient to the deviations of the dependent variable(s) away from the periodic solution. Thus, the generating solution becomes the conventional variational equation. The zero order equation in the present perturbation scheme will contain peribdic time dependent coefficients, in contrast to the usual perturbation solutions which are characteristically contrived to produce constant coefficient forms.
METHOD OF APPROACH
Consider a system governed by equations of the form 3), = fi(Yl, Y2. . . . . Yn, t),
i = 1, 2 . . . . . n,
(1)
where the f / a r e periodic in t, with a period P. Also, the f~ are analytic in Yl, Y2 Yn" We assume the existence of a periodic solution Yi = Oi, of period P; and seek another solution near 0;, expressed as yi(Oi +/~txi)./~ is a quantity which may be described as an . . . . .
1
2
RICARDO CHICUREL
amplitude parameter. If initial values of the xi are specified, then the nearness of the solution (0~ +/~x~) to 0~ depends on the value assigned to ~. Treating/~ as a perturbation parameter, we put i = 1, 2 , . . . ,
Xi = Xio + ]lXil -~- ~/12Xi2 "t- . . . .
(2)
n.
Since the initial values of x~ are specified independently of/t, we require that X~o(O) = xi(O),
(3a)
xo.(O) = O,
j = 1, 2, 3, ..
(3b)
Substitution of y~ = (0~ + #xi) = [0i + ~tXio + It2Xi -t- . . . ] pansion in powers of/~, produces the following equation :
into equation (1), and ex-
P:~io + P25qx + . . . . . . .
t) + It
/
ao(t) Xjo j=l
n
+lt2(~
j=l
(4)
n
ao~t) xja +
j,
~= biat(t) XjoXko) + . . . i = 1
1,2 . . . . . n
where a~, blj k ..... are periodic functions of time, derivable from the f~. The first term on the left-hand side and the first on the right-hand side of equation (4) cancel out on account of the fact that 0~ is a solution of (1). Thus, the power of # may be dropped by one in each remaining term of equation (4). This leads to a set of differential equations associated with ascending powers of # :
(It°)"
Xio = ~ aqxjo
(5a)
j=l
(,//1) "
Xil
a o x j l d-
= j=l
j,
2
bijkXjoXko.
(5b)
1
Equation (5a) is the conventional variational equation for the first variation, xi0, whereas (5b) is an equation for the second variation, xil, and so on. The solution of (5a) may be expressed through the use of Floquet theory [2], in the following form : Xil =
~ Aj
j=l
Cij e hit
Fj(t),
(6)
where the h~ are characteristic exponents*; the Fj are periodic functions of period P: % constitute the elements of a non-singular matrix : and the Aj are arbitrary constants. For the solution of (5b), a variation of constants technique will be used. Thus, the xn will be expressed in the same form as the xi0, except that variables zj(t) will now replace the constants Aj:
x . = ~ zj cij e h~t Fj~t). j=l * It will be a s s u m e d t h a t the hj are all distinct.
(7)
Stability o f periodic solutions based on higher order variational equations
3
Substitution of (7) into (5b) produces the following result: j)" + j
hkt Fk)
~,j (cij e h~ F j) = j=l
j,
+
~ j, k,t,m= l
bijk At cj t Am Ckm e(h,+h.OtFtFm.
(8)
The cancellation shown in equations (8) occurs by virtue of the generating equation (5a). Equations (8) may be solved simultaneously for the Fs, so that if we let c=O ~ represent the typical element of the inverse of the matrix of the c~j, we obtain: 11
1 X
C~ii 1 b u k Cfl Ckm A t A m e (h*+h"-hr)t F I F m
z" - F,
i,j, k,l, ra= l
=
~
gtr,,re (h'+h'-h~)'
(9)
t,m = 1
where g(t)tm, =
~
c,~ 1 bok cjl CkmA~Am(Ff m/F,). The g's are periodic functions of time,
i,j,k= l
which may be expressed as Fourier Series: +ao
gzm, =
flZm,~exp
t.
s= -or)
Consequently, Equation (9) becomes fl,.,~ expL~--}-~- h t + h m - h , - t
2, =
(10)
Imm= l s= - oo
All terms in the expression for ~, for which the quantity [(2srci/P) + ht + hm - hr] :/: 0, retain their exponential form when integrated, however, if the bracketed expression is zero, then integration will result in a term of the form (constant) × t. Thus, xil may be expected to contain terms of the form H(t) exp[(2sT~i/P) + hi + hm]t and secular terms of the form tH(t), where H(t) represents a periodic function. The first type are bounded for all time if the real parts of all the characteristic exponents are nonpositive. When secular terms are found, we have a situation which suggests the use of a procedure to suppress those terms in the manner of a classical method such as that of Lindstedt [3]. In this way, the form of the Floquet Solution, from which stability is readily ascertained, is imposed on xzl as well as xt0. It will be seen in the example presented below that this operation produces a connection between the initial values of xt and the characteristic exponents, which reflect the stability of the system. On the other hand, if no secular terms arise in the second variational equation, it is then necessary to proceed to successively higher order variational equations which can be dealt with in a manner similar to that described in connection with equation (5b) until, and il, secular terms are indeed encountered Rather than attempt a formulation of the procedure to eliminate the secular terms based on the general system of the form given by equation (1), the approach will be described through a specific example which has received considerable attention in the literature:
4
RICARDOCHICUREL
the parametric excitation of a pendulum due to an imposed vertical oscillation of the support point. M o t i o n o f a p e n d u l u m with o s c i l l a t i n 9 s u p p o r t
The physical system to be considered has been examined by various authors. Skalak and Yarymovych [4] analyzed the subharmonic oscillations associated with a small amplitude sinusoidal vertical motion of the support of a simple pendulum and applied the method of Andronow and Witt to investigate stability. Struble [5 7] presented a solution based on an asymptotic expansion in powers of a parameter representing the amplitude of the vertical oscillation of the support, and incorporating a principal part term containing slowly varying amplitude and phase parameters. Chhaptar and Dugundji [8], and Dugundji [9] determined theoretically, as well as experimentally transient and steady state responses of the parametrically excited pendulum. The equation of motion appropriate to the motion of a simple pendulum whose support point oscillates harmonically in a vertical direction may be written as 0 +
(-/~ ~092 t) L coso) sinO
= 0.
(11)
Introducing a dimensionless time, z = cot~2, and letting a = (49/o92L), q = 2(ct/L), equation (11) becomes 0" + (a - 2q cos 2z) sin 0 = O.
(12)
We investigate the stability of the solution 0 = O, by putting 0 = p x , and then expressing x as a power series in #: x = Xo + I t x l + p a x 2 + . . . .
(13)
In order to suppress the anticipated secular terms in the higher order variational equations, the parameters a and q are also expressed as power series in p: a = a o + lla I + p e a 2 -t- . . .
(14a)
q = qo + /Llql q- /12q2 -t- ...
(14b)
Thus, a generating solution corresponding to a neighboring point in the a~/space is implied, that point being (ao, qo)Substitution of the series given by (13), (14a), (14b) into equation (12) leads to variational equations corresponding to ascending powers of p: (po) x~ + (ao - 2qo cos 2z) Xo = 0
(15a)
(pl) x'l' + (ao - 2qo cos 2z)xl = - (al - 2ql cos 2z)Xo
(15b)
(It z) x~ + (a o - 2qo cos 2z) x2 = - (a2 - 2qz cos 2z) Xo - (al - 2ql cos 2z) xl + ~ (ao - 2qo cos 2z) x 3.
(15c)
Stability o f periodic solutions based on higher order variational equations
5
Equations (15) are in the form of the Mathieu equation. The first of the group is the conventional homogeneous variational equation with a general solution which may be given as x o = A o e ip' F x + B o e -ip' F z
(16)
where Ao, Bo are arbitrary constants, ip and - i p are the characteristic exponents, pure imaginary in this case, and F1 and F2 are periodic functions of z. Letting G1 = e ip" F1, G2 = e- ip' F2, we have: x0 = Ao GI + Bo G2.
(17)
For the solution of (15b), we write: x 1 = ul(z ) G 1 -k- pl(Z) G2
(18)
in accordance with the method of variation of constants, which also requires that the derivative of xl be of the same form as for the solution of the homogeneous equation: X'x = Ul G'x + 1)1 G[.
(19)
G1 u'l + G2 V'l = 0.
(20)
From (18) and (19), we obtain:
Substituting equation (18) into (15b), with due observance of equation (19) and using the fact that G 1 and G2 are solutions of the homogeneous Mathieu equation, results in the relation G'I u't + G~ v'l = - (al - 2ql cos 2z) Xo.
(21)
The simultaneous solution of (20) and (21) for ut, vI yields: u'~ =
G2(a I -
2ql
G1G'2 -
cos 2z) Xo
,
v'~ =
- Gl(al
G'IG2
-
2ql
GAG'2 -
cos 2z) Xo G'IG2
If the expressions for Xo, G1, G2 are examined, it is apparent that u] and 1)'1will contain the periodic terms: AoF1F2(a I - 2ql G 1 G '2 -
cos 2z)
G'IG 2
and
- BoFxF2(a I -
'
GIG, 2 -
2q1 cos z)
(22)
G'IG 2
respectively. To avoid secular terms in x 1, we then require that the average values of the expressions (22) from z = 0 to z = n vanish. This leads to the choice a 1 = ql = 0. The initial values of x, as represented by Ao and B o, have not, up to this point, had an effect on the solution, and it becomes necessary to consider the next higher order variational equation (15c). Repeating the procedure for the solution of (15b), where, instead of (19) we now have t
P
(23)
X2 = U2G'1 + 1)2G2 leads to the following periodic term in the expression for u~ : A__o0[ 2 ( 0 2 2
2q2cos2z)FiF2
-
AoBoF2F2(ao-
GiG' 2 -
G'IG 2
2qo cos 2z!]
(24) "
The expression for v~ contains the same periodic term, except that the factor Bo replaces
6
RICARDOCHICUREL
A o. Thus, only one condition for the suppression of secular terms exists. This permits us to choose arbitrarily the value of either a2 or q2. For simplicity, set q2 = 0. a2 may be expressed according to the second order approximation: ]A2a2 = a - a o
-/ta
1.
The value of p at this point may be set equal to unity, since in the solution of equation (12), written as px, the factor p can be absorbed by adjusting the function x by application of a multiplicative factor. Additionally, al = 0. Therefore, a 2 = a - a o. From these observations and the requirement that the periodic term (24) have a zero average value, we arrive at the following result:
AoB o =
[
,i
2(a - ao) ! G -~2 ~ G'IGz.J/ d
G,G'2 -- ~-1G2
oz.
(25)
F r o m equation (16) it can be seen that, since x o is real, the functions AoF 1 and BoF 2 are complex conjugates of one another, and because G1, G2 are known only within an arbitrary constant factor, we can assume G1 = G2, F , = F 2, and Ao = Bo. Thus we may put
G2=qb-i~,~ Bo ~ 3
G~ = ~b + i~k,
A o = o r + ifl,
(26)
where the functions q~, ¢ and the constants a, fl are all real. From these definitions and equation (16), we find that xo(O) = x(0) = 2 I ~ ¢ - 3¢)~=o, x~)(o) = x'(o) = 2 ( ~ ¢ ' - 3~"),=o,
and in inverted form: \
/~=\
2w
J,=o
2w
/,=0
where W = (~b~b' - ~b'~k). Thus, A°B°=~z+fl2
= { ~ [1( @ , 2
+ ~,2) x2 _ 2(~b~' + ~b~b')xx' + (~2 + ~b2)x,2] } ,=o.
In the appendix it is shown that ~k(0) = ~b'(0) = 0. Therefore, =
~b
2
The manner in which equations (25) and (27) are used is this : a value of a o, close to a, is assumed. The functions F 1, F2, GI, G2, ~b, ~ are then determined by solution of equation
Stability of periodic solutions based on higher order variational equations
7
15(a). This yields numerical values for the right-hand side of equation (25), and for the coefficients of x2(0) and x'2(0) in equation (27). Equating the right-hand sides of equations (25) and (27~, results in a relationship between x(0) and x'(0). This relationship corresponds to an ellipse in an x(O~-x'(O) phase plane, and its significance is that, for points on the ellipse, the characteristic exponents of the infinitesimal stability analysis govern the character of the solution provided a is replaced by ao. As ao is allowed to differ more from a, i
I
r
0.9 0"8 0-7 0.6 0'5 0"4 0"3 0,2 0.1
-x o -0.1 -0"2 -0'3 -0"4 -0"5 -0'6 -0'7 -0"8 -0.9 I I -0-6 -0-5
I
-0.4 - 0 ' 3
-0-2
-0.1
0
O'l
0-2
0'3
0"4
0-5
0,6
x FIG. 1. Phase p l a n e plot for z = 0, n, 2n . . . . for p a r a m e t r i c a l l y excited p e n d u l u m w i t h a = 3.42, q = 2.00.
the ellipse becomes larger. The solid-line ellipses of Fig. 1 are a family obtained in this way for an example in which a = 3.42, q = 2.00. Consider a point on an ellipse corresponding to a particular value of ao. Such a point represents a set of initial values of x and x' which generate a solution x(t). The values x(rO, x'(n) from that solution can be regarded again as initial values by translation of the
8
RICARDO CH1CUREL
time origin. This new set of initial values should belong to the same ellipse as the original set since the long time character of the solution is determined by the value of ao. In order to determine how well this condition was being satisfied in the example chosen, numerical solutions of the non-linear equation (12) were obtained by computer. Values of x and x' at intervals of n of the independent variable for each such solution are plotted as points in Fig. 1. As can be seen, there is excellent agreement with the approximate solution for small x(0), x'(0), and, as expected, a deterioration of this agreement as x(0), x'(0) increase. Now suppose that a o approaches a in the approximate solution. Then from equations (25), (27) it is apparent that the ellipse in the x(O)--x'(O) plane associated with a0 shrinks to a point and that the limiting ratio of the lengths of its axes is I{[-Ip'(0)J/l-t~(0)-l}a=aol. It is shown in the appendix that the values of x and x' at "c = 0, n, 2n . . . . from a solution of the first variational equation (15a), with a = ao, as in a conventional stability analysis, plot as points on an ellipse of those same proportions, but in this case, the proportions are independent of the size of the ellipse. For small values of x(0), x'(0), no appreciable correction on the solution for x based on the variational equation (15a), is introduced by the present method. The correction will become noticeable when the ratio of axes of the phaseplane ellipse ceases to remain essentially constant as the ellipse size is increased. This can be detected by means of a simple graphical construction in the phase-plane consisting of drawing a curve through a number of points, one for every ellipse, with coordinates equal to the ellipse semi-axes. Beyond the point on this curve where a straight line approximation is no longer acceptable, the conventional infinitesimal analysis is not reliable. From Fig. 1, it is seen, for example, that if x'(0) = 0, then infinitesimal stability may be assumed to remain valid for Ix(0)l ~< 0.15 approximately. In the same figure are also shown, by means of a brcken line representation, the fixed proportion ellipses of the infinitesimal stability analysis. In the problem examined, the solution of the first variational equation was corrected by altering the value of the parameter a in a manner dictated by the second and third variational equations, using the criterion that secular terms not be generated by the latter. The solution can thus be described as a third order approximation. Its deviation from the solution of the non-linear equation (12) is therefore due to fourth and higher order corrections. Although no bounds on these corrections are available from the third order approximation, it can be said that, as long as the latter is in good agreement with the first order approximation, the fourth and higher order corrections are expected to be negligible.
CONCLUDING REMARKS
The procedure of the present paper constitutes a refinement of the method to determine the stability characteristics of an oscillation by examination of the variational equation. By introducing higher order variational equations, but requiring that they have solutions of the same form as for the first equation, the notion of characteristic exponents is retained. However, the latter are now linked to the initial values of the solution representing a disturbance of the oscillation. As shown in the example analyzed, a comparison with the results of a conventional infinitesimal stability study reveals the range of initial values within which those results may be considered valid. The procedure does not depend on a small parameter in the governing equation(s), as is the case, for example, in the method of Kryloff-Bogoliuboff [10], which when applied
Stability of periodic solutions based on higher order variational equations
9
to the problem studied in this paper, would require that the parameter q be sufficiently small. Application of the procedure to study the stability of rotational locks of satellites is being attempted by Abbitt [11]. Publication of his results is expected in the near future.
REFERENCES [1] N. MINORSKY,Nonlinear Oscillations, D. Van Nostrand Co., Inc., Princeton, N. J. (1962). [2] G. FLOQUET, Sur les equations differentielles linears a coefficients periodiques, Ann. Ecole Norm. Sup., Paris 12, 47 (1883). [3] A. LINDSTEDT,Mere. de rAc. Imper. de St. Petersburg 31 (1883). [4] R. SKALAKand M. I. YARYMOVYCH,Subharmonic oscillations of a pendulum, Trans. ASME, J. appl. Mech. 27, 159-164 (1960). [5] R. A. STRUBLE,Oscillations of a pendulum under parametric excitation, Q. appl. Math. 21, 121 (1963). [6] R. A. STRUBLE,On the subharmonic oscillations of a pendulum, Trans. ASME. J. appl. Mech. 39, 301-302 (1963). [7] R. A. STRUBLE,Oscillations of a pendulum under parametric excitation, Q. appl. Math. 22, 157-159 (1964). [8] C. K. CHHAPTARand J. DUGUNmI, Dynamic stability of a pendulum under coexistence of parametric and forced excitation, AFOSR Report 68-0001, M.LT. ASRL-TR-134-5, (1967). [9] J. DUGUNDn, Dynamic stability of a pendulum under parametric excitation, AFOSR Report 69-0019, M.LT. ASRL-TR-134-4 (1968). [10] N. KRYLO~ and N. Boc_,OLiUaOrr,Introduction to Non-Linear Mechanics, Princeton (1943). [11] M. ABBITX,Unpublished Study. (Received 11 July 1969)
APPENDIX
Initial values of ~b, ~p' The two linearly independent solutions of equation (15a) used to express Xo are:
Gl(Z )
G2(z) =
---- e ip~ f , ( z ) ,
GI('L') = e-ip,: FI(T).
G l ( - z ) is also a solution, as may be seen by replacing z by - z in equation (15a). But Gl(-Z) = e-ip~Fl(-z). The only solution of the form e -ip~ × (periodic function) is 62, and hence we must have F l ( - z ) = Fl(z ). Therefore, G2(z ) = G l ( - z ). From equation (26), ~5 = ~ (G1 + 62) --i
~, = - ~ - ( 6 ~
:
[GI(T) + G 1 ( - z l], i
- 62) = - ~ [G~(r) - 6 1 ( - ~ j 3 .
Thus, ~b is an even function, and ~b is an odd function, which implies tk'(0) = ~b(0) = 0.
10
RICARDO CHICUREL
R e l a t i o n between x o, x'o when z is a multiple o f n If t is r e s t r i c t e d t o t h e v a l u e s 0, n, 2 n . . . . . G 1 = e ip~ GI(0), G', = e ' ~ G',(0), B u t G~(0) = G2(0 ) = ¢(0), G'~(0) =
t h e n , G ~, G 2 a n d t h e i r d e r i v a t i v e s a r e g i v e n b y : G 2 = e - i P ' G2(0), G i = e - ' p ' Gi(0).
- G ~ ( 0 ) = i~b'(0), s i n c e ~b'(0) = ~b(0) = 0. T h e r e f o r e ,
X0(Z) = A o G 1 "b B o G 2 = (A0 eip' + B0e-ip~)~b(0),
]
(A.11 t = 0, rc, 2rc . . . .
X'o(Z)
A o G ' ~ + B o G '2
(A o e ~p" - B o e-~P') i~'(O).
(A.2)
D i v i d i n g e q u a t i o n (A. 1) b y 4>(0) a n d (A.2) b y ~'(0), s q u a r i n g e a c h o f t h e r e s u l t i n g e q u a t i o n s , and adding, produces the following result: [~b(0)]2 + [~b,(0)] 2 -
4 A o B o, z = 0, rt, 2 n . . . .
(a.3)
E q u a t i o n (A.3) r e p r e s e n t s a n e l l i p s e w i t h r a t i o o f a x e s e q u a l t o I ff'(0)/~b(0)[. T h i s r a t i o is i n d e p e n d e n t o f t h e size o f t h e e l l i p s e if ao in e q u a t i o n (15a) is set e q u a l t o a, as in a c o n v e n tional stability analysis.
R6sum~-Dans un syst6me non lin6aire on exprime un 6cart par rapport ~ une solution p6riodique connue par px. oh t~est un param6tre petit. Le d6veloppement en puissances de p des 6quations r6gissant le syst6me conduit une s6rie d'6quations, la premi6re 6tant l'6quation variationelle classique de la stabilit6 infinit6simale, justiciable de la th6orie de Floquet. Les solutions des 6quations suivantes contiennent des termes qui sont des produits de fonctions p&iodiques et d'exponentielles, ainsi que des termes s6culaires. Ces derniers ne sont pas compatibles avec un crit~re simple de stabilit6, mais par une m6thode semblable h celle de Lindstedt on peut les 61iminer, comme le montre un exemple 1i6 A l'excitation param6trique d'un pendule. En comparant avec une analyse classique utilisant seulement une 6quation variationelle on d6termine l'ordre de grandeur des 6carts pour lesquels on peut supposer valable la stabilit6 infinit6simale.
Zusammenfassnn~In einem nichtlinearen System wird eine st6rungsbedingte Abweichung von einer bekannten periodischen L6sung als laX ausgedrfickt, wobei Ix einen kleinen Parameter darstellt. Eine Stfrungsentwicklung der Hauptgleichung in Potenzen von tt ffihrt zu einer Folge von Gleichungen, von denen die erste die konventionelle Variationsgleichung ffir unendliche Stabilit/it darstellt und die mit der Floquet-Theorie behandelt werden kann. L6sungen der nachfolgenden Gleichungen enthalten sowohl Produkte von periodischen Funktionen und Exponentialgr6ssen als auch s/ikulare Ausdrficke. Die letzteren sind mit einem einfachen Stabilitatskriterion unvereinbar, kfnnen jedoch mit einem dem von Lindstedt ~ihnlichen Verfahren entfernt werden, wie in einem Beispiel, das mit der parametrischen Anregung eines Pendels in Beziehung steht, gezeigt wird. Dutch Vergleieh mit der normalen Analyse, bei d e r n u r eine Variationsgleichung benfitzt wird, kann die Gr6ssenordnung der St6rungen bestimmt werden, innerhalb derer unendliche Stabilit/it als giiltig angenommen werden kann.
A H H O T a l [ H J t - - ]~ lleJlllli0~illOJi CHCTeMt~ CMelL[elllle OT II:H~eCTIIOI'O ll(+])HOiI,HqeC HOI'O p(!lll(~lllUI BbI[)aH{cIIO qODP;3 /~X, F;[e /z HB~qHeTC}I MaJIBIM napaMeTpoM. Bo:~Myttienne y t l p a B ; l n l O L K e r 0 ~'[)aBftt}ltll;I B CTe[IeHflX /x B e ~ T 1{ IlO.qOqetlllH HOC2Ie~OBaTezqsIIOCTII y[)aBHl
Stability of periodic solutions based on higher order variational equations
11
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