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Stability of the periodic solutions to Duffing's equation and other nonlinear equations of second-order

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DELFT

THE COLLEGE OF A E R O N A U T I C S

C R A N F I E L D

n

STABILITY OF THE PERIODIC SOLUTIONS TO D U F F I N G ' S EQUATION

AND OTHER NONLINEAR EQUATIONS O F SECOND-ORDER

by

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March 1965.

THE COLLEGE OF AERONAUTICS

CRANFIELD

Stability of the periodic solutions to Buffing's equation and other nonlinear equations of second-order.

by

p . A. T. Christopher, D . C . A e . , A . F . I . M . A .

SUMMARY

The stability of the periodic solutions of Buffing's equation is discussed in t e r m s of the asymptotic stability of the corresponding "variational equation"

and, thereby, in t e r m s of the characteristic exponents of this equation. Two methods of evaluating the characteristic exponents a r e considered. The first is based on Whittaker's form of solution of Mathleu's equation and the second, due to Hale, Is based on a general iterative method for determining periodic solutions of systems possessing a small p a r a m e t e r . The methods a r e compared and it is shown that the r e s u l t s for the two methods a r e In agreement.

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Summary

I ; Introduction 1 2. Mathleu's equation 2

3. Hill's equation 6 4. Another equation of second-order 11

5. Stability of periodic solution to Buffing's equation 12 6. Evaluation of characteristic exponents by the method of Hale 14

References 20 Figures

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1. Introduction

The equation of Buffing

X + bx + c,x + Cjx'= Q Sin ut , (1.1) where the dots indicate differentiation with respect to t, may be considered as a

prototype, in many r e s p e c t s , to other nonlinear differential equations which a r i s e in the theory of oscillations in nonlinear systems. Examples of such systems are given in Ref. 1 and Ref. 2. In particular, the author has shown in Ref. 3 how the "short-period" motion of a pitching airframe may be described by an equation of this type. It is also t r u e that systems of higher order than the second often exhibit characteristics which, although more complex, are basically sinnilar to those of systems described by Buffing's equation.

Considering the case where b >0, c, > 0 , C3>0, i . e . a hard system, it is well known that the asymptotic solution of this equation as t ••'«' i s , in the first approximation, a sinusoid

x = F Sin (ut + <P), (1.2) where the amplitude F is given by

(c, - <o^+|c3F^)==+«^b^ = ( Q / F ) " (1.3)

and the phase angle 4> by

T a n ^ = '"^/(c^ . „2 +fc3F'=) (1.4) See Ref. 1, Chapter 4, Ref. 4, Chapter 14, Theorems 3. 2 and 3 . 3, and Ref. 5,

Chapters 7 and 8.

The associated graphs of u, F (the frequency response curves) take the form shown in Fig. 1. It is well known from analogue computer solutions of (1.1) that the region R, bounded by the locus of vertical tangents of the response curves, of Fig. 1 is a region of asymptotic instability in the sense that "jumps" of amplitude occur, both up and down, at the points of vertical tangency of the response curves. In Ref. 1, Chapter 6, Stoker discusses this instability problem and demonstrates, for the case b = 0, that the boundary of asymptotic instability associated with the periodic solution (1.2) corresponds to the locus of vertical tangents of the response curves. One demonstration of the instability associated with the region R for the case b > 0, based upon Minor sky's stroboscopic method, has been given by the author in Ref. 3. A more rigourous proof of this is given by Hayashi in Ref. 6 and will be discussed. The motivation behind the present study is first to show how the previous result may be rigourously derived from the theory of Mathleu's equation, following Hayashi, Ref. 6, and then to show how this same result may be obtained by a method due to Hale, Ref. 7. This latter method has an important advantage over the p r e -vious two in that it is equally valid for systems of any order and not r e s t r i c t e d , as they a r e , to systems of second-order.

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The results to be presented in this and the next section a r e , in the main, well known and it is questionable whether they need be given in this detail. However, the author thinks that this exposition will be welcome to many r e a d e r s and, t h e r e -fore, feels justified in its inclusion.

It will be seen in Section 5 that the stability of the periodic solutions to equation (1.1) is determined by a variational equation which may be reduced to the form of Mathleu's equation, which is itself a degenerate form of Hill's equation, A summary will, therefore, be given of the stability theory associated with Mathleu's equation. The equation will be taken in the form

X + (a - 2q Cos 2t) X = 0 , (2,1) with a and q r e a l . This equation has the equivalent vector form

0 1 -(a - 2q Cos 2t) 0

where x and x a r e column v e c t o r s .

X = X , ( 2 . 2 )

From Floquet's theory for periodic systems (see Ref. 4, Chapter 3, Theorem 5.1) it follows that a fundamental solution matrix for (2, 2) has the form

[ x . y ] = P(t)e*^ , (2.3) where P(t) is a 2 x 2 periodic matrix of period ir and R is a 2 x 2 constant matrix.

Further, R may be taken in diagonal form, which implies that (2, 3) may be written [x,y] = P ( t ) d i a g ( e ' ' ' * , e'^'^) ,

provided that r, ^ r j . It follows that the general solution of (2.1) may be expressed as

x = A , e ' ' ' % , , (t) +A,e''»*p,^(t), (2.4) where A, and A^ a r e a r b i t r a r y constants, p„ and p,j are periodic functions of

period wand r, and r^ a r e the characteristic exponents. The asymptotic stability of (2.4) will depend only on the characteristic exponents r, and r^, and it is clear that when the r e a l part of r, or r^ is zero this gives r i s e to periodic solutions. Such periodic solutions, which can only occur for certain pairs of values of a and q, are known as "Mathieu functions". In the present context, these periodic solutions are not of interest. However, the Mathieu functions of integral order are solutions of (2.1) which lie on the boundaries between the stable and unstable solutions and,

therefore, the loci of a and q, which correspond with the existence of these functions, define the boundaries of asymptotic stability of (2,1) in the a, q plane. See Fig. 2. Methods of determining the s e r i e s which define the Mathieu functions and the c o r r e s -ponding loci in the a, q plane a r e set out in great detail in Ref. 8,

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More insight into the values of r, and r^ may be obtained in the following way. Bisregarding the a r b i t r a r y constant, a first solution to (2.1) may, from (2,4), be taken in the form

X = e*^ ^ (t)

F u r t h e r , since (2.1) will be unaltered by writing -t for t, then

X = e-"^ <p(-i)

will be another solution. These solutions will constitute a fundamental set p r o -vided their corresponding Wronskian determinant is not zero. See Ref. 4, Chapter 3, Theorem 6 . 1 , In the present case the Wronskian is

W >tit,

it)

e-'^Ul-t)

e''^[^<t.{t) + <p(t^

•''*[-p*(-t)+^(-t)]

= -2n<l>{t)</>(-t) +<l>a)i>{-t) -#(-t)<6(t). When t = 0, W(0) = -2^<f{0).

Using Abel's identity (see Ref. 4, p. 83, equation 6, 5), then W(t) = -2^f(Q) exp I 0 dt

Jo or

W(t) = W(0) = -2n<t>'(0). (2,5) It follows that provided is ^ 0 or 0(0) ^ 0 then W ^ 0 and the two solutions a r e linearly independent and constitute a fundamental set. The complete solution of (2.1) is then

x = A,e''%(t) + A , e ' ' ' % ( - t ) , where u is function of a and q,

(2,6)

The asymptotic stability of (2. 6) depends only on ti, and a method of determining n is required. Being periodic, and generally possessing complex coefficients, ^(t) may, as a result of F o u r i e r ' s theorem (Ref. 9, pp 175-6) and Laurent's theorem (Ref. 9, p. 100), be expressed in the form

0 ( t ) =

r = - «0

^2r« 2rti (2.7)

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equating powers of exp (2it) to zero gives r i s e to a system of equations involving ji, q, a and the coefficients c^j.. The elimination of these coefficients Cgj. between the equations gives r i s e to an eliminant which is an infinite determinant known as Hill's determinant. This determinant may be evaluated (Ref. 9, pp 415-7) and, even-tually, an expression derived for u.

Using this result, which is only suitable for numerical evaluation when q is small (the case when q is large is discussed in Ref. 8, Chapter 5), it is possible to demonstrate numerically (See Ref. 10) that the unshaded regions of Fig. 2 c o r r e s -pond to values of ji which a r e r e a l or complex, whilst the shaded regions corres-pond to values of H which a r e imaginary. In a shaded region, therefore, the form of solution will be

X = A , e ^ % ( t ) + Aj,e'^^%(-t) (2,8) Now e ' ^ and e"^^ a r e periodic functions of period 27r//?, and the products

ei/9t ^(t) and e ' ^ ^ ^(-t) will, when /3 is non-integral, be near-periodic functions of finite magnitude. When /S is integral, which it is only on the characteristic c u r -ves aco , ag, , etc. of Fig, 2, then (2,6) and (2.8) a r e no longer complete solutions; these now consist of one of the Mathieu functions as a first solution together with another function fe^^^ (t,q) or g e ^ (t,q) as the second solution. See Ref. 8, Chapter 7. These second solutions a r e unbounded a s t - «>, except in the case q = 0. It follows from Ref, 4, Chapter 13, Theorems 1,1 and 1.4 that the solution (2.8), with /9 non-integral, is "stable" in the sense of Lyapunov, but not asymptotically stable.

In unshaded regions of F i g . 2 the form of solution will be

X = A,exp (oc+ j/9)t. <t>{t) + A.exp - ( « + i/3)t. ^(-t) (2,9) which from the considerations above, may be written a s

x = A,e"*, «,(t)+A,e~"^ * / t ) , (2,10) where t^and f^are near-periodic functions when/9 is non-integral and periodic

functions when/3 is integral. In either case and whatever the sign of « the complete solution is asymptotically unstable. It follows that the region between a^^ and a = -"> is a region of instability, and so are the regions lying between ag^ and acn respectively. See Ref. 8, pp 76-79 and Ref, 10. F u r t h e r , except in the case q = 0, the boundaries of these regions a r e included in the unstable regions.

As will be seen later, particular interest will a r i s e in the evaluation of

H in the unstable regions defined above. F o r this purpose it is particularly

conven-ient to use a method suggested by E. T. Whittaker in Ref, 11. When considering the solution in the unstable region between ag^ and a^, it would be advantageous if the form of solution could be chosen to be that of (2.10), thereby ensuring that the index of the exponential t e r m is r e a l . This may be done by writing the solution in t e r m s of a parameter a so that

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w h e r e

^(t,cr) = Sin (t -<j) + a, Cos (3t - a ) + b . Sin (3t -<T ) + a , C o s (5t - a ) + b . Sin (5t -0-) +

(2.12) and 0(t, -c) ie a s i m i l a r e x p r e s s i o n . In t h e s e e x p r e s s i o n s ^, a , , bj , a, , b^ , e t c . a r e functions of q and <r, and so i s a. T h e p a r a m e t e r oris a s s u m e d t o v a r y between 0 and -»r/2 such that when cr = 0 then<^(t,0) = s e , ( t . q ) and when <r= - t r / 2 , ^ ( t , -w/2) = c e , ( t , q ) . Since <r,a and q a r e i n t e r r e l a t e d , then for s m a l l v a l u e s of | q | it m a y be a s s u m e d t h a t a and n m a y be w r i t t e n in the f o r m

a = 1 + qh,(o-) + q«h,(cr) + ( 2 , 1 3 ) and

ti = qgr,(a) + q'g;(or) + ( 2 , 1 4 )

Substituting e''* 0(t,<r), t o g e t h e r with the e x p r e s s i o n s ( 2 . 1 2 ) , ( 2 . 1 3 ) and ( 2 . 1 4 ) , into ( 2 . 1 ) and equating coefficients of the s a m e p o w e r s of q to z e r o , g i v e s , a f t e r s o m e ntianlpulation (See Ref. 8, p p 7 0 - 7 3 ) , the r e s u l t s

a» = ^ q" Sin 2cr - ~ q» Sin 4o-+

V = - i q +

a = 1 - q C o s 2 a + i - q* (-1 + | - C o s 40") + ( 2 . 1 5 ) and

H = - i - q Sin 2<r + ^ q ' S i n 2 o - - ( 2 . 1 6 )

A s i m i l a r t e c h n i q u e i s valid for t h e o t h e r u n s t a b l e r e g i o n s . Again ( 2 . 1 1 ) i s a s s u m e d t o be the f o r m of s o l u t i o n , w h e r e in the nlll u n s t a b l e r e g i o n

0(t,o-) = Sin (nt -<T) + a p p r o p r i a t e t e r m s In q, q', q ' , e t c . , (See Ref. 8, p . 78 and Ref. 10, pp 8 5 - 8 6 ) ; giving r i s e to the e x p r e s s i o n s

^ = * - f e - I Sinv)+....

( 2 . 1 7 )

]

in the s e c o n d (n = 2) u n s t a b l e r e g i o n and

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„ '. (2,18) -3-84 ^' ^'^ 2 a .

i

in the third unstable region, and so on,

By varying q and a, values of a and /i may be calculated and curves of con-stant /i and o" plotted in the unstable regions of the a , q plane. Fig. 3 shows an approx-imate diagram of the curves in the first unstable region.

3. Hill's Equation

In the particular case of Buffing's equation the variational equation associated with the solution (1.2) i s of the Mathieu type. However, this is not always the case with other equations of second-order and the resulting variational equation often takes the form of Hill's equation

eo

*x + [a + 2 ƒ öj, Cos 2nt] X = 0, (a, 6 real) (3.1)

where the s e r i e s ) fl_ is taken to be absolutely convergent. The r e s u l t s of Floquet's

E«"'

n»i

theory apply also to this equation and it follows that the complete solution may be taken in the form

x = A^e^'\<l>,W + A^e'^^ ^^(t), (3.2)

provided the Wronskian Is not z e r o .

In the case of equations of second-order with periodic coefficients the theory of Floquet may, by the use of equation (5.11), Chapter 3 of Ref. 4, be made to yield more information than just the form of solution. Consider the equation

X + [ a + b.p(t)] X = 0, (3.3) where p(t) is periodic of period T, Clearly (3.1) is a particular case of (3.3), The

complete solution may be written

x(t) = A,x,(t) +AgX,(t), (3.4) where x, and Xg a r e linearly independent. F r o m Floquet's theorem it follows that

X (t + T) = A,x,(t + T) + Aj,Xjj(t + T) =Xx(t) (3.5) and upon differentiation

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w h e r e X Is a • c h a r a c t e r i s t i c multiplier". P r o v i d e d the condition W /^ 0 i s s a t i s f i e d the choice of i n i t i a l v a l u e s for x and x i s a r b i t r a r y and for t h e p u r p o s e of d e t e r m i n i n g X it i s convenient t o c h o o s e t h e s e to be

x,(0) = i^iO) = 1 and X2(0) = 3c,(0) = 0,

which give W(0) = 1. E q u a t i o n ( 3 . 5) and ( 3 . 6 ) then b e c o m e A,x,(T) + Aj,X2(T) =XA,x,(0) + XA2X2(0) = XA,

and

A , i , ( T ) + A,-i^(T) =XA,x,(0) +XA^x^(0) =XA^ E l i m i n a t i n g A, and A^ between t h e s e e q u a t i o n s g i v e s x,(T) - X 3^(T) x,(T) x,(T) - X o r o r X' - X[x,(T) + Xj(T)] + [x,(T) x , (T) - A,(T) x,(T)] = 0 X" - X[x,(T) + ia(T) ] ' + W(T) = 0 F r o m A b e l ' s i d e n t i t y t h u s W(T) = W(0) exp f Odt = W(0) = 1 . o X''-Xf(T) + 1 = 0 , w h e r e f(T) = x,(T) + Xg(T), and ( 3 . 1 0 ) h a s the r o o t s

\a=|[f<T) i[f*(T) - 4 ] ' J

Now ( 3 . 3 ) m a y be w r i t t e n a s the e q u i v a l e n t v e c t o r equation X = A(t)x , w h e r e 0 1 A(t) = - [ a + b p ( t ) ] 0 (3,7) (3.8) (3.9) ( 3 . 1 0 ) ( 3 , 1 1 ) ( 3 . 1 2 )

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and whose t r a c e i s zero. Therefore, from equation (5.11), Chapter 3 of Ref. 4, T

W = exp f t r a c e A(t) dt = 1 (3.13)

^ 0

When the roots X, ,X, a r e distinct then two cases a r i s e ,

(i) f*(T) > 4, X, > 1, X, < 1, both roots r e a l . Now the characteristic multiplier

X = exp/i T, (3.14) where li is the characteristic exponent, therefore,

/i, = ^ l o g e X , >0 and

It follows from (3,2) that this case yields an unstable solution.

(ii) f*(T) < 4, X, and X, a r e complex conjugate of modulus unity. F r o m (3.14) /J, and /jjinust be purely imaginary and the solution (3. 2) is stable in the sense of Lyapunov.

F r o m the continuity of f(T), it follows that the stable regions and the unstable regions of the a , b plane have their boundaries composed of points for which f"(T) = 4, or

f(T) = 2 or f(T) = -2 (3.15) At these boundary, or transition, points equation (3.11) has the repeated roots

X, J = 1 or - 1 . It is clear from (3.5) that when X = 1, the corresponding solution is'perlodic of period T, When X= -1 the solution may be shown to be periodic of period 2T, (See Ref. 4, p. 219).

The nature of the regions and boundaries in the a , b plane have been discussed by Haupt in Ref. 12. F o r each fixed b there exists an infinite set a^ of isolated values of a, bounded on the negative side of the a-axis but unbounded on the positive side, that satisfy (3.15). Upon moving in a positive sense along the a-axis the points &{ fall, with the exception of the first point, into pairs of adjacent points in such a way that one pair satisfies f(T) = 2 whilst the succeeding pair satisfy f(T) = - 2 . Starting with the first point, which corresponds with a periodic solution of period T, there follows a pair of points corresponding to periodic solutions of period 2T, then a pair with period T and so on alternating in p a i r s . The region to the left, i . e . a •• -•», of the first point corresponds to unstable solutions and so do the regions between pairs of points of the same type. I . e . corresponding to solutions having the same periodicity. The points aj^ themselves, by reason of their associated unstable second solutions lie in the unstable regions, with the exception of the points a^, b = 0. It may readily be demonstrated that the points a^ a r e continuous functions of b and, therefore, the points make up a s e r i e s of continuous boundaries in the a, b plane.

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T h e above d e s c r i p t i o n c l e a r l y f i t s , a s it should, the c h a r a c t e r i s t i c d i a g r a m in the a , q p l a n e of the Mathieu e q u a t i o n . A s in the c a s e of M a t h l e u ' s equation i n t e r e s t i s a g a i n c e n t r e d on e v a l u a t i n g 12 in the u n s t a b l e r e g i o n s . F o r t h i s p u r p o s e it i s a g a i n convenient to e m p l o y a v e r s i o n of W h i t t a k e r ' s m e t h o d . (See Ref. 8, pp 1 3 4 - 5 ) . F u r t h e r , it m u s t be s t r e s s e d that in a p p l i c a t i o n to the v a r i a t i o n a l equations which a r i s e in

r e l a t i o n to the p e r i o d i c s o l u t i o n s of n o n l i n e a r e q u a t i o n s , the v a l u e u t o be d e t e r m i n e d w i l l be c o n s i s t e n t with the a s s u m p t i o n t h a t 6, , 0, , . . . e t c . in ( 3 . 1 ) a r e s m a l l .

C o n s i d e r ( 3 . 1 ) , and t a k e the f i r s t solution in the f o r m

x = e^^^(t). ( 3 , 1 6 )

S u b s t i t u t i n g ( 3 , 1 6 ) into ( 3 , 1 ) g i v e s

as

0 + 2^^ +la+^' + 2 \ Q„ C o s 2 n t ] 0 = 0 ( 3 . 1 7 ) T h e function ^(t) in the m ^ u n s t a b l e r e g i o n m a y now be a s s u m e d to have the f o r m

^ ( t , a ) = Sin (mt -cr ) + 6 , f,(t,o-) + m = 1, 2, 3 ( 3 . 1 8 ) i n which <r i s a p a r a m e t e r to be d e t e r m i n e d . F o r %, Q^ , • • • e t c . sufficiently s m a l l ,

which t h e y will be u n d e r t h e a s s u m p t i o n s u s e d to d e r i v e the v a r i a t i o n a l e q u a t i o n , ( 3 . 1 8 ) m a y be a p p r o x i m a t e d by <p (t,cr) = Sin (mt -cr) ( 3 . 1 9 ) S u b s t i t u t i n g ( 3 . 1 9 ) into ( 3 . 1 7 ) g i v e s -m* Sin (mt -cr) + 2/:<m C o s (mt - a ) + [ a + ^* + 2 \ on Cos 2ntjSin (mt - a ) = 0 ( 3 . 2 0 ) n ' 1

It will be o b s e r v e d that the t e r m 26„ Cos 2nt Sin (mt - a ) g i v e s t e r m s in Sin m t and C o s m t only when n = m . Now

2 e „ C o s 2mt Sin (mt -Ö" ) = on Sin (3mt -<r) - Sin (mt -a)

and if t h e coefficients of Sin m t and C o s m t r e s p e c t i v e l y , in ( 3 . 20), a r e equated to z e r o , t h e r e a r i s e t h e r e l a t i o n s

2/im Sino- + (a + /J^ - m*) C o s cr-ö^ C o s cr = 0

and ( 3 . 2 1 ) 2;i,m Cos a - {a. + ^' - m") Sin a + e„ Sin a = 0

Multiplying the f i r s t of t h e s e e q u a t i o n s by Sin a and the second by C o s a and adding t h e p r o d u c t s gives

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or

W = ^ Sin 2 o- (3.22) Zm

Multiplying the first equation of (3. 21) by Cos <r and the second by Sin a and subtrac-ting the lower product from the upper gives

(a + /i* - m*) - Q„ (Cos*a- Sin*<r) = 0 or

a = m* + 6„ Cos 2 a - n' which from (3.22) becomes

a = m « + e „ Cos 20-- (6n/2m)' Sin* 2<r (3.23) The parameter c m a y , of course, be eliminated directly between the equations of

(3. 21) by squaring the sides of each equation respectively and adding, giving 4/i*m*+ ( a + ^*- m')' = e„»

or

tt* + [4m* + 2(a - rrP)] ^^ + {& - m«) -e^" = 0

which has the roots

U* = -(a + m«) ± (4m«a + e^*)^^ (3.24)

A second solution of (3.1) which is linearly independent of (3.16) i s , to the same degree of approximation as (3.19),

-Mt

x = e Sin (mt +0"), giving a complete solution

X = A^e*'* Sin (mt -cr) + A,e''^* Sin (mt +or) (3.25) In an unstable region n is r e a l , thus M* > 0. The boundaries between the

unstable and stable regions will, therefore, correspond to M* = 0, or from (3.24) /i« = -(a + m») ± (4m*a + e„»)*^ = 0

or

(a + m»)* = 4m*a + e„* or

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giving

a = m2 ± e„ ( 3 . 2 6 ) which r e p r e s e n t s t h e s t a b i l i t y b o u n d a r i e s a s s o c i a t e d with the mÜI u n s t a b l e r e g i o n .

A l t e r n a t i v e l y t h i s r e s u l t m a y be obtained by putting cr = -ir/2 and cr = 0 in ( 3 , 23).

W h i t t a k e r ' s m e t h o d m a y a l s o be used t o e s t a b l i s h s o l u t i o n s for an extended f o r m of H i l l ' s equation

X + [ a + 2 ) e^g Sin 2nt + 2 ) Q^^ C o s 2 n t ] x = 0 ( 3 . 2 7 )

n = 1 n ati

T h e s o l u t i o n , to the s a m e d e g r e e of approxinaatlon a s ( 3 . 2 5 ) , i s

X = A^e*^* Sin (nt - a , ) + A ^ e ' ' ' * Sin (mt -&, ), ( 3 . 2 8 ) w h i l s t the e x p r e s s i o n for /i* i s i d e n t i c a l t o ( 3 . 24) p r o v i d e d 6^* i s i n t e r p r e t e d a s

4 . A n o t h e r e q u a t i o n of s e c o n d - o r d e r

T h e v a r i a t i o n a l e q u a t i o n s which a r i s e f r o m the c o n s i d e r a t i o n of the s t a b i l i t y of p e r i o d i c s o l u t i o n s of n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s of s e c o n d - o r d e r m o r e g e n e r a l l y contain a t e r m in i . Such v a r i a t i o n a l e q u a t i o n s often have the f o r m

X + 2/9x + [ a + 2 \ 6^ C o s 2nt] x = 0, ( 4 . 1 ) n = i w h e r e a,/9 ,6 a r e r e a l and /9 > 0 . W r i t i n g x--e'^^y ( 4 . 2 ) t r a n s f o r m s ( 4 . 1 ) into y + ["(a -/3* ) + 2 V e „ C o s 2nt] y = 0, ( 4 . 3 ) which i s c l e a r l y of the s a m e f o r m a s ( 3 . 1 ) . T h e c o m p l e t e solution of t h i s e q u a t i o n

m a y b e t a k e n in the f o r m

y = A^e''* ^^(t) + Ay^^ 0^(t), ( 4 . 4 ) w h e r e 0, and ^^ a r e p e r i o d i c , giving

X = A , e x p ( A ( - / 9 ) t . 0,(t) + Aj, exp - (*i -/9 ) t . ^^(t) ( 4 . 5 ) F r o m Section 3 it i s known t h a t ( 4 , 4 ) will be a s y m p t o t i c a l l y u n s t a b l e only when

M i s r e a l . It follows that ( 4 . 5 ) will b e c o m e u n s t a b l e when M - /9 o r -H-fi. w h i c h e v e r ie the l a r g e r , b e c o m e s g r e a t e r t h a n z e r o . T h u s for a s y m p t o t i c s t a b i l i t y

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Since n and /9 a r e r e a l , t h e n condition ( 4 . 6 ) b e c o m e s

^ > n' .

o r on t h e s t a b i l i t y b o u n d a r y

fi' - U^ = 0 ( 4 . 7 )

F o r t h i s c a s e t h e v a l u e o f / i ' , to the f i r s t a p p r o x i m a t i o n , i s given by ( 3 . 2 4 ) , w h e r e a i s to be r e p l a c e d by a - fi', t o be c o n s i s t e n t with ( 4 . 3 ) . Substituting into ( 4 . 7 ) g i v e s

/9' = - [ ( a -y9*)+ m ' J ± [ 4m*(a -fi') + 6,»] ^ o r

(a + m2)2 = 4m*(a -^ ) + Q^ o r

(a - m«)» + 4m»/?«- 6^" = 0 ( 4 . 8 )

5. Stability of P e r i o d i c Solution to Buffing's E q u a t i o n

T h e object of t h i s s e c t i o n i s to d e t e r m i n e the b o u n d a r y of a s y m p t o t i c s t a b i l i t y of equation ( 1 . 1 ) with r e s p e c t to the p e r i o d i c solution ( 1 . 2 ) . F o r t h i s p u r p o s e c o n s i d e r f i r s t the d e r i v a t i o n of the equation of " f i r s t v a r i a t i o n " o r " v a r i a t i o n a l " equation defined in Ref. 4 , p . 322. W r i t i n g y = x + 6 , w h e r e x i s given by ( 1 , 2 ) , the v a r i a t i o n a l equation in the p r e s e n t c a s e b e c o m e s g + b i + (c, + 3c3x'')é = 0 o r 4'+ b i + l c, + 3c3F*Sin''((üt + *) J 5 = 0 ( 5 . 1 ) E x p r e s s i n g Sin*(üjt + <6) in t e r m s of C o s 2 ( u t + ^ ) and w r i t i n g z = u t + ^ r e d u c e s ( 5 . 1 ) to S"+ 2 ^ t ' + ( 6+ 2 E C o s 2z)5 = 0 ( 5 . 2 ) w h e r e p = b/2(o , 8 = ( c , + f c , F » ) / « « , e = - I c , F ' /u* and

5' = de/dz.

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F r o m Ref, 4 , C h a p t e r 1 3 , T h e o r e m 2 . 1 , it i s known that the p e r i o d i c solution ( 1 , 2 ) will be a s y m p t o t i c a l l y s t a b l e a s t - * » p r o v i d e d the t r i v i a l s o l u t i o n , g = 0, of ( 5 . 2 ) i s a s y m p t o t i c a l l y s t a b l e . Now equation ( 5 . 2 ) is a p a r t i c u l a r c a s e of t h e equation c o n s i d e r e d in Section 4 in which P i s identified with P, 6 with a, e with Sj^and Ög, Ö3 . . . e t c . a r e z e r o . It will be r e c a l l e d t h a t the a p p r o x i m a t e e x p r e s s i o n s for [i given in ( 3 . 2 4 ) , and i m p l i c i t in ( 4 . 8 ) , a r e b a s e d on the a s s u m p t i o n that Q^*^^ , , . . e t c . a r e s m a l l c o m p a r e d with 6 3 . T h i s condition i s m e t by equation ( 5 . 2 ) , t h e r e f o r e the r e l a t i o n ( 4 , 8 ) m a y be used in o r d e r to obtain the a s y m p t o t i c s t a b i l i t y b o u n d a r y of ( 5 . 2 ) , in t h e f i r s t a p p r o x i m a t i o n . T a k i n g m = n = 1, t h i s b o u n d a r y b e c o m e s

( 6 - l ) * + 4 / ? * - £ *= 0 ( 5 . 3 ) S u b s t i t u t i n g for 6 , ^ and £ in ( 5 . 3 ) and m u l t i p l y i n g throughout by u* g i v e s

( c , + ^' C3F* - u 2) 2+ (j^b^ - - c | F ^ = 0 ,

' 2 16 •"

which upon e x p a n s i o n of the b r a c k e t e d t e r m and r e - g r o u p i n g g i v e s

( c , - <j'= + f c3F')(c, - u ' + l C 3 F ' ) + «*b* = 0, ( 5 . 4 ) a s the a s y m p t o t i c s t a b i l i t y b o u n d a r y in t h e w , F p l a n e , i . e . on t h e f r e q u e n c y r e s

-p o n s e d i a g r a m .

C o n s i d e r now the equation ( 1 . 3 ) , which defines the r e s p o n s e c u r v e s in the « , F p l a n e . Multiplying by F * and d i f f e r e n t i a t i n g i m p l i c i t l y with r e s p e c t t o F g i v e s

2 F ( c , - u)==+| C3F') j (c, - u . ^ + 1 C3F") + (-2tü ^ + 7 C 3 F ) F J

+ 2b* wF (w + F | | ) = 0 d F

Upon i n s e r t i n g the condition for v e r t i c a l t a n g e n c y , ck) / d F = 0, t h i s equation b e c o m e s 2F [ ( c , - J- + | c 3 F * ) ( c , - u* + ïCjF*) + w^b'' J= 0

S i n c e , in g e n e r a l , F ^ 0, t h e n t h i s e q u a t i o n m u s t r e d u c e t o equation ( 5 , 4 ) . T h e c o n c l u s i o n m a y , t h e r e f o r e , be d r a w n t h a t the b o u n d a r y of a s y m p t o t i c s t a b i l i t y of the p e r i o d i c solution ( 1 . 2 ) of Buffing's e q u a t i o n ( 1 . 1 ) , c o r r e s p o n d s with the l o c u s of v e r t i c a l t a n g e n t s of the r e s p o n s e c u r v e s , given by ( 1 . 3 ) , in the u , F p l a n e .

It i s not i m m e d i a t e l y c l e a r that the r e g i o n R of F i g . 1 i s the r e g i o n of a s y m p -t o -t i c i n s -t a b i l i -t y , h o w e v e r , -t h i s b e c o m e s a p p a r e n -t by r e f e r e n c e back -t o -the i n e q u a l i -t y ( 4 . 6 ) . T h u s the r e g i o n of a s y m p t o t i c s t a b i l i t y i s defined by

/9* -/^* > 0 , which c o r r e s p o n d s with

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Befining a p a ra m et er K by the equation

(c, - u' + 1 C3F*) (c, - u* + Ï c,F*) + w^b* = K (5.5) it may readily be seen that the various regions may be characterized by this p a r a

-m e t e r . Thus when K = 0, (5. 5) defines the stability boundary. When K < 0, (5.5) defines a family of curves in the region R of Fig. 1, and when K > 0, this equation defines a family of curves outside of R. The region where K > 0 is associated with asymptotic stability, and that where K < 0. instability.

Finally, it is of interest to observe that when b = 0, the stability boundary may be broken up into two distinct equations

c, - « " + I c,F* = 0 (5,6) and

c, - w* + • CjF^» = 0 (5.7) which a r e asymptotes to equation (5.4). Equation (5.6) defines the response curve

for the free undamped oscillation, whilst (5, 7) defines the locus of vertical tangents of this free oscillation. Thus the case b = 0, discussed by Stoker in Ref. 1, emerges a s a special case of (5.4).

6. Evaluation of characteristic exponents by the method of Hale.

It is clear from the preceeding sections that the principal problem in d e t e r -mining the asymptotic stability of the variational equation is the evaluation of the characteristic exponents. In the case of the second-order equations considered this is done by using the known form of the solutions of (3, 3) or (4.1). More recently, Hale in Ref. 7, Chapter 8, has given a method for determining the characteristic exponents of a general class of linear periodic systems which does not depend upon any very detailed knowledge of the solutions, other than that given by Floquet theory. It should, perhaps, be s t r e s s e d here that much of the detailed information concern-ing the stability of Hill's equation comes from equation (3.13). It is clear that for systems of higher ordei the corresponding result to (3.13) i s l e s s useful and permits almost no detailed discussion of stability. The value of Hale's method is thus immediately apparent. In the present section the method will be used to evaluate the c h a r a c -t e r i s -t i c exponen-ts of (5.2) and -the resul-ting s-tabili-ty boundary compared wi-th -tha-t given by (5.3).

Consider equation (5.2). Befine p and cr by

2/9 = P E (6.1)

and

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and s u b s t i t u t e t h e s e e x p r e s s i o n s into ( 5 . 2 ) , giving 5 ' + 5 = - £ I P 4' + (0-+ 2 C o s 2z)5 A l t e r n a t i v e l y ( 6 . 3 ) m a y be w r i t t e n a s t h e s y s t e m 5' = C 6 + S « ( z ) 5 , w h e r e 5 = col (5 ,^2 ), the c o l u m n v e c t o r , ( 6 , 3 ) ( 6 . 4 ) C =

:

i]

and $ ( z ) = 0 - ( a + 2 Cos 2z)

-:]

T h e f i r s t s t e p i s t o t r a n s f o r m ( 6 . 4 ) s o that C i s in diagonal f o r m . T h e c h a r a c -t e r i s -t i c equa-tion of C i s det (C - XE) = 0, ( 6 . 5 )

which h a s the r o o t s X, ^ = ± i. R e d u c t i o n of C t o diagonal f o r m i s then a c h i e v e d by m e a n s of the s i m i l a r i t y t r a n s f o r m a t i o n

T C T - = diag (+i, -i) = B , w h e r e T = 1 -i 1 +1 and T 1^ "21 -i -1 1 1 T r a n s f o r m i n g ( 6 . 4 ) by m e a n s of t h i s t r a n s f o r m a t i o n gives y ' = B y + £ * ( z ) y, w h e r e * (z) = T « (z) T ' ' F r o m ( 6 . 7 ) and ( 6 . 9 ) * (z) 1 21 1 1 -1 +1 0 -(0-+ 2 Cos 2z) 0 p -1 1 -1 -1 o r

<^)

= T i

(o-+ 2 Cos 2z) + i p - ( o - + 2 Cos 2z) - i p (a-+ 2 C o s 2z) - i p -(0-+ 2 C o s 2z) + i p ( 6 . 6 ) ( 6 , 7 ) ( 6 . 8 ) ( 6 . 9 ) ( 6 . 7 ) ( 6 . 1 0 )

T h e next s t e p i s t h e r e d u c t i o n of ( 6 , 8 ) t o H a l e ' s s t a n d a r d f o r m . See Ref. 7, C h a p t e r 6, e q u a t i o n s 6 . 1 and 6 . 2 , and C h a p t e r 8, equation 8 . 5 . F o r t h i s p u r p o s e let y be any finite c o m p l e x n u m b e r , t h e n in t h e p r e s e n t c a s e , t h e r e d u c t i o n t o s t a n d a r d

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f o r m i s a c h i e v e d by m e a n s of t h e t r a n s f o r n a a t i o n

y = exp (X,+ Ê 7 ) z . e x p ( B - X , E ) z . w , ( 6 . 1 1 ) w h e r e w i s a two v e c t o r . It w i l l , of c o u r s e , be o b s e r v e d t h a t t h e f i r s t e x p o n e n t i a l

t e r m i s a s c a l a r m u l t i p l i e r w h i l s t the s e c o n d i s a m a t r i x . Now

(B - X , E ) = diag (+1, -1) - diag (+1, +1) = diag (0, - 2 i ) , ( 8 . 1 2 ) t h e r e f o r e ,

exp (B -X, E ) z = e x p . diag (0, - 2 i ) z = diag (1 ,e"^^^,) ( 6 . 1 3 ) E q u a t i o n ( 6 . 1 1 ) b e c o m e s

y = exp (X, + 6 7 ) z . e x p . diag (0, - 2 i ) z . w ( 6 . 1 4 ) It w i l l b e r e c a l l e d t h a t

^ ( e x p . Az) = Aexp (Az), ( 6 . 1 5 ) w h e r e A i s a c o n s t a n t m a t r i x . T h u s d i f f e r e n t i a t i n g equation ( 6 . 1 4 ) g i v e s

y ' = ( X , + f T) exp ( X , + £ 7 ) z . e x p , d i a g (0, - 2 i ) z . w

+ exp (X, + 6 T ) Z . diag ( 0 , -21), exp, diag (0, - 2 i ) z . w

+ exp ( X , + 6 7 ) z . e x p . diag ( 0 , - 2 i ) z . w ' ( 6 . 1 6 ) S u b s t i t u t i n g for y and y* in ( 6 . 8) and c a n c e l l i n g t h e c o m m o n f a c t o r exp (X, + £ 7 ) z

g i v e s e x p . d i a g (0, - 2 i ) z . w ' = d i a g (+1, - i ) . e x p . diag ( 0 , - 2 i ) z . w - ( X , + £ T ) e x p . d i a g (0, - 2 i ) z . w - d i a g ( 0 , -21). e x p . diag (0, - 2 i ) z . w + Ê * ( z ) e x p . diag (0, - 2 i ) z . w = d i a g (1 - X, - E T , -i - X, -fcT + 2 i ) e x p . diag (0, - 2 i ) z . w + 6"l'(z) e x p . d i a g (0, - 2 i ) z . w =-Ê7 e x p . d i a g ( 0 , -21) z . w + £ «(z) e x p . diag ( 0 , - 2 i ) z . w T h u s w ' = £ ' ) ' w + £ e x p [ - d i a g (0, - 2 i ) z ] »(z) e x p . diag ( 0 , - 2 i ) z . w ( 6 , 1 7 )

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Using (6.10) and (6.13), (6.17) becomes w' = -£YW £ ,, ., 2iz. 2 | d i a g ( l , ) (0-+ 2 Cos 2z) + i p -(o'+ 2 Cos 2z) - ip (0-+ 2 Cos 2z) - i p

-(cr + 2 Cos 2z) + i p diag (1, e'2^^ )w

which reduces finally to w' = -£YW +£r(z)w, where 1 (6.18)

r(z)-

21 ( a + 2 Cos 2z) + ip 21 Jiz f (o-+ 2 Cos 2z)

-.]

e"^^^ (cr+ 2 Cos 2z) - (o-+ 2 Cos 2z) + i p

-"]

(6.19)

Equation (6.18) is now in the required standard form appropriate to the "totally degenerate" case where all the characteristic roots of C a r e imaginary,

Following Hale, if 7 can be determined in such a way that (6.18) has a periodic solution of period T = 2w/2 =ir, i . e . the period of the term 2£Cos 2z and hence of • (z), then this solution h a s , from (6,11) , the forna

y = exp (X, + E7)z. p(z), (6,20)

where p(z) is periodic of periodw. This implies that X,+£ 7 is a characteristic expon-ent of (6.8). The characteristic exponexpon-ent is unchanged under a similarity transfor-mation such as (6.6) so that X^ + £7 is also a characteristic exponent of (6.4) and hence o f ( 5 . 2 ) .

The solution for 7, in the first approximation, comes readily from Hale's general iterative method for the periodic solutions of differential equations containing a small p a r a m e t e r , in the present case £. The present problem is a particular case of this method and the result is summarized in Theorem 8 . 1 , Chapter 8 of Ref. 7. In order to use this theorem, define the matrix

G (7,e = 0)

. T

Y I r(z)dz,

(6.21)

where T is the period, then, according to the theorem, 7 may be evaluated from the determinant

det [G(7, £ = 0) - YE] = 0

F r o m (6.19) the coefficients of the matrix G a r e

(6.22)

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1 r"" T / ( - 0 - + i p - 2 Cos 2z) dz = ( a - i p ) / 2 i , 2nr 2tri L e^^^(o-+ i p + 2 C o s 2z) dz 2wi = 1/21 , and s i m i l a r l y

—r / (ö-+ ip)(Cos 2z + i Sin 2z) + 2 Cos* 2z + 21 Sin 2z C o s 2z jdz

—rl I (cr+ ip)(Cos 2z + i Sin 2z) + 1 + C o s 4z + i Sin 4z + i Sin 4z je

1 /"^ ~ 2»ri / • o e"^^^ ( o - - i p + 2 C o s 2z) dz = - 1 / 2 1 , giving G(7.e = 0) = - ^^ g into (6.22) gives - ^ ( a + i ) - T 1/21 r-cr + -1 "• ip 1 - a + i p -1/21

-h ^-^^"'

) - 7 (6.23) = 0. o r a f t e r e x p a n s i o n and m u l t i p l i c a t i o n t h r o u g h o u t by -4 = (-21)* this b e c o m e s (o-+ ip + 271) ( - a + i p + 271) + 1 = 0 o r 7» + P7 + i {cr'+ p' - 1) = 0, which h a s the r o o t s 7, J = i l - p ± (p^ - 0-2- p* + 1)^ J= i j - p ± (1 -cr«)

* ] •

(6.24) ( 6 . 2 5 )

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il

One characteristic exponent of (5.2) i s , therefore, in the first approximation

X , + e 7 = +i + i [ - E p ± 6(1 - o - * ) M = +1 -/9 ± i ( l -a^)^ (6.26) A second characteristic exponent of (5, 2) i s given by the complex conjugate of X^+£7,

as shown in Theorem 8 . 1 , Chapter 8 of Ref. 7. Since only the r e a l part of the c h a r a c t e r i s t i c exponent determines the stability of (5. 2), then this stability will be d e t e r -mined solely by fi when CJ* > 1. In this case the stability is dependent only on the "damping t e r m " , b. This is consistent with combinations of frequency and amplitude which a r e away from the region R of F i g . 1. However, when o* < 1, the r e a l part of (6.26) is

£ 7 = - / 9 ± i 8 ( l - o * ) ^ ' ,1

The asymptotic stability boundary corresponds to | t 7 | = 0 , or, since 7 and E a r e both r e a l in this case,

^ = i 8 » ( l -a*)

or

4 / 9 * + e ( o - ' - 1) = 0 .

Now from (6.2),£o- = 6 - 1 , which upon substitution gives an asymptotic stability boundary defined by

( 6 - 1 ) * + 4/9*- £•= 0, (6.27) which agrees exactly with the result of equation (5.3), The subsequent development

of the boundary in the form (5,4) and the associated implications then follow a s in Section 5.

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References 1. Stoker, J . J .

2. Minor sky, N.

"Nonlinear Vibrations in Mechanical and Electrical Systems". Interscience Publishers, New York, 1950. "Nonlinear Oscillations" VanNostrand, Princeton, N . J . , 1962. 3. Christopher, P . A . T . 4. Coddington, E . A . and Levinson, N. 5. Struble. R . A . 6. Hayashi, C. 7. Hale, J . K . 8. McLachlan, N.W. 9. Whittaker, E . T . and Watson, G.N. 10. Young, A.W. 1 1 . Whittaker, E . T . 12. Haupt. O.

"The Stability of the Short-Period Motion of an Airframe Having Nonlinear Aerodynamic C h a r a c -t e r i s -t i c s and Subjec-t -to a Sinusoidal Eleva-tor Oscil-lation". College of Aeronautics Report 176, 1964. "Theory of Ordinary Bifferential Equations". McGraw-Hill. New York, 1955.

"Nonlinear Bifferential Equations". McGraw-Hill. New York, 1962.

"Forced Oscillations in Nonlinear Systems".

Nippon Printing and Publishing Co. , Osaka, Japan, 1953.

"Oscillations in Nonlinear Systems". McGraw-Hill, New York, 1963.

"Theory and Application of Mathieu Functions". Oxford University P r e s s , 1951.

"A Course of Modern Analysis". Cambridge University P r e s s , 1963.

"On the Quasi-Periodic Solutions of Mathleu's

Bifferential Equation. " Proceedings of the Edinburgh Math. Soc. Vol.32, pp 81-90. 1914.

"On the General Solution of Mathleu's Equation". Proceedings of the Edinburgh Math. Soc. Vol. 32, pp 75-80, 1914.

"Über Lineare Homogene Bifferentialglelchungen Zweiter Ordnung Mit Periodischen Koeffizienten" Mathematische Annalen, Band 79, p. 278 (1919).

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FIG.I. FREQUENCY RESPONSE C U R V E S -DUFFING'S EQUATION,

FIG.3, CONTOURS OF CONSTANT /i AND a IN THE FIRST n . | UNSTABLE R E G I O N .

CHARACTERISTIC N U M B E R - a

FIG.2.CHARACTERISTIC CHART FOR THE SOLUTIONS OF MATHIEU'S E Q U A T I O N .

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