On parametric excitation

ON PARAMETRIC EXCITATION. BY

N. MINORSKY, Ph.D, E.E., David Taylor Model Basin, Washington, 1). C : i. I N T R O D U C T I O N .

The possibility of exciting a dynamical system (electrical or mechanical) by means of a periodic variation of its parameters such as capacity,-inductance, spring constant and so on, has been known for a long time. Lord Rayleigh (I), for example, analyzes Melde's experiment (2) in which one end of a stretched string is attached to a prong of a tuning fork vibrating in the direction of the string. In this experiment the oscillations of the tuning fork produce periodic fluctuations in the tension of the string which, in turn, cause transverse vibrations of the string. M. Brillouin (3) and H. Poincar6 (4) investigated similar effects in electric circuits. More recently these phenomena have been studied by a group of Russian scientists under the leadership of L. Mandelstam and N. Papalexi (5) who correlated these phenomena with the theory of subharmonic resonance. In view of the fact that these phenomena of self-excitation are generally associated with the periodic variation of certain parameters of the system, the term parametric excitation is commonly used in the current literature on this subject. The investigations of Mandelstam and Papalexi resulted in the development of an interesting electric machine, the parametric generator (6), capable of delivering a considerable power output. This machine transforms the mechanical power absorbed in the process of a periodic variation of its parameter into electrical energy of an alternating current of a certain frequency. The mathematical theory of these phenomena is closely related to the theory of the differential equations with periodic coefficients, more specifically, to the equations of the Mathieu-Hill type (7). Most of the literature on this subject deals, however, with the periodic solutions of these equations and very little is known about the unstable solutions which are of considerable interest from the point of view of parametric excitation. Moreover, only linear differential equations with periodic coefficients have been investigated so far. In applications one always comes across non-linear equations of this kind. In fact, the unstable solutions of the linear equations increase indefinitely and can not, therefore, characterize a stationary periodic process. Mandelstam and Papalexi obtained an interesting confirmation of this faet in their experiments. They report that if the circuit of the parametric generator is devoid of non-linear elements, the voltage 25

26

N. MINORSKY.

IJ. l". I.

builds up to a high value at which the insulation is punctured. By providing a non-linear element, on the contrary, it is possible to obtain a stable performance of the generator. Unfortunately there exists no theory of non-linear differential equations with periodic coefficients and the most one can do is to study the conditions of stability of the solutions in the linear region'. I m p o r t a n t contributions to these studies were made by E. Meissner (8) and by B. Van der Pol and M. S. O. Strutt (9). These authors have explored the zones of stable and unstable solutions of the Mathieu-Hill equation although the actual form of the unstable solutions and the effect of the relative frequency o f the parameter variation have received very little attention. It will be shown in w h a t follows t h a t by transferring the problem to the phase plane, it is possible to extend the qualitative study of solutions of these equations so as to be able to approach the problem from a more general standpoint. 2. EQUATIONSOF MATI-IIEffAND roLL.

The differential equation of Hill is q- F(t)x = o,

(I)

where F(t) is a periodic function with period 27r. The Mathieu equation is 2 + (a 2 + b 2cost)x = o. (2) A particular case of t h e Hill equation is the so called Hill-Meissner equation in which the function F(t) is a rectangular "ripple" shown in Fig. I. The Fourier expansion in this case is F(t~

JF dz tj

0

F(t)

Figure

I

02 + -4b02 ( c o s t - - - cIo s 3 t 7r ;3

+ I cos St . . . . 5

) "

(3)

T h e theory of Hill's equation is very similar to t h a t of the Mathieu equation so t h a t both equations, to some extent, are studied together. Although Hill's equation appears to be more complicated than Mathieu's equation owing to the presence of the Fourier series, in some cases the discussion is facilitated by the particular form (3) of the function F(t) as was pointed out by Meissner, Van der Pol and Strutt (loc. tit.).

July, I945.]

ON PARAMETRIC ]~.XCITATI()N.

27

\Ve shall make an extensive use ()f this remark. The zones of stal)ility ()f solutions of the Mathieu equation and of the Hill-Meissner equation are of the same general c h a r a c t e r as Was pointed out b y S t r u t t (IO) and it seelns likely that on the average the unstable solutions of both equations are not much a p a r t from each other. T h e experimental evidence seems to corroborate this conclusion although no direct quantitative comparison of the unstable solutions of equations of both types has been a t t e m p t e d so far. We shall first analyse the q u a n t i t a t i v e aspects of the integral curves, the phase trajectories of the Hill-Meissner equations, by transferring the problem to the phase plane of the variables (x, ~) and will return to the Mathieu equation in a later section. It is convenient to write the Hill-Meissner equation in the form

+ (a ~ ± b2)x = o,

(4)

which means t h a t we consider alternatively the two equations + (a ~ + b~)x = o ;

~! + (a ~ -

b~)x = o ,

(5)

during each half period rr of the ripple with the u n d e r s t a n d i n g t h a t solutions have to be continuous on physical grounds although not necessarily analytic at the points at which the changes from (a 2 + b ~-) to (a ~ - b2) occur. We will assume t h a t a 2 > b2 i n a s m u c h as we will be concerned with the problem of modulatibn of the q u a n t i t y a ~ b y a rectangular ripple b2. 3. P A R A M E T R I C E X C I T A T I O N OF A

NON-DISSIPATIVECIRCUIT.

It is useful to approach the problem of p a r a m e t r i c excitation in tile simple case of a non-dissipative circuit whose differential equation is

d2q

I

L0d~g+C00 q = o,

(6)

where L0 and Co are the inductance and the capacity respectively and q is the q u a n t i t y of electricity stored in t h e c o n d e n s e r . If a certain charge q0 exists initially and the circuit is closed an u n d a m p e d oscillation of frequency oa0 = t/x~LoC0 will be established. T h e energy will oscillate indefinitely between the potential form CoVo2/2, Vo being t h e potential difference across the condenser when the c u r r e n t i = dq/dt is zero, and the electromagnetic form Loio2/2 when the condenser is discharged and the current is a m a x i m u m . T h e energy stored in the s y s t e m is Eo = (I/2Co)qo ~. The m a x i m u m values q0 and i0 are given b y t h e expressions q0 = CoV0 and i0 = q#0. We shall now assume t h a t the capacity Co is " m o d u l a t e d " b y a rectangular ripple ± AC between the limits C ..... = Co + AC and C,~i,. = Co - AC, (AC << Co). Let us start from the instant w h e n t h e whole energy is electrostatic and reduce the capacity ( - AC) suddenly.

28

N. MINORSKy.

[J. F. I.

To accomplish this, impulsive mechanical work will be required to overcome the electrostatic forces. Since the system is conservative by our assumption, an equivalent increment, &El, of electric energy will be added. Assuming AC small, this increment is obtained by differentiating the expression E = (I/2C)q0 2 with respect to C which gives AE1 = -- (qo2/2Co2)AC. The increment is positive if ~C < o, that is, if the capacity is decreased. After this operation the energy stored in the condenser will be '

a'

(7)

where a = I + ( [ A C ] / C o ) > I. A quarter period later when the energy is purely electromagnetic (q0 = o; i = dq/dt maximum) we can reestablish the former value of the capacity by increasing it by + AC without doing any work. At t = T/2, when the energy is again purely electrostatic, the capacity is again reduced ( - AC) which adds another increment of energy AE2 = qo2a/2Co ~, so that the new energy content is now L Cl q02 a ~ (8) E2 = E1 -~- AE~ = ~ 0 0 I -]- Co I = ' ~ 0 0 and so on. It is seen that by timing the discontinuous changes of capacity in the manner shown the energy stored in the system will gradually increase as the result of the operation of the ripple ± AC. One verifies easily, that if instead of decreasing the capacity ( - ~C) at the instants o, T/2, T , . . . the capacity is increased ( + ~C) at these instants, the operation of the ripple ± AC will withdraw the energy from the system. In practice instead of discontinuous changes ± AC of capacity a continuous modulation C = Co + ~C sin 2wt, (w = 2~r/T) is employed. The results obtained are found to bo in qualitative agreement with the above conclusions obtained on the basis of a discontinuous variation of capacity. This shows t h a t on the average the solutions of the Mathieu equations are not far apart from those of the Hill-Meissner equation. The latter is simpler to discuss, as will appear later, so we will proceed with the discontinuous method. It is to be noted also that in both cases (continuous and discontinuous) the frequency of modulation is twice t h a t of the oscillatory process in the circuit. We shall later consider a more general situation. The argument remains the same if instead of the capacity variations ( ± AC) the inductance variations ± ~L are used. The timing of the ripple in this case is exactly the same as before, viz: the coefficient of the inductance is decreased ( - ~L) at the instants o, T/2, . . . and increased ( + AL) at the instants T/4, 3T/4, " " . To the same timing there will correspond, however, a diametrically opposite effect, viz. at

July, t945.1

ON PARAMETRIC EXCITATION.

29

the instants T/4, 3T/4, when L is increased there will be an a d d i t i o n of energy since the whole energy is electromagnetic at these instants, whereas at o, T/2, ... when L is decreased no work will be done since the electromagnetic energy is zero. 4. T O P O L O G Y OF T H E

HILL-MEISSNEREQUATION.

We shall write equation (6) t a k i n g into a c c o u n t tile step-wise modulation of capacity as I

-J¢- L oC0(I -4- ~c) q = o,

(9)

where 3'c = AC/Co is the index of modulation. P u t t i n g I/LoCo = wo=' and considering 3',, << I w i t h o u t a n y loss of generality e q u a t i o n (9) becomes + w02(I 7= 3"c)q = o. Introducing a new independent variable r = wot we have d2q + (~ ~= .yc)q = o.

dr 2

(~o)

This equation should be considered as an a l t e r n a t e sequence of the two equations

2d2~q + a~2q = o dr

and

d2q

~

+ Oz22q = O,

(I I)

where a2 2 = I + 3"c; c~22 = I - ~ , replacing each other at the frequency of the ripple 4- AC. Since the solutions q(r) of these e q u a t i o n s have to be continuous, the problem reduces to fitting the solutions of both equations by c o n t i n u i t y a t the end of the intervals ( + AC, -- &C, + AC • • .). T h e solution q(r) as a whole is t h u s a continuous f u n c t i o n of time although not a n a l y t i c at the points where C varies discontinuously. In some cases this loss of a n a l y t i c i t y appears as a d i s c 9 n t i n u i t y in the second derivative, the first derivative being c o n t i n u o u s ; in o t h e r cases the discontinuity m a y occur in the first derivative of the solution q(r). We will now transfer the problem to the phase plane of the variables q, dq/dr. The solutions q(r) will then be represented by t h e integral curves, or phase trajectories of equations ( I I ) , and the d y n a m i c a l process described by these equations will be represented by the motian of the representative paint R on the trajectories as shown in Fig. 2. For 3"~ = o, ~1~ = ~3 = I the trajectories of equations ( I I ) form a continuous family I'0 of concentric circles with the origin 0 as center. If ~ ¢ o, the trajectories form continuous families I~ and 1~2 of concentric homotethic ellipses shown in Fig. 2. T h e family F~ corresponding to a 2 > I has a c o n s t a n t ratio b/a = a 2 = I -Jr- 3"c of semiaxes

30

.~'. ~'[INORSKY.

IJ. F. I.

and for the family 1'~ this ratio is b'/a = a2~ = I - 7~. The family I'~ corresponds to the reduced value Co - AC of capacity and F~ to the increased value Co + AC. The origin 0 is the singular point of the differential equations (II). T h e two families F1 and F2 thus serve as a kind of reference systems determining the motion in the phase plane. For instance, if for t = o certain initial conditions, say q0, 0 are given and the value of C is prescribed, e.g. C = C o - AC, the process is depicted by the motion of starting from the point A corresponding to the initial conditions and moving along the ellipse of t h e family I'1, passing through t h a t point. If a later instant t = tl, to which corresponds the point B on the ellipse, the capacity is changed and is C = Co + AC, the representative point will pass on the elliptic trajectory belonging to the family F2 passing through B and will continue d'r

Figure

2

to move on t h a t trajectory until the next change (C = Co - AC) and so on. We shall see in the following sections t h a t this geometrical representation of the solutions q(r) of equation (9) is very convenient and permits obtaining an account of the various features of the dynamical process. It is apparent t h a t whenever a piece-wise analytic trajectory formed by the elliptic arcs is closed and the path is reentrant * we encounter a periodic phenomenon. As an example of the application of this method, we shall consider the case of the parametric excitation discussed in Section 3 in connection with the capacity ripple. Let us start from a point A(q0, 0) after the capacity has been reduced (C = Co - AC). The representative .point, * It will be shown in Section 6 t h a t under certain conditions the path may be closed after one revolution (2~r) of the radius vector without the reentrance of the path. In such cases the p h e n o m e n o n is not periodic with period 27r b u t it may be periodic for 4~r, 67r, . . . .

Jtfly, 1945.]

ON PARAMETRIC }?.XCITATION.

31

Fig. 3, will move on the arc A B of the elliptic t r a j e c t o r y of the family I'1. At the point B (q = o, d q / d r max.) the c a p a c i t y is increased (C = (',, -F AC) and the arc B(" of tim family l'e is followed. At the point C the capacity is reduced and tim next }tl-C (i'D is of the family I'i, and so on. After o n e period 27r one reaches tim point E corresponding to q2, > q0 which shows that the energy c o n t e n t of the system has been increased. If at the point A the capacity is increased (q- AC), at B' decreased, at C' increased and so on, a convergent piece-wise analytic spiral AB'C' • • • would be followed, in which case the ripple 4- A C w i t h d r a w s the energy from the system. dq C

d'r

q"

D Figure

5

Similar conclusions can be obtained when the inductance L undergoes a step-wise modulation L0 4- AL taking into the a c c o u n t the observation made at the end of Section 3. W e note that t h e loss of analyticity at the points B , C, D , • • • is due to the fact t h a t the second derivatives arc discontinuous, although the first derivatives are continuous. In the general, case the discontinuities occur in the first derivatives (e.g. point B in Fig. 2). 5. RIPPLES OF DIFFERENT FREQUENCIES A N D P H A S E ANGLES.

The preceding s t u d y was limited to a v e r y special case when the " f r e q u e n c y " of the ripple is twice t h a t of the circuit and w h e n the changes 4- AC (or 4- AL) occur at the instants O, T / 4 , T / 2 , 3 T / 4 . . . of the oscillation as shown in Fig. 4. W e shall now investigate a more general situation. For this purpose, it is necessary to establish certain general relations governing the motion of the representative p o i n t on elliptic arcs of the two families 1"1 and I'2. Suppose we s t a r t a t the

32

N. MINORSKY.

[J. F. I.

point A Fig. 5, on elliptic trajectory E~ (family F1). r is given b y the equation

The radius vector

n0 I.

(I2)

cos -~~ + ~12 slnZ q~,

where q~ is the angle of the radius vector, a0 = OA the semi-axis (on

! l i i i

!

O

"i

.

.

.

.

.

.

4

Figure

the x axis) and al ~ = I At-"~/c as before. If instead of changing the capacity (C = Co + AC) at B as is Fig. 3, this change ociurs at a point M (Fig. 5) whose coordinates are xl = rl cos ~1 and yl = rl sin q~l, !dr

IrI

~'%

4._.Oo._2

Figure

-

[q

5

where rl is given b y equation (x2) in which 4) = qh, the representative point will describe an elliptic arc M N of the family F2. The x - semiaxis al of the ellipse E2 is obtained from the equation

x1_I' + l--2-~ Y = i ,t,

0.,12

~ 12Ct~2 2

which expresses t h a t the ellipse E2 passes through M and has the ratio

July, 1945.]

ON

PARAMETRIC

of semi-axes a2 2 = I - yc.

33

EXCITATION.

This gives al = 4 x , 2 + y'-f

(I3)

~2 2 '

where xl ~ and y 2 has been previously calculated. At the point N of coordinates x~, y2 another elliptic t r a j e c t o r y E11 (family r~) starts and continues to a point P with coordinates x3, y3 and with p a r a m e t e r c~2 = I + "r~ and so on. Carrying o u t this procedure (we omit the calculations) one obtains the following general relations expressing aN, the x-semiaxis of the ellipse after N changes of capacil~y, depending upon whether N is even or odd. ax = as, = ao f l f 3 ": f2f4 • "f2, ' aN =

a 2 v + l ~--- a o

eqJ~f3 .." f2,+,

where

v ----- I, 2, 3, "'"

(I4) (I 5)

~,

•/a•

~ + tan ~ ¢~ fi(¢i) = ~ - ~ l2 + tan2 ¢i"

(I6)

It is a p p a r e n t that f,(¢i) = L ( -

+

¢,) =

= L(-

¢, +

(i7)

The only case of practical interest is when all intervals are equal and are fractions of 2 K r where K is an integer. W e shall call this m o d e of subdivision the equiphase intervals inasmuch as the phase plane is subdivided into equal sectors. The general form of subdivisions for N = 4 for instance will be ¢0, ¢1 = ¢0 + 0r/2), ¢5 = ¢0 + (2~r/2), ¢:~ = ¢0 + (3r/2), ¢4 = ¢ 0 + (4r/2) = ¢0, as is obvious from Fig. 6.

Figure

6

The actual phase relation between the ripple and the u n d a m p e d oscillation in the circuit is shown in Fig. 7. W e m a y call the angle ¢0 the phase angle of the ripple and the n u m b e r N of changes + AC contained in the period 21r as the relative frequency of the ripple. A

34

N. MINORSKV.

/J- t'-1.

complication arises, however, due to the fact t h a t the equi-phase intervals as above defined are not the intervals of equal time, the equi-time intervals, owing to the non-uniformity of motion of the representative point on the elliptic trajectories. We shall come back to this question in Section 7. It is sufficient to mention here t h a t the .conclusions derived from the consideration of equi-phase intervals permit obtaining the principal features of phase trajectories of the Hill-Meissner equation in a v e r y simple m a n n e r and the introduction of the equi-time intervals, while complicating the calculations, does not reveal a n y additional features of interest, except in some special cases. 6.

EQUI-PHASE

INTERVALS.

We shall examine now the various cases of parametric excitation in terms of the two p a r a m e t e r s N and 40. It. is convenient to consider separately the following groups of numbers (I) N = 4v; ~--

*[

2."

I Figure

?

(2) N = (2u + I)2; (3) N = 2u + I; (4) N = p/q; where v = i, 2, 3, • • • and p and q are relatively prime. In the first three groups N is an integer and in the last it is a rational fraction. This covers all cases of interest in practice. (I) First group N = 4, 8, z2 . . . . Let us consider the first case N = 4 which has been studied in Section 3 by an e l e m e n t a r y m e t h o d ; we shall now apply the general m e t h o d of Section 5. T h e intervals are ¢0, $1 = ¢0 + (7r/2), $2 = ¢0 +(2~r/2), ¢3 = ¢0 + (37r/2), $4 = ¢0, shown in Fig. 6. Hence by the properties (I7) of the functions f we have f0 = f2 = f4; f~ = f3. Using equation (14) we get l"jij8 ~'

fl 2

(I8)

h f , = a°fo2 T h e condition for p a r a m e t r i c excitation is clearly f12 /f o2 ) 1.

Replacing

July, ~945.]

ON

PARAMETRIC

EXCITATION.

35

.1': a n d j'o by their values (I6) this c o n d i t i o n is a2 2 t a n s ¢o + I ~2s + t a n s ¢o al s t a n s ~b0 -t- I >/ O~'1s + t a n s ¢0

(I9)

T h e equal sign corresponds to the closed trajectories. T h e c o n d i t i o n for the closed trajectories is given b y the e q u a t i o n t a n 4 ¢0 = I, t h a t is for ¢0 = 7r/4, 37r/4, 57r/4 a n d 77r/4. T h e lines A C a n d B D in Fig. 8

t J Figure

8

correspond to the closed trajectories, hence t h e y are the t h r e s h o l d s a t which the p a r a m e t r i c e x c i t a t i o n a p p e a r s (or disappears). In o r d e r to d e t e r m i n e the zones of ¢0 w i t h i n which it exists, let us p u t ¢0 = 0 (or ¢0 = 7r) a n d c o m p a r e b o t h t e r m s of the i n e q u a l i t y (19). T h e left h a n d t e r m is one and the right h a n d t e r m is c~22/al2 so t h a t in these zones. shown in shading, the p a r a m e t r i c e x c i t a t i o n t a k e s place. In t h e nons h a d e d sectors no p a r a m e t r i c e x c i t a t i o n occurs; one verifies this b y p u t t i n g ¢0 = 7r/2 (or ¢0 = 37r/2) in the i n e q u a l i t y (19) which gives t h e left h a n d t e r m c~22/al2 a n d r i g h t h a n d t e r m is u n i t y which m e a n s t h a t a4 < a0 so t h a t no p a r a m e t r i c e x c i t a t i o n exists in this case. T h e " p u m p i n g " of the e n e r g y i n t o the s y s t e m b y t h e ripple is m a x i m u m a t ¢0 = 0 or ¢0 = ¢. S u b s t i t u t i n g these v a l u e s i n t o e q u a t i o n (18) one finds ael2 I + at . . . . ;a0 - - - a o . (20) a2-

I - - "y

Defining 5 as average d e c r e m e n t , clearly a4 = aoe 2~ whence I log a'" t~ =

-

-

271"

o

Oe,.,-

I h)g(l -

-

27r

(21)

'~- ~" ) -

-

-

-

1 - - "y

"

One can c o n s t r u c t the t r a j e c t o r i e s for d i f f e r e n t values of ¢0 following the graphical procedure o u t l i n e d in c o n n e c t i o n w i t h Fig. 2. A few s u c h trajectories are shown in Fig. 9 for N = 4 for several v a l u e s of ¢0. F o r N = 8, 12, I6, • • • the conclusions r e m a i n the s a m e b u t t h e n u m b e r

36

N.

M1NORSKY.

[J" }'" I.

of a l t e r n a t e zones in w h i c h t h e t r a j e c t o r i e s d i v e r g e (or c o n v e r g e ) is accordingly greater. (2) S e c o n d g r o u p N = 6, 1o, x4, I 8 , . . . . As an example, c o n s i d e r t h e case v = I, N = 6 a6 =

(22)

O,0 f-' f ~ - f,~

f2f,f6

t h e i n t e r v a l s are ¢0, ~bl = ~b0 + 2 r / 6 , . . . so t h a t f0 = f3, f l = f 4 , f 2 = fs, w h i c h s h o w s t h a t as = a0 for a n y v a l u e of ~0. T h i s m e a n s t h a t o v e r t h e p e r i o d 27r t h e r i p p l e does n o t a d d a n y e n e r g y to the s y s t e m so t h a t n o p a r a m e t r i c e x c i t a t i o n is possible. N=4,

A ~ , ~2

2n 8

Figure

9

(3) T h i r d g r o u p N = 3, 5, 7, 9, " ' " C o n s i d e r the case v = I, N = , 3 ; t h e c o n c l u s i o n s are a p p l i c a b l e to N = 5, 7, • " ". W e n o w h a v e t o use e q u a t i o n (I5) w h i c h gives a3 = a0 0/-2xfff---da 0/2 f 2

(23)

"

T h e i n t e r v a l s are h e r e : q~0, ~, = q~0 q- (27r/3), ~2 = ¢0 q- (47r/3), Ca = ¢0, a n d t h e c o n d i t i o n for p a r a m e t r i c e x c i t a t i o n is 0/~f , f o a2 f2 >~ I.

(24)

July, I945.1

ON PARAMETRIC EXCITATION.

37

One sees at once t h a t the closed trajectories correspond to the values ¢0 = o, ¢0 = r, in which case f l = f2 and

fo = ~

+ tan2 ¢° = ~ +

t a n 2 ¢o

al

which gives a~ = ao i n equation (23). By s y m m e t r y there are four other values ¢o = r/3, ¢o = 4rr/3, ¢o = 2rr/3 and ¢o = 5r/3 for which

F'i~lure

I0

the trajectories close. It is to be noted, however, t h a t a l t h o u g h the trajectories close for these values of 00, the p a t h is not r e e n t r a n t so t h a t for the second revolution (27r, 4 r) of the radius vector the representative point describes a t r a j e c t o r y different from t h a t which it has followed during the first revolution (o, 27r). Figure IO shows a trajectory of this kind for N = 3, 00 = - 3o °.

38

N. MINORSKY.

[J. V. I

One ascertains this easily by comparing at, with a4 which gives al = ao(al/a2)fl and a4 = ao(flf3/f2f4). From the size of the intervals it is a p p a r e n t t h a t of = f3, f l = f4, so t h a t f0

a4

0~2

ax

alflf2

"

In general al/a2 ~ fo/flf2 although the equality m a y take place for special values of 40 which proves the statement. After two revolutions (4~r) the path is always reentrant. This is to be expected, of course, since such a case falls within the second group just analyzed. Hence no parametric excitation is possible in the third group either. (4) Fourth group N = p/q. One finds t h a t a great majority of fractions p/q are either of no interest from the standpoint of parametric excitation or fall within the scope of the previously studied cases. The only cases of interest are N = 4/3, 4/5, 4/7, " ' ' , 8/3, 8/5, " ' , corresponding to the intervals A4 = 3r/2, 5r/2, 77r/2, . . . which fall into the first group. T h u s for example for N = 4/3, A4 = 3r/2, the intervals are 40, 41 = 40 ~- (3~r/2), 42 = 40 d- (67r/2), 43 = 40 -t- (97r/2), 44 = 40 + (I2~r/2) = 40, so t h a t the injections of energy occur at the analogous points of the oscillatory process but the operation of the ripple is delayed each time b y the angle r as compared to the case shown in Fig. 3. 7. EQUI-TIME INTERVALS.

The difference between the equi-phase and equi-time intervals is due to the fact t h a t the motion of the representative point R is not uniform on the elliptic trajectories as was mentioned in Section 5Only in v e r y special cases N = 4, 4/3, 4/5, " " , 8, 8/3, " " and 40 = o, r/e, r, 3r/2, . " both intervals coincide. As was shown, o n l y t h e s e special cases are of practical interest. We shall now formulate the problem in t e r m s of the equi-time intervals. Figure 1I shows two trajectories EI and E2 of the previously defined families I'1 and I'2 and C is the principal circle for the two ellipses. Each of the two ellipses can be considered as a projection of C on a plane inclined to the plane of C a b o u t the axes of coordinates and, as known from kinematics, a non-uniform motion of the representative point on the ellipse m a y be obtained by projecting the corresponding uniform motion on C. In view of this to a position R of the representative point on E1 defined by angle 4 there corresponds a point /~ on C defined by the angle q~; one obtains a similar relation for the other ellipse E o. Since in the phase plane representation the ratio of the semiaxes along the y and the x axes for the E1 and E~ ellipses is a l 2 = I -I- "It and c@ = I - ~, respectively, one finds the relations connecting 4 and ~ of the form tan4

= al 2tanq~

(forE1);

tan¢ = a22tan~

(forE2).

(26)

July, ]945-}

ON

I)ARAMETRI(:

39

FXCITATION.

Both angles 0 and 4; coincide when the radius vector coincides with the coordinate axes. This is the reason w h y for the most interesting case of parametric excitation shown in Fig. 3 there is no difference b e t w e e n the two angles ¢ and q~. For the intermediate positions of the radius vector there exists in general a difference between the two intervals, the equi-phase ¢ and the equi-time 4;. In this manner the equi-time intervals on the elliptic trajectories are reducible to the equi-phase angles 4; on the principal circle. Y !

x

II

Figure

Replacing ¢i in terms of 4;i from equations ( 2 6 ) o n e o b t a i n s the following expression for the x-semi-axis a,, of the elliptic t r a j e c t o r y after n equi-phase intervals 4;¢ of the ripple i=n

a,, = aoHg,,,

(27)

i=1

where g,, = g2,-~ =

F 2 ~'cos 4;~,-t +

~2

s i n 2 4;~,-~, •

¢h

g,, = g2, =

cos 2 4;2, + ~ sln 2 ¢2,,

(28)

where v is an integer and X2 _

cq 2 _

_ _ I4 - 3' >

~2 2

I --

It is a p p a r e n t t h a t g2,-: > I and g2, < I. also written in the form @

--

I ~/a - - COS 4;2v-1 ;

g2v --

I.

Expressions (28) can be

x=gf-Ti_i ~-

~,fa Jr- COS 4;~,,

(29)

40

N. MIn0rsKv.

[J. F. I.

where ~k2 --1- I a-

k2

>I. --

I

As an example let us apply equations (29) to the previously studied case N = 2v = 4. In terms of the equi-time intervals one has now: 24;1 = 24;0 -~- 71", " ' ' which gives (X 2 -

a4 = aoglg2g~g4 = ao

I) 2

(a + cos 24;0)2.

4 x2

For ~b0 =

O,

a4 =

a 0 Gk2

I)2

X2 -J¢- I Jl- I I

4

-

=

a0X 2 =

-

a0-a2 2 '

which gives the same expression (2o) which we obtg.aed by means of the equi-phase intervals as is to be expected since in this particular case both types of intervals coincide. For the intermediate conditions when the intervals do not coincide with the axes of coordinates, this generally is not the case and it is necessary to operate with the equi-time intervals. T h e calculations are more complicated but do not yield anything interesting from a practical standpoint as previously investigated. T h e r e are certain zones in which parametric excitation occurs followed b y zones in which it does not occur; conditions for closed trajectories are more complicated and so on. All this, however, is of relatively little practical interest because the really i m p o r t a n t cases are those where both types of intervals q~i and 4;¢ coincide. 8. PARAMETRIC EXCITATION OF A DISSIPATIVE CIRCUIT BY A CAPACITY RIPPLE.

Let us consider the problem of excitation of a dissipative circuit whose equation is I

Lo~ + RoO + Cooq = o.

(30)

Dividing by Lo and putting I / L o C o = 00o2; R o / L o = 2po this equation can be written as -]- 2 p o q

Jr- wo2q =

O.

"

(31)

I n t r o d u c i n g a new variable Q defined by the equation q = Q e - J "re' = Q e - f "°at

(32)

equation (3I) becomes O -~- 0022Q =

(33)

O,

where ¢.022 = 0~o2 __ p 2

dp

dt

=

°~°2

-

p°2 = °~12"

If one of the p a r a m e t e r s L or C is a periodic function of time, o)12 is also periodic and equation (33) is generally of the Hill type. With our

July, x945.1

ON

PARAMETRIC I?~XCITATION.

41

assumption of a rectangular ripple it is of the Hill-Meissner type which we will consider. Replacing in equation (3o) Co by Co -t- A C =

Co

I ±

Co

=

Co(I ±

"t'e),

the equation (31) becomes 0 q- 2p00 -q- ¢00z(I q: "Y~)q = o.

(34)

The change of the variable (32) transforms this equation into the form 0 +

:F a)Q = o.

(35)

Introducing the independent variable r = wit it becomes

d~Q d r 2 -~- ( I =[z •)Q

(36)

= O

which is equivalent to the two equations occurring alternatively at the frequency of the ripple ± AC, viz. :

d2Q d& + ~12Q = o;

d2Q

(37)

d& + a22Q = o,

where GI 2 =

I q- 6 =

I q- coos y~ -

-

0012

and

Or22 =

I -- ~ =

I

6002 -'Yc. 6012

(38)

T h e plus sign in the first equation (38) corresponds to the value C = Co - AC. Equations (37) have the same forms as equations (xo) so that the previous conclusions are applicable here with the difference however, that equations (37) contain the d e p e n d e n t variable Q whereas the equations (Io) contain the variable q, the two variables being related by equation (32). The trajectories of equation (37) as previously, are either convergent or divergent piece-wise analytic spirals, formed by elliptic arcs, the closed trajectories appearing as a threshold between the two forms of spirals. For a closed trajectory, clearly, Q is bounded, hence q is monotone decreasing. This means that to the closed trajectories in the (Q, dQ/dr) plane correspond convergent spirals in the (q, dq/dr) plane, so that no parametric excitation is possible in this case. It is obvious that for a parametric excitation the amplitudes of q m u s t be either monotone increasing or, at least, constant, which requires

Q = Qoe+fmat,

(39)

where Pl ) P0. This means that the trajectories in the (Q, dQ/dr) plane must be divergent spirals With negative decrement * - Pl having the aboslute value Pl greater than, or, at least, equal to the positive * Negative decrement, or increment, because the spirals are divergent.

42

N. MINORSKY.

IJ. 1,'. 1.

d e c r e m e n t Po = Ro/2Lo of the dissipatory circuit (Ro, Lo, Co). Physically this means t h a t the energy injections into the circuit by the ripple :t: AC m u s t on the average be greater than the energy dissipation. This condition is obtained by transforming the differential equation of the dissipative circuit I

LoO q- Roo q- -Cooq = o into the form

d2q + 2Po dq d& ~ -1- q = o,

(4o)

where r = o00t; o00 = I/L~L~0c~and P0 = Ro/2Loo0o. As is well known the trajectories of equation (4o) are convergent logarithmic spirals and the ratio of the amplitudes q2~ and q0 after one period is q2_z = e-~02~, q0

(4 I)

On the other hand in the o p t i m u m c a s e ( N = 4, q~0 = o) of the parametric excitation it follows from equation (20) t h a t for the divergent spirals this ratio is ~2x --

0~12 -

-

q0

e +~'2~,

(42)

0/2 2

which defines the average increment Pl

-- ~

•2~r

log a1-~2

0:2 2 "

(43)

Expressing the condition for the parametric excitation P1 ) P0 and substituting for al 2, a2 2 and P0 their values one gets ~/c )

o002 - -

o0o2

p02 e"° l , e~o + I = ~c,

(44)

where u0 = rrRo/Loo0o. Since ax 2 and a22 are positive and a 2 > a22 one obtains the o t h e r limit for ~c by expressing t h a t a22 is positive. This gives ~c ~< 00°2 - P°2 - ~ / ' .

(45)

O002

F r o m equations (44) and (45) it follows t h a t ~ must be in the interval ( W , ~c") viz. "Yc' < "Y~ < ~/~" (4 6) in order to obtain the parametric excitation. For R0 = o, #0 = o, P0 = o hence the interval is (o, I) and decreases w i t h R0 increasing. For o002 -- po2 = co~2 = o, t h a t is for Ro = 24Lo/Co the interval (TJ, ~/c")

July,

t945. I

ON

43

EXCITATION.

PARAMETRIC

reduces to zero with 3'/ = 3'~" = o. One ascertains that this is the condition for critical clamping. It is, therefore, impossible to o b t a i n a parametric excitation of a critically d a m p e d , or o v e r - d a m p e d circuit. 9. PARAMETRIC E X C I T A T I O N OF A D I S S I P A T I V E CIRCUIT BY AN I N D U C T A N C E R I P P L E .

In Section 3, it was mentioned t h a t for a non-dissipative circuit the effects of the capacity variation ( ± AC) and of the inductance variation (-4- AL) are the same from the s t a n d p o i n t of the excitation. This is due to the fact both L and C enter s y m m e t r i c a l l y into the expression ~002 = I / L o C o for the frequency. For a dissipatory circuit the situation is different in t h a t the capacity enters only into the expression for the d a m p e d frequency oa~2 = w0" - P02 (through ~0o"~) b u t does n o t appear in the d e c r e m e n t P0 = R o / 2 L o . As r e g a r d s the inductance, it appears both in the expressions for the frequency and for the decrement. A priori, one m a y expect different results in both cases. We shall now- consider the effect of a step-wise variation ± AL of the coefficient of inductance L and will define the modulation index "1~. = A L / L o . The difference with the preceding case is in t h a t the decrement R0 R0 P

--

2L

-

2L0(I

:t: TL)

is not constant now b u t also undergoes a modulation. M o r e o v e r , in the transformation (32) P # P0 so t h a t in equation (33) the v a l u e of (.022 is now * (47)

dp dt "

w~'-' = Oao~- -- p'a

For a step-wise variations ± ~XL, the derivative d p / d t = o t h r o u g h o u t except at the points at which the j u m p s -4-AL take place; a t these points d p / d t has no meaning. It is possible, however, to o b v i a t e this difficulty by surrounding the j u m p s in the (L, t) plane b y the infinitesimal intervals ± ~ parallel to the L axis. Since the trajectories are continuous curves, although not analytic at the points at which the j u m p s occur, one can disregard w h a t h a p p e n s in these intervals. T h e expression (47) is then o~'.,°" = o~02(I q: yl.) - p0'-'(I q: 2 y L ) =

( ~ o -~ -

p 0 ~-) :F v,.(~0'-'

-

2po"-)

=

~-~

:F v ~ ( ~ , - '

-

p02)

(48)

neglecting the term with yz-~. The corresponding Hill-Meissner equation is here (2+oa~ 2 * \ V e will r e t a i n t h e n o t a t i o n we'2 = w0-" - p2, w h e r e p ~ P0.

I =FyL w~z = ~0 2 -

¢012

Po"-.

O = o. In Section

8 we had

(49). we2 --= w(-'.

Here

44

N.

MINORSKY.

[J, F. 1.

Introducing the " a n g u l a r time" r = (01t this equation becomes d2Q ~dr + a12Q = o;

d2Q dr ~ -~- a22Q = o,

(5 O)

where ~2 = I + 3 , L

0012 - - p02

and

(012

~¢-' = I - ~'i,

(012 - - p02 (012

Moreover c~,2 > ~22 equations (5o) have the same form as equation (37) but the values of ~12 and ~22 are different. Using equation (43) and expressing t h a t P1 /> P0 we obtain (012

vL >/ (012-

e u0 - - 1 p 0 2 e "o +

= ~L'

I

(50

in the previous notations. It is easy to show t h a t the second limit 7 L " does not exist here. In fact, expressing the condition t h a t a22 > o one finds vL~<

(012 (012

-

-

po 2

(002 _ p0, (002 2po 2" -

-

On the other h a n d it has been assumed t h a t ~'L << I so t h a t the preceding inequality does not give a n y useful information. Hence in the case of the excitation by the inductance ripple only one condition (51) exists instead of the two as previously. I t is to be noted t h a t in all preceding discussions, it has been a s sumed t h a t the modulation index ~c or ~'L is small which permits writing 1 / ( 5 4 - ~,) ~_ 1 :F % If one waives this condition more terms in the expansion of I/(I + -y), i / ( I 4- ~,)2, etc., must be retained which leads to more complicated expressions. I0. APPROACH TO THE MATHIEU EQUATION.

The preceding graphical m e t h o d does not applly to the Mathieu equation in which case instead of the two elliptic families r , and 1% there exists an infinity of such families varying continuously between these two limiting families. One m a y conceive that the representative moves continuously f r o m one family to the other so t h a t at the limit it has only one point on a curve of each family. In spite of the difficulty of representing such a motion in the phase plane a certain approach to this condition can be m a d e by approximating the continuous variation of the p a r a m e t e r along a sinusoidal curve instead of a rectangular ripple by a step-wise ladder function following the "curve in the m a n n e r shown in Fig. 12. It is clear t h a t if one makes the intervals A~ sufficiently small there will be a discrete sequence of families rl, r2, . . ' , rn of elliptic trajectories on which the representative point will move during each interval A~i passing from a curve of one family, say, Fi to the

July, 1945.]

ON

45

PAP.AMETRIC E X C I T A T I O N .

following F,.+~ by continuity. It is a p p a r e n t also as the size of the intervals A¢ = 2rr/n decreases the motion will approach the motion which is obtained when the p a r a m e t e r varies continuously as in the case of the M a t h i e u equation. If one applies the reasoning of Section 7, one ascertains easily t h a t the equations (28) still hold provided we define the quantities V by the relation a2

X2~:,~-+~ -

c~2~.+1

-

~ -t- 3"0 sin ¢i "~ I - 3"0 cos ¢ , A ¢ I + 3"0 sin (¢.i + ~¢) --

(53)

assuming t h a t 3"0 << I and A¢ very small. Introducing the factor 13 taking into a c c o u n t the relative frequency of the parameter variation and an a r b i t r a r y phase angle ~ the general

J I

}

¢

Figure 12

expression for this coefficient is V¢,~+1 = I - 3'0 cos B(¢i - ~)zX4.

(54)

The expression for the function g,. (compare with equation 28) becomes g,: = ~/I - 3"0 cos B(¢~ - ¢) sin2 ¢~.A¢ and the condition for parametric excitation is i=n

i=n

IXg~ = I I ~ I i~l i=l

-

3"0 c o s ~(~bi -

¢) sin 2 ¢¢A¢ )

~.

(55)

This condition is clearly equivalent to the following I(¢) = ~ l o g ( I

- 3"0 c o s 2 ( ¢ , - ¢ ) s i n e¢iA¢) ) o.

(56)

i=l

Introducing the n o t a t i o n f ( ¢ i ) = cos ~(¢i - ¢)sin 2 ¢, it is ot)vious t h a t ~,02

-

3"0f(¢,)a¢

< log (i -

-y0f(¢D~¢)

<

-

3"of(¢,)~¢ + TY(¢~)zx¢~.

(57)

By letting the index i run through the values 1, 2, . . . , n the sum of

46

N. MINOICSKV.

[J. l:. :[

the left hand terms of (57) clearly converges to the limit .l,(~b) :

- ~0

.f(4~)dO.

(58)

T h e sum of the right hand terms of (57) converges to the same limit i~n A ¢ 2 for n = ~ is zero. since the limit of the series Ei=~)a(¢~) Hence the limit of the sum (56) exists and is given b y equation (58). The condition for parametric excitation for a continuous variation of the p a r a m e t e r is then J(~) =

cos B(4~ - ~) sin 2 4~d4, ~< o.

(59)

As an example consider the previously investigated case (N = 4, ~ = 2). F o r ¢ --- o the value of the integral (59) is - re~2 which means that parametric excitation occurs in this case. For ~, = ~r/2 the value of the integral is + rr/2 which indicates the absence of the parametric excitation. One verifies easily t h a t for ~ = ~'/4, 3~'/4, 5~r/4 and 7~'/4 the value of the integral is zero which shows that the trajectories are closed curves. These values of ¢ are therefore the limits at which parametric excitation appears (or disappears). W e have thus obtained exactly the same situation which has been already discussed b y the Hill-Meissner method. It is to be noted, however, t h a t this discussion has been conducted b y assuming t h a t 3, << I which enabled us to use the simplified expression (53) for the coefficient X~.~+t. B y waiving this restriction calculations are more complicated b u t the qualitative picture of the phenomenon remains substantially the same. The writer is indebted to Prof. S. Lefschetz and Mr. M. L e v e n s o n of the T a y l o r Model Basin staff for valuable discussions of this matter. BIBLIOGRAPHY.

(1) LORD RAYLEIGH, "On Maintained Vibrations," Phil. Mag., 5th series, April I883, pp.

229-235. (2) MELDE, Pogg. Ann., Io9, 5, I859. (3) M. BRILLOUIN, "Th~orie d'un alternateur auto-excltateur," Eclairage Electrique, Vol. XI, April 1.897, pp. 49-59. (4) H. POINCAR~, "Sur quelques th~or~mes g~n~raux de l'~lectrotechniclue,' ' E c l . El., Vol. L, March I9O7, pp. 293-3oi. (5) L. MANDELSTAMAND N. PAPALEXI, "Expos~ des recherches r~centes, etc.," Journal Tech. Phys., U.S.S.R., I934. A complete bibliography on parametric excitation is appended to this reference. (6) See reference (5), PP. I23 to I27. (7) E. T. WHITTAKERAND G. N. WATSON', "Modern Analysis," Chapter XIX, Cambridge U. Press, I927. (8) E. MEISSNER, "Ueber Schfittelerscheinungen, etc.," Sehwelz. Bauzeitung, Vol. LXXII, I918, pp. 95-98. (9) B. VAN DER POE AND M. S. O. STRUTT,"On the Stable Solutions of Mathieu's Equation," Phil. Mag., 7th series, Jan. I928 , pp. 18-38. (IO) M. S. O. STRUTT, "Lam~'sche, Mathieu'sche nnd verwandte Funktionen," Springer, Berlin, 1932.