Gorelik - Resonance Phenomena in Linear Systems with Periodic Parameters - 1935.pdf

Gorelik - Resonance Phenomena in Linear Systems with Periodic Parameters - 1935.pdf

135

Original Title: Phenomenes De Resonance Dans Les Systems Lineaires A Parameteres Periodiques

Published In: Technical Physics of the USSR, Leningrad; Volume 2, Number 2-3, pp. 135-180, 1935

Translated, Transcribed and Edited by

:

Alissa Pesavento, Peter Pesavento, and Philip Pesavento

RESONANCE PHENOMENA IN LINEAR SYSTEMS WITH

PERIODIC PARAMETERS

By G. Gorelik

INTRODUCTION

Section 1: Object of this work

In all the branches of science and technology where one deals with oscillations, resonance phenomena are the most important. Let us briefly examine their characteristics.

The action exerted on an oscillatory system by a time-varying force of given intensity, which we shall write as f(t), is, under certain conditions, extremely sensitive to small variations of the form of the function f and the value of certain parameters describing the oscillatory system; sometimes it is enough to very slightly vary one of these parameters in the form of f in order to change the magnitude of the oscillation signals.

In the past we have occupied ourselves with a very particular case of resonance phenomena: the classic case where the system, driven by the force f, can be described by a linear differential equation with constant coefficients. If this equation has two degrees of freedom, and if the system is slightly dampened, it is a “harmonic resonator”. It physically divides the force f into two components; one is of the type

A

cos

ω

0

t

+

B

sin

ω

0

t

, and the other is orthogonal to the functions cos ω

0 t, sin ω

0 t* (A, B are arbitrary constants, ω

0

is the resonator’s natural frequency). If the constants A and B are not null

(so the force isn’t orthogonal to at least one of the functions cos ω

0 t, sin ω

0 t) then there is resonance; the primary component of f excites an intense oscillation, whereas the effect on the secondary component is negligible; the resonator “rejects” it and, to use Professor

L. Mandelstam’s phrase,

1

it has an “existence independent” from the sinusoidal force A

cos ω

0

t + B sin ω

0 t.

For some time, physicists and engineers have become more and more interested in resonance phenomena that are present in certain systems where the properties differ essentially from those of linear systems with constant parameters. They are: 1) nonlinear systems; 2) linear systems in which the parameters are time-varying functions of time

(arbitrary periodic functions, for e xample).

*

Throughout this article the function f is assumed to be periodic or quasi-periodic. The notion of orthogonality is defined in Section 3.

136

The goal of this work is to give a general theory of the effects of resonance which takes place in linear systems with periodic parameters possessing a degree of freedom or can be considered as a superposition of systems with a degree of freedom (see Section 2). We will make use of the term periodic resonator to concisely describe a linear system with periodic parameters possessing one degree of freedom and which are weakly dampened.

As examples of systems which, under certain conditions — of which the first is the stability of the state of equilibrium existing when f = 0 – can be considered, at least approximately, as a periodic resonator or some groupings of periodic resonators, we cite:

In radio-technology: a) Super-regenerative receivers, where the effective resistance of the oscillating circuit varies with a smaller frequency in relation to is “natural frequency”; b) The circuits of “parametric regeneration” studied at the Central Radio Laboratory in Leningrad.

2

The resistance, the capacity, or the effective self-induction of

these circuits is modulated by a related frequency that is double the “natural mean frequency”.

In electronics: c) The “parametric machine” of L. Mandelstam and N. Papalexi

3

. It contains an

oscillatory circuit in which one periodically varies the capacity or self-induction by a mechanical process; the problem which interests us here arises when one introduces an alternating EMF in this circuit.

In electro-technology: d) Oscillating electronics in certain inhomogeneous electronic fields that can exist inside vacuum tubes.

4

In acoustics: e) All the analogue devices with Melde wires.

5

This type of device, proposed by L.

Mandelstam, has been studied in our laboratory by H. Sekerska: It is a vibrating wire traversed by an alternating current which is made to vary its temperature periodically and, consequently, its tension; the problem of forced oscillations comes up when the wire is placed in a magnetic field.

In molecular optics: f) The electrons making up part of an oscillatory molecule which, via the action of a light wave, produces some phenomena of combinatory diffusion (Raman’s effect); if one uses some classical notions one can consider these electrons like some vibrators in which the tension varies with the rate of the molecule’s oscillations.

1,6

In industrial mechanics: g) The connecting rods of electric locomotives, in which the effective tension varies periodically, and, furthermore, which are subjected to the action of exterior periodic forces.

7

137

The generalized theory of linear resonance should not just give an account of phenomena that present themselves in some systems, such as those that have just been enumerated.

The theory should also contribute equally to expand the same notion of resonance and certain ideas in physics that are its tributaries; notably it should provide a clearer concept of the physical role of sinusoidal functions of time.

Throughout physics and technology these very special types of time functions possess an exceptional importance. The fundamental notions of monochromatic light and basic color are connected to these types of functions. In a great number of cases one finds it useful to represent non-sinusoidal oscillatory phenomena by summations of sinusoidal oscillations.

Where does the exceptional role of the sin and cos functions come from? Nothing could be clearer than to state, as one might sometimes say, that these functions are privileged because they describe the simplest oscillations. The particular role of sinusoidal functions in physics, and the fact that they appear often as the quality of “simple oscillations”, are closely related to the properties of certain systems to which we have been accustomed in physics and technology, and of primary importance in some of the most current instruments available for analyzing oscillatory phenomena: harmonic resonators.* These resonators, as we recall, filter the non-sinusoidal force f(t) to extract certain sinusoidal components from it.

This property disappears if one substitutes a periodic resonator for the harmonic resonator. Some other functions — they will be specified during the course of this report

— come to supplant the sine and cosine in their role of privileged functions and simple oscillations. A linear resonator with periodic parameters naturally attenuates the force f(t), even if it is sinusoidal, and makes obvious its certain functions as determined by its intrinsic properties.

Hence, from a general point of view, there are at least as many types of “simple oscillations” as there are types of periodic resonators. The sinusoidal “simple oscillations” are only a limited case, which corresponds to resonators with constant parameters.

The general ideas that guide this work come from the thoughts of Professor L.

Mandelstam and from his report on them given in 1931 at the Conference for the Study of

Oscillations.

1

The action of an external force on a system with periodic parameters formed the subject of the research covered by this Conference.

In conclusion, let me express my profound gratitude to Professor L. Mandelstam for his valuable advice and to Professor N. Papalexi for the interest he has give to this work.

*The prism and the diffraction grid system are, in a certain sense, equivalents of harmonic resonators.

138

Section 2: Mathematical Formulation of the Problem.

The differential equation of the phenomena that interests us is

y

&&

+

2

δ

y

&

+

2

σ

(

t

)

+

ρ

(

t

)

y

=

f

(

t

) (1) the functions

σ(t) and ρ(t) being periodic and f(t) being periodic or quasi-periodic; δ is a constant;

σ has a mean null value; ρ takes only positive values; the dots replace the operator d/dt. We will suppose that

σ and ρ have the same period of T seconds, and that t =

τπ/T, τ being the time measured in seconds. The quantities of t, δ, σ, ρ then do not have dimension;

σ and ρ have period π in the parameter t.

Equation (1) describes an electric or mechanical system with one degree of freedom. y is the charge of the condenser or the displacement. The coefficient of the self-induction or inertia is constant; the ohmic resistance of the friction coefficient varies in proportion to

2[

δ + σ(t)], and the reciprocal of the capacity or tension is in ρ(t); the function f is proportional to the exterior electromotive or mechanical force.

In the case of variable self-induction, the equation of forced oscillations is

d dt

L

&

+

R

(

t

)

+

C y

(

t

)

=

F

(

t

)

(R—resistance, C—capacity, L—self-induction, F—applied EMF) or

L

(

t

)

+

R

(

t

)

+

dL dt y

&

+

C

(

y t

)

=

F

(

t

)

L(t) being a positive function. Dividing by L(t), one returns to the form of (1).

The study of a linear system having several or even an infinite number of degrees of freedom, where the parameters vary periodically and which allows normal coordinates, reduces to a series of finite or infinite equations of the type (1). We take, for example, a p(t). Let us submit it to the action of a force of linear density

Φ

(

x

)

F

(

t

) ,

x

being the distance between a reference point and one of the wire’s extremities. The transversal displacement, u, satisfies the equation

p

(

t

)

2

x u

2

=

q

2

t

2

u

− Φ

(

x

)

F

(

t

)

+

r

u

t

(q—density, r— coefficient of friction). Take as boundary conditions the limits: u = 0, for x = 0, and x = 1. The transformation

u

=

s

=

1

y s

sin

S

π

l x

139 gives an equation resembling (1) for each of the normal coordinates y

2

. In order to shorten the equation (1), we write it in the form

Ly

+

2

δ

y

=

f

(1a) y

1

, y

2

being two independent solutions of the homogenous equation

L y

&

+

2

δ

&

=

0 . (2)

The solution of (1) which corresponds to the initial conditions t = t

0

y

.

, y = 0, = 0 can be expressed as:

y

= −

y

1

(

t

)

t t

0

f

(

τ

)

D

(

y

τ

2

)

(

τ

)

d

τ

+

y

2

(

t

)

t t

0

f

(

τ

)

D

(

y

1

τ

)

(

τ

)

d

τ

(3) or

D

=

y

1 2

y

2

y

&

1

(4)

The character of the forced oscillation y depends on one of the free oscillations y

1

, y

2

Now, according to the choice of

δ, σ, ρ these possess some very different properties.

.*

That’s because there are advantages to distinguish several cases and to study them separately.

We know that the theory of differential equations with periodic coefficients allows us to distinguish the following three cases (see for example

8,9

)

1) The system described by the equation

Ly

=

0 (5)

In the “ideal” system there exists, in the absence of an external force, our “real” resonator when

δ

0 ; it becomes a “stable region” where equation (5) allows only some quasi-periodic solutions, or eventually, periodic solutions.

2) The “ideal” system Ly = 0 is found at the limit of instability. Equation (5) then possesses a periodic solution of period

π or 2π, y = u, and a solution of the form

γ

u t

+

v

, where

γ is a constant and v is a periodic function of period π or 2π.

3) The “ideal” system is found in an unstable region. Equation (5) possesses an increasing solution e

kt

ϕ

(t ) and a damped solution

e

kt

ψ

(t ) ; (k is a positive constant, and ϕ ψ

π or 2π).

Our report is divided into three parts which treat, respectively, cases 1), 2), and 3).

*Expression (3), which is obtained by the method of variation of parameters, can be interpreted as follows: The oscillation y is the sum of the free oscillations produced by all the impulses f(

τ)d(τ) received by the system up to the instant τ = t.

5

140

Part One.

The system Ly=0 is stable.

Section 3:

Intrinsic properties of the Resonator.

If the ideal system described by the equation

Ly

=

0 (1) is found in a stable region, the general solution of (1) is

y

=

Au

+

Bv

A and B being some arbitrary constants, and the particular solutions u, v being some real quasi-periodic functions (they can in certain cases degenerate into periodic functions).

The term Hill functions, introduced by Strutt,

9

can be used to designate u, v and their linear combinations.

We agree to say that a quasi-periodic (or periodic) function f is normalized if its mean square (ff) is equal to:

(

ff

)

=

T

lim

→ ∞

1

T

+

2

T

T

2

ffdt

=

1 and that both quasi-periodic functions f, g are orthogonal if their product has a null mean value (fg):

(

fg

)

=

T

lim

→ ∞

1

T

+

T

2

T

2

fgdt

=

0 .

Since the derivative of a quasi-periodic function has a null mean value, it follows that every quasi-periodic function f is orthogonal to its derivative:

(

f f

&

)

=

T

lim

→ ∞

1

T

+

T

2

2

T

1

2

d dt f

2

dt

=

0 .

One can always construct (by an infinite number of ways) two particular solutions u, v normalized and orthogonal; this is because we can allow that

(

uu

)

=

1 , (

uv

)

=

0 , (

vv

)

=

1 . (2)

Furthermore

(

u u

&

)

=

0 , (

v v

&

)

=

0 . (3)

We supposed that

σ is identically null. Following Liouville’s theorem

u v

& &

=

2

ω

141

ω being a constant different from zero; we make the mean of the two members of this equation:

(

u v

& ) (

v

& )

=

2

ω

(4)

Making the mean of the two members of the identity

u v

&

=

d dt

one obtains

(

u v

&

)

It follows from (4) and (5) that when

0

(

u v

σ

, Liouville’s theorem gives

&

)

=

(

v

ω

,

&

(

)

v

=

u

&

)

0

=

uv

(5)

ω

(6)

2

ω

S

(

t

)

1

(7)

u v

&

v u

& where

ω is a constant, different from zero and

=

S

(

t

)

=

e

2

0

σ

(

x

)

dx

(8)

S(t) is a periodic function of the period

π.

If L is an even operator, that’s to say, it doesn’t change when one replaces t by –t, one can choose u, v in such a way that u for example can be even and v odd.

The general theory indicates that, in its stable regions, equation (1) allows some solutions of the form

y

=

e iht

ϕ

(

t

),

y

=

e

iht

ϕ

(

t

) (9)

h

being a real constant and ϕ

(t ) a function of the period

π

= −

1 ; the bar indicating the complex conjugate. These solutions are naturally linear combinations of u and v and vice versa. If u is even and v odd, the trigonometric series of u, v have the form

u

=

a n

cos(

h

+

2

n

)

t

,

v

=

a n

sin(

h

+

2

n

)

t

. (10)

The frequency spectrum of the sinusoidal components of u and v form a series of equidistant lines; their distance is equal to the modulation frequency (fig. 1).

It is convenient to use the term natural oscillations for the solutions of (1) and to call the solutions of the equation

Ly

2

δ

&

=

0 (11)

“free oscillations”. We here apply to this the transformation

y

=

e

− δ

t Y

(12)

142

We obtain for Y the equation

LY

2

δσ

Y

δ

2

Y

=

0 (13)

If the system Ly = 0 is sufficiently far from the unstable regions and

δ sufficiently small,

(13) allows some stable solutions of the form

Figure 1: Schematic of the frequency spectrum of the sinusoidal components of u,v

Y

=

e iHt

Φ

(

t

),

Y

=

e

iHt

Φ

(

t

) (14)

Φ periodic function of the period

π.

In the case where

σ is identically null

H and

Φ distinguish themselves from

h

, ϕ

by some quantities of the order of magnitude

δ

2

:

H

=

h

+

δ

2

h

Φ = ϕ

+

δ 2 ϕ ′

(15a)

In the contrary case

(

σ

0

)

, the differences between H and ,

Φ and ϕ

are of the order of the magnitude

δ:

H

Φ

=

=

h

ϕ

+

+

δ

h

δ ϕ ′

(15b)

The formulas (12), (14) show that the free oscillations are damped*; the law of damping is the same as in the harmonic oscillator that one would obtain by making

σ = 0 and ρ = constant. From (15a), (15b) friction causes a spectral shift on the order of magnitude

δ

2 or

δ — according to whether σ = 0 or

σ

0 — of some frequencies of the quasiperiodic factor Y of the free oscillation relative to those of natural oscillations. A method of calculating some of the corrections

h

′, ϕ ′

is indicated in

10

.

Section 4: Resonance.

1. The equation of forced oscillation in a periodic resonator is

Ly

+

2

δ

y

=

f

. (1)

We begin by studying a particular case. Let

Ly

=

f

=

P, Q, being arbitrary constants. By virtue of the identities

P u

&

P u

& +

+

Q v

&

Q v

&

2

(2)

δ

y

&

Lu

0 ,

Lv

0 the equation

(1a) evidently allows the solution

y

=

Pu

+

Qv

. (3)

2

δ

*We suppose throughout that

δ > 0.

143

All of the solutions of the homogeneous equation

Ly

+

2

δ

y

&

=

0 being damped, the motion (3) is the forced stationary (steady state) oscillation in which the initial conditions become established. This motion is identical to one of the natural oscillations. At each instant, the applied force f and the force of friction 2

δ

y& mutually cancel each other. If

δ is very small, a small force of the form (2) is enough to excite a very intense oscillation. In a more mathematical form: the forced stationary oscillation produced by a force

δ

(

P u

& +

Q v

&

)

remains finite when

δ tends towards zero.

Let us go to another particular case. We suppose that that in ( 2

1 ) *

σ is identically null and that the applied force is represented by a function g(t) orthogonal to u and v:

Ly

+

2

δ

y

&

=

g

, (

gu

)

=

0 , (

gv

)

=

0 (4)

The stationary oscillation due to the force g stays limited when

δ tends towards zero. Or, returning to the same thing, the stationary (steady state) oscillation due to a force

δg cancels itself with

δ. In effect, in availing ourselves to (2, 3) and arbitrary solutions of

(3,11) given by (3, 12) – (3, 15a), we obtain the expression

z

= − Ι + Ι for the forced stationary (steady state) oscillation due to the force g, where

I

=

e

δ

t

Y

(

t

)

+ ∞

− ∞

e

(

δ

i

δ 2

h

)

τ

{

[

g

D

0

+

δ 2 ϕ

D

]

+

δ

2

g

ϕ ′

}

d

τ

(5)

d

the constant

D

0

+

δ 2

D

being equal to

Y

(gu) = 0 and (gv) = 0, g is orthogonal to

d

Y dt e iht

ϕ

Y dt

Y

. By virtue of the conditions

, the linear combination of u, v. Thus, the expressions between brackets have a null mean value. The integration doesn’t introduce small divisors (of the order of

δ) and z stays finite when δ = 0.

One can show in the same manner that if

σ

0 , a force g such as

(

gSu

)

=

0 , (

gSv

)

=

0 produces a forced oscillation which stays limited when

δ

0 . Thus, in physics parlance, the oscillation excited by a small force orthogonal to Su, Sv — or if

σ

0 , to u, v — remains weak, if the mean friction is small — or, if

σ

0 , the friction completely disappears.

We can now show the action of an ordinary force f. One can write

f

=

P u

.

+

Q

.

v

+

g

*(m, n) designates formula (n) of section m.

Technical Physics, V. II, No. 2-3.

144 and choose the coefficients P, Q in such a way that the function g is orthogonal to Su, Sv, or to u, v if

σ is identically zero. For that, it is necessary and sufficient, if

σ

0 , to make or if

0

P

= −

(

fv

)

ω

,

Q

=

(

fu

ω

)

(7a)

σ

, and, in order to simplify L, to assume u is even and v is odd*,

P

=

(

u

&

fSv

)

( )

,

We superimpose the forced oscillation due to

Q

=

(

fSu

)

. (7b)

(

v

&

Su

)

P u

& +

Q v

&

and g; the stationary oscillation excited by the force f is

y

=

Pu

2

+

δ

Qv

+

z

0

(8) z being a quasi-periodic function which remains limited when

δ = 0.

If

δ is sufficiently small, the forced oscillation will have an amplitude which is an order of magnitude different, for the same intensity of a force f, according to whether P and Q are null or that one of the least of these coefficients is different from zero. If

Q

0 , or

PQ

0

P

0 , or

, the forced oscillation is intense. If P = Q = 0, it is weak. One can say that the first of these cases is a case of resonance.

It is desirable to construct a more precise definition of resonance. We consider the stationary oscillation excited by a force

δf:

y

*

=

Pu

+

Qu

+

δ

z

(8*)

We will say that the force f produces the phenomenon of resonance if the forced stationary (steady state) oscillation due to the force

δf remains finite when δ tends towards zero. For it to have resonance, it is necessary and sufficient, according to (7), that the force f may not be orthogonal

σ

0 , to a solution of the equation Ly = 0.

S.

In other words, for resonance it is necessary and sufficient that the force f contains —in the sense indicated by (6) and (7)— a component proportional to the speed of one of its natural oscillations. The resonator “magnifies” this component

P u

& +

Q v

&

and “attenuates” the component g: it gives to the term

P u

& +

Q v

& an “independent physical existence.” If

δ component

P

& +

Q v

&

—so that it is not damped by the resonator— it should be of the type say that the functions

P u

& +

Q v

&

are simple oscillations for our resonator.

*In this case S is even. Therefore

( )

=

0 ,

( )

=

0 .

. One can

145

The formulas (8), (8*) show all the more that, when

δ is sufficiently small, the oscillogram of y (charge on a condenser, displacement) is one of the solutions of Ly = 0; thus, it replicates the form of a Hill function.

2. One can measure a resonator’s sensitivity to a normalized force, f, by the mean square (yy) of the forced oscillation that it excites. In cases of resonance the latter is practically equal to

y

=

(

fv

) if

δ is sufficiently small.*

By virtue of (3, 2) the sensitivity relative to f is

2

u

+

ωδ

(

fu

)

v

S

[

f

]

=

(

fu

)

2

4

ω

+

2

δ

(

2

fv

)

2

.

We seek the forces to which the system is most sensitive. We write

f

=

(

fu

)

u

+

(

fv

)

v

.

Multiplying the mean of the two members of this equation by u, v we see that

( ϕ

u

) ( ϕ

)

=

0 .

Thus

(

ff

)

=

1

=

(

fu

)

2

+

(

fv

)

2

+

( ϕϕ

),

Being given that

( )

0

S

[

f

]

=

1

4

ω

(

2 ϕϕ

δ 2

)

, the sensitivity attains an absolute maximum when ϕ

=

0 , that is to say, when f is a solution of Ly = 0.

3. Let us write

e

(

t

)

=

ω 2

0

+

qm

(

t

) where

ω

2

0 and q are arbitrary positive constants and m(t) is a periodic function of null mean value. In the particular case of

σ = 0, q = 0 the functions

u

,

v

,

u

&,

v

& become respectively 2 cos

ω

0

t

, 2 sin

ω

0

t

,

ω

0

2 sin

ω

0

t

,

ω

0

2 cos

ω

0

t

.

Introducing these functions into the preceding statements and formulas, one falls back on the well-known properties of the harmonic resonator: It puts in evidence, in an ordinary force, a sinusoidal “simple oscillation” of frequency

ω

0 itself with the help of the functions cos

ω

0

; the condition of resonance manifests t and sin

ω

0 t. But, in the general case the sinusoidal functions do not play any privileged role and are not simple oscillations. An adequate mathematical device [tool] with resonators with periodic parameters should make use of not the cos and sin, but the functions u ,

v

,

u

&,

v

& of the considered resonator.

In our report, the properties of the harmonic resonator appear as a limited case as those of periodic resonators. When one follows

*We suppose in this case that

δ = 0.

5*

146 the opposite course, generalizing the statements about the harmonic resonator so as to render them applicable to systems with periodic parameters, one encounters a minor difficulty. When

q

0 the functions

u

,

v

,

u

&

,

v

&

,

Sudt

,

Svdt

etc. combine in groups to give only two independent functions: cos

ω

0

t and sin

ω

0 t. That’s because if we want to substitute, for the example, the statement: “f = cos

ω

0

t gives the forced oscillation y = sin

ω

0 t/2

δω

0

“ by a statement applying itself to all the resonators which interest us here, we don’t know a priori if it is necessary to say “a force u gives a forced oscillation proportional to v/2

δ” or “f = u gives a forced oscillation proportional to Sudt/2δ, or f = gives y = u/2

δ” etc. The third of these propositions, which possesses an immediate

v& physical sense (the applied force and the force of friction mutually compensating each other) is exact. The functions cos

ω

0

t, sin

ω

0 t ensue in a multitude of applications in the theory of harmonic resonators. When one goes to the generalized theory, one must substitute for them, in each of their applications (simple oscillation, the maximum given force of sensitivity, etc.), by a different function.

4. We examine by virtue of example the action of a sinusoidal force f = cos pt on a linear system with periodic parameters.

Suppose for simplicity that S and u are even and v odd. In that case (fSv) = 0 whatever p may be. For there to be resonance, it is necessary and sufficient to satisfy the condition:

(

uS

cos

pt

)

0 (9)

This condition being satisfied, the resonator decomposes the force f so as to extract a proportional to v.

We translate these results from the adequate language that we have used up to the present into the usual “sinusoidal” language. The non-orthogonality of cos pt towards Su signifies that the frequency p is equal to one of its frequencies

h

±

2

n

(n = 0, 1, 2) which figure in the spectrum of Su’s sinusoidal components. We recall the condition of resonance that G. Hintz and I discovered theoretically (by a more primitive process) and

verified experimentally, several years ago, for the super-regenerative receiver.

11

*

From the sinusoidal viewpoint, the harmonic force cos pt excites a spectrum, composed of the frequencies p 2

n

(n = 0, 1, 2…), in the resonator.

When one tries to vary the frequency p of the applied force, without altering the resonator, the latter goes through a series of resonances. They are especially noticeable, experimentally, since the value of (Su cos pt) is considerable. If

σ

0 , and if,

*It is wrong to believe that M.G. Hassler was the first to have explained this condition of resonance (V. Hochfrequentztechnik u. Elektroaskustik, Sept. 1934 [from High

Frequency Technology and Electroacoustics, Sept. 1934). I reserve the right to return elsewhere to the discussion of M.G. Hassler’s work.

147 furthermore, u is even and v odd, the resonance’s intensity corresponds to p – h = 2n and is proportional to

a

2

n

[see (3, 10)]; Figure 1 can serve as a diagram of the relative altitude of the various maxima of intensity.*

If

δ is small different frequencies excite some sinusoidal forces of the same form as forced oscillations, provided that they fulfill the condition of resonance. The forces cos ht, cos (h + 2)t, cos (h + 4)t etc…. make the same spectrum appear as those of the function v (in order to speak of developed spectra). This observation can be useful for the classical interpretation of certain particularities that are present in cases of resonance in the combinatory diffusion of light (Raman’s effect).

We remark, in concluding this discussion, that if two sinusoidal forces

a

1 cos

p

1

t

and

a

2 cos

p

2

t

, which, taken separately, satisfy the condition of resonance (9), their sum does not always give a phenomenon of resonance; in effect it can happen that

a

1

(

uS

cos

p

1

t

)

+

a

2

(

uS

cos

p

2

t

)

=

0

5. We have examined only stationary oscillations. The solutions of the homogeneous equation

Ly

+

2

δ

y

&

=

0 being known (see Section 3), the study of transient phenomena presents no difficulty at all. In cases of resonance the “amplitude” of the forced oscillations of a system, which can be found prior to its coming to rest (

y

=

0 ,

y

&

=

0 ) , begins increasing proportionally to t as in harmonic resonators**.

Section 5: Resonance Curves.

The definition given above, based on proceeding to the limit

δ = 0, separates resonance from nonresonance in an absolute way. We now turn to cases that are of physical interest, such as when the decrement,

δ, although very small, is still nonzero and exists as a “Soft” transition between resonance and non-resonance***. It is this transition that we are now going to study.

*I have previously published a calculation of the resonance intensity of a super-regenerative receiver under

the action of a sinusoidal EMF, and the form of its oscillations

12

. I take some nonlinear terms into

consideration in this calculation, and I indicate under what circumstances they can be omitted. — The same article indicates the existence of “multiple resonances” in a frequency-modulated system subjected to the action of a sinusoidal EMF.

** If

σ

0

, the equation

Ly

=

Pu

+

Q u

&

allows the exact solution y = ½ (P u + Q v) t, which describes the start of a transitory phenomenon.

*** Let us suppose, for example, a harmonic resonator of natural frequency

ω

0

, on which the force

δ cos (ω

0

+

ε)t acts. If ε = 0 the forced oscillation is y = sin ω

0 t/2

δ for whatever δ is. But if

ε

0

, the forced stationary (steady state) oscillation tends towards zero with

δ, however small

ε

is. We suppose now that

δ is very small but fixed. If

ε

is very small, the forced oscillation y differs very little from what it is when

ε = 0.

148

We begin by restricting the problem. We will suppose that after having chosen the force f so as to fulfill the condition of resonance, we will gently alter the resonator, but not f.

In other words, we will examine what happens when one de-tunes the resonator.

There is only one way to de-tune a harmonic resonator, and that is by varying its natural frequency

ω

0

.* But, if one wishes to de-tune a resonator with periodic parameters, one has the embarrassment of choice between an infinity of different procedures.

The collection of all the harmonic resonators can be represented by that of a semi-right triangle’s points, their abscissa being equal to

ω

0

2

. But to “geometrically” represent the collection of periodic resonators one must resort to the set of points of a functional space.

In effect, all systems described by an equation of the form

Ly

+

ξ

(

t

)

y

+

2

δ

=

f

(1)

[ (t )

ξ being any periodic function of period

π] can be considered as a de-tuned periodic resonator relative to the initial system

Ly

+

2

δ

y

&

=

f

.

The set of “de-tuning procedures” for a periodic resonator are the directions of the space of functions

ξ

.

The strength of the harmonic resonator’s oscillation is a function of

ω

0

; its graphic representation is the curve of resonance. If one considers the set of all the “virtual detunings” of a resonator with periodic parameters, the intensity (yy) is a function of

ξ

(t ) and one would not know how to represent it by a resonance curve in the usual sense of the term.

Giving equation (1) the form

Ly

=

f

ξ

(

t

)

y

2

δ

(1a) the total perturbing force which impedes the resonator’s movement from being one of the proper oscillations of the system, Ly = 0, is the sum of the applied force f, the “force of de-tuning” —

ξ

( t)

y

, and the friction — 2

δ

y& . If there exists a natural oscillation y* of the system Ly = 0, such that the perturbing force

f

ξ

(

t

)

y

*

2

δ

y

* exerts a negligible effect, then the de-tuned system’s oscillation will differ little from y*. Now, one easily sees that a natural oscillation of Ly = 0, possessing the requisite property, always exists when

δ and

ξ

(t ) are small. In effect, we write:

f

=

ξ

(

t

)(

P

u

+

Q

v

)

+

2

δ

(

P

u

+

Q

v

)

+

g

(2)

*We make a distinction between the case where one varies the capacity and where one modifies the self-induction.

149

The coefficients P’, Q’ can be chosen in such a way that the function g is orthogonal to

Su, Sv. We suppose that

σ

0 , S = 1: The generalization of the results that follow for the case where

σ isn’t identically null presents no difficulty.

We multiply (2) successively by u, v and form the mean of the two members, in posing the conditions

(

ug

)

=

0 , (

vg

)

=

0

By introducing the notation

ξ

uu

=

(

u

ξ

u

),

ξ

uv

=

(

u

ξ

v

)

= ξ

vu

,

ξ

vv

=

(

v

ξ

v

) (3) and in applying the formulas (3, 2) (3, 3) (3, 6) (4, 7a), we obtain the equations

ω

Q

ω

P

=

=

P

P

ξ

(

uu

ξ

+

uv

Q

(

2

ξ

uv

δω

)

+

+

2

δω

Q

ξ

)

vv

They give



P

=

ω

Q

ξ

vv

+

ω

P

(

ξ

uv

+

=

2

The stationary solution of (1) is

δω

ξ

uu

)

ξ

y

,

vv

=

Q

ξ

P

u

2

uv

=

+

+

4

ω

P

δ

Q

v

2

+

ξ

ω 2

z uu

ω

Q

(

ξ

uv

2

δω

) 



(4)

(5)

P’, Q’ being defined by (4) and z satisfying the equation

Lz

+

ξ

(

t

)

z

+

2

δ

z

&

=

g

A rational parallel to the ones in Section 4 demonstrates that if

ξ and δ are sufficiently small* then z is negligible in relation to P’u + Q’v. Therefore, approximately

y

=

ω

{[

Q

ξ

vv

+

P

(

ξ

uv

+

2

δω

)]

u

[

P

ξ

uu

+

Q

(

ξ

uv

2

δω

]

v

} (6)

(

yy

)

=

ω 2

2

{[

Q

ξ

vv

+

P

(

ξ

uv

+

2

δω

)]

2

+

[

P

ξ

uu

+

Q

(

ξ

uv

2

δω

)]

2

} (7)

In these formulas y and (yy) are the explicit functions of

ξ. In certain interesting physical cases in which

ξ depends on only one parameter, (yy) is then a function of this parameter, and one returns to the usual notion of a resonance curve**.

*And if, as we have admitted since the beginning, there is a natural resonance when

ξ = 0.

**If

ξ (t) depends on two parameters one can construct a “resonance surface.”

150

We take two examples. a) When f = 0 and

δ = 0, one can vary the modulation rate of a system described by

Mathieu’s equation:

y

&&

+

[

λ

+

(

q

+

q

1

) cos 2

t

]

y

+

2

δ

y

&

=

f

(

t

)

In this case

ξ(t) = q

1

cos 2t. One easily shows that u being even and v odd

ξ

uu

=

ξ

vv

=

q

1

.

k

,

ξ

uv

=

0

(8) k being a constant. Therefore

(

yy

)

=

4

δ

P

2

2

+

+

k

Q

2

ω

2

q

2

2

1

(9)

The intensity is maxima when q

1

= 0*. b) One modulates the ‘average stiffness [capacitance]” of a resonator, whose applied force is the “simple oscillation” Pu:

Ly

+

ξ

y

+

2

δ

y

& =

P u

&

(

ξ is a parameter independent of t). We then get

ξ

uu

= ξ

vv

=

ξ

,

ξ

uv

=

0

y

=

ω

P

( 2

δ

4

δ

ω

u

2

ω

2

+

ξ

ξ

v

)

2

(10)

(

yy

)

=

4

δ 2

P

+

2

ξ

ω

2

2

(11)

The de-tuning makes a component appear in v that is a phenomenon analogous to the

phase-shifting in a de-tuned harmonic resonator.

The resonance curve (yy) = F(

ξ) is similar to that of a harmonic resonator, except that formula (10) is exact if and only if

ρ

=

ω

0

2

=

const

.

, and is only approximate if

ρ

=

ρ

(t ) **.

We remark that

ξ being an ordinary constant, the equation

Ly

+

ξ

y

=

Pu

+

Qv

allows the exact solution

y

=

Pu

+

Qv

ξ

.

The oscillogram of the development is similar to that of the force.

*This result can be generalized. It is exact only if the characteristic exponent h is not an integer

10

.

**The decay (2) is similar to that of a harmonic oscillation in two harmonic oscillations of the same frequency of which the phases differ from

π/2 and of which the amplitudes are in a given relation. But, whereas this final operation occurs exactly, the decay (2) of

P u

& +

Q v

&

generally necessitates the correction g.

151

Part Two.

The system Ly = 0 occurs at the limit of instability.

Section 6: Natural oscillations and free oscillations.

In this second part we will study resonance phenomena that occur when an “ideal” system (that one could obtain by eliminating friction in the resonator itself) is found near a limit of instability. Furthermore, we will suppose that (see, conditions (A)), for a given friction, the modulation is sufficiently strong.

We will begin the study of forced oscillation through the case where the ideal system is situated exactly on one of the boundaries separating the regions of stability from the regions of instability (Section 7). But, naturally, in making the friction of the actual system tend towards zero we can not physically obtain a system situated exactly on one of these mathematical lines. That is because afterwards we will have to take into consideration some corrections that produce a gentle de-tuning, which is sufficiently small so that the actual system is stable (Section 8); one knows that, on the contrary, the theory of stationary phenomena can only be a nonlinear theory.

When the system described by the equation

Ly

=

0 (1) is situated on one of the boundaries of instability, the equation possesses a normalized periodic solution, of period

π or 2π,

y u

and an increasing solution

y

=

γ

ut

+

v

(3) with

γ being a constant, and v a normalized periodic function (of the period π or 2 π) that is linearly independent of u.

In order to simplify the theory a bit, we will henceforth suppose that L is an even operator and that

σ is identically null. In this case equation (1) can be expressed as

y

&&

+

[

λ

+

qm

(

t

)]

y

=

0 (1a) where q and

λ are some constants, and which possesses an even solution and an odd solution. One easily sees that either u is even and v odd, or u is odd and v even.

(

uu

)

=

1 , (

uv

)

=

0 , (

vv

)

=

1 (4)

Furthermore one has

(

u u

&

)

=

0 , (

v v

&

)

=

0 (5)

152 and, by forming the mean of the two members, one obtains the identity

(

u u v

&

v

&

)

=

=

(

v u

&

d dt

)

uv

=

ω

(6)

In all the cases that interest us

ω

0 .

This results in the identity

L

(

γ

u t

+ v

)

0 that

Lv

=

L

γ

ut

=

γ

tLu

+

2

γ

u

&

=

2

γ

u

& (7) v/

γ is then a periodic solution of the equation

Ly

= −

2

u

&

.

If one knows u, for example in the form of a series ordered according to the powers of q, one can make use of this equation to obtain v/

γ (in the form of a series ordered according to the powers of q) by the usual methods studied in periodic solutions. One will then obtain

γ by applying the condition (vv) = 1. Note 1 gives u, v, γ for the boundaries of the primary region of instability of Mathieu’s equation.

Once u, v,

γ are known, the methods of calculating perturbations allow us to approximately solve the equation

Ly

+

ξ

y

+

2

δ

y

&

=

0 (8) for some sufficiently small values of

ξ and δ. Τhat is to say, to determine the free oscillations of a system that distinguishes itself from (1) by a small friction and a gentle de-tuning*. For the correction due to the de-tuning to be of the same order of magnitude as that of friction,

ξ should be the same order as δ

2

; furthermore, so that the corrections are small, one must satisfy the conditions

δ

γ

<<

1 ,

γ

ξ

<<

1 (A)

The calculation is indicated in note 2. It gives

y

1

=

e

[

δ

( 1

e

)

+

...]

t

.(

u

+

y

2

=

e

[

δ

( 1

+

e

)

+

...]

t

.(

u

δε

γ

δε

γ

Or

ε

= + Γ

1

δ

ξ

2

v

+

...),

v

+

...)

(9)

(10)

Γ

2

=

γ

+

γ

2

ω

(11)

*This de-tuning is of a very special type (see Section 5). The results can be easily generalized for a small periodic function

ξ(t).

153

The functional determinant of our solutions is

D

= γ

1

γ

&

2

− γ

&

1

γ

2

=

2

δε

γ

e

2

δ

t

(

u v

&

v u

&

+ γ

u

2

)

+

...

The expression between parentheses is

(

u u d v

&

)

(

γ

(

ut v u

&

+

)

v

)

+

γ

(

(

γ

uu

)

ut

=

2

v

)

ω

du

γ

dt dt

that is to say, a constant different from zero, equal to the value of the mean

+

Therefore

D

= −

2

δε

e

2

δ

t

+

...

(12)

Γ

2

The dots at the end denote terms of the order of

δ

2

. We will often omit these dots, replacing the solution with some approximate expressions.

We proceed to a discussion of the solutions given by (9)

When

δ = 0 they become

y

1

=

e

Γ −

ξ

t

− ξ

(

u

+

Γ

γ

γ

ξ

ξ

v

),

(9a)

y

2

=

e

− Γ

t

(

u

Γ −

v

)

We suppose that, when

ξ = 0, we find a point A at the left boundary of an unstable

Figure 2: Schematic of the boundaries of an unstable region (in a system without friction).

region (Fig. 2). In this case the solutions (9a) should quasi-periodic if

ξ < 0; if, on the contrary,

ξ > 0 one of the solutions will be more damped. This final result comes from the fact that Γ is imaginary:

Γ i Γ or, from (11),

2

ω

γ

< −

1 .

If the operator L corresponds to one of the points B of the right boundary of one of the regions of instability, the solutions (9a) should be quasi-periodic when

ξ > 0 and either increasing or dampening when

ξ < 0. This is the final result when Γ is real.

Now suppose that

δ

0

. Near the boundary on the left of an unstable region

y

1 , 2

=

e

δ

1 m

i

Γ

1

δ

ξ

2

t t



u

±

δ ε

γ

v



.

154

Figure 3 shows the real parts of

k

δ

1 , 2 = −

1 m

i

Γ

1

δ

ξ

2

 as a function of

ξ/δ

2

. When

ξ < δ

2

the real part of the two exponents are equal to -

δ. All the motions decay at the same rate as in the absence of modulation. When

δ 2

<

ξ

<

δ 2

( 1

+

1

Γ

2

)

= −

2

ω

γ

δ 2 the two exponents are again real and 0 > k

3

> -

δ > k

1

. One of the oscillations is less dampened, and the other more dampened as in the absence of modulation. Finally when

ξ

>

δ 2

1

+

Γ

1

2

= −

2

γ

ω

δ 2

Figure 3: Real parts of the characteristic exponents in the de-tuning function.

Left boundary of an unstable region

the two exponents are again real and k

2

> 0, k

1

< -2

δ . One of the oscillations increases indefinitely; we are in the region of instability of the frictional system. The other solution decreases and the exponent of damping is at least double what it would be if the modulation was suppressed.

Near the right boundary of an unstable region

y

1 , 2

=

e

δ

1 m

i

Γ

1

δ

ξ

2

t t

(

u

±

δ

γ

ε

v

)

Figure 4 shows the real parts of

k

1 , 2

δ

= −

1

± Γ

1

δ

ξ

2

as a function of

ξ/δ

2

. When

ξ > δ

2 the real parts of the two exponents are equal to -

δ. When ξ < δ

2

the two exponents are real, when

ξ is decreasing, the damping of oscillation y

1 y

2

is augmented. Finally whenever

decreases and that of oscillation

ξ = δ

2

1

Γ

1

2

= −

δ

2

2

γ

ω we will enter into the region of instability. Naturally 2

ω/γ > 0,

Figure 4: Real parts of the characteristic exponents in the de-tuning function.

Right boundary of an unstable region

because otherwise friction would render unstable a system situated in a stable region when

δ = 0, which is

155 which is physically impossible for a system with one degree of freedom.

Figure 5 resumes the discussion for the first region of instability of an equation that becomes Mathieu’s equation if

δ = 0.

y

&&

+

(

λ

+

q

cos 2

t

)

y

=

0 (10)

The dotted lines are the boundaries of the unstable region in the absence of friction. The solid lines schematically indicate the boundaries of the different regions of the frictional system: The “normal” region of damping [without hatchmarks]; the region where there exists some solution of partially neutralized damping

[spaced-out hatchmarks]; and finally the region of parametric excitation [closely drawn-together hatchmarks]. The two dotted lines traverse some regions where all the solutions of the friction system are damped; when

ξ = 0 an infinitely small friction is enough to stabilize the system. We observe this qualitative asymmetry; if the frictionless ideal system is at the right boundary of an unstable region, the friction of the corresponding real system is partially neutralized, which is not the case at the left boundary. We will say that u is a natural oscillation of the

Figure 5: Regions of normal damping, partially compensated damping, and parametric excitation

(in a frictional system with strong modulation).

resonator, and that y

1

, y

2

are free oscillations.

Section 7: Two “degrees” of resonance.

In this paragraph we will study the oscillations produced by a periodic force, or a quasiperiodic force, f(t) in a system which, when the friction tends towards zero, places itself exactly at one of the boundaries of instability:

Ly

+

2

δ

y

=

f

(1)

First, we take the particular case of

f

= :

Ly

+

2

δ

y

& =

The stationary motion is

u

& (2)

y

=

2

u

δ

(3)

156

It coincides with a proper oscillation of the resonator and, from the definition of Section

3, corresponds to a resonance phenomenon.

We go to another particular case:

f v

&

Ly

+

2

δ

y

& =

v

&

(4)

We make the substitution

y

=

v

2

δ

+

y

*

From (6,7) it proceeds as

Ly

*

+

2

δ

y

*

= −

1

2

δ

Lv

=

From (2), (3) the stationary solution of this equation is

y

*

=

2

γ

δ 2

u

γ

δ

u

&

As a consequence the stationary solution of (4) is

γ

=

2

γ

δ 2

u

+

1

2

δ

v

(5)

If

δ/γ is very small, the first is by far the most important; the forced oscillation is practically inversely proportional to the square of the coefficient of friction. The system is much more sensitive to the force

f v

&

f

= .

We generalize the definition of resonance as follows: the force f produces a resonance of degree n (n > 0) if, when

δ

0 , the stationary oscillation excited by the force

δ n f tends towards a finite limit and is different from zero.

Harmonic resonators, and those studied in Part One, present only some resonances of the first degree. Here the force

f

=

u

& produces a resonance of the first degree, and the force

f

=

v

&

a resonance of the second degree*. In the limit

δ = 0, the forced oscillation

y

=

γ

2

u

+

δ

2

v

(due to the force

f

=

δ

2

v

&

) is indistinguishable from the natural oscillation y =

γu/2.

If we diminish the modulation factor, q, of equation (6, 1a) while at the same time modifying

λ, then, as that the system descends along the instability boundary (Fig. 2), γ will diminish and,

δ remaining fixed, the resonance of the second degree will be less and less evident. It will disappear when q and

γ annul each other.

A part of the force of friction developed by the oscillation (5), namely 2

δ

v&

2

δ

, is

*The existence of the second-degree resonance has been pointed out by myself and by

Professor L. Mandelstam.

157 caused by the force of friction due to the principal part of the oscillation, they are compensated by the energy coming from the device which goes to vary the parameter.

An ordinary force f can be written in the form

f

=

P u

&

+

Q v

&

+

g

(6) where the coefficients P, Q are chosen in such a way that the function g is orthogonal to u and v:

(

ug

)

=

0 , (

vg

)

=

0 (7)

From (2, 5) (2, 6), P, Q must have the following values:

P

= −

(

v

&

ω

f

)

,

Q

=

(

u

ω

f

The stationary forced oscillation due to the force f:

y

=

γ

Q u

+

Pu

2 δ 2

+

2

δ

Qv

+ z being the stationary solution of the equation

Lz

It will be shown that z remains finite when

δ

+

0

2

δ

z

=

g

)

z

(8)

(9)

; the force g therefore doesn’t give resonance (see Section 9).

For there to be resonance of the second degree it is necessary and sufficient that the force f be orthogonal to u and non-orthogonal to v.

If a resonator in which the functions u, v are expressed by the formulas (5), (8) of Note 1 is submitted to the action of the force

f

= cos(

t

− ϕ

) one has, to within a constant factor,

P

= sin ϕ

,

Q

= cos ϕ

.

If one varies the phase ϕ from 0 to π/2, one weakens the resonance of the second degree; when the phase ϕ = π/2 the force f produces only a resonance of the first degree.

Under the action of a force such that

Q

0 but otherwise arbitrary, for example sinusoidal, the stationary oscillations are practically equal,

δ/γ being sufficiently small:

y

=

γ

2

δ

Q

2

u

that is to say, one of its natural oscillations. If Ly = 0 is Mathieu’s equation, then the oscillogram of y will be that of a Mathieu function. The same observation applies to the case where

P

0 , Q = 0 and

δ is sufficiently small.

158

Section 8: Detuned resonator.

Up to now we have supposed that when

δ = 0 the resonator is situated exactly at a boundary of instability. We see now what happens if the resonator is gently detuned in relation to this boundary. We will examine only the most simple case where in the equation of the de-tuned resonator

Ly

+

ξ

y

+

2

δ

y

=

1 (1)

ξ is a constant.

We write

f

=

[

ξ

(

P

+

γ

δ

Q

)

u

+

Q

v

]

+

2

δ

(

P

u

&

+

Q

v

&

)

+

g

(2) and choose the coefficients P’, Q’ in such a way that the function g is orthogonal to u and v. Taking into consideration (6, 4) – (6, 6) and (7, 8) we obtain

P

+

γ

δ

One can give to equation (1) the form

Q

Q

=

=

Ly

+

ξ

y

+

2

δ

&

y

=

ξ

P

+

γ

δ

Q

2

4

u

δ

δ

4

ω

2

+

2

δ

ω

2

P

2

ω

ωξ

P

+

+

Q

v

2

+

ξω

+

2

+

2

2

δ

ωγξ

2

Q

ω

δ

ω

[

γ

2

+

+

P

ξ

2

γ

Q

ξ

u

&

+

2

+

ω

ξ 2

2

Q

Q

v

&

(3)

]

+

g

One verifies directly that it allows the solution

y

=

(

P

+

γ

δ

Q

)

u

+ z being a solution of the equation

Lz

+

ξ

z

+

2

δ

z

&

We define “relative detuning”

η, by the equality

ξ

=

δ

2

η

.

Q

v

=

g

+

.

z

One easily shows that

( )

η

=const

=

0 (

δ

) .

(4)

That is to say, that for

η being fixed, δz is of the order magnitude of δ (see Section 9).

Granting that

ξ = 0, if y is a resonance of the second degree,

Q

0 . In the case of the solution of (1) corresponding to a force

δ

2 f, one has

y

=

( 2

γω

2

Q

+

δ

2

ηω

Q

4

+

ω 2

2

δ

+

ω

2

2

P

)

u

ωγη

+

+

δ 2

( 2

η

δω

2

2

If when

ξ = 0 the resonance is of the first degree

(

Q

Q

=

0 ,

P

ωδ

0 )

2

η

P

)

v

+

δ 2

z

(4a)

, a force

δf excites the forced oscillation

y

=

4

2

ω 2

ω 2

+

Pu

2

ω

γ

ωηδ

η

+

δ

Pv

2

η 2

+

δ

z

(4b)

159

We begin by discussing the last formula. Neglecting terms on the order of

δ, and dividing by

δ, we obtain the “zero approximation” formula for the oscillation due to f:

y

=

2

Pu

δ

+

γ

ω

ξ

δ

.

To influence the zero approximation it suffices that the detuning be of the order of magnitude of

δ

2

. When the detuning is included linearly in the denominator, the

“resonance curve” will be asymmetrical. At the boundary of instability (

ξ

= −

2

ω

γ

δ

2

), the forced oscillation becomes infinite.

As long as

ξ is small, the terms of zero approximation remain proportional to u.

There is nothing analogous to the phase-shifting that detuning makes appear in the principal expression of oscillation strength for a harmonic resonator.

We go to the resonance curve of the second degree. Neglecting terms on the order of

δ and

δ

2

in formula (4a), we have a zero approximation for the oscillation due to f:

y

=

γ

δ

2

δ

Qu

+

γ

ω

ξ

δ

.

In this approximation everything occurs like with first-degree resonance, since the factor is nearly

γ

δ

Q

P

.

Re-establishing terms on the order of

δ in (4b), we obtain the more exact formula

y

=

2

δ

+

1

γ

ω

ξ

δ

[

γ

δ

Qu

+

Pu

+

Qv

] which shows the existence of a relatively weak component proportional to v.

Note that, when

ξ is varying, the resonator passes the region where friction is partially neutralized, and, surprisingly, it produces no singularity.

Section 9: Action of a force orthogonal to u and v

In order to support the results of Sections 7 and 8, it remains to be demonstrated that when

η = constant (in particular η = 0) the magnitude δz (η, δ) is of the order of δ, and consequently a force orthogonal to u and v doesn’t give resonance.

*One can obtain the solution (3), (4) by the method of varying constants, in using (6, 9).

The integration vanishes along the boundaries between these regions.

Technical Physics, V. II No. 2-3. 6

160

A force

δg produces the stationary oscillation

δ

z

=

δ

D

0

y

1

− ∞

t g y

2

e

2

δτ

d

τ

+

y

2

t

− ∞

g y

1

e

2

δτ

y

1

,

y

2

,

D

=

D

0

e

2

δ

t

being given by the formulas (6, 9) – (6, 12).

Neglecting the terms of the order of

δ

2

we can replace (1) by

d

τ

(1)

δ

z

=

Γ

2

2

ε

e

− δ

( 1

− ε

)

t

(

u

+

δε

γ

v

)

t

− ∞

e

δ

( 1

− ε

)

τ

g

(

u

First, we examine the integrals

δε

γ

v

)

d

τ

e

δ

( 1

+ ε

)

t

(

u

δε

γ

v

)

t

− ∞

e

δ

( 1

+ ε

)

τ

g

(

u

+

δε

γ

v

)

d

τ

(2)

t

− ∞

e

δ

( 1

ε

)

τ

g

δε

γ

vd

τ

,

t

− ∞

e

δ

( 1

+

ε

)

τ

g

δε

γ

vd

τ the function g being orthogonal to v

gv

=

b n e i p n

τ where all the p n

are different from zero. Executing the integrations we obtain some functions of the order of magnitude

δ.

The terms of the order of magnitude

δ

2

, which had been omitted in formula (2), will, after integration, give at the very most some terms of the order of

δ. Therefore

δ

z

=

Γ

2

ε

2

e

δ

( 1

ε

)

t

(

u

+

δε

γ

v

)

t

− ∞

e

δ

( 1

ε

)

τ

gud

τ

e

δ

( 1

+

ε

)

t

(

u

δε

γ

v

)

− ∞

t e

δ

( 1

+

ε

)

τ

gud

τ

+

0 (

δ

)

Therefore, when the force g is orthogonal to u

gu

=

a n e i p n

τ for values of the p n that are different from zero. Introducing this expression into the preceding formula, we obtain through integration,

δ

z

=

Γ

2

2

ε

u

a n e i p n t

i p n

+

δ

1

( 1

ε

)

i p n

+

δ

1

( 1

+

ε

)

+

0 (

δ

)

The terms between brackets are of the order of magnitude

δ. Our assertion is thus demonstrated.

161

Section 10: Action of a gently detuned force

Consider a system in the state of first or second degree resonance. What will occur if one gently modifies the function f? We will consider ideas for determining the particular case f = cos (1 +

ε) t, ε being a small “detuning” parameter, and we suppose that when δ = 0 the resonator is described by Mathieu’s equation (6, 10) and occurs at the left boundary of the first region of instability. In this case, according to Section 8 and the development

(5) of Note 1, when

ε = 0 there is a resonance of the second degree.

One can consider the force, f, as the sum of two modulated oscillations:

f

= cos

ε

t

cos

t

− sin

ε

t

sin

t

(1)

A force a cos t – b sin t, a and b being arbitrary constants, could produce the forced oscillation

α β

y

=

a

α

2

γ

δ

u

2

+

2

1

δ

(

b

β

u

+

a

α

v

)

+

z

; z remains finite when

δ

0 .

If the detuning,

ε, is very small and, as a consequence, the modulation indicated by the functions cos

εt and sin εt is done very slowly, one can use formula (1) for the forced oscillation excited by the force f by replacing with cos

εt and

b

by sin

εt. One obtains

y

= cos

ε

t

α

2

γ

δ

u

2

+

2 or, approximately, if

δ is sufficiently small,

1

δ

(sin

ε

t

β

u

+ cos

ε

t

α

v

)

+

z

(2)

y

− cos

ε

t

αγ

2

δ

2

u

. (3)

This formula represents an “oscillation of frequency 1 modulated by a frequency

ε”.

Equivalently, one could say that a stationary oscillation is formed by beating two periodic waveforms, one of which has the frequency (1 +

ε), and the other the frequency (1 – ε).

This phenomenon gives a physical reality to the decomposition (1) of a sinusoidal oscillation into the form of two modulated oscillations, because it allows us to physically separate the oscillation cos

ε

t

⋅ cos

t

from the oscillation sin

ε

t sin

t

.

Section 11: Transitory regimes

neglect the term 2

δ

y& preceeding f and use the equation

Ly f y

=

ut

2

6*

162

A first degree resonance does not change the system’s type of instability. It is different from this for the phenomena of second-degree resonance. The equation

Ly

=

γ

u

+

2

v

& allows the exact solution

f

=

a

(

γ

y u

=

+

2

γ

v ut

)

2

+

+

g

,

vt a

.

2

All forces producing a phenomenon of second-degree resonance can be put into the form

0 , (

gu

)

=

0 .

Therefore, in cases of second-degree resonance the increase of the oscillations is not uniform, but is uniformly accelerated.

Part Three.

The system Ly = 0 exists in an unstable region

Section 12: Natural oscillations and free oscillations

We cannot render an account of a resonator’s stationary regimes for the case where parametric excitation phenomena are manifested without going beyond the limits of linear theory. Addressing the case where the system is ideal: the equation Ly = 0 exists in an unstable region; therefore we suppose that friction is sufficient to stabilize a resonator in the actual real world.

If the equation

Ly

=

0 (1) describes a system situated in an unstable region, its solutions are of the form

e

kt

ϕ

(

t

),

e

kt

ψ

(

t

) k being a real constant, and ϕ and ψ being periodic functions of period π or 2π.

The equation

Ly

+

2

g y

&

=

0 (2)

--the coefficient of friction is designated here by g — allows solutions of the form

e (

k

′ −

g

)

t

ϕ ′

,

e

(

k

′ +

g

)

t

ψ ′

; where k is a constant, g an arbitrary function, ϕ’, ψ’ are some new functions of period π or 2

π, which contain g in the value of the parameter.

For the resonator to be stable, it is necessary that the condition k’ < g be satisfied.

If k’ = g equation (2) allows a periodic solution; the resonator is then at the limit of instability.

We let h designate the value of g for which k = g. We introduce the operator

£

=

L 2

h d dt

and the notation that

δ = g – h. Equation (2) then becomes

163

£y

+

2

δ

y&

=

0 . (2a)

δ is the excess of the coefficient of friction over the value that it should have so that equation (2) possesses a periodic solution. Throughout this discussion, we assume that

δ << h, that is to say the excess of friction is only a small part of the total friction h + δ; the latter is typical.

The equation

£

y

=

0 (1) which corresponds to

δ = 0, possesses a periodic solution u, of period π or 2π, and a damped solution e

-2ht v, v being periodic of period

π or 2π. The functions u, v will be assumed normal:

(

uu

)

=

1 , (

vv

)

=

1 . (2)

Furthermore

(

u u

&

)

=

0 , (

v v

&

)

=

0 (3)

The identity

u v

&

=

d dt uv

furnishes the relation

Figure 6: Schematic of the boundary of an unstable region of a frictional system

(

u v

&

) (

v

&

)

=

0

The determinant function of the solutions u, e

-2ht

. (4) v is

D

By virtue of Liouville’s theorem

=

e

2

ht

(

u v

&

v u

&

2

huv

) . (5)

D

=

2

ω

e

2

ht

(6)

ω being a constant different from zero, *and consequently

(

u v

&

)

(

v u

&

)

2

h

(

uv

)

=

2

ω

. (7)

In the case where the system described by equation (1) exists at the lower point M at an unstable boundary (Figure 6) — an interesting case by many accounts —

(

uv

)

=

0 (8)

(see note (5)) and consequently

(

u v

&

)

According to (4) and (7a), one then has

(

u v

&

)

=

(

v

&

)

ω

,

=

(

u

&

v

)

2

ω

=

. (7a)

ω

. (9)

164

v is a periodic solution of the adjoint

£ *

13

y

equation (1), which we will write as

=

0 . (1*)

(See note 3.)

The functions u, v have a similar application here, as that assumed in the first and the second part of the functions that appear in the solution of Ly = 0. They play an essential role in the theory of resonance and serve the zero approximation for the oscillations of a system possessing a small surplus of friction

δ.

The equation

£

y

+

2

δ

y

&

=

0 (10) allows the following solutions

y

1

=

e

k

2

δ

t

(

u

+

δ ϕ

1

),

y

2

=

e

[ 2

h

+

( 2

k

1

)

δ

)

t

(

v

+

δ ψ

1

) (11) where

k

1

=

1

+

1

h

(

uv

)

=

1

1

h

(

uv

) /(

u v

&

)

(12)

/(

u

&

v

) and ϕ

1

, ψ

1

are some functions of the period

π or 2π (see note 5).

We will say that y

1

, y

2

are the free oscillations of the system and u a natural oscillation.

By decreasing

δ we can reduce the damping of the type y

1

oscillations to zero, but the damping exponent of the type y

2

oscillations cannot become less than 2h. If the modulation is suppressed (q = 0) all the solutions contain the same factor of damping e

-(h+

δ)t

. The modulation parameter decreases the exponential damping of y

1

by the quantity h + (1 - k

1

)

δ, as required by Liouville’s theorem, to that of the type y modulation doesn’t modify the sum of these exponents.

2

. The

The general solution of equation (10) is

y

=

A y

1

+

B y

2

If we give the system an initial perturbation

y

0 at an instant t = t

1

, chosen in such a way that B = 0, the oscillations will dampen themselves slowly. If we give the same initial perturbation at an instant t = t

2

such that A = 0; the oscillations will die, but much more quickly. The choice of the instant where one applies a given perturbation to the resonator, then, is not arbitrary. This property — it can only appear in a system which the parameters contain time explicitly — is of a primary importance for the resonator’s behavior under the action of an exterior force. One can consider this as a continuous succession of impulses (see Section 2). Now, the effect of each one of those depends essentially on the moment when it occurs. That is because the action of a force f (t - ϕ) will depend strongly on ϕ: ϕ varying, all the impulses are displaced in time. If for

165 example the force is sinusoidal, under certain conditions one can try to change the forced oscillation’s order of magnitude when modifying its phase.*

The modulation doesn’t act only on the exponential factors of the free oscillations; it equally alters the periodic factors. We write equation (10) in the form

y

+

[

λ

+

qm

(

t

)]

y

+

2 (

h

+

δ

)

=

0 .

When q = 0 we have — in language not very correct — a damped oscillation of frequency

λ

. But if q is big enough so that there is partial neutralization of friction, we have some oscillations — one strong, the other weakly damped — of frequency 1 or 2

(see note 6).

Section 13: Resonance

The equation for forced oscillations in a resonator that has an excess of friction, 2

δ, is

£

y

+

2

δ

y

=

f

. (1)

First, we will consider the case where

£

f y

=

δ

+ 2

δ

u

&

y

&

. The equation

=

δ

u

&

(2) allows the exact periodic solution

y u

2

The general solution of the homogenous equation being absorbed, solution (3) represents a stationary motion when the applied force and the excess of friction reciprocally adjust themselves at each instant and towards which all other motions tend asymptotically as

t

→ ∞

.

δ doesn’t figure in solution (3); when δ = 0 this one remains as a periodic solution of the

resonance, generalizing the definition of resonance given above (see Section 7) in such a way that

δ is the excess of the coefficient of friction over the value that it possesses when the resonator can execute free periodic oscillations or quasi-periodic ones.

All quasi-periodic functions f’ can be put into the form

f P u

+

g

(4) where P is a constant and g a function orthogonal to v. It is evident that

P

=

(

fv

)

(5)

(

u

&

v

)

If (uv) = 0 (see Section 12) then

P

= −

( fv )

ω

. (5a).

*It is the same from there for the systems studied in the second part.

166

In making use of the formulas (2, 3), (12, 10)—(12, 13) one easily shows that the oscillation excited by the force g remains limited when

δ

0 . There is no resonance then if P = 0. The condition of resonance is

(

fv

)

0 (6)

It requires that the force isn’t orthogonal to the periodic solution of the equation £ *y = 0.

The condition (6) is more restrictive than the conditions of resonance of the systems studied in Part One.

The structure of the forced oscillation is

£

y

=

Pu

δ

+

z

(7)

2

The smaller that

δ is, the nearer the forced oscillation is to one of the periodic solutions of y = 0.

Here, for example, f = cos (t - ϕ), the functions u, v being given by the formulas (6) of note 4. P is proportional to cos ( ϕ − π/4) and the forced oscillation is approximately proportional to

u

cos( ϕ

π

4

) .

The “amplitude” depends essentially on the phase of f. When

π

4 ϕ

= −

π

4

or

3

π

4 the resonance disappears.

With the help of (2, 3), (12, 10)—(11, 13) one easily obtains an approach to a formula describing the transitory phenomena in cases of resonance. We suppose that at the instant t = t

0

one applies a force f non-orthogonal to v to the resonator, which exists in the state of equilibrium

(

y

=

0 ,

y

&

=

0

)

. For the principal term the solution of (1) that satisfies these initial conditions has

Y

= −

(

vf

)

2

ω

k

1

δ

u

[ 1

e

k

1

δ

(

t

t

0

] where, according to (12, 7), (12, 12)

Y

=

(

vf

2 (

u

&

v

)

)

δ

u

[ 1

e

k

1

δ

(

t

t

0

)

]

. (8)

In particular when (uv) = 0, k

1

= 1

Y

=

(

2

vf

)

ωδ

u

[ 1

e

δ

(

t

t

0

) ]

. (8a)

The law of growing oscillation doesn’t depend on t

0

. The smaller that

δ is, the longer one must wait for the “amplitude” to attain a given percentage of its ultimate value (as in harmonic resonance).

167

Aside from the case of

f

=

u

&

, there exists another case where an exact solution is obtained without difficulty. It is that of

£

y f

= . In effect the equation

+

2

δ

y

=

v

&

(9) can be written as

£ * y

+

(

4

h

+

2

δ

)

=

v

(see note 3). By virtue of £ *v = 0

y

=

v

4

h

+

δ

(10)

2 is a solution. Naturally, this solution does not correspond to a resonance phenomenon in the sense given to this term above.

The case where

δ<<h<<1 presents certain interesting particularities. The forced oscillation due to a force g, orthogonal to v but non-orthogonal to u, possesses the structure

y

=

Qv

4

h

+

z

1

,

Q

=

(

gu

)

(11)

(

u v

&

) z being limited when h = 0.

Its order of magnitude depends essentially on the value of Q. If Q

0 and h is sufficiently small, the forced oscillations are intense and differ little from one of the

For

δ << h << 1 it is easy to obtain an approximate formula for the transitory phenomena that occur when a force for which P = 0 but Q is sizeable begins to act at the moment t = t

0

where the system exists in the state of equilibrium. A calculation analogous to formulas (8), (8a) gives the principal term the expression

y

=

(

ug

)

v

[ 1

. (12)

4

h

(

u v

& )

e

2

h

(

t

t

0

)

]

The steady state regime establishes itself much more quickly than in the case of resonance.

We write the equation of the forced oscillations under the form

y

&&

+

[

λ

+

qm

(

t

)]

y

+

2 (

h

+

δ

)

=

f

.

In the absence of modulation (q = 0) the force f

1

= a cos

ω

0

t + b sin

ω

0 t,

(

ω

2

0

=

λ

)

gives a phenomenon of resonance and the force f

2

= a cos t + b sin t doesn’t give one (we suppose of course that

λ

1 ). But if the modulation is strong enough, the force f

2 produces the phenomenon of resonance (except when the ratio a/b takes certain exceptional values), whereas the force f

1

doesn’t produce it. The modulation can then modify the frequency that should have a sinusoidal force to produce an effect of resonance.

The modulation can equally modify the system’s sensitivity, as we have defined it in

Section 4.

168

The maximum sensitivity that can be had in resonance cases of a harmonic resonator (q =

0) is

S

o

m

=

1

.

4

λ

(

h

+

δ

)

2

If the portion h of the friction is neutralized by modulation, and

δ << h, the forced oscillation in case of resonance differs little from

y

=

(

fv

)

u

.

Therefore

(

u

&

v

) 2

δ

S

[

f

]

=

4

δ

(

2

fv

)

2

(

u

&

v

)

2

.

The greatest sensitivity is attained when (fv) is maximum. We write

f

=

(

fv

)

v

+ ϕ

It is evident that (v ϕ) = 0,

(

ff

)

=

1

=

(

fv

)

2 +

( ϕϕ

)

(fv) is maximum and equal to 1 when ϕ = 0, f = v. The maximum sensitivity is then

S m

=

4

δ

2

(

1

u

&

v

)

2

.

Consequently

S

S m

0

m

=

ω

2

0

(

u

&

v

)

2

h

δ

2

2

(approximately).

If the modulation is strong, the maximum sensitivity is much larger than when q = 0

(“parametrical regeneration”).

The preceding results show that the phenomena due to “parametric regeneration” differs noticeably from the usual phenomena of “autonomous” regeneration such as that occurring in feedback (regenerative) receivers.

Section 14: Detuned Resonator

We will consider the equation

£

y

+

ξ

y

+

2

δ

&

= f (1) of a system that distinguishes itself from that studied in Section 12 by the “detuner”

ξ, which we will suppose is independent of t and sufficiently small. Furthermore, we will suppose that the operator £ characterizes a system that exists at the minimum point of an unstable boundary.

1. First, we will assume that

δ = 0.

In this case (see note 5), the higher order terms in

ξ being neglected, the free oscillations are

y

1

=

e

− βξ

2

t

(

u

+

ξϕ

1

),

y

2

=

e

( 2

h

− βξ

2

)

t

(

v

+

ξϕ

1

)

(2)

169 where and ϕ

1

,

ψ

β

= −

(

u

2 ϕ

ω

1

)

=

1

are periodic of the period

π or 2π.

(

v

2 ϕ

ω

1

)

(3)

β > 0, because the free oscillations of the detuned resonator are damped, since that is the mark of detuning.

Proceeding to forced oscillations, we observe a circumstance that seems paradoxical at first glance.

Here f = u. The function u being orthogonal to v, the equation

£ y +

ξy = u (1a) possesses a family of restricted periodic solutions when

ξ = 0, in particular the periodic solution

y

=

4

1

ω

v

+

w h

(see note 5 and formula (12, 2)).

But when

ξ

0 equation (1a) possesses the

(4) obvious solution

y u

ξ which increases without bound when

ξ

0 . All the solutions of the homogeneous equation being damped, those being the initial conditions, the system tends towards the motion (5).

We take the initial conditions that dampen the motion (4) when

ξ = 0. The intensity of the stationary (steady state) oscillation is represented, as a function of

ξ by the discontinuous curve of Fig. 7; the ordinate of the isolated point P is the intensity that

Figure 7: Intensity of the stationary oscillations in function of

ξ(δ = 0) exists when

ξ = 0.

Is this result in conflict with the well-known theorem which says that when one tries to vary a parameter in a differential equation, in a direct continuous way,* the solutions corresponding to some given initial conditions equally undergo a continuous variation? To affirm it would be, at the least, premature: The theory only applies to a finite time interval; we have compared the limits to those that have solutions when

t

→ ∞

. To respond to our question, then, one must examine transient phenomena.

*Subject, naturally, to certain conditions satisfied under the circumstances.

170

According to formulas (2, 3) and (2), the solution of (1) that corresponds to the initial conditions t = 0, y = 0, y&

=

0 is,

y

=

2

ω

+

1

ξ

D

1



e

− βξ 2

t

(

u

+

ξϕ

1

)

t

0

e

βξ 2 τ

u

(

v

+

ξψ

1

)

d

τ

+

e

(

βξ 2 −

2 )

t

(

v

+

ξψ

1 where D is a constant and the higher degree terms in

ξ being neglected.

The most interesting term is

)

e

(

βξ 2 −

2

h

)

τ

u

(

u

+

ξϕ

1

)

d

τ

Y

= −

1

2

ω

e

βξ 2

t u t

0

e

βξ 2 τ

ξ

(

u

ψ

1

)

d

τ or carrying out the integration and taking into (3) consideration

Y

=

u

ξ

( 1

e

βξ 2

t

) . (6)

If

ξ is small but different from zero, Y is the principal term of the solution.

The approximate solution (6) confirms what we know about the stationary (steady state) regime. For

ξ

0 , t

= ∞

Y u

ξ

On the other hand, for all finite values of t,

ξ

lim

0

Y

=

0

How does the character of the transitory regime vary with

ξ? For different values of ξ/ξ

0

,

ξ

0 being an arbitrary parameter, Figure 8 gives the rate of change of the time function

F

=

1

e

βξ 2

t

ξ

, which indicates the law of increasing “amplitude”.

When the detuning decreases, the stationary

(steady state)

“amplitude” increases

(as 1/

ξ), but, in return, the time necessary for the “amplitude” to attain a given value increases also.

ξ being small enough, we will not ascertain any effect

Figure 8: Characteristics of the transitory phenomena for different values of

ξ/ξ

0

(f = u).

171 even after having waited for a very long time. As

ξ

0 , at the end of a fixed time t = T, the “amplitude” reaches zero in a continuous way, as demanded by the theory of continuity.

The conflict between the magnitude of the stationary (steady state) “amplitude” and the rapidity of transient phenomena is much more dramatic than in ordinary resonance. If the force f = u acts only during a limited time it is disadvantageous to decrease the detuning below a certain value.

We will now consider the action of the force f = v.

With the aid of formulas (2, 3) and (2), one easily obtains the solution determined by the initial conditions t = 0, y&

=

0 , y = 0. When

ξ is small, the principal term is

ξ

and

t

Y

= −

u

2

ω

1

e

βξ 2

t

βξ

2

. (7)

, it tends towards the stationary oscillation

Y

= −

u

2

ωβξ

2

. (8)

ξ

, the forces u and v excite steady state oscillations of practically identical form.

In the case of f = u the oscillation is proportional to 1/

ξ, and in the case of f = v the oscillation is proportional to 1/

ξ

2

. There is a certain analogy here to the two degrees of resonance studied in Part Two.

Formula (7) has the same structure as (13, 8a). The detuning is almost equivalent to an excess of friction

δ = βξ one obtains

2

. Incidentally, in formula (7), at the limit

ξ = 0, t being fixed,

Y

= −

ut

ω

+

z

(9)

2 where z is a periodic function. (By the way, at the limit

δ = 0 in the formula (13, 8a), one arrives at the same result).

2. We re-establish the term 2

δ

y& in the equation (1). In a real system,

δ is never rigorously null; now, if

δ is small, this term ceases to be negligible when the detuning is small enough. The solutions of equation (1) are approximately

y

1

=

e

(

δ

βξ 2

)

t

(

u

+

ξϕ

1

)

,

y

2

=

e

(

δ

βξ 2

+

2

h

)

t

(

v

+

ξψ

1

)

(10)

(see note 5). They are damped for all the values of

ξ.

If (fv) = 0, the solution of (1), which satisfies the initial conditions t = 0, y = 0,

Y

= −

2

ω

(

δ or, in the steady state regime (

t

→ ∞

Y

= −

2

ω

ξ

(

)

δ

(

ξ

(

f

ψ

+

f

ψ

+

1

)

βξ

1

)

βξ

2

)

2

)

u

u

1

δ

e

(

δ

+

βξ 2

)

t

+

βξ

2

= −

2

ξ

ωβ

(

(

δ

f

ψ

+

1

)

ξ

(11)

2

)

u

(12)

172 where

δ ′

=

δ

β

. (13)

When f = u , formula (3) shows that

Y

=

δ

ξ

+

ξ

2

u

. (12a)

The friction excess (as with friction in the resonators briefly considered above) “softens” the curve, which imparts a detuning function to the steady state “amplitude”. The discontinuous function

A = 1/

ξ

, which, for

ξ

2

ξ

2

>>

δ’, becomes practically identical to 1/

ξ ; but for

<<

δ’ diminishes with ξ as ξ/δ’.

For certain values of detuning, the intensity of the steady state oscillation attains a maximum.

Figure 9: Resonance curves of the degree ½.

These values are

ξ

= ±

δ ′

. Upon introducing them into the expressions (12a), (14) one obtains

A = 0,

ξ = 0 is replaced by the analytic expression

A

=

δ ′

ξ

+

ξ

2

(14)

Y m

= ±

1

2

δ

1

u

,

A m

2 =

1

4

δ

1

. (15)

Figure 9 shows the aspect of the curves A

2

= A

2

(

ξ) for different values of δ.

One can say that when

ξ

= ±

δ

the force u exhibits a resonance phenomenon of degree

½, because the force

δ

1/2 u excites the forced oscillation

δ

β

δ

1

y

= ±

u

+

0 (

2

)

2 which, when

0 , tends towards a free oscillation different from zero for the system

£y = 0. Indeed, passing to the limit

δ = 0, we should, at the same time,

173 try to stretch the detuning function

ξ towards zero so as to maintain the relation

ξ

= ±

δ ′

.

£y = 0.

When

δ is small the forced oscillation is nearly a periodic solution of the equation

All forces orthogonal to v and non-orthogonal to u equally give a resonance phenomenon of degree ½ (see (12)).

We now assume that

fv

0 . When

ξ = 0 there is a first-degree resonance.

y = 0,

The principal term of the solution that corresponds to the initial conditions t = 0,

y&

=

0 is

Y

= −

1

2

ω

(

e

(

δ − βξ 2

)

t

δ

+

βξ

2

)

(

fv

)

u

. (16)

The transient phenomena present the same appearance as in the harmonic resonator. The steady state oscillation is (approximately)

Y

= −

2

ω

(

δ

(

fv

+

)

βξ

2

)

u

(17)

In particular, if f = v

Y

= −

(

u

2

ω δ

+

βξ

2

)

(17a)

And, if

f

=

u

&

Y

= −

u

2

(

δ

+

βξ

2

)

(17b)

When

ξ = 0 these formulas become those of Section 13.

Whereas, in the harmonic resonator a detuning on the order of

δ exhibits a marked influence on the amplitude of the oscillations, here the detuning only acts in a prominent manner if it attains the order of magnitude of

δ

1/2

. In this sense the resonance curve is less sharp than that of the harmonic resonator. But, let us not forget that when the modulation is strong enough,

δ is much smaller than the exponential damping of the harmonic resonator that our system becomes when the modulation is suppressed.

If (

f

ψ

1

)

0 and 0 , the resonance phenomena of the first degree and the ½ degree are superimposed. According to formulas (12) and (17), the steady state “amplitude” is approximately

A

=

(

fv

)

2

ω

(

δ

+

ξ

(

f

+

ψ

βξ

2

1

)

)

(18) and the resonance curve becomes asymmetrical. Figure 10 shows the appearance of the resonance curve

A

2

=

(

fv

)

2

ω

(

δ

+

ξ

+

(

f

βξ

ψ

2

1

)

)

2

(19)

174 which corresponds to

δ

=

0 , 01 ,

(

fv

)

2

ωβ

=

0 , 01 ,

(

(

f

ψ

1

)

fv

)

=

0 , 1 .

If the force f is sinusoidal:

f

= cos(

t

− ϕ

) the ratio (fv)/(fψ

1

) and, therefore, the form of the resonance curve, is dependent on φ.

3. In studying the oscillations excited by a force slightly detuned relative to resonance, one can reason analogous to Section 7. Here also, the steady state regime is characterized by a slowly varying modulation or, if one prefers, by “beats”.

In conclusion, I take this opportunity to express my profound gratitude to Professor

L. Mandelstam for his valuable advice and Professor N.

Papalexi for the interest he has brought to this work.

Moscow, at the University’s

Institute of Physics/Oscillations

Laboratory

Received 8 February 1935.

Notes

Figure 10: Asymmetrical resonance curve.

1. To calculate the boundaries of instability and the periodic solution of Ly = 0 one can make use of developments following the powers of q. For example, we take Mathieu’s equation

y

&

+

(

λ

+

q

cos 2

t

)

y

=

0 . (1)

To find the boundary of the first unstable region and a periodic solution U, proportional to u, we pose

λ

=

1

+

q

λ

1

+

q

2

λ

2

+

...

(2)

U

= cos

t

+

qU

1

+

q

2

U

2

+

...

(3)

Introducing these series into (1) and, canceling the coefficients of the like powers of q, we obtain a series of inhomogeneous equations with constant coefficients of which all the known terms are functions of

λ

….

1

,

λ

2

Requiring that the corrections U

1

, U

2

…be periodic we obtain

λ

=

1

q

2

q

2

32

+

q

3

512

+

...

(4)

U

= cos

t

+

q

16 cos 3

t

+

q

2

256

(

− cos 3

t

+

1

3 cos 5

t

)

The equation

v

&&

+

(

λ

+

q

cos 2

t

)

y

= − allows a periodic solution V, proportional to v. To calculate it we pose

2

q U

&

+

...

(5)

(6)

V

=

b

sin

t

+

qV

1

+

q

2

V

2

+

....

(7) and introduce this series, as well as the series (4) and (5), into (6). Canceling the coefficients of the powers of q we obtain a series of inhomogeneous equations with constant coefficients. Requiring that V periodic we find

1

, V

2

…be

175

V

=

2 sin

t

+

q

sin 3

t

+

q

2

( 5 sin 3

t

+

1 sin 5

t

+

3 sin

t

)

+

...

(8)

8 128 3

One can obtain

λ, U and V for the other boundary of the first unstable region by a simple transformation.

We replace q in (4) by –q:

λ

y

&&

=

1

λ

+

q

q

2

2 32 512 which is an expression that gives one of the boundaries of the first unstable region of

+

(

q

cos 2

t

)

y q

3

=

0

+

...

.

(4a)

(1a)

This equation does not differ from (1) for the phase of modulation; its unstable regions can not be different from those of (1). Therefore (4a) corresponds to a boundary of the first unstable region of (1). It is evident that (4), (5), (7) correspond to the left boundary and (4a) to that of the right boundary

.

Replacing q in (5), (7) with –q, one finds functions U, V belong to the right boundary of the first unstable region of (1a). Next, replacing 2t with 2t +

π one obtains for the right boundary of the first unstable region of the original equation (1):

V

=

U

2 cos

t

=

+ sin

q

8

t

+ sin 3

q

16 sin

t

+

q

2

3

t

128

+

q

2

256

(sin

(

5 cos 3

+

3

1

3

t

+

1

3 cos 5

t

sin

+

5

t

)

+

...

3 cos

t

)

+

...

(5a)

(8a)

For the two boundaries

u

=

2 ( 1

q

2

512

+

....)

U

,

v

= ±

1

2

( 1

7

512

q

2

+

.....)

V

,

γ

= ±

q

2

( 1

3

256

q

3

+

...).

Moreover,

ω

=

2.

The equation

(

u v

&

)

=



 m

±

( 1

( 1

+

+

3

128

3

128

q

2

q

2

Ly

+

+

+

...)

...)

(

(

left boundary right

)

boundary

)



 q not being too large

ω/γ > 0 on the right boundary and ω/γ < 0 on that of the left boundary.

ξ

y

+

2

δ

&

=

0 allows two independent solutions of the form

y

=

e kt

ϕ

, ϕ being periodic of period π or 2π; ϕ is a solution of

L

ϕ

+

2 (

k

+

δ

) ϕ

+

(

k

2

+

ξ

+

2

δ

k

) ϕ

=

0

Technical Physics, V. II, No. 2-3

(1)

(2)

(3)

7

176

Here

δ

=

µδ

1

,

ξ

=

µξ

2

,

2

k

2 ϕ

0 ϕ

δ

1

) ϕ

&

1

0

µϕ

1

+

µ

2 ϕ

µ being a small parameter. We introduce these expressions into (3) and cancel the coefficients of like powers of

µ:

L

ϕ

0

=

0

(3a),

L

ϕ

1

= −

2 (

k

1

+

δ

1

) ϕ

&

0

(3b)

L

ϕ

2

(3a) allows the periodic solution

=

k

=

µ

k

1

+

µ

2

k

2 (

k

1

2

+

+

...,

=

u

ϕ

+

= ϕ

(

k

1

2

0

+

+

ξ

2

+

2

k

1

δ

1

) ϕ

2

0

+

...

(4)

(3c)

(5)

We introduce it in (3b):

L

ϕ

1

= −

2 (

k

1

+

δ

1

)

u

&

(6)

This equation allows the periodic solution ϕ

1

=

(

k

1

+

δ

1

)

γ

v

(see Section 6). We introduce (5), (6) into (3c):

L

ϕ

2

= −

2

k

2

u

&

2 (

k

1

+

δ

1

)

2

γ

v

&

+

(

k

1

2 +

ξ

2

+

2

k

1

δ

1

)

u

so that ϕ

2

is periodic, the second member, according to a known theorem, should be orthogonal to the periodic solution u of L ϕ

2

= 0, in which the condition

2 (

k

1

+

δ

1

)

2

ω

γ

+

(

k

1

2 +

ξ

2

+

2

k

1

δ

1

)

=

0 or, multiplying by

µ 2

and neglecting the terms of the order of

µ 3

2 (

k

+

δ

)

2

ω

γ

+

k

2 +

ξ

+

2

k

δ

The roots of this equation are

k

= −

δ

± Γ

δ

2 −

ξ

,

Γ 2 =

γ

=

0

+

γ

2

ω

' which gives for ϕ the expression ϕ

=

u

±

Γ

γ

δ

2 −

ξ

v

+

...

3. The adjoint of the operator L + 2h d/dt is L – 2h d/dt. Equation (12, 1*) can then be written as

Ly

2

h

&

=

0

(1*)

The identities

0

=

Le

2

ht v

+

2 show that v is a solution of (1*). Another solution is e

h

2ht

d dt e

2

ht v

= u. In effect

e

Le

2

ht u

2

h d dt e

2

ht u

=

e

2

ht

(

Lu

2

ht

+

(

2

Lv h u

&

)

=

2

h v

&

)

0

4. It is easy to obtain u, v, h in the form of an ordered series following powers of q. For example, we take the equation

y

&&

+

λ

y

+

q

cos 2

t y

+

2

h y

&

=

0 (1) and assume that

λ differs from 1 by a quantity of order q 2 periodic solution. Let us set

(first boundary of instability). Here is U as a

U

= cos

t

+

h

B

=

λ sin

qh

1

=

t

1

+

+

+

qU q

3

h

3

q

1

2

h

2

+

+

q

...

2

U

2

+

...

(2)

177

(q being given, there exist two values of the coefficient of friction, h and –h, for which (1) allows a periodic solutions. These values don’t change if one replaces q with –q. Then h can only be even or odd in q. In our case the calculation shows that

h

1

0

; that’s because the h series contains only odd powers of q).

Introducing the series (2) into (1) we obtain, successively, for U parameters can be chosen in such a way that U

1

, U

2

1

, U

2

… a series of inhomogeneous equations with constant coefficients for which all the known terms are some functions of B, h

… are periodic. The calculation gives:

1

, h

3

… These

U

= cos

t

− sin

t

+

q

16

(cos 3

t

− sin 3

t

)

2

q

λ

2

+

1

32 sin

t

+

+

q

2

{

24

1

32

(cos 5

t

− sin 5

t

)

1

8

λ

2

+

1

32 sin 3

t

8

3

32 sin 3

t

(3)

8

3

32 cos

h

=

3

t q

4

16

1

32

q

3

4

 16

+

7

32

λ

2

8

+

+

λ

8

3

2

λ

2

2

+

2

 sin

λ

2

2

t

}

+

...

+

...

(4)

L y

&

2

h

&

=

The operator L being even, one passes from the equation

Ly

+

2

h y

&

=

0

to the adjoint equation

0

upon replacing t by –t. The adjoint of (1) possesses the periodic solution V(t) = U(- t) or

V

= cos

t

+ sin

t

+

q

16

(cos 3

t

+ sin 3

t

)

+

2

q

λ

2

+

1

32

 sin

t

+

...

(5)

In order to obtain u, v, it suffices to normalize U, V. q having a fixed value, h is a function of

λ

2

; this function attains a maximum when

λ

2

is the abscissa of the bottom point of an unstable boundary graphed on the plane

λ, q. The condition

λ

h

2

=

0 gives for this point the relation

λ

q

2

=

1

32

Therefore, at the bottom point of the first unstable boundary one has

U

= cos

t

− sin

t

+

q

16

(cos 3

t

− sin 3

t

)

+

24

q

2

32

(cos 5

t

− sin 5

t

9 cos 3

t

9 sin 3

t

)

+

....

V

= cos

t

+ sin

t

+

q

16

(

c

0

s

3

t

+

+ sin 3

t

+

)

+

24

q

2

+

32

k

(cos

+

5

t

+ sin

+

5

t

9

= cos 3

t

3

t

)

h

=

q

3

4

According to the general theory, the equation

1024

Ly

+

q

3

2

δ

y

&

+

+

...

ξ

y

=

0

(6)

(1) possesses, on account of the small values of

δ, ξ, some solutions of the form equations

y

1

=

e kt

ϕ

,

y

2

=

e

[

k

+

2 (

h

+

δ

)

]

t

ψ

, (2) k being real and ϕ, ψ some periodic functions of period π or 2π. These functions satisfy, respectively, the

£ ϕ

2

δ ϕ

& 2

δ

k

ϕ

2 ϕ

& 2

hk

ϕ ξϕ

0

+

9 sin

+

...

(3)

£ *

ψ

2

δ ψ

&

+

2

δ

k

ψ

+

k

2

ψ

2

k

ψ

&

+

2

hk

ψ

+

ξψ

=

0 (3*)

7*

178

Their study will reduce down to those inhomogeneous equations of the type

Ly

=

f or L

*

y

=

f

f being periodic with the same period as u, v. The general solutions of these equations are, respectively,

y

=

2

1

ω

u

t

0

fvd

τ

+

e

2

ht v

t

0

fue

2

ht d

τ

+

Au

+

Be

2

ht v y

*

=

2

1

ω

v

t

0

fud

τ

+

e

2

ht u

t

0

fve

2

ht d

τ

+

Av

+

Be

2

ht u

An examination of these formulas shows that some periodic solutions exist when, and only when, f is orthogonal to v (equation 3) or u (equation 3*).

The periodic solutions form continuous ensembles of the type

y

=

1

2

ω

v

(

fu

)

2

h

+

w

+

Au

or

y

*

=

2

1

ω

u

(

fv

2

h

)

+

w

*

+

Av

.

The periodic functions w, w* remaining finite as a) We first suppose that

ξ = 0 and write

h

Since the corrections ϕ

1

,

ψ

L

ϕ

1

= −

2

u

&

+

2

k k

ϕ

ψ

1

u

&

=

=

=

+

u v

2

k

+

1

+

δ

hk

δϕ

δψ

1

+

u

&

1

,

1

k

+

2

+

δ

...

...

2

0

L

+

,

*

...

1

which satisfy, respectively, the inhomogeneous equations

ψ

1

=

2

v

&

2

k

1

v

&

+

2

hk

1

(

uv

)

=

0 are periodic, it is necessary, according to what has been said, to choose k in such a way to fulfill the conditions

2 ( 1

k

1

)(

u v

&

)

2

hk

2

(

uv

)

=

0 , 2 ( 1

k

1

)(

v u

&

)

+

2

hk

1

(

uv

)

=

0 which give

k

1

=

1

+

h

1

(

uv

)

=

1

h

1

(

uv

)

.

(

u

&

v

) (

v

&

u

)

The periodic corrections ϕ

1

,

ψ and remain finite when

h

ϕ

1

1

have the form

0

=

.

2

1

ω

{

k

1

v

+

w

}

, b) For the time being, let

δ = 0. We write

k

=

a

ξ

ψ

+

1

βξ

=

2

1

2

ω

+

...

{

k

1

u

+

w

*

} ϕ

ψ

=

=

u v

+

+

ξϕ

1

ξψ

1

+

+

...

...

The equations which determine ϕ

1

,

ψ

L

ϕ

&

1

=

2

α

+

1

are

2

h

α

u

+

u

;

L

*

ψ

1

= −

2

α

v

&

+

2

h

α

v

+

v

179

The condition of the periodicity of ϕ

1

,

ψ

1

gives

α

=

(

uv

)

2

[

h

(

uv

)

)

]

= −

2

[

(

uv h

(

uv

)

+

)

(

u v

&

)

]

(

u

&

v

If the system Ly = 0 exists at the lower point of an unstable region, k should be positive for all the values of

ξ different from zero; therefore in this case α = 0, (uv) = 0, β > 0, ϕ

1

*,

ψ

1

* remain finite when ϕ β

4

h

1

ω

h

v

0

+

β ϕ

1

*; ϕ

ψ

. In general

u

1 ϕ

=

1

v

4

ψ

u

ω

h

0

+

,

ψ

ψ

β

1

1

u

*

0

When

α = 0, the equations which determine the latter corrections are

L

2

= ϕ

1

=

2

u

&

2

h

+

1

;

L

2

=

2

v

&

+

2

h

.

β

v

+

ψ

1

They give c) Finally, let

δ

0

β

=

, 0

(

v

ϕ

1

2

ω

)

= −

(

u

ψ

2

ω

1

)

ξ

, and (uv) = 0. The terms that depend on

δ are comparable to those which depend on

ξ, thus it is required that δ be of the same order of magnitude as

ξ

2

. Let us pose that

ξ

=

µξ

1

,

δ

= ϕ

ψ

µ

=

2

u v

δ

+

2

,

µξ

µξ ϕ

ψ

k

=

µ

2

k

2

= +

1 1

+

...

1 1

+

...

µ being a small parameter.

The first corrections are the same as for

δ = 0. The second corrections satisfy the equations

L

ϕ

2

=

2

δ

2

u

&

+

2

k

2

u

&

+

2

hk

these give terms on the order of

µ

3

2

u

+

ξ

1

2 ϕ

1

;

k

=

L

*

ψ

δ

2

=

βξ

2

2

δ

2

v

&

2

k

2

v

&

+

2

hk

2

v

+

ξ

1

2

ψ

1

6. We suppose that, in the equation

y

&&

+

[

1

+

η

+

qm

(

t

)

]

y

+

2

g

&

=

0 (1) q,

η, g are small positive quantities of the same order, and that m is a normalized function of period

π. There exists a solution of the form

y

=

e xt

ϕ

(t ) (2) ϕ being a periodic solution (of period 2π) of the equation ϕ

&&

+

2 (

x

+

g

) ϕ

&

+

(

x

2 +

2

xg

+

2

g

2

) ϕ

+

[

1

+

η

We pose a solution of the form ϕ

=

A

cos

t

+

B

sin

t

+

q

ϕ

1

+

...

In requiring ϕ

1

to be periodic, one obtains the following formula

+

qm

(

t

)

] ϕ

=

0

x

1 , 2

= −

g

±

(

a

2

+

b

2

)

q

2

1

4

η

2

a

=

1

2

π

2

π

0

m

(

t

) cos 2

tdt

,

b

=

1

2

π

2

π

0

m

(

t

) sin 2

tdt

180

Let g = 0; the solution (2) is then quasi-periodic if

a

2

+

b

2

q

<

η

2

and unstable (parametric excitation!) if

a

2

+

b

2

q

>

η

2

. The lower part of the unstable region of a frictionless system is included between the straight lines

q

= ±

Now let

g

2

η

a

2 +

b

2

0

; when

.

a

2

+

b

2

q

<

η

2

, equation (2) represents a modulated oscillation for

gt

η

2 2

η

2 which the damping factor is

e

; when

<

a

+

b

2

q

<

g

+

2 4

0 < x

1

< g; the parameter’s modulation reduces the damping; finally, when

, the exponents x are real and

a

2 +

b

2

q

>

g

2 +

η

2

, x becomes positive: we are in the unstable region of a system with friction.

4

The space between the straight lines

q

= ±

a

2

η

b

2

and the hyperbola

(

a

2 +

b

2

)

q

2 =

g

2 +

2

+

— a territory where the parametric instability has been overcome by the friction — is the region of

“parametric regeneration.”

η

4

2

In all our formulas q is affected by the factor

a

2

+

b

2

, which is a function of m(t). If one accepts the definition

a

2 most efficient is sinusoidal.

+

b

2

as the efficiency of the modulation m(t), then the modulation which is

References

*

1

L. Mandelstam: “ Some Related Questions on Electrical Oscillatory Systems and of Radio-

Technology”, (Compendium of the Conference for the Study of Oscillations), Moscow-Leningrad,

1933 and [Successes of the Physical Sciences] 13, 1933, No. 2, p. 161.

2

A series of works concerning “parametric regeneration” will appear continuously in [News of the

Weak Current Electro-industry]).

3

L. Mandelstam and N. Papalexi, Journal of Technical Physics, 4, 1934, No. 1, p. 5. –V. Lazarev, ibid., p. 30, ---V. Gouliaev and V. Migouline, ibid., p. 48.

4

G. Gorelik, Physics Journal of the Soviet Union, 4, 1933, Vo. 3, p. 569.

5

Rayleigh, Theory of Sound, 1, London, 1929, p. 66, 68b.

6

M. Born, Optics, Berlin, 1933, p. 82.

7

Timoshenko, The Theory of Oscillations in Industrial Mechanics, Moscow-Leningrad, 1931 (in

8

Russian.) (There exist some English and German versions.)

A. Andronov and M. Leontovitch, Journal of the Physical and Chemical Societies, Physics

9 section/part, 59, 1927, No. 5-6, p. 429 (in Russian).

M.J.O. Strutt, Lame-Mathieu and Applied Functions, Berlin, 1932.

10

G. Gorelik, [Journal of Technical Physics], 4, 1934, No. 10 and 5, 1935, Nos. 2 and 3.

11

G. Gorelik and G. Hintz. Journal for High Frequency Technology, 38, 1931, Vol. 6, p. 222.

12

G. Gorelik, Journal of Technical Physics, 3, 1933, No. 1, p. 110.

13

Courant-Hilbert, Mathematical Methods of Physics, 2 nd

edition, Berlin 1931, p. 238.

*

The list of references is far from being a complete bibliography on the subject. One will find abundant documentation on linear systems with periodic parameters in the work of M.J.O. Strutt

9

.

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