ASYMPTOTIC METHODS IN THE THEORY OF NONLINEAR OSCILLATIONS

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STAT
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ASYMPTOTIC IIET OUS IN 1NE TNLONY OF
1O?ILINE'Ap OSCILLATIONS
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This book is devoted to the approxiwate asymptotic methods of
soliiap the problems in the theory of eosiinesr oscillations met in
Maur fields of physics end enRineerinR. It is intended for the wide
circle of engineering-technical end scientific workers who are con-
cerned with oscillatory processes.
STAT
0
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Preface .,,...,,.,1 .,...H...
TABLE OF CON'TE$TS
? t ro ~ tia~ P V . . i . . . . . . . . . . . > . . . . . . . . . . . . . . ? < . . . . . . . . . . . . . . . . Y . . . . . , ,
Chapter I Natural. Clscillationy in Qussi-Linear Syxt.ea~s .. 42
Section I. C nstruction of Asymptotic Solutions ,....?....?.
Section 2. Conservative (quasi-Linea Syatews ...................... 60
Section 3. The Case of Nonlinear Friction 75
Section 4. Self-Sustained Oscillatory Syatess ....................... 83
Section S. Stationary A+epl i Ludes and Their Stabi 1 i ty .............. 90
Section 6,?Construction of Stationary Solutions ................... 105
Section 7. Equivalent Linearization of 111onlinear Cscillatory
Systems ..... ......... 116
tkrspter I.i The Method of the Pt-ase Pane ...................... 135
Section B. Patk on the Risse r'lane. Sinqultr points ............. 135
Section 9. The Lienard Method .................................... 161
Section 10.. Relaxation Oscillatory Syste.a 178
Section 11. A.Ai1Orodnitayn' x Iriethod for the ran der l'o1 Equation .. 184
Qiapter'III The Influence of External Periodic Forces ,...... .. .19?1;
Section 11,. A*yazptotic Expansions in the "Nonreson+rnt" Case 194
Seettou 13. The "Heaonant'` Cases ...................................
~ectiof .14. Influence of ? Sinusoidal 'Force on a Nonlinear
Cscillntnr ............. ....... ...
Section 15= Influence of a Sinusoidal Force on a Nonlinear System
eith a Q-aracteristic Cooposed of llectil igear Segents ..
Suction 16. Pers~etric Resonance .......... . ... . ... . . . . .
Section 17. Action of Periodic Forces on m Relaxation Systea .,....,.
Section 18...Nonli,ne$r Syateo$ with Slo.ly Varying Parameters .......
STAT
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f'ae
Ospter IV 'the Method aI the MeM .............................. 319
"Sertion 19. Eipuauona.of Mint and Nig1,er ApproxiNation in he
Method of ttie 11eaa ........ ....................... 319
Section 20. T1-e Case uI a f*p-dIy 1latating Phase .................. 348
Chapter V Junl.ification of the Asyirptotic Method .................... 356
Suction 21. Ju*ti$icetion of the 1e hod of the can ............... 366
Section 22. Neighburhoodn of points of hyuilibriue~ and of
Closed Orbi ta ......................................... 374
Section 73. f'criodic and Quasi-Periodic Solutions ................. 414
E3ibliornpl,y .......................................................... 44S
STAT
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PLEf ACE
time the questions of nonlinear oscillations are attractia
At the present t~,e
great attention in eidely varyinr fields of engineering a+~~ vy??~_`
'The methods of aaympwtic expansion in pores of s mall paraaeter are re
fect;ve means for studying nonlinr~sr oscillations.
1fY their aid, in a large number of cases of practical importwce, it i$%
to obtain reatively ornaut$tion layouts and detailed interpret$tioa $j
l s,.-T 1--e - c
character of the course of the oscillatory process.
In this rconnection, there is a definite need for a book dei~cribine tbiS;
i rin an excessive
odo.logy in the simplest possible fpm, erthout reW g
background of the reader.
li
ne*X
N. Krylov and NN.{loogolyu ov, '"Introduction to Mon
The book by
recisely these questiw
acs' (Bib1.2Q)- published in 1937 and devoted top o-?
toda, the eethods eorked out by these authors, he y
bxblrogrephxs acstclty y
41
. t
Q
sa der?bl y expanded
__~ :. 1 LLed t0 the reader's scrrtl
In this sense, the present back s -.-
se is to describe the method of asyeptptic expen$ih&
urpo
b
i
c p
as
Ltr
. of sNell parameter' in its modern forty, with respect to the problem` Mf
rmechsnic$.
'f'or this reawn, the examples given are *415 1Y of an 11luktrati11i
the body ,skes no claim ihstsoer,r to complete coverage of the theory
. ~'
.
red in it
id
e
`." or physical phenooen# aona
and five chapters.
hook eststs of an introd
"the
iii
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 11
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In Qiapter 1, natural oscillstiona in quasi linear aystrrs eith one degree of
,freedor are diacuaaed.
Chapter Z contaigs basic elerentsry infomat, r>b4 on Method of the phase
plane. FrRe oscillations in relaxation type s~yst+ 4FW discussed.
F'or the understandin~t of the question of the tr-~r+?s.wion to the discontinnuou$
trast.ent of relaxation oscillations, we see Riving the fundarental propositions o
the aethod of the large psraMeter developed by kA.Clorodnitsyn.
O apter 3 is devoted to a study of oscillatory syntesas under the influence of
exter.nel periodic forces.
C apR+aa I describes the auethods of the ,ean, by the and of which istewa with
racy degrees of frrador can be coraidered.
These four chapters are written for a reader faeiliar with matheimatics to tlrt
extent covered b=g the norm/ course of Polytechnic Institutea,
Chapter S is intended for .atheaaticiaus who are interested in questions of
theory of differential equations with a wall parawteter. In it questions of the
jUstificatioa of asyrptotic tethods are discussed and a series of theores~s on thin
existence and stability of periodic and quasi-periodic solutions are eatablished.~
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I NTIiOIIDCT l DN
1. The study of oscillatory processes is of fundamental importance for the
widely varying branches of mechanics, physics, aid engineering, The vib-rstionsof
structures and machines, the electromagnetic oscillations in radio engineering and
optics, self-sustained oscillations in systems of automatic control and servosyst
sonic and ultrasonic oscillations all these oscillatory processes, seemingly dif
ferent, with no resemblance whatever to e'*ch other, are correlated by the methods
mathematical physics into one general doctrine of oscillations.
It should be noted that, as science sad engineering develop, the role of the.;;
doctrine of oscillations is also rapidly expanding Disregarding such disciples
as radio engineering and acoustics which have been completely "covered" by .he
trine of oscillations, let us take something like s~achine building as a typical
ample. It is not so long ago that no particular importance ass attached in this
field to the study of oscillatione, and stress calculations *ere based on static
concepts on the relation between deformation and load. However, with the tend
tosard increasing rotational speeds and decreasing dimensions, the role of oaci
tions in. the transition to high-speed machine building canna longer be disre
,:The ammeroas accidents, due to the increased actual loads produced by the exci
.of oscillations, have made it imperative for designers and engineers to study,
scare, the possible vaoratrona of machine IZarts, and to estimate their intensac
The sources of the am theory of oscillation are clearly defined in
classical, mechanics of tb. times of Ga111t o, lluygens, Newton,
lrlot~ion of the pemdulua.. The works of La raage contain already formulstion i-
theory of small oscillstross. In its further development, the term "theory
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STAT
;=?"RM~rit- wr ? ~.~ Y- .w4. .w.w.u rArx.,as,urai,.
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11
'liaesr oscillations" was useda i,~-, oacillstiona characteris.d by linear differea?
tisl equstions rith'onatent coeffiientai .ith either homosenaous or tier terms,
Leis* known functions of time,
In the work of many scientists, linear differential equations have been an ef-
ficiast tool of research. Thus, A.1Lkrylav and his students, developing the theory
of 11h.sroscillstions, spplied.it successfully to the solution of the problem of
roll of s ship, to the theory of the Gyroscope, and to artillery problems.
The siaplicityof the basic principles of the theory of liner &,fferentisl
equations with constant coefficients has resulted in a larKe aMount of .ork done o%
0 and
di.
fi
t
minus sign ford c 0, since the force of friction is always directed opposite to
the velocity o f the body.
Let us asslsr further that, this same body suspended on the spring, is not
jected to the square-la, frictioa but to Coulomb friction.
Then the equation of Lotion will be
1
to !nexample, illustrating the estimation of the forces of internal f
,rich
Zia consider the vertical oscillations of a.certain maaa m auspended frog.
a
STAT
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g g an arr,
and assuming that -the force of friction is proportional to the reloci.
the absolute value of the force of friction,and
dx
dl
dx
'I'
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?lsagth l sad cross section F, with a modulus of elasticity p (fig,6)?.
ht ma sssutse that the assa a can execute only vertical oscillations, while the
mess of the rod, which in our cans plays the role of the spring, is small by compari -
see'with the suspended mass a, Then the system may be considered as a system with
one degree of freedom. In setting up the differential equation of motion of the os?
`cillating mass m, we will take into consideration the losses of energy of oscills,
lion due to internal di f fusion in the material o,f the rod, but, we sill neglect the
mass of the rod itself.
Let *be the rlstiwe elongation of the rod. Ne then obtain an equation in the
dissipation in the material of the rod, is in f
pressed by two differential equations.
~e will assume that, the deviation of the relation between the stress a
deformation x in the asterial of the rod differs little from the linear /look*.
a This example has been considered in detail by G"S.Piaarenko (Bib1.34).
study
he dev
l
d
l
e
ope
a genera
,
method persittinthe influence of interaej~
ties to...be. a y
twlesdan s sirs with either a finite or en infinite numb
yew of freed..
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rirl'f ~'t
the natural frequency of the linear system, 2$(*) a function
allowing for the demping in the material of the rods;
its rslue being different for the ascending and de=
scending motions. In the equation, this is indicat
-~ by the two opposite arrows.
w
Thus the process of oscillation of the s st
r~. ?~ under consideration, in which energy is being loo
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n this respect, .q.(38) will be clone to linear,
adic` force co sting of a single hsr bnic, then we obtsiia the"folluwing egsr
ootioa:
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b sow present sn explicit expression for the function W24(x), Let us assume
that, for vertical oscillations of the mass m, suspended on an elastic rod, the
hysteresis loop constructed in the coordinates: x ? relative elongation and a
al stress, will be symmetrical (Fi gj ?) .
Io this case, the true modulus of elasticity will he variable, and according to
N.llavidenkov's hypothesis, the expression for the function w24(x) will hive the
fallowing fare:
where v and n are constants determined experimentally, while a is the amplitude of
oscillations,
In the above-considered oscillatory cases, the dissipative forces (forces of
friction) have sometimes not been takes
into account; however, in reality the
as a result of which the oscillations
damped in time.
Undamped oscillations can, in pre
tice, exist in a case, where there is
certain source of energy in the sys
which ii able to compensate the expenditure of energy created by the presence of
dissipative forces,
Srch a source of energy may be a periodic force acting on the oscillatory~
tea. For example, if, an ordinary linear vibrator is subjected to an external,
 Self sustained oscillatory syater~s are widely encountered, and are rarq
STAT
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rcoordiag to which uadamped oscillations will exist in the oscillatory system, In a
giien,,caaa, the losses due to friction csysed by the presence of the disaipstive
~
A
!teas will be cowPeWasted because of the externally generated enerwy and charsc? n
at C
tensed by the periodic tens t'. sin n.
The source of enemy in itself aay hive no definite periodicity, while its ac-
tion on the oscillatory syste~a plays a role similar to that of negative friction,
.hich co.penaates the ordinary positive fric-
Lion introduced by the diaaipative forces.
Oscillations of this type, .hick differ;
substantially fros the above case of the
presence of a periodic source of energy, are
called ad f?sustained oscillations.
1
In self?austained oscillatory systema,
poder certain conditions, the equilibrium
position loses stability, and motion occurs
bringing the system into a state of stationary periodic oscillation (i.e., oscilla
- tiona having a constant amplitude and phase).
For the realization of a stationary periodic state, the system iuat con_ist
three parts: an oscillatory system; a certain source of energy controlling the o
dilatory system, .hose action on the system compensate: the losses due to frig
wakes the position f' unstable, and causes the oscillations to iac]
,~ad o certain limiter,' .hich bringa these increasing oscil lationa into a statii
The first two parts of the system may be linear, but the oscillstioa his>f
lshays naaliaesr; . therefore, say self?auatained oscillatory systa~ is descri,
jaoalinesr differentia cation.
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portent in physiaaand enineeria~,
To et a clear idea on the character of the excitation of oscillations in a
self-austaiaed oscillatory rysta., let us consider the oscillations of ? systr~ with
ONe depree of freedon,
If the oscillation. are of asall amplitude, re can use the linear differential
equation
kx o. (41)
.t' ,.. (fr r i OS (~uf
where a and a are integration constant,,
G . _. I
?112
Therefore, if > U, the w~plitude of the well oacillstiona se'bt will obwi?
ously Ge duped by an exponentioi law, If, on the other hand, < 0, the aaall oa
cillstions rill continue, and their amplitude will increase try an exponential 1ar,?
In ri+ew of the fact that the +waplitude of oacillation cannot increase rithout
iimit, it is natural to postulate that, be~inniny at &OIse instant, the coefficaen'
of dwpinsuit change its and become positive.
Tim property of an oscillatory syate?o say be reflected in the differential
equation of the oscillationa by replacing the constant coefficient by a raritb
cwaifciant, for exwpie, of the following forty:
iron ibich it foilows that the daopin1 is ne~stire for small abaolute values,.
aPC
and "pooit re for far a ebaolute values. Tea, oscillations of ssall ampllt
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coatinue, sad those of large amplitude will be damped,
'Ibis iadicatea that there exist undamped aelf?auatained oacillationa toward
which oscillations with both small and large amplitudes tend.
Equation (15) is called the Naylei~ equation and is of major importance in
thaorq of self-sustsined oscillations,
Equation (45), by subatitution of the variables
may be reduced to the folloring fore:
The equation of self-sustained oscillations, represented in this form, is
called the van der Pol equation.
We now present concrete examples of self-sustained oscillatory syste~us.
consider the electronic oscillator schematically shown in fig,8.
Let it, and ig be, respectively, the currents of the inductance L, thi
capacitance C and the resistance 11 of an oscillatory circuit.
Let b'a be the constant voltage in the plate circuit, Vs be
the total vol
on the plate of a vacuum electron tube; Vg the grid voltage; is the plate c
r
M.the coefficient of mutual inductance between the grid circuit and the o?cj
according to the diagram in Fig.8, .neglectin~c the grid current, we hair
~_~?._..._ __._.,,r._..~
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rllJ 1 ,its
L: .ii ?.' ~ -I.- ~._ ,f,. i
rll,~. .~~. _~tt 1,
As is we11 known, the plate current is is a definite function of the control
roltaRe u'' V + 1)V1, i.e.
J(ii) 1 ~V,,'f.I)t(52)
where t is a conatsnt called the grid through of the electron tube. The numerical
slue of I is usually nail with respect to unity.
(bi subatituting eq.($2) in e,.(51) sad taking account of eq.(48) and eq.(49),
Obviously V ii the alternating coaponent of the control voltage u, excited,
the oscillations of the current in the oscillatory circuit, while En is the c
composeAt, excited by the source of direct current.
Th.,
n for n unk e-~! V , eq'twv~ ~a I53) y:.1d c+YxG/71 QI ~ 6n ` following
nn t~1. R o
(;41 ! 1))~i'(If
,,...3are obtain the following equation for the electronic oscillator in di~ear
for.:
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opd, .oeiequeatly, we can write the diffetential equation
r
1 dI f~
~
_-.
As is. well known; the plate current is is a definite function of the control
voltage u ' Vg + DVa, i.e.
where D is a constant called the grid through of the electron tube. The nus~erical
value of U is usually small with respect to unity,
substituting eq.(52) in eq. (51) and taking account of eq, (48) and eq. (49),
dl Li))!!...
Obviously V is the alternating component of the control voltage u, excited
;the oscillations of the current in the oscillatory circuit, while Eo is the coii
compoa.Nt, excited by the source of direct current.
"Then, for an unkaowe %, eq:(53) yield; an equation of the following fox.
?we obtain tho following equation for the electronic oscillator in di.ljfl$L
.. foes
rr.1t
dya
.W?{WWIW4Yrt'Y4'?~iCl%WW W.:r'.::
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we then obtain
li (/' ,
l1 ~-
Applying eq,(2,21) to this equation, we find (confining ourselves
two terns in the expansion of the sine):
' *e note here that the difference
+H) ,r ? .r
(2. 31)
to the first
does not exceed 0,000326 in absolute value, if x oscillates between - 30 and
+ 30', while the difference
x'- .;~
alt/ X -- X - ,..... -t.. "MI""
A
daea,..not exceed 0000002.
STAT
 A
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~
"' I
whence
xli
l1 COS 'i
ns
ili
It is obvio'.a from eq.(2.33) that an increase, in the m7plitude of the oscilla-
tion of the pendulum causeS a decrease in the frequency, while the period of the
natural oscillations
_I
increases. (Here 70 ? -- ? 2+~ -- )?
wo g
To construct the solution in second approxisation, let u5 utilize eq.(2.23);
z
then, takintt account in the expansion for sin x of the tern
whence
(a)
(2. 32)
(2.33)
(2.34)
U' rt' 1 .! COS 3'J
TI'
0
a COS
we obtain
.. r
214(i i()S J'',
p3 ,~1
ii; + 1112
. a~ ~a4
Tr 1024)
For the max mum deviations, eq.(2.24) yields
(2.35)
(2.36)
The esultsnt lotiuiules can be used for calculating the frequenciesperiods,
and oaxium,deviations fur a number o,f aalues of a (amplitudes of the first har-
70
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2n
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I
I lI
,a
.,.
;-,IN~I
1),(n
,
I 17)
2
1
,~1M
11,!11)
il,lrr, l
,
)
?
32
U ,7S
0,h22I
E 1?11(
I,:l;;l-I
Il,hi
tt,:,Ili(i
1,"+~-t
I
I
A comparison of Table 2 with Table 1 shows readily that, for deflections of
the pendulua not exceeding t 35' (within these liai ta, the frequencies and waxiaun
deviations spree with the exact values to the fifth and fourth decimal place in-
clusive), we may successfully use eq,(2.31) and the corresponding simpler approxi-
mate solutions (2.32) sad (2.35) instead of the exact equation (2.25). At greater
angles of deflection, of the order of t 1iO the relative error of the first ap-
proxiaation amounts to 13% and of the second, to only about 3%.
In considering tit free oscillations of the pendular, we have disregarded the
' forces of friction.,
Assuming that the oscillations of the pendulum are darped under the action of
. forces proportional to. the velocity, we arrive at the investigation of the followia
s
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Table 2
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Let us now consider the oscillstioas of a system in ehich the characteristic o
the restoring elantic force has the for.
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According to the teaerel foruiula of Section 1, the solution of eq.(2.40) will
L. in first approximation
X - It CO 'i, (2.41)
there a and w Bust be deteriuned froe, the system of equationa of the first approxi-
^ation
rl(1
+a(1,
'U
where re have introduced the synLol & ' .
2
On inteRratin~ the first equation of the systec (2.42) at the initial relues
of t ? 0, a ? ap, we find
After this, the second equation of the systess (2.42) yields
where 6 is the initial value of the phase.
Ci substituting the values of the aaplitude (2.43), and of the phase (2.44) in
eq?(2:41), ee obtain the first approxisation in the for*:
a`
x ii' ;' C:OS u41 1 (,, : r I 4. 1J. (2.45)
. Thus, in. first approximation, the osti 11ations .i11 Le dasped, and will hare.
a frequency depending on the .litudc W ? W(e); with increasing tiaae, the gradual
~drpiag sill cause the instantaneous frequency to increase, spproach*ng,. as a liait
the constant"linen" value of the frequency nr ?
 In this caae, we obtain the nonlinear differentia) equation
ehich way be inteprstcd in explicit for. lay elliptic functions. Q nsequastly, here
too the approximate solutions ansy be conpared rith the exact solutions.
Since, for this case, the expwnsiun (2.17) will be
((1 LOS ~If COS'a ...... , , COS ?;'t
the introduction of the diarension1eas coa~binations and the use
a
of egs.(2.21) and (2.23) will Mire
in first approxi*Ition.
in second approxiaration:
tiv
.1~1) ~ ~ 11 lO5'i
"~ fff
N1' w
3' ( ?::~
Me find the exact value! of O '~ x^ax and W for given values of (I)1' a f
a era at rose
'fables of elliptic functions, by using the forsiuls
X ('b, a) .. Xq~=+ r ' cit
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Their numerical value will
STAT
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Here cn,
and q denote respectively the elliptic cosine, the a-odulua, the total
0,29927
II
1.4592
1,7 h43
2,0'11
?,.3140
1,19!)1
2,$S II
1
e i, ' 3i
+
1IrmL lirra~
O,;jf1(Nl.i
0,1
(I,599M
11,1 i
0,899.!
U,! )
I,l!)I
12
I,1!)(ili
I,;'i
1,794$
I M
2j4P931
2I
1,3!11 2$
2,9$77
1,113;1
1,0331
1,0331
I,I2IM
1,12,i$
1,1259
I
I1,2641
11~';di ;
I,1;3;i3 1,?4;310
ILi
I,t 241 1,(i257
I,,+lllli
2,022
l,h2!I'7
2,(h$T9 2,0)9:3
10
2,2ii97 : 2,2710
2,6,11) 1
2,731M 2,73s,
0esrin in mina tI*e simplicity of egs,(2.48) and (2.49) we must also recolpnize
in this exarple that the degree of approxiastion obtained is entirely satiaafactory.
It cad. also be shown that. these formulas are still valid even when a (1) lj2
a
?1~2 ;t \i
where q is t.. for the modulus It ' 1 , while the dote denote a teas whose ratio
11 fi
to the ibope first tern approschea zero as a (Y)~2 . Analogous asymptotic forms*;
las are obtained Eras egs.(2,4$) snd (2,49) for MI(a) and reapT ectirely1 w'
L u ~ nvx
equation (2.51) lead. to the follorinK asymptotic for*ule
 1
uwsI1I" ).M~ r
y _ 'rJ
co.psrinp the litter expansion with eq.(1.16), +. find
gig (a) ,.. Fri (uu) cos ' ~d d (a) ,.
or this . reaaoa,' eq. (1.I7) sill yield
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1
Tu., at the halt .hen a ()/2 ? ?, the relative error of the first epproxi...
a
tion of frequency count. to 2.4%, and that of the second a+proximation to only 0.6%.
Section 3: The Cese of !Nonlinear friction
As a second specisl case, let u? consider an equation of the tore
+l ~'.r rLt'
J
ni k.t- (ft
which say be interpreted as the equation. of oscillation of the saaa m un1er the ac-
tion of the linear elastic force Mx end the nonlinear .e.k friction of (d"), de-
pending on the velocity.
This equation obviously belong to the type of the Reneral eq.(1.1), .bile here
.~ , 1 I. L ) .
In order to Mahe use of eq,(1,21)-(1.28) for detez inin~ the wanted approximate
solutions, let us.conair the expansion
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Thus, taking egs.(1.23) aid (1.21) into account, we obtain the first approxias=
ties in the follawiwi faro;
This readily shoes that, for these systems described by an equation of the type
of eq.(3.1), the esplitude of oscillation in first approximation is daaMped by a law
expressed'by the first equation of the system (3.5). As for the instantsneous fre-
.quucy, it is eonstaat and equal to the ordinary linear frequency rr, so that
? =wt f, li,
'here 8 is the initial value of the phase q.
Thus, in. first approximation, the oscil!:tions are found to be hu!wwaic, at a
constant frequency rr.
ge have already had an opportunity to prove that nonlinear oscillatory aystems,
peserelly speaking, are not isochrosous.
The exsrple under consideration, however, is one of the important cases when,
is fiat approximation, the syatem is isochronous. Such cases will be denoted as
quasi ?isochroa0ua~ .
Let: us discuss the construction of the second approximation: Equations (1.28)
d (3.3) yield
J F'( ? aw sin j sus ek sfn .dir...
?? 1. add the" quasi" oscasse" the correspsdin oscillations of the
system, as wi
be shown below, wt~ "l l
.. be isockroaous only is ftrat approximation,
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 4t / 1)
? ', .
I t" ( uw Sin
21'?af'1
IP
e a aubatitrtion of e - for in the integrals, *ires.
u
Further, integrating by parts, .e get
~R
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f' Slli$ (
ii,,, in
say thus be written ii .follows;
i),, (u) :': _ 0,
:.1' (uw) d1 ' 1(a.') 1' i (aw)
sin 1l'(1
the second approxi.atioei has the. ion
t 1. 1(aw)co(+
-
t)
Dl. f
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I rl!'~,(d4b) X
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ehere ,(s) is d.termined from eq. (3.8).
Before diicusiing the analysis of the equations for the amplitude as a function
of time for various Is.. of the force of friction, i.e., for various forma of the
.function F (dx).,ce note that, for these formulas to be applicable, a general limi-
tation is necessary, namely that the force of friction must be sufficiently amsll.
In describing the analysis of concrete exasples, let us first consider the
linear equation
For this equation
and, therefore,
. w 'X
d ~'y AI
r(aw) 0 (u . U, w
As indicated by, he first equation of the syater (3.12), full a~ree.ent is ob-
4taiited bebreei the lam of despisi of the amplitude and the exact formula
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while, for the oscillatia frequency, we hafe, the approximate formula
which corresponds to the first two suar#anda in the expansion of the exact expreuion
for the frequency
in powers of-., .hich, hoaever is entirely natural, since we are diaregarding terms
of in order of amx]heaa hither than the second.
dx
,y'-v~. 0. tc:iii Ptr U,
It.
where. a always .~;~? deaater the absolute value of . (we resort to this no
d
tton to rq i,cate that the terra $ ( ) represents a resistance to motion).
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To get an idea of the degree of accuracy of the resultant approximate eq.(3.13),
let us take, for example, ' )R2 . Me note that this value of the coefficient
cocreapoc~ds to a considerable damping. Thus, in one period, the amplitude of the
oscillations diminishes to half, in absolute value, the "perturbation term' X
: dt
still is about 4 of the "principal terns" i or w x. In spite of this, the rela-
dt
tine error of eq.(3.3) is less than 0.01.
Let us consider another simple example leading to an equation of the type of
eq.(3.1) namely a haraosic, or any small oscillation, of a pendulum in a medium
whose resistance is proportional to the square of the velocity and is anal!.
Is chit case, the equation of the oscillation will be
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Tea eq,(3.1S) will be an equation of the fora of eq.(3.l), where
K~rrY'~
Thus egs.(3.8). (3.9),
Ii ll)S'e
and (3.10) will give
7"(7' 'p
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d.r ~
u
d1 i'
Let us find the e!cpreaaion of the nth term in the Courier expansion for
i'(a cos
whence
Coaaideria~t the damping sufficiently week, we use
the second approxiaation in the
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Let us now coipare the resultant sppp?oximate solution with the exact solution:
81
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here,
for abbreristion, we introduce the ayaibola
00
?F )2
n,
... 0,0.401. , .
Ly integrating the first equation of the systew (3,11),
12150
r!
whence we find the law of drpinR of the
the oscillation:
(3.20)
In this way, the amplitude of the oscillations, for the square law of daaping,
is damped approxisately inversely proportional to the increase of the linesr func-
tion of time.
On subatituting eq.(3.20) in the second of the equstiona (3.17), and integrat-
ian, we obtain the law of rotation of the phase angle
-
TI . I
t~~P 'I~ar1? I ;
xRa~p+i .
i
(3.21)
Thus, we have explicit expressions for the representation of the oscillatory
process, in second approximation.
pe note that tie correction terms of the second approximation are very saa11,
even at considerable damping. 'Ihw, if we take ~-s~ ' , i.e., if we consider the
case in which the *.plitude a, one cycle after the beginning of the oscillations,
is reduced to one half, then the sam of the amplitudes of all first harmonics of th
oscillations will suount to less than 1% of the amplitude of the fundeeental her-
monic; but the correction of the second approxi. ation for the fregsency of the os-
cillationa will. be less than 0.25%..
we have
(3,18)
1., -. t (3. 19)
amplitude of the fundrental harmonic of
 l ,.
I Pit,
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Although eq.(3.15) can be inte*rated to the end, as a result, we would arrive
at the transcendental quadrature
ri r1 z ,.,
S 1 `:: .M, ..~..,. a., (3.22)
(where x denotes the distance between the extreme positions of the oscillating pendu?
lum) and thus the function required can;iot be represented by the aid of the eleaen-
tary function.. It is not, however, difficult to establish the equation for two suc-
ceasive aiaplitudes dasped by the presence of a friction proportional to the square
of the velocity. In accordance with I.hrasil? we have
or, in our notation,
(I7o,, 1) In ()"u,, ~..!) (Ixu~ I~ iii 1 ',
I), (3.24)
where a? is the initial value of a.plitude and a~ the value of the awplitude after
one period of oscillation has elapsed.
In order to coapare the results obtained by the exact formula (3.24) with those
of the approxiuate forwula (3.19), let us transforr eq.(3.19). It may obviously be
represented in the followin for.:
Q- substituting, in the right-hand side, the value of the period in first approxioa-
tion, we obtain the following relation which connects two successive amplitude*'?:
I l
Ii~J1
The Table given below shoos the food apres,ent of the successive ??plitudes
calculated by the exact. eq.(3.24) and the approximate formula. for 4aso
? h.Praail (Bibl.46)
" We note that the awe approximate formula was empirically found by A,de Ca1i~a
(Bibl..4?).
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 dx
'it
b1 caparinp eq. (', l) 'pith eq. (1.1)', re have
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that eq.(3.I). previously
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for the csae whnn the rw.plituda decresaea to 0.6 of its value after one cycle, the
results obtained Eros ea.(3.26) (s ich characteri:ea only the first approximation)
I,INNN)
0,54 :i?)
O, I24)
0.,33111
0,`1701
O,22 K)
0,I~)S(i
O, I i.-:3
0,1,'i70
0,11?0
rl, I'~1f3
0,11?~1
0,1 I(Wi
0,10,31)
O,(M 4i4
II,IhNNi
(1zal (4xix)
i,
0, t i7R
(I, I 't 24
0,1301
0,11(1)
0,IIII
0,10:3.1
O,(1Ni
n,rr.N~K
differ by only 1% from the exact results of eq.(3.24).
hoeeve r, it differs only by 0.4%.
rr,rMc-1
O.O$0S
O O7l-7
(()7:1O
Section 4. Self-Sustained Oscillatory Systesls
Let us consider another oscillatory syaten, vhich is described by an equation
of the fon
r4iich also is a special case of eq.(1.1), ~e note
sidered, .ay :be reduced to the for. of eq.(4,11.
So., psttiA~
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f (o ~'~-~ '~) fIs' i;
Therefore,, in order to s&ke use o.f ega.(1.21)-(1.28), the expression
sust be expanded into a. Fourier series.
To siaplify this operation, let us consider the function
(x a
and its expansion into a Fourier series
J Ala
I~ (a COS 'J) P,4(i) cos n?'.
? N
d at
'U
4'{
f7n differentiating eq.(4,3) rith respect to w, we obtain, on the basis of equs.-
dl
f(x) dx _rt;
-;;- Pr.(11), r
(4.2)
(4.3)
(4.4)
(4.6)
The results of the preceding Section say be used in conatructiag the second ap
roxiastion.
Starting Eros egs.C3;2), (3.6), (3.1), (3.8). and (4.4), and bearing in rind
at
tf(aos.') IUS'i SIIi'~i d:=
STAT
nA
av
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tion (4.2),
/(u Lo1S 'I) OW Sin
rlr
It
~f-,1?(~t) sin
ly cosparing eq.(4.4) rith egs.(1.16) and (1.1?at find
rl~ (ro) . ~ Fi (o, III (u) .. O, (4.5)
ehooce, in first spproxisation, ee have
X a COS',
There sand y suet satisfy the equations
.r
f J I .~' - (IX
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-y wtl to
11 1'41' '
+dlera a and are defined by the equations
Jr
and Br(a) has the followin~c form:
N
Qs comparing the resultant approiisate equations rith the solutions of eq.(3,1)
?ivee in the precedisR Section, we convince ourselves of their complete 1Ientity.
Thus, the system described by eq.(4,1) is likewise quail-synchronous.
As an example, let us consider the van der II equation
and, therefore,
according to which we obtaip
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11'
r (u) U, it
a~Uj~..
n .1. ii, 3.
(4. 11)
Thus, taking eq.(4.6) into consideration, we have in first approximation
X !e Cos (4.12)
where a and must be determined from the system of equations
(It ..~... ) I
SI!
dl
(4.13)
Thus in first approximation we obtain a harmonic oscillation having a conatsnt
frequency a " 1, whose amplitude varies according to the first differential
equation
of the system (4.13). To find the law of dependence of the amplitude of oscillation
on the time in explicit form, this equation must be solved. On multiplying both
sides of the first equation of the system (4.13) by a, we have
en cc
rf a`~ ~+ e '
dt
dal
....., ~t
.1
W= ; fa~3
4 _~... rgw a"a
p3
011 u'4
ere a, is the initial value of amplitude.
Frou eq, (4.15) we finally find
Bb
'I,
= dl,
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(4. 14)
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On substituting eq.(4.16) and eq.(4.12), we have an expression for the first ap"
1
roxtoation in explicit form
p
I ~ -1
l
v i'OS (wt f Ili (4.17)
As will be clear from eq.(4.17), if the initial value of the amplitude aA is
equal to zero, then the amplitude remains equal to zero for any c, and we obtain
x ' 0, i.e., a trivial ~uolution of the van der Pol equation. Thi3 trivial solution
obviously corresponds to the static state, i.e., to the absence of oscillations in
the system.
However, starting from this save foraula, it is easy to dra? the conclusion
that this static state is unstable. However amall the initial value of the ampli-
tude may be, it will still monotonously increase and approach a value equal to 2.
In this way, since accidental small shocks are unavoidable in practice, oscillations
with increasing amplitudes are automatically excited in the oscillatory system under
consideration, in a state of rest, i.e., the system is self-excited.
Equation (4.11) also shows that, if ao ? 2, then a ' 2 for any values of t. > 0.
This solution corresponds to the stationary (steady) dynamic state
X 2 Cass (l
J
In other words, any oscillation with increasing t will approach the stationary
In contrast to the static state, the dynamic state possesses excellent stabili-
ty, due to the fact that, no matter whether the value of as / 0 is larKe or amall,
a(t) ? 2 will alwnys occur while t y ~.
oscillation of eq.(4.18).
$e note that only in first approximation is it possible to represent the sta-
(4. 18)
tionary state (4.18) as a harmonic oscillation of a frequency w ? 1 and amplitude
~equal to 2. In reality, however, the stationary state is not. harmonic
Let us pass now to the construction of the second approximation. From equa
tions (4.7), (4,8)- and (4.11)- we find
(4.19)
X a :4{;' ~11i -_~,
87
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STAT
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where a sad w must be determined from the equations
P
t/il 111(1 i/
For stationery oscillations, in second approximation,
we obtain
The above simple example of an oscillatory self-excited system, described iy
the ran der Pol equation, shows the fundamental difference between this system and
the oscillatory conservative systems described by an equation of the type of equa-
tion (2.1).
More specifically, in conservative oscillatory systems, as demonstrated store,
oscillations at any constant amplitude are possible, while in self-sustained oscil-
latory systems oscillations at constant amplitude are possible only at a certain
value of this amplitude. Physically, thin is clear from the following obvious con-
sideratioas. Since, in a conservative system neither dissipation nor any source of
energy exist, the oscillations once excited can neither increase nor be damped, so
that their amplitude will remain equal to its initial value.
in self-excited systems dissipation of energy and sources of energy exist. The
;amplitude of the oscillations will therefore increase if the quantity of energy de-
livered by the source exceeds the quantity of energy dissipated by the dissipative
forces. Olt the other hand, if the quantity of energy supplied by the source is less
than the quantity of energy dissipated, then the oscillations will be damped.
However, the amplitude will remain constant only if these quantities of energy
are in exact balance.
! 0 ~i rs n wwo
??~~ .w ceustruct approximate solutions for the van der Pol equation, util-
1 zinthpe
~ method of the mesa. For this pyspore; eq. (4,10) court be reduced. to the
;standard for. This it easily done, if the unknown function x is replaced by two
STAT
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X :.:,.. a t'OS (t -f-Il ),
dx
a &n (t f I'),
qtr
giffereatiating eq.(4.22) and comparing with eq.(4.23) yields
i
,...~1) ~rAli ~iU(t '~) O.
Differentiating eq.(4.23), and taking egs.(4.22) and (4.10) into consideration,
tl~~ u { a~ , Ci)S 2(t
(hJ f-h) sin1(t 4. r1)
f- uaing the method of the wean, we obtain in first approximation
where.
M ;cc)s'2(1
S.
new fuactions a and 0, using the followiag fonwlaa for the substitution of vari?
On solving the eq.(4.24), (4.25) with respect to the derivatives, we arrive.
,.? (I) -.. It rtl ~~-S~d k
tt l
z 11 '1 It ; ~)) Ii aiU (t (1)
at the syatea of two equations in standard foes
11 '1)1tIS1l~(t y1).
t11
.
~lh w ~1i It~~t)SSIt f..~))1tilri(t F i) )t'tISIJ ~f-il.
ttl
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Obviously, the equation of first approximation (1.28), agrees with the above
obtained equation (4,13).
The refined first approximation will obviously lie
as
.s
cu
.1 (1 -+? f-, ),
~. w
cud
)cos2(t 1.-" )-+-
COS .} (t _-f. 1) ).
(4.29)
above, we have
(1(1)-I 2 at 1-' . '-
and consequently, for a steady oscillatory regime at al * 2, the formulas (4.29) of
the refined first approximation will yield
for the stationary state,
e
si
2
1
I
I1
n
1)-.,
(
-f-
sin4(1~f
J~),
X11. - cos 2 (t 4- I)~) .. cus 4 (1 .{- IJ t ),
On substituting these values in eq.(4.22), we get
[2._4sIn2(t+i)+
-
+` sin 4 (t~-fii) cos t+-"L- cos 2 (tw f M1) +- - cos 4(1
4
(4.30)
(4.31)
or, neglecting the terms of the second order of saullneas, after elementary trens-
formations, we obtain the refined approximation
x - 2 cos (t 4, IJ1) ? Sk113 (t rv' IJt) (4.32)
which agrees with the expression for the refined approximation found earlier.
Section 5. Stationary Amplitudes and Their Stability
In the preceding Sections we have obtained approximate solutions determining
the law of variationeith time of the amplitudes of the fundamental oscillation har~
!manic.
s
for any nth approxaaton, this equation will have the form
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91
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da :~ iII, (a),
dd
('5.1)
4'(a)= s 4 (a) + s2 4 (a) -}- , , ,
Mad therefore can be integrated in qusdraturea.
lbwerer, even without integration, it is possible to investigate the behavior of
the solution a ? 1(t) as s function of the properties of f(a), which will be done
below:
Let us first asra.ae that no positive quantity a' exists for which
4 (a) > 0 for any a >
It is evidently necessary to adopt this condition for purely physical consider-
it ions.
P w, if iuch a quantity a? did exist snd assuaging that the initial valuc of the
euplitude as is heater than s?, it follows that
which would mean that, in sccordaace with eq,15.I). the saplitude would increase
without limit
a (t) -s cc as t -r cx,
i.e., the oscillations would brasden without limit, which is physically ispoasible.
For this resaon, it will always be assu* ed in the following that this condi-
tioa (which msy be called the condition of limitation of Mpittudea) is satisfied.
Equstion (5.1) indicates that the amplitude increases when 4(s) ' o and de-
ers when ?(s) ' 0
The unchaagiastationary values of a ire detez*ined by the equation
(5.2)
which is obtaiaed by equating the right side of eq. (5,1) to zero.
Equation (5'.1) shoo. that, if the tntttsl value of the plitnde is not eta-
; tionsry (does not satisfy 'eq. (5.2)1, then, with increasing time, the auq-litude a(t
^onatoao sly incTOsiag, (if ?(sa) > 0) or decreasing (if 4(s?) < C1, will eoproach
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 'the question of the stability of stationary
Let us sow pass to
oscillations.
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92.
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Tux every aonatstionary oscillation, with the psassge of time, rill approech a
atstioaary atate. Nonstatianary oscillations are asuslly called unsteedy oscills-
tions, or oscillations' in a traesitioaal Mate. The fact of the appruach of any os-
cillation to $ stationary stato reveals the special role of the atationary oacilla-
tioaa, in particular,, for hi~1h?freq,epcy oscillatory processes, for which, in view of
the brevity of the period of oscillation, the transient state very rapidly ap-
proaches the atstiogary state. For this replan, oscillations of this kind any be
considered in prsctice ee being stationary almost iu.edistely after the beginning of
the oscillatory process.
Its note that there exists a case of degenerstioa, when the function i(a) is
idtatically egrsl to zero. In this case there are no transient states, and every
osciliption is atptionaryo. This cue occurs, for exaap le, when f(x, ) depends
only on x but aot on Then eq.(1.1) takes the fora
dt
d
shich has beep diseusaed in detail store,
(5.3)
This equation ^ay be integrated as the equation of oscillations of a material
point under the fiction of a force depending only on poailion and, therefore, orig ::?
ating in the potential
F (x)
dU
dX '
fF(x)dx.
u
Equatioa? (5. 3), ie'the equatioa of a conaervptive oscillatory syst+aw hprins an
enemy 1nvsr1aat throe hoot the oscillations.
In practice; however, no ordiasry oscillatory syate~i is conservptire and al-
wsys contsias dissijstira forces ceaainp the dissipation of energy; likewise,
sel[?austazUd oscillatory syst~,Ray slso contsia energy. sources.
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Aaawae that a~ is a certain root of eq.(5.2), i.e., a constant stationary nobs.
~~.+tlon of rq.(5.1). Conaidet solutioaa of eq.(3.1) infinitely close to ad. Ten,
?.~ariag
for ea isliaitely email. increment 6. (neglecting tens of s higher order of small-
ness), we obtain
which givea
Thus, the value of the amplitude under consideration is stable,
spends to a stable stationary oscillation, if
4' (AU) 11, .
i.e.,
tie-
(5.4)
4" (n,,) Q,
the corresponding stationary oscillation is obviously unstable.
In particular, since the value a, ' q, correapondin~r to the state of equilib-
rium (static state), is always a root of eq.(5.2) [by virtue of eq.(L 33)], the in-
'1" (()) . ()
fill represent the condition of self-excitation of the oscillations.
Sriting tbia in the expanded fore
l , N'
(U) _ FZ e1.(0) . f ..... f.' it, (O) ,11,
and disregarding the case when the function A1(s) may have ?eltiple roots, we see
?+ thet, for sof.fic_iently mail values of a (which however are always assuaed either
'explicitly or implicitly), the problem of self-excitation ,is solved by the sign of
a single term, namely, eA(0), i.e.,
as i
the same way as though we were desling with
iirMrrer, 'ia eccordeace with eq. (5.2), the stationary rpl tude% mu
t snttify
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A1 (at?')
~; taj0~1
Since a liven stationary state will be either stable or unstable, if respec-
i
sAe(a).f-t'A-(a) 4....4 . "A-,(a) u,
and since, because of the relation
eA i (a) 4- t=A: (a) 4- ...:. ~ :'+A (u) zA; (a'?) w~ . a ... ,
which results from eq.(5.S), the sign of its left side is determined (if a is suf-
ficiently small) by the sign of eA'(a~?~), we see that the problem of the stability
of the stationary oscillations is solved,
for sufficiently small t; by the sign of
tA'(a~'~), i.e., as though we were dealing
with en equation of the first approximation.
It must be emphasi zed that, except for
certain special cases, equations of the
first approximation lead to the sane quali-
tative results as those of higher approximations. The transition to equations,of
;hider approximations usually leads only to corrections of a quantitative character,
for example corrections is the value of the stationary smmplitude, etc.
e note that the condition of self-exeitation of the oscillation is not neces-
sary for the existence of a stable stationary state of oscillation. For this put-
`pose it is obviously sufficient for the equation of the stationary state (5,2) to
have at least one nonzero root, aatiafyinp the condition (5.3).
STAT
the equation
A~ (a)-E zA4(a) 4- 4- z"'4(a) := 0,
For this reason, disrepardinp the alave-Mentioned cases of multiple roots, ^e
can expand into a power series of the parameter s
a a(?)-f - sa"l -{- t'a+') . } ... (5.5)
where nl'1 is the root of the equation An(a) ' 0 (stationary amplitude in first ap-
proximation)
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9,
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Ia addition to as analytical study of the function 4(e), the chsrscter of the
oscillstory process cab, in ?any cases, be conveniently detersined by using graphs
of the type in liigs.25.28, which piv* 1(a) na a functio. of a.
The stationary aoplitudea here sre determined by the points of intersection of
the carve 9(a) Nith the abscissa. It is obvious that the pointa at which the de~
sceadia. branch of the curve intersects the Os .iis correspond to a stable aapli-
tude of the oacillstioas, while the points at which the ascending branch of the
Fig. 27
curve intersects the Os axis yield unstable amplitudes.
In fact, in the former ca;, ttac iapair+sent of the stationary aeplitude leads.
to $ sabsegnent ?sriatioe, canning the aaplitude to return to. its atationsry value.
In the latter cane le' hsre the opposite picture. In II a.25.28, the arrows show
the direction of the vsriation,of a. Figure 25 corresponds to the dissipative case,
Fin. 26 to the case of self-excitation wits one possible stationary asp1itude, and
}f?i*.29 to the osse of aelf-excitation with several stationery asiplitudes: nt,
ilLatrans with amplitudes a, ae, are obviously unstable).
(the osc
In`,peaeral, i'f, the function $(a) has a root s? satisfying the inequality
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ax dx
+- r,+ s x'?%4x?+r. x+ydt j (X,
Qt
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~~(?a) < b, then ? stationery stet. of oscillations with a oonstaat amplitude equal
to s? is possibls. we not. that the stationary arplitude of self?excited oecills-
Fip.28
tiona (le, the limit of a monotonously
increasia~t amplitude of oscillations for
^hich a? would be very ama11) is equal
to the aaallest of all possible ata-
?LY
tionary amplitudes. This fact becoaes
clear from the logical physical consider-
ation that an amplitude, on increasing,
cannot juwp over the stable root of the
equation 4 (a) ? 0, i.e., over the root
of this equntioe that satisfies the condition $'(a) r 0.
lifture 28 aho.s the case when the system is not aelf-exciting, but ^ay still
contain stationary oscillations. In this case, if the initial value of the ampli-
tude. a? is less than al, the oscillations will decay; if the initial value is
greeter than at, the oscillations will build up and, at the limit, rill be trans-
formed into stationary oscillations of an amplitude as.
A. an ezaa~pls, lot. is consider the equation
(5.6)
which is encouatered in the theory of vacuum-tube oscillators.
To assure the applicability of the results obtained by us fro. this equation,
let us nssuue:that the perturbrtioa term, is aufficieatly amall, and let ua put.
L ?Pa4'O
W
?
?4 T1
4Contiq""n~tly, eq.(t f') will yield the fotlowia equation of first wpproxiuation
+
1I '
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for the .Mplitede
4o
dt ' 2 8
~i c 0 the right?hMd side of this equation is positive for all
is note that, x f , s
es of a, 'Thus, in this cue, oscillations with ? sufficiently
sufficiently lame vela. which is ob-
~eat amplitude Mill broaden sithout limit, i.e., .(t) ., s t ,
~.riously i.poasible~Or physical reasons.
e that > 0, Ne further note that the. condition of
Let. us, therefore, sssua s
w`ll be ~+ < 0. Un considerifF the case of the al,sence of self-
self-excrtattoa s t
excitation, we put ~t > 0. On solving the equation
Let, on the other hand,
ampe
,
ill
will
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~. i.'' --
T -r T -r -
addition to the 'static' solution a ' 0, we also find
in
s eX
Since ~1 > 0, ) > 0, then, if Xs > 0 or if (!) < - eq.(5.7) has no posi-
tive solutions. The graph obtained by plotting against a will have the form
shown in leig.2S, indicating that oscillations of any amplitude will be draped.
. fora versus a, as obtained for this case,, is presented in h .29? Ob-
The graph
.
" a corre da to rnstable oscillations and a to stable types.
~rtously t ~
d
1.~
two possible solutions for the ss-plitude of the stationary oscillations
slder than 'I w
be d
ii.lsti0as with as initial ,wplitude i
r. w*..".` `.....mob-
97
(1V'. l/i
,t
 ` while orcill+ttiona with an initial aaaplituda greeter than ?~. will approach a arable
I
ststioasry state.
tha ease when an oscillatory systa coataina a certain
1,et Ws now consider as desired
a certain group of parameters) which may vary as slowly
paraNeter ua for
(adiabatically). In this cane, the right aide of eq.(5.1) w111 depend on ? and may
be represented is the form $(a,u).
low by comparison
Ne will conaider a variation of the parasete.r that is w s
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Fiq.29
the tranaitnt state (i.e., by ccar~pariaon with the
with the effective duratiop,of '~ lly stationary), that for
time during which an arbitrary oscillation becomes Pradica
course of this variation, the cacillatian ^ty be assumed
each value of ? during the to be statioPary?
s we aaawae that for values of -- arallcr than s
For the sake of defxnitenes
certaia u,? A
4'(O. !s) < O,
. valves,
acrd for cnlusa of ii larger than these ll.
creasing. it from a certain value ?1. leas,
,,_~1 sow rarer a-diebatical ly, rs
Is
than Ni?
n e i libriuN: a ' 0. 'then, mace for ? t p,,, the '
Let the ~ a~rateM be ~att~rrlly i +9n
ureeter
'tisg, it will also regain in equilibria until the p
t i
system s not heyf ea~c
..~i. ~ is reached, for the transition throu this
Ie of the critical value equal to ~-
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Thus the dependence of the asgplitude on the pars.eter is represented in the
r
y = 'I- (a), .
I-en tha stationery rrplitudsa rill ba found frog the intersections of this
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;critical value, ss1f?excitattion r111 appear, vhile equilibrium beco-es impossible,
aid the aoplitude a passes fro. zero to the value a(W) equal to the agaal(est stable
root of the equation
a = 0 for )L < tLO,
a ~?) t~r L ' !Lu.
If the curve of the dependence of a and u so obtained is continuous, we will
say that ve have a case of soft excitation of oscillations (with respect to the
given paraseter). In the case of soft excitation, on transition across the critical
value u the systeggw vi l l begin to
generate oscillations whose ar.plitude,
increase fro. zero.
However, if there is a discontinuity
at the point IL ? uo, then, on transition
through the critical value, the agsplitude
viii igrsaediately jump fro. the zero 'slue
to the value ?(ua ' U). 'lhia case is
called the case of hard excitation.
For exa.ple, let the right fide of eq.(5.1) have the followin lore:
4'(a, ?) = 4- (a) .__}T (a. ~i4, (5.8)
;.here ! (a,u) a 0, and There f(s) is s ceitain function, not dependent on ii..
In this case, the, question of the chsrscter of the excitation may be decided
the aid of one of the tre f_I1_e : g g ph:e costructiog?s:
Let us coA trVct the curve (cf. Fir. 30)
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cwrve with the straight liner of the form y If the elope of the straight line
u
is greater than the slope of the tangent to the point of inteisection, the station-
cry *pl.i tulle will be stable.
In Fig.30 we have a case of soft excitation since, as soon as u passel through
the critical value, equal to
1 1
the amplitude begins to increase from zero.
In fig.31 we have a cage of hard excitation. In transition through the criti-
cal value, the amplitude jumps from zero to a.
Let us consider in detail the case represented in fig.31. Let the parameter u
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Pig. 32
be gradually increased from zero, and the angle y of the slope of the straight line
!a be decreased in the same way, and let the system be initially in the state
y
i-
of rest a ? 0?
Then obviously the amplitude will remain zero until the moment when y becomes
to the value
less than f. After the transition through , the amplitude will jump
and will rhea begin to increase continuously. If we no begin to decrease u
(increase v)1 beginning from the value y1. > then the amplitude will decrease and,
ll b
e
will break off, in this case, a state of rest wi
? L`
h
i
y
begining wit
n tablished in the system. when plotting the dependence of a,on u in the course of
such process we obtain the curves tfrig.3$a, having a characteristic hysteresis
f
l
ue o
The, value of the stationery amplitude will depend lot only on the va
-t "?" viritPr.
the pare
 1
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d
0)
to .
f ~ (a) = l 1f(a cos 0) dt
,
0
. .r
f? (a) = f (a cos 0) cos n0 dD.
sR
t-
e now note that the harmonic component of the roltate
f ~(a) co: (nub! + n44, (n = d, 2, 3,..)
...'.producea,...in thu_lineac element, the current
. Z-1(p)fn (a) cos (nwt + n).
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Let us introduce he sl solute value and the phase at the coeplex resistance
immediately obtained. It is necessary that the tern a cos a be the principal term,
and the other. terms only correction terms. For this reason and to make this method
applicable, it is necessary that
f(', cos ~) 4- ~),
In this wsy, the more exact variant of eq.(7.48) will become
11 ..; li, w, 4, ... )
Whence, amonp~ other thinR>, a criterion of the applicability of this method is
(t=
U, 2. 3, .. , ).
Let us make use of eq.(7.51) to render eq.(7.50) more precise.
2(1(J) =R(Q)E.
r f'(t~? t.us(-t1 1(-iay)~? (7.51)
fir (a)
R-j tun (ni)
iM.O,2,1,...)
'However., since must be small by comparison with the first term, the ro1tstte V will
be
V = _ I>? ta) cos ran -.}-
Is>of
'.i` t ! t4-n (a) cos of ? O;; (al sin of +4'0(a), (1.52)
IK>1).
. wl Niia
i
d
unct os
yielda the eioension into a Fourier series, of the
132
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= f (u cos. ))? : f '(a cos ))).
[ expanding the rewltaat expressioe into a Fourier series, we find
(nw)).
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$f' (a cos I)).
Me. will be prrticulsrly interested in the values of ?i(a) and G1(s). We there..
fore present the rr*_sp??diep ferref
ftp(ucos;;)sn i'i d)).
,z
a
Oa consideria the stationary oscillations s~sin, we equate the first harwonic
of the voltage fro. eq.(1.52) to the first h rnonic of the voltage across the liner
el anent
sad then find
f t (a) cos (wt + 1+) + 4, (a) cos (wt +'.) + 0, (a) si n (wt ..}..'!#) =
R (o) a cos (wt -~- ~ -}-~ (w)},
whence we obtain the refined equations of harmonic balance
R (w) o cos ~ (w} :: h (a~ -~- ~', (a)-
R (o3) n sin ? (u-) = = -01(a),
us no siuplify the expressiona for ~1(a} and G1(a). We have
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4'~ (a) = (a cos A cos $ d~-,
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Thus the refined equations of harannic balance for stationery oscillations will
(In coaparin~ thao with the equations of the first spproxiiation of eq. (7.49),
+ie see that the influence of first. harmonics of the oscillations is reflected here.
The equation (7.54) so obtained caa also be uaed for ? core detailed elucidation of
the 1i~its of applicability of the equations of first spproxiaation.
Nye note also that these results could have been obtained by the ?ethod of
auyoptotic expaasioas., For this prrpose,it is expedient to represent the funds-.
:seatal equation of the oscillatory process in eq.(1.47), for exaaple,in the form
(7.55)
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Section a. Paths on the phase Plane. Sin4ular Points
The ave-described asymptotic methods are limited in their application by the
requirement that a pine!! parameter be present in the equation. In many cases, how-
ever, me have to do with equations of s more general type, to which these methods
are inapplicable.
If the equations describing the motior. of the dynamic system under study can
be reduced to the form
then qualitative methods of study can be applied to them. All equations considered
in this book can be reduced to equations of the type of eq.(8.1). In addition, as
ae shall see later, in studying oscillatory systems that are weakly nonlinear but
are under the influence of external periodic forces, equations of the type of
eq.(8.1) are also obtained as equations of first approximation.
For. a, qualitative investigation of the solutions of eq.(8.1) it is expedient
to consider *; y'a: the coordinates of a point on a plane. fins plane, as generally
known, is termed the phase plane, and the point x, y the phase point. The Action
x (t), y ' y (t) is performed along a certain line which in called a phase path.
` ibe , coestruc4iou of he phasu path of
Carrs.expressing the velocity as a"function of the displacement, for the assigned
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notion.
The phase pl.aae, with the phase paths on it, iarediately shows the totality of
sll motions that can occur in the dynaaic system under consideration.
To construct the phase paths we must find the solution of the system of eq.(81)
x (t),.y = y (t), repreaeuting the equation of the phase path in a parametric
form, or find the characteristics, i.e., the integral curves of the equation
dy __ Q (.Y. Y)
",. -- n,.. .~1
ax l~~ 71
which directly yield the relationship between x and y.
Let us discuss the siaplest cases first.
Consider the equation of a linear osviil.tur:
putting
dx
we bring eq.(8.3) into ti:e form
d~x+,Zhd
d~
- 2ky-- kx,
-~-
dl
I
(8.2)
(8.3)
Assume that the friction is small, i.e. h2 < k, k > O; then the solution of the
system (8.4). is written in the form
- cos wi a ,
y=
There ~-~
1
iaitial values.
Equation (8.5) is the equation of the phase path in pat Ctric for. By it
aid the character of the motion of the phase point on the phase plane can be
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ao e' ht sin (wit +
tan 1 a, a and are arbitrary constants determined by the
wt
0
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analysed without difficulty.
we note that
single tangent to
of the tangent is
the system (8.4) determines, at each point of the phase plane, a
the integral curve, except at the point x a 0, y ? 0. The slope
defined by the expression
dy
dx
At the point x ? 0
Ey
Such points are called critical or singu-
lar points. For the simplest singular
points (singular points of the first
order, or elementary points) either no in-
tegral curve at all or more than one in-
tegral curve pass through the singular
point.
Fig.41 Let us assume first that h ' 0. Then
the solution of eq.(8.5) assumes the form
x = a Cos (wt -?- a),
c8.7)
y = _ aw sin (wt? x). On the phase plane we obtain a family of similar ellipses
( Fig 41) , end i this
case, only one ellipse, corresponding to definite initial conditions, passes through
each point of the phase plane.
By eliminating the time t from eq.(8.7) as obtain the equation of the family of
ellipsee in the ions
III
.~ J.._.._. -.
Al
. II 1 t t. .1 1. 1. J ?- _ w
wnlcn we COULD u ao nave oocaiaea uy direciir integrating eq.18.~1 at h u.
Not a single integral curve passes through the origin of coordinates. Such a
singular point, in whose neighborhood the integral curves are closed and aurroand
the singular, point, is called a center.
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-21sy--hx
Y
(P.6)
the direction of the tangent becomes indeterminate.
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Since, in this case, all the phase paths are closed curves or ellipses, (ex-
ceptthe pith thn degenerates to the point x 0, y 0), the motion will be
he singular point x " U, y 0 corresponds tv the
of v'.;:i I h. i? Ckw
oscillatory system under consideration.
It is entirely clear that, in the general case of eq.(8.1), the states of
equilibrium of the system correspond, on the phase plane, to the points for which,
simultaneously, % 0 and -~ b, i.e.,
the singular points of eq.(8.2); to the
periodic motions taking place in the system
there correspond, on the phase plane, the
closed phase paths of eq.(8.2).
In the following, we will have occa-
sian to make use of the concept of stabil-
ity of the equilibrium state.
>ke here present only its definition,
without going into detail. A state of
equilibrium is stable, if, for any
assigned region of allowable deviations from the equilibrium state (the region ri),
we are able to indicate a region b(n), surrounding the state of equilibrium and
possessing the property that no motion commencing within b(n), ever leaves the
region n (Fi6.42).
Aealytically, tais definition of stability may be expressed as follows: The
state of equilibrium x xo, y * yo is called stable, if, for any n a~aeigned in
.u,.a.e, ao matter has smells b(n) can be found, so that, for t = to,
but also for any values of t such that tb 0, which corresponds to a doped oscillatory process. In this
case, ee obtain fro eq,(8.5), on the phase plane, a faiailyof spirals for which the
origin of coordinates is the asymptotic point (Fig. 43); in Chia case; the smalls
the ratio h/cal. i.e., the smaller
the friction, the closer will the
spiral approach the form of an
ellipse during the course of a
single revolution. The phase ve-
locity in this case wilt not
vanish anywhere, except at the
origin of coordinates, but will
continuously decrease as the
representative point approaches
the origin of coordinates.
The phase paths correspond in
this case to oscillatory but
duo ed notions, while the singular
point x ' 0, y,y 0 corresponds to
the equilibriw position.
The singular point under consideration in this case, being the asyaptotic point
of all integrsl curves baring the form of spiralis called the focus; for h >
0,
this foci. will be stable.
Hoe let .b < 0, Ia this cam ere again obtain a fwily of spirals (Fi.4 b
g 4) ut
the,paa,.e point, in -tine, vii! leave the origin of coordinates. The velocity aSTAT
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r4
^ 0, y 0, will',oaotonoualy increase as the point moss sway frog the origin of
point z *O, y R 0 is an unstable focus,
?ceordinates. In this case the position of equilibrium it unstable, and the aingular
Let us now consider the case when h2 > k, which corresponds, for h > 0, to a
Fia.44
drped aperiodic process.
:represented in the form
In this cue, the solution of the system (8.4) may be
Cie-'4it--~ C9e_Sit,
(8.9)
(8.10)
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D
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of the representative point slung the integral curse, which becomes zero at
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q2, and add to eq.(8.10). Os raising the results so obtained to the powers q1 and
q2 respectively, ye find
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k. j
T obtain the ies~e on the phase plane, aultiply eq.(8,9) first by ql, then by
(y + qlx)
C '+qx>",
dr
y ? q1x = C(y-~- g4x)t,
-r j
\\\\\ \ \\\\\ \
Fig.45
141
On the phase plane we obtain a family of defored parabolas (Fig.45),
(8, lU
(8.12)
(8.13)
which
ti~..~~t to o t`.7 ~ rte.e bt 'i!e y ? ?- At11 if the or `~r1AY' YRwY s;$,n~ .... / 71 2gtl: of Y cooWrLigKCa. 14 li
{rW
(V'
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The paint x y 0 will be a singular point, and all, integral curves will
puss through it. X singular point of this type is called a nude, In the case under
Fin. 46
consideration the equilibrium position will be stable, and a stable node will cor-
respond to this position.
Consider now the character of toe ia~c6s i cu;; v.. the phase pte e fr the
case of a'higb negative friction A k. In this case, making use of
egL(8.13), a family of parabolic-type curves (Fig.46) with a singular point of the
node type is obtained on the phase plane. M analysis.of the motion of the repre-
I fl
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not difficult to natabliah the direction of the representative point along the in-
te rsl curve n. The representative point will mare along the integral curves in the
on Fig.45 by to arrows, i.e. it will always approach the origin
of coordinates.
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cc t er~ang the direction of motion of the representative point on the phase
?plwne, we come to the conciuskon that no matter where that point may be at the ini-
tial instant (except for the origin of coordinates and the asymptotes y kx) it
Fig, 47
will ultimately always more away from the
origin of coordinates, and this motion will
be of an aperiodic rather than an oscilla-
tory character. The equilibrium position
corresponding to a saddle-type singular
point will always be unstable, in view of
the fact that the motion along the
asymptote y a - irk_x can never be exactly
realized, since the probability of an ini-
tial state corresponding to motion toward
the singular point is zero.
Let us pass now to a consideration of
the general case. First of all let us
study the equilibrium points - the singular
points of eq.(8.2) in which
P (x, y) z U, Q (x, y)=0. (8.18)
Let p (x, y) and Q (x, y) be real
analytic functions. Assume that the equilibrium points, i.e., the solutions of
are isolated and, thus, that the number of singular points in any bounded
regica it finite.
Tea, for the szalys s of the behavior of the dynamic nystea in the neighbor-
hood of a given singular point x = xo, y = yo, let us use
x=
x0+ %x, .y
(8.19)
*ithout disturbing the form of eq.(8.1), we may take the singular point x
yo as the origin of coordinates. Ten, by substituting eq.(8.19) in eq.(8.1),
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x ....- C1e' 4- C/,t,
y = C1 11& `t+ C y &.,r
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N! obtain
where, for sibpiieity, the syabois for the variations 6x, 6y have been replaced by
x y swd the follomiag notation has been introduced:
c . P- (p, p), d --? p(0, 0), 0), h -_ Qr1(0. 0),
W p
while P, (x, y) and Q2 (x, y) are functions having continuous partial derivatives of
up to the sec;Nd orae= inclusive and vanishing, together with their parti?_1. derive
tires of the first order, at the origin of coordinates.
Ne 1ectiag, in eq.(8.20), the terns of higher order +aith respect to small de-
viations from the equilibriua point, we obtain the following system with constant
coefficients:
which, ss is cosronly known, are termed "equations of variation" about the equilib-
riusi point.
The characteristic equation of the system (8.21) will be
A~ -- {b -}- c)1. (ud bc) 0. (8.22)
Ne aril/ consider oily the cases where the characteristic roots X1, Xy are not
equil to zero, so that nd ? be # 0. The corresponding critical points are termed
first-order criticil points, or eleaentary points.
The solution. of the systesr of eq. ( 8.21) will be
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where 1 and n2 vaatnau by the Ivrtewia` expressioaa;
I 4- c ? V44I
lb ? G - (b -}w lal~~
while xl and x2 are the roots of the equation
dx + (h - c) x - a = 0.
(8.24)
(8.25)
On. analyzing the right-hand aides of eq.(8.24), we can easily find the ratios
between the coefficients of eq.(8.21) a, b, c, d, at which the right-hand sides of
eq.(8.23) will approach zero or will res~ain bounded as t M , and at which, con-
sequently, the corresponding critical point will be stable.
For the critical point to be stable, it is necessary that b + c < 0 for
(b - c)2 + tad 0; in the case where (b - c)2 + 4a4 > 0, it 'ii also necessary, for
ensuring stability, that ad - be < 0, otherwise the critical point will be unstable.
If b + c 0, it is necessary, for stability, that (b - c)2 + tad 0, the critical point will always be unstable.
If ad - be # 0, the character of the critical point of eq.(8.1) will be de-
tersined essentially by the character of its first approximation (except for the
case when b + c * 0), i.e., by the character of the solution of the system (8.21)
obtained when P (x, y) and Q (x, y) are replaced by their terms of the first order.
It is obvious that the presence of the higher-order terms rejected by us will
not modify the character of the motion in the neighborhood of the equilibrium point
only cause small additions to the "effective" values of the damping decrement. In
the cue where Re [Xi] 0, or lie [x2] u, the small rejected terms may affect the
character of the motion,, since its stability will be determined precisely by these
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In order to study the character of the equilibrium point x 0, y 0, let us
obtain the equations of variation
according to which the equilibrium point x 0, y = 0 is stable, since b + c
4ad? -4 0). The
circle with the radius p - 1 will, in this
case, represent the limit cycle of eq.(8.44);
see (Fie.58).
Section 9. The Lienard Method
In many important special cases, it is convenient to investigate the nonlinear
differential equation (8.2) by means of a graphical construction of integral curves
on the phase plane. A very elegant method of graphic construction of the integral
curves is the method proposed by the French engineer Lienard (Bibl.24). By this
method, all types of motion permitted by the given equation can be studied and the
lihit cycles found.
Lienard inve.tigated an equation of the form
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162
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o
Equations of tljs type, as is commonly known, include the ran der fool equation,
the Rayleigh equations, and others that could be cited.
After Lieuerd, the problem of establishing the existence criteria and the
uniqueness of the lidit cycle for equations of the type of eq.(9.1) were investigat-
ed by several authors. Me may mention, for exarple, the work of V.S.Ivanov, of
Levinson and Smith, ~-nd of A.V.Dragilev.
A formulation of kV.Cragilev's theorem is presented here. Ne introduce the
notation
F (x) = ff(x)dx, 0(x) = g(x) dx.
(9.2)
1) if g (x) satisfies the Lipschitz condition
xg (x) O, x = Q; 0 (ao) = ro;
2) if F (x) is uniquely determined in the interval m < x < m and, for each
finite interval, satisfies the Lipschitz condition, while, for sufficiently small
CxlF(x)0, and F(x)>0atx M,
then eq.(.9.1) bust least one, limit cycle.
It is clear' that, under very general conditions, the existence of at least one
limit cycle is thus, established.
The question"of the. uniqueness of the limit cycle is the subject of the
Levinson and Smith theorem.
Let:
C
) be' an odd function so that g (x) > 0, for x > 0;
1
x
. g
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3. F'(x) ? m while
4. The equation F (x) 0 has a single positive root x a and, in addition,
for x ? a, the function } (x) increases monotonously.
A. readily demonstrated, these conditions are satisfied by the van der Pol
equation and also by the Rayleigh equation.
le will prove that, when the above conditions are satisfied, eq.(9.1) has a
unique closed cycle which will be stable. The proof will be furnished by means of
the very clear and elementary method given in the book by Lefschetz (B1bL23). Ne
with this natation, Z may be interpreted as the kinetic energy, and the above-
mentioned function G (x) may be interpreted as the potential energy.
Let us now determine the energy dissipated by the system under the oscillations
defined by eq.(9.1). Ne have
, (' +- cI x) -4- P + 0 (X4 =
?r ) ~ dt 1 wit
f( x) (tt f g () + (x) -- (!. P (x))
dt dt
? 1be proof for these theorems is given by V.V.Nemytskiy and V.V.Stepanov (Bibl.31).
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F (x) be an odd function with a value xo so that F (x) < 0 for 0 < x ' xo~
and F (x) > 0 and increases monotonously, for x'y xo;
4, f (x) and g (x) satisfy the Lipachita conditions over any finite interval.
In this else, eq.(9.1) has a limit cycle and, at that, a unique limit cycle.
No proof for these theorems will be given here'.
Consider 'the simpler case when the following limiting conditions are satisfied:
1. f (x) is an even function, g (x) is an odd function and, in addition,
xg (x)' 0 for any values of x, while f (0) < 0;
2. f (x) and g (x) are analytic functions;
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or, tnk n~ eq.(9.l) uno_the notation of egn.(9.2) and (9.3) into consideration,
find,. after c*nceling out_dt,
d?: F(x)d-R. (9.4)
In this way the energy dissipated by the system will be expressed by the nag-
nitude of the integral j F (x) dy talon along the integral curve.
Passing to the variables x, y, eq.(9,1) will yield the equivalent system
d.c
di
dy
Thus we have to prove that the system (9.5) has a unique and stable cycle.
The syste. of eq.(9.5) has the following obvious properties:
1) if x (t) and y (t) are solutions of the system of equations (9.~), then, by
virtue of the imposed limitations, -x (t), -y U) will likewise be solutions
(since F (x) is an odd function); consequently, the curve symmetric to an integral
curve with respect to the origin of coordinates will also be an integral curve
of eq.(9.b);
c the
2) the single critical point of the system (9.5) on the phase plane is the
origin of courdinstea, and therefore the limit cycle must be circuwacribed about
origin of coordinates;
3) the slope of the integral curie I' is determined by the following equation:
dy _ a Y R. (9.6)
Since g (O) 0, .11 the tangents to the path r at points lying on the axis O
(except the origin of coordinates), are horizontal.
On the other hand,..af we consider the curve A, whose equation will be
y - F (x) _ 0 (Fig.59, broken line), then it is not hard to see that all tangents
- to I' at the points of its interaection with A are vertical except at the origin of
coordinates (since on A, y - F (x) " G and, consequently, d ? m). Moreover, since
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]f5
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g (x) is odd, it follows that xg (x) > 0, then, according to eq.(4.5), y decresses
along the curve C to the right of the axis Qy and increases to the left of the
axis Oy. In addition, x iacresses when r lies above A [since, in this cue,
y - F (x) > 0) and. decreases, when r lies below A. Consequently, the curve i aaa
the foray shown in Fig. 59.
Fig.59
Let a denote the abscissa of point.B and let r~ replace r.
we will now establish the conditions under which r~ will be a closed cycle.
It is obviously necessary that OA' OA, since, if this is not the case a
repetition of our srgument will show that a prolongation of r beyond the point A',
because of the fact that the cycle cannot intersect itself, will sive the point A"
i !yang below A' (Fin 60), sad so on. Iii twi: -.y, if OA' OA. the curve ra can-
not return either to the point A nor to the point A' and, consequently, cannot be
closet, fleece, r s~ust intersect each axis at two and twn points only. It follows
ho. this that OA= - OC.
Now assure that O A f - O C, and let the points A' and C' be syaretric to the
points A and C with respect to the origin of coordinates. According to the first
property of the iyateh of eq.(9.S), s curve sya etric to the curve ra aith respect
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 to the origin of coordinatea ,ill be the closed integral curve,r1, p assinq through
the points C'. Since the axis Oy is perpendicular to we axrive
at the
position shown in Fig.61 where the curves l'a and 1'1 intersect, which is impoaaible.
Therefore OA" -OC.
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Thus, for ra to bs a closed cycle, it is necessary and sufficient that
Fig. 60
coordinates rill be an arc of the cycle joining the point A to the point C at the
left of the axis Oy. Together with the arc AC, this will form a closed cycle.
Oa the other hand, aasume that
OA ? - O C. Then a curve symmetric to the
arc A C with respect to the origin of the
Fig. 61
OAS -OC.
Since, according to the symbols in eq.(9.2), X (0, y) the latter condi-
tion may be ferawl ted is the following way:
For ca to be a do 4 cycle, it is necessary and sufficient that
A)=A(~. (9.7)
i can ahow that, if tae conditions satisfied by the functions f (x) and g (x)
are wet, q.(9.7) will be valid so that. eq.(9.1) will have a limit cycle.
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As a proof let us consider the following curvilinear integrals taken along
tho curve _I`.:
Let
A~~' ~ Ab0
If a ~.a (cf. fig.59), then dy < 0 and, according to the fourth condition
oscillations can exist in the system.)
Lhu anerpv disaipatrd by thr system is positive and obviously; no undaeoped
(cf. above), alsof,(x) < 0; in Chia way, (a) > 0, i.e., X (C) > X (A).
Consequently, Pa cannot be a closed cycle. (In this case, $ F (x) dy
ARC
For this reason, assume that Q j a, i.e., that the curve PQ has the form shown
In fig. 59. Let us denote
cp (a) -.. fd+ fdA 2(z)= di,;
Then,
A(C)-,.(,4) = fdA - fF(x)dy.
At) lT nBF.
(2) =(7)+9).
~ (a
On the basis of egs.(9.4) and (9.6) we may mrite
d1. = P(x) d}dx = !'tX)A(X) dx (9.9)
Since f (x) < 0 for x < a, then d X is positive when P1 is described in the di-
rection from A to D or from E to C so that ~1 (a) > 0. On the other hand, along
DBE we have dX < 0 and, consequently, ~2 (a) < 0.
It is obvious that when n increases the arc AD will rise, while the arc CE
will sink so that, for a fixed x, lyl will increase. Since, for ml (a), the limits
of integration, bearing in wind eq.(9.9), are fixed (from x 0 to x = a), then an
iacrease in a will cause a decrease in +rl (a) since d X - (xi dx decreases
~tl
with increasing y'.
Let us pass now to an evaluation of the character of the variation of a2 (a)
with increasing a. Let al and.. a2 be tw@ successive values of a, and let a2.
*re can slow that
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(from a higher to a lower value).
On the other hand, for a given y, the
abscissa x of the point of the curve DE1 will be
greater than for the corresponding point of the
Drop the perpendiculars 01Di and E1E~ to the straight line D2E2 (Fig,62). Then,
J P(x)dy f f'(a)d'. (9.11)
7
From the very construction of Di and E we see that y varies along the.
curves D1D1E1 and within the sane limits
ictually 02 (aZ)
curve D1D1E1?
For this reason, for a given y, F (x) on
1D1E1 will be smaller than F (x) on D1B1. Con-
sequently, since dy 0,
F(x)dy4- f
~~e Far J'?r I''
1 f 1 f
(x) d y. (9. 10)
It follows that
I' .r
I . t t ,
.f F(x)d v is F(x)dy, (9.12)
F d la
and, from eq.(9.11) we find
F(x)dv< f F(x)dy;
< 02 (e1) for a2 > a1.
(9.13)
Thus, ?(.a) ` ?(i)+ 1q (e) for a ? 0 is a monotonously decreasing function of a.'
*e note that, in the case a a, we have
Ne will show that.
(~c).+co f a -- oo. .
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For this purpose,_, we establish some value of xl such that
and lay the axis PP' parallel to the axis Oy through the point x1 ou the axis "Ox
(see Fig, S9).
We then have
dA < fd,=fr(x)d.v.
1h8R Ply?/ PBP,
for the arc PBP1 a have :1 and, cy'uic`yventlj,
F(x) >P (x,).
bl..rwrr
^qw~r ? v? ,
e, therefore,
(9,14)
pdF.' Pill,'
whence
find that
d' < F (x1) dy = -- F (x1)1 PP1
.. a ~ .~ fd#>2.kP.
PBt'
(9.15)
It is clear that the segments KP and KL may be taken as large as desired for
sufficiently large values of a.
Thus, in reality,
1.1
In this way we have shown that (a) is a function monotonously decreasing
from the values of (a) > 0 to (a) . - m while a m, Consequently, (a)
vanishes once and once only for a ao, and ra will be the required unique closed
0
characteristic since, for it, the condition (9.7) will be satisfied.
~e will show that ra is a stable limit cycle.
0
- if a < ao, then a (a) > 4 and, consequently, k (C) > A (A),
If a > ao,.the? (a) < 0, and, consequently, A (C) < A (A).
Let the poia:ts & and Co correspond to the intersection of ra with the y axis;
0
then, as is obvious, the point C is closer to ra than the point A if' a < ao, so
0
that the point A' is -closer to ra than A.
0
By analojous,.reasoning for the case a.) ao, we arrive at the conclusion that
the. limit cycle ra is stable.
u
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Li.dnard'a grsphic method is generally used where the elastic force g (x) is
linear with respect to x. In this case, by an sppropriate selection of the new
variables, we may, without interfering with the generality, reduce eq.(9.6) to the
0
i
N
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we will low discuss the method of actual construction of the integral curves
on the pbe+re plane,
Fig. 63
proposed by L e'nard, is as follows: On the phase plane, let us construct the
curve A, whose equation is (Fig.63)
dy r. -- _ x
(' 16)
The method of graphic integration of equations of the type of eq.(9.16),
?
y ---- F (X) = Q (9.17)
After eoastructing this.curve, the direction of the tangent to the integral
curve of eq.(9.16), passing through any point of the phase plane, can be graphical
.1y determined:Frosi the point M (x, y), for which we seek the direction of the
tangent, let us drop the perpendicular to the abscissa MO and extend it to its in-
tersection with the curve A at the point D.. From the point D let us drop a perpen-
dicular to the ordinate ON. Then, the line NA will be perpendiculir to the
integral curve of eq.(9.16) passing through the point M. If the phase point of
.4(9.16), at the tile t ? 0, coincides with the point M (x, y), it will be dis-
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dx=(y-F(x))dt=Ml)dt
Since he tri angles N D M and M C' M' are similar, it follows that
ens dt) song the ordinate by the segment
;.pieced (after the time.-xep
dy---xdt--NUdc=MC?
and, consequently, M'M 1. w
order-to produce. to a given point M (x, y) in the x,, y plane, the
'thus, to tangent to the integral curve t, passing through this point M (x, y), it is auf-
-- t 1 straight,
a
line D N , and to join
points
ne M C p and the port von
The required cangcnc co the curve r will
be perpendicular to the straight line NM,
whence it follows that, having the arbitrary
curve and the arbitrary initial conditions
xo, yo sapped by the point Mo (xo, yo), it is
easy to find the direction of the tangents,
and, consequently, to construct the approximate
integral curve.
Thus, to construes the integral curve C,
passing through an assigned point of the phase
plane M (xy.), we proceed as follows: from
the abave?described construction we find the
tsn~ent to the given point and replace the
in the neitbborhood of this point by a small segment of the tangent.
end of the resultant segment, we again determine the direction of the
'11ten, at tMe
in the neighborhood of the new'point we replace the integral curve by a
,tangent, rd
segment of a strs bt-line, giving the approximate integral curve in the ford of
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a broken line. The degree of accuracy Will depend on the d.gnitude of the.
individual links.
For the U sniformition of eq.(9.1) it is in many csaes convenient, not to sub
atitute the variable$ by eq.(9.3) but.to pertorm the auba'itution_by the.tormula
t 1 .: f .L rlt
and to_consider ho equation in the 'form
rlfx ' . IC
U
dxl
or, using the notation ___ * y and eliminating the time t,
jf ?FW?x --0 (9.19)
In thla cue, the equation of the.auxili,ry curve wit.! be
xt -f 1: (tea,) : 0, (9.20)
so that, on the phase plane, we obtain the construction shorm in Fig.64. Dropping
perpendiculars from the point M to the abscissa 1VV and to the ordinate MC, and
also dropping a perpendicular from the point U to the abscissa, eq.(9.19) will yield
"V
t
Consequently, eq.(9.19) ^ay be written in the form
;Vm == C It
iince CM? xl, CD. F (y)
UI),
(9.21)
(9.22)
Thu, in our?csae, as well, we may perforw the construction of approximate
integral curves according to the above ace.
If the curie A is symmetric with respect to the origin of coordinates, then
the intcgrs1 c.*ve! r o constructed viii 11 rlong closed curves - limit cycles -
corresponding to the periodic state whose existence and stability have been proved
above.
s Tr.n1 ator's note: See errata sheet.
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note that the igraphic construction, proposed by Lienard, does not postulate
that the curve A.*uIt necesaarily be symmetric. This graphic method is also appli-
cable to the case where A is care or less close to a syeimetric curve, ajpticable,
r.
,
Tor example, io the c;irve determined by tae characteristic of a neon tube, etc In
case the curve 1 need not necessarily be represented by an algebraic equation
this
and may, instead, be obtained experimentally. This feature is very ii.portant from
the practical point of vier.
Belos, we preaent a few exampbes to illustrate the e`+ove-described graphic con-
struction of integral curves.
that for certain special cases; the Licnard construction iaaiedietely
Ne note
yields the integral curve, eliminating the'neces$ity of constructing an approximate
broken line.
For example, in the case of free Linear oscillations described by the equation
In this case, toe equatio
dx y
(9.24)
the curve A will be x ? 01 snd the point N will
coincide vitb the origin of coordinates for all assigned values of the point Q. Con-
entl the integral curves will be circles having their center at the origin of
stqu Y~
coordinates.
If the oscillations of the system take place under the influence of a. linear
elastic force in the presence of Coulomb friction, the equation of motion asy be
presented in the form
d--- + it sigh x ? x =- fJ. (9.25)
dt .
In this case, .e obtain the following equation for the curve A:
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In this way, the integral curve 1' will conaiat of
arc; o; circles aith their centers at the
points S1 and S2. These sres will merge at the
intersection of the integral curve with the
axis OY. In this case, it is obvious that the
s~plitude of the damped oscillations will de-
crease by the quantity 2 A with each passage be-
tween two successive rest positions of y = 0,
until the oscillatory system finally reaches a
state of rest.
Let us no^ integrate the van der Pot equation by the Liinard oethod, taking it
in the form
x; O.
(9.26)
the integral curve (the point N.111 coincide in the upper half-
point S1, and in the lower half-plane with the point S2 (Fig.G5),
regardleaa of the aaaigned values, of the point U.
i~
( der \j der
_..~wdl)Jrlt.~,.
The equation of the curved on the
phase plane will be
-x+s(I --- V)y -= O, (9.28)
where t ix a certain par r+eter..
The properties of the function
t (1 - y2) y are as follows:
1) sty~0andy? i1, x?0;
2) at y ? t , x will asaua~e the
1?4
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ACC4rtliAj_to this, an increase in i.ceui A tNe.IGop to elongate along the
+. `
axis and to approach the p air of straight lines y t 1 (Fig.61)?
x
In the- case where e " 0, the integral curves of the equation.
dy --~(1--y--y -x
dx
(9,29)
fora a f.Nily of concentTIC circles with the center at the origin of coordinates;
then, this equation .i U correspond to simple harmonic oscillations.
c_?
I-? .vt
For e f 0...e will study the behavior of the integral curves of eq.(9.27) by
ueans.of Li enird's graphic method. According to this method, let us construct the
field of directions for the carve of eq.(9.28) and let us find the limit cycles.
~Figure 6'1 gives the curves of eq.(9.28) constructed for three different values
0.1 e ? 1, and e t 10, reapectively. in thrae sane graphs, the limit cycles of
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the system have been constructed by the l.i& erd eethod, Since, an generally known,
0 in the case under consideration, the origin of coordinates is an unstable
equilibrium position so that all the integral curves leaving the origin of coordin~
ates will descrbbc expandifg spirals about it. 'However, the spirals unwinding about
the origin of coordinates cannot extend into an indefinite distance, since, for
Fi g. 68
e/8
great values of y, the dasyring in the oscillatory system described by eq.(?.27) be-
comes positive.
As each spiral expands, its successive loops approach more and more, and all
spirals asymptotically wind from within toward a closed curve, the limit cycle.
Along this limit cycle, spirals close to the origin of coordinates and spirals
far from the origin will wind. The closed integral curve, the limit cycle toward
which all integral curves of eq.(9.29) tend, corresponds to the periodic solution
of eq.(9.27).
*e note that the closed cycle contains one singularity with the index
176
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for a * 0.1 and t * 1.0 this point will be an unstable' focus, and that for a 10,
im here an. unstable node.
On the basis of Fig.67btbe variation in the character of the motion within the
system with any variation in the parameter e. can be eat mated. For any values of
e, is the system, ielf-sustained oscillations take place, but the width and form of
these self-sustained oscillations and the character of their build-up differ. For
comparison, Fig.68 shows the results of the numerical integration of eq.(9.29)
Fig. 69
the sane values of the parameter c, as well as the curves characterizing the varia-
tion of x with time (Fig.69).
In conclusion we note that Hensuki Usui (B bl.3O); cowbining the Lienard method
(developed in detail by the author for the case of a syumet:ic characteristic) and
the Kirstein method (which is exceedingly difficult for practical application), has
developed a standard ~csphic method for solving the nonlinear differential equations
that describe the processes in self-excited systems. The method developed by him
may be wed for the consideration of the oscillatory processes in complex circuits
;and also is'connected circuits.
This method will not be further discussed here, and the interested reader is
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referred to epeci8l literature on this subject.
Section 10. ~,laxition Oscillatory Syntem5
considered the van der Pol equation mainly for small values
(% to aow we have !acre in the
ke
t
p
a
in Suction 8 have we pointed out the changes that
of e, and only
Let us now consider the van der Pol equation
at large values of t, and, in particular; let us
P try to find the hsyaptotic fora of the solution as
For this inveatigation, it is convenient to
take the van der Pol equation in the fora
c~w.r ~rx t d.r '_ x (10.1)
.dl=...~. t rlt ;~ ~t!t
and to have a small parameter in front of the
second .derivatitC.
Putting, in eq. (10.1)
(10.2)
~.! ... dY~ 1 dr, ?? U (10.3)
s' dti dt, 3 dt~
(10.4)
1 4JI is eq. (10.4)
For e ' 1, we may in first approximation neglect the smamand L2.dr1
after which we obtain the following relationship between and r\:
the cheiracter of the motion on the phase plane can be
118
(10.5)
readily investigated.
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Let us construct the curve of eq.(10.5); (Fig.7O). a note that. according to
eq.(10.4), the field of directions on the curve of eq.(lq.S) is horizontal, since
- . ? 0 for all values of r, i.nd n s isfying eq. (10. S). In the remaining points of
the phase plane, however, except for points very close to the curve of eq.(1U,5),
the field of directions approaches the vertical as a since, according to
eq.(10.4), .. d -- as a -? ? for all points not satisfying eq. (10. S) . Starting f rons
~
,
s o
this premise,
for large values of e, the integral curve of
it i
bvious that
eq.(10.4), lesving,the arbitrary point P (cf. Fig.70). will be very close to a ver-
tics! straight line almost up to the point P1, lying on the curve of eq.(10.5).
Further, the integral curve runs along the curve of eq.(10.5), remaining below it
until it reaches the neighborhood of the point P2, after which it runs vertically
upward until it again reaches the point of eq.(10.5). The integral curve will then
follow along the curve of eq.(10.5), remaining above it; after reaching the point P4,
the integral curve will turn vertically downward. As a result, we obtain the limit
cycle which, as e - will have the farm shown in Fig.70.
Ne obtain such a picture because of the fact that the segments P3P4 and P2P5
of the curve of eq.(10.5) possess the property of attraction; in this case, the
greater the value of e, the stronger will be the attraction. Since the field of
directions is vertical at. any point while it is horizontal on the.curve of eq.(10.S),
each point will first asymptotically tend toward the curve of eq.(10.5) and will
then move away from it, since the field of directions is horizontal along the curve;
after this, the point will again tend to approach the curve of eq.(10.5). If a is
sufficiently great, these deviations will not be detectable so that, in practical
cases, the picture shown in fig.70 is obtained.
Let us.find the asymptotic value for the period of oscillations in the approx-
isation undo: consideration, by calculating the integral along the limit cycle.
For eq.( 10. 4) ye have
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d"
[',y
( Iii ?r1 -
r 1") ~?
T _ dr1 (10.7)
vl
Since on the 'vertical parts of the cycle, dr 0, we may replace eq.(10.7) by
According to eq. (10, 5)
i~
(10,6)
(10.8)
1, M
2 and, consequently, obtain the
following equation for the period Tl at large ralues of e:
T1 w 1,614
or, using the old variables, the following.aaymptotic fonrula
(10.9)
T z 1,614.. (10.10)
'Thua, for the case a ~- 1 and using the asymptotic treatment, the oscillatory
roceas gill proceed as follows: Nhen n increases, beginning with the values of
P
no, the 'velocity will be positive and the representative point on the phase
apace will Bove along the curve I'3P9 (cf. Fig.70). Nhen ri reaches its maximum value
of +n0, tbe'represent$tive point jwm4a from the position Pa to t5, which corresponds
to an instantaneous reversal of sign of the velocity n. Nhen n then decreases, the
'velocity rti will remain negative, and the representative point will aore along the
curve PcP2. M the point I'2, the velocity will sgain change its sign, while the
representative point On the phase p[sn. will jusp to the position P3.
Thus during one period of oscillation, th 'velocity rj will undergo a dig-
continuity twice, st the instance of reaching the maxiswa and uniniaua ralues of rt.
Of course, in ceslity the velocity is continuous (although fluctuating rapidly)
since, even though a is great, it La
still a finite value, and, in speaking of e
discontinuity, aye introduce a certain siuplificstion corresponding to the asyaptotic
spproxirnItion adapted by us.
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we find ~1 s
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(10.12)
Wkere 4) (x) represents a certain many-
value function of the type schematically
sham in Fig. 74.
Vie now present anotber exsxQle of a
concrete relaxation oscillatory ayatem
described by an equation of the type of
eq.( 10,12).
Consider a circuit (Fig.73) consist-
ing of the inductance L, the resistance N,
and a nonlinear element with an S-type volt-aapere characteristic, connected in
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After haying obtained the relation between the velocity wid the displacement
on the phase plane mad having found the period of oscillation, there will be no
difficulty
of the van der Pol equations presented here
for large values of r Way also be applied,
in the general caae, to the investigation of nonlinear oscillatory systems with
e 1. In such a treatment, we neglect the inertia term in the equation, so that
the relaxation oscillation will be chsracteri:ed by the first-order differential
dx
It is convenient to invert this equation with respect to d~ and to write it in the
in con ti,ucting curves representing n and r~ as functio s of t(Figs.71-72).
The oscillations se have just con-
aidered acre called relaxation oacillations
and are widely found in nature.
The idealised discontinuous treatment
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 series S to the source of direct-current voltage E. Here, for the S element, the
volt-arpexe characteristic has the form roughly shown in Fig.74. M a concrete
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mode1 of such ?_ nnn1k~a element; an electron tube in the dyneiron slate can be used.
Qn aetting up a voltage balance for this type of circuit, we arrive at a dif-
ferentia! equaiion of tea tyre
Rd _f_ 1:1.
f lu. 13)
Since the only source capable of supplying oscillation energy in our system is the
inductance, the energy stored in it will be equal to Li`. Since the energy must
L
Fig. 73
Fig.74
vary continuously during the course of the oscillatory process, it is evident that
the Magnitude of the current i must likewise vary continuously, smoothly increasing
and decrossing.
Oo the other hand, Fig.74 shows clearly that, with a smooth increase in
current land, accordingly, *ith a smooth decrease, the voltage v will very as shown
this diagraa (taking into account only the segments of the solid line), by the
functional relation
in Fig.75. Let us denote the relation between the voltage and current, plotted in
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183
Ite cor-
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.I,
in`+rhich f (i) has two values for a current i varying in the interval
the pars~etera of the oacill*tor be so selected that in the interval
values of the function
'(I)
tl
corresponding to the lower branch of f (i) are positive, while those for the upper
branch are negative. Then, in our disgraa relaxation oscillatory process is
excited, under whose effect the current i will fluctuate over the range frow
io to il. The differential equation describing the
oscillatory process will be of the type of
eq.( 10.12).
Obviously, in this exaaple we did not carry the
process ss far as the construction of a second-order
differential equation but isroedistely adopted a lay-
out which led to a discontinuous equation of the
in which the inertia terL is
12)
(10
f e
t
.
,
q.
ype o
ig i
Fig. 75
disregarded.
Up to now we have considered the case of the
presence of a aingle closed cycle. If the curve
characterizing the relation (10.12) has the fora shown in Fig.76, then we obtain
two closed cycles.
Ne note that the relaxation oscillatory processes under consideration take
place without external periodic forces; for this reason, it is natural to call
eq..(10.12) the equation of free relaxation oacil! ations,.
The ~' 0 (? 213) while, on approach to the boundary of convergence, the value y
ii eq.(1L15) will be of the order of t' 11a.
Next. the solution in region IV *111 be derived. This region is defined as
follows:
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It is easy to sho, that the series (11.6) maintains its asyey totic character
For conjugation with the solution y of eq.(11.IS), determined for the
region III, it is necessary thet the quantity E'2r`3 Q1 (u) is bounded for uz Q (e').
An ws1ysia of the expression for Qn (u) shoal that the series (11.20) main-
tsins its asyaptotic character up to values of u limited by the condition Q (u) <
< Q (e213), i.e., st values of x satisfying the condition 0 (x - 1) < 0 (1), and,
thus, of the region in which the suitable solutions of eq.(11.20) and (11.15)
overlap.
It now remains for us to conjugate the solutions for the regions I and IV. For
this purpose we must. conjugate the solution of eq.(11.6) with the solution of
eq.(11.2()), taking into account, in the latter, the substitution of rari:bleb(11.16).
Ne note that since y > 0 for x k -- 1, the constant c suet be more than 3.
Let u? pat c ? 3 + y, and let ua determine the order of y. Since y (- 1) =
U (e'''), follows that t Y vita iixewise De oI Gne otutt OI e-- and, con-
sequently, y ' 0 (e').
to va u , of x satisfying the condition 0 (x + 1) > 0 (e't'3); in this way, the
regions in wbich the solutions (11.20) and (11.6) are valid o,erlap, pith an asymp-
totic coATergaace of these. expansions being ensured for x
_ 11f
E- ?j
Thus, the illte$rstiun constant c can be deter*l.ned by equating, for x - 1 +
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A A O uc c ~,aa..vv
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+ a'1/3, the values of y obtained from eqs. (11.20) .And (11.6):
`s-4nf,(-1+
(U. 22)
~I determining y from this re!etion with an accuracy to tens of the order of
where bo bo (a) is a known quantity.
On deteraining the constant c
Y, It
(11.23)
is easy to find xl (tbe root of the
equation fo (x1) ? 0) after which, making use of eq.(11.12), we find the a. litude
of the self-sustained oaci!lations
(11.25)
For this purpose, we divide the entire interval of integration into fire parts
corresponding to the different regions:
1) fro. - ato -x2, in the region 11, where x2 is the value of x obtained
fro. eq.(1.10), when
(11.24)
The period of the self-sustained oscillations may be calculated according to
2) fro.-z2 to -(1k t*1I3), in the region III;
Z) ~.~. - (1 + -I/3) to -(1- e"3) in the region IV;
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formula
(1126)
S) from ~? to a in the region 11.
Then the total period T will be equal to
T=2 Tj+T.~+ lit?T4+Tip, (11.27)
where T is a part of the integral of eq. (1L 2S) ~.aken with the itt~ interval of
1
integration.
By performing the integration for the total period, we obtain
t
221n e
}- 3 In 2 In 2d :. t 4- Q fr~'~) (11.28)
ti u
or, on substituting the numerical values of the coefficients,
22 hi
. -- - +
T - 1,C 1370~h + 7,014322_.. ~
?0,0087 a-1 ?O (s'1). (11.29)
For a sufficiently large value of e, all terms except the first in this equa-
tion may be neglected, resulting in an asyaptotic expression for a period coincid-
ing with eq.(10.Y0) given in the preceding Section.
For the asymptotic solutions (11.6), (11.10), (1i.1S), and (11.20), it may be
proved without difficulty; by the method of successive approximation, that they are
convergent (asymptotically) in their respective regions.
A consideration of the above asymptotic equations clearly indicates that the
case of a large a is considerably more complex than the case of a small e. For
w.had net power asymptotic formulas while, for a 1, fractional powers or
192
in a x,.. a_+ .M ??
$ '. 1= i )i
x
-......g..... _,.__ ._.
(x.-1)3j
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logsritbsic teraa enter the equation. In the case of e ' 1, ae hire a higher sen-
ec1i fic ntr. of the eq to then 1!1 the t*** of e
ne!???:R!~ ~?? c .
p
"therefore natural that, at high nonlinesra~ty; the actual constructioiw of approximate
solutions should require a higher degree of concreteness in the differential equa-
tions under study.
Ne note that for investigating this ieportant and difficult problem of finding
the asysptotic approximation for a large paraaeter (or for a small parameter before
higher derivative) the effective aaymptotic methods developed b~+ A.N.Tikhonov
(Bibl.40) and his students may be used successfully.
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CHAPTER III
IBE. INFLUENCE ON EXIEBNAL PERIODIC FORCES
for a system dasctlbed _by eq.(I3.1),.iet as discuss again the analysis of the is-
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1 - - ? JI _~..rriww A/ t6..ethnd of _finding .. a$yaptotic.$O1ULAVp .
(12.1)
J?*'xtf(t.
where a 1s a wall posttire parareter and`f(Vt, x, dx) is a function vhich is peri-
eft
'odic with respect to rt and has the period 2x; this can be represented in the fore
nr~ . .
in absolute value, each term of the series (12.46) will be, respectively,
? indeed, let us fix a certain positive b and positive r?, as small as desired.
Let us take a'positive value of C such that
2C
I
Ih +(psi l>l
and let us construct the set of intervals dan (where a and n are any desired posi-
tive and negative .integers) with centers at the points ~ and lengths ( lnl+i`I)2+b
C the one hand, it is clear that for any number x not belonging to even a
single one of the intervals ln.a. the inequality
~I (Inir~. !mf)=+'
is satisfied for, any integers n,s.
(e)
( the other hand, the, set x, which does belong to one of the intervals
has a measure seller than
me l? ,m .2 (12.54)
the amplitude a, with increasing t, tends toward zero no th:t asynchronous extinc-
tion takes place in the system.
Let us now find the solution of eq.(12.50) in second approximation. %king use
of egs. (12.39) and (12.3?), we have
a)
x a cos 'e + . !_t . _ ~U~ ~ ~,. 2+
~ cos ~1 + cU cis 3`i -~"
4(1 y ( )
.here a sad w must satisfy the system of second approximation
(12.56)
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Its +ssa, to be expected, in our second ;approximation, the forced oscliaiIOas
with fxeggaaciea of~v and 3v, equal to the fregaencies of the external farce, are ac,
co.toenied by components with. multiple frequeaeiea.3& and with compound frequencies
v t" tai, 2r t ~, which are characteristic only of nonlinear systems.
Moreover, on the basis of the above statements in satisfying the condition
(12.53), the heteroperiodic oscillation state is unstable and thus physically im-
possible. In the case where it satisfies the condition (12.54), a heteroperiodic
state will be the only stable stationary state, pith the passage of time, hetero-
periodic oscillations of the forty
.L' . t(1v ( 9) t'OS ri 4.- ` COS 31. (12.57)
will become established in the system.
Section 13. 11~e "flesonant" Cases
Let us.nox discuss the resonant cases.
r
where p and q are certain relstively prime numbers.
"Then, depending on the character of the problem involved, two different
~~IgtiAn m.y eriae: 1) in investigating the resonance it is su~fi-
cient to confine the consideration to the resonant tegion itself; 2) besides study-
the resonant region, it is also necessary to study the approaches to this region
from the aonresonant :one.
-' -
In flew
of the fact that, in this case, we sssume that we are considering values of v suf-
ficiently close to w it is natural to put
}where sA represents the detuning between the squares of the natural sad external
frequencies
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This the initial eq.(17.1) rill be .rittan in the form
`IM
\a
"
f (.,, .Y, d x __x}. . (13.7)
ry ~i
hen the "det~tnin " s& in view of its ~aallnees, .i11 L related to the pee-
turbatioo, after which the solution of eq.(13.2), as in the nonreaooant case, can be
x t! ills ' _;., cl(~l!
there a and.y are certain functions of tiae.
detuaiep (at exact resonance, d." coast).
Here we here a ` P Vt ;
q
phase
In the cases preciously considered, the instantaneous frequency and arplitude
of the oscillations did not depend on the phase difference; however in the resonant
cue, as shown above, the phase difference .ill here a sub$tantial influence on both
amplitude and frequency of the oscillations, so that .e are forced to introduce the
dependence cm the phase detuninR
r ,r.
'1'
in the s ht sides of the equation deter.ininp a end r.
Thus, in eq.(13.3) we dust substitute a and a as functions of tide, determined
fro. the follo.in~t aystem of equation:
cll
d,
(13.4)
Since not the full phase a but the phase angle a enters the right ^ides of the
expresaioaa for da and de of eq.(13.4), it rill be expedient to eliminate a fra.
r at
eqs. (13.3) mad .(13.4).
Than, putting a- ? Vt + 9
q
tioa (13.3):
/A
we obtain the following expression instead of equs-
:~~ n, .N t) 4_a'u., ((l, fl--: t) _ f . (I3.5)
Ii wltiieh th. functions of ti.. a and 9 sett satisfy the equation
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(13,4)
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(13.6)
As ie the precedi'ag ease, for the deterrinstion of the functions u1(s, 6, q t),
Y Q ~i
(s, 8, a t),..:i A1(s. )o A2(s. 9),..., D1(a, ~), B2(s, 8),... we find at first
q
'-sin p ~?t + ;')+s' "' f, (13.7)
q q a~ a~
dux Cos P ?~~ + ~I ~? t ~~, saw, ..~., ~'? -~-
... as -~-~ t as , . ? dr
dt~ ~ q
+= j + da 4()'+
a+a db
+ sin P ~d+ f + i du ai + std + dt d~ +
4
.. a+
2 ..- P ti sin D yl -~}-1- -~- '~+`~ -+- =i ......4'
+ .'._. a si n -- ~t + ' + sj? s + ... a -~-
9
db ~
-~- ~ ~- n dos 4/ -~ ~ ~ . ~ dt
d5
y cos ~t -~- ~1 -}? s -}- to day +' ~t ?"
+21_a
9
(13.8)
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217
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.a .
t~tt dD
111 dl
``-A i 8t f ~s . .
J
substituting egs?(13.'S) and (13.8) in the left aide of eq.(13.2), taking ac-
and arranging the result by powers. of
cou*t in this case of eqe. (13.6) and (13.9),
the psrreter e, xe Ret
Expanding the right side of eq.(13.2) in powers of the small psrs+ceter, we find
a cas f'- t .-I_ i1 , _ u P i sin A (..:? if u1 +
9 9 9.
t, a cos P ''t ? it , - a P v sin ~' yt Q))><
9 q q
cos P .t ,-~- o). aB, sin A vt ti dui _
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Equating the coefficients of the se powers of c in the right sides of equs-
tians (13.10) and (13.11), we obtain the following syata of equations for deter-
.ining the wanted function.
As in the previously diacussed cases,. fh(ai vt, ~~-- Yt + A) are periodic functions
with. the period 2n in both angular ranables `n, vt + 6, Ui;(a, 6) and I3 (s, 8),
as will be clear ho. the subsequent calculations, being periodic functions with re
218
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+
n ?, f j ~ --i-? ii ---~1 A i -~- fi ~ ~` ,~I -j-- `:`~ i ~~ c i I! P Yt i~
q da d q
B. ' Al _a' ~ Hi~f aBy cos P '.t? i- , (13.13)
q cn r q
where the foliowinp notation has been ini.r6o ced:
Jo (a, vi, P vi --}-1) ::
9
f t, a cos p Yt + B), --a P '?sin 9 vt?11 (13014)
q q
-.v1?n)=
11 sin L t-- i1' + 2a ,P YBt cos p 'st + i- ----
q q ~ 9 9
-a cos P vt + 1l , (13.12) .
_- q
+L:,
n cos l Ptit -}- it --
.~_ q
(t. a cos P vi + II ,
q
P ti sin (- '#t? 1')) ui+
q
L
O
t+
in (
))x
-a sv 9 q
A~ cl~s p vt x.-11 --- aBi stn P Yt-~- ~- .~. t)u~1
q q ?,~ ,tt
d'-u. ,..,
--- D ut lA~ aA dt -: ii wi ?
STAT
 ~Bl cos p vt -- n)_- u cos C? ~t +1- , (13.18)
9 4 4
whence, by equuat nr ti. coefficiata for the se hanonica, we find
vt + 9), A1(a, 6) and n1(s, 0), obaerv-
R
illiL ,)
I
for all veluea of n and ^ satisfying the condition
P ;+rn\ r
91 9
0,
,iq (rat -t i) p * 0;
? also obtain the relation for deter iniag A1(a, 6) and Bl(a, e):
2 P vAisin P qvt ? 11 ? 2a P9v8t ~- du cos P 9'4
q
+
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epee to 9, ri tb the pariod 2x1.
Let vs find lEor.eq.(13.12) u,(s, vt
ing the condition that terra whose den ir:tor: mid .wish must be absent from the
v
_ vt ? 6).
exp essione for u1(a, vt,
q
On representing u1(a, vt, pt 6),
and also the right aide of eq.(13.12) in the
i(nit+i. yc(13.16)
no!
n n
-~ ,.,1 p s
r,rfra
P~,u, AtQ
'4, vt -+- t " j,(u) e - . .. (13.17)
ti ,H
Let na nor substitute the right aides of ega.(13.16) and (13.17) in eq.(13.12).
This yields , P v i'w~t},a lp ,t+!} _.
4
9
N N,
( .s~)) ~. j101(u) a + 2A L sin - vi ? D +
tl NI g
7
ftaNr (a) (13.19)
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(13.20)
(13.23)
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220
tier (13.16). ae find
o, salt :tutin~ the value of j)(.) ffU~ eq+(13.19) in the right side of egaa-
t,1 ~rt } Mu (!: F 1)
t
t
w
tt, vt, p vt }- !! ry ..., .-:--.---'----Y
X i , vl , w ~-i 4 D) F,*
i" ') d yt d - +r +-
Let us aor turn to eq.(13.2)). The satiation in this case, as already indi-
cated, proceeds for all inte~era n, a (positive, neRstiYe, and zero), for .hick
nq-f(in ? 1) j, = o. (13.22)
For this reason, the sua will contain cosplex exponents of the torn
= cos vt ? O :~ i slu . l) et4,uit n.y,
4
Q
Nye also note that, by virtue of ea.(13.22); ^ t 1 is divisible by q, so that
*e any write this factor it the fora qa(- ? < o < ~).
Qn.. atin1 the coefficients of cos(! yt + 8) and sin(p Vt + 8) in eq.(13.20)
,1
I?+iUueFt~1/ot q q
q q
 (13.26)
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Thus, is first appro*uoatlon for tlereaonant cue, the volution o
.111
(13.24)
I 0.10
Since, in the resonant cast we assuage the detuning eA to be a quantity of thr,
Lint order of srnsllness, we ^sy with the se degree of accuracy represent the sys-
te# of eq. (13.2 4) in the fora
der "_ . \~e+q~b
.~ A
dt
N G
(13.25)
, y
d+ __ p y ._ f( a, vt, Y) a cos' d4 d'.
dl 9 qr.~~?~~n i* a
~
knowing the expressions for ui(a, rt, q Vt ` 8), A1(s. ~) and 1(a, 8) we can
in accordInce with eq.(13.1S), find an explicit expression for fl(a, Vt ?after
`which eq.(13.13) 'iii yield the expressioas for A2(a, ?) and B2(a, 6); which are
necessary for constnictinp the second approxieation:
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q v is not necessarily amsll.
Let us then deter~ine the functions ui(!, vt, ~), A1(a, 8), 8(s,), fur i
2,.... For this`purpoae, we first ese the systa
tl~ p dA~~;4 dAi~ d~l~C~ w
f
+-
i
Q ,
+z'{1A dl's . ~ q ~ ~-8 aa t? ~ t+
du s
(;u-) ._.. s A~ -~,- a
9.
222.
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(13.28) to
obtain
p an. i
-.e ...__
-a
J4JT
9
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e1Q't) ) fi (a, t,
)e_d,1 cos l t l d$.
iot. u?' now consider the post general 'case.
Let it be required to investigate the bdhavior of a quasi-resonant system for
the appraachea to the resonant :one fror a nonresonant zone. for thin purpose, it
is necessary to coaitruc.t approximate solutions that penait studying the behavior of
the systen for a sufficiently large frequency interval and which, in special cssea,
will yield the above-presented foriaulae for the resonant as sell ss for the non-
resoesnt case.
Here the detuninK can no longer be considered a.all so that the approxiMate so-
lution aust be deterained directly for eq.(12,1). l~oreorer, the dependence on the
aa*ie of phase shift rust be introduced in the expreasiona for the instantaneous
ssiplitude and frequency.
T us, the solution, as done above, can be obtained in the fora of the series
x , a Cos (1. a4- ll -{-..u1(a, `4,'')'-f -z u~ (u, 4, 4')+?. .. , (13.27)
. +l
where a and a uuet be detenained fro. the following syateea of differential equa-
tions:
da
dl
dil
dl
!h I
-~ ~r- .. .
(rr, ()) .4- .'1il. (o,
Ilk
and where, in this case, the difference ~.
? ( rLJI {{ { r,.
-t-- d.J3 ~{d F
-
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,rNx ~ u~8.r ~ W .-: p v dot _..: 7wi, ~3, cos ~
rtry _ q da ~
aw sin P ~t _}. SI ul
4
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(13.31)
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;after ehich, on aubatitutinp the value of `x fro. eq.(13,5) and that of dt3 fro. equa
Lion (13.8) in the rijht side of eq.(12.1) and bearing in mind egs.(13..28), mad
(13..29), we obtain the following expression after orranginp the results by powers of
the a.11 pacauter.
du + Z~d~t, aI .. ~, ? w - p + w9,
dt d8 dt q 9
s8 w P v `~'~ -- 2u0131 cos P vt -}_
+ . q dl q
.- w --~ A v u d By 2wA. sin P ~r -}- ~~ -~- d~auy
~ ~ -~' ~ y dls
a:x
(13.3)
clU ~l q d8~ 9
d8o At +B, H, + 2A1B, sin P ~t + ~y +
da d8 9
du d~, du p ~~ dB, -4-
q
+ 2 d~uN a P v )B1+2A1+2jB
sluff w _ P v )A}+s8..
+ 2d
da ~ y
he right side of eq.(12.1), accordia~ to egs.(13.7), (13.77), (13.28), and
(13.29), can be represented in the foci
--- a. sin P tit +
9 .
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ll (A1cos p ~f-f41. ---
t, a cos p tit- -i- , -? nW sin P tit 1
-
q q
---- sin ' (13.31)
By equating the coefficients of the ass~e powers of a in the riot aides of equa-
tions (13.30) and (13.31), we obtain the fo11owinfc system of differential equations:
_.. -.?.~:..~. to -v ..~_r~8~ g P 1 ~-
L' LIU
_~-
. ~' 11
wt 4-. .... ~ v (1.!~-~.2w~11 -
t~ 11, rt, ~ -~., .~, ~
(13.32)
I~la.o teal;, n t C1t: ~, wIl:
) .-~ - 1 J
N c)1
Ir. ?, tt ~~~~
U ~ 8t 4.- ` :1if31 sin (?- P 4 --~_
' aC3, -4-- (i'' -1- A ~ H, - uB cos tit 4-- i
the following
4}111
_(O..
ynlwls have been introduced:
f ?~~, a cos '.t -i.-- ---aw sin (.?- vt-j--~l)).
i1 q p ?.tawl! f ?~t; a cos ('.t+'M) -a i sin (": ?~t-}-i-~~
x
tlu1
y vA,
~ W--
-
q tw
r~tt~
1 i)ri rl~t A
+t a cos P vt+=N ; -_.vu, sail p vt.SI x
,
.~ 4
(. 4 ? t.~ - aB, in (-? ' ) j- (o
Q
()tif
2w.%. 4- (a
t)u
(13.34
STAT
where
au, W .~.p y dB, -
_
va or i vu vo ~
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(, w-..p,) o! r fo'_?"1?'1 cos ?, d4
.
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225
4 (4'+ ?( '1
~~ -- (a'i Ill tt) )
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For determining u1(a vt, vt + 6), A~(?, 0) and I1(a, ) from eq.(13.32), +e
q
proceed as
..
in the resonant c.sse; the condition that the expreasi.ons for ux(a,vt,gvt
+ 8) muat contain no terms whose denoalinator$ rsi,ght vanish
(at . q v),
permits a
unique selection of the corresponding expressions for A1(a,
8) and B1(a,
9)?
After a series cf calculations analogous to those 'iven for the resonant case,
we find, for ul(a,
F' ? i
= 11 111
~ngf{w u1) p/Ol
P vt .e, the expression
q
and for detersining A1(s,
ra~r~
b
) ~
y. _
log
ate ~ ~.
ta,t h H' ( t 9 ~
rl ?v! c1'~
(13.35)
e) and t31(a, Q) we obtain the following systems of equa-
J tit, q yt-~--0) ~'-lye0 io5
5)'
9
(13.36)
Since the expressions for A1(a, 9) and L1(a, 6) a-uat be chosen so sa to satisfy
the condition that u1(a, vt, p vt + g) must be finite, we can select, as the wanted
q
P*~ eisioes, a certsin partial solution, periodic in 6, for the aysteu (13.36),
which gives us no difficulty.
The explicit expressions for A1(a, @) wd S1(a, 8) will have the following
S U 0
-
v It
.,t+)e-"i sin 'd.l ''??
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(13.33)
Aft r s3:i-g fo;~.Zd ? ? ,.1(e, vt, vt + ei, A,(a. 6), H,(a, 8), eq. (13.33), can be
e~a~c. q , -
used for determining all quantities necessary for the construction of the solution
in second approximation.
Thus, for determining, A2(a,
6) and bi (a, 8) we obtain the system
dA, ?' t
y1, p r14._ ill e"''I?D cos' d4 "i.
tIR.
'vi -{- e- j'r,;l 5111 'e 1!'.f (h!.
(13.38)
It it obelous that the formulas derived by us for studying the resonant region
a: r!l as the epproecha to it. will yield s11 formulas found earlier. Thus, by
putting cr,- p v ' eD in eq.(13.36), we find with an accuracy to tens of the first
order of rallnesa, expressions for A1(a, 8) and I3(a, 8) of eq.(13.23), which were
obtained in the case of resonance.
2w , ,?4 3& In Y 1vf d4
cc e
`
(q to pv) l f fc ' w 1q'g In d'4
J
4,
4,
I
/ f 4
4 _ (qw _ PY)s 3~
!lseu~sing that the oscillatory system is out of resonance, we bare Wt - q vt '
coast, so that eq.(13.36) will yield expressions for A1(a, 8) and l3(a, ~) coins
ciding with those of eq.(12.35).obtained for the nonresonant case.
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To suarsrixe; we present a scheme for the construction of a solution
in first end uund rux1wations
for the most. general case.
a cus! 'wl -- s,I ),
rl .
(13.39)
where sand 0 aunt be determined from the equations
... (13.40)
-- ? Z13i (i,, :u,
rtf ' ? to ~~
(hi
..~1 ... a.11 (a, ~I), 1
in which A,(a, 8) and B,(a, 0) are partial, periodic solutions of the system (13.36).
In second approximation we put
.~ - a cos 1...,, .4_ su, u ..i, {- ?~l ~., ;~ , (13.41)
where a and 0 are determined by the equations:
rft
(13.42)
rl t q
in which A,(a, 6), l(a, 0), A,(a, 0). h2(a, 0) must be found from the systems
(13.36) and (13.38) and u,(a, vt, p Vt + 0) from eq.(13.35).
y
Ne note again that the equation of second approximation (13.42) taking account
of the expressions for A2(a, 0) and B2(a, 8) of eq.(13.38) appear to be rather com-
plicated only because they are written in the moat general form. For concrete ex-
smples, even in second approximation, we obtain relatively simple equations de-
termining the amplitude and phase of the oscillation Icf., for example, egs.(12.56)
and (14.38)).
Let us consider the first approximation.
In contrast to the nonresonant case, here the variables are not separated in
the equations of first approximation (13.40), and we have a system of two mutually
'connected..equations.for determining the two unknowns a and 8.
Ie note. hret that,, for sufficiently large values of p and q, in view of the
assumption made earlier that the functions f?(x, .-) are of polynomial character,
227
of eq. (12.1)
As the first approxi-
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? It must be borne in mind that, for every solution, the quantity a remains finite.
Fro, the physical point of vier this limitation is always satisfied, since the
oscillation imp/ tude cannot increase without limit.
STAT
the first approximation in the resonant case will not differ from the nonresonant
case. ha_fact , fuc sufficiently large values of p and q only terra corresponding to
Let us return to.a consideration of the equations of first approxiasation (13.40).
Since the right sides of these equations depend on sand 0, we are unable, in
the. general case, to integrate them in a closed fo?rm ~. ~~ ri-?
~V ?~ qualitative character of
the solutions can be investigated in the general case, by means of the Poincare the-
0 will remain in the sums on the right sides of eq.(13.40), which terra will co-
incide with the expees.ions (12.35) obtained in the nonresonant case.
Thus the effect of resonance will be manifested, generally speaking, at small
values of the numbers p and q.
ory, since here we have to do with two first order equations.
According to the fundamental results of this theory (cf. Q spter 11) we may as-
sert that any solution' of eq.(13.40), with increasing time, approaches either the
constant solution -
(I 1, i) ,
determined from the equation
(a, 0,
~o --- v -~- $I~~ (ct, '~) ~- (13.43)
or a periodic solution.
'thus, we obtain two basic types of stationary oscillations: oscillations cor-
responding to.a constant solution or, as they say, to the "equilibrium point' of
eq.(13.40), and hose corresponding to a periodic solution.
In the former case, the oscillations (in first approximation) take place at a
frequency exactly equal to q v which is, consequently, at a simple ratio to the ex-
citation frequency. For this reason such a state of oscillations is called syn-
chronous.
In the higher approximations [cf. e.g, .eq.(13.21)) the expression for
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(a, vt, vt + 0), Kenerally speakiatt, contains in addition to the fundwnental fre-
pv
y
quency V other:overtonea of the subfrequrncy -.
q
In the case where a constant solution of the type a ?'q exists in the Byateo,
corresponding tot a absence of natural Oscillations, the expression for u~(a, vt,
P vt + 0) of eq.(13.21) will 1x the aaMe es In the nonreaonant case of a .(12.34
q q ),
~cnd trill represent a heteroperiodic state of oscillation.
Let us study the question of the stability of the stationary synchronous state.
To deter*ine the stability of the constant solutions ao and 8p determined by equa-
Lion (13.43), the correapondin~t equations of variation roust be Bet up.
zfj =~~t~-
1=0
the
(13.44)
.4.. z13t.t) ', -'= (.l t~Qj3t,- 13~~s,,1 t,-) -.. 0, (13.45)
frow eq. (13.45) we obtain the following conditions for stability of the synchronous
stationary oscillations under consideration:
,4;rJ (a0, D0) +I (a0,
I1~u (ao; )~q)B1~ (a0, o) - ilv (sir,, ''n;~r (any
(13.46)
(1.3.41)
In the latter case, correapondinK to the periodic solution of eq.(13.40), the
oscillation viii occur in first approximation with two fundamental frequencies, i.e.,
the frequency w.or v ? bw and the beat frequency Dw, where G~ 2('[ period of
q T
the Riven periodic solution). These oscillations are called asynchronous.
A. an sa~ple'of-illuatrating the character of the synchronous and asynchronous
ststes, let us consider the vacuue?tube oscillator under the influence of an ex-.
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Un the basis of eq.(13.40). the equation* of vitiation
:ri tteu in
11 ail? + I
r- .1 ids (a,l, %c? . f. a.1 ~, (u~- e) %'t),
ill
("u, ~,ir j? :J3~~-(rr~,, ;1) Gil.
aVt
characteristic equation for the systee (13.44) w111 then be
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the di f fwre't ii c ji tion deacribing the oscillatory process will be
where L is the inductance; M the coefficient of wutual inductance; Ii the resistance;
o tie constant component
control Vo
tags,
of the total control voltage; F cos vt the
component of the nonlinear control voltage due to external excitation; a the com-
ponent of the control voltage due to oscillations in the circuit.
Let us consider the case when f(E, ? u), where u ? e + F cos vt, is a cubic
polynomial
~(if f-u) -'fUT) --Su f- S1& ..,-,S,tv, (13.49)
in which S2 J 0.
Assume that the terms on the right aide of eq.(13.48) are small; then the os-
cillations will be close to harmonic, and we will be able to construct the approxi-
mate solution of eq.(13.48) by making use of the formulas presented above.
matie,
(13.50)
where a and @ must be determined from the equations
3 ~y
dt --- o -- u5 ~."' SEr a - "' cos ~))~ (13.51)
In the case .here the oscillator is set up in accordance with the diagram given
II C the capacitance; d the transconductance
F
r of the tube; w - 1 - the natural frequency
Foes of
-- j-o of the line circuit; i~ ? f(ho + F cos vt +
1 R e) the tube characteristic (ia ??plate
current); E ? E, + F cos vt + e the total
'-'~---
l
F t
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where 6a
(Sa
dl
510 S~II 2)1,
2' Cr
In riew of the fact that there is only one unknown function, namely R in the
second equation of the system (13.51), it asy be integrated by quadrature.
Let us discuss still another question.
Let us find. the relations between the frequencies and v, and likewise between
the coefficients of the polynomial (13.49) in the oscillator, at which stationary os-
cillationa will exist.
Ibsuie first that
1
Then, by integrating the second equation of the system (13.51); we set
w) (1) -- l0, (13.53)
t -3- ao
where
v
1
- , aresiu 1 'Scr.
t
From the first equation of the system (13.51), when eq.(13.53) is
we find
a (1) .-+
lru,
where aa.is deteesined from the equation:
4
?2 ,
(13.52)
(13.54)
satisfied,
(1i,cc
U? (13.56)
i StP COS 2;'u 0, (13.57)
Thus, when the conditions (13.52) and (13.57) are satisfied, a h_eteroperiodic
231
scr-..tiff'2? 4 .SiFcos 2Da r
r
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stets +rill be established in the osciliat~r; w d only the st:titnary state is pos-
sibli in this case.
Let now
exceed a certain value in this case. In other wards, a
aisle et sufficiently siu ll values for the detuning,
u
,! Sri' cos 2
Thea, the solution ao * 0 is unstable, and the ayatr+o will be self-exciting.
hum eq.'(13.56.)we find the value of ap a: follows:
~.`~_i L . -.S.V._ '`~i ~`.- ~ - - 1 st~CQS 28, 1 ? (13.59)
tlr~ -... ~ fit,' L ,.~~ -~- ~ i 2 "~
Thus, when the._conditions (13.52) and (13.58)
are satisfied, a synchronous
state is established in the oscillatory xyatex1 under c nAidlfatlon. and in first ap-
proxiaatiai%
synchronous state is pos-
e = i10 COS I `1" l~p -
where a :nd a are dete*eined according to eq.(13.54) and (13.59)?
Stationary oscillatioaa with constant aep1itude and phase and with a frequency
equal to half the excitation frequency, are estaLlished in the oscillator.
According to condition (13.52), the detuning of the resonance - 2 lust not
het us now consider the case when the syatee is far fros resonance, so that
the condition
1 1 8nS1F I
(13.58)
(13.60)
(13.61)
is satisfied.
Then, on inte,rating the second equation of syateo (13.51), 8 way be repre-
sented in the fore
11_.. Ai t .hr;t..,I;.t..t
U . . 4JW( -~ ?I `a. W, T- 'J,
(13.62)
`.here a is an arbitrary constaaE, 4(9) is a periodic function of 8 with the per--
oa sn
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2!t
7'
~,. -. .,, ..~. sIn 20
tt "cr
(1363)
On substitutin~t the value of 8 ?tu +?(13.62) in the first equation of the
syste~s (13.61), ee obtain a first-order cquatior+ with a periodic coefficient for the
deteraination of the stationary ar+plitude values.
This equation wi11 allow the solution a ? 0, corresponding to the heteroperiodic
.:.... ? n a.p.-,-dg on the sign of
state The question of the stani1 ,1.t of the aoIutio
. y
the expresaion:
Se,, . "* ? S1F COS dli, (13.64)
:. 1
where c s 20 denotes the averaged value of cos 28 over the period T:
COS 1;- di.
(13.65)
1-
-l
(13.66)
If the expression (13.66) is negative, there is no self-excitation and the
heteroperiodic state a ? 0 is stable, but if the expression (13.66) is poaitire,
*elf?excitation occurs in the aystea and, conaeque~tly; the heteroperiodic regime
is unstable.
Thus, ,hen the condition (13.61) and the condition
By virtue of eq.(13.63) we have cos 20 ? 0, so that eq.(13.64) takes the form
Scr -
(13.67)
are satisfied, it ^ay be shown that
!1(t) -.1(. uul _f11), (13.68)
ahere A(~t18) i, the cnrmpondiat reriodic solution with period T `~of the
first equation of the systaa (13.51) after aubstitution of the value of 0 from equa-
tioo (13.62) . in it.
In particlar, for aufficiestly great values of the detuning Ii - we ob-
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(13.69)
i.e. the value of the stationary amplitude in the nonresonant case.
Thus, when the conditions (13.69) and (13.47) are satiafied, s stationery two-
frequency state ? asynchronous oscillations - is estsbliahed in the oscillator. In
first approximation, for the stationary oscillations, we obtain the expression
v'
in which ? Aw it the fundamental frequency, while the amplitude A(6iwt
;, .
phase 8 +w?.(A wt ' 8) oscillate with the beat frequency &a.
on analyzing expression (13.74). it is not difficult to note that,
sway from resonance, the oscillations (13,70) approach the nonresonant oscillations
of the fore
e=acos( t-~-~)?
This obviously occurs in the most general case as well. On studying the general
equations of first approximation, eq.(13.40). we may show that when the detuning
1w - v~ increases, the "resonant first approximation" is continuously transformed
into the "non resonant' type.
Section 14. influence of a Sinusoidal Force on a aline!r Oscillator
As a special case of the oscillatory ayst d ~ Y.an aea..., h.d by eq. (12.1) , consider
the nonlinear oscillator under the action of a harmonic force. The oscillations of
such a system. as stated above, are described by the following differential equa-
.,
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) sin (~.1 ?f (~) r1(?.1.+ )
.
(14.3)
r.
E /o (a. 4+ D) COS (?~
.~ .z ,2fwam.
ma (w + ?r)
/o("- .,l f- ) _ /(a Cos (?.1-j-- ), -- au+ site (.1
In second approximation we obtain
~? ~.. a COS (~.1 _f i-) .+-
r 1 [cos in (?,t + ')X
2r.
,
X jfo(a,
-
(14.4)
-.- il) cos nt (i 4 )d (tit + I)) -{-
0
in this case that the amplitude of the external harmonic force is
? It is aaswed siible because of physical considerations, we
mall. If this conclusion is i~po
have the equation
d-x-
m ''{{ -
u i~
..? 4 L: i.... 0.:ii?inn Y ' Y
-allCR, VT
type of eq.(12.1),
ci.r' J iu i,
.1$ d
(T
, is reduced to an equation of the
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a1;aipa thia equation in omr introduction, we ce~ae to the conclusion that,
first spproxipation,'un1Y the fundamental reeuiiWCe cstn he detected,
?.~
to -,~~ _~
oximatC s
-
--
Thus, using the formulas already derived, let us conucwct spr.
'o a of eq:(14.1), for the ceae of fund ntg1 resonance (p" 1, q ? 11t1 n field
. fiat .rn~nY116stion, yield (13.39) and (13.40)., ja Fi apps--_
(14.2)
X =U COs ('4 ?4- ),
where a and e ?uat be determined from the systew of equations
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approximation, which we will not write out.
Let us discuss in more detail the investigation of the first approxicuation.
As in the case of a nonlinear system subjected to a disturbance not explicitly
depending on time,.we put, for abbreviation [cf. eq,(7,4)J,
~!K
+ sin m ~~-~" )tl
where a end d are functions of time, and are determined by equations of the second
I l0 (a, ',t f . ) sIfl (J ..,.i rJ ('J
I.tu
and note that the
I
r~ E I fo (t1, 'J i t!) i()S ('4 f i) ) 1! (?J
l .
U
parameters Ae(a), ke(a), so introduced are the equivalent coeffi-
dent of damping and the total equivalent coefficient of elasticity for the oscil-
latory system under consideration in the "free" state, in the absence of external
excitation, i.e., for a system described by an equation of the fora,
rn : f kx aI(x. :). (14.6)
Jl
After this, eq.(14.3) may be written as follows:
drt
~i Ctl 1 II
wiiair bUe atabAH6wij aiitC. 6f il~ie vsciiiGtiirui.
vw.s
In(o+v)
ti:
~....f _.... sin ti,
ma (b)
(14.1)
/leis) ke(a)
where. 6e(a) ye(a) m are the equivalent decrement of damping and. ?: V
d8
dl
tion (14.6).
the equivalent frequency of the nonlinear natural oscillations described by equa-
obtain the stetionsry
values of the ariplitode a and the phase 8, in first approximation, the right sides
of eq. (14. i) must be equated to zero, after which we obtain the relations
(u)u
_f,. fl) situ iii ('a --F ~i) d (61
s1;
m ~ :.4. v) COS it
236
u (14.5)
=0,
(14.8)
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2ns' a ,, (n) ::- - d: cos D,
sea (a) - - zt sin ~t,
whence, on eliminating the phase 0, we find the relation between the
the stationary oscillations and the frequency of the external force:
The resultant equation: (14.9) and (14.10) coincide with the equations used in
the classical linear theory for determining the amplitude and phase of forced os-
cillations
X tt Cos ('44.. f)
in a system with the mass m, the coefficient of elasticity ke(a), acrd the coeffi-
cient of damping Ae(a) (and, respectively, with the frequency 'e(a and
) m
the decrement be(a) ), under the influence of the external sinusoidal force
tf; sin rt.
we may therefore formulate the following rule: Given a certain nonlinear sys-
tem under the influence of an external sinusoidal farce with a frequency close to
the natural frequency of the system. Iequired, to find the values of the
-
and phase of the stationary synchronous oscillations of eq.(14.2).
For this purpose, by linearizing the given oscillatory system in the free state
(i.e., not taking into account the external force of sin vt), we determine the func-
tions of the aaaplitude, the equivalent decresent and the equivalent frequency of the
aatura! oscillations;
( substituting the resultant value in the classical reistioni
theory of osciliationr (14.9) and (14.10), we obtain an equation for determining the
recirea goanc _ ~itics, ..__~:t..Lo and n phs.t~.
ii d
This rule has been formulated for the special case of the oscillatory :ystms
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or, with an accuracy to terms of the second order of smallness, the following re-
let.ionri
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In view of the notatcn introduced ie eq.(14.5),
we may write
(14.16)
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described by tine d,fferential equation (14 1). lut it may be extended to more eener'a
caaea of oscillatory syt:terwa.
Let us introduce the conditions of stability for the synchronous stationary os?
cillatioae under.-.consideration.
For the. resonant cs.e, the equations of first approrcitation (14.7) ray 13e r?pre?
sented, with an accuracy to tetras of the second order of s+aallneas, in the fora
as
di
and the equations of the stationary synchronous states, in the form
k(u, i) U,
11' (n, ~l) u,
(14.12)
(14.13)
where 14(ad) and $(a,e) denote, respectively, the right aides of eq.(14.12).
Let a and a be any solutions of the eq.(14.1.3). To investigate the question
of their stability, let uis uske use of the conditions previously derived (cf.equs?
tion (13.46), (13.47)), As applied to our case, they will have the following form
a!h, (a, ) 'I' (u. ) C U. (14.14)
1? (u, i!)11,' (rr, ;') _ I ,(o, ;i) > 0. (14.15)
Let us discover the meaning of these inequalitici.
) ~-'I',u, ~l} = --- F`at~i(a) _w !~atr il,~ ~~ n1
whence, bearing in mind the first equation of the ayst'e (14.12), we find
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2ya' f(w(a) - .~.)
(14.19)
(14.20)
(14.21)
follows:
(14.22)
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Under the ordieary 1.. of friction, '4(s) increaites with the swplitude, so that
U?~qu) l),
In this lay, if we confine ourselves to the consideration of systems obeying
npresmts the aesn poeer dissipated by the force ef(x, Q-) under the oscillations
X :,. (1 U;S (wt 4- H),
the ordinary is. of friction, then, condition (14.14); according to eqa.(14.16) arid
(14.17), will always be satisfied.
Consider nor the. condition (14.15). For this purpose, let us investigate the
dependence of a and 8, the solutions of eq.(14.13), on the frequency Y.
r dt1 ? t~~~1
?
Ru d,ja ~-- R d,4 =_
/~
, dry
--
~
'' ~~
'
4-
-
~-
-
0, ?e obtain tie resonance curve shosn in Fig.101. CM
Y analy:iaR this curve re see that as r incieaaes, beginning with saall values, the
' `b fh ttLt$s .ta sill' be absent anti l r reaches values correapond nR to
i= . the-point .A~ .., Nhet 1- reaehes-the point A, oaci l l.tiona .ill aeia. in the iyatoo,_ and
descringd by eq.(16.68) are close to harmonic.
eq.(16.68) that corresponds to the presence of a
in the ryste. in the fora:
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X12
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v inrreeees further, the. amplitude of these oscillations will vary along the up-
analo~oua picture, except that
-,=slues of v (bit. 105).
e,
Ihzs_rragust:on, as staL o~
his satisfied.
modulation necessry for parametric resonance at a given damping.
the expression for a must be equated to zero.
.`so that the width of the resonance zone will become
At
the point B,,
the oscillations lose their
st zility aau decay.
At v decreasing from large values, the
oscillationa are abruptly excited at the
point C (hard excitation), and as v then de-
creases, the amplitude of the oscillations
will vary along the curve A13.
In the case where y < 0,
we obtain an
the resonance curve will be inclined toward the small
To determine the boundaries of the zone of synchronization.
In first approxim:tion, the zone of resonancewill be
(16.73)
1e note that the presence of damping reduces the interval AC within which pars-
;metric resonance takes place.
the right side of
1
w= -{ ;i- It~kw4 .~ 16w-~-
'
(16.75)
`' deteninea the minimum depth of
-
'
It is obvious that d will be real.if the inequstion
Let us consider still another case of parametric resonance in an oscillatory
aystr with ooalinear friction.
In the case' of parametric exci tstion E o f . a circuit with a vacuum tube
the'egnation of oscillation will be
~x
II)' 4
d :..~_ (i,u+'.') di
w''(1 li CoS .l)X
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k
f.
where, (xis u iu ~pt,cr i, represents * certain t o?ralued function over the ia?
~ tarral (a,b)
11,1) it is expedient first to transfon
4
frog rnnttructinf the solution of eq?
.. ..~.
.q.(17.1) ... is order.aa elisinat+r . the awiti 6valrea iunctAoat) .fr it. Eor
?bi:_
we will start fro. a certain partial periodic solu-
tion of the equation of free relaxation oscillations:
eq,(17.2) in which the value x assuses its ainisus
Fig.1Q1
value at t 0.
tenoting the frequency of the free relaxation oscillations by w, we write this
periodic solution in the fora
w.7
'1 3
.
M` where :(t) is a certain periodic function of c with a period of 27t
two-valued
4O.....
then the solution of eq. (17.2) say be represented in the fors (Fig.107)
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and that,
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Taking into consideration the results of Section 10, it becoses obvious that
the derivative :'(m) during a single period undergoes a discontinuity twice
lin absolute value, it is always greater than sole positive constant.
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Qe differentiating eq.(17.10) and substituting in eq.(17.1), we obtain
) ? is (f)1 ? COS.
i'
11
11
(
,,
)
.
(
Y
a
for, in view of the identity (11.9):
.
a ~
Q .... s nos rt
(1712)
.
The tresforwed equation (17.12) no longer contains multi-valued functions in
.M the right side.
!a t~.,.
then tbe.periodic, solution xiej may be writLea as fu11oea;....
fnon i) muat iaenucally aatialy the following relation:
(17.9)
e now perfow a lubstitution of variables in the equation describing the
forced oscillations, eq.(17.1).
Ne replsce the unknown x by a new unknown tD, using the formula
x=z(~)? (17.10)
j nor convenience iii. constructing the approximate solutions of differential equa-
tions, it is usually desirable that the right aide be a regular function. In equa-
tion (17.12), the right aide, in view of the presence of the discontinuous function;
'6._.':'(+) in the denominator; does not satisfy the condition of regularity.
In order to regularise eq.(17.2), it is sufficient to interchange the roles of
i
i; the variables t and s? and subsequently to consider the independent variable, and
E (,) =a.+ p~ 0 < 'p po.
z (,) = a + +h x =_ ,j, f~ < 'p < 2sc.
(17.8)
e note that, since eq.(17.8) representa a solution of eq.(17?.2), the func-
t an unknown function of q, determined by; the differential equation
d--
+tt~ cos ~t .. __ ......
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(17.1
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Lii
14 (x) t > 'r a < x < h. (17.14)
Aaauwe that the snplitude eE of the external disturbing force is leas than y
Then the denoa~1nat4r in the right side of eq.(17.13) is positive, and the right side
of eq.(17.13) is itself an analytic function of the unknown t, Eaations of the
p type of eq.(17:13) have been investigated' by Poincare and Denjoy. Their results,
however, clarify only the qualitative character of the solutions. To obtain a tech-
. ;
1. Iaigae allowing quantitative calculationa, let us make use of the method of the mean
{..m`which has been briefly described in Qmapter I.
If we denote by Y a positive constant, such that
In order to apply. the results of Section 1 to eq.(17.13), let us expand the
aright side of eq.(17.13) into a power series of e. Me have
v
investigate eq.(i7.16) for the resonant case. Assume that the ratio
'iiI"jia close to some rational number p where, as before, p and q, generally speaking,
-_fare small prise numbers.
i
Then, putting
_and introducing the new variable t from the formula
Me have agreed above to denote an equation of the type of (1718) R*
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(17.16)
(17.17)
(17.18)
an equation of I
STAT
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t~! 1orinrt.
Cot' St
*,
' raxias j o!' , ft Obt,a n_ . _ ,
(17.22)
STAT
be constructed an the beds of
,,..... u 'tion ay
Thspprosioatewlutlon of this eq
A
-~
~tt*:.pr~s'i .
In first spproxlast10n~ Mczvcv...e,
(17.19)
(17.20)
?
` d'
t aide of the
eratiaa of taking the ?ean in the rips
Let us further discuss the op 1_.
and the function into a
i necessary to exp
~:
Ne note
~ 4 ' n? to this cane we n??~?
only where
from zero
w.-tcaa be different )~ (E --- iii ?
p Al i cos (n + ?)pus (E + n~ , cos a.., er .nn, ieatlOf
:j6: Thus, if
q
33t1720) de eaeretea iaw
42
irae` which we find.
s
i-e.
413
is
-
For this, It
ri r -T ter
the results of SectiOn 1e p
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cwt'~1.
(17.24)
p
h is, we obtin for forced oscillations in first ap-
in the care when
Thus
,
q
pproxioetion the same kind of expression as for free osozllrtions when the external
-{?1force sE cos vt does nit act on the syste :
t
Thus, in first ipproxiuaation, the influence of a gill external force on the
for. and frequency of the oscillations is negligible unless. its frequency is suf-
'ficiently close to one.uf the overtones of the natural frequency.
Then the derivative ;d will be an alterne(ing function.of of the, type shown in
Fig 108,
0
From eq.(17.70) we find
Thus it. is obvious that there exist
the equation
STAT
(17.25)
The equation so obtained is considerably simpler ts the corresponding equs-
fz.
{
+1tion of first approximation (13.40) for the ayaterr considered in Sect.13 of the
Consider now the case when q is equal to
Ito the aubharmonic resonance w
preieot Chapter, where we obtained a system of two differential equations in two
1unknowni, namely the rplitude and full phase of the oscillation. In the case under
?4discuaaion (that of relaxation oscillations) we have only one differential equation.
_.~
with respect to the phase angle , which, in addition, is integrated by mechanical
dquadrature.
The character of the solution in eq.
a certain integer m, which correaponda
.25) may be found directly, without first
=a
integrating it.
4Uu.J
Let, for instance,
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' ~.~._`~ ~_.... , sat-zone, ao t at
o ide
Conaxder ~e the case when v. ,4i4
STAT
tab1~ tM_,o1utio.s _fnr. i14h
F, * c?iA,a sin (; i1fe) < 0
Since, in the cue under conaideratidn, ? ? vt - rnW, we hate
4-
X. __,_- Z .....-. I ,
?t J
'shence it. is obtiaun that forced relaxation oscillatioati, with the passage of tine,
approach steady periodic oscillations, corresponding to the different roots of equs-
~.=
3",
Em/la,
1U-
?V
`tion (17,27), and occurring at a frequency exactly equal to the aubharmonic - of the
,.;frequency v of the external force.
Thus, for valves of the frequency v lying inside the resonance band detenained'
4t Z,
Eby the inequatron (17.26)
The xidth of the resonant zone, in first apprcxi.ation,
I z n:4,,, $,
~1zis obviously proportional to' the a?plitud of the external fora.
h,.108
r - ... L (. ___
(11.28)
(17.29)
the phenowenon of synchronisation occurs.
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w~u i f 19 9&1 iuiI .lam=elv PL~f .i:a
vow, v1~ L1.ly/ ?_,. ...~~
{ atut ei which is the same ss the sign of the difference
On intern,itias ea.(17.23), we set
(17.31)
(17.32)
in ihich 6s is r arbitrary constant, f(~) is a periodic function of with the pen-
(17.33)
Jrhere 1(0) is a periodic function of a vi h the period 2s.
;god of Zs:
's
No.inj that, in the approxiration.? ted,
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the frequency Qp which can be called
1
Thus, in the case under consideratioj, the oscillations are aultiperiodic and
occur with tro fuadrmeotal frequencies: w th
the variable natural frequency,
;where G(8) is a periodic function cf 9 ei h the period 2.
MI substituting the value of 8 fresi eq.(17.37) in the right side of eq.(17.31)
the following notation is used:
r-- m a' s--JH'
pl = const
m +a
IS. aQiastiiuLid~ cgq:(iiu we find .1.. fiuaisypivxiiatc cxp?va=
son for the forced relaxation oscillatio*s in the for.
z =Z g/ ~1? lm(Q,--Q,,)t-milf. (17.39)
f
repreaentia8.the diffsrence tone between he frequency of the external forte v.
and tba ^th overtone of the variable natual frequency.
solving this equation with respect to 8, we find:
since +' for c ? 4, and with the best frequency
' a rI
(17.36)
(17.37)
(17.40)
aY ?I _ 1 s ~n t1,N
in .j..a I - . 4
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*e sate that, as v approaches the bo4ndsry of the resonance tone, a Q, an
t
1t to dhow that, on departure from resonance, e
ft
.:.Ore: Lba beat Ir.II$ also Leads toeerd sero.
- -- _
h
Next, we construct the second approximation; for this purpose, let us first
~1atens~
ral fregaeilCY approaches its value ' correspondis to free oscillations.
,able .nat~w
:
aw
In addition, it rs not dr
of the belts, deteratned by the function o, diexnishes sthat the vari-
find the expression for the refined first approximation.
Let. us consider fiat the Reneral ca$e of an arbitrary rational value for the
;a . J ratio p.
i Making use of the Fourier expansion of eq.(17.21), we obtain
,,,,"?pressios for the coefficient of the first power of t in the riot aide of eq.(17.20):
is vutasw _f wi a =.ivw ~ . aRE _--
~ and mroximation:
11
~ .A sin n i 1 } ? 4 u11'
P
q2
r_ sb
q
JQ
On ssbstitutin$ thi value of t from iq.(.13.41) in eq.(i7.it)) ad
(17.42)
averagiag
STAT
The refined first approximation will, therefore, have the followinn foj:
- e~ (17.41)
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0
Coi. dei fist th a nrt onant case when the ratio. D is not equal either to a
q
?whole nu.ber or to.haU..a whole nunber,`and when, consequently, the frequency of the%
?external force does not lie close to the gvertones of the natural frequency; nw nor
?
half of it, nw.
we note that in the adopted nonresonant case the following inequation are ob-
r fl..
(17.44)
For., this reason, takia. account of eqs. (17.42) and (1744), we .ay write
STAT
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! , A jA.
,ir.i71
.. ,~.. ,, ~_r.4.... ...._
es, rn tTie nonceaonant case ?eobain the~fohlo+nn~t expresstons oI.thie ?econ
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{'t ?.... t - I I ~
-
(17.45)
whereo is an arbitrary constant. Bearing in mind eq.(17.41). we obtain the fol-
w. lowing formulas of second approximation
y h
-~ + .u a, - m
4frow which, with an accuracy to tenma of the second order of a.alneas inclusive,
v1 S
we obtain
Mom,putting
(17.43) then takes the following form:
(17.47 )
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'. 0
x Z ("p),
sinfn(a,t,f p)+14+h I+
(17.50 )
i" n Q t 4 aA sin yt,
where, according to eq.(17.48), with the degree of accuracy adopted by us, it fol-
lowa that
4 p~
n#o
(17.51)
fmm the formulas of second approxiewtion so obtained it is not difficult to
eliminate the auxiliary quantity, the ratio P. Since the difference - is of
the first order of rallnesa, eq.(ll.50) is valid with an accuracy to terms of the
third order
i th
e
Jisecond order of smallness, while eq.U7.51) holes to quantities u
of s~aallne... pith the are degree of accuracy, we may write
sir
j,t+?u+ 2 nw" y sif4rt(?p1+?u-F1+il? +
(17.52)
-f sin in( r-f p)---.tit-_Daj4-:Asasin =.t,
t~A!I - raa) , /ate
The solution thus found corresponds to the asynchronous
state of oscillations.
_11$ere the oscillations will. be quasi-periodic with t.o fundamental frequencies v and
The variation in the phase snple v ii represented here as a rotation at con-
stant angular velocity equal to gip, on wh)ch oscillations of small amplitude, with
,
frequencies V, s - v, nQ,, T . sic superkmposed.
fr?
Let us now construct the second apprpximation for. the resonont case.
-~otim-wse~' tie _teao~~. eaae.~e wst.taire -the ratio -~ as
I
.d..4 -
`y-'rteprr, o to a /ra t-int r. If .we--put p '- m, the_.reaultant..equatioa of...second. _ap-
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STAT
 ture
,..l frequency of the asynchronous
saw ~~~~. _--~._
proxi.ation will differ fro. equation (17.;25) by terse of the second order of se ll?
;sass. With this equation, we can refine the position and width of he teeoneait wii -
oscillations,
Without d,elling-,on this, let us consider the case when the ratio q is
To elucidate the operation of sreraging in the equation of second
/1. substituting the value of eQ. (i7. 53)
--nu.ber of calculations, we find
r_ 3
J' Equation (17.54) so obtained differs from eq.(17.45) for
by the presence of the su aaad
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._ s9 i ern < _..... _,.. - +1
4
(17.56)
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& S~, cos (2E --
4
a~.
As in the cue of the first approxi.*tion, it is obvious that the resonant tone
` ix defined t;
~r
(17.53)
after a
(17.54)
(17.55)
the nonresonant case
-
a half
approxi.a-
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.n. intraidcin.$ the variable natural frequency ,
the.,s edegree of accuracy by the_inequlUan
2m+I __ 2m(1
a
this tone is defined with
Thus, in _onsiderin~t this first sppr4xiaat1.01 , we have found the resonant tones,
es, of v lying in the neighborhood of m- their width being proportional
valu
f
l
or
on
y
wer of t. In the second approximation we discover additiu""- resonant
~?, .. to the first po
v l in in the neighborhood of _-~- . where the width of these
tones far values of y g
~
dart' zones is proportional to the square of c.
resence
h
e p
dicate t
s of the higher approximatiioua would likewise in
l
M ana
ysi
a for v ti P ~" q ` 3, 4, .. with a width of the order of ???
of 'resonant canes q
,y .
Section 18. Nonlinear steps with Slowly Varying Parameter.
2
In the preceding Sections
;which the time t entered under
periodic influence).
efficients
we have considered nonlinear osci.llstory systems in
the sign of the trigonometric functions (external
the equation' describing an oscillatory
that also depend on tine.
with time ("slowly' with respect to the natural unit
a34-time, the period of natural oacillstions)
lowing nonlinear differential equation w)
r, t
, 0-i
dx
0 18.2)
d'xs. ''. e "F.(; :, x, (
'' -(ii)--
,
being, in turn, certain poly-
f this finite aim Fn(t' x' fit)
M
fficients o
~
the coe d..,,,,a of t, we will assume, in addition, that
enaae cAQltf3cro _r----
r
paANtsln x, ~
~-,r
d8 or encX. of the extecn~l periodic force
? s ~tegy .
tnst~asaneou
slowly with time
... spas yaw's -
is the'"alow" time,
~in,wnicu,
' d 2n which may ke repre-
F(c, 9, x= ail is 'a functiOA periodic in t$ with the per&o "
d
sented in the for. of the sum
we arrive at a consideration
h slowly varying coefficients:
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'The cw f:tk?cttii.o a of th e; p o ieate olutlona of eq, (18.1) znrnlwes no adda,-
tionrLfw 4a .entrl..difficultie* and esy bo. effect*d. by means of the esya~ptotic meth?
god described above, Nye note'that for the cet_raction.of the asymptotic aeries it
dx
iii necei'ory that the coefficients of eq.(IOll,' i('s), k('t), as ,tell as F(T,
and v(t) have a sufficient numter of. derivatives with respect to t for all finite
;values of t and, in addition, for any raluea.of t over the interval 0
. i ~ ele
`y6 the oscillations will decay.
s a slow iacruae in pendalu..lea tb. as was to be expected, favors damping'
~6 9hM
E t ,then the anpli wde wi 11 ia'?
,q8,_ of the oscillations. if 11 c 0, n > 0, aid I ~
arplitude, in
,
crease, while for f 1 ' ' the. s.pli taode will decoease
`
_ t.!
' 0. the a.plitud~1wi11 increase
' 0 sad a
1
t
~
S? Eor:
a the absence of denpi*g (a ? 0), tie ~Plitude of oaclllations will increase
1 itb decraniag lee,. th and decrease with 3acreasrng length,
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(18.29)
..
( ____
= w t? - lj i, ~.
and s is the initial phase Value.
The latter .,_ i.th _ ....(2.u) already found.
equations coincide . ~
ua+e that the length of the pendulUm varies by the linear law 1(T) '
pe. sow P ssa
r where 1 is the length at t ' 4. ell is the rate of change of the.pendu?
+ 1
1
lua lea th (for a short time interval we can always aasuLe, with a sufficient degree
of accLracythat the length varies by a linear law). In this use, we have the fol-
%,
 ao::,tbe.,sba*cs of dMpi-et,_ the nstanteneoiue fr, ppeacy..decreases uith
y
a
is the psadelwe length and increases with 4a decrease in length.
At ~w Calt*tite` tho airs d pprowi ?`icn for this .:.~A ;,le: ccarding
}bona (18.18). (18.19), and (18.20), afte a nuaber of calculations, we have
1
X = a COS'' --- X92 cos 3',
t2.- where a and q- must be determined from the equations
ii 4'(t)
W b (S)
5..'3!"( s1 (i)a4 3s~lJ"~ ds)
'" ' -dl..
;which may likewise be totally integrated. 'Thus, the first equation of the sys-
4tee (18.33) furnishes the following re1at>4on
r.
analysis may be off, . _...re....
e for the oscillati.en frequrcy. Tor esamale,
S
~r Ott 3
ne :(!(O)\1
`"lf + a9 a Y i6 -a a; ?
also integrate the second equation of the systo
consider the differential
between a and t:
(18.33).
equation of the oscilla-
of a sinusoidal force, whose asp-
1r this case, we have the
litude sad instantaneous frequency
- following differential equation:
4~d ft jdf +aF(:)sinU, (18.35)
4where d0 ` e(t); t ' it, while s and is a constants.
46J The oscillatory r stems described by equations of this type play an important
8_jrole in machiae?baildias, electrical enpieeriag, etc.
"ice tact only the fuadaraital resonance; Sher` fore making use of eq. (18.6),
ro mst`oo >(oe the case of the fuadreatnl resonance
1 ..W......,a : W ....... ,.. _ ....:_ ...,. .
increase
(18.32)
(18.33)
-iNg1-
w '
(18.34)
let us set
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au.1Y.
-T-v.+
(18.38)
decrement and the
aeacr:b'd by ea. (1.1).
from eq.(18.10) we find the expresaiin for ut(~- a, 8, 8
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a nber of ca1c-,lationa for the solution of e,;.(18.35) in first approx
After. .
X = U COS (+ tf)~
where $ and 8 swat he deterwined from the ey+rtea of equations
..!_..-..', f(",
hr.mda, .
it
+
?
stn i'
,
ma (u
f 8 (a, 1J + 1)) = f (a cos (b -}-;-), _ o w sin (~i - f . t))).
--i1)sin('t }-i )d('l-4.11)
cos =),
Making use of the notation of eq.(14S) (cf. Section
f
or?.j
may be representee in the fo11owinF
da
dt
tE (z)
14). the
dO y "..) 4. t1 sin
jequivalest frequency for the nonlinear o?ta...y~?,
i
_!I) f i1 =)s,,n . (rh ..-, ,
_sbin(1I o(a, . ~4.11)d(t- ~, -~-
does aut depend on the slow tine t ~nd coincides
(la. 39)
with the second susaad in
N'e will nat develop the expressions for A2(t, a, e) and F32(v, a. 6), and merely
1~ + iI) I 1=1 cos U (ol +;.) X
1
(o, ' J _{- ~i) cos n (', + %))4 (~ + I-) +
X fo
hi-f- ')cos (i -- )d(~J-4--
I
(18.36)
(18.37)
aystem (18.37)
respectively, the equivalent daupang
(a) are
d y
)
s
(
`I
,
e
a
an
e
ehere
,
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it is si.plesL. to :tea use of tho oquctioaa of-harrwaic ba1+iicu (18.12), taking into
acnt_the above resiark on thiaubject:
{hsou*b a resonance in a concrete ?ystem.t To aske it easier to compare the results
{obtained from the ststron?ry state studied, let us conaraer, as. in Section 14, a none
F
Lot us apply these foriulaa in inves;igating. the oscillations on paaenpe
E
linear vibrator with a hard characteristi of the nonlinear. restoring force (f
cx + dx3), under the influence of an external sinusoidal force of constant ampli
Jtude and variable frequeacy.. Let the oscillations of this iibr?tor be described by
the following equation
where x denotes a coordinate determining the position of the system; t the ti.e;
}.~ the Via; b rssf1ici'~nt of rAlist* ; 1 * CY + dx3 the nonlinear restoring
" elaatic force; E the amplitude of the disturbing force; e(t) ? certain function of
:. i,..._ ~
!time. To sipns 1ify the calculations, let Us introduce, sa ?bore, the dimension-
1mm , eq. (18.40) ,i11 be written in the Form
Aawre that the friction, the Mplitide of the externsi force, and the tern
charscteri ainp the :zonl iaearity are su f fi elatly mall by r_.ampari son
ral lrequaacy df the systiem, i.e.,
4syat*, snd put:
with.the natu-!
s clone-to-linear conservative
STAT
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h,, o pp
ea eq.(14.5) yields 8-~(a) ~ - '; ke(a) ' 1 + ~3a ,after which, in first a roxima
tion, , rnm have
da
`where a and 8 oust be deterni.ned frog tl-d ayst~ of equations
-'dl
dO
ti cos (4 _}-
1
I ?4
(18.44)
'where v(t) _ is a certain function of time characteriiin the law of variation of
!the iostantanpoua frequency of the external force with time.
In Section 14 we examined in detail the stationary state of systens described
by an equation of the type of eq.(18.40), constructed resonance curves, investi-
gated the stability of the various branches of the curves and considered the hys-
teresis phenomena arising in connection with the nonlinearity.
Here we will consider the behavior of the curves of the oscillation
amplitude
as a function of the frequency of the external force, during a slow variation in
`J frequency with time, we will assume that during this variation, the frequency of
z;2 the external force passes through resonant valuea?. In order to construct the reso-
;..-.1nance curves on pasaage through resonance, the system of equations of first approxi-
_*stiore (11.44) must be numerically integrated by some method of numerical integra-
ar; time. for the equations (18.44) under investigation, the method of numerical inte-
t)_.Fgration developed by A.?4.krylov is convenient, *e note that there is no need of
^2 userically inte~tratin~t eq. (18.44) over the whole time interval during which the
(ifregaency of the external force varies. 'lo obtain a complete picture of the process
f itakiaR place on passage through resonance,: it is sufficient to integrate
310
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the sys-
tm .(18,44) from an instant of time at which the frequency of the external force is ?
s
sufficiently close to the natural frequency of the system, but has not yet reached
the resonance zone directly. Practical cdnstructions of resonance curves, on passage
i
through $ resonance show that, for values of the frequency of the external force at
'This question has been discussed by as in an earlier paper (8ib1.29)
,.... 4 CS) cos _~,
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tig.;l04
when the rate of change of the frequency of the external force is fairly high. In
F addition, the initisl values have almost Io effect on the character of the resonance
which the stationary resonance curve is dome to a horizontal line, the, curves of
pssase through resonance differ little frosa the stationary resonance curves, even
Q8 09 1.4 i i %z
Ii 110
-S. o1s.Naa it-psaaos tbrovgk z.aonance (on the.. value.. and.. position of: the.oaxi ue. etc.),
C-,ptuudd...ex.,us the resonance :one itsel f (i. e., in the
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v() ..yam ? X71
(18, 45) __..w.I
i
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312
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sons of fsquencies :ere the amplitude sharply rises), For tha reason, .for a nu-
.e~errct..tntcicwtun`oi iinc eyate* (18.44) it is expedient to adopt, as the initial
;values, the valises of a, 8 and v that satisfy the stationary state close to the reaof
A#n "7{~31~j Wig ^[e 4c ass _3 the avnc of ;r;p~ury i f?aai1i iiLuves.
*e note thateq.(18,d0) could be directly integrated by means of nuaerical meth.
i
ods; however, this Would be a coaaplicsted task, requiring an extraordinary *aunt of
Fig,1.11
tise and presenting such difficulty because of the possibility of cua~ulation of a
'large syateaatic error. However, a nuaerical integration of the equation of first
(or second) approxiaation involves no difficulty, becauae of the fact that the varies
sbles in these equations are the aaplitude and the phase.
To obtain a coaplete picture of the process, it is sufficient to calculte a
Z-aia11 nuaber of points located on a relatively"aaooth' curve, which substantially
L aiaplifiea the nuaerical integration, while the direct integration of eq.(18.40)
r3..aauld require us to find the sinusoid directly, instead of the envelope.
L! For limp ~ 11ci ty a will coniider the 'a:e ~~ew t! e iw:t t.atoi Ireucncy of
?? ?1 M? u~ VY Y
i
?;~the external force deoenda linearly on the tiro
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STAT
Fig 112
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Fig, 114
cospsriaan ritn
tiU?? ary reaofance curves constructed Ly the formulas of Section 14 are presented in these
vase disgraaa.
d 113 give both the atitionary resonronce curves
2
1
and the curves of
figures 1
ance for a case ,hen:the characteristic of the nonlinear
..:,,passage throagh rewn
!storinfc force has the fora given
? (f , . . . ' especial literature (Bibl.29)
} Me will further discuss the investi-
gstion of certain examples of the rnlsc
- Fig?1.15oscillstory aystdrs with variable coe#-
~~:~ y ?...? or..nll; con?
ssibie. As oui f;.. --~-
?~4 ficients, in *hich r more couplex rescnande is po
d depending on the ,
ki
`
n
f the
? in Te
nonce o cider the behanoc of the aeplrtude
of.l.latorv--airort-with...sa~-fete--teed- .
ie.. as -+
? -.
, o~
~ ..?acanat~oa~ ,? atata..o~ a
a ..2 , has al ready been
asth cpaateat . detuninA,
`C~~,~..w:~l~;--~-w.llL..th4.:..Caa~..O~.- ..
STAT
ab4olute ra ue a
aceiall pai? thrbi,~ .eeon
sy
ea
ratios of the syst
w Qn nwerical integ
For Q, ~ Q, it decreaaea with time.
, iaC ~, rus~l s w
> . tie trek i ^ith tisa and for
arAie*ge tbrough the. resonance `dopondr, ~aa the...wolue of ~? 1)c target' the
the rate of 1 aE the sore rapidly the
Analysis
of the resoaanc!
E; re attain
~.... of
at vetiONa iwa~~?-
eq~oatians
~ nuNb~r of cartes of passage through resonance,
110 ~ '
ehich are presented in Figs.109, 111. For
in Fig. 114.
cur
4onstructed on their Passage
~nance allows a nuaber of characteristic
features of this coaplex phenoaenon to be
elucidated, as well as the influence of
the i0nlinearity of the systea on it.
call not, hoMever, go into this question
/7 1 \ here, since it has been considered in
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r1 n[: sin 2)
~ ,
36
(18.52)
(18.53)
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If the detunin*.
we will obtain the
?hich, by means of the aubstitutioa of vstisbles
0.016 h n n h'
k,, -= 0,05.
(18.47)
(18.50)
r: .
1!,!)lli ~. k() _-2(z sin !1 -~
t: = ds 2t: ~~ .~.,
+i (z
bf~
Making use of egs.U8.5) and (18.6) end Putting p
Jculstioos ~tiree, in the first approxi.ssti4n,
. z u cos (1 --F- }),
4
sin 2t
3
varies rith time, then it is obvious ths.t instead
equation
` ()f(x , (`~ j.. E S1n 2t, (1&46)
r.{..z=() f ?!_?~ cos 2t, (,j)? (18.48)
,~- 3- stnfit' V! 3
tlf ?
Aasuse, as in Section 14. that
where s and O suet be determined from thelsystes of.equstions
r (18.49)
it sore definite, let us put;
sin 21).
3 J
(18.51)
2, a number of cal-
r (z
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For constructing traphs characterizing the variation in oscillation nplitude
at the "resonance of the second kind under various states of variation of to de-
Substituting the value of (:) of eq.(18.54) in equa-
tion (18. 53) and performing nuaerical integration, we obtain
curves characterizing the relation between a and E for vari-
ous values of a (Fzg,115).
3
For comparison, the same diagram contains the resonance
curve (heavy line) in the stationary statf, constructed from eq.(14,12).
M analysis of the graph so obtained peraita a number of conclusions: As usual,
'--ehen the rate of passage through resonance increases, the maxima of the resonance
curves are lowered and shifted. There isa striking difference in the behavior of
I
t, the oscillation on passage through a resonance of the second kind by comparison with
?tbe above considered example o1passage through ordinary resonance. Chile, in pas-
sage through ordinary resonance (cf. Fig. 110), the first maximum of the resonance
curve is followed by a few sore maxima of ;smaller values so that the fluctuations
are of th
nature of damped bests,
the ampli rude
maximum value, continues to decrease stead4ly and
As our second example, consider the aasage
in our case, after
tends toward zero.
reaching its
through parametric resonance.
the rod of a.length 1. with hinged ends (Fig. 116) be
longitudinal force
aub jec+ted
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r~s'6
tuning ~(i), we must, as usual, numerically integrate the
system (18.53).
'lo make the ptablem more definite, assume that the de-
tuning varies as a;function of the variation in the natural
frequency of the original oscillatory systen ir, and assume
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y'
t
r cin st
Ce ..."
can 'be reduced to the following.
d~
{ reuoA~ce p ' 1, 4 ' 2.
sire a and 6 must be determined from theIsystem
37..
:S111111
=
9(
7
(18.5?)
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i
Fig. 117
iTerefore, by suns of the substitution
:
lt'.
]eq.(18.56)
_H
1
Jwhere
4
to be tl>rouuph the doubled first critical value).
1hq altfor . ti 1 equation of the transverse
0,
0,
(18.57)
d12 + u (l __.. h CAS (l) x = O1 (18.58)
vibrations of the rod say be written in the form
EJa~y TM ~~`~ :a COS ~) dz' = u, t le. ~o)
There, is in Section 16, A is the cross sectional
area; E1 the rigidity; y the density; and the
acceleration of gravity.
The boundary conditions will be
W
Assuming than ' v(t), varying with time, passes through the doubled values
`'of the frequency , let us construct the ~irst approximation corresponding to the
d6 ? v(t)- .hi h varies slowly with tine and passes
r a ublad critieel ,slue (to ask. `th. formulatios a-ors daiiaita, sssu~we it
Using eq. (18. S) and (18.6), we have
I
x a cos (-,' ,1
l
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` srotricel values to h, ? at and . .
ee
thm reson+wnce shown in Fij.11 .
obtoia_the curves of pess~e
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CHAPTER IV
THE METHOD OF Tll E MEAN
Section 19. Equstiona of First and Higher 4proxiaations in the Method of the Mean
At the beginning of this book se had briefly discussed the reduction of a non-
linear differential equation (containing a small parameter) to the standard form and
had described the construction of an approximate solution by the principle of aver-
aging or the method of the mean.
This question..111 be discussed in mare detail in this tiapter.
It is well moan that the form of nonlinear differential equations containing a
(will `-iri%GtCr, and il1V tuv viiilcFcr Vf tiu aiil ~leti~Ct4r ltAFlf, 1_N:y iorl
widely.
In many cases, however, by means of sioople substitutions of variables, the dif-
krential equations of oscillations may be reduced to one general form, in which the
right sides are proportional to the small parameter. Ne decided to call this form
of differential equations the standard form.
The reduction of differential equations to the standard form by application of
the principle of averaging is an effective method, especially in studying nonlinear
oscillatory systems with many degrees of freedom. Thus, for instance, in the case
there a nonlinear oscillatory system with n degrees of freedom is characterized by
the following expression for the kinetic and potential energies
STAT
319
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-where ql~ q2, qn are general .,zed cool'dinatea, aka, bk) are constants, and the
;quadratic fora, land V are definitely poati:e,
`linear transfOrustion
can be used for introducing the normal coordinates 11, X2, ..., for which
r~
T X I! - ~uxk, (19.3)
Ten, the Lagrange equation for unperturbed motion takes the following form:
!see now that our system is exposed to a small disturbance of the form
where Q* denotes the frequency of the disturbing forces and E is a a*all parameter.
Ten, changing to noraal coordinates in eq.(19.5) we obtain the following
is determined from the condition of equivalence of work by the formula
STAT
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(k
= euht Xk~ (xl, x2,
Y
=1
(19.14)
(19.15)
STAT
321
2, ..., n)-
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iatio (19.6) by swbatitutinn of Lhe rsrirsbles
(19.8)
(19.9)
;where s and a= are complex conjugate unknown functions of time, may be reduced to
the standard dorm,
Differentiating eq.(19.9) and substituting in eq.(19.6), we obtain
Using, for s siaplification of the notation,
1'). Ii))
(19.12)
(19.13)
Equations describing the oscillation of systems under the influence of forces
of high frequency and of other systems, may al.o be reduced to equations of the
Let as, therefore, describe the forms! method of constructing approximate
solution; for equations in the standard form
where a is a asall parameter and X11 may be .epreaented by the sues.
In facts differentiating eq.(19.8) and cowpsr.ing it with eq.(19.9), we get.
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Section 19. equations of First and Higher Approxiaations in the Method of the Mean
At the beginning of this book we had briefly discussed the reduction of a non-
linear differential equation (containing a small parameter) to the standard form and
had described the construction of an approximate solution by the principle of aver-
aging or the method of the mean.
This question sill be discussed in arore detail in this Chapter.
It is sell known that the form of nonlinear differential equations containing a
.11 ~aii p~& b a, a1W Ql.v ti~C t lei L!vi I.1,1 tilt aii ~larwicttii itacla, wok gory
widely.
In many cues, however, by mews of simple substitutions of variables, the dif-
k rential equations of oscillations may be reduced to one general fora, in which the
right sides are proportional to the small. parameter. Ne decided to call this form
of differential ~quationa the standard form.
The reduction of differential equations to the standard form by application of
the principle of averaging is an effective method, especially in studying nonlinear
oscillatory systems with many degrees of freedom. Thus, for instance, in the case
where a nonlinear oscillatory system with n degrees of freedom is characterized by
the following expression for the IMinetic and potential energies
STAT
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(k - I, 2, ..., n). (19.4)
Assume now that our system is exposed to a small disturbance of the form
.where Ra denotes the frequency of the disturbing forces and E is a small parameter.
Then, changing to normal coordinates in eq.(19.S) we obtain the following
system of nonlinear equations:
where eXk is determined from the condition of equivalence of work by the formula
STAT
where qT, q2, q ere generalsad coordinates, aka, bkj are constants, and the
quadratic f!f** T end 'r are
linear transfOrsition
... t6hew ~y ~,ya1V1i, wnvu+j ,??? nF the
definitely .:.G _ it is co kno t
can be used for introducing the normal coordinates xl, x2, ..., for which
V-_-$Xk, 19.3)
2 ..r 2 r
Then, the Lagrange equation for unperturbed motion takes the following form:
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x,~) = e~' Xky (x,,
Y
(k = 1, 2, ... it),
xK)
(19.14)
(19.15)
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321
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(19.9)
conjugate unknown functions of time m`y be reduced to
the standard fora.
Differentiating eq.(19.9) and substituting in eq.(19,6), we obtain
Using, for a simplification of the notation,
iatf (19.6) by $ubatiwution of the ariablca
( 19. LU )
(19.12)
(19.13)
Equations describing the oscillation of sya~tems under the influence of forces
of high frequency and of other systems, a~ay also be reduced to equations of the
type of eq. (19.13).
In fact; differentiating eq.(19.$) and conpar.ing it with eq.(19.9). we get.
Let. us, therefore, deacribe the forss1 method of constructing approximate
solutions for equations in the standard fora
(k
;whore t is a.sash l paraaeter and Xk may be represented by the sums,
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n
rIb r (f, xt, ..., xq) a '?k dhk dxq
dt ~t dxn dt
d!' dl" dt dx a? 'dx n p
rlx dt dt 'd ax)
rlt rit
in. this way, ax is treated as the ratrin
(1917)
.
(19.19)
(19.20)
STAT
to which Y daotea constent'frequenciee.
region, and the caap ex for. of representing the sinusoidsl oscillations used in.
aq:(19:15) is lntroduced aerely for simplicity of notation. In considering the high
apprexi^ations, it is often expedient to consider terms of higher order with respect
to a lathe differential equations. In this case we obtain, for exasiple,
(k ":.. I, 2, ... ,
(19.16)
where Yk is a function of the sane fora as X. This type of equation will also be
denoted as the standard fora. In applying the theory of perturbations, no sub-
stantial changes are introduced here.
It should be aeatinned that eq.(19.11) is considered exclusively in the real
Before proceeding to a descriition of this theory, we will introduce a number
of abbreviated notations. Thus, the set of n quantities xl, xa, xn will be
`I,
X (t, x) = etX, (x).
Y
The forawlas for differentiation of complex functions
') nn
Jii
RLaI L.. i.IIYMGa in Y?I ?V&?I'
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Let, further, F (t, x) he a sua of the fora
Coosider by the systaa of differential egnationa (19,17), there s is a steep
araaetsr aid *re the expressions X, as functions of the time t, are repreaented
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ebile 14 xJ is treated as the scalar operator product
V d.rf1 a_
ai df izq
(19.21) .
that the use of this aatrix-rector systea of notation requires
no particular explanstioa and has crn:iderabie adv?_ntage? in. shortening the forualaa.
(19.23)
(19.24)
9e trill denote the operator " as the inte$ratin6 operator, the operator M as the
c
" operator of ariraliag for coaatsat x or.tle operator of srergin* over expiicity
P (f, r) .. #-, e1'e4 F, (x). (19.22)
STAT
 U
note that th? ton of the approxiaate solution aay..be found, or rather
~!e!hedby eatir.d intuitive considerations, namely: Since the first derivatives
qu nt ities, Let us represent x as the superposition of a smoothly vary-
and a sn~a of smell `vibrational te~w?; in vi6u
term'
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latter, e assume in first approxi.ation that x ? . Ten,
are proportional to the slash parereeer, it is natural to consider all x as slow-
() 4-~ moll ainusoidal oscillatory teas
Considering that these sinusoidal oscillatory teas are due only to the uall
fl
u oI
a Ac
.., ....
? _..,...,
uence on .,ne syate
vibrations of x about ^nd exert no in
obtain the equation of first approxi^atlon in the for.
expression for x; assuming .not the ten eelvtXy(!)
in' eq. (19.26) cssaes,. in x, oscillation. of the for.
d?
;=zX0()=M {X(r, E)j.
To obtain the second approximation it is also necessary to take the vibrational
ee reach the folio.i*g approximate expression:
+t V Iv#('~~:++~~~ ;)~ (19.28)
ran,
ubstit tiny .q. (19.2S) in .q.(19. 17), is have
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~X---i'~) _.. X (t, )
dI
dx
-M tX(t~ 4w) ) 4 small sinusoidal
I11 osci liatom ierm,a
t?
atic variation of ', ve. obtain the equations of second `Fproxiestiof?
d:
r11
'J_a11 'isne
whence, neglecting the tntiuenCe of the sinuuodi oscillti- ~ terms nn the BV9tem-
Ie
and so on.
This reasoning obviously cannot claim to be at all convincing; the objection
can be raised, that, in setting up the approximate equations (19.27), terns of the
same order of smallness as the retained term eXo were rejected in ega.(19.17).
dx
dl
i; not .6.:rd, however; to put than into a more justified fore.
zM ;X (i. ~+- :X ), :.t1 X (f, :) -.4_ xr); X (t' ') I
~ t
For this purpose let us perform a substitution of variables in eq.(19.17)
J11% 911
Differentiating eq.~i,. ~.,
wire the terns are regarded as new unknowns.
w I.av.
.c .. .v
M ~
? 3 __ ~f _ .
d .C d. 1 , &V (I, ) d ~
'
dl dt ~ ~ dl dd
Xo(:~.
Substituting Qqt. (19.31) and (19.32) in eq. (19.17), we obtain
a d~(1,~~) do ?~' aX (t
+
-
a: dt
dt
r,
:X t, .,
(19.29)
(19.30)
(19.31)
(19.32)
) sXp (:). aX {l, E +&(1, I)).
STAT
Howwerer, in view of the properties (19.24) of the integrating operator,
325
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(19.38)
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{ X (t, ? sX) - X (t, ) r (1933)
,
inhere l is regarded as theunit matrix.
VI'
c ,J
(19.34)
we note that the new unknown terns satisfy equations of the fora
e JI _L~d~l~`r
(t -f-iX)- X(I, = i? (19.35)
On the other hand, expanding eq.(19.34) into a power series of e, we have
~v.tM
1 ?jj'
ar ur :~ +
v;
where, in general, the symbol e" denotes quantities of the order of smallness of
Mtltipiying eq.U9.33) from the left by
Equation (19.35) thus yields:
dt axo l?1+ s'
or, in yore detail,
(19.36)
d;.=zX0(t -s _ _ 4X0(-)-3 (X(1,, zX)_X(t. .
de a~
r
. aXA(t) a~ ~X (rr _!X0()?$2( X a X (t, E) 4.. tv ... (19.37)
dC d~
Thus if satisfies eq.(19.36). whose right side differs from the right aide
of the equation
326
d'
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represents the exact solution of eq. (19,17) under consideration.
For this reason, sae may take as our first approximation
(19.39)
(19.40)
taking as the solution of the equations of first approximation (19.38).
quadratures; at u 2 the famous Foincare theory may be used for their investigation.
For any value of n, if X (!) vanished at. a certain point o, we may con-
sider the "quasi- static" solution
of the equations of first approximation; To investigate the stability of thin
solution, ye. cam proceed. in the usual way, by setting up the equations, for small
STAT
39) in which satisfies these same equation*, will be caned the
Equatior. .,(1o.~
refined first approximation.
Substituting the relined first approximation in the exact equation (19,11), it
is obvious that this approximation satisfies them with an accuracy to terms of the
second order of smallness.
It is clear that, for an effective construction of the approximate solution,
it is primarily necessary to solve the equation of first approximation; the fact
that these equations (like the exact equations) are differential equations, imposes
however, be ea~hasi:ed that, for a large number of cases of practical interest, the
equations of first approximation are found to be far sialer and far more amenable
to investigation. In many cases in which it is impossible to obtain a general
solution, we may find at least iapo*tant partial solutions, for example, the cor-
responding steady oscillatory processes.
For example; at a ' I the equations of first approximation are integrated in
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 Aeturoins to eq.(19.3$), we note,that, by definition of the averaging operator,
STAT
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(l_ ~)rt',-(~r~) :.
dl ?
If afl the reef part: of the roots of the chiracter.istic equation
(19.41)
I +~t R I* p __. ~ n Cc,~a (1 (19.42)
are negative, then the quasi-static solution under consideration will be stable.
Every aoluticn of the equations. of first eprror! tion; atsrting from initial values
rte ', ._ 5 ~ ~ .' ~ ~... ~ tL
auIIIclendy vit4e to z, w iat he quasi-static solution as
If, even for one of the roots of the characteristic equation, the real part
is positive, ee hire a case of instability. We nay also represent the critical
case, when all the real parts are equal to tero. This case can sometimes be re-
duced to the **u preceding casea by considering higher approximations.
As shown by the refined first approximation for the quasi-static solution
under consideration, x is represented in the fore of the sum of a constant term and
s1as11 ainusoidal oscillations with the "external" frequencies v. The higher ap-
proximations would likewise reveal the presence of terma with compound frequencies
consisting of the frequencies v.
These conclusions, formulated on conaideration of the approximate solutions,
1 lP _ ?L_ r r ... -
can uue confined for Uie c*act soiuciona of eq.` 1.9.17), on the basis of rigorous
mathematical theory. It has been shown (I3ibL6) that, in the case where the real
parts of the rooti of the characteristic equation (.19.24) do not vanish, we may
" establish, for very general conditions, that the exact equations (19,17) have a
quasi-periodic solution x x (t) (with base frequencies v), lying in the vicinity
This area *ay be taken as small as desired for sufficiently
small slue. of e. Thiz qaa:i-periodic solution will be stable or unstable, ec?
cording to the signa of real parts of the roots of the algebraic equation (19.42).
328
 ,i=
..X0 j) ... . S .
To obtain the second approxiaation we find the analo6ous substitution of
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X0()
t
Cane quutly; the "equations of first approxiastion ay be represented in the foci
do
dt
tw
X \i, .J J?
(19.43)
Thus the equstion* of, first approximation (19.43) are obtained from the exact
equations (19.17) by averaging the tatter over the explicitly contained time hi
fora(ng the wean, the f terms are treated as constants.
This tonal process, consisting in the replace*ent of exact equations by aver-
aged ones, is aometimes called the principle of averaging or the aethod of the sean.
As de onstrsted later, for. justifying the uethod of the wean it is not required
that X (t, ) must be representable by tht iuum (19.18); what is of substantial is-
portance here is only the existence of the mean value.
r
X0()= Iir ~ X (t, ,) cit.
T-~oo~~
a
(19.44)
It must be noted that, in one form or another, the nethod of the Sean has long
been used for obtaining approximate solutions. Thus, as far back as the method of
secular perturbations" developed by the founders of celestial sechanics, practical-
ly the sale aethod of the mean had been in use. However, it naa only recently that
astheaaticians here begun to coacern the~aselves with the problem of justifying
Let us describe belos the construction of the second approximation.
Ne note that, in the construction of the first approximation by substitution
of the varisble!~ (19.31) eq. (19.17) was transformed into
this principle.
 To apply this`wubatitution of ,ariablet by what (in our opinion) is the most
ehich, for ssluea of f. satisfying an equation of the type
(19.49)
Od he otiber nod, f riluea of C determined from eq.(l9.47) , we find by
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r-striable+t, trsnsforiing the rsriable x to & satisfying an equation eL the tora~
would satisfy eq.(19,_17) with an accuracy to teras of the order of smallness of rte.
Since, at f determined from the equation of first approximation
the expression.
mill satisfy eq.(19.17) with an accuracy to terma of the order of sa,allnesa of e2,
ae e111 look for the solution of eq.(19.46) in the form
 ! .J (19.53)
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~1. r1 _K1 a-,~, j
tl~ di
Thus eq. (19,50) will be equal to eq. (19.51) with an accuracy to terra of the
order of smallness of t3, if ?e select the available P (~) and F (t, ~) in such a
manner that the follo.in~ relation is satisfied:
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R
=M1X(t, +e2k1{(X)x(& )1
(19.55)
(1.9.56)
..,(t,E)?_ ~ Xa X(t~ )__2 )
?-)Xo()
:+s
. accuracy to tens: of the ;order of c3.
111 ..ti sEy eq} (19.111 Kith as f 1 o f
fe~-
eiwt(l)t (:)I
l1 t ~ (J)X(t. E)ax (teE) Xo (;)
I'I
-(4)x -~~-- ax
)- '
?e ,ran affirm that, for values of f determined from the equation
To suawartie,
Next, iNe rill shore that, if eq. (19.51) so obtained is copuderea a onu a
the uakaarn x dEtermined by the exact
-`?~ssbat?itatioa of ~rari~lea tciaafonsia$ . -- 1.___
eq.(19.17) into the aew uakdora C, then it rill satisfy an egeacion of ?~.-
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the wsraatioa is extend to ali p:ir:
the sums of eq~ (1.9, S3).
$stioa (9.54) asy,
Yl
?
ad eq;(19.52) viii be satisfied if ae a`suue that
" ) of the frequenrie8 r figuring in
consequeatiy, be represented by s aum of the
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Then we obtain
there 1 denotes the unit matrix.
Nnaever, the eery definition of the integrating operator yields
virtue,. of eq. (19.17). this expression *ust be equel to the foiiotind:
By
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For this purpose.. let us di f%erentiate eq: (19.5?),
eq. (19.55) to ?~rten the f o raut ?s.
rM (X(t,
I
(19.59)
ad therefore it fol fora f rom eq. (19.59) that
.. X d X (t t) Z ....."?M !(X k) X ltr
(19.58)
uai~-~ the notations of
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r1i
This :home that the variw:e
so that eq. (19.60) may be presented in the fora
(19.60)
coinciding with eq.(19.56).
Thus, if satisfies eq.(19.5R) whose right side differs from the right side
of eq.(19.56) by terms of the order of smallness of c3, then eq.(19.59) is the
exrct solution of eq:(19.17 )?
Thus, as our second approximation, let us take:
(19.61)
where a is determined by eq.(19.S6). In other words for the second approximation
we adopt the form of the refined first approximation in which a satisfies an equa~
Sion which is no lower of the first approximation, but already of the second.
fke wilt cili eq.(19.57) in which f is determined from eq.(19.56), the refined
iecond approximatioa.
As we have seen, the refined second approximation satisfies eq. (19.1') with iii
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(19.67)
M (19.68)
This ~seans that the equations of second approximation may be obtained directly
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etrnlr of th. nag of aalineoa of t"?
which contain terar of the second order of saallneas,
(19.62)
.il dX(t, );-f-s M y (t )) 3n~~ t ~) } (19.63)
and the expreaaion of second appeoximstioa Trill be
x = E + EX (t, .)
In this came;-the equations of second approximation take the fore
(19.64)
fiadily, for the refined second spproximstion e find
x ;+aX(t- ~)+s2Y(t, ~)+=9 X ; X(t, )__X0( .' ~~ .)? (19,65)
(19.66)
=M{$X(1-E)+a Xa; X(,.)3 t
Therefore, since the Beras of the order of smallnesa.of i.3 are diaregsrded in the
equations of second spproxieat,ion, eq.(19.63) may be +rritten indifferently either
AU above statenenta csa_be directly enerslixed to equations of the type
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a, ii order to set up the .th approximation, let us consider the expression
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rat the e*ect a9uatione (19. if the tern x ;n their riftbt Bides is substituted
b the foam of the refined First tpproxisittion (or, whet is the sere. thing, the.. fora
of the second approximation) and the mean is taken over the e* licitly ccntailsed,
the"rsrisbles j in the process of averaging as constants, chile
tine t, treatiag
terms of the third order of smallness are're)ected.
This principle cf tversging .sy be formulated as follows. Equations of the
second apProx.iustion are obtained by taking the..ean of the exact equations (19.62)
on both sides *here the refined first approximation with respect to explicitly con-
tained titre has been substituted. Indeed, the equations of second approximation
follofro. the relations
1 = M { sX (t, x) + s'' Y (t. x); (19.69)
(.here, in both,sides eX (t, ) is substituted for xl, while, during the process
of averaging, ~- are treated ss constants and terms of the order of smallness of
a[
t3 can be neglected. It may be vorth .entioning that, with this interpretation of
the operation M, we obviously will have
I.
v
dX
d~ 4- zM
(; 1 ar t
so tbit eq. (19.69) is 'trsnsfor*ed into eq. (1968).
In coeciusion, a few rairks on setting
1-e
bigher dpproxi.ations will be
Let the general equat on is the standard form be
ax = 8X(1, x)? a9X1(t, x) _I , - f amX,.,_1(t, x),
dt
(19.70)
rrbere f~ (t, ii) are certain trigon.etric seas of the are type as X (t, x).
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are suns of the for.
while the variable will be a solution of the equation
(19.71)
sP~ (;) + e9P ) -I- -}- i'*Pm (~)? (19.77)
substituting eq.(19.71) in eq.(19,70) and equating the coefficients of the
sage polera of a to the nth order inclusiare, we select F1, ..., Fs, and .P1, ... Pa
in such a wanner that eq.(19.71) will satisfy eq.(19.70) with an accuracy
of the order of swallness of ew' 1.
In this case, we obtain
= (X.) X(t, t) . d {X(`, 4-X( t, E); .. .
to tet*S
If, after baring deteruined Fl, ..., Fw and P1, ..., Pw, we consider eq.(19.71)
as a certaie forwul.a for substitution of variables, transforsing the unknown x into
the new uaksnwn this will be deternIned by an equation of the for.
. . -F- ~mPm (t) _ _ tw+ t .. (19.73)
'11uus if the . rariable asti of ies eq. (19.73) , differing f row eq. (19.72) by terwa
of the order of swallwss of tw 1, then eq.(19.71) represents the exact solution
F ` . ,,for. eq. (19.70).
For _thi reason, the expression
337
(t) Al X (I, z)j
r
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 in, ch is deternifled by the equation of tRa as`" .,proximation (19.72), may be
adopted as the ntb.approxi*stion. For such a , eq.(19.71) miit yield a refined
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::japproxiwation aatisiying the exact. equation (19.70)
an error of the order of
k note that if the form of the iaprorred (0-1) th approximation is knoan, the
euation oCth aPProximation can be directly found from the exact equation (19.74)
,. q a-
.the ahole, in the apj-lications of the above theory of perturbations, we use sainly
the fir$t approximation and only occasionally the aecond. The higher approximations
;are rarely eaployed, because of the rapid increase in the coplexity of their
construction.
As an example illustrating this theory, let us consider the oscillations of a
;physical pendulu~ which is a solid body, freely rotatable in a certain vertical
plane abot aits point of suspension. Let the point of suspension perform, in the
vertical direction, sinusoidal oscillations of small asplitude a, at high
,,Hfrequeacy in such a way that
(19.74)
It mill be found that the unstable upper position of the pendulum can be made
To consider this interesting phenomclon, let ua.aet up the equation of oscilla-
`tion of a pend4ui with a vibr:tine point of suspension. Considering the ding
;proportional to the velocity", me have
quency.of the mall oscillations.
Ins f.ct1.;t e. qu*tios of oscillations of a pendulum with its point of suspension
.t rast will become, as eenerslly knoua, (continued on next page)
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d l d? d3_1 d2
i e1 dt'
d`e - sin 0,
df dl l
(19.16)
(19.77)
(a)
Ho,ever, fro. the point of view of the principle of relativity, the motion of a
pe the litb i vertically vibrating point of suspension is the equivalebt to
the motion of a peadrlnm nth a point of surpbhsioa at rest in a field of
"gravity" Mitb an rceeleration + . Oa replacin` g sa eq(a) by g $ yn we
arrive at eq. (19, 7h).
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where a ie the angle of deflection .ensured from the loeec position of equilibriu~a;
y a sin at?is the vertical dirplscernent of the point of suspension; is the co-
efficient of ding. Kith renpect to thesgnitude of dating *e affisue+e that, at
the fixed point of suapension, the motion of the pendulua at sash deflections from
the lower position of equilibrium has an cscillatotY character. Ten, as is gener-
coat'd)
To find the,aaall psrsaeter to eq?(19.75), it is expedient to introduce"di-
aensionless" tiuue. Mare specificallY, the time t measured in seconds is replaced
by the tine t foe which the unit of aessureaent sill be related to ?n which is the
period of oscillation of the point of suspension, i.e., .1. le have, obviously,
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Taking the ratio of the aaplitude of oscillation of the suspension point to the
reduced length of the pendulum as our snail parameter r, ac have finally
where, according to egs.(19.74)- (19.75), and (19.79), the constants a~ and k will
:be leas than unity
Since the equation so obtained, containing the small parameter e, is not an
equation of standard for., it must first be transformed into this for. before direct
appiic.Llon of the above theory becomes possible.
It turns out that, by means of a simple substitution of variables, this dif-
ferential equation f the second order can be transformed into two equations of the
;first order in the tandard forma For this purpose., we replace one unknown function
,
(1981)
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Differentiating eq. (191 AQ) and coaparing it with eq. (19. A1) we have
 N.
(sin (~ -a sin - sin ') --- sin '; silt .
ec- cn ~ c~~s
-~k`3asin(-:sin ~sin)-
dew -
i a COS' COS P d
e d ---
asintF~Sinf asin.Sinl:p)--- sine
~ gC1 :
-_ k~a~ sin (p -- a sin ; sin +2(Q Q ___ 8 cos SUl '$,
ati a and taking eq:(19.82) into consideration, we obtain
+dreace, elisia flS
-a-
s sin'c c?us? d~
sQ`_ d COS '- sill n:
. COs d Sin +=
(i - t sin t cos ~) dt = ifl,
dot!
dc
COSC :c)S ` Sill ~ Sin!:
an
(a sin ; __ k3 i) sin iy ---Zia 1- ,
- 2n(Q - cos sln ).
(19.82)
(19.83)
This indicates that, because of egs.(19.82) and (19:83). the variables ~, Q
.:.._. in
sstsfy the differeatial Cglaiti in r6~ standard torn
d =sg+
dt
I
341
(19.94)
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a _..1..r_itutifR is eq. (19.?9) , we get
DI f tereatiating eq: t ' . o u -u .,.------- ..
I
-sin sin Cog , - kss1n 'p.?
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CVs' cos ? ..." 2r''?2 ~ col "G X711) 1!1 4 y ... .
1Yitg the method of the scan to these a ;sessions .nd'taking account of the
`
gip
we obtain the equations of first approximation in the form
(19.95)
These two equations of the first order (19.85), are obviously equivalent to
the single equation of the second order -
-cos )s!U =0. (19.86)
The resultant equstion of the first order is far simpler than the exact
eq.(19.79), already by virtue of the fact that it does not contuin the time ex-
plicitly. This equation constitutes the oscillation equation for a system similar
to the pendult~ math a fixed suspension point in ?hich the "restoring force" is
cos sin c. It is interesting to note,
asgng other things, th.t for exaapie certain gyroscopes (Bibl.54) :re syte- of
kind.
In the absence of damping (a 0), eq.(19.86) is completely solved in elliptic
functions. However, for considering the question in which we are interested, we do
not need expressions of the general solution. Equation (19.86) indicates directly
that this equation sdmita of the quasi-static solution z ii, corresponding to the
upper position of equilibriuM of the pendulun.
For studying the stability, consider the small deflections 6~ = m - rt from this
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po:i t ivu. ~r1;_u
Siuce here eat > 0, the condition of stability will be
1;e., bearing In w;id the definition of k:
w~V`lWoa
0, (19.87)
(19.88)
Thus, if the frequency of vibration of the point of suspension is great enough
to satisfy the inequality (19.88), then the upper position of the pendulun becomes
stable.
For Instance let 1 ' 44 cagy, a 2 cm. In this case, the condition (14.88) will.
yield -?--
20 140
V Au
The upper position of the pendulum Mill consequently be stable if the number of
cycles of the oscillation of the suspension point is more than 2n i.e., eiore than
22.3 cps.
If we consider analogou$ly the quasi-static solution ? 0 cprresporda?E to "
-_ pcsItien of equilibrium, it will become obvious that this will remain stable
tv,rcz P -
for any ,alues of It and that the frequency of oscillations at. small deflections,
without allowing for damping, will be equal to + k2 for the time ti and cor-
respondinily to
for the tine .t.
the above concrete e~reee.ls, at a frequency of oscillation of the suspension
For
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the equation of variation for 6~ takes the form
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+ 2h2 ds __ s stn h10
s1
d
(19.89
The angle 8 ~eaaures the deflection of the axis of the pendulum from a certain
pout equal to 60 cps t 3?3 L \ the frequency of rho small oscillations of
sec
the pendulum will be w~ * 142 -, chile in the case of a suspension point at
rest, this frequency will be equal tov
4,.94...L. The effective restoring.force
Sec
is here increased by s factor of !. 8.2. This force, at snail deflections,
v
will thus be the name as in a corresponding ordinary pendulum which is 8.2 times
as heavy.
Ne aote finally that the equation of first approximation (19.86) permits a
consideration of the question of stability not only at small but also at large
deflections.
Next, we will discuss the oscillations of a pendulum in second approximation.
It is obvious that the equations of second approximation coincide with the equations
of first approximations.
For this reason, in constructing the second approximation, another possible
type of motion of the pendulum will be investigated. It has been found that a
pendulum can rotate synchronously at the angular velocity w, expending work for
overcoming the resistances, provided that these resistances do not exceed a certain
quantity. Here oscillations of the pendulum are possible about an axis rotating
uniformly at an angular velocity exactly equal to .
In order to simplify the calculations slightly we exclude the action of
gravity, assuming that the motioa of the pendulum takes place in a horizontal plane.
Then, putting k ? O in eq.(19.19), we obtain
fixed axis and, since we intend to study the oscillations of a pendulum about en
axis rotating at constant angular velocity W, it is expedient to replace the angle 8
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Since
the refined first approximation (second approximation) will be
by the angte
or for the dimensionless time t, used in eq. (19r. e9),
For the angle the equation of oscillation will obviously be
To reduce this eq.(19.90) to the standard /ors, let
d'+
Y .;~ Y, S ~R v,
d
(19.91)
As a result, we obtain two equations of the first order with respect to the unknowns
in which f may be taken as the small parameter.
Substituting eq.(19.93) in the right aides of eq.(19.92) and taking the mean
?iti respect to * with constant ~, Q we arrive at the equations of second approxi-
nation:
(19.94)
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or
If .e return to the time t measured in seconds (t. z ) , then the resultant
equation of second approximation can be represented in the form
dl~ -TT `dt 2C
(19.95)
Ne note, among other things, that, in the notation adopted, the equation of
first approximation .ould be
It differs fro. the equation of second approximation by the absence of the term
A d due to the doping.
On considering the equation of second approximation, re see that it admits the
quasi static solutions
? = f0, whir? - COS'0 =%W, (19.97)
corresponding to the rotation of the pendulum (0 ! t * Apo) at constant angular
velocity ~a, provided only that
a+~
0
is positive, the real parts of the roots of this equation are negative; at
W2 sin ~o < 0. this equation has a root with a positive real part.
21
Thus the solution (19.97) is stable for tin ~o > 0 and unstable for sin ~o < 0.
*e have, consequently, one stable quasi-static solution 0 < so < n and one unstable
it