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TIGHT BINDING BOOK
64440 > CQ
^ CO >
OSMANIA UNIVERSITY LIBRARY Call
No.^/^.^ Ace. No. 3260}
HltO
Osmania Call
No
sS~/7
M
University
3
Ubrarq
Accession No
Author Title This book should be
jtturned
on
or before the date last
marked berew
ORDINARY NONLINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING
AND PHYSICAL SCIENCES BY N. W.
McLACHLAN
D.SC. (ENGINEERING),
LONDON
OXFORD AT THE CLARENDON PRESS 1950
Oxford University Press, Amen House, London E.C,4 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALCUTTA MADRAS CAPE TOWN Geoffrey Cwnberlege, Publisher to the University
PRINTED IN GREAT BRITAIN
PREFACE THE
purpose of this book is to provide engineers and physicists with a practical introduction to the important subject of nonlinear differential equations,
and to give representative applications
in
engineering and physics. The literature, to date, exceeds 300 memoirs, some rather lengthy, and most of them dealing with applications in various branches of technology. By comparison, the theoretical side of the Subject has been neglected. Moreover, owing to the absence of a concise theoretical background, and the need to
book for economical reasons, the text is confined methods employed A wide variety of these is included, and practical details given in the hope that they will interest and help the technical reader. Accordingly, the book is not an analytical treatise with technical applications. It aims to show how certain types of nonlinear problems may be solved, and how experimental results may be interpreted by aid of non linear analysis. The reader who desires information on the justification of the methods employed, should consult the references marked with an
limit the size o^this
chiefly to the presentation of various analytical in the solution of important technical problems.
asterisk in the
Much work
list
at the
end of the book.
involving nonlinear partial differential equations has
fluid mechanics, plasticity, and shock waves. The and physical analytical aspects are inseparable, and more than one treatise would be needed to do justice to these subjects. Accordingly, the present text has been confined (apart from Appendix I) to ordinary nonlinear differential equations. Brief mention of work in
been done in
plasticity, etc., is will
made
in
Chapter
be found in the reference
list,
I,
while the
titles
and particularly
of
many
in [62].
papers
Appendix
has been included on account of the importance of the derived formulae in loudspeaker design. A method of using Mathieu's equation as a stability criterion of the I
solutions of nonlinear equations
I
am
is
outlined in Appendix II.
particularly indebted to Mr. A. L. Meyers for his untiring checking most of the analytical work in the manuscript,
efforts in
and
for his valuable criticisms
and suggestions. Professor W. Prager
PREFACE
vi
very kindly read the manuscript, and it is to him that I owe the idea of confining the text to ordinary nonlinear differential equations. I am much indebted to Professor J. Allen for reading and commenting 5.1703 also to Mr. G. E. H. Reuter for doing likewise with upon 4.1968, the material in which is the outcome of reading his paper on subharmonics [13 la]. My best thanks are due to Professor S. Chandrasekhar for per;
mission to use the analysis in 2.302 from his book [159] to Professor R. B. Lindsay for facilities in connexion with 7.22; and ;
to Sir Richard V. Southwell for permission to use the analysis in
3.1803 from his book [206]. am much indebted to the following for either sending or obtaining papers, books, and reports: Sir Edward V. Appfeton, Professor I
W. G. Bickley, Drs. Gertrude Blanch, M. L. Cartwright, and L. J. Comrie, Mr. B. W. Connolly, the Director of Publications Massachusetts Institute of Technology, the Editor of Engineering, Professors
N. Levinson, C. A. Ludeke,
van der
J.
Marin, N. Minorsky, and Balth.
Pol.
Finally I have pleasure in acknowledging permission from the following to reproduce diagrams in the text: American Institute of
Physics (Journal of Applied Physics), M. Etienne Chiron (UOnde filectrique), the Director of Publications M.I.T., the Editors of the Philosophical
Physics
Magazine,
and the U.S.S.R. Embassy
(Technical
of the U.S.S.R.).
N.
LONDON May 1950
W. M.
CONTENTS I.
II.
III.
V.
VII. VIII.
.
.
EQUATIONS INTEGRABLE BY ELLIPTIC INTEGRALS AND FUNCTIONS .24 .
.
.
EQUATIONS HAVING PERIODIC SOLUTIONS
.
.
.
41
METHOD OF SLOWLY VARYING AMPLITUDE AND PHASE
VI.
1
EQUATIONS READILY INTEGRABLE .
IV.
.....9
GENERAL INTRODUCTION
" .
.
.
THE EQUIVALENT LINEAR EQUATION
.
.
.
.
EQUATIONS HAVING PERIODIC COEFFICIENTS
GRAPHICAL AND NUMERICAL SOLUTIONS
APPENDIX
I.
HORN APPENDIX
.103 .
113
.
145
SOUND WAVES OF FINITE AMPLITUDE IN A LOUDSPEAKER .
.
.
.
.
.
.
MATHIEU'S EQUATION AS A STABILITY CRITERION
II.
REFERENCES A. SCIENTIFIC PAPERS
.
REFERENCES B. BOOKS AND REPORTS
INDEX
.
.87
.
.
.
.
.
.
.
.
.
.
.
.
.
.180 .
188
.191
.197
.200
CHAPTER
I
GENERAL INTRODUCTION IN the socalled
classical theories of different
differential equations are
mainly
branches of science the
linear in type.
They have been the
subject of intense study, and the existence of wellknown forms of solutions is now established beyond doubt. If anyone skilled in
mathematical analysis encounters a linear differential equation of standard type, the formal solution is usually not difficult to obtain. The comparatively simple nature of such equations is due to the 'characteristic' relationships of the systems, \vhich they describe symbolically, being assumed to be linear. For instance the characteristic relationship used in developing the theory of sound propagation in air is the adiabatic law pvY = a constant. Now the graphical relationship between p and v is a curve, no finite portion of which is linear. To overcome this difficulty from a mathematical viewpoint, the theory is based upon infinitesimal pressure variation, so it is
assumed that the adiabatic curve may be replaced at the workingpoint by its tangent. In practice all audible sounds necessitate finite pressure amplitude. Fortunately, however, there is no need to depart from the linear theory based on infinitesimal (and, therefore, inaudible) vibrations, until the sound is fairly intense. In modem science certain phenomena cannot be explained on the classical linear doctrine, and it is then imperative to resort to nonlinear differential equations in order to deduce the desired information. For example, consider a lowly damped vibrational system whose is represented by ay {by* (y the displacement), the driven we have the system being by a force fcosajt. When 6 2 co ). In linear case, and the singlevalued amplitude is A //(a
restoring force
the nonlinear case, where b 7^ 0, the second approximation entails a cubic equation for A^ the amplitude of the fundamental vibration of
frequency
W/ZTT.
Here a *)A l f=0,
(1)
*JLAl+(a
, all the roots of (1) are real. Outside this range only one root is real. By introducing a viscous damping term into the
from which amplitudePhenomena arising from the
differential equation, formulae are obtainable
frequency curves may be plotted. multivalued nature of A l may be explained by aid of such curves. One arresting and important feature of the analysis reveals that the
motion is nearly sinusoidal. Formulae for the nonlinear case, obtained merely to a second approximation, are adequate to enable a satisfactory explanation of the behaviour of the system to be given, whereas on a linear basis it could not be explained at all!
Some
thirty years ago our knowledge of nonlinear differential equations might have been compared with that of linear differential
equations at the time of Newton, Leibnitz, and the elder Bernoullis (James and John), i.e. about the beginning of the eighteenth century. Until recently the subject of nonlinear differential equations has been a happy (!) huntingground populated almost exclusively by the
Research into methods of solving these equations has been neglected by the pure mathematician. Like our resources of
technologist.
coal, those of the
hausted in a
mathematician in the
finite
Interest has been
and
in Russia,
field
of linearity
may be
ex
time!
shown in nonlinear equations,
and research has now started
chiefly in
America
in Britain, with very
promising results [269, 131 a]. Although the analytical difficulties to be encountered in setting the subject of nonlinear differential equations
on a firm foundation are formidable,
it is
here precisely that the
pure mathematician can help the technologist. Such assistance will ultimately be mirrored in new technical devices destined to benefit the community in general and, therefore, the pure mathematician in particular.
When a problem involving a new kind of nonlinear equation arises, or
new
nonlinear boundary conditions are encountered, the technologist may be puzzled as to the proper form of solution to be assumed. Usually these equations cannot be integrated explicitly in terms of
known
functions. Thus the solution must be adequate to account for
will
be an approximate one, but
the salient physical features of the problem. Moreover, theoretical knowledge formulated by the pure mathematician in the guise of suitable forms of sol/tion, existence
it
all
theorems, and methods of solution would prove invaluable. So far as questions of periodicity, stability,
and
instability are concerned,
ORDINARY NONLINEAR DIFFERENTIAL EQUATIONS IN ENGINEERING AND PHYSICAL SCIENCES
GENERAL INTRODUCTION
CH.I
we may turn profitably, but not exclusively, [14],
Liapounoff
[87],
and Poincare
A
to the
3
work of Bendixon
[116].
knowledge of the experimental aspect of a problem may give guidance as regards an appropriate form of solution, e.g. 7.230.
But it will be realized that the accumulation of experimental data
may
in certain cases
be either too costly or impracticable, so that other must then be invoked. Graphical or numerical methods, procedure although tedious, are sometimes useful. Better still is the employment of a differential analyser, or an electronic computing machine. From a purely utilitarian point of view, difficult and troublesome nonlinear differential equations involving much numerical work are best solved by a machine. Just as the expert craftsman in various branches of industry has been replaced largely so in the
by machines,
course of time the technical mathematician
may be replaced by other calculating machines. Then the intrinsic interest in mathematical problems will have ceased to exist. At the moment, however, these machines are so rare and the differential
analysers
or
price so high, that they are beyond the reach of the average technologist,
and from
his standpoint
may, therefore, be counted out. a concrete idea of the growing importance of nonlinear give differential equations, the following typical (but not exhaustive) branches of science where they occur may be cited: acoustics, aero
To
dynamics, astronomy, cable telegraphy, circuits,
elasticity, electrical
power
electrical
machinery, electronics, engine governors, fluid jets, hydraulics, hydrodynamics, naval architecture (stabilization of ships), plasticity, wave motion of finite amplitude on fluids and in solids.
S.
One of the earliest nonlinear equations of acoustics was given by Earnshaw in 1860 [32], and pertained to the propagation of plane
sound waves of finite amplitude in air. A general equation for expanding sound waves of finite amplitude, of which the above is a given by the author in reference [183]. Solutions of and exponential loudspeaker horns have been obtained by S. Goldstein, N. W. McLachlan, and A. L. Meyers [41, 100, 101]. Nonlinear equations were encountered by Lord particular case,
is
this equation for conical
Rayleigh in connexion with an electrically maintained tuningfork [131], and by C. V. Raman in his experimental work on vibrating strings [127, 128].
An
important nonlinear equation which occurs in astronomy
GENERAL INTRODUCTION
4
CH.
i
concerns the gravitational equilibrium of gaseous configurations (stars ) It originated with J. Homer Lane in 1870 [78], and has been discussed .
some length by S. Chandrasekhar [159], V. R. Emden [165], R. H. Fowler [36, 37], and Lord Kelvin [68]. Of other nonlinear equations
at
in celestial mechanics, those pertaining to the white
and the pulsation of cepheid
may
variables, treated
dwarf starsj [159]
by A.
S.
Eddington,
be mentioned [164].
About twentysix years ago the speed of signalling on long submarine telegraph cables was increased some fivefold by 'loading* the inner copper conductor with thin nickeliron tape or wire. This alloy > 0) of the order 4,000, and has an initial permeability (dB/dll as
H
with normal sending voltages (40 to 60) it becomes saturated magnetically, thereby introducing nonlinearity and consequent distortion of the signals. Using a method of G. Riemann for the propagation of impulses in a gas, H. Salinger showed that a wave front, vertical at the sending end, tilts backwards (on a time basis) with increase in
distance along the cable. If the sending voltage exceeds a critical value, the wave front along the cable has both vertical and nonvertical parts [135].
There are a variety of nonlinear problems in the theory of elasticity
An
.
5
early problem
is
that of the 'elastica or flexible rod bent in one
plane so that the two ends approach each other. It attracted the attention of the Bernoullis, Euler, and Lagrange. R. V. Southwell
has given the solution for a uniform bar in terms of elliptic integrals [206], while W. G. Bickley has studied the problem in which fabrics
bend under their own weight [16]. In connexion with large deflexions of beams, solutions have been obtained by H. L. Cox, K. 0. Friedrichs, Th. von Karman, J. J. Stoker, J. S. Way [62], and others. Finite deformation of solids has been discussed analytically by M. Biot, J. Boussinesq, G. Kirchhoff, and F. D. Murnaghan [62]. The author has studied the deformation of steel istics,
due to impulsive forces
(
shells,
with nonlinear character
8.208.23).
In the realm of vibra
tional mechanical systems having nonlinear restoring forces, analytical and experimental work has been done by E. V. Appleton [11],
G. Duffing [162], J. P. Den Hartog [479], and C. A. Ludeke [91, 92, 92 a]. In certain cases the control stiffness decreases with increase in The radius of such a star is much smaller than that of one of th/main stars. Thus same luminosity the former will have a much higher effective temperature than the latter. Hence the smaller star will be much 'whiter' than the larger one. This is t
for the
the origin of the
name
'white dwarf.
OH.
GENERAL INTRODUCTION
i
5
amplitude, resulting in instability which is evinced by a 'jump' or discontinuity. Appleton obtained this effect with a 'magnetic' vibration galvanometer [11], and solved the appropriate nonlinear equation. The decreasing stiffness characteristic occurs also in con
nexion with a simple pendulum and a synchronous electrical motor. Nonlinearity arising from ironcored apparatus in electrical power circuits may introduce oscillations, whose frequency is a submultiple of the supply frequency,
when the
circuit switch is closed.
frequencies must not be confused with sub harmonics circuits
These sub
in electrical
under different conditions or excited parametrically as in J. D. McCrumm and are
The former have been discussed by initiated by 'shock' [99]. 7.12.
An
experimental investigation into resonance effects in LCR having ironcored inductances has been made by C. G. Suits
circuits
A variable potential difference (50 c.p.s.) was applied in series with the LCR combination. Provided R < a certain value R Q when [140].
,
the potential difference reaches a certain magnitude, the reactance vanishes and the current suddenly jumps to many times its former
The matter has been investigated analytically by E. G. Keller [64, 65], R. J. Duffin has discussed the behaviour of electrical networks having positive nonlinear resistors [31]. value.
In connexion with transient effects in suddenly loaded synchronous have been
electrical motors, solutions of the nonlinear equations
given by H. E. Edgerton, P. Fourmarier, W. V. Lyon, and F. J. Zak [33, 34, 93]. The solutions were obtained in graphical form by means of a differential analyser. In the field of electronics the triode oscillator
the outstanding The characteristic is curved, the is
example of a nonlinear device. curvature changing from positive to negative, there being a point of inflexion. During a period of the oscillation the damping of the associated electrical circuit is sometimes negative (maintenance or growth) and at others positive (limitation and loss). The nonlinear differential equation of the circuit,
namely,
was first studied by E. V. Appleton and B. van der Pol [8], Later van der Pol obtained solutions by the isocline method (see 8.12) with f(y)
==
e (l
2
y ),foT
=
01, 10, 10.
In the latter case he found that
GENERAL INTRODUCTION
6
under certain conditions a triode
oscillator
OH.
I
executed relaxation
being a limiting case when the parameter is large also showed that relaxation phenomena occur in a number
oscillations, this
[119].
He
of branches of science, e.g. physiology, the heartbeat being a relaxation oscillation. B. van der Pol and S. van der Mark constructed an electrical
model working on the same principle as the heart and
exhibiting similar characteristics [121]. V. Volterra has investigated the variation in the numbers of individuals in animal species which
time
FIG.
1.
in
live together,
The ordinates in represent the number of soles, and the number of sharks (different vertical scales).
(2)
one feeding on the other [147, 209]. The curve of varia
a relaxation type. The problem may be illustrated by the simple case of two species offish in the sea, e.g. sharks and soles. The sharks eat soles, and the latter live on food in the sea, of which an ever
tion
is
If the soles existed alone, they would an ever increasing rate. The sharks, however, multiply in number at take care that this does not happen, by devouring large quantities of soles, so the latter diminish rapidly in number. Ultimately there is an inadequate supply of soles to sustain the sharks which commence
present supply
is
assumed.
to die off at a high rate. Thereafter the soles again begin to increase in number, so the sharks now have more food, thus entailing a growth in the shark population, which reaches a maximum. Then the cycle is
repeated indefinitely. The relationship between the two populations is depicted in Fig. 1, being in the nature of a relaxation
and time
although the changes from maxima to minima and vice versa are less precipitous than those in an electronic relaxation oscillation,
oscillation (Fig. 63).
parasites
show
The recurrence of epidemics and the problem of when a time base is used [182].
similar characteristics
Other aspects of nonlinearity in valve
circuits
have been treated by
W. M. Greaves, and B. van der Pol [710], e.g. the 'silent interval' when a valve oscillator is driven by an external source.
E. V. Appleton,
The problem of parametric excitation and of oscillations in electriand other systems having nonlinear elements has been studied
cal
by A. Andronow,
S.
Chaikin, N. Kaidanowsky, L. Mandelstam,
CH.
GENERAL INTRODUCTION
i
7
A. Melikian, N. Papalexi, H. Sekerska, S. Strelkow, K. Theodortschik, and A. Witt [94, 95, 139, 142]. Methods of solving the types of non
above kind have been N. developed by Kryloff and N. Bogoliuboff The procedure is one of successive approximation, being based on rational assumptions relating to such applications. The method may be classified with linear equations occurring in researches of the .
Lagrange's variation of parameters. It is described in [175] by S. Lefschetz, and in [187] by N. Minorsky. The Poincare perturbation
method, developed primarily for astronomical problems,
may
be
applied (with limitations) to nonlinear equations for various types of oscillatory system.
Mandelstam and Papalexi have extended the
procedure to cover the equations for a selfoscillatory system when acted upon by an external source. The extended analysis enables certain resonance
explained, e.g.
phenomena
peculiar to nonlinear circuits to be
subharmonics in a thermionic valve circuit into \vhich
an e.m.f. is injected from an external source [94]. Problems in hydrodynamics involving nonlinear equations occur in connexion with rivers, artificial channels, and hydroelectric
They have been studied by J. Boussinesq, R. Forchheimer, and others [30, 62, 166]. systems.
S. Cole, P.
Nonlinear problems in the science of plasticity have received attention by various authors, of whom we cite J. Boussinesq, Th. von
Karman, A. Nadai, W. Prager, and G.
I. Taylor [62, 125, 126, 189]. Extensive researches in connexion with viscous and with compressible
fluids
have been conducted by many authors during the past century.
Of these W. G. Bickley, A. Busemann, S. Goldstein, D. R. Hartree, Th. von Karman, C. W. Oseen, L. Prandtl, 0. Reynolds, and G. I. Taylor
may
be mentioned
[62].
The theory of ship stabilization by means of anti rolling tanks and auxiliary mechanism involves nonlinear differential equations. This subject has been studied by N. Minorsky [10811]. Wave motion of finite amplitude on fluid surfaces has been treated
by T. H. Havelock, T. LeviCivita, Lord Rayleigh, G. G. Stokes, D. J. Struik, and others [62]. For additional information on the subject of nonlinear equations
analytically
in general, the reader may consult the references at the end of the book, and also those in [62], which contains a bibliography of 178
The importance of the purely mentioned already. During the past few
items classified under nine heads. theoretical aspect has been
GENERAL INTRODUCTION
8
CH.
i
years the theory of nonlinear equations for (a) mechanical vibrators, (b) triode oscillators (including relaxation oscillations), has been studied by M. L. Cartwright and J. E. Littlewood [269], G. E. H. Reuter [131 a], H. J. Eckweiler, D. A. Flanders, K. O. Friedrichs, J. J. Stoker, F. John [163], N. Levinson and O. K. Smith [826]. Although solutions of a number of the problems mentioned above were obtained either by graphical means or by a differential analyser, the majority of cases to date have been solved approximately by analytical or numerical processes. In some problems the amount of computation is considerable, calculating machines being needed. The methods of solution used in this book are summarized below:
Chap.
II.
Integrable exactly, using suitable transformations.
(with some exceptions) in terms of either Jacobian or Weierstrassian elliptic functions. IV. Approximate periodic solutions by (a) successive approxi
III. Integrable exactly
mation
(iteration), (b)
perturbation method,
(c)
assuming a
Fourier series, and determining early coefficients therein. V. Approximate periodic and nonperiodic solutions by method of slowly varying amplitude and phase. VI. Method in V applied to derive equivalent linear equations. VII. Approximate periodic solutions, assuming Fourier series, as based upon theory of Mathieu functions. VIII.
Appendix
(a) Isocline
graphical construction,
construction, (c) Maclaurin series, As at IV (a).
I.
Lienard graphical numerical methods.
(b)
(d)
CHAPTER
II
EQUATIONS READILY INTEGRABLE 2.10. Definition. If in
dent variable y and
an ordinary
differential
equation the depen
derivatives are of the first degree only, there 2 3 no like being products yy' y'y'', y y the equation is said to be linear. But when the degree of y and/or its derivatives differs from unity, or its
,
',
if
,
they occur as products, the equation
said to be nonlinear in y.
is
For example
is
a linear equation of the second order. The presence of # 3 x 5 does not ,
constitute nonlinearity in y.
But
dx 2 are nonlinear equations of the second order, y dy/dx, y 2 being nonlinear terms.
Examples. In
2.11.
this chapter
we
shall deal
which are integrable exactly without recourse to
We
commence with those
of the
(dy/dx)*
with equations
elliptic functions.
order.
= o, ^+? dx y
1. Solve
the initial condition being y t/ The equation may be written
x so the second
first
,
term
is
(1)
when x
dx+y dy
=
=X
Q.
0,
(2)
we have
nonlinear in y. Integrating,
= A, a constant. The initial condition gives A = x%\y% = a say. is #
2
f 7/2
2
,
(3)
Hence the
solution
2
(4)
,
the equation of a family of concentric circles solution may also be written in the form
radius a
y=(as)*, so y
is
doublevalued, and
integration.
is
also a function of a
(variable).
The
(5) 2 ,
the constant of
10
EQUATIONS READILY INTEGRABLE Comparison may be made with the linear equation
whose solution
A ^
where
is is
l
y
=
CH.
AJx,
n
(7)
an arbitrary constant multiplier dependent on the
initial condition.
=
2. Solve dx the
initial
condition being x
the form
(x+y) dx
the term
=
y
=
y dy
=
1.
(x
If the equation
is
written in
_ y) ^
(2)
Let
is nonlinear in y.
x
rcos6,
rsind,
(3)
dO,
(4)
dr+r cos 6 dd.
(5)
y
being variable. Then
r,
dx
and
dy
Substituting from (cos
= =
cos 9 dr sin 6
(5) into
(4),
(2),
r sin
we
get
drr sin 9 dd) = (cos0 sin 0)(sin0 dr+r cos rsin dO = sin rfr+rcos ^ d0, = r. rfr/d0 logr = 0+A,
0+sin 0)(cos d cos 2
so
2
2
dr
giving
Thus J.
1,
xy
being a constant of integration, so in terms of $log(x
Since y
=
1
when #
implicit form
For
initial
i
I,
+y*)t&n
A=
l
=
y/x
(loge 2
77),
2
1
#,
XQ y
?/
,
,
(8) (9)
y we have (10)
and the solution takes the

the solution
0.
(11)
is
=
2
+y )/(*+^
0.
Here again we
see that the solution of a nonlinear equation
function of the
initial condition.
+? = ^ dx y
3. Solve which is nonlinear and (1) becomes
Write y
in y. 7
,
(6) (7)
A.
og (a: +2/ 2)_2[tan (^)7r]
condition x 2
log[(*
=
2
dO),
2
is
a
(1)
2,
vx,
(12)
then dy/dx
= &(dv/dx){v
EQUATIONS READILY INTEGRABLE
2.11
fj?)
xv
or
(tx

r
Thus
y

J (V 1
(y>
f 2 .
+
$=!?
C
=
2
1)
=
0.
Cdx
dv
+ (^11)5 ,
J
(3)
log A, a constant,
X
J
L)
Cd(vl)
2 J
/*
fti)
+ (v
11
^
J
(4)
==
and, therefore,
=
Iog(t;l)+loga;log4
Hence
log[(t;
=
1)/4]
l/(vl).
l/(v
(6)
1),
(7)
which yields the implicit form
(yx)
A
=
Ae*l v x \ (
dy/dx+x/y
=
(8)
The equation
being the constant of integration. 2/c
> 0)
(ic
(9)
should be solved by the reader, using the substitution y 2.12.
Bernoulli's equation. This
may
=
vx.
be written
C
tjc
+f(x)y==g(x)y,
(i)
^
where/, g are continuous functions of x, but not of y, and /x l which entails nonlinearity in y. Write u y ~^ and we get
=
= +00. This equation occurs hydrodynamics
The
first
=
CH. (1),
is
(D
0.
when #
=
0,
w is
or
,
we
so
write x
=
2 )
3u,
(3) (3)
may
be
*
=
~ (w"+2ww')
du
so
t^"
=
2ww'{A
0,
(4)
a constant.
)
> 0,
virtue of the above conditions, with when ^ > +00, so 0, and w
A
1
w"+2ww'
=
= K+w; du
0.
(5)
w = w"
Thus, from
when (5),
0,
(6)
0,
(7)
2
or
)
w'
so
B=
=
0,
the accents denoting differentiation with respect to u.
=
when
a thrice differentiate function of u. Then
w"'+2(u>w"+w'
written
By
=
y'
in the theory of the plane jet in
remove the constant
to
%w(u), where y(x) (1) becomes
u
the general
[17].
step
n
= a ^ 2
2
(8)
,
a 2 being a nonzero constant of integration. Hence j
dw/(a
w+(7,
w*)
tanh~ 1 (^/a)
giving
Since
2
d
w
when ^
=
0, it
satisfies
= u{C.
follows that
w= which
a constant,
C
=
atanhaw,
(9)
(10) 0,
so (11)
the three conditions above.
Restoring the original
variables leads to the solution
y
which
=
2eatanh(a#/3),
B=a
(12) 2
is a function of the constant Since t^e differential equation is of the third order, there are three constants of integration, C 0, and B. namely, A .
EQUATIONS READILY INTEGRABLE
2.27
Solution of
2.27.
with
2.26
(1),
when A,
B,
C
21
^ 0. We commence
2.26, then
(5),
^(w'+^^A, Writing
z
__
dw
dz __ ~~
dz
du
du so (2)
(2)
= Au+B, we get dw
This
= Au+B.
w'+w 2
so
(1)
,
dw '
dz
= o. *?+^!_l dz A A
becomes
(4)
= =
the Riccati type of equation at (4), 2.130, with a b I/A, 2.130. In the present 1, so mutatis mutandis the solution is (12), \L case a 0, jp 1/2, v 1/3, q 3/2, Z 2/34, which gives is
=
=
=
=
=
C being a constant of integration. Finally by f
=
z
in (5) above,
is
2.30.
Some
[4(a;/3)+5]
we obtain y
which
2.26 with
=
2tw[A(x/3)
+ B],
(6)
a function of the three constants of integration A, B, and C.
astronomical equations.
[159].
These arise in con
nexion with the gravitational equilibrium of a gaseous configuration
Here the total pressure is due to the usual gas in virtue of radiation. It is given by the formula that pressure plus
in stellar structure.
p
= &T*+RT/v,
(I)
where p = pressure, T = absolute temperature, a = radiation constant, v = volume of unit mass, R = gas constant. The pressure p the relationship and density p = l/v both vary with the radius r,
between the two former being
p
m

Kp l +
l
IP,
(2)
is the mass of matter within a sphere of K, ^ being constants. If radius r, and G the gravitational constant, namely, 667 x 10~ 8 dynes cm. 2 gm." 2 the equations of equilibrium for the configuration are ,
EQUATIONS READILY INTEGRABLE
22
_4wr = 2
and
/>
aT
From
(3)
p ar
and by aid of
+ 0w =
CH.
n
0.
(4)
o
(5)
f
becomes
(4), this
O.
(6)
p dr)
Write p = A0J1 where A is, for the time being, an arbitrary constant. Then from this and (2) we get ,
p Substituting from
= KW+^6^.
(7)
into (6) yields
(7)
?
=
with & 2
4arGX l WI(ii+l)K. Putting
=
fer
in (9) leads to
+ This
is
known
as the
at r = sity
,
0,
LaneEmden
equation. Unless
and we have 9
is
=
I,andd0/d
/*
=
or
1, it is
by making 6=1 = the central denOwhen^ = 0. A solution which
arbitrary, but we now fix the centre of the sphere, which gives A
nonlinear. So far A
it
/o c ,
these boundary conditions is termed a LaneEmden function of index //,. Tabular values of some of the functions are given in
satisfies
reference [105].
2.31.  =
Transformation of
i/x,
d/dg
=
(10),
2.30 [159]. The substitution
x 2d/dx transforms the equation to
*g + * = We
=
shall consider the integrable case
0.
(1)
where
/x
=
5.
Substituting f
(J)*aty into (1) gives
'ly(iy*)
=
o,
(2)
EQUATIONS READILY INTEGRABLE
2.31
while with x =
eu
,
(2)
23
becomes
O, which
is
2.32.
Solution of
nonlinear
(3)
by virtue of the term %y 5
Write v
2.31 [159].
(3),
and we get
.

dy/du, du
=
+00, r > 0. From (4) we have
u Let sin
2
^
\y*,
= 2 J
and
(5)
u
(dyly)l(lW+Bi.
(5)
becomes
=
(
dw/smw\B 1
(6)
(7)
tan^w =
so
.#e M
jB^.
(8)
to/(l+tan \w^ = so (9) y = [12JS2fa/(i + J82^)2]t. = = But = (\) x*y, and I' so ^ (i)ty~*> ^ n(i on substituting for from (9), we get Q = 3jB + B 2a)2p (10) ~ 5, we take 0=1, For the LaneEmden function of index = = the function of 5 which Hence at 0, 1/3. gives d0/dg
Now
sin 2
= l
4 tan 2
4
2
J?/
a;
,
1
,
[
2/( 1
t
JJL
2
index 5
is

(11)
CHAPTER
III
EQUATIONS INTEGRABLE BY ELLIPTIC INTEGRALS AND FUNCTIONS 3.10. Solve y+ay+by* = 0, (1) = = a initial 6 and the are conditions where > 0, ^ 0, 0, y y y when = 0. This equation refers to a massspring system of the type ,
'
but the characteristic' of the spring is i2/+s 3 ?/ 3 no damping. The differential equation of the system is
illustrated in Fig. 2,
and there
is
,
my+s^+Szy*
= sjm, and b = s
=
0,
(2)
being the mass. A spring control of the form Siy+s^y* may be obtained by using a flat bar and suitably 0. The effective shaped blocks, as shown in Fig. 4 A, provided s 3
so a
3 /ra, ra
>
length of the spring decreases with increase in the amplitude of vibration. The stiffness is defined to be the derivative of the restoring force, so
s
Thus the
= 5l +3^
2
3 ?/
stiffness increases or decreases
(3)
.
with increase in the displace
s 3 > or < 0. A case where s 3 > is illustrated in We shall see later that in the case of a simple pendulum
ment, according as Fig. 12 B, c. 53
0,
so in (1) b
>
0.
Another example of a nonlinear restoring force is that of a mass m at the centre of a taut uniform vertical wire, as illustrated in Fig. 4 B. It is left to the reader to show that if y/l
2/
,
(1)
(2)
displacement y Q
Thus equating the
/ y dy+s 3 j
we have
y*
dy
=
K2/g+i*32/o
=
9^+yo)
=y
Q,
the motion
3.13, 3,14, so the yt curve
is
the
is
identical with that discussed in
elliptic cosine
en t. Since damping to y Q is obtained by
absent, the initial part of the curve from y reflecting the first quadrant of cut in the yaxis, so that
is
behind the
origin.
In
=A Thus if we commence at = for
a>t,
it
comes
we assumed
3.14
y
\TT)
(3)
(3) yields
After reaching y
(cot
may be
strain energy
I/O
1/0
si
0,
l
cos wt+As cos
3^.
(5)
t when the impact occurs, we must write which gives the approximate solution
y
=
^4 1
sin^
A l and A^ are calculated as in
.4 3
sin3co.
(6)
3.14, the value of y Q being that in
(4)
above. 3.160.
The simple pendulum.
of motion
Referring to Fig. 6 A the equation
is
mass X acceleration f restoring force or so
m(ld 0/dt*)+mgsm6 =
=
0,
(1)
2
d 2 9/dt 2
+ sin0 =
0.
0,
(2)
(3)
EQUATIONS INTEGRABLE BY
32
Since
sin0=:0
03 3!
05
OH. in
)..., the degree of the equation differs
from
5!
unity, so it is nonlinear. Owing to the shape of a sine curve, the control 'stiffness' decreases with increase in (Fig. 6 B).
mq sm0 (restoring)
(8)
placement)
Vmg
cos0
(stiffness)
\
FIG. 6 A. Schematic diagram for simple pendulum. * B. Restoring force and stiffness* curves for simple
pendulum, showing nonlinearity.
Write v
dd/dt y g/l
= a, and (3) becomes v dv ... + asm0 = 0, ,
do
f
so
giving If the
v dv v
maximum
= 2
a
sin0 d9\%A,
f
2a cos 0+4
swing (amplitude) t;
=
0,
and
is
6
A=
(4)
(5) (6)
.
= ^, 2acosi/r.
Thus Put
AND FUNCTIONS
ELLIPTIC INTEGRALS
3.160
v
=
(2a)*(cos0 k,
sini/r
cos^)*
sin0 = ksiny,
and
(7)
2
^sin becomes
2a*(sin
33
2
(7)
0)*.
the negative sign bei^g chosen, since decreases with increase in as reckoned from the instant of maximum deflexion. Also i cos
W d0 = k cos 9 d^, =
2&COS9 dq>/(l~k sm y)*. the for d0 from (8), (9) gives Equating expressions d0
or
t
=
0,
v
2

2
=
Accordingly on integrating
between the
(10)
(9)
2
2
d9/a*(l& sin 9)*. and 9 = TT, and when Z = JTO dt
When
tt
limits
(10)
= =
,
9
0,
9
=
0.
we
(TT, 0),
obtain the quarter period, so a~*
T O
\
fc
dy/(I
2
sin 2 9)*
(11)
(l/g)*F(k,%7r),
o
where F(k, k
=
sin
\\fj.
a complete elliptic integral of the first kind, modulus Hence the time for a complete period is given by
TT) is
From a table of elliptic integrals we find that F increases with increase in k, and, therefore, the greater the amplitude the longer the periodic
Consequently the motion is not isochronous, i.e. the time of swing a function of the amplitude, which is to be expected, since (3)
time.
is
is
nonlinear in
3.161.
0.
Approximate solution of
write sin0
^
3
J0
.
3.160. If
(3),
Then the nonlinear equation
0+a0+60
3
=
0
0,
=
but small enough, and
(2) is satisfied,
the equation
to be solved takes the approximate equivalent linear form
Sf+2fcy+a>fy
For the initial conditions y = V = y
T/ O ,
an approximate solution of
is
Q
y e
=
= Kt
0.
at
0, it
follows that
cosa)t,
Since
(1).
t
(5)
=
(6)
A ^ yz~
Kl
l
>
from
(3)
we
obtain (7)
so
co
> a* as
t
> +00.
qualitatively when to motion is oscillatory at hold. that the (6) Suppose the start with by% :> ay Q When the amplitude decreases to a value y,
The behaviour of the system may be considered
K
is
too large for
.
inonosay, the subsequent motion will be nonoscillatory, and y > a as t > {co. if the control For still /c, tonically (ay {by*) is larger
monotonically from the start.
inadequate to promote oscillation, y > If
we change
the signs of K and
y
and
o>
^
= yQe
b, (6), (7) Kt
become, respectively,
cosa>t,
(a3fo/ge
(8)
2
(9)
'/4)*.
Hence as
t increases from zero, the amplitude builds up, but the rate of oscillation decreases by virtue of the decrease in stiffness with increase in amplitude. The function ay by 3 has a zero value when K* y (a/6)*, so to avoid instability, y Q e (a/b)* in (8), (9).
=
2
=
0,
A
nonperiodic term in the particular so the complete solution of (7) is
A
2
integral,
we
2
(8)
.
a>Q
With the above
A =^ Vl

(^
2
2
/a>
find that
2
2
l
so
we
initial conditions, /3co
J?!
,
=
0,
(9)
)[^Hcosa; ^+icos2a;
(10)
^.
l 2/o+ 2 2/o 2/i)
A* 3oj [A*
J3
1
(12)
To avoid a
nonperiodic term in the particular integral of (12), the coefficient of cos o> 1 must vanish, so o>
Then the complete 2/2
=
=
5A 2 l6ajl
solution of (12)
is
(13) f
1
n 9
A3
1
cog
48
A* co
OS3c 0^ (14)
EQUATIONS HAVING PERIODIC SOLUTIONS conditions above, we obtain
4.131
Using the
A2 = Hence by of
(!),
y

51
initial
(1), (5),
4.130,
29,4 3 / 144w o>
and
JB 2
=
0.
(15)
(10), (14), (15) to the second order in 6, the solution
is
[
o
^
(16)
bA /6a)%, i.e. bA/6a
2
cog
If a
>
56 2 4 2 /6 Q decreases with increase in A. For a spring control ay+by*, the stiffness is (a+36?/ 2 ) and increases with increase in y, whether i.e. it is an even function of y (see ,
3 Fig. 12 c). The graph ofay\by is antisymmetrical, so cu increases with increase in amplitude. The minimum ordinate in Fig. 1 1 A occurs when y a/26, and for stability the negative swing must be
less
than
a/6, i.e. ymin

< a/6.
apposite to remark in connexion with the foregoing analysis that a tone of double frequency (the octave) is audible when a tuningIt
is
EQUATIONS HAVING PERIODIC SOLUTIONS
52
CH. iv
fork vibrates with a large amplitude. As the latter decreases, the pitch of the fundamental tone rises. This is in accord with the analysis.
EXAMPLE. The reader should solve (1), 3.10, by the perturbation method using the initial conditions y = A, y = 0. The result to order three in 6
is:
bA 3
 ^^* ,
4
IbA*
\
,
1
1
,
cos 3o)
.
.
(20)
4.132. Effect of viscous
Write y
= e~'
c/
u(^),
damping. The equation
y+Zxy+ay+by* and we get 2
Q
for
(a/c 2 ),
(1) is

16^ 2e^\
bA 2e' ,
+^3
2
cog
6
/
cog
(3)
When
=
0, y =
A,y =
KA which is small enough to be neglected.
A better approximate solution of (1) may be obtained using the perturbation method. This
is left
as an exercise for the reader.
4.140. Solve
y+ay+by* =fcosa)t
(a
>
0,
b
%
0).
(1)
the equation for a lossfree massspring system, with control proportional to ay+by*, driven by a force/cos a>t as shown schemati
This
is
cally in Fig. 12 A.
The
of the driving force at t
=
inevitable transient follows the application 0, but we may suppose that there is sufficient
damping to extinguish this and enable the periodic gftate to be attained after a short time interval. The damping is supposed to be small enough to have negligible influence on the amplitude of the motion. When a, 6, co, / have appropriate values, a subharmonic of
4.140
EQUATIONS HAVING PERIODIC SOLUTIONS
Fcoswt
53
(A)
m
A. Schematic diagram of mass and nonlinear driven by an alternating force /cosorf. B. Antisymmetrical curve showing combination of linear and
FIG. 12. spring
s,
cubical springs ay, 6i/ 8 , respectively. The restoring force is an odd function of y, o. Stiffnessdisplacement curve for case B: af36t/ a is an even function of y.
frequency CO/GTT occurs. This aspect is considered in 4.190 et seq. For the time being we shall assume that the conditions for the existence of a subharmonic are not satisfied. In analytical work herein,
it is
tacitly
presumed that the reaction of the system on the
EQUATIONS HAVING PERIODIC SOLUTIONS
54
CH. iv
driving agent may be neglected. Thus the amplitude and functional form of the applied force is invariable.
For a periodic solution we may assume a Fourier series. Since ay {by* is an odd function of y, the corresponding forcedisplacement graph in Fig. 12 B is antisymmetrical about the force axis. It follows that the solution may take the form
where
=
ifj
ay
=
We shall restrict ourselves
a)t.
stituting into
(1),
we
to
two terms
in
(2).
Sub
get
a(A l cosi/j{A 3 cos3ifj)
by* J == (3)
Equating the have
or
coefficients of cos
on each side of
(1),
*
or*
=
by
aid of
=/,
(a+lbAlf/A l )+lbAl
+ 2Yl
(3),
we (4)
(5)
This may be regarded as an approximation to the amplitudefrequency relation.
For the
coefficient of cos
3*)Az+lbAl+lbA\Az+lbAl so
In
.4 3
(5)
=
assume \A^A^\
0,
lbAH($a>*albAllbAl).
\A*IAi\ 0, b < 0. Hence if (13)
so
if
1

1,
(13)
is satisfied,
the forced motion of the system is nearly cosinusoidal, the displacement of the fundamental being in phase with the driving force. In the foregoing procedure, the nonlinear equation is solved directly, whereas
the methods of iteration and perturbation entail the solution of linear differential equations.
Energy considerations. Writing
4.141.
takes the form so
v dv/dy +ay+b v dv\a
I
By
4.
(2),
j
y dy\b
dy/dt, (1),
^ = /cos ^ =/
3
\
=
v
y dy
J
cos cut dy.
4.140, (1) (2)
140,
dy Substituting
(3)
=
into
=
oj(A l sin
(3) ^+3^4 3 sin ty+...)dt. the righthand side of (2), and integrating over a
(0, ^TT/OJ), the lefthand side vanishes by virtue of periodiand the righthand side by virtue of orthogonality of the circular city, functions. Hence during steady motion, no energy is supplied to the system from the driving mechanism, as we should expect, since there is no dissipation.
period
t
4.142. Equivalent linear differential equation. By virtue of the motion being almost simple harmonic, the original equation may be replaced by what tion.
Then by
may be regarded as an equivalent linear approxima
(8)
4.140, if f/A l
is
we have
negligible,
y+(a+I64)y=/cosarf,
(1)
the particular integral being
y =/cosa>^/(a+6^4f
co
2 )
A
l
cos wt.
(2)
Experimental illustration of analysis in 4.140 [92]. The apparatus shown schematically in Fig. 13 A, B has been used to obtain
4.15.
EQUATIONS HAVING PERIODIC SOLUTIONS
56
wave forms (1),
4.140.
for
CH. iv
a system akin to one represented symbolically by
A beam is mounted on a fulcrum so that oscillation occurs
about the latter in a vertical plane. The mechanical construction is such as to reduce friction to a small amount. One end of the beam is constrained by a nonlinear type of spring while a mass m rotates at Rotating Unbalance
Fulcrum Timing motor Slotted time disc
Parabolic mirror fastened to ax le v
Axle fastened to beam
s
s
\
Point sources ,
of
light
;yxNs.
Springs and supports rfot
Shown
Base/ Beam Concrete pier (approximate weight 15 ton)
Revolving
drum Slotted isc which rotates with unbalance
Fio. 13. A. Schematic diagram of apparatus for investigating system with nonlinear restoring force. B. Plan view of apparatus.
radius r with angular velocity o> about the other end of the beam, thereby causing unbalance. If I is the moment of inertia of the oscillating parts about the fulcrum, f(0) the springcontrol torque,
distance of the centre of rotation of
of motion
I
the
m from the fulcrum, the equation
is
2 where m(a) r cos ojt) is the accelerational force due to the rotation of m, and the righthand side of (1) the corresponding driving torque.
The motion of the lever was recorded photographically, and a record is 14. The torquedeflexion curve /(#) for the spring is depicted in Fig. 15, being such that f(0) = 9), i.e. it is an odd /( reproduced in Fig.
4.15
EQUATIONS HAVING PERIODIC SOLUTIONS
57
1y
3o sec.
FIG. 14. Record of wave form obtained from apparatus in Fig. 13 using loaddeflexion curve of Fig. 15. 60
h
50
O
0_
30 &>
3
cr
10
02
01
Deflection
03 in
04
radians
FIG. 15. Loaddeflexion curve pertaining to Fig. 14.
function of
0,
and the curve
is
antisymmetrical about the torque
Despite appreciable departure from linearity everywhere in Fig. 15, the wave form in Fig. 14 is almost a simple harmonic type. axis.
4.16.
Amplitudefrequency relation for
y+2 Ky+ay\by* = fcos(ajt+y). This equation is for a dynamical system of the type considered in 4.140, but with a term 2/cy representing viscous damping. The
68
driving
EQUATIONS HAVING PERIODIC SOLUTIONS force will now be out of phase with the displacement
CH. iv
corre
sponding to the fundamental vibration, so to simplify the analysis have introduced the constant phase angle 9. may assume as
We
adequate approximation that with
=A
y
= 2/cy
Then while
an
a)t
iff
cost/j+A 3 cos3i/j.
(I)
2a)K(A l sin *fj+3A 3 sin
/cos(o)+9)
.
l
we
= /(cos 9 cos
(2)
3 0, 6 0, and 6 a*, \A^\ 0, is
>
=
0, and
1^1
It is evident
from
=
( a, the amplitudefrequency curves for b are asymptotic to the straight line I4J b
0,
to
=
>
(B)
FIG. 16.
Amplitudefrequency curves for driven massspring system, with spring characteristic of the
form
ay} by
3 .
For
6
increased gradually. The operating point will travel to Q, where \A\ drops suddenly to R and continues to move along to S with increase in
From S suppose the frequency is decreased gradually On reaching T \A!\ jumps suddenly to U and then follows the curve down to P.
a>.
.
y
Thus a form of hysteresis
when b
is
exhibited.
/ >/
The point there are three amplitude curves for forces /3 x 2 a If is lies on such w an increase constant, f'2 P! that/3 2 curve/g
>
>/
.
.
EQUATIONS HAVING PERIODIC SOLUTIONS
60
M
(B)
f = O.K =
V
b>o
CH. IV
b.
A when A =
=
A when A ^ 0. When
(4) is
9a+27(6/
2
(18)
/4)*,
then
(1) is
y=(4//6)*costarf.
.
(19)
These results enable us to obtain an equation which has a subharmonic solution A cos at. Writing 3o> for w, 4//6 = A* in (18), gives a = ( so (1) becomes ,
3
= /cos 3^.
(20)
The other subharmonic solutions are A cos(a>+277/3),4 cos(a>^+ 477/3).
A
similar equation is given in [27]. a subharmonic solution of 2
2/+(4o> /3.4
3
2 )i/
=
If 3/
= 4o>
2 ,
then
A Goscot
is
(21)
EQUATIONS HAVING PERIODIC SOLUTIONS
4.191
4.191.
a
=
2, b
Example. To illustrate the analysis in 4.190, we take 005, f = 2. When the subharmonic starts
A l ~ f/8a = Using
(9),
4.190, co
2
=
given in Table
2.
A = \A^ =
and
0125,
we
Assigning values to
A^
and using
424
~
(1)
4.190,
(5), (6),
3a*.
we
(2)
obtain the data
These are exhibited graphically in Fig.
A =
Data
is
relationship between tion of the case when b
g+6cof+6 o;l+...
being determinable constants. Here we expand in ascending powers of 6 (see 4.110, 4.131).
the
y'(Q)
o>
2
(not a)
EQUATIONS HAVING PERIODIC SOLUTIONS
68
Substituting from
(2)
CH. iv
into (1) gives
(co8+&co+&i+..0(y;;+^^
+b(y*+2by Q yl +...) Equating the
coefficients of 6
6:
r ,
r =
"g^+a^^O,
For period
27r
s
in
if/,
or
so the complete solution of (4) 2/
2
y"Q +(a/a> )y Q
=
(3)
side of (3) yields: 0.
(4)
=^
= a,
(5)
is
sini/f+JS
(6)
cosi/r.
4.110, the initial conditions entail
in
y
on each
bFcos
f
i
,
n
n v/OS OyJ
(17)
Hence
to the second order in b
(18)' v
2
\2
Also to order two in 6
In virtue of the constant term in 'centre of oscillation'
is
(18),
on the negative
what may be regarded
as the
side of the origin of the force
2 displacement curve ay^by (Fig. 11 A). We showed in Chapter II that the solution of a nonlinear differen
equation depends upon the initial conditions. If (1) is solved such that the coefficient of cos ^ is A for all t, the coefficients of the various
tial
terms will differ from those in y(Q)
= A.
(18),
and the initial condition will not be
Also the last term in (19) will not appear. The reader should
verify these remarks as an exercise.
4.195. Amplitudefrequency relation. (18), at t A, y' subject to the conditions y
=
must be
satisfied also,
=
which means that
=
4.194, 0.
But
was derived (19),
for fixed values of a, 6,
4.194,
and/,
EQUATIONS HAVING PERIODIC SOLUTIONS
70 co
2
CH. iv
complies with this equation. Written in the form
a cubic equation for A, which
it is
The consequences of such a
may be compared with
relation are
examined
4.140.
(9),
4.16 et seq.
in
From the analysis therein it appears that a physical system operating in accordance^ with the equation
y+2 K y+ay+by 2 = /cos wt
(2)
should exhibit the jump phenomenon described in 4. 1 70. Consideration of the signs of the terms in (1), where b 0, shows that it is of the
>
same type K
>
4.140, with b
0, 6, K > turbation. Let wt = z, 2/c small,
for
and
becomes
(1)
oj
2
2
y"+eb y'+ay+by y = y
Assume that
=
2
bF cos 2z.
(2)
2
Q
+by 1 +b y 2 +...,
(3)
the yr being periodic twice differentiate functions of
where the
o> r
are to be determined. Substituting from
z,
and that
(3), (4)
into (2)
gives
^2) + ..K + %I + 6
2
2/2+...)
+ (5)
Equating
coefficients of b
r
6:
,
r = 0, uW+ay 
1, 2,...,
so for a solution with period
yQ
A,
13,
shall
we must have co
b, /, K, to;
yields (6)
=
a,
and, therefore,
4sinz47? cos z
(7)
to effect simplification of the
keep them the same throughout.
stage) differ slightly b:
0,
(5),
the equation nonlinear, the ultimate initial conditions (unknown at the present
algebra is
=
are functions of a,
we
2?7,
on each side of
from
Sftice
= J5, y'(Q) ~ A. =  2^^ yl~yl+F cos 2z,
j/(0)
wgyl+ay 1
(8)
EQUATIONS HAVING PERIODIC SOLUTIONS
4.196
71
so
(l/a){(A*+B*)/2+ABsm2z[(A*B*)/2+F]cos2z}. (9) To avoid nonperiodic terms in y l9 we must have a) l = 0. Thus with g
 (A*+B
2
)/2,

h
F+(A*B*)/2, we
get
sin 2z
"2 cos z
.
J3 sin
2)+ 2a>
a> 2 (^4
(A/3a)cos 20.
(10) (11)
2/o2/i
sin
2+ JB cos 2)
so
(12)
(13)
where P, Q are given below. To avoid nonperiodic terms and Q must vanish, and with A, B, nonzero we have
= 2(o> K) + 2(^/a )(A/3a Q/B = 2(^> + 2((7/a + (A/3a 2
P/A
2
Subtracting
w2
=
)(^ /3a
)
2
+ (^^
)~M/co B)
(14),
=
= 
P
0,
(14)
0.
(15)
a^/2, gives
5(A +B*)/$a+(Ka> Q /b*)(A/BBJA). (15),
obtain
(16)
and multiplying throughout by
=
(A/B+B/A)
Solving (17) yields
A/B
=
6 2 co /2/c,
=
6//3/ccog
 0.
(17)
{)8(j34)t}/2,
K Mo (A/J3~BfA)
so
2
2
j/ 2 ,
2
o> /e
we
2
)(J5 /3a
and multiplying throughout by a/2
(14), (15) 2co Q
2
)
)
Adding
2
2
in
=
(18)
2 {(6//3a) 4*c a}!. 2
(19)
a. = a>2+26 WoWl +& K+2a> a> )+..., By (4) = 0, by aid of (16), (19), we get and since w^ = a, w = a56 (4 +^ )/6a{(6//3a) 4/c a}i. 2
2
2
(20)
){(6//3a) 4/c a}*]}i
(22)
which gives the amplitudes. of the subharmonics. To order one in 6 (small) the solution is y = yQ Jrbyv so
y
= .4sin2+Bcosz+(6^5/3a)sin22 = (6/2a)F2 +rcos(z6' )(6/6a)r 1
(6^/3a)cos 2 2
(bg/a)
(23)
cos(2z+6 2 )(//3a)cos22, (24)
EQUATIONS HAVING PERIODIC SOLUTIONS
72
= AB/{(A B )/2}, = where tan = A/B, tan when there are tion is valid only for Y real > 0, 2
2
2
2
l
i.e.
The
CH. iv
20^ This solusubharmonics.
term represents a unidirectional displacement, and as in 4.195 the 'centre of oscillation' is on the left of the origin. The second and third terms represent, respectively, the subharmonic and its first overtone, while the last term gives the forced oscillation which first
has the same period as the driving force.
4.197.
Subharmonics and
The motion
stability.
defined to be
is
stable (unstable), if after being subjected to a small disturbance
returns to (moves
away from)
its
former
In [131
state.
a],
by
it
aid of
stability criteria given in [85], it is shown that the upper (lower) internal and the two external signs in (22), 4. 196 correspond to a pair
of stable (unstable) subharmonics. Thus there are four subharmonics, two stable and two unstable. By virtue of the two external signs in 4.196, the
(22),
components of each pair
differ in
phase by
TT
radians
Under
certain conditions specified in [131 a], the forced of oscillation driving frequency becomes unstable (see remarks in
or
7T/o) sec.
4.198). Nevertheless, there is still the oscillation due to the overtone of the subharmonic. The analysis pertaining to stability is beyond our present purview, so we shall merely state what results may be
expected on the basis of reference [131 4.198.
Forms
of a and
to,
of oscillation. For
a].
7 to be real,
whatever the values
we must have /
>
(I)
6i
we must have
(w*a),
(2)
~4* 2a}*.
(3)
for the unstable pair,
(aaj
2 )
>
2
{(6//3a)
Variation in f.
For
(i)
co
2
a,
as
/
is
forced oscillation of amplitude (ii)
>
For a
6/ca f
>w
2 ,
when /
=
increased from zero, there
X C2 //(4co
6/ca*/6
2
a),
is
a stable
but no subharmonics.
the threshold
is
reached, and for
a pair of stable (unstable) subharmonics having amplitude given by (22), 4.196 with the upper (lower) internal sign.
/
When
/6 there is
2
a
[aco {(6//3o) 4ic^}]
=
0,
the unstable subharmonics vanish while the stable ones remain.
(4)
At
4.198
EQUATIONS HAVING PERIODIC SOLUTIONS
this point, however, the forced oscillation
73
becomes unstable and the
amplitude jumpsf to that for one of the stable subharmonics, as given by (22), 4.196 with the upper internal sign. Increase in /is accompanied by that of
Y
and
also in the amplitude of the
subharmonic
now
decreased, the stable subharmonic persists but with decreasing F, until the threshold/ 6*a*/& is reached. It then
overtone. If/
is
=
stable forced oscillation; CA = unstable subharmonics; CB A. OA subharmonics. At A, B, /8 = (3a/&){(ao> 2 ) a f 4* 2 a}i; at C, ^ = 6*a*/6. For / increasing from O, the forced oscillation becomes unstable at A, and there is a jump to one of the stable subharmonics at B. B. OA = forced oscillation AB = stable forced oscillation which is unstable between B and subharmomcs. c. ABDF ; BC = stable subharmonics DE unstable subharmonics.
FIG. 21.
A
;
D
;
vanishes and the amplitude of the motion jumpsf down to that for the 2 forced oscillation (now stable), namely, a). The cycle c//(4o>
X
is portrayed in Fig. 21 A, and there is a type of 'hysteresis' as in Fig. 16 A. In the first case two frequencies are involved, but only one in the second.
of changes
system were started suddenly by applying a force / (/!
2
a,
7
4.196 cannot be real with the lower
in (22),
internal sign, so there are no unstable subharmonics. The stable pair cannot occur unless (1), (2) are satisfied. As /in (2) increases from zero, there is the forced oscillation
f=/
the stable subharmonics
now
with
X ^//(4o>
is
tone. Increase in
/ is accompanied by that
Variation in If (1)
(i)
There
a),
and when
X
commence with amplitude
unstable, but there
forced oscillation
illustrated in Fig. 21 B.
2
is
is
.
The
the subharmonic over
These remarks are
in Y.
no hysteresis
effect [131 a].
CD.
not
is
satisfied, there are
no subharmonics, but there
is
a
stable forced oscillation. (ii)
If (1)
is
and w 2 > a+{(bf/3a) 2 4* 2a}*, there is a stable but no subharmonics, since Y is imaginary. But
satisfied
forced oscillation
when \aaj 2
2 If a{(bf/3a) 2 4:K 2 a}* to there is a stable forced also two of the oscillation, subharmonics, provided a is not too pairs
to
co
2
{
nearf to
}*.
co
2
+{
}*.
The trend of these remarks
is
illustrated in Fig.
21 c. Subharmonic resonance does not occur.
In general the forced oscillation is unstable when accompanied by a single pair of subharmonics, but remains stable if another pair of subharmonics is present to 'counteract the instability. This remark 5
is
exemplified in Figs. 21 A, B, o [131
a].
In Fig. 11 A the slope of the forcedisplacement curve changes sign to the left of the minimum, and if
Amplitude
limitation.
Iftninl
>ab l
>
the displacement increases continually with increase in time. Usually the characteristic for a physical system has no minimum, but it may
have a point of inflexion. In the case considered restoring force occurs.
pneumatic, neither a
Example. Imagine a hollow
4.199. length
is
I,
closed
in
4.199,
where the
minimum nor an
circular cylinder, of
inflexion
working
by a rigid disk of mass J m which is driven axially by a
The approximate analysis does not hold if a is too near to these values, j This includes 'accession to inertia' due to the external air [183].
t
EQUATIONS HAVING PERIODIC SOLUTIONS
4.199
75
such that the displacement  is 'finite'. If p Q is the atmospheric pressure, and v the corresponding undisturbed volume,
force /cos
2a)t,
we have
for adiabatic operation
pv7
p Q v%
=
a constant.
If the displacement during expansion
p(Z+f )? so
i>
Thus the
=
jp
IY,
(2)
(i+^) y ^i>o[iy^+y(y+i)^/2P...].
'spring control' per unit crosssectional area,
within the cylinder,
Accordingly disk
then
is )
= PO
(1)
if g/l
is
is
^
air
given by
1,
the approximate equation of motion of the
= /cos 2co,
2
fw"+2*ffaf +6
=
(3)
due to the
=
(5)
yp Q Ajl, b y(y+l)p ^4/2Z ^ being the crosssectional area of the cylinder. Here the slope of the pv curve is negative, and what may be regarded as the 'centre of oscillation' moves down this with a
curve (see
minimum large
(18),
,
4.194 with b
value, so
would
2
+00. As example of the latter we cite the familiar case of the free oscillation
of a linear electrical
LCR /
Here A(t)
=
Ce~ K* >
circuit in
=
which the current
is
given by
(7e*<sin(a>+9).
(2)
{co, while 9 is a constant phase angle. Consider a differential equation of the form
as
t
>
2
y+*g(y,y)+< y
=
o,
(3)
which the nonlinear term eg(y,y) is relatively small, and g is a function J of both the displacement y (or its equivalent) and its first time derivative y. Neglecting this term, the equation reduces to the in
linear one 2y
of which the complete solution
y
=A
l
=
Q?
(4)
is
coQa>t{B 1 sma)t
(5)
i
A!, B! being arbitrary constants. In a sense this may be regarded as a first approximation, although it possesses no nonlinear characteristics. (5) is expressible in the form (1), with A (A\ ^i)*, and
=
=
+
1
tsm (A l /B 1 ) these being constants. 9 Writing (a)t+q>) == x> the solution takes the form )
= Asmx; y = AOJ cos xy
so
(6) (7)
t Strictly 'phase' pertains to sine waves, but it is convenient to use it here. 3 J If terms of the form 6y occur, they are to be included in g, i.e. g comprises damping and nonlinear control terms.
METHOD OF SLOWLY VARYING
88
We now tiating (6)
A
suppose that
we
and 9 vary slowly with
CH. t,
so
by
v
differen
get [77, 175]
y
=
Substituting for y from
Asmx+A(a)+^)GOs x (7)
0.
(9)
we have
(7),
AOJ cos x~ ^4co(co+9)sin ^.
y Substituting from
(8)
into (8) leads to
A sin x+^9 cos X = Differentiating
^
(6), (7), (10)
(10)
into (3) yields
A GOSX~~ ^4
0.
From
increases monotonically with increase in value 2 as t > +00. If we suppose that
decay and > 2 as
t
> +00.
t,
(11) it follows that
and tends
AQ >
2,
A
to the ultimate
the amplitude will
METHOD OF SLOWLY VARYING
90
By
(2)
above, with
x and
for
v
5.10,
(15),
,
OH.
27T

9 == Since
where (11),
o>
cp
t
(l^ 2 sin 20)cos