THE TEMPORAL PASSAGE OF MECHANICAL
SYSTEMS THROUGH RESONANCE
By
RICHARD LEE FEARN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
June, 1965
ACKNOWLEDGiENTS
The author wishes to express his thanks to Professor
Kronsbein and Professor Millsaps for serving as cochairmen
of his committee, and to Professor Millsaps for suggesting
the problem which is treated in this dissertation. The
author also wishes to thank the Physics Department, the
National Science Foundation and the Graduate School for
financial support during the past four years.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ..................... ........... ...... .. ii
LIST OF SYMBOLS ...................................... iv
CI:'APTER
I. INTRODUCTION .... ............................. 1
II. CONSTANT ACCELERATION THROUGH RESONANCE ....... 9
APPENDICES
I. THE METHOD OF VARIATION OF PARAMETERS ......... 18
II. THE CASE OF DAMPING ........................... 21
REFERENCES ......................................... 24
BIOGRAPHICAL SKETCH .................................. 26
LIST OF SYMBOLS
0 constant
an Fourier coefficient
A integration constant
b constant
bn Fourier coefficient
B integration constant
c constant
CM Fresnel integral of argument v
E energy
T amplitude of an harmonic driving function
T general driving function
S acceleration due to gravity
S spring constant
Length of a simple pendulum
un mass
n integer
,p/2ir natural frequency of a simple vibrator
P amplitude of the driving force
0 constant; inversely proportional to the angular
S acceleration of the driving force
R nondimensional dependent variable in the equation
of motion for a simple vibrator
Re envelope of the maximum amplitudes of R
Rm the maximum amplitude encountered in passing through
resonance
Scv) Fresnel integral of argument V
t time
T period of a simple harmonic motion
U nondimensional independent variable in equation
of motion for a simple vibrator
Um the value of ( where the maximum amplitude of Re
is encountered
v argument of the Fresnel integrals
X displacement
Z integration variable
a /27T constant angular acceleration of the driving
force in cycles per second per second
Q integration variable
Snondimensional damping coefficient
C phase angle
6 initial phase of the simple vibrator
Y impulse
f time at which the frequency of the driving force
is equal to the natural frequency of the simple
vibrator
P angle
w/2TT frequency of the driving function
CHAPTER I
INTRODUCTION
The study of vibrations begins historically and
naturally with the simple pendulum. Consider a particle
of mass 1n suspended from a fixed point by a negligibly
light rigid rod of length If the particle is con
strained to move in a vertical plane, and if the angle the
string makes with the vertical is ( then the equation of
motion (1) is
m.Vd'(/dt' + mi2 sinQ =0
This is a nonlinear equation whose solution involves elliptic
(2)
integrals of the first kind. For sufficiently small os
cillations Sin P == P The equation of motion is then
linear
d"x/dt2 +4 x =0
where X= ( and /. The complete solution (3) is
X = a cost E
The particle performs simple harmonic motion with a period
T=27T/p which is independent of the amplitude and is deter
mined by the nature of the system itself. This observation
was made by Galileo in 1583, the pendulum being a lamp hanging
in the cathedral of Pisa. A theorem to this effect in
Newton's Principia states, "Supposing the centripetal force
to be proportional to the distance of the body from the
centre; all bodies...which move in right lines, running
backwards and forwards alternately, will complete their
several periods of going and returning in the same times." (5)
The concept of a dynamical system being perfectly
isolated and free of dissipative forces is an ideal one.
To represent the effect of dissipation, whether this is
due to causes internal to the system, or to the communication
of energy to a surrounding medium, a force of resistance
proportional to the velocity is introduced into the equation
of motion (6)
d'x/dt' + .pdx/dt + 'X = 0
The solution is
x = a e t cos( pt e)
where = (1O/l/) If the friction is so great that
6/4 > the solution changes form and is not an oscillatory
function of time. For 1/4 < the solution may be
regarded as expressing a vibration whose amplitude is not a
constant, but decreases exponentially in time. The expression
for the frequency involves only the second power of X ,
so that to the first order approximation the friction has
no effect on the frequency of the vibration. The vibrations
(8)
considered above are called free vibrations. They are
f those executed by the particle when disturbed from equilibrium
and then left to itself. A system obeying the above equation
of motion is called a simple vibrator. Another illustration
of the simple vibrator which is commonly used in physics
and engineering is a single mass particle suspended from a
fixed point by means of a spring. The "stiffness" of the
spring is denoted by its spring constant J which is
defined as' the force necessary to extend the spring a unit
of length. The period of vibration is T=2 TVi/ .
A significant part of most textbooks on mechanical vibrations
is concerned with methods of approximating more complicated
mechanical systems by the simple vibrator and calculating
an effective spring constant.D)
The most commonly treated disturbing force is one which
is a simple harmonic function of time. The equation of
motion for an undamped simple vibrator driven by such a
force is
d'x/dt2 + p'x = fcos ot
The complete solution of this equation is
X = A cos pt + B sin pt +[F/up2w)] cos wt
The first two terms with the arbitrary constants represent
the free vibration of the particle with a frequency deter
mined by the system. On this is superposed a forced
vibration (11) represented by the last term. This is of
simple harmonic type with a frequency equal to that of The
disturbing force. In the case of exact coincidence between
the frequency of the driving force and the natural frequency
of the system the amplitude of the forced vibration becomes
infinite.(12) Physical examples of the simple vibrator
such as the pendulum and the mass on a spring are restricted
to small amplitudes in order that the above description be
valid. We are thus led to conclude that for WL= this
solution becomes unmeaningful. An intelligible result may
be obtained, however, if we examine a particular case in
which the initial conditions are definite.(13) Suppose,
for example, that the mass starts from rest in the zero
position at t= 0 The solution is then
x = [/(w'p')] costt cos ut)
which may be written
X F sin /2(Lpt Sin t(uo+p)t
W +'P VZ (W CP)
Thus when the frequency of the driving force is very nearly
equal to the natural frequency of the system
x +/(2p) tsin pt
This may be interpreted as a periodic vibration whose amplitude
increases linearly with time, and is a valid representation for
(14)
the early stages of the motion.4)
The effect of friction may be examined by including a
velocity dependent dissipative term in the equation of motion.
d'x/dt' + 6P dX/dt + 'X = fcos Et
Because of damping, the free vibrations will eventually become
negligible. The particular solution is
x =[ sin /(A pw)] Cos(wtE)
where
tan E = E 'W/pLj 'w1)
The forced vibration has the same period as the applied
force. The phase, C goes from zero at L~=0, to TT as
L approaches infinity.SThus the phase of the oscillation
always lags behind that of the driving force. In particular,
when the frequency of this applied force is equal to the
natural frequency of the system the phase of its oscillation
lags behind that of the driving force by a quarter cycle.
An early application of the forced simple vibrator was in
the theory of tides.(1617)
Suppose that f is given and consider the effect on a
given system of a variation in the frequency of the disturbing
force. The kinetic energy of the system as it passes through
(18)
equilibrium is (18)
E= ir Sin'6
This is a maximum when Sin 6 = I i.e., when the frequency
of the driving force is equal to the natural frequency of the
system. To note the effect of damping on the maximum amplitude
of vibration the particular solution is written as
r
Tr^ 
[(p2~ 1~)&'~ 6 P2 13'~
COS wlZ6)
Thus when = the amplitude is f /f,) which approaches
infinity as approaches zero. The maximum amplitude
occurs at
Consider now the equation of motion for a simple vibrator
driven by a more general disturbing force
d'x/dt' + , dx/d& + p2'x = ft
The solution is easily reduced to quadratures by the method
of variation of parameters.(19)
x= sin St frt) e cost dt
cos spt f e sins t dt
where g= (/''/) It is unnecessary to include explicitly
the free vibration terms since they are already present by
virtue of the arbitrary constants implied in the indefinite
integrals. Lamb considers the case when there is no damping
and f(t) is sensible only for a certain finite range of t .(20)
If the particle is initially at rest and in the equilibrium
position
X = Slinpt f cos pt dt COStftj Sinpt dt
The vibration which remains after the applied force has
become negligible is
x = A cos pt + sin pt
where
A = W1 fit) Sinpt dt and B= E jf])cspt dt
P r Ioco
(21)
For example let (21)
TV, = t'i
This represents a force which is sensible for a certain in
terval on both sides of the origin of the time axis, depending
on the value of C, the integral amount of impulse being p.
An instantaneous impulse can be approximated by making C
sufficiently small. For this f(t) one finds that
X = (p es:t/p) sin pt
The exponential factor shows the effect of spreading out the
impulse. This effect is greater, the larger the frequency of
the natural vibration.
A frequently mentioned method of treating a general
disturbing force f(t) is to expand it in a Fourier series.(22)
S== a,/2 + (a,, cos nott + b, sin nwt)
ftl
where 2n/w is taken to be the period of the disturbing force.
Since the equation of motion is linear, the total forced
vibration is a superposition of those caused by each term in
the series. The remaining problem is to evaluate the coefficients.
For certain (t) such as a square wave,(23) these integrals
can be easily performed, and only a few terms of the Fourier
series will represent the function to a suitable degree of
8
accuracy. However, for many physically interesting ()
the evaluation of the Fourier coefficients poses as formidable
a problem as performing the original quadratures. In fact
for the disturbing force considered in the next chapter it
poses precisely the same problem.
CHAPTER II
CONSTANT ACCELERATION THROUGH RESONANCE
Many mechanical systems capable of vibration are normally
operated above their critical speed and must pass through this
critical speed in being started and again in coming to rest.
The response of a simple vibrator to a force whose frequency
sweeps through the natural frequency of the system in the
simplest manner would yield information about this process.
Such a driving force can be written
(f) = P coS L)t
where P is the amplitude and uL):) is a function of time.
The equation of motion for an undamped vibrator driven by
such a force is
md'x/dt' + ax = Pcos })t
where rn is the mass and A is the spring constant. The
integration of this differential equation is easily reduced
to quadratures by the method of variation of parameters
X = Sin pt cos wo t cost dt
COS jpt cos Wnt sinpt dt]
10
where P= ~ /m is the natural frequency of the vibrator.
Some difficulties in performing these quadratures are
immediately evident. The large number of cycles that a
mechanical system may perform before reaching the interesting
region near resonance discourages the use of either analog
or digital computers to perform the quadratures of the
motion. If one expands the driving force in a Fourier series
COS Ljtt = a./2 + a. cos nwt + E bo Sin nwt
n=l ne
the task of evaluating the coefficients
n = W cos wit cos nwt dt
.bn = J CoS wLt Sin nwt dt
is equivalent to performing the original quadratures.
In a classical paper F. M. Lewis (24) analyzed the
dynamics of a simple vibrator for the case when the frequency
of the applied force depends linearly on time. His quantitative
results were obtained by graphical contour integration and
presented in graphical form, only for widely selected values
of the parameters. The figure which shows the summary of his
conclusions for zero damping is frequently reproduced.(25)
Additionally, Lewis noted that the quadratures in the case
of undamped motion could be expressed as Fresnel integrals;
however, he did not develop this procedure. In view of the
general interest in the analysis by Lewis and, in particular,
in the design of high speed turbines and propellant pumps
it may be worthwhile to show that a systematic treatment by
Fresnel integrals leads to simple algebraic expressions for
the approximate location of the maximum displacement of the
vibrating system and also for its approximate amplitude.
If the initial frequency of the applied force is zero
then
f(t) = Pcos (9 + ,tL)
where 0 is the initial phase and c(/27i is the constant
angular acceleration of the applied force in cycles per
second per second. The equation of motion may be put into
dimensionless form by the following transformations:
U = t/T and R = ~A/P
where T is the time at which the frequency of the applied
force is equal to the natural frequency of the vibrator, i.e.,
W =) =o Z =C D The resulting nondimensional equation of
motion is
d lR/diu + 4Tr'q'R = 4Trr cos( + 7Tg
where = 4~/(2TD) is the number of oscillations the free
vibrator would perform in time In terms of the angular
acceleration of the driving force = P2/(2NT). The com
plete solution of the differential equation is
R(u. .,) = A cos 277u + B sin 2qu + R,
12
where A and B are the usual integration constants and
where Rp the particular solution, is
R, = 2ng cos(0+Tr p') sin[2Trng p)]dp
If the vibrating system is initially at rest and if it is
in a configuration such that the elastic force is in equilib
rium with the applied force, then R(o.,e) == COS 6 and
(R/5Q),=. == 0 and the complete solution may be put
into the form
R(~u..,) = (u~.) COS e + 2 (u, ) Sin
where
R.(u, = 2Tifi [cos Tr p in 2inL(p)]dp + cos 2 u
and
RI(,s = 2nf [sin rTp' sin 2Trj(up()]dp
The amplitude of the maximum displacement and its location
may be found from the equation for the envelope, Re which
is obtained by eliminating 9 between the equation for
Rt(u,,ea) and ~R/DO=0. Obviously, Re =R, +R.
If
C(v =f COS (TrZ/2)dz and Sam = Sin (TU'/2)da
denote the Fresnel integrals,(26) some manipulation shows
that
13
R, (u.e = Ti\/ HCbil ,vy] Sin 7n(2u)
4[ St, S ] cos 7(2u )
Q;[C C Vn2 C[vc] Tinsinc (2 u ')
[QS, o) Sc ]cos nq( 12u +1) +cos 2nqLu
and
R2(u" j= Clov? + C5 1 cos 71T ( 2 L, 1)
[ Scu..)] + Sti21] s in T2 (2t0
[ C )u, l C ]Jcos Trc6(2uL +i)
[ SD. S S ] s in r (2 u+i)
Interesting values of o0 which is inversely proportional
to the angular acceleration of the driving force, are of the
order of ten and greater for mechanical systems. For
smaller values of 0' the frequency of the driving force
is in the vicinity of the natural frequency of the vibrator
for such a short time that there is no significant build up
of the amplitude. Hence, the complexity of the analysis can
be reduced by introducing asymptotic expressions for the
Fresnel integrals with argument 0i0 and (L+i) V The
(27)
appropriate asymptotic expressions are
S i sin (TTv/2) cos (nrrv2)
vC ~ V+ V3
2 ;rv Va
14
and
S_ cos(TTv2) in(77nv1.)
(,v) 2 TT 12 V
where V is positive. If terms of order are con
sistently neglected, the condition DRe/DUL 0 will locate
the maximum amplitude at LAr where
[C .i)Lj +aj cos Tir(u.l)if+[S[u[.i ++ ] sin Ttq(Lm)O
(28)
Using a table of Fresnel integrals, one numerically
solves this transcendental equation and obtains
LU A i + .8606 qgV
and the corresponding maximum amplitude, Rm Relu1n,) is
found to be
R,, 3. 679 .250) + .0085O /
The analogously interesting case of constant de
celeration from a steady state forced vibration of frequency
Ws may be treated by noting that as time runs from cD
to 0 the initial conditions must be derived from the
requirements that the displacement and its first derivative
have continuous values when the decelerating frequency takes
place along the lines LdWs and w=a
being at =ct, Since the asymptotic expansions for
the Fresnel integrals are valid for positive arguments, it
is convenient to reflect the temporal dependence about the
origin. If U,= t,/ /T 2 the accuracy of the asymptotic
expansion of the Fresnel integrals of argument (U.l) ~C is
15
consistent with those of the Fresnel integrals of argument
2 One then finds that the condition on LU for a
maximum in Ra is
[Cu..,]j Y2]cos nygurj + [Suu.on t] sin TrL(ul)'= o
The numerical solution to this equation is
u. = 1.8606q2
and the corresponding maximum amplitude is now
Rm 3.679 "^ + .2501 +.00854'
For =0IO the error in locating L4n is .013, or about
0.6 per cent. This error becomes smaller as Q becomes
larger.
Summary
The quadratures of motion for ah undamped simple
vibrator driven by an applied force whose frequency varies
linearly with time have been performed in terms of Fresnel
integrals. Furthermore, by using the asymptotic expansions
for the Fresnel integrals of large argument simple algebraic
expressions have been derived for the time at which the
maximum displacement is encountered, and for the approximate
magnitude of the maximum displacement. The explicit results
are
t n.ax C ( l+ .86c06')
16
and
x.. (P/.) (3.679q h .2501 + .0085 ')
where t.,. is the time of occurrence of the maximum dis
placement, r is the time at which the frequency of the
applied force is equal to the natural frequency of the
simple vibrator, 9C is the number of cycles which the free
vibrator would perform in time C X,nx. is the maximum
displacement of the simple vibrator, P is the amplitude
of the applied force and A is the spring constant of the
vibrator. The upper signs refer to an acceleration through
resonance, and the lower signs refer to the analogous decel
eration through resonance.
APPENDICES
APPENDIX I
THE METHOD OF VARIATION OF PAPRAETERS
Consider an inhomogeneous linear differential equation
of order n .
L (y) = f,
where
L(y) = (> C)
Suppose that a fundamental set of solutions Lj(x), u .)
..., An~x) of the homogeneous equation
L(iu) =o
are known. The method of variation of parameters,(29)
which is due to Lagrange, can then be applied to determine
a particular solution of the inhomogeneous equation. Let
y = V, u, + \V + ... , V Un.
where V, VA ..., V are undetermined functions of X ,
and y is assumed to satisfy the inhomogeneous equation.
The problem then is to determine the functions Vi explicitly.
As an example which can be generalized, consider a
second order equation
y" a,mi y'+ / . y = a ))
18
19
,where the primes denote differentiation with respect to X .
Suppose that LA,[X) and VL() are" knoun and fonr a fundamental
set of solutions of the homogeneous equation. Assume a
particular solution of the inhonogeneous equation of the
fom
y= V,u, + V, u
Then
y'= V, u: + V, L; V, "., + V, U,
Since the differential equation itself is equivalent to a
single relation between the functions V, and /2, it is clear
that one other relation may be set up. This is chosen to be
V,'u, + VL'u. = 0
Thus
y' = V, ,A + V, L"u
and a second differentiation gives
y" = V, u.," V, u + V," + V.
Substitutia^A these expressions for y, Y and y into the
inhonogeneous equation leads to the relation
V,'.L, VU. V = F .
Since U,, LMi, A,', U1' and F( are kno,~n, this equation
together with
V,'u, + V;.' =0
20
constitute a pair of linear algebraic equations which may
be solved simultaneously for V, and V.
uL ^ \/ u L, 
where A(L,,U. is the Wronskian of U, and UL. The general
solution of the inhomogeneous equation is thus
y= uW A((A.U) dx + U ;Cx) Au.,u) dx
The generalization to the ordinary linear differential
equation of arbitrary order is evident.
APPENDIX II
THE CASE OF iDA PING
When a velocity dependent damping term is included,
the equation of notion for a simple vibrator driven by the
force described in Chapter II is
d'x/dt't 'pdx/dt 'p'X =(P/m') cos(i +a')
where Q is a Cdiensionless damping coefficient. If the
same variable transformations are made as in the case of
no damping one finds that the dimensionless equation of
motion is
d' R/du' + 277Pn dR /du +4 R = TTCos (+Ti )
If the _aitial conditions appropriate to the case of
acceleration through resonance are applied, the complete
solution may be written in the form
R = R, cos J PR Sin9
where
Rf == fJe" COS Ts yp' sin27iC8(up)d(
C e Cos 2 cTB Su
and
e= Sin; p7 sin2TrrS(Lp)d
21
22
and where = () The equation for the envelope of
the family of curves defined by 0, obtained by eliminating
9 between the equation for R and DR/Zi= may be written
once again as R~R'+R2. The maximum amplitude is encountered
at iUn where LU satisfies the condition DR/ODuq and
the maximun displacement is then
Rm = Reu.,y) = [R+(u,, + R( Cu.i,]6
The first task then is to perform the integrals in
R, and IR. The author has not been able to reduce these
integrals to any functions which have been studied and
tabulated, or to shonw that the integrals may be approxiated
by simple expressions. For 6 sufficiently close to unity
one can manipulate Pe and R into a form where the primary
contributions are due to Rosser's integrals.(0)
R, )= ecos f a
and
R^ = /e 5sm ds
which have been tabulated, and to integral of the form
I, = e e Cos
and
S= 773 "sl3
iL = e e Sin s d
O
23
for which no useful expressions could be obtained. In
fact. even if one possessed useable expressions for R,
and R) there would remain another serious problem: The
expression for m obtained froc the condition De/Du 0
would be complicated by the appearance of R, and F
themselves because of the exponential terms.
By noting the results obtained by Lewis one can mke
other qualitative remarks about the effect of damping on a
simple vibrator which is accelerated through resonance.
For a given value of the acceleration, the effect of in
creasing the value of the dampinS coefficient, ) is to
decrease the maximum amplitudes encountered, shift the
location of the resonance to slightly smaller values of U ,
and to broaden the resonance peak.
REFERENCES
(1) Sounerfeld, A., Mechanics (Academic Press, Inc., 1952),
Eglish transl. of 4th ed., Chapt. III, pp. 8790.
(2) lo. cit.
(3) Lord Rayleigh, The Theoryof Sound (Dover Publications,
Inc., Ne: York, 1945), 2nd ed., Chapt. III, p. 44
(4) Lab, H., The Dynamical Theory of Sound (Dover Publications,
Inc., Ner York, 1950), 2nd ed., Chapt. I, p. 10.
(5) Ne.wton, I., .athenatical Princioles of Natural Philosophy
(University of California Press, Berkeley, California,
1934), English transl. of 3rd ed., Book I, p. 149.
(6) Rayleigh, oP. cit, Chapt. III, p. 45.
(7) Scmmerfald, A., op. cit., Chapt. III, p. 104.
(8) Rayleigh, op. cit., Chapt. III, p. 46.
(9) Denartog, J. P., Mechanical Vibrations (NcGrawHill
Book Co., Inc., New York, 19OT 27ind ed., Chapt. II, p. 34.
(10) Jacobsen, L. S., and Ayre, R. S., Egineering Vibrations
(McGrawHill Book Co., Inc., Newi York, 1956).
(11) Rayleigh, op. cit., p. 46.
(12) DenHartog, op. cit., p. 59.
(13) Lamb, op. cit., p. 19.
(14) loc. cit.
(15) Sommerfeld, op. cit., p. 105.
(16) Young, T., "A Theory of the Tides, Including the Consideratior
of Resistance," Nicholson's Journal, 1813; Miscellaneous Works,
London, 1855, Vol. II, p. 262.
(17) Airy, Tides and Waves, Art. 328.
25
(18) Fayleigh, oo. cit., >. 51.
(19) Reddick, H. U. and miller, F. H., .l'v.nced E:atheratics
for Eninoers (Jonn Wiley & Sons, Inc., iew York, 1955),
3rd ed., Chapt. I, pp. 6365.
(20) Lanb, on. cit., pp. 1920.
(21) ibid.
(22) Timoshcnio, S., Vibrat.on Problems in Engineerij.n (D. Van
Nostrand Co., DIntc. Pinrceon, Nc.. Jersey, 1955), 3rd ed.,
Chapt. 1, pp. 99100.
%23) Jacobson, on. cit., pp. 4748.
(24) Lew s, F. P :Vibration durin acceleration through a
critical speed," Trans. A.S.H.E. 54,253261 (1932).
(25) E.g., Tinoshoni!o, S., op. cit., Chapt. I, p. 110.
(26) Gautschi, W., Fandbokl of uathenatical Ynctions, ed.
Abramo:.itz, 'I. and S;eSun, I.A. (iJatlon2al Bureau of
Standards, Uashington, D.C., 1964), Chapt. VI, 7.3.1
and 7.3.2, p. 300.
(27) Gautschi, op. cit., 7.3.9, 7.3.10, 7.3.27 and 7.3.28,
p. 3012.
(28) Pearcey, T., Table of the Frenel Integral (Cambridge
UL'iversity Press, N ou York:, 195), p. 15.
(29) Lagrange. J. L., Nouv. IMem. Acad. Berlin, 5 (1774),
p. 201; 6 (1775), p. 90 OBEvrees, , pp. 9159.
(30) Rosse, J.B., Th'eory and Apolication of )e dx
and t 6 y _t.x (xapleton House,
Brooklyn, ew Yfork) Section 25, pp. 165191.
BIOGPAPHICtL SKETCH
Richard Lee Fearn was born .:arch 24, 1937, at iMobile,
Alabama, In June, 1955, he ia.s graduated from Murphy High
School. The folloTring September he enrolled at Auburn
University, receiving the degrees of Bachelor of Science in
1960 and Itaster of Science in 1961. In September, 1961, he
enrolled in the Graduate School of the University of Florida
where he has pursued his work to..ward the degree of Doctcr of
Philosophy until the present time.
26
This dissertation ras prepared under the direction of
the chairman of the candidate's supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Arts and Sciences and
to the Graduate Council, and ..as approved as partial ful
fillment of the requirements for the degree of Doctor of
Philosophy.
June 22, 1965
Dean, Co ege o Arts
and Sc inces
Dean, Graduate School
Supervisory Committee:
Ch airman
Cochatlrn n
C. V rI
i4 1 ICur
