Title Page
 Table of Contents
 List of symbols
 Constant acceleration through...
 Biographical sketch

Group Title: temporal passage of mechanical systems through resonance
Title: The Temporal passage of mechanical systems through resonance
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097899/00001
 Material Information
Title: The Temporal passage of mechanical systems through resonance
Physical Description: v, 26 leaves. : illus. ; 28 cm.
Language: English
Creator: Fearn, Richard Lee, 1937-
Publication Date: 1965
Copyright Date: 1965
Subject: Vibration   ( lcsh )
Resonance   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida,1965.
Bibliography: Bibliography: leaves 24-25.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097899
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000568373
oclc - 13648995
notis - ACZ5104


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Table of Contents
    Title Page
        Page i
        Page i-a
        Page ii
    Table of Contents
        Page iii
    List of symbols
        Page iv
        Page v
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Constant acceleration through resonance
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Appendix I: The method of variation of parameters
            Page 18
            Page 19
            Page 20
        Appendix II: The case of damping
            Page 21
            Page 22
            Page 23
        Page 24
        Page 25
    Biographical sketch
        Page 26
        Page 27
        Page 28
Full Text







June, 1965


The author wishes to express his thanks to Professor

Kronsbein and Professor Millsaps for serving as co-chairmen

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.


ACKNOWLEDGMENTS ..................... ........... ...... .. ii

LIST OF SYMBOLS ...................................... iv


I. INTRODUCTION .... ............................. 1




II. THE CASE OF DAMPING ........................... 21

REFERENCES ......................................... 24

BIOGRAPHICAL SKETCH .................................. 26


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 non-dimensional 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

Scv) Fresnel integral of argument V

t time

T period of a simple harmonic motion
U non-dimensional 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
Snon-dimensional 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

P angle

w/2TT frequency of the driving function


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
integrals of the first kind. For sufficiently small os-
cillations Sin P == P The equation of motion is then

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
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/up-2w)] 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(-L-pt 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
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(wt-E)

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
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
Tr^ -

[(p2~- 1~)&'~ 6 P2 13'~

COS wlZ-6)

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

X = Slinpt f cos pt dt COS-tftj Sinpt dt

The vibration which remains after the applied force has
become negligible is

x = A cos pt +- sin pt


A = W1 fit) Sinpt dt and B= E jf])cspt dt
P r Io-co
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)
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


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.



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]


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

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 non-dimensional equation of
motion is

d lR/diu + 4Tr'q'R = 4-Trr 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 2-qu + R,


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

R.(u, = 2Tifi [cos Tr p in 2inL(-p)]dp + cos 2 u


RI(,s = 2nf [sin rTp' sin 2Trj(u-p()]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.

C(v =f COS (TrZ/2)dz and Sam = Sin (TU'/2)da

denote the Fresnel integrals,(26) some manipulation shows


R, (u.e = Ti\/ HCb-il -,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
R2(u" j= Cl-ov? + C5 1 cos 71T ( 2 L, -1)

[ Scu-.-.)] + Sti21] s in T2 (2-t-0

-[ 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
appropriate asymptotic expressions are

S i sin (TTv/2) cos (nrrv2)
vC ~ V+ V3
2 ;rv Va



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
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 Ld--Ws 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


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 nygu-rj + [Suu.-on -t] sin TrL(u-l)'= 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


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

t n.ax C ( l+ .86c06-')



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.



Consider an inhomogeneous linear differential equation
of order n .

L (y) = f,

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 ))



,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

y= V,u, + V, u

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

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


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.


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

Rf == fJe" COS Ts yp' sin27iC8(u-p)d(

C- e Cos 2 cTB Su

e= Sin; p7 sin2TrrS(L-p)d



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 approxi-ated
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


R^ = /e 5sm ds

which have been tabulated, and to integral of the form

I, = e e Cos


S= 773 "-sl3
iL = e- e Sin s d


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.


(1) Sounerfeld, A., Mechanics (Academic Press, Inc., 1952),
-Eglish transl. of 4th ed., Chapt. III, pp. 87-90.

(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) Den-artog, J. P., Mechanical Vibrations (NcGraw-Hill
Book Co., Inc., New York, 19OT 27ind ed., Chapt. II, p. 34.
(10) Jacobsen, L. S., and Ayre, R. S., Egineering Vibrations
(McGraw-Hill 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.


(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. 63-65.

(20) Lanb, on. cit., pp. 19-20.

(21) ibid.

(22) Timoshcnio, S., Vibrat.on Problems in Engineerij.n (D. Van
Nostrand Co., DIntc. Pinr-ceon, Nc.. Jersey, 1955), 3rd ed.,
Chapt. 1, pp. 99-100.

%23) Jacobson, on. cit., pp. 47-48.
(24) Lew s, F. P :Vibration duri-n acceleration through a
critical speed," Trans. A.S.H.E. 54,253-261 (1932).

(25) E.g., Tinoshoni!o, S., op. cit., Chapt. I, p. 110.
(26) Gautschi, W., Fandbokl of u-athenatical Y-nctions, ed.
Abramo:.itz, 'I. and S;eSun, I.A. (iJatlon2al Bureau of
Standards, Uashingto-n, 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. 301-2.
(28) Pearcey, T., Table of the Frenel Integral (Cambridge
UL'iversity Press, N ou York:, 195), p. 15.
(29) Lagrange. J. L., Nouv. I-Mem. Acad. Berlin, 5 (1774),
p. 201; 6 (1775), p. 90 OBEvrees, -, pp. 9-159.

(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. 165-191.


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.


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


June 22, 1965

Dean, Co ege o Arts
and Sc inces

Dean, Graduate School

Supervisory Committee:

Ch airman

Co-chatlrn n

C. V rI

i4 1 ICur-

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