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- https://ufdc.ufl.edu/UF00097899/00001
## Material Information- Title:
- The Temporal passage of mechanical systems through resonance
- Creator:
- Fearn, Richard Lee, 1937- (
*Dissertant*) Kronsbein, John (*Thesis advisor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1965
- Copyright Date:
- 1965
- Language:
- English
- Physical Description:
- v, 26 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Acceleration ( jstor )
Amplitude ( jstor ) Damping ( jstor ) Equations of motion ( jstor ) Fresnel integrals ( jstor ) Mathematical maxima ( jstor ) Natural frequencies ( jstor ) Sine function ( jstor ) Spring constant ( jstor ) Vibration ( jstor ) Dissertations, Academic -- Physics -- UF Physics thesis Ph. D Resonance ( lcsh ) Vibration ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- .
- Thesis:
- Thesis--University of Florida,1965.
- Bibliography:
- Bibliography: leaves 24-25.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000568373 ( alephbibnum )
13648995 ( oclc ) ACZ5104 ( notis )
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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 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. 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 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 resonance 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 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/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 (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(wt-E) 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 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 position 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 where A = W1 fit) Sinpt dt and B= E jf])cspt dt P r Io-co (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 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, -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(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. 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\/ 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 and 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 (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 Ld--Ws and w=a 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 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 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 fo-m 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(u-p)d( C- e Cos 2 cTB Su and e= Sin; p7 sin2TrrS(L-p)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 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 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. 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. -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. 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. 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 Co-chatlrn n C. V rI i4 1 ICur- |