- 作者: 袁定培
- 作者服務機構: Department of Electrical Engineering; Taiwan Provincial Cheng'-Kung University
- 中文摘要: The study of oscillatory processes is of basic importance in various branches ofphysics and engineering. Apparently diverse and dissimilar oscillatory processes likevibrations of plants and machinery, electromagnetic oscillations in radio engineeringand optics, autcmatic oscillations in control systems and servomechanisms, sonic andultrasonic oscilations, etc. have been brought together through the methods of ma-thematic physics into the important science of oscillation. Generally speaking, there are two distinct types of oscillations or vibrations:simple harmonic or sinusoidal cscillation resulting from the solution of linear differ-ential equation with constant coefficients and relaxational oscillation as a naturalconsequence of nonlinear differential equation which is of by far the more frequentoccurrence in nature. Since harmonic or linear oscillations may be expressed interms of elementary functions by classic methods well known to us and nothingmore is left unexplored, it is to the discussion of the latter type of oscillation thatthe present treatise is chiefly devoted. There are three major avenues of approach to the solution of physical or engine-ering problems formulated by Nonlinear Differential Equations, to wit:(1) Graphical methods or topological methods of graphical solution.(2) Analytical methods or asymptotic methods of analytical solution.(3) Numerical methods or iterative methods of numerical solution. All methods obtained so far are only approximate in nature. The topologicalmethods are based on the study of the representation of solution of differential equa-tions in the phase plane which is mapped by means of point singularities or singularlines so as to obtain certain topological domains in which integral curves in theform of phase trajectories can be investigated by relatively simple methods. Themain advantage of the topological methods lies in the fact that insofar as they dealwith trajectories rather than laws of motion, they make it possible to obtain a bird'seye view of the totality of all possible motions which may arise in a given systemunder all possible conditions. A topological picture of a domain of integral curvespermits ascertaining at once in which regions of the domain the motions are periodicand in which they are aperiodic or asymptotic. Likewise, the critical thresholdswhich separate the regions of stability and instability can be easily ascertained bythese methods. The principal limitation of topological methods is that they do notlend themselves readily to numerical calculations. The graphical methods adapted inthis treatise are those of isoclines and Lienard construction which are applied to thetypical cases of relaxation oscillations derived from the Van der Pol's and Rayleigh'sequations respectively. The analytical methods as well as the numerical methods possess the advantageof leading directly to numerical solutions of a quantitative nature which are of specialimportance in physical and engineering applications. These quantitative methods,however, inevitably narrow the field of vision to a relatively limited region of thedomain. This frequently limits the grasp of the situation as a whole, particularlyif the system possesses critical thresholds such as the separatices, singular andbranch points at which the qualitative features of the phenomena undergo radicalchanges. All analytical methods developed so far for the solution of nonlinear problemsare well applied to a category of so called "quasilinear" differential equations conta-ining a nonlinear term which is proportional to a small parameter. When the par-ameter is zero, the equation is reduced to the idealized linear case which can behandled quite easily. If the nonlinear term or the parameter is getting larger andlarger as in our case of relaxation oscillation, the accuracy of solution decreasesquite appreciably with increase of that parameter. The most important of analyti-cal methods well known to us include the following:(1) Ranscher's iteration method(2) Poincare's perturbation method(3) Van der Pol's method and(4) Kryloff and Bogoliuboff method It is the Last one of the above listed methods which is due to Russian analystsKryloff and Bogoliuboff that we are most interested in this treatise since the writerconsidered it as the only method which can be carried over partially although notcompletely from the "quasi-linear" to the "non-linear" range as required of therelaxation oscillation. Basically it assumes a solution of the product of an amplitudeenvelope function a(t) and a harmonically oscillating function sin [ωt+ψ(t)] conta-ining a phase function ψ(t). When the parameter is increased to such an extentthat the actual oscillation or vibration emerges from the harmonic (sinusoidal) tothe nonharmonic (relaxation) type, the function a(t) for the envelope of amplitudestill holds true but the function sin [ωt+ψ(t)] ceases to be able to represent theactual jerky type of relaxation oscillation which is radically different from theoriginal harmonic type of motion. As to numerical methods of solution we are in favor of the new tool of contiin-uous analytical continuation rather than the routine method of finite difference. Themethod of continuous analytic continuation is based upon the existence theory ofCauchy's calculus of limits involving only a few derivatives of the function inquestion. Its flexibility, its ready adaptation to mechanical computers and the factthat its application is not only limited to real segments, but may be extended to theevaluation of solutions over paths in the complex plane make it a method of unusualpower. All three kinds of methods are introduced for solving the same simple Van derPoI equation whose solution is typical of relaxation oscillation when the parameter becomes large. The reader may be interested in checking the results by com-parison. Finally a discussion on the period of relaxation oscillations has been made.The asymptotic formula for the computation of such a period worked out by A. A.Dorodnitsyn has been given for reference which is rather elaborate for moderatevalues of the parameter with its first term of highest order to be approached by thelimiting case as ∞. Since the work of Dorodnitsyn is too complicated to bereproduced here a simplified version by linearization has been presented for the sakeof clarity.
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