UNIVERSITY OF ILUNOIS ENGINEERING EXPERIMENT STATION. Bulletin Series No 421, ALMOST SINUSOIDAL OSCILLATIONS IN. NONLINEAR SYSTEMS, Part III Transient Phenomena, UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION. Bulletin Series No 421, ALMOST SINUSOIDAL OSCILLATIONS IN NONLINEAR SYSTEMS. Part III Transient Phenomena, J S SCHAFFNER, formerly Research Assistant. Professor of Electrical Engineering, Published by the University of Illinois Urbana. 3050 11 53 53455F L S, This bulletin is the third in a series of three bulletins. dealing with a number of oscillatory problems In the. first two bulletins emphasis was placed on steady state. oscillations while in this bulletin the discussion is con. cerned with nonsteady state or transient oscillations. The oscillatory phenomena discussed here are para, metric excitation synchronization simultaneous oscil. lations and amplitude limitation by means of lamps. The reader should understand however that the meth. ods presented in this bulletin can also be applied to a. number of other oscillatory problems, I INTRODUCTION 7. II AUTONOMOUS OSCILLATIONS WITH ONE, DEGREE OF FREEDOM 9. III ALMOST SINUSOIDAL OSCILLATIONS 10, IV PARAMETRIC EXCITATION OF NONLINEAR SYSTEMS 12. V SYNCHRONIZATION 23, VI SIMULTANEOUS OSCILLATIONS 27. VII THE LIMITATION OF THE AMPLITUDE OF, OSCILLATIONS BY LAMPS 32. VIII CONCLUSIONS 37, APPENDIX BIBLIOGRAPHY 39, 4 1 General Oscillatory Circuit with Excitation by Variable Capacitance 13. 4 2 Oscillatory Circuit with Variable Capacitance Excitation R and L Constant 13. 4 3 Equivalent Linearized Circuit of Fig 4 2 14, 4 4 Trajectories 15. 4 5 Oscillatory Circuit with Variable Capacitance Excitation Inductance Nonlinear 17. 4 6 Region of Stable Oscillations for a 0 18, 4 7 Trajectories for p 0 z 2 Large Detuning 18. 4 8 Trajectories for p Y2 z 2 Large Detuning 18, 4 9 Oscillatory Circuit with Variable Capacitance Excitation Resistance Nonlinear 20. 4 10 Equivalent Linearized Circuit of Fig 4 9 20, 4 11 Region of Stable Oscillations 20. 4 12 Dependence of the Amplitude I on Detuning 21, 4 13 Trajectories for Circuit Nonlinear Resistance z V2 p 1 A 1 21. 4 14 Trajectories for z 1 05 p 2 1 22, 4 15 Oscillations Corresponding to Drift Curve Shown in Fig 4 14 22. 5 1 Equivalent Circuit of a Tuned Plate Oscillator with External Synchronizing Voltage 24. 5 2 Equivalent Linearized Circuit of Fig 5 1 24, 5 3 Regions of Stability and Variation of X as a Function of po for p 3 q 1 25. 5 4 Trajectories with Stable Singular Point 25, 5 5 Trajectories with Unstable Singular Point 26. 6 1 Oscillator with Two Degrees of Freedom 28, 6 2 Equivalent Circuit of Fig 6 1 28. 6 3 Trajectories for Asynchronous Oscillator and i av ov 2 yv 3 30. 6 4 Trajectories for Asynchronous Oscillator and i av v yv3 v 4 Ev 5 30. 6 5 Trajectories for Synchronous Oscillations wc 3w 2 31. 7 1 Oscillator 32, 7 2 Oscillator with Triode Replaced by Negative Conductance 34. 7 3 Equivalent Circuit of Fig 7 2 34, 7 4 Trajectories Describing the Transient Behavior of the Oscillator 35. 1 INTRODUCTION, One of the most significant trends of the twentieth century is the. departmentalization of science The broad field of human experience. has been divided into innumerable sections and subsections each mas. tered only by a small group of specialists One of the consequences of. this division of science is that structural similarities between different. fields are often overlooked The recognition of these similarities can. frequently be a great source of inspiration a source closed to the. specialist who fails to develop any interest outside his own field. A typical example of such a similarity is the occurence of oscillatory. phenomena in such diversified fields as electronics aerodynamics eco. nomics biology etc In recent years a uniform theory of oscillations. applicable to all of these fields has been developed However the main. applications of this theory at present are in electrical and aeronautical. engineering Mathematically oscillatory phenomena lead to nonlinear. differential equations Because of this the theory of oscillations is often. called nonlinear mechanics, Most important for the study of an oscillatory system are the steady. state oscillations They may be stable or unstable depending on whether. the oscillator will or will not return to its original state after being. subjected to a small disturbance noise etc An oscillator may have. several possible steady state oscillations In this bulletin particular. emphasis is placed on the nonsteady state or transient oscillations. These terms nonsteady state and transient are used synonomously. The study of these transient oscillations is often important for the. complete understanding of the circuit behavior For example in a. system having several possible steady state oscillations the transient. study predicts which one of these will be reached from a given set of. initial conditions It also determines the manner in which oscillations. will build up and describes the behavior of the system if a circuit para. meter is changed abruptly Generally a study of the transient oscilla. tions results in a more complete understanding of the oscillatory system. One important problem to be overcome by the investigator is that. of representing the transient oscillations Analytical representation is. not practicable since even if an approximate formula could be found. it would be so complicated that it could not be interpreted easily The. ILLINOIS ENGINEERING EXPERIMENT STATION, most common graphical method of representation makes use of a plane. with time as the abscissa and the dependent variable as the ordinate. This method is however limited to the cases in which the differential. equation can be reduced to, This reduction is possible for only a few oscillators. Autonomous systems with one degree of freedom can often be. described by the differential equation, d 2x dx dx dx. dt2 f x dt dt g z xdi 0 1 1, The corresponding oscillations can be represented in a plane with x as. the abscissa and dx dt as the ordinate the phase plane Such a repre. sentation is shown in Chapter II, No general method for the graphical representation of oscillations. in more complicated systems is available If it can be assumed however. that the oscillations are almost sinusoidal then a large class of oscillators. can be represented in some sort of a phase plane In this new plane an. oscillation with a certain amplitude phase angle etc will correspond. to a single point, II AUTONOMOUS OSCILLATIONS WITH ONE. DEGREE OF FREEDOM, The study of the autonomous oscillators with one degree of freedom. takes a rather special place in nonlinear mechanics insofar as these. oscillators can be treated conclusively and in all generality by topological. methods Equation 1 1 corresponding to autonomous oscillators with. one degree of freedom can be transformed into two simultaneous dif. ferential equations of the first order by the substitution dx dt y. dt x y y g x y x, The time can be eliminated from these two equations by dividing. one by the other, dy f x y g x y X 2 2, The solutions of this equation can be represented by curves in the x y. plane which can be found by graphical construction method of iso. clines If the system is at rest then x and y are constant in time. and the differential dy dx at such a point is indeterminate singular. The x y plane may also contain a number of closed curves or limit. cycles corresponding to periodic solutions of Eq 1 1 The singular. points and the limit cycles determine the structure of the x y plane. The state of the oscillator is completely described at any one time by a. point in the x y plane If the system is not disturbed then this repre. sentative point will move along a curve defined by Eq 2 2. Typical oscillatory systems that can be treated by this method are. for example the ordinary triode oscillator the Prony brake and the. multivibrator It is possible but impractical to treat more complicated. systems by this method because spaces with three or more dimensions. are required to describe them, III ALMOST SINUSOIDAL OSCILLATIONS. A large group of oscillatory phenomena can be represented in a plane. entirely different from that discussed in Chapter II provided that it. can be assumed that the oscillations are almost sinusoidal In electrical. circuits for example oscillations are nearly sinusoidal if the quality. factors Q s of the resonant circuits are high Then if it is assumed. that the oscillations are purely sinusoidal the circuits can be described. completely by a few parameters for example by the amplitude of. oscillation the phase angle relative to an external sinusoidal voltage. etc For steady state oscillations these parameters remain constant. for transient oscillations they will change A representation in a plane. is possible if the system can be described by two parameters only say. X and Y When one of these parameters is a phase angle the plane. lies on the surface of a cylinder so that the lines 0 0 and 2 7r are. identical Many oscillatory phenomena can be described by two para. meters and can hence be represented in this plane, The rate of change of X and Y is determined by the state of the. system and hence depends on X and Y only, dt A X Y 3 1. Equations 3 1 can be obtained by various analytical and experimental. methods The method used in this bulletin is that of equivalent lineariza. tion The mathematical formulation of this method has been described. extensively in Buls 395 and 400 and hence is not presented here. The time can be eliminated from Eqs 3 1 by dividing one by the. The solutions of this differential equation can be represented as. curves in the X Y plane If the system is not disturbed then the. representative point corresponding to an oscillation with parameters. X and Y will move along one of these curves For steady state oscilla. tions X and Y remain constant in time, Bul 421 OSCILLATIONS IN NONLINEAR SYSTEMS. This corresponds to a singular point in the X Y plane The symbols. X 0 and Yo are used for the parameters of the steady state oscillation. A singular point and the corresponding steady state oscillation are. stable if the oscillation and the corresponding point will return to their. original position after a small disturbance In order to discuss the. behavior of the system in the neighborhood of the steady state oscilla. tions it is necessary to expand Eq 3 1 around Xo and Yo Letting. then the first terms of the expansion are, d 6X affX a Sy. It is assumed that SX and 5Y are small and therefore that the higher. terms of this expansion can be neglected Equation 3 4 is a set of two. simultaneous linear differential equations which can be solved by the. usual methods The variables 6X and 6Y will approach zero from any. initial conditions in the neighborhood of the steady state oscillations. provided that both the roots Ki and K2 of Eq 3 5 have negative real parts. a 2 af 2 a, Necessary and sufficient conditions for this to exist are that. 2 af f f2 0 3 6, aX aY aY aX, Like the limit cycles of the x y plane described in Chapter II the. X Y plane may contain closed curves which are called drift curves. Physically these drift curves correspond to oscillations with periodically. system having several possible steady state oscillations the transient study predicts which one of these will be reached from a given set of initial conditions It also determines the manner in which oscillations will build up and describes the behavior of the system if a circuit para meter is changed abruptly Generally a study of the transient oscilla tions results in a more complete

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