# Nonlinear Oscillations in Physical Systems

Chihiro Hayashi
Pages: 402
https://www.jstor.org/stable/j.ctt7zv6k6

1. Front Matter
(pp. v-vi)
2. Preface
(pp. vii-viii)
Chihiro Hayashi
(pp. ix-xiv)
4. Introduction
(pp. 1-8)

Basically, all the problems in mechanics are nonlinear from the outset. The linearizations commonly practiced are approximating devices that are good enough or quite satisfactory for most purposes. There are, however, also certain cases in which linear treatments may not be applicable at all. Frequently, essentially new phenomena occur in nonlinear systems which cannot, in principle, occur in linear systems [15, 66, 111].¹ The principal aim of this book is not to introduce methods of improving the accuracy obtainable by linearization, but rather to focus attention on those features of the problems in which the nonlinearity results in distinctive new...

5. PART I. PRINCIPAL METHODS OF NONLINEAR ANALYSIS
• Chapter 1 Analytical Methods
(pp. 11-32)

There is usually considerable advantage in finding an analytical solution for a differential equation when that is possible. The analytical solution is obtained in algebraic form without the necessity of introducing numerical values for parameters or initial conditions during the process. Once the solution is obtained, any desired numerical values can be inserted and the entire possible range of solutions explored. Because of this flexibility, it might be well to attempt first to seek a solution of a specified differential equation in analytical form. However, it should be recognized that only a very few equations that arise from actual physical...

• Chapter 2 Topological Methods and Graphical Solutions
(pp. 33-68)

The topological method of analysis is one of the important means of investigating various phenomena of nonlinear oscillations, and it is applicable to the study of autonomous systems. By this method solutions of differential equations are sought not as explicit functions of the time, but as solution (or integral) curves in a phase space or, more generally, in the state space. Considerable insight into the qualitative aspects of the solution, and some quantitative information as well, can be obtained through a study of integral curves. If, however, we rely on graphical methods for the representation of solutions, the applicability of...

• Chapter 3 Stability of Nonlinear Systems
(pp. 69-98)

The question of stability is concerned with what happens if a system is disturbed slightly near an equilibrium condition. In general terms, any disturbance near an unstable equilibrium condition leads to a larger and larger departure from this condition. Near a stable equilibrium condition, the opposite is the case. An equilibrium condition may be either stationary or oscillatory. When it is stationary, the variables of the system remain constant; when it is oscillatory, the variables are undergoing continuous periodic change.

The question of stability is relatively simple in a linear system in which a single equilibrium condition exists. If the...

6. PART II. FORCED OSCILLATIONS IN STEADY STATES
• Chapter 4 Stability of Periodic Oscillations in Second-order Systems
(pp. 101-113)

When a periodic force is applied to a linear system, the resulting motion is obtained by a superposition of the transient- and the steady-state components of the oscillation. The former is due to the free oscillation of the system, and the latter is related to the forced oscillation which arises from the action of the external force. When the system is (asymptotically) stable, the free oscillation is damped out after a sufficiently long period of time. Hence only the forced oscillation having the same frequency as the external force would be observed. Thus, as far as linear systems are concerned,...

• Chapter 5 Harmonic Oscillations
(pp. 114-127)

In this chapter we deal with the harmonic oscillations [19–21, 94] in which the fundamental component having a period the same as that of the external force is predominant, so that the higher harmonics may be neglected. In order to get the physical idea concretely, we consider an electric oscillatory circuit and derive the nonlinear differential equation of the form (4.10) [31, 33].

The schematic diagram in Fig. 5.1 shows an electric circuit in which the nonlinear oscillation takes place owing to the presence of a saturable-core coil of inductanceLunder the impression of an alternating voltageE...

• Chapter 6 Higher-harmonic Oscillations
(pp. 128-141)

In the preceding chapter the stability of harmonic oscillations was investigated by making use of variational equations of the Hill type. As mentioned in Sec. 5.1c, the stability condition (4.6) forn≧ 2 must be considered when the amplitudeBof the external force is very large. Oscillations under this condition are worthy of consideration, since anomalous excitation of higher-harmonic components [31, vol. 29, p. 670; 94, p. 778] results if the above stability condition is not satisfied. Since very few investigations have been made in this field, this section will be concerned with such oscillations.

Referring to the...

• Chapter 7 Subharmonic Oscillations
(pp. 142-180)

In the preceding two chapters we have investigated the harmonic and the higher-harmonic oscillations whose fundamental frequency is equal to that of the external force. We are now to deal with the subharmonic oscillations whose fundamental frequency is a fraction 1/v(v= 2, 3, 4, …) of the driving frequency. As we have briefly noted in Sees. 3.5 and 4.1, these oscillations are of another important type in the field of nonlinear oscillations, and they frequently occur in various branches of engineering and physical sciences [35, 65, 72, 76, 80, 96, 100].

We first take the fundamental equation$\frac{{{d}^{2}}\upsilon }{d{{\tau }^{2}}}+2\delta \frac{d\upsilon }{d\tau }+f(\upsilon )=B\cos \, \nu \tau \caption {(7.1)}$...

7. PART III. FORCED OSCILLATIONS IN TRANSIENT STATES
• Chapter 8 Harmonic Oscillations
(pp. 183-208)

It was mentioned in Sec. 4.1 that various types of periodic solution may exist for a given nonlinear differential equation, depending on different values of the initial condition. In Part II we have concentrated our attention on the periodic solutions and discussed their stability. We shall now investigate the transient state concerned with the oscillations until they get to the steady state. When we have done this, the relationship between the initial conditions and the resulting periodic oscillations will be made clear. But it is usually not possible to solve nonlinear differential equations with arbitrary initial conditions, except for special...

• Chapter 9 Subharmonic Oscillations
(pp. 209-237)

In the preceding chapter we have considered the transient state of the harmonic oscillation and discussed in detail the stability of equilibrium states correlated with singular points. We are now to deal with the subharmonic oscillation whose fundamental frequency is a fraction 1/v(vbeing a positive integer) of the driving frequency. As mentioned in Chap. 7, such oscillations may occur in nonlinear systems, particularly in the case where the system is described by Duffing’s equation$\frac{{{d}^{2}}\upsilon }{d{{\tau }^{2}}}+k\frac{d\upsilon }{d\tau }+f(\upsilon )=B\cos \, \nu \tau \caption {(9.1)}$and the periodic solutions have been assumed, to a first approximation, by [see Eq. (7.4)]$\upsilon =z+x\sin \tau +y\cos \tau +w\cos \, \nu \tau\caption {(9.2)}$where$w=\frac{1}{1-{{\nu }^{2}}}B$

In this chapter we consider...

• Chapter 10 Initial Conditions Leading to Different Types of Periodic Oscillations
(pp. 238-261)

In this chapter we shall be particularly concerned with the relationship between the initial conditions and the resulting periodic oscillations of systems governed by the differential equation$\frac{{{d}^{2}}\upsilon }{d{{\tau }^{2}}}+k\frac{d\upsilon }{d\tau }+f(\upsilon )=g(\tau )\caption {(10.1)}$wherekis a constant,f(υ) is a polynomial ofυ, andg(τ) is periodic in the timeτ.

Before going into the present investigation, we shall briefly review the method of analysis which has been used in the preceding chapters. For the sake of brevity, we confine the problem to the analysis of harmonic oscillations under the impression of the external forceg(τ) =Bcosτ. We write the solution...

• Chapter 11 Almost Periodic Oscillations
(pp. 262-282)

When a periodic force is applied to a nonlinear system, the resulting oscillation is usually, but not necessarily, periodic. When it is periodic, the fundamental period of the oscillation is the same as, or equal to an integral multiple of, the period of the external force. To these phenomena the terms harmonic oscillation and subharmonic oscillation have been respectively applied. There are also certain special cases in which the response of a nonlinear system is not periodic even when subjected to a periodic excitation. This chapter is concerned with a type of oscillation such that the amplitude and phase of...

8. PART IV. SELF-OSCILLATORY SYSTEMS WITH EXTERNAL FORCE
• Chapter 12 Entrainment of Frequency
(pp. 285-308)

The phenomenon of frequency entrainment occurs when a periodic force is applied to a system whose free oscillation is of the self-excited type. A typical and important case is the system governed by van der Pol’s equation with an additional term for periodic excitation [86]. The frequency of the self-excited oscillation falls in synchronism with the driving frequency provided that the two frequencies are not too different. If their difference is large enough, one may expect the occurrence of an almost periodic oscillation; in other words, a beat oscillation may result. However, the entrainment of frequency still occurs when the...

• Chapter 13 Almost Periodic Oscillations
(pp. 309-338)

Almost periodic oscillations may occur in some sorts of nonlinear systems which are subject to periodic forces. It has been shown in Chap. 11 that oscillations of this type may arise in electrical systems which contain saturable iron cores as nonlinear elements. As mentioned in Sec. 11.1, the amplitude and phase of an almost periodic oscillation vary slowly but periodically even in the steady state. However, since the ratio between the period of the amplitude variation and that of the external force is in general incommensurable, there is no periodicity in the almost periodic oscillation.

This chapter deals with almost...

9. Appendix I: Expansions of the Mathieu Functions
(pp. 339-340)
10. Appendix II: Unstable Solutions of Hill’s Equation
(pp. 341-342)
11. Appendix III: Unstable Solutions for the Extended Form of Hill’s Equation
(pp. 343-348)
12. Appendix IV: Stability Criterion Obtained by Using the Perturbation Method
(pp. 349-352)
13. Appendix V: Remarks Concerning Integral Curves and Singular Points
(pp. 353-358)
14. Appendix VI: Electronic Synchronous Switch
(pp. 359-361)
15. Problems
(pp. 362-377)

Some of the problems listed here are not difficult to solve and may be useful as exercises,¹ but there are others that may be considered more appropriate as subject matter for research work. It is highly recommended that the reader be acquainted with various phases of nonlinear oscillations by both experimental observation and computer analysis before he gets into an elaborate mathematical treatment. For this purpose a number of problems pertaining to experimental study and computer analysis are also included in the list.

1. Using the perturbation method, obtain the periodic solution of Rayleigh’s equation$\frac{{{d}^{2}}x}{d{{t}^{2}}}-\mu \left[ 1-\frac{1}{3}{{\left( \frac{dx}{dt} \right)}^{2}} \right]\frac{dx}{dt}+x=0$to the second-order approximation, that...

16. Bibliography
(pp. 378-384)
17. Index
(pp. 385-392)