An Introduction to Nonlinear Oscillations 0521222089

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An Introduction to Nonlinear Oscillations

An Introduction to Nonlinear Oscillations

RONALD E. MICKENS Professor of Physics, Fisk University

CAMBRIDGE UNIVERSITY PRESS

Cambridge London New York New Rochelle

Melbourne

Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

© Cambridge University Press 1981 First published 1981

Printed in the United States of America Typeset by Progressive Typographers, Inc., Emigsville, Pennsylvania Printed and bound by The Book Press, Brattleboro, Vermont Library of Congress Cataloging in Publication Data Mickens, Ronald E

1943—

An introduction to nonlinear oscillations. Bibliography: p.

Includes index. 1. Differential equations, Nonlinear - Numerical solutions. 2. Approximation theory. 3. Nonlinear oscillations. I. Title. QA372.M615 515.3'55 80-13169 ISBN 0 521 22208 9

To my wife Maria and my son James Williamson

Contents

Preface 1

1.1 1.2 1.3 1.4 1.5

2

Nonlinear Physical Systems

Introduction Examples of Nonlinear Physical Systems Dimensionless Form of Differential Equations Exact Solution for Period of a Pendulum Exact Solution of d2∙y∣dt"l∙ + y + ey3 = 0 Problems References The Perturbation Method

2.1 Introduction 2.2 Secular Terms 2.3 Lindstedt-Poincare Method 2.4 Worked Examples 2.5 Shohat Expansion 2.6 Existence of a Periodic Solution Problems References 3

Method of Slowly Varying Amplitude and Phase

3.1 Introduction 3.2 First Approximation of Krylov and Bogoliubov 3.3 Worked Examples Using the Method of Krylov and Bogoliubov 3.4 Method of Krylov-Bogoliubov-Mitropolsky 3.5 Worked Examples Using the Method of Krylov-Bogoliubov-Mitropolsky 3.6 Stationary Amplitudes and Their Stability 3.7 Equivalent Linearization

page xi

1 1 2 15 20 23 27 30

32 32 34 36 38 45 47 52 53

55 55 56 61 68

75 79 84 vii

Contents 3.8

Nonlinear Oscillations with Finite Damping Problems References

4 Multi-Time Expansions 4.1 Introduction 4.2 Two-Time Expansion 4.3 Worked Examples Using the Two-Time Expansion 4.4 Derivative Expansion Procedure 4.5 Worked Examples Using the Derivative Expansion Procedure Problems References

5 5.1 5.2 5.3 5.4 5.5

5.6 5.7 5.8

6 6.1 6.2 6.3 6.4 6.5 A.l A.2 A.3 A.4 A.5

B.l B.2 B.3 B.4

viii 88 93 95 96 96 98 101 111 112 116 117

Forced Oscillations Introduction Forced Oscillations of Linear Systems Combination Tones Subharmonic Oscillations Iteration Methods for Harmonic Oscillations without Damping Perturbation Theory Applied to Forced Oscillations Worked Examples Using the Perturbation Method Duffing Equation: Resonance Curves and Jump Phenomena Problems References

118 118 120 124 126

Advantages and Disadvantages of Various Techniques Introduction Perturbation Method Method of Slowly Varying Amplitude and Phase Multi-Time Expansion Procedures for Solving Nonlinear Problems

156 156 156 157 157 158

Appendix A: Mathematical Relations Trigonometric Functions Factors and Expansions Solution of Quadratic Equations Solution of Cubic Equations Differentiation of a Definite Integral with Respect to a Parameter References

159 159 161 161 162

Appendix B: Series Expansions Uniform Convergence Weierstrass M Test for Uniform Convergence Properties of Uniformly Convergent Series Power Series

164 164 165 165 166

129 132 135 145 153 154

163 163

Contents

ix

Taylor Series of a Function of a Single Variable Taylor Series of a Function of Two Variables References

167 169 169

C.l C.2 C.3

Appendix C: Fourier Series Definition of Fourier Series Convergence of Fourier Series Expansion of F(A cos x, ~ A sin .v) in a Fourier Series References

170 170 173 174 175

D.l D.2 D.3 D.4 D.5

Appendix D: Asymptotic Expansions Gauge Functions and Order Symbols Asymptotic Expansions Uniform Expansion Elementary Operations on Asymptotic Expansions Examples References

176 178 179 179 181 182

Appendix E: Basic Theorems of the Theory of Second-Order Differential Equations Introduction Existence and Uniqueness of the Solution Dependence of the Solution on Initial Conditions Dependence of the Solution on a Parameter References

183 183 184 184 185 187

B.5 B.6

E.l E.2 E.3 E.4

F.l F.2 F.3 F.4 F.5

G.l G.2 G.3 G.4 G.5

H.l H.2

Appendix F: Linear Second-Order Differential Equations Basic Existence Theorem Homogeneous Linear Differential Equations Nonhomogeneous Linear Differential Equations Linear Second-Order Homogeneous Differential Equations with Constant Coefficients Linear Second-Order Nonhomogeneous Differential Equations with Constant Coefficients References

188 188 188 190

192

193 195

Appendix G: Existence of Periodic Solutions of Certain Second-Order Differential Equations Limit Cycles Lienard-Levinson-Smith Theorem Levinson-Smith Theorem Cartwright-Littlewood Theorem Levinson Theorem References

196 196 198 198 199 199 200

Appendix H: Stability of Limit Cycles Introduction Stability Condition References

201 201 202 208

Contents 209 209 209 213 215 218

221

Preface

This book is concerned primarily with nonlinear oscillations of one­ dimensional physical systems, which may be represented by the har­ monic oscillator equation, with the addition of a “small” nonlinear term: + +β~dr^ (r ~ R0)3

+ ∙ ∙ ∙

where, at the equilibrium position,

dV(R.) . dr

(1.35)

If we define a new variable y = r - Ro

(1.36)

and let

,.

d2V(R0) dr2

= 1

Λ3V(fl0) dr3

(1∙37)

then equation 1.34 becomes

V(R0 + y) = V(R0) + iky2 - ik1y3 + ∙ ∙ ∙

Figure 1.8.

(1.38)

Potential energy of interaction between two atoms.

Nonlinear Physical Systems

12

where

Λo V(*o) k k1

= = = =

21'6B A 72A∕2ll3B2 6048∕21'2B3

(1∙39)

Note that k and k1 are both positive. Under these conditions the equation of motion becomes

d2y 1 m ~dt2 = ~ky + 3k'y2 + ’ ' ,

(1.40)

Thus, for small displacements from the equilibrium position, the oscil­ latory motion of a diatomic molecule can be represented by a nonlinear differential equation.

Nonlinear Oscillators with Damping Consider a particle of mass m constrained by a nonlinear re­ storing force - g(x). If the particle moves in a viscous medium, such as air or water, it will experience a resistive force —f, which we assume is a function only of the velocity. Under these circumstances the equa­ tion of motion is 1.2.7

(1.41)

where we have taken g(x) to have the form

g(x) = kx + g1(x)

(1.42)

k is a constant, and g1(x) is a nonlinear function of x. What can be said about the resistive force/? To tackle this equation, we make the following assumptions: 1. We assume the object to be spherical in shape with radius R. (More generally, R may represent a “typical” linear dimension of the object.) 2. The frictional force depends only on the viscosity of the medium, η, the density of the medium, p, the radius of the ob­ ject, R, and the velocity of the object, V = dx∕dt.13 3. The frictional force is zero when the velocity is zero. 4. The frictional force is an analytic function of V at V = 0. These four conditions allow us to write the frictional force as ∕(Λ,p,η,V) = ∕1V +f2V2 + ∙ ∙ ∙ +fnV" + ∙ ∙ ∙

(1.43)

where the coefficients fn are functions of R, p, and η. If we further as­

1.2 Examples of Nonlinear Physical Systems

13

sume that these coefficients have a power-law dependence on their variables, then they may be written as

fn = CnτγepuRz

(1.44)

where Cn, x, }∖ and z are pure numbers. The use of dimensional analy­ sis allows us to determine the values of x, y, and z.14 We find that a=

2 - n y = n - 1 z = n

(1.45)

and consequently the coefficient ∕nCR,p,η) is

(1.46) The constant Cn cannot be determined from dimensional-analysis con­ siderations alone.14 It can be shown, for a spherical body of radius R, that the viscous frictional force, in the limit of small velocities, is exactly15 f= 6π,ηRV

(1.47)

Retaining the first two terms in the expansion (equation 1.43) leads to the following nonlinear differential, for oscillations of a particle in a viscous medium: + kx + gι(x) = 0

A final comment on this problem: For an object moving in a viscous medium, the frictional force is dissipative. Physically, this means that the frictional force on the object is in a direction opposite to the veloc­ ity. This requirement can be met by writing the frictional force as dx

(1.49)

where f1 and∕2 are positive. Consequently, equation 1.48 becomes dx }^t + kχ + dt

= θ

As a second example of a frictional force, we consider so-called Cou­ lomb or dry friction. This type of damping arises when an object slides on a dry surface.15,16 For motion to begin, there must be a force acting on the body to overcome the resistance to motion caused by friction. The dry frictional force Fd is parallel to the surface and proportional to the force normal to the surface, Fn. The force Fn is equal to mg in the case of the mass-spring system of Figure 1.9. The constant of propor­

Nonlinear Physical Systems

14

tionality is the static friction coefficient μs, a number whose value is between 0 and 1, depending on the surface materials. Once motion has started, the force drops in value to μfc∕ng, where μk is the kinetic fric­ tion coefficient. In general, μk < μs. The friction force is opposite in direction to the velocity and remains approximately constant in magni­ tude as long as the forces acting on the object are larger than the dry friction. In Figure 1.10 we give an idealized functional form for Coulomb or dry friction. It should be emphasized that for actual physical systems the frictional force is generally more complicated than that given in Figure 1.10. However, for mathematical purposes, a functional form similar to that of Figure 1.10 is often used. Mathematically, it has the following representation: (1.51)

Fd = -μkrng sgn

where the function sgn (vv) is defined to be , , f +1 Sgn(w)=t-1

for for

w > 0 ⅛, < 0

(1.52)

Finally, the equation of motion of the oscillatory system of Figure 1.9 can be written in the form

d2x

, ∣xkmg sgn

∕ dx∖

+ kx = 0

Figure 1.9. Mass-spring system with Coulomb or dry friction. The frictional force is constant in magnitude and has a direction oppo­ site to that of the velocity.

Fd = μ∣c mg

◄-------kx ◄-------

V --►

T7

1.3 Dimensionless Differential Equations

15

1.3

Dimensionless Form of Differential Equations We have seen that the formulations of many physical problems lead to nonlinear differential equations that have the following form:

(1.54)

where α represents the set of constants necessary to specify the non­ linear function F. In general, m, k1, k, and a are dimensional con­ stants, y is the dimensional dependent variable, and t is the dimen­ sional independent variable. (A good discussion of units, pure numbers, dimensional and dimensionless constants, and variables is given in Chapter 1 of the work by Isaacson.14) The function F repre­ sents a nonlinear force that is usually stated to be “small” in some sense. However, the magnitude of a dimensional constant or variable depends on the system of units.17 Consequently it is difficult to know exactly what being “small” means in this context. The way out of this dilemma is to convert the original differential equation (equation 1.54) into an equivalent equation in which all the new constants and vari­ ables are dimensionless. Generally, if we have chosen our dimen­ sionless variables wisely, a “small” dimensionless parameter will nat-

Figure 1.10.

Idealized functional form for Coulomb or dry friction.

Nonlinear Physical Systems

16

urally appear, multiplying the nonlinear terms. It is at this stage that we can apply our various methods of approximation to obtain a solution to the nonlinear problem. In this section we give the rules for carrying out this procedure and illustrate the technique with a number of examples. Let us interpret equation 1.54 as an equation describing a mechanical system. Thus, this equation represents a mass m acted on by a har­ monic restoring force -ky, a linear damping force -kl dy∕dt, and a nonlinear force -F(a,y,dy∕dt). We take the following for our initial conditions:

^Γ = 0

y(0) = A

(L55)

(These initial conditions are sufficiently general to cover all physical systems of interest.) Note that y has the dimension of length, m that of mass, and t that of time. We now give the steps for converting equation 1.54 to dimensionless form: First, list all dimensional physical constants associated with the problem. From equation 1.54, we have

m, k, k1, a, A

(1.56)

Note that we have included the initial amplitude among the dimen­ sional constants associated with the system described by equation 1.54. Second, from the various dimensional physical constants given in equation 1.56, form other dimensional constants having the dimensions of time and length. These new dimensional constants will be used as time and length “scales.” Thus, from the constants of equation 1.56, we have

T1 = (ψ)1'2 L1 = A

T2 =

T3 = g1(a,k1,k2)

L2 = g2(a,k1,k2)

(1.57) (1.58)

where g1 and g2 are functions of a, kl, and k2. In the usual case, where the linear damping is small, we have Tl « T2. Third, using the time and length scales, form new dimensionless dependent and independent variables: (1.59)

If several length and/or time scales are available, then usually only one particular set of dimensionless variables will lead to a dimen­

1.3 Dimensionless Differential Equations

17

sionless differential equation where the nonlinear terms will be multi­ plied by a small parameter. This will be illustrated in the examples that will follow. Fourth, substitute the dimensionless dependent and independent variables into equation 1.54 and simplify. For most problems the natu­ ral time scale is given by T1. This corresponds, in the absence of damping and a nonlinear term, to essentially the period of the free har­ monic motion. Thus the dimensionless time variable can always be chosen to be

i = T1

(i.60)

To illustrate this procedure we apply the preceding rules to three ex­ amples.

1.3.1

Linear Damped Oscillator Our first example will be the linear damped oscillator:

d2y dy m~d^ + kidt + ky = 0

(L61)

with initial conditions

=0

y(0) = A

The following two time scales can be formed:

'r _ (m∖l'2 T1~\k)

t - HL τ*-k1

Note that T1 is the characteristic time for oscillations and T2 is the char­ acteristic damping time. The initial amplitude A will be taken as the length scale; that is, L1 = A Thus, we can form the following dimensionless variables:

y=⅞

' =⅛

If equation 1.65 is solved for y and t and substituted into equation 1.61, we obtain

¾÷ R⅞ + (~h = ⅛⅛ + 7k⅜ + ⅛v dt2 ∖m∕ dt ∖m∕ dt2 T2 dt 7χ

A∖d2y l T21) dt2 = 0

T1T2J di

Nonlinear Physical Systems

18

Simplifying, we get dp

(1.67)

∖Ti∕ di

If we let e = l∖∣T∙l, then equation 1.67 may be written as

dtλ

(1.68)

dt

In the case where the damping is small, then ∈ ≪ 1. Thus the small parameter in equation 1.68 is the ratio between the time associated with the free oscillations and the time associated with damping. 1.3.2

Nonlinear Oscillator Consider the following differential equation:

(1.69)

+ ky + kιy3 = °

m

with initial conditions y(0) = A

(1.70)

= 0

From the physical constants m, k, k1, and A, we can form the time scale = (τΓ

(1∙71)

and two length scales ∕ k \1/2 L1=y

L2 = A

(1.72)

Thus we have two sets of time and length scales: (T1, L1) and (T1, L2)∙ Consider the first set of scales T1 and L1. We may form the dimen­ sionless variables

V = i ' Ll

f=— Ti

(1.73)

Substituting these variables into equation 1.69, we obtain

® ⅛ ÷ (⅛) * ÷

r - θ

(1.74)

or (1.75)

Using the definition of Li from equation 1.72 and the definition of T1 from equation 1.71, the coefficient of y3 is

1.3 Dimensionless Differential Equations

-1--*71 = 1

19 (1.76)

m

Thus equation 1.75 becomes

^‰y + y3 = 0 df2

(1.77)

We conclude that as presently formulated, with scales 7,1 and L1, equa­ tion 1.69 cannot be put in dimensionless form with a small parameter multiplying the nonlinear term. Consider now the second set of scales T1 and L2, with associated di­ mensionless variables

(1.78)

With these variables, equation 1.69 becomes (1.79) or

(1.80) The coefficient of the y3 term is

m=(⅛)2

(1.81)

where we have used the fact that LI = k∕k1. Defining ∈ as

equation 1.80 becomes dt2

+ y + ey3 = 0

(1.83)

For the differential equation 1.83 the dimensionless parameter ∈ is the square of the ratio of the initial amplitude to the characteristic length Ll = {k∕kl)1'2 associated with equation 1.69. If the initial amplitude is small compared with the characteristic length L1, then e is also small.

1.3.3

Rayleigh Equation Consider the Rayleigh equation

(1.84) with initial conditions

Nonlinear Physical Systems

y(0) = A

dt

20 (1.85)

There are two length scales

τ

_ (am]112 1 “ ∖βk)

(1.86)

and two time scales (1.87)

The appropriate dimensionless variables for this problem are

In these variables the differential equation 1.84 becomes

⅛2>'- Jl - 1 ≡21⅛ + J = o dt2 L 3 ∖diJ J dt

(1.89)

where e = T1∕T2 ■ The associated dimensionless initial conditions are (1.90)

We have shown in this section how to transform equation 1.54 into a form in which all the variables and constants are dimensionless; that is, in dimensionless form, equation 1.54 may be written as follows (we have absorbed the linear damping term of equation 1.54 into the func­ tion F): ^+y + fp(a,y,^∖=0 dt2 ∖ dt∕

(1.91)

where all the barred quantities are dimensionless and e is a dimen­ sionless parameter that characterizes the “smallness” of the nonlinear term. In the rest of this book, unless it is stated to the contrary, it will be assumed that the given differential equation of interest has been put in dimensionless form.

1.4

Exact Solution for Period of a Pendulum In Section 1.2 we obtained the equation of motion of the free oscillations of a pendulum:

where θ is the angle of deflection, g is the acceleration due to gravity, and L is the length of the pendulum (see Figure 1.1).

1.4 Exact Solution for Period of Pendulum

21

If the pendulum is displaced by an angle θ from its position of equi­ librium, then the potential energy of the system, which is the work done against gravity to lift the mass of the pendulum a distance h, is given by V(θ} = mgh = mgL(↑ — cos θ)

(1.92)

The kinetic energy, in angular coordinates, is

(1.93)

Energy conservation gives ÷ V(θ) = t-0

T

(1.94)

where Eo is a constant representing the total energy. Substituting equa­ tions 1.92 and 1.93 into 1.94, we have (dfi∖2 ∖dt} + mgE = 0

(1100)

This is a linear equation with constant coefficients whose general solu­ tion is θ(t) = A cos (ωZ + φ)

(1.101)

where A and φ are constants and the period of the oscillation Po is given by

Nonlinear Physical Systems

22

(1.102)

Po

We see that the period is independent of the amplitude of oscillation for this case. Let us now determine the period of oscillation for large amplitudes. We write equation 1.99 in the form

d0= + ∕⅛ dt ~ ∖L

1/2

(cos θ - cos 0o)lβ

(1.103)

Solving for dt, we obtain dt = ±

L ∖1/2 dθ 2g ∕ (cos θ - cos θ0)1'2

(1.104)

The period of oscillation is four times the time taken by the pendulum to swing from θ = 0 to θ = θ0. Thus M1'2 fe° do P(0o) = 4 2g∕ Jo (cos θ - cos 0o)1'2

(1.105)

If we use the trigonometric identity

cos 0=1—2 sin2

(1.106)

then equation 1.105 may be written as L∖1'2 p___________ dθ___________ Λ0o) = 2 2g) Jo [sin2 (0o∕2) - sin2 (0∕2)]1'2

(1.107)

Introducing new variables k and φ, (1.108)

and (1.109) equation 1.107 becomes

'~ ■ ‘ ©" ΓiT→⅛.-7>≈

(1.110)

or P(0o) = 4

∕L∖ιz2

K(k)

(1.111)

where K(k) is called the complete elliptic integral of the first kind. Its values have been tabulated for various values of A .18 Equation 1.108 determines k in terms of the maximum angle of de­

1.5 Exact Solution of d2y∕dt2 + y + O'3 — 0

23

flection 0o. We may determine the period P(0o) from tables of the func­ tion K(k) and equation 1.111. We see that the period of a pendulum de­ pends on its amplitude. For example, for 0o = π∕3 or 60o, we have

P^) = l.O7Po

(1-112)

where Po is given by equation 1.102. Consider now the differential equation

where ∈ is a small parameter and∕(y) is a nonlinear function of y. Based on our experience with the pendulum, we expect the period of oscilla­ tion to depend on the initial amplitude A and e; that is, P =

2π ω(A,e)

(1.114)

For e = 0, equation 1.113 describes a linear oscillator with period 2π. Therefore, if ω(A,e) is an analytic function of e, near e = 0, then ω(A,∈) = 1 + eg(A,e)

(1.115)

In the chapter that follows, we shall consider a number of equations having the form of equation 1.113 and shall show that our guesses, given in equations 1.114 and 1.115, are, in fact, correct.

1.5

Exact Solution of d2y!dt2 + y + ey3 = 0 Consider the nonlinear differential equation19 ⅛+ y +

= o

(1.116)

with e > 0 and initial conditions

y(0) = A

(1.H7)

⅛o dt Defining √y v=~di

(1.118)

and using the fact that d2y _ dv _ ctydv _ dv dt2 dt dt dy v dy then equation 1.116 can be written as

(1.119)

Nonlinear Physical Systems

24

v~dy + y + ey3 = 0

(1.120)

Integration of equation 1.120 gives l>2

y2

y4

(1.121)

y + 2 + € 4 = constant

The value of the constant may be obtained by evaluating the left-hand side of equation 1.121 at t = 0. We obtain v2 + y2 + θ-) y4 = A2 +

A4

(1.122)

Solving equation 1.122 for v(y) gives

v2 = (A2 — y2) + (∣) (A4 - y4) = (A2 - y2) + (∣) (A2 - y2)(A2 + y2)

= (A2 - y2)

(1.123)

1 + (∣) (A2 + y2)

and

υ(y) = ±{(A2 - y2)[l + (√2)(A2 + y2)]}1'2

(1.124)

The graph of equation 1.122 is given in Figure 1.11. The closed

Figure 1.11. Graph of the equation υ2 + y2 + ey4∕2 = A2 + eA4∕2', y1,2 — — A and t>lj2 = ±A[1 + (e∕2)A2]1,2. The arrows indi­ cate the direction of “motion” along the path C.

1.5 Exact Solution of d2y∕dt2 + y + ey3 = 0

25

curve, C, is a so-called phase trajectory in the v-y “phase space" (see Section G.l of Appendix G and Section H.l of Appendix H). From equation 1.122 it is easy to see that C is symmetric with respect to both coordinate axes. A closed path in phase space corresponds to periodic motion. Consequently, equation 1.116, withe > 0, has aperiodic solu­ tion for all positive values of ∈. Equation 1.124 may be rewritten as ⅛ = ±{(Az - y≈)[l + (e∕2)(A≈ + yz)]}>'2

(1.125)

Solving for dt gives dt = +______________ ⅛______________ “ {(A2 - y2)[l + (e∕2)(A2 + y2)]}1'2

(1.126)

The time to go from the point (A,0) to the point (y,υ) in the lower half-plane (Figure 1.12) is ______________ dy______________ {(A2 - y2)[l + (e∕2)(A2 + y2)]}ιz2

(1.127)

Let y = Az; then equation 1.127 becomes dz {(1 - z2)[(2 + ∈A2)∕eA2] + z2}112

(1.128)

The integral on the right-hand side of equation 1.128 may be written in terms of an elliptic integral of the first kind, F(ψ',k)ι20,21

Nonlinear Physical Systems

t(y) =

F[arc cos (y∕A)∙,k] (1 + ∈A2)1'2

26

(1.129)

where ∈A2 2(1 + cA

(1.130)

Equation 1.129 can be solved for y; we obtain y(t,e) = ACN[t(l + eA2)ll2',k]

(1.131)

where CN is the Jacobi elliptic function.20,21 A moment’s reflection will show that the period of oscillation is given by p = 4' a > 0 1 > b> 0 4b2 > a2 AΛt = 0 the capacitor has a charge Qo. It is then put in series with a coil of inductance Lo. Obtain the differential equation for Q

+ ωgβ(l + aQ + bQ2) = 0

ωo = l c

and put it in dimensionless form. 1.16 The following nonlinear differential equation occurs in the problem of the rolling of a ship fitted with stabilizing equipment:

{where a, β, y, and k are positive constants.23 Obtain the corre­ sponding dimensionless form of this equation. 1.17 Consider the physical system shown in Figure 1.13. The mass m is restrained by four linear elastic springs, each of which has a stiffness constant k and a length, under no tension, of L. (a) Determine the non­ linear equation of motion for large displacements of the mass in the horizontal direction, (b) Obtain an approximate nonlinear equation of motion for large displacements, (c) What is the linear equation that describes the motion for small displacements?

Figure 1.13.

29

Problems d 2x m~dP +

= 0

(a)

d2x

2*' + (p) ', - 0

(b)

Λr

2kx = 0

(c)

1.18 Consider the differential equation (72v —— + y + by3 + cy5 = 0 a,b,c > 0 dt2 Show that for any initial conditions y(0) = A and dy{Qi)∕dt = Vo the solution is periodic. 1.19 The equation d2 u l l du , , —-r + 2k [ul — + au = d a,k > 0, d > 0 dti

dt

occurs in connection with a hydroelectric power system; u is the velocity of the water.24 Rewrite this equation in dimensionless form. 1.20 A particle moves in a rough horizontal straight groove under the ac­ tion of a spring attached to it and to a fixed point on the groove. The tension F in the spring when it is extended a distance y is T = ay + by3 a,b > 0 If the frictional force is proportional to the square of the velocity, show that the equation of motion is dy — + ay + by3 = 0 k dt Put this equation in dimensionless form. 1.21 The equation d2y _ ∕ λy ∖(dy∖2 + ay = 0 dt2 ∖1 + Ky2)∖dt) (1 + λy2) has an exact solution of the form y = A sin (ωt + θ) Determine the conditions on A and ω that will allow the preceding so­ lution to exist.25 1.22 Consider the motion of a simple pendulum, moving through a resistive medium, in the linear approximation. The equation of motion is

where 5 = Lθ, L is the length of the pendulum, θ is the angular dis­ placement, and m is the mass of the bob. The coefficient ki is equal to 6πR-η for a spherical bob of radius R moving through a medium of viscosity η. Consider a pendulum where the bob is a lead sphere of mass 1 kilo­ gram, the length L is 1 meter, the initial angular displacement is θ0 = 7r/10; let the medium be air at a temperature of 5oC. Solve the

Nonlinear Physical Systems

30

equation of motion for θ(t), and determine the time it takes for the am­ plitude of the motion to decrease to one-half its initial value. 1.23 In Section 1.22 we considered the problem of a mass attached to a stretched wire. Consider the case of small horizontal displacements, where the length of each part under no tension, a, is equal to d (see Figure 1.2). Under these conditions the equation of motion, given by equation 1.8, becomes

where we have neglected the remaining terms. Show that this equa­ tion may be solved exactly for x(t), and obtain this solution. What is the period of the oscillation?

REFERENCES 1. T. L. Saaty and J. Bram, Nonlinear Mathematics (McGraw-Hill, New York, 1964). 2. H. L. F. Helmholtz, Sensations of Tone (Longmans, Green, London, 1895), 3rd ed. See page 158 and Appendix XII. 3. E. M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1967). 4. W. A. Edson, Vacuum Tube Oscillators (Wiley, New York, 1953), pp. 408-412. 5. W. J. Cunningham, Introduction to Nonlinear Analysis (McGraw-Hill, New York, 1958). See example 5.3 in Section 5.8 and example 6.7 in Section 6.5. 6. Y. H. Ku, Analysis and Control of Nonlinear Systems (Ronald Press, New York, 1958). See Section 6.4. 7. B. van der Pol, Phil. Mag. 2, 978 (1926). 8. B. van der Pol, Phil. Mag. 3 , 65 (1927). 9. J. W. S. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. I, pp. 79-81. 10. J. W. S. Rayleigh, Phil. Mag. 15, 229 (1883). 11. J. E. Lennard-Jones, Proc. Roy. Soc. (London) Al06, 463 (1924). 12. G. R. Fowles, Analytical Mechanics (Holt, Rinehart & Winston, New York, 1962). 13. A. B. Migdal, Qualitative Methods in Quantum Theory (Benjamin, Reading, Mass., 1977), pp. 2-4. 14. E. de St. Q. Isaacson and M. de St. Q. Isaacson, Dimensional Methods in Engineering and Physics (Wiley, New York, 1975). 15. G. H. Duffey, Theoretical Physics (Houghton Mifflin, Boston, 1973). See Section 13.10. 16. L. Meirovitch, Elements of Vibration Analysis (McGraw-Hill, New York, 1975). See Section 1.7. 17. See reference 5, example 4.3, pp. 67-69. 18. See reference 14, Chapter 1. 19. B. O. Peirce, A Short Table of Integrals (Ginn, Boston, 1929). 20. F. Dinca and C. Teodosiu, Nonlinear and Random Vibrations (Academic, New York, 1973). See Section 9a. 21. P. F. Byrd and M. S. Friedman, Handbook of Elliptic Integrals for Engi­ neers and Physicists (Springer-Verlag, Berlin, 1954).

References

31

22. E. Jahnke and F. Emde, Tables of Func tions with Formulas and Curves (Dover, New York, 1945), pp. 41-106. 23. N. Minorsky, Proc. Natl. Acad. Sci. U.S. 31, 346(1945). 24. N. W. McLachlan, in Wave Motion and Vibration Theory, edited by A. E. Heins, pp. 49-61 (McGraw-Hill, New York, 1954). 25. P. M. Mathews and M. Lakshmanan, Quart. Appl. Math. 32, 215 (1974).

2 The Perturbation Method

Introduction Several concepts that may be new to the reader are introduced in this chapter (e.g., asymptotic expansion, uniform convergence, limit cycles). We urge the reader, at this time, to read all of Appendixes B and D and Section G.l of Appendix G. In this chapter we present the simplest and perhaps the most useful of all approximation methods: the expansion of a solution in a power series in a parameter. This tech­ nique is known as the perturbation method. This method will be ap­ plied to obtain periodic solutions to second-order nonlinear differential equations of the form 2.1

0

(2.1)

where ∈ is a small parameter and F is assumed to be an analytic non­ linear function of y and dy∕dt. We assume that a periodic solution to equation 2.1 may be written as a power series in e: y = -Vo(O + ey1(t) + e2y2(0 + ∙ ∙ ’ + ∈nyn(t) + ∙ ∙ ∙

(2.2)

where the coefficients of the powers of the parameter € are functions of the independent variable t. The justification for the form of solution given by equation 2.2 lies in a result first obtained by Poincare,1 who showed that if a differential equation contains terms with a parameter, the solution is an analytic function of this parameter. Thus, if e is suffi­ ciently small, the series given by equation 2.2 converges. The functions yn(t) are found by substituting equation 2.2 into equation 2.1 and equating the coefficients of like powers of e. This leads, in general, to an infinite set of linear inhomogeneous differential equations that may be solved recursively.

33

2.1 Introduction

As an example of these ideas, consider the following nonlinear dif­ ferential equation:

0

This case corresponds to linearly damped oscillations.

(2.129)

51

2.6 Existence of Periodic Solution

One easily finds for P(A) the equation P(A) = nCA = Q

(2.130)

Since C > 0, then A must be zero. Thus, for the case of linear damping, the only periodic solution is the state of equilibrium, A = 0. F a Function Only q/’dy/dt We assume F to be a function only of dy∕df, that is, F = f2{dy∕dt). Let u = - A sin t; then du = -A cos τ dτ and u = 0 when τ = 0or 2π. Putting this transformation in equation 2.119, we obtain 2.6.4

2πω1A=0

(2.131)

This means that for A ∕ 0, ω1 = 0. Thus equation 2.118 becomes Γ2π

P(A) =

f2(-A sin τ) sin τ dτ = 0 Jo and may be solved for the amplitude A.

(2.132)

Example 3 Let f ( 2), then the amplitude will monotonically decrease and approach, asymptotically in time, the value a = 2. Note that the sta­ tionary oscillatory state of the van der Pol equation does not depend on

Slowly Varying Amplitude and Phase Method

68

the initial conditions, but depends uniquely on the parameters of the system, which means that it is determined solely by the differential equation itself. In the (y,dy∕dt) phase plane, the previously isolated periodic solu­ tion of the van der Pol equation gives rise to a closed curve that is a circle of radius two. The term limit cycle is used to denote such an iso­ lated closed path in the phase space corresponding to a periodic solu­ tion.8

Discussion We have worked out a number of examples using both pertur­ bation (Section 2.4) and Krylov-Bogoliubov techniques. On comparing the two methods of obtaining solutions, we conclude that the Krylov-Bogoliubov method gives the frequency correct to order ∈ and the solution correct order e°; that is,

3.3.9

ω = 1 + ∈ω1(A) + O(e2) y = a(t) cos {[1 + eω1(A)]t + φ0} + O(e)

'

'

where φ0 is a constant and A = α(0).

Method of Krylov-Bogoliubov-Mitropolsky In the preceding section we considered the Krylov-Bogoliubov first approximation to the solution of differential equations having the form 3.4

⅛ + y = eF(y,⅛)

0 0 da

(3.174)

then the state of equilibrium (i.e., a = 0) is unstable. If

dK(G) < 0 da

(3.175)

the state of equilibrium is stable. Equation 3.174 represents the condition for self-excited oscillations; that is, if equation 3.174 holds, then a small perturbation away from equilibrium will cause the amplitude to increase to finite values with in­ crease of time. Note that the condition of self-excitation is not essential for the existence of stable stationary oscillations. All that is required is that the equation of stationariness, equation 3.164, have at least one nonzero root satisfying the condition of equation 3.174. If e is sufficiently small, then it is clear that in equation 3.162 we need only use the first approximation; that is,

= K(fl) = eΛ1(β)

(3.176)

The addition of terms of higher approximation does not modify the qualitative character of the solutions, but merely modifies their quanti­ tative nature slightly. In many cases one can determine the character of an oscillating process by considering the (a,K) plane. Stationary amplitudes are

Slowly Varying Amplitude and Phase Method

82

found at the points of intersection of the curve K(a) with the a axis. Stable amplitudes correspond to points where the curve intersects the a axis from the upper side, and unstable amplitudes correspond to points where the curve intersects the a axis from the lower side. In Fig­ ures 3.1 to 3.4 we illustrate these points. The directions of the changes in amplitude a are shown with arrows in these figures. Consider Figure 3.1, where K(a) is negative for a > 0. Here the mo­ tion is purely dissipative, and the amplitude decreases to zero with in­ crease of time. Figure 3.2 corresponds to the case of self-excitation with one stationary amplitude at a = a1. In Figure 3.3 we again have the case of self-excitation, with one stationary amplitude at a1 and an unstable stationary amplitude at a2 ∙ Let us discuss the case of Figure 3.4 in somewhat more detail. Note that the state of equilibrium (i.e., a = 0) is stable; al corresponds to an unstable stationary amplitude, and a2 is a stable stationary amplitude. If the initial amplitude α0 is less than a1, then with increase of time the amplitude decreases to zero. If a0 satisfies the condition a1 < a0 < a2, then it will increase until it reaches the value a2. If a0 > a2, it will de­ crease with time to the value a2. From general considerations explained in connection with Figures 3.1 to 3.4, the following theorems result: Theorem I. Assume that a system possesses stable limit cycles forming a sequence a1, a3, a5, . . . . Between each pair of consecutive stable limit cycles there is always one unstable limit cycle; these unstable cycles form another sequence a2, a4, a6, . . . .

Figure 3.1. The amplitude function K(a) for the case of a purely dissipative system.

83

3.6 Stationary Amplitudes and Stability

Theorem II. The limit cycle reached spontaneously by a system starting from rest is always the one that corresponds to the smallest root aι of the sequence. Theorem III. The stable limit cycles corresponding to larger roots a3, a5,a7, . . . , of the stable sequence can be reached only if the system is given a shock excitation carrying it beyond the corresponding un­ stable limit cycles a2, a4, .... We illustrate these principles with an example, the van der Pol equa­ tion. From equation 3.159 we have the following equation for the time variation of the amplitude:

⅛ = (τ)0-⅞)

(3.177)

The function K(a) is (3.178)

and has the following nonnegative roots: a1 = 0

a2 = 2

(3.179)

The derivative of K(a), evaluated at these points gives, dK(Q) _ e da 2

dK(2) _ _ da

Figure 3.2.

(3.180)

The amplitude function for a self-excited system.

Slowly Varying Amplitude and Phase Method

84

Thus we see that the oscillation is self-excited and the stationary ampli­ tude at a = a2 = 2 is stable. The function K(a) is plotted in Figure 3.5.

Equivalent Linearization We have shown that the equations of the first approximation give the same qualitative results as are obtained from the equations of higher-order approximations. Thus, in view of the general complexity of the calculations connected with the equations of higher-order approximations, it is usually sufficient, in applied problems, to limit discussion to the first approximation only. Krylov and Bogoliubov13 have developed a method, the method of equivalent linearization, in which a given nonlinear differential equa­ tion can be replaced by an equivalent linear differential equation with the property that the solutions of the two equations can be made to differ from each other by terms of the order of e2. This method and its generalization, often called the describing function method, have found important uses in the theory of modem control systems.14,15 In this section we give a brief discussion of the method of equivalent linearization. Consider the nonlinear differential equation

3.7

Figure 3.3. A self-excited system with a stable stationary ampli­ tude at a1 and an unstable stationary amplitude at a2.

3.7 Equivalent Linearization

85

We have shown that in the first approximation equation 3.181 has a so­ lution of the form (3.182)

y = a cos ψ where a and ψ must satisfy the following equations:

= —

J

F(a cos ψ,-a sin ψ) sin ψ dψ

(3.183)

= ωe(a)

in which

ω∣(α) = [1 = 1 —

J

f

F(λ cos ψ,-a sin ψ) cos ψ dψj

F(a cos Ψ,-α sin Ψ)

(3.184)

• cos ψ dψ + O(e2)

We now define two new functions of the amplitude, Ke(a) and λe(α), as follows: Ke(a) = 1 — (-~I f X^TCl/ Jo

λe(α) = (~~j J

F(a cos ψ,-a sin ψ) cos ψ dψ (3.185)

F(α cos Ψ,-« sin ψ) sin ψ dψ

Thus the equations of the first approximation may be written Figure 3.4. The amplitudes a = 0 and α2 are stable stationary am­ plitudes, whereas a 1 is an unstable stationary amplitude.

Slowly Varying Amplitude and Phase Method

da _ _ ^λe(a)^ a dt ~ 2

86

(3.186)

= ωe(a) = [K"e(α)]1'2

Let is now differentiate equation 3.182 twice and use the results of equation 3.186. We obtain = -aωe sin ψ - (y) a cos ψ

(3.187)

and

= - αω∣ cos ψ + keaωe sin ψ ÷

a cos ψ

÷⅛p⅛s.nψ÷⅛⅛)^c0sφ (3.188)

= -^→4√> + (⅛k⅛sin*

+⅛ (⅛)ay Equation 3.188 may be rewritten in the form

+ ⅛ (⅛) °y Figure 3.5. tion.

(3.189)

The amplitude function K(a) for the van der Pol equa­

3.7 Equivalent Linearization

87

From equations 3.184 and 3.185 we have

λp(fl) = O(e) dωe(a) = O(e) da

(3.190)

dλe = O(e) da Hence, all the terms on the right-hand side of equation 3.189 are of order e2, and equation 3.189 may be written

⅛ + λf(α)^+ Kc(a)y = O(e2)

(3.191)

Thus the first approximation to the solution of the nonlinear differential equation 3.181 satisfies also the linear differential equation 3.191 within terms of order e2. Note that this is precisely the accuracy with which the first approximation of Krylov and Bogoliubov determines the solu­ tion of equation 3.181. From this point of view, the first approximation to the solution of the nonlinear equation 3.181 and the solution to the linearized equation 3.191 are equivalent. We shall call λe(a) the equivalent coefficient of damping and Ke(a) the equivalent coefficient of elasticity. Comparing equations 3.181 and 3.191, we find that the latter may be obtained from the former by replacing the nonlinear term by the fol­ lowing linear term:

eF (y, ⅛)----- > - {[*.(«) - l]y + M) ⅛}

(3.192)

where Ke(a) and Ke(a) are defined, respectively, by equations 3.185. We illustrate the application of the method of equivalent lineariza­ tion with an example. The book by Minorsky16 gives a number of worked examples and should be consulted for further details. Consider the differential equation ⅜Γ + y + ey3 = 0

(3.193)

Here, F(y,dy∕dt) = -y3, and Ke(a) - 1 =

λe(α) = -

∫Jπ cos4 ψ Jψ = ½p J

∞s3 ψ sin ψ dψ = 0

The linearized equation corresponding to equation 3.193 is

(3.194)

Slowly Varying Amplitude and Phase Method

d2y , Λ , 3€«2\ n ~df + √ +~Γ) y = 0

88

(3.195)

with equivalent frequency given by ωβ(α) = [Ke(α)]1'2 = 1 + ~ + 0(e2)

(3.196)

Since the system is conservative (i.e., the function F depends only on y), the amplitude α(t) is a constant, and the frequency is a function of the amplitude. The solution of equation 3.195 is

y = a cos

(3.197)

where 0 is a constant. Comparing equation 3.197 with the solution of equation 3.193 given in Section 3.3.2, where the first approximation of Krylov and Bogoliubov was used, we see that they are in agreement.

3.8

Nonlinear Oscillations with Finite Damping Thus far in this chapter we have considered nonlinear differen­ tial equations of the form

g + > =

(3∙1W

where e is a small positive parameter. If this equation contains any damping terms, then they must be small, since they are multiplied by e. It is of interest to extend our approximation methods to the case where there is finite damping in addition to small nonlinear terms.4,5

3.8.1

Technique In this section we show how to extend the method of Krylov and Bogoliubov to obtain approximate solutions to the equation

W∙>∙4S We assume that 0 < y < 1, so that the system is underdamped in the linear approximation (i.e., when e = 0). We follow closely the method of Mendelson.5 For e = 0, equation 3.199 has the solution y = a0e~yt cos (ω0∕ + ψo)

(3.200)

where aQ and ψ0 are constants, and

ω0 = (1 - γ2),'2

(3.201)

3.8 Nonlinear Oscillations with Damping

89

We seek a solution of equation 3.199 that reduces to equations 3.200 and 3.201 in the limit as e→ 0, Following the methods of Krylov, Bogoliubov. and Mitropolsky, we look for a solution (3.202)

y = y(w,ψ)

such that y is periodic in ψ with period 2π and where

⅛=w

(3.203)

⅛ = ω If we substitute equations 3.202 and 3.203 into equation 3.199, we obtain

where the following relations have been used: d2a _ d (da∖ _ dξ~da _ .. dξ dt2 dt ∖dt) da dt * da d2ιb d ( dιb∖ dt2 ~ dt UJ

dω da _ da dt

dω da (3 205) { ’ f

dy_dyda,dyd^ dt ~ ∂a dt ∂ψ dt

d2y _ dza ∂y dtz dt2 ∂a

d2ψ ∂y dt2 ∂ψ

∕da∖2 ∂2y ∖dt ∕ ∂a2

(dψ∖2 = -G)t1>

(3.214)

Solving for Ajtn and Bjkυ, we obtain Λ(1) = _ Γ(* - l)ynn + 2ω0G{∣cυ k L 4ω% + (k - l)2γ2 β - (k - ∖)yG^' k L 4ω20 + {k - l)2γ2

(3.215)

Let us now determine y1. yNe write it as a Fourier series in ψ and ex­ pand the Fourier coefficients in a power series in the amplitude a , that is,

>,ι(α,ψ)

+ jζ [ynG0 cos nψ + zn(a) sin nψ]

=

n=2

yn(a) = ∑ A^alc

(3.216)

k=0

zn(a) = ∑ B^ak k=0

Note that the Fourier series for y1(α,ψ) does not contain cos ψ and sin ψ terms. If we substitute equations 3.216 into equation 3.209 and use equations 3.210 and 3.211, then the following equations are ob­ tained: [(1 - n2)ω2 0 + (k- l)2γ2]Afc0 - 2n(k - l)yω * [(1 - n2)ω02 + (k- l)2y2]2 + 4n2(fc - l)2γ2ωg

This procedure can be easily extended to calculate higher-order terms. The first approximation is

Slowly Varying Amplitude and Phase Method

92

y = y(fl>Ψ) = a cos ψ j7

= ~ya + eξ1(a)

(3.219)

= ω0 + eω1(α)

where ξ1(a) and ω1(α) are given by equations 3.213 and 3.215. Note that ξ1(fl) and ω1(α) are determined completely by the Fourier coeffi­ cients of the nonlinear function F(a cos ψ,-ω0a sin ψ — ya cos ψ). An Example As an example of this procedure, we consider the following equation:

3.8.2

(3.220)

The function F is (3.221)

with Fourier coefficients F^ = -f F(33) = -⅛

(3.222)

All other coefficients vanish. We easily obtain

(3.223)

_ 3ω0 β3 ~ 8 d(d

Substituting the results of equation 3.223 into equations 3.213 and 3.219 gives

(3.224)

The first of equations 3.224 has the solution oy2 + J2 = -2D0y1 - 2D1y0 ~ 2D0D1yl - D[y0 - 2D0D2y0

(4.128)

The solution to equation 4.126 is yo(^o√ι√2) ~ Aq(jl,t2) cos rθ + Bo(ti't2) sift fo Substituting equation 4.129 into equation 4.127, we obtain

(4.129)

113

4.5 Worked Examples

Z>oJι + J1 = 2 (~f^Γδ + Λ>) sin t0 - 2 (j⅛ + Bo) cos t0

(4.130)

Elimination of secular terms in the solution for y1 requires

Λ4fi dt1

Ao = 0 (4.131)

and gives

(4.132)

+ Ji = 0 The solutions to equations 4.131 and 4.132 are A9(t1,t2) = a0(t2)e~tl ~ β0(t2)e tl

(4.133)

and 3,ι(to,,t2)

A1(t1,t2) cos rθ ÷ $2(^15⅞) sin rθ

(4.134)

The result given in equations 4.133 allows us to write the solution yo(W1√2) as

Jo(⅞√ι,⅛) —

6 cos ∕0 + β0(t2)e t' sin t0

(4.135)

Substituting equations 4.134 and 4.135 into equation 4.128 gives £>o?2 + y2 = (-2

- 2B1 + α0e~'1 - 2

e ,1) cos t0

(4.136) + (2

+ 2A1 + ^°e~tl + 2

β^*1) δin t°

Elimination of secular terms for the solution y2(t0,t1,t2) gives the fol­ lowing equations:

(4.137)

Likewise, yι(t0,ti,ti) will contain secular terms unless the right-hand sides of equations 4.137 are set equal to zero; that is,

dβ0 _ o⅛ ^2

2

(4.138)

da0 = _ βo dt2 2

Equations 4.138 have the solutions

Multi-Time Expansions

114

β0(t2) = C1 sin (⅛ + φj (4.139)

α0('2) = C1 cos

÷ φj

where C1 and φ are constants. If equations 4.139 are substituted into equation 4.135, and use is made of the fact that cos (x — y) = cos x cos y + sin x sin y

(4.140)

we obtain

(4.141)

y0(WιΛ) = C1e tl cos

Making use of the initial conditions, given in equation 4.123, and the definition of the tn, given in equation 4.118, we find that the first approximation to the solution of equation 4.124 is

y(t,e) = Ae et cos (1 - y½ + O(e) 4.5.2

(4.142)

Example B For our second example we consider the van der Pol equation d 2y . Λ^ +

∕ι

2λ dy

(4.143)

-y = t" ~'⅛

To terms of order e2, we have

Dly0 + y0 = θ

(4.144)

T>o)>ι + yi = -2Po^o + (1 - >,o)A>,o

The solution to the first of equations 4.144 is Jo(Wi) = A0(r1) cos r0 + B0(r1) sin t0

(4.145)

Substituting y0(∕0√ι) into the second of equations 4.144 gives

°°y' + * = {2⅛^ “

[1 ^ (^4 gj)]} : (4.146)

cos r0 + terms that do not produce secular terms Elimination of secular terms gives the following equations for Ao(∕1) and B0(∕1)ι

(4.146a)

4.5 Worked Examples

115

The initial conditions give

Ao(O) = A

Bo(O) = 0

(4.147)

Evaluating the second of equations 4.146a at t1 = 0 gives

dBo(O) dt1

(4.148)

The solution to equation 4.148, with the initial condition of equation 4.147, is B0(r1) ≡ 0

(4.149)

Using this result, the first of equations 4.146a becomes (4.150)

Integrating equation 4.150, we obtain A0(r1)

2A μ2 + (4 _ A2)e-+(-)]

(e)

d2y , 1 the forced oscillation is 180o out of phase with the external force. The oscillation is a superposition of two simple harmonic motions, one with frequency of value 1 corresponding to the free oscillations of the system, the other with the frequency of the external force. In the case of resonance, where ω = 1, the motion due to external force is not periodic; however, it is oscillatory, with the amplitude increasing linearly with time.

Linear Systems with Damping Consider now the equation of motion for a linear system with damping acted on by an external harmonic force,

5.2.2

(5.11)

5.2 Forced Oscillations of Linear Systems

121

where we assume that e is small and positive. The solution of equation 5.11 may be written as the sum of the solution to the corresponding homogeneous equation and any solution of the inhomogeneous equa­ tion. The solution of the homogeneous equation is vh

= Cie~tt cos [(1 - e2)ιz2r + φ]

(5.12)

where C1 and φ are constants. For the inhomogeneous equation we assume a solution of the form

yl = D1 cos ωt + Di sin ωt

where Z>1 and Z>2 are constants. These constants may be determined by substituting equation 5.13 into equation 5.11 and setting the coeffi­ cients of cos ωt and sin ωt equal to zero. We obtain (1 - ω2)D1 + 2∈ωD2 = F - 2eωD1 + (1 - ω2)Z>2 = 0 The equations 5.14 may be easily solved, giving

1

(1 - ω2)F (1 - ω2)2 + 4e2ω2

d =______ 2eωj7 2 (1 - ω2)2 + 4∈2ω2

Thus the inhomogeneous solution can be written in the following form: yι~

{1 — ω2)F cos ωt + 2eωF sin ωt (1 - ω2)2 + 4e2ω2

(5.16)

Defining the “phase shift’’ δ as χ_________ 1 — ω2_______ COS 6 [(1 - ω2)2 + 4c2ω2]ιz2 (5.17)

c, 2eω sin δ = [fl - ω2)2 + 4e2ω2]ιz2

and using the relation cos (a — b) = cos a cos b + sin a sin b

(5.18)

equation 5.16 becomes

Feos (ωt - δ) yι ^ [(1 - ω2)2 + 4e2ω2]ιz2

(5.19)

Consequently, the general solution of equation 5.11 is y = Cle~et cos [(1 - e2)ιz2r + ι and 2ω2 would appear. The occurrence of combination tones may lead to certain undesired effects in acoustic instruments. Since the characteristic of a loud­ speaker is nonlinear, two periodic electromagnetic forces with fre­ quencies ω1 and ω2 acting on a loudspeaker will excite forced oscilla­ tions having not only the frequencies ω1 and ω2 but also combination frequencies. Consequently, if the intensity of the primary frequencies is high enough, the intensity of the combination tones will be above the threshold of audibility. This may lead to serious, unpleasant distor­ tions. Helmholtz, in his study of the sensations of hearing, used the nonlin­

Forced Oscillations

126

earity of the human hearing system to explain the fact that tones of fre­ quency ω1 ± ω2 are often heard when two notes of frequency ω1 and ω2 are sounded.8 The book by Rayleigh gives a good summary and critical discussion of various nonlinear acoustic phenomena.9 We can calculate to higher orders in ∈ by assuming a solution of the form y = A1 cos ω1∕ + A2 cos ω2t + eu1(t) + e2u2(t) + ∙ ∙ ∙

(5.31)

and substituting it into equation 5.25. Setting the coefficients of the various powers of e equal to zero, we obtain a set of equations that may be solved in succession. If ω1∕ω2 is rational, then the excitation force F1 cos ω1t + F2 cos ω2t is periodic, and the methods to be given in Sections 5.5 and 5.6 may be used to obtain periodic solutions of equa­ tion 5.25. In addition, it may be shown that for e sufficiently small, the series converge.10 If, however, ω1∕ω2 is irrational, then the excitation force is an almost periodic function of the time, and application of the preceding methods leads to an interesting result: Determination of solutions of equation 5.25 by either the iteration or perturbation method leads to series that diverge.11 This result may be understood by considering the nature of the terms obtained by calculating to higher orders in ∈. The higherorder expressions will contain more and more terms whose denomina­ tors contain higher and higher powers of nω1 ± mω2, with n and m integers. However, a theorem of Kronecker12 tells us that expressions of the form nω1 ± mω2, with n and m integer, will be arbitrarily close to zero for infinitely many different n and m if ω1∕ω2 is irrational. Thus there will be higher-order terms that have coefficients of arbi­ trarily large magnitude. This problem was first pointed out by Poincare and is known as the “difficulty of small divisors.” Since, for the systems of interest, ∈ is assumed to be small, we need only calculate to low orders of e. Thus the “difficulty of small divisors” does not arise.

Subharmonic Oscillations In Section 5.2 we saw that a periodic external force with fre­ quency ω acting on a linear oscillator may excite periodic oscillations of frequency ω∕q, where q is an integer greater than 1.13-15 These oscil­ lations are called subharmonic. We shall show that similar results ob­ tain for nonlinear oscillators. 5.4

5.4 Subharmonic Oscillations

127

Classification of Subharmonic Oscillations We begin our discussion of subharmonic oscillations by stating the classification of such motions given by Rosenberg.16 Assume that the subharmonic oscillation can be represented by the following equation:

5.4.1

,y = A, cos (^ I + φ.) + ∑ Ak cos (f t + φt)

(5.32)

where An and φn are constants. If ∣AJ » ∣Afc∣ for all k ≠ q, then we have a strong subharmonic of order ↑∕q. If Aq ≠ 0 and Ak = 0 for all k ∕ q, then the oscillation given by equation 5.32 is called a pure sub­ harmonic of order 1/q; if, in addition, φq = 0, then it is said to be a simple subharmonic of order ∖∕q. Subharmonic Oscillations and Chebyshev Polynomials Let us now consider the conditions that must be satisfied by the nonlinear differential equation

5.4.2

+ g(y) = F COS ωt

(5.33)

in order that this equation will have a simple subharmonic solution of the form

y = A cos

This problem has been solved by Rosenberg16 and Kauderer.17 We follow the procedure of Kauderer,17 who used the so-called Che­ byshev polynomials to solve this problem. These polynomials are de­ fined by the relations18 Tq(x) = cos {q arc cos x)

q = 0,1,2, . . .

(5.35)

Equivalent definitions are Tσ(cos a) = cos qa

(5.36)

Tq(x) = ⅜{[x + z(l - x2)1'2p + [λ - z(l - x2)1'2]9}

(5.37)

and

We may calculate the higher-order Chebyshev polynomials by using T0(x) = 1 and T1(x) = x and the recursion relation

Tn+1(x) - 2xTn(x) + Tn_i(x) = 0

(5.38)

For equation 5.34 to be a solution of equation 5.33, it is necessary that

128

Forced Oscillations

y + s(y)= rcos ωt

-

(5.39)

However, from equation 5.36 we have cos ωt = cos q

= Tq cos

= Tq

(5.40)

consequently,

y + FT, (⅛)

g(y) =

(5.41)

Thus we conclude that the equation

⅛+(^)y + F7∙,(f)=fcosωr

(5.42)

has a simple subharmonic solution, given by equation 5.34, for any integer q greater than 1.

Two Examples To illustrate the use of equation 5.42, we consider two ex­ amples. The first example concerns the conditions under which the Duffing equation

5.4.3

d2y 1 , o 7τ + y + ey3 = F cos ωt

(5.43)

has the simple subharmonic solution (5.44)

y = A COS

From equation 5.42 we have, for q = 3, the result

(5.45) Comparing equations 5.43 and 5.45, we obtain

ω2 _3F 9 A

1

4F = eA3

(5.46)

Thus, in order that equation 5.43 have a simple subharmonic solution of order 1/3, given by equation 5.44, A and ω must, for given F and e, satisfy the following relations:

Λ - (5 = F cos ωl

(5.50)

Comparing equations 5.48 and 5.50, we obtain the following condi­ tions that must be satisfied if equation 5.48 is to have the simple sub­ harmonic solution of equation 5.49:

20F A3

a

(5.51)

For given α, β, and F, the amplitude A and the frequency ω are given by the following expressions:

—[∙-(ix-≡nι For A and ω to be real, we have the additional requirement aβ < 0. In Sections 5.6 and 5.7 we shall again consider subharmonic oscilla­ tions using the perturbation method. The references13-17 give addi­ tional worked-out examples and other techniques for obtaining subhar­ monic oscillations of nonlinear differential equations. Iteration Methods for Harmonic Oscillations without Damping In this section we consider the harmonic solutions of the Duffing equation

5.5

d 2y -j-ζ + y + ey3 = F cos ωt

(5.53)

Forced Oscillations

130

Harmonic solutions are those periodic solutions that have the same fre­ quency as the external exciting force F cos ωt.

5.5.1

Duffing Technique Our first iteration method is based on a technique devised by Duffing.619 The first step is to write equation 5.53 in the form (5.54)

—rf∙ = -y - ey3 + F cos ωt

This iteration method consists of assuming that the nth approximation to the harmonic periodic solution of equation 5.53 is given by = -yn-1 - eyl-1 + F cos ωt

(5.55)

where we start with (5.56)

y0 = A cos ωt

In addition, we require that yn(t) be periodic with period 2π∕ω. If we substitute equation 5.56 into equation 5.55, we obtain the equa­ tion for the first approximation:

A+~eA3 - F∖

cos

ωt + ∣ eA3 cos 3ωt

(5.57)

Integrating equation 5.57 twice gives

yι = (⅛)(a +1eA3 ~ f) cos ωr + ⅛ (¾^) cos 3ω*

(5.58) (5.58)

The integration constants have been set equal to zero in order that y1(r) and the next approximation y2(t) be periodic. Note that in order to have this iteration procedure converge we must require that ∈, A, and F be sufficiently small. Thus any results we obtain are subject to the con­ dition that the differential equation 5.53 be weakly nonlinear with a small (amplitude) external exciting force. Our next step is to set the coefficient A1 of the cos ωt term in equa­ tion 5.58 equal to A; that is, (5.59) We make the following justification for this procedure: If y0(t) = A cos ωt is a good starting approximation, then A1 should differ very little from A. Note, also, that this procedure gives the exact result in the linear case (i.e., e = 0) and thus should be expected to give good results for ∈ small (compare with equation 5.9, with appropriate change of notation). This line of reasoning leads to the following relation

5.5 Iteration Methods for Nondamped Case

131

between the frequency of the exciting force and the amplitude A of the first Fourier coefficient of the response oscillation: ω2 = 1 + 7 ∈A2 - £ 4 A

(5.60)

A Second Iteration Technique Our second iteration procedure starts with the differential equation

5.5.2

t∕2v + ω2y _ (ω2 _ l)∙y _ ey3 _|_ e∕τθ cos ωt

(5.61)

which may be obtained from equation 5.54 by adding the term ω2y to both sides and replacing F by eF0 ∙ Since the amplitude F of the excit­ ing external force is of order e, this means that our method gives the development in the neighborhood of the linear free problem. The nth approximation to the harmonic periodic solution of equation 5.61 is to be given by

~fj~Γ + ω2yn = (ω2 - l)yn-1 - ey⅛-1 + ∈F0 cos ωt

(5.62)

where our starting solution is

y0 = A cos ωt

(5.63)

We require that yn(t) be periodic with period 27r∕ω. Since equation 5.61 is invariant under the transformation t → — ∕, we conclude that y(r) is an even function of t and thus contains only the terms Am cos mωt, where m is an integer. Equation 5.61 is also in­ variant under the transformations

t

y

-y

and consequently y(r) contains only terms Am cos mat, with m an odd integer. If we substitute equation 5.63 into equation 5.62, we find the follow­ ing differential equation fory1(r)ι d2y1 + ω2yι = dt2

(- 1 + ω2)A - ∣ eA3 + eF0 cos ωt

4

cos 3ωt

The solution y1(f) will contain a secular term unless the coefficient of cos ωt is zero; that is,

Forced Oscillations

132

(- 1 + ω2)A - ∣ ∈A3 + ∈F0 = 0

(5.66)

or

(5.67)

Thus the solution of equation 5.65 can be written as

y1(t) = A1 cos ωt +

cos 3ω∕

(5.68)

At this point the value of A1 has not been determined. Following the previous procedure of Duffing, we choose for A1 the value A of the am­ plitude of the starting solution given by equation 5.63. Consequently, the first approximation is

(5.69)

In general, the nth approximation to the solution of equation 5.61 will obey a differential equation of the form

d2yn

+ ω2yn = Pn cos ωt ÷ Qn cos 3ωt + Rn cos 5ωt + ∙ ∙ ∙

(5.70)

where Pn, Qn, Rn, and so forth are functions of A and ω. Elimination of secular terms in the solution for yn requires that Pn = 0. This gives an improved relation between ω and A. Finally, once Pn has been set equal to zero, equation 5.70 may be solved with the coefficient of the cos ωt term taken to fie A. Note that the solution to the homogeneous equation gives us a term An cos ωt. The significance and interpretation of the relations given in equa­ tions 5.60 and 5.67 will be discussed in detail in Section 5.8.

5.6

Perturbation Theory Applied to Forced Oscillations In this section we consider the application of the perturbation method to obtaining periodic solutions of the differential equation (5.71)

where e is a small parameter and, in general,/is a nonlinear function of its arguments. Proofs and justifications of the perturbation method, ap­ plied to forced nonlinear oscillators, are given in the references.20 We seek a periodic solution y(r) that has the same frequency as F cos ωt. We may avoid working with functions of unknown period by introducing a new independent variable θ'.

5.6 And Perturbation Theory

133

(5.72)

0 = ωt

Thus equation 5.71 becomes

ω≈⅛ + y + √(y.ω⅛) = Fcos(i

(5.73)

In the variable 0, the solution y(0) has period 2π. The perturbation method consists in developing the desired solution y(t) in a power series with respect to e where the coefficients are func­ tions of θ∖ that is,

y{θ,ε) = y9{θ) + ∈y1(0) + ∈2y2(0) + ∙ ∙ ∙

(5.74)

For e = 0, equation 5.73 becomes

+ ?o = Feos 0

(5.75)

There are two cases to consider: ω 1 and ω ≈ 1. The second case corresponds to a resonance situation. We consider these two cases separately.

Nonresonance Case For the nonresonance case, ω ≠ 1, the procedure for obtaining periodic solutions of equation 5.73 is as follows: Equation 5.74 is sub­ stituted into equation 5.73, and the coefficients of the various powers of ∈ are set equal to zero. This leads to a system of recurrent linear dif­ ferential equations. At the (n+l)th step we have 5.6.1

ω2-^ + yn = Fn(0)

(5.76)

where Fπ(0) is a periodic function of 0 with period 2π that depends on the functions yo(0), y1(0), . . . , yn-ι (0) already determined at the ear­ lier stages. Since ω is not equal to 1, each of the equations 5.76 has ex­ actly one periodic solution of period 2ττ. The first term of the expansion given by equation 5.74 is called the generating solution and is, for ω ≠ 1, given by

yo(0) = (γ⅛) ∞ »

(5.77)

In the nonresonance case, only one periodic generating solution exists, and it is given by equation 5.77. Resonance Case In the resonance case, where ω is equal to or close to 1, the preceding procedure does not work, since equation 5.75 has no peri­

5.6.2

Forced Oscillations

134

odic solution of period 2ττ. In fact, for ω = 1, y0(θ) contains a secular term of the form (FΘ∕2) sin θ. To overcome this problem, we assume that ω has the following expansion in powers of e:

ω = 1 + ∈cι>ι + e2ω2 ÷ ∙ ∙ ∙

(5.78)

where the ωj are to be regarded as given; in addition, we assume that the coefficient of the external exciting force vanishes together with e; that is, F = ∈F0

(5.79)

With these assumptions, the equation for the generating solution is

(5.80)

⅛ + y, = 0 This equation may be easily solved to yield

(5.81)

y0(θ) = Ao cθs θ + Bo sin θ

Note that unlike equation 5.75 in the nonresonance case, equation 5.80 has not only one periodic solution with period 2π but infinitely many such solutions given by equation 5.81. Therefore the procedure for obtaining a periodic solution of

ω2

+ y + √ (j,ω

= ∈F0 cos θ

(5.82)

in the case where ω is close to 1 is to substitute equations 5.74 and 5.78 into equation 5.82, where yo(0) is given by equation 5.81. The con­ stants Ao and Bo are to be determined by requiring that y1(0) be peri­ odic with period 2π. This last requirement may be written as + Fo cos θ - 2ω1 -⅛]

sin θ dθ = 0

(5.83) + Fo cos θ - 2ω1

cos θ dθ = 0

where the equation fory1(0) is

⅛÷^ = -∕(^⅛) + Focos0-2ω1⅛

(5.84)

Thus the constants Ao and Bo are obtained by setting to zero the coeffi­ cients of the cos θ and sin θ terms in the Fourier expansion of the right-hand side of equation 5.84. As noted previously, in the nonresonance case only one periodic generating solution exists, and the periodic solution of the nonlinear equation may be found in the neighborhood of this generating solution. However, in the resonance case, we have a different situation. There

5.7 Worked Examples

135

are infinitely many periodic generating solutions (equation 5.81), but only one of them, the one satisfying equation 5.83, is in the neighbor­ hood of the periodic solution of the nonlinear equation 5.82. The solution of equation 5.84 is of the form y1(θ) = Aι cos θ + B1 sin θ + ψ1(θ)

(5.85)

where ψ1(0) is a particular solution of equation 5.84. The constants A1 and Bi are then determined by setting to zero the coefficients of the cos θ and sin θ terms on the right-hand side of the differential equation for y2(θ). The same procedure is applied for the higher-order terms.

5.7

Worked Examples Using the Perturbation Method In this section we apply the perturbation method given in Sec­ tion 5.6 to obtaining solutions for three forced oscillation problems. Since the interesting phenomena for forced oscillations show up for resonance situations, we consider only this case.

Forced Linear Damped Oscillator For our first example we apply the perturbation method to the linear oscillator with weak viscous damping:21

5.7.1

⅛ + 2h ^dt + y = F c°s ωt

(5.86)

The general exact solution of this equation, for h < 1, is

y = A0e~ht cos [(1 — h2)ll2t + ]

(1 — ω2)F cos ωt 2hωF sin ωt + (1 - ω2)2 + 4∕ι2ω2 + (1 - ω2)2 + 4∕ι2ω2

(5.87)

where Ao and φ are constants. The only periodic solution of this equa­ tion is

(1 — ω2)F cos ωt 2hωF sin ωt y = (1 - ω2)2 + 4∕ι2ω2 + (1 - ω2)2 + 4Λ2ω2

(5.88)

If we make the transformation θ = ωt, in equation 5.86, we obtain

0. (In Chapter IV, Section 2, of Stoker,2 a graphic technique is given for constructing the response curves.) In both cases the portion of the response curve to the left of the skeleton curve corresponds to the forced oscillation being in phase with the exciting force, whereas that portion to the right of the skeleton curve corresponds to the forced oscillation being 180o out of phase with the exciting force. One sees that the response and skeleton curves in the nonlinear case may be thought of as arising from those for the linear case by bending the latter to the right for e > 0 and to the left fore < 0. Further consideration of Figures 5.2B and 5.2C gives us insight into why the amplitude A should be given in advance, whereas the fre­ quency ω is to be determined as a function of A. We see that for certain values of ω there are three corresponding values of A; thus, neither the iteration nor perturbation method can give all the branches of the response curves. (Note that equation 5.162 is a cubic equation in A.) In addition, we began our iteration or perturbation procedures with the

146

Forced Oscillations

free linear oscillation (see equations 5.63 and 5.139), and since in the linear problem A is arbitrary, its value must be assumed at the start of our calculations. Note also that unlike the case of the linear oscillator, the amplitude A of the nonlinear response is finite for any finite values of F and ω, even if ω = 1 (see Figures 5.2B and 5.2C). It may be shown that the stable oscillations correspond, on the left of Figure 5.2. A typical response curve for the Duffing equation without damping. Cs is the skeleton curve, and δ is the relative phase between the forced oscillation and the exciting force.

e>0

e< 0

147

5.8 Duffing Equation

the curve C,, to those portions of the response curve on which A is an increasing function of ω2 and, on the right of the curve Cs, to those por­ tions of the response curve on which A is a decreasing function of ω2.27 The transition from stable to unstable periodic oscillations or unstable to stable oscillations occurs at those points on the response curves where the tangent is infinite. Let

H(ω2,A) = -ω2 + 1 +

4

+ £ A

(5.164)

Thus dA _ dω2 ~

∂H∕∂ω2 1 ∂H∕∂A ~ 3eA∕2 ± F∕A2

(5.165)

and the points where the response curves have a vertical tangent are given by (5.166) Eliminating ±F∕Aλ between equations 5.162 and 5.166 gives the equa­ tion of the locus of the points where the response curves have a vertical tangent: ω2=ι+Z^L

(5467)

The graph of this function is represented in Figures 5.3A and 5.3B by the dot-dash lines and is denoted by Cv. It is easy to show that the response curves given by equation 5.162 attain no local maximum value for the amplitude A for finite ω2. Thus at a local maximum of the response curve, we have √A = 1 _ A2 dω2 3eA∕2 ± F∕A2 3eAi∕2 ± F

(5.168)

However, equation 5.168 is satisfied only if A = 0 or ∞, and neither of these values of A corresponds to a local maximum. Let us now examine how A varies with ω2 by using the response curve given in Figure 5.4. The dash line, Cs, is the skeleton curve, and the dot-dash line, Cυ, is the locus of the points where the various response curves (obtained by varying F) have a vertical tangent. Ac­ cording to arguments given earlier, the point D, where Cv intersects the response curve, separates the stable oscillations from the unstable os­ cillations. Thus the stable branch of the response curve corresponds to that portion to the left of Cs, labeled ABC. Also, that portion of the response curve labeled DE is stable. The unstable oscillations are given by the curve labeled DF.

Figure 5.3. Plots of the response curves for two values of the forcing amplitude F and the skeleton curve Cs for the Duffing equa­ tion without damping. Cυ is the locus of points where the response curves have a vertical tangent.

e< 0

5.8 Duffing Equation

149

Let us start with a value for the frequency that is small. In Figure 5.4 this means that we are at the point A on the upper portion of the response curve. As the frequency ω is increased, the amplitude A in­ creases continuously along the stable branch ABC of the response curve. If for some large value of ω we are at the point C on the upper portion of the response curve, then a decrease in ω will cause a corre­ sponding decrease in the amplitude A. In summary, if we start out on the stable portion of the response curve, we remain there; an increase or decrease in the frequency ω only moves us up or down the curve ABC. Along this curve the harmonic response oscillation and the forc­ ing terms are in phase. Consider now the case where we start at point E on the lower portion of the response curve. At such a point the amplitude of the harmonic response and the forcing term are 180o out of phase. If we increase ω, the amplitude decreases. However, if we decrease ω, then the ampli­ tude increases until it reaches point D, where the response curve has a vertical tangent. At point D the stable zone ends, and the amplitude A jumps up to point B, with the phase between the amplitude of the harFigure 5.4. A typical response curve for the Duffing equation without damping. ABC and EDF define the upper and lower branches of the response curve. The point D is where the response curve has a vertical tangent. All points on ABC and all points on the segment DE correspond to stable oscillations. The points on the segment DF correspond to unstable oscillations.

Forced Oscillations

150

monic response and the forcing term changing discontinuously from 180o to zero. If the frequency is further decreased, the periodic oscilla­ tion remains in phase with the external force, and the amplitude de­ creases continuously along the curve BA. However, if at point B the frequency is increased, the amplitude increases continuously along BC. In any case, the future motion will be confined along the upper portion of the response curve ABC. The behavior just discussed is often called the jump phenomenon, and it is one of the interesting char­ acteristics associated with nonlinear oscillations. Note that for ω > ω1, where ω1 is the position of the vertical tangent of the response curve, for each value of ω there are three values of the amplitude A. The intermediate value of the amplitude corresponds to the unstable oscillation (Figure 5.4).

Duffing Equation: Damping We now consider the case when the Duffing equation contains a linear damping term (i.e., equation 5.131). The response curves may be easily obtained for fixed F = cF0, from equations 5.153, by elimi­ nating the phase δ. Doing this, we obtain,

5.8.2

(1 - ω2)A +

3e43^∣2 + 4e2*2A2 = F2 4

(5.169)

A careful study of equation 5.169 shows that the corresponding response curves differ only slightly from those of equation 5.162. The

Figure 5.5. damping.

A typical response curve for the Duffing equation with

1

151

5.8 Duffing Equation

major difference is that the response curves given by equation 5.169 are rounded off in the vicinity of the skeleton curve, Cs, given by set­ ting F = 0 in equation 5.169. This result is to be expected, since when e is sufficiently small, equation 5.169 represents curves only slightly dif­ ferent from the response curves of the linear damped oscillator (see Section 5.2 and Figure 5.1). A typical response curve for e > 0 is illustrated in Figure 5.5. For e < 0, the curves are bent to the right. Again, we see that these curves may be considered as arising from the response curves of the linear damped oscillator by bending the latter to the right for e > 0 and to the left for e < 0. In Figure 5.5 we have indicated the locus of the vertical tangents of the response curves by the dot-dash line, denoted by Cυ. The interior of the region bounded by Cυ is where the response curves turn over on themselves. Note that there exists a value of F = F1 such that for F < F1 the response curves do not turn over on themselves. We now discuss the jump phenomenon associated with the Duffing equation with damping. F = eF0 is considered to be fixed while we vary the frequency ω (Figure 5.6). As ω is increased from zero, the response amplitude increases along the curve ABCD. At the inter­ secting point D of the resonance curve with the skeleton curve Cs, the amplitude A takes its maximum value, and then decreases a little up to Figure 5.6. The portions ABCDE and FGH of the response curve correspond to amplitudes of stable oscillations. The segment FE corresponds to amplitudes of unstable oscillations. The points E and F are points where the response curve has vertical tangents.

1

Forced Oscillations

152

point E, at which the response curve has a vertical tangent. A further increase in ω causes a jump in the amplitude to point G and a decrease of A along the curve GH. Let us now start at point H, corresponding to a large value of ω. As ω decreases, the amplitude A increases until point F (where the response curve has a vertical tangent) is reached. The amplitude then jumps to point C, where if ω is further decreased, the amplitude A decreases along CBA. In Figure 5.7 we give the behavior of the phase δ as a function of ω2. The points marked by letters correspond to the same points in Figure 5.6. For a good discussion of this topic see Dinca and Teodosiu.28 The jump mentioned earlier does not take place instantaneously, but requires a few cycles of oscillation to establish a steady-state oscilla­ tion at the new value of the amplitude. Thus, when ω approaches a value ω1 where the response curve has a vertical tangent, the response amplitude increases or decreases at a finite rate, and transitory oscilla­ tions occur until the new steady state is attained. The exact character of the transitory oscillations depends, in detail, on how rapidly ω passes through the critical value ω1.28 The jump phenomena discussed earlier have been observed experi­ mentally, by a number of persons, in both electrical and mechanical systems. We refer the reader to the references for details.6’29*30

Figure 5.7. The behavior of the phase δ for the forced Duffing equation with damping. The points marked by the letters corre­ spond to the same points as in Figure 5.6.

153

Problems

Problems 5.1 Consider a system described by the Duffing equation and acted on by three external harmonic forces: ^y2γ,

—r⅛- + y + ey3 = Fi cos ω1∕ + F2 cos ω2∕ + F3 cos ω3t dr Assume a solution of the form y = A1 COS ωit + A2

cos ω2t

+ A3 cos ω3t + e(7(t) + O(e2)

and determine the coefficients Ai, A2, and A3 and the function {7(r). What combination tones occur? 5.2 Let d2y —jp + y + ey2 = Fl cos ω1t + F2 cos ω2t and assume a solution of the form

y = A1 cos ω√ + A2 cos ω2t + cUi(t) + e2U2{t) + O(∈3) Substitute the assumed solution into the preceding nonlinear differen­ tial equation, and determine the coefficients A1 and A2 and the func­ tions Ul(t) and t∕2(r). What combination tones appear, and what are their relative intensities? (The intensity is proportional to the square of the amplitude of the term in which the combination tone appears.) 5.3 Consider a nonlinear system, with linear damping, acted on by two external harmonic forces; that is, d2y dy + y + ey2 + ek — = F1 cos ω1t + F2 cos ω2t with k positive and 0(1). What combination tones, if any, exist? Can you think of an actual physical system that this equation might repre­ sent? 5.4 Prove that the definitions of the Chebyshev polynomials given by equations 5.35, 5.36, and 5.37 are identical. Calculate T2(x), T3{x), T4(x), and T5(x). 5.5 Does the nonlinear differential equation

-j- + y + ey2 = F cos ωt dt2 have a simple subharmonic of order!? What about the equation

where F1 and F2 are constants? 5.6 Given the nonlinear differential equation

d2y —rr + y + ey3 = F cos 5ωt dt2 z study the existence of a subharmonic of order ⅛ (i.e., a contribution having angular frequency ω).31 5.7 Apply either of the iteration techniques of Section 5.5 to obtain a solu­ tion to the nonlinear equation

Forced Oscillations

154

d2y ~dp + y + ey2 = F cos ωt

Plot the response curves, determined from the second approximation, and compare them with the corresponding response curves of the Duffing equation. 5.8 Consider the equation

5.9 5.10 5.11 5.12

5.13

d2y dy -77 + 2ek ~ + y + ey2 = F cos ωt dt2 dt where k > 0. Use the perturbation method to discuss both the reso­ nance and nonresonance cases. Plot the response curves, and discuss the jump phenomena. Show that the higher approximations to the resonance perturbation solution of equation 5.93 are identically equal to zero. Verify equations 5.105. Verify equations 5.124 and 5.127. Determine the equation of the curve giving the loci of the vertical tangents for the response curves of the Duffing equation with damping. Consider the following nonlinear differential equation with both linear and Coulomb damping:

⅛ + kl ^dt + y + pθ (dr) = F° sin ^ωt + where k1, F, Fo, and φ are constants and θ(x) is + 1 for x positive and - 1 for x negative. Obtain an approximate expression for the periodic steady-state solution of the equation.32 5.14 Show that 1 y = —— sin ωr 2√⅛ is the exact solution of the differential equation33 J2v dy 4αω2 - 2αω(l - by2) + 0, there exists a posi­ tive number N that depends only on e, such that ∣∕nW -∕(x)∣ < e

(β∙2)

for all n > N for every x such that a ≤ x ≤ b. Consider the infinite series of real functions

⅜ WnU) M=l

(B.3)

Series Expansions

165

each of which is defined on a real interval a ≤ a ≤ b. Consider the se­ quence [∕π(a)] of partial sums of this series defined as follows: ∕1(x) = ∕∕Ja)

∕2(x) = Z∕1(.v) + u2(x)

(B.4)

fn(x) = u1(x) + m2(a) + U3(x) + ∙ ∙ ∙ + un(x)

The infinite series is said to converge uniformly to f(x), on a ≤ x ≤ b, if its sequence of partial sums [∕n(x)] converges uniformly to f{χ) o∏ a ≤ x ≤ b. Weierstrass M Test for Uniform Convergence The following test, known as the Weierstrass M test, is ade­ quate for determining the uniform convergence of a large number of series. Let [Mn] be a sequence of positive constants such that the series of constants 2 B.

(B.5)

∑ Mn n=l

converges. Let

⅜ un ,λx

Theorem IV. If the series ∑∏=1 wn(x) and ∑n=0 υn(x) are uniformly con­ vergent for a ≤ x ≤ b and ∕ι(x) is continuous for a ≤ x ≤ b, then the following series are uniformly convergent for a ≤ x ≤ b: ∞

y [«»(*) ± ^nU)]

(B.ll)

n=l

y [A(x)wn(x)]

(B.12)

n=l

4 B.

Power Series A power series in powers of x is a series of the form ∑ Cnxn

(B.13)

n=0

where Co, C1, . . . , Cn, . . . are constants. A power series in powers of x - a is a series of the form ∑ Cn(x - a)n

(B.14)

167

Series Expansions

The power series given by equation B.14 converges when x = a. This may be the only value of x for which the series converges. If there are other values of x for which the series converges, then the values of x form an interval, the convergence interval, having a midpoint at x = a. The interval can be either finite or infinite (Figure B.l). The main properties of power series are summarized in the following theorem: Theorem V. (1) Every power series

(B.15)

∑Cn(x-a)n n=0

has a radius of convergence R such that the series converges absolutely when ∣x - α∣ < R and diverges when ∣x - a∖ > R. (2) The number R can be zero (in which case the series converges only for x = a), a posi­ tive number, or infinite (unbounded, in which case the series converges for all x). (3) If R is not zero and R1 is such that 0 < R1 < R, then the series converges uniformly for ∣x — a∖ ≤ R1. (4) The number R can be evaluated in the following two ways: R = lim

n→∞

(B.16)

if the limit exists

1 if the limit exists (B.17) ∣On∣1'w Finally, we list a number of other important properties of power series: (5) A power series represents a continuous function within the interval of convergence. (6) A power series can be integrated term by term within the interval of convergence. (7) A power series can be differen­ tiated term by term within the interval of convergence. R = lim

Taylor Series of a Function of a Single Variable Let∕(x) be the sum of a power series with convergence interval a—R 0, and let, for every positive number δ, independent of ∈, the following condition hold:

D.l .2

∣∕(e)∣ ≤ δ ∣g(∈)∣

for ∣e∣ ≤ e0

(D∙8)

then ∕(e) = o∣⅛((e) C>(e) O(e2)

cos cosh cot 1 - cos

e e e €

= = = =

0(1) 0(1) O(e~1) O(e2)

and

sin (x + e) = 0(1) sin ex = O(e) D.5.2

uniformly as e → 0 nonuniformly as e → 0

The Symbol o As e → 0, sin e = o(l) sinh ∈ = o(l) coth e = o(e~3,2)

cos € = o(ell2) 1 - cos e = o(e) sin e2 = o(e)

182

Appendix D and

sin (x + ∈) = o(e-1'3) e~ix — 1 = o(∈ιz2)

uniformly as e → 0 nonuniformly as ∈→ 0

The Function sin (x + e) Let us consider the function sin (x + ∈) in more detail. For e → 0, we may expand it in a series of powers of e:

D.5.3

sin (x + ∈) = sin x cos e + cos x sin ∈

∕

∖

g5

^3

+ (f-3! + 5!-7! + ' ■ ■ )c05λ

∈2 .

.

= Sin X + € COS X - yj Sin X -

∈3

cos x + ∙ ∙ ∙

For all values of x the coefficients of all powers of e are bounded. Con­ sequently, the expansion is uniformly valid.

The Function exp (—ex) For a nonuniformly valid expansion, consider the expansion of exp (-ex) for small e:

D.5.4

00

(crVn

exp (-ex) = χ (-)m¾m=0

(D.34)

m’

The function exp (— ex) can be represented by a finite number of terms only if ex is small. Since e is small, this means that x = 0(1). If x is as large as O(e-1), then ex is not small, and a finite number of terms cannot give a good representation of exp (-ex). To obtain a satisfac­ tory expansion for all x, all the terms in equation D.34 must be re­ tained.

REFERENCES 1. N. G. de Bruijn, Asymptotic Methods in Analysis (Interscience, New York, 1958). 2. A. Erdely, Asymptotic Expansions (Dover, New York, 1956). 3. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973). See Chapter 1. 4. J. G. Van der Corput, Asymptotic Expansions (Lecture Notes, Stanford University, 1962). 5. W. Wason, Asymptotic Expansions for Ordinary Differential Equations (In­ terscience, New York, 1965).

Appendix E Basic Theorems of the Theory of Second-Order Differential Equations

l E.

Introduction Consider the differential equation (E.l)

If we make the transformation

?i = y

(E.2)

then we obtain the following system of first-order equations, which are equivalent to equation E.l:

(E.3)

dy2 = f(y1,y2d) dt

A more general system of first-order equations is given by the following expression,

⅛jf = fAy

2√)

(E.4) ⅛=∕2(y1,W)

In this appendix we state, without proof, a number of theorems con­ cerning the system of first-order equations E.4. Proofs and further dis­ cussions of these theorems can be found in the references listed at the end of this appendix.

Appendix E

184

The following assumptions and definitions apply to all the results of this appendix: (1) We assume that the functions fι(yi,y2,t) and ∕2(yι,y2√) are defined in a certain domain R of real three-dimensional (yι,y2,0 space, are continuous in this region, and have continuous partial derivatives with respect to y 1, y2» and t. (2) A point having coor­ dinates (yx,y2,t) will be denoted as P(jι,y2√). 2 E.

Existence and Uniqueness of the Solution

Theorem I. Let P0(yt ,y% ,t0) be any point of R. There exists an interval of t (∕1 < t < t2) containing r0, and only one system of functions )>ι = Φι(0 (E .5) y2 = 02 (0 defined in this interval, for which the following conditions are satisfied: (1) φ1Oo) = y? and Φ2 (t0) = y2∙ (2) For all values of t in the interval ∕1 < t < t2, the point P[01(t),02(r),r] belongs to the domain/?. (3) The system of functions given by equation E.5 satisfies the system of dif­ ferential equations E.4; that is,

= Λ[Φ1ω,⅛ω√] (E.6)

= ∕2[01ω,02ω√] (4) The solutions given by equation E.5 may be continued up to the boundary of the domain R∖ that is, whatever closed domain R1, con­ tained entirely in R, we may choose, there are values t' and t", where t1< t' < t2 and t1 < t" < t2, such that the points Pι[φi(t'),φ2(t'),t'] and P2[φι(t"),φ2(t"),t"] lie outside R1.

Dependence of the Solution on Initial Conditions The solutions of equation E.3 depend on the initial values (yι,y2√o)∙ Consequently, we may write the solutions as

3 E.

y1 = φ1(t,t0,yι,y2) y2 = Φ2(t,t0,y01,y02)

~

with

yt = Φ1(todo,yo1,y2) j2 = Φ2(Jo,to,y1,y2)

,E8)

The following theorem gives information concerning the dependence of the continuity of the solution on initial conditions.

Second-Order Differential Equations

185

Theorem 11. Let

y1 = φj(Lr*,y*,y2) y2 = Φ2(Λ∕*,y*o,2)

(E.9)

be a solution of equation E.4, defined for t in the interval t1 < t < t2 and having the initial valuesyl(t*) = y* andy2U*) = y2 . Let Tl and T2 be arbitrary numbers satisfying the condition t1 < T1 < T2 < t2. Then, for an arbitrary positive e, there exists a positive number δ = δ(e,T1,T2) such that for the values of t0, y↑, and y2 for which

∣r0 - r*∣ < δ the solutions

∣yι-y*∣ c

where each integrand is evaluated at x = φ(f), y = ψ(t). For nonlinear equations of the form given by equation H.l, the con­ dition for stability becomes

€ f2π aHΦ(∕)>Ψ(t)] dt < 0 Jo

∂y

If the period of oscillation is T rather than 2π, the condition given by equation H.41 becomes

€ F iff½L⅛Σ∣ dt < 0 Jo

(h.42)

∂y

where we have used the fact that y = x = φ(t). If we consider the first approximation to equation H.l, where φ(t) = A cos t φ(t) = ψ(t) = -A sin t T=2ττ

(H.43)

then the stability condition, equation H.42, is €

f2π ∂F(A cos t,-A sin t) Jo θy

< θ

(H.44)

We leave it as an exercise for the reader to prove that the condition for stability given by equation H.44 is the same as that derived in Section 3.6. We illustrate this procedure by considering the van der Pol equation

+ x = e(l - x2)

0 < e « 1

(H.45)

Appendix H

208

where

F(x,y) = (1 - x2)y

(H.46)

In Sections 2.4.4 and 3.3.8 we showed that the solution to the van der Pol equation, in first approximation, is

x = 2 cos t

(H.47)

Thus, y = -2 sin z, and = 1 - x2 = 1 - 4 cos2 r = - 1 - 2 cos 2z

(H.48)

Substituting equation H.48 into the right-hand side of equation H.44, we obtain

e

1—2 cos 2z) dt = — 2-7re < 0

(H.49)

Thus the periodic solution (limit cycle) of the van der Pol equation given by equation H.47 is stable. REFERENCES 1. T. V. Davies and E. M. James, Nonlinear Differential Equations (AddisonWesley, Reading, Mass., 1966). 2. Reference 1, Section 3.7.

Appendix 1

Numerical Examples

Introduction In this appendix we consider a number of nonlinear differential equations that might occur in the analysis of actual physical systems. In several cases we compare the approximate solution, previously found in the text, to the exact solution.

1.1

1.2

Simple Pendulum The equation of motion of the free oscillation of a simple pen­ dulum1 is

+ (£) sin θ = 0

(1.2)

where θ is the angular deflection from the position of equilibrium (θ is measured in radians). If we expand the sine function and retain only the first two terms, then equation 1.2 becomes (I∙3)

where (I∙4) The initial conditions are taken to be

0(0) = 0o ±≡ = o dt

Let us define new dimensionless variables

(I∙5)

Appendix I

210

(I∙6)

t = ω0t

Substitution into equation 1.4, and simplifying, gives (I∙7) where the dimensionless parameter e is (1.8)

and the initial conditions for y(t,e) are

y(0) = 1

(1.9)

dyW) = 0 di Note that the requirement ∈ ≤ 0.1 implies that the initial angular dis­ placement θ0 should be less than π∕4 radians. Consequently, equations 1.3 or 1.7 should be good approximations to the exact equation 1.2 for fairly large angular displacements. The solution to equation 1.7, with the initial conditions of equation 1.9, can be determined from the calculations of Section 2.43. We obtain

y(f,∈) = cos ω(e)f +

[cos ω(e)t - cos 3ω(e)f] + O(∈2)

(1.10)

where

ω(∈) = 1 - y -

+ O⅛3)

(1.11)

In terms of the original variables, θ and t, equation 1.10 becomes

θ(t,i) Γ ∕ ∖.ι —— = cos [ω0ω(∈)t] "o +

{cos [ω0ω(e)r] - cos [3ω0ω(e)t]}

(1.12)

+ O(e2)

Let us now numerically evaluate the parameters occurring in equa­ tion 1.12 for a pendulum of length L = 1 m. (The acceleration of gravity2 is taken as g = 9.802 m∕sec2.) Substitution of these values into equation 1.5 gives us the angular frequency g

1/2

= 3.131 sec 1

(1.13)

211

Numerical Examples

In Table 1.1 we give values of ∈ and ω(e) for a number of initial ampli-ides θ0. Figure 1.1 gives graphs of equation 1.12 for e = 0 and = 0.103. The case ∈ = 0 corresponds to the free vibrations of a linear scillator angular frequency ω0. The last two columns of Table 1.1 give the angular frequency obtined, respectively, from perturbation theory, equation 1.11, and the xact result, equation 1.111. Note that even for initial displacements as ιrge as θ0 = ττ∕4, the difference between the exact and perturbative alculations is only about 15%. Consider now the following nonlinear differential equation: ⅛+v + ∈y3 = 0

(1.14)

'here e > 0. This equation differs from equation 1.7 only in having an pposite sign for the cubic nonlinear term. With the initial conditions of quation 1.9, the perturbative solution of equation 1.14 is

(t,e) = cos ω(e)t - ⅛ [cos ω(e)f - cos 3ω(e)f] + O(e2)

(I-15)

here

(I∙∣6)

+ O(e3)

ω(e) = 1 + y -

In Section 1.5 we calculated the exact value of the period for the onlinear differential equation 1.14. The last two columns of Table 1.2 ive the angular frequency calculated, respectively, from the perturbaon technique, equation 1.16, and the exact result, equation 1.132.3) lote that in this case, even for values of e as large as 0.5, the difference ≥tween the two values for the angular frequency is less than 1%. Finally, it is easily seen that very accurate approximate solutions to able 1.1.

Parameters for the simple pendulum Angular frequency

00

)egrees)

(Radians)

e

5o

0.087 0.175 0.349 0.785

1.26 5.10 2.03 1.03

y )o 50

× × × ×

IO~3 10-∙3 10~2 10~1

Perturbation technique value, equation 1.11. Exact value, equation I.111.

ω(e)

P.T.α

Exact6

0.999 0.998 0.992 0.961

3.128 3.125 3.106 3.009

3.128 3.106 3.037 2.652

Figure 1.1. Plots of θ(t,e) for the simple pendulum. See equations 1.11 and 1.12. The amplitude units are arbitrary.

213

Numerical Examples

equations 1.7 and 1.14 may be obtained, for 0 < e < 0.5, by keeping only the first term in the expansions of equations 1.10 and 1.15. This follows from the fact that the terms of O(e), in equations 1.10 and 1.15, have relative amplitudes of less than 2% of the first terms. Nonlinear Spring A light asymmetric spring that has a 0.25-kg mass attached has the following relation between its restoring force and its displacement from the equilibrium position: L3

F(x) = -kx + kixz

(1.17)

k = 12.25 kg∕sec2 kl = 0.50 kg/m ∙ sec2

(1.18)

where

The force equation is m

d2χ

+ kx — k1xz = 0

(1.19)

The initial conditions we take as follows: x(0) = x0 (1.20)

dx(0) = n dt

Defining the new variables i = ω0Γ y = χ∕χo

ωo = k/m

(1.21)

Table 1.2. Parameters for the equation d2y∕dt2 + y + ey3 = 0. Angular frequency e

P.T.α

Exact6

0.001 0.010 0.050 0.100 0.500

1.000 1.004 1.019 1.038 1.188

1.000 1.004 1.019 1.037 1.171

a Perturbation technique value, equa­ tion 1.16. h Exact value, equation 1.132.

Appendix I

214

and substituting into equation 1.19 gives d2y + y dt2

(1.22)

ey2 = 0

where the dimensionless parameter e is €

= k1x0 k

(I∙23)

The angular frequency ω0 is (1.24)

ω. = (⅛f = 7 sec-

This corresponds to a frequency of 1.11 sec-1. The perturbation solution of equation 1.22 is given in section 2.4.1; it

y(t,e) = cos ω(e)t + (jθ [3-2 cos ω(e)t - cos 2ω(e)t] +