Theory Of Oscillations 1245194003, 9781245194006

Table of contents :
Front Cover
Title Page
PREFACE TO THE ENGLISH LANGUAGE EDITION
CONTENTS
Introduction
1. Linear Systems
2. Non-Linear Conservative Systems
3. Non-Conservative Systems
4. Dynamical Systems Described by a Single Differential Equation o f the First Order
5. Dynamical Systems Described by Two Differential Equations o f the First Order
6. Dynamical Systems Represented by Two Differential Equations of the First Order (Continued)
7. Discontinuous Oscillations. Parasitic Parameters and Stability
8. Systems with Cylindrical Phase Surface
9. Quantitative Investigation of Non-Linear Systems
APPENDIX A Structural Stability
APPENDIX В Justification o f the van der Pol Approximations
APPENDIX С The van der Pol Equation fo r Arbitrary Values o f the Parameter1
Supplementary Reading List
INDEX OF PHYSICAL EXAMPLES
INDEX OF MATHEMATICAL TERMS

Citation preview

THEORY OF OSCILLATIONS

THEORY OF

OSCILLATIONS Ry Л. A. ANDKONOW and С. E. CHAIKIN

English Language Edition 10 0). .Setting kjm = i — 1 ) is deformed into a similar grid (the angles are preserved) without rotation. In the "ac tiv e ” sense the deformation is produced by stretching the lines whose slope is h/(wi — 1 ) by a factor «i. The lines parallel to the у-axis are not stretched. In the case an = 1 , the points on the line x = c are dis­ placed a distance he. Thus under the reversal of this deformation the family of log­ arithmic spirals in the u,t'-plane will go into a family of spirals which wind toward the origin in a clockwise direction. Since lines through the origin go into lines through the origin, the turning of the radius vector to the path with increasing time has the same conditional period 7\ = 27t/coi in both planes. The angular velocity of turning is constant in the u,v-plane and in general not con­ stant in the phase plane. As we know, the radius vector is stretched uniformly and hence the logarithmic decrement of the decrease of its length, the same in both planes, is d = hTi. The family of logarithmic spirals covers the w,i’-plane and one and only one spiral passes through each point. Clearly this must also be true of the family of spiral paths in the phase plane. The origin x = 0, у = 0 is again an exception. It is a degenerate path at the origin. The phase velocity is zero at this point and does not vanish elsewhere. It does decrease with each revolution and can be shown to have the logarithmic decrement d = hTi. 4. Direct investigation of the differential equation. The pre­ ceding results may also be derived directly from (13) without referring to the solution (14). To that end, replace the initial equation of second order (13) by two equivalent equations of first order: (24) and hence ,OK\ (25)

7 t= y'

f = - 2* y - " S * .

dy 2 hy + co2 0.r — -------------------dx у

We can see immediately that this equation, like (9), determines a

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LINEAR OSCILLATOR WITH FRICTION

21

certain field of tangents on the phase plane, and together with (24) it will determine a vector field having just one singular point x = 0 , у = 0. By means of isoclines we can easily make an approximate stud}' of the character of this field. If the paths are to have the slope X, the equation of the corresponding isocline will be 2

hy + w\x =

У

or

X,

у = ax

where (26)

< j

cc20 X + 2h

i.e. the isoclines will be straight lines through the origin. If we give to x a sufficiently large number of values (Л and co0 being fixed by the system), we obtain a family of isoclines which will enable us to con­ struct a field with a suitable degree of precision. Fig. 15 represents such a field constructed by means of a few isoclines. This figure enables one to visualize the nature of the paths. One may also easily integrate (25) by a well-known procedure for homogeneous equations. We thus obtain here (со2 > Л2) for the paths the equation y- +

2

hxy +

coqX

O—

= Ce

tr

arc e "ix

derived without knowing the solution of (13). phase velocity v can be deduced from (24):

The expression for the

V- = «о* 2 + 4has\xy + (1 + 4h2)y2. By this method we can see at once that the phase velocity only becomes zero at the origin. It decreases steadily as the representative point approaches the origin. Given the character of the paths and the expression of the phase velocity, what can we say regarding the motions? First, we may affirm th at all the paths (other than x = 0 , у = 0 ) correspond to oscillatory damped motions tending-to the state of equilibrium. In fact, all the paths are spirals; when the representative point moves on one of the spirals, the coordinate and the velocity pass many times through zero and therefore the spirals on the phase plane correspond to an oscillatory process. Furthermore, the radius-vector of the representative point moving on a spiral decreases with each revolution. This means that we are dealing with a damped process. The maxi­ mum values of x and x decrease from one revolution to the next. It is also clear that the origin corresponds to the state of equilibrium.

22

LINEAR SYSTEMS

[Сн. I

To sum up then: Whatever the initial conditions the system undergoes a damped oscillatory motion around the state of equilibrium x = 0, у — 0 with the exception of the case when the initial conditions coincide exactly with the state of equilibrium. У

A singular point such as above, which is the common asymptotic point of a family of paths consisting of concentric spirals, is called a focus. Let us now examine whether the focus is stable. Since along any path the representative point will move towards the singular point, it becomes clear that the condition of stability for the equilibrium state, formulated above, is fulfilled. In fact, we can always choose such a

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LINEAR OSCILLATOR WITH FRICTION

23

region 5 (doubly shaded in Fig. 1 0 ) that the representative point does not cross the boundary of the region e (singly shaded in Fig. 10). Con­ sequently the equilibrium is stable and the singular point is a stable focus. The stability of the focal type of singular point is apparently related to the fact that the paths are winding or unwinding with respect to the movement of the representative point. Since the direction of the movement of the representative point is in a sense determined by the choice of coordinates (the point must move clock­ wise), the stability of the singular point is also determined with a sense (since the direction in which the time is measured cannot be changed). Q onversely, if the spirals should unwind, the singular point would be unstable. Equa­ tion (23), for example, shows that the winding of the paths is con­ ditioned by h > 0 , since it is only in this case that the radius-vector decreases for a clockwise motion (obviously we consider an positive and real). Thus, generally speak­ ing, a focus can be either stable or unstable, while the center type of singular point, as we have seen, is always stable. In the present case, the focus is stable because h > 0. The physical meaning of this condition of stability is clear; friction must be positive, i.e. it must resist the motion and dissipate energy. Such positive friction resists motion; its overcoming requires work to be spent, and it cannot induce instability. If the equilibrium of the system is already stable in the absence of friction (in the har­ monic oscillator), then it will remain stable in the presence of positive friction. In the sequel we shall also consider unstable foci. The stable focus possesses a “ stronger” stability than the center examined in the previous paragraph. Actually, in the case of the stable focus, not only the condition of stability according to Liapounoff but even a stronger condition will be fulfilled. Namely, for all initial conditions, the system, after a sufficiently long period of time, will return as close as one wishes to the state of equilibrium. Such a stability, for wrhich the initial deviations do not increase but on the contrary dampen, will be called asymptotic stability. In the case of the linear oscillator that we have studied, the focus is asymptotically stable.

24

LINEAR SYSTEMS

[Сн. I

. Damped aperiodic process. Let us now examine the case when the roots of the characteristic equation are real, i.e. when h 2 > wj;. Setting q2 = h2 — we obtain the solution in the form : 6

x = e~ht(Ae9t + Be~qt) or, introducing the notations Xi = —h + q = —qi)

X2 = —h — q = —q2

(so th at q2 > qi > 0 ), in the form x = Ae~9i‘ + Be~9,t. Here A and В are determined by the initial conditions. for t = 0 , x = Xo, and x = x0, then (27)

x

Xo ~h q2X0 9 2 — qi

+

Namely, if

Xq + qiXp qi — 9 2

It is clear th at whatever the initial condition the movement is damped, since qi > 0 and q2 > 0 , and therefore, for t —>+ 0 0 , x(t) —> 0 . In order to learn more about the character of the damping, we have to find th at ti and t2—the times (i.e. the intervals of time after the initial moment) for which x and x respectively become equal to zero. From (27) we deduce the following equations for h and t2: (28) (29)

=

1

_

Xo +

Xo + 92X0

e(w- ei)t, = 92(^0 + giXo) = J + qi(x 0 + q2x 0)

M q 2 - 91) , qi(x 0 + q2x0)

It is easy to see th at each of these equations has only one root, so that a damped oscillating process cannot occur—we are dealing with a so-called aperiodic process. Let us examine the situation when the equation (29) for t2 has no positive roots. In that case the damping of the motion is monotonic, tending asymptotically to zero. This will take place when in the expression (29) for t2 x0 < 0. Xo + q2Xo The region of initial values satisfying this inequality (region II) is represented in Fig. 17. For all other initial values Xo

Xo +

92

X0

>

0

§4]

LINEAR OSCILLATOR WITH FRICTION

25

and the equation defining t2 has positive roots. It means that the dis­ placement does not decrease monotonically, but its absolute value first increases and then, only after having reached its extreme position, does it start to decrease, tending asymptotically to zero. Here it is necessary to distinguish two cases according to whether under the initial conditions the equation determining ti does or does not

have a positive root. If there is no positive root, then the displace­ ment during the entire motion ( 0 < t < + °o) retains its sign; the system leaves the state of equilibrium, reaches a certain maximum deviation and then approaches monotonically the position of equi­ librium, but does not go through it. According to (28) this takes place when Xq + qix о The regions I of Fig. 17 correspond to this situation. If the equation determining fi has a positive root, the system will first approach the position of equilibrium; at t = h it will go through the position of equilibrium; later 1 at t = t2 it will reach a certain 1 From (28) and (29) we deduce the relation g (* 2 —J l)((2 —

1,

Hence the exponent at the left is positive and, since q2 — qi > 0, also t2 — h > 0, or 0 . 6. Representation of aperiodic processes on the phase plane. We have found the general solution x = Ae~qi' +

x = у = —Aq\e~4il — Bqze~qil,

from which follows by a simple calculation (30)

(y +

q x x ) q* = C i ( y +

q z x ) qK

Since q\ ^ qi it is no restriction to suppose that q\ < qi.

To analyze

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LINEAR OSCILLATOR WITH FRICTION

27

the family of curves (30) we shall use again a linear transformation of coordinates У + qix = v, у + q*x = u. After this transformation, (30) will have, in terms of the new variables, the simple form: v = Cm0,

a = —> qi

1

.

Let м and v be rectangular coordinates. We can say that after the transformation we shall obtain the family of "parabolas” whose type is determined by the value of the ex­ ponent a = qi/q\. Independently of the character of the exponent, how­ ever, (i.e. whether it is an integer or a fraction, odd or even, etc.) we may assert the following: 1. All the paths, with the excep­ tion of the curve corresponding to С = со, are tangent at the origin to the Maxis. For dv/dic = Сам0 - 1 and hence (dv/du)u=0 = 0 . 2 . The paths degenerate into straight lines for C = 0 and C = « . When (7 = 0 we have v = 0, i.e. the м axis; when С = со we have и = 0, i.e., the v axis. 3. The paths turn their convexity towards the и axis (since v increases faster than u), and the absolute value of the ordinate increases monotonically with increasing u. A family of such parabolas is represented in Fig. 19. Let us return to the x,у plane. To the v axis on the u,v plane there corresponds the straight line у + qix = 0 ; to the м axis there corre­ sponds the straight line у + q\X = 0. The other paths or, more precisely, the other curves of the family (30) on the x,y plane represent deformed parabolas tangent to the straight line у = —qix. In order to draw this family of curves it is convenient to take into account the following properties which are readily deduced from (30): 1 . The curves of the family have unlimited parabolic branches parallel to the straight line у = —qix.

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LINEAR SYSTEMS

[Сн. I

. They have tangents parallel to the x axis at the points of inter­ section with the line Q1Q2 ( ?1?2 ^\ у = ---т ~ x> (— z—