125 lines
6.4 KiB
Markdown
125 lines
6.4 KiB
Markdown
---
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title: 28-anharmonic-oscillations
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---
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which includes terms of the first two orders. To determine the limits of instability, we again
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treat the coefficients as constants, obtaining
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cos(wote)t-
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+[-2woebo+thwo861] sin(wo+e)t.
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+[-30002a1+thanoPao] cos 2(wote)t+
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sin 2(wote)t+[c1wo+thwo2ao] 0.
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Hence a1 = hao/6, b1 = hbo/6, C1 = -thao, and the limits aret € = -5h2wo/24, € = h2wo/24.
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PROBLEM 3. Find the conditions for parametric resonance in small oscillations of a simple
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pendulum whose point of support oscillates vertically.
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SOLUTION. The Lagrangian derived in §5, Problem 3(c), gives for small oscillations
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( < 1) the equation of motion + wo2[1+(4a/1) cos(2wo+t)) = 0, where wo2 = g/l.
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Hence we see that the parameter h is here represented by 4all. The condition (27.11), for
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example, becomes |
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§28. Anharmonic oscillations
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The whole of the theory of small oscillations discussed above is based on
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the expansion of the potential and kinetic energies of the system in terms of
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the co-ordinates and velocities, retaining only the second-order terms. The
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equations of motion are then linear, and in this approximation we speak of
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linear oscillations. Although such an expansion is entirely legitimate when
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the amplitude of the oscillations is sufficiently small, in higher approxima-
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tions (called anharmonic or non-linear oscillations) some minor but qualitatively
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different properties of the motion appear.
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Let us consider the expansion of the Lagrangian as far as the third-order
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terms. In the potential energy there appear terms of degree three in the co-
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ordinates Xi, and in the kinetic energy terms containing products of velocities
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and co-ordinates, of the form XEXKXI. This difference from the previous
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expression (23.3) is due to the retention of terms linear in x in the expansion
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of the functions aik(q). Thus the Lagrangian is of the form
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(28.1)
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where Nikl, liki are further constant coefficients.
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If we change from arbitrary co-ordinates Xi to the normal co-ordinates Qx
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of the linear approximation, then, because this transformation is linear, the
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third and fourth sums in (28.1) become similar sums with Qx and Qa in place
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t
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Generally, the width AE of the region of instability in resonance near the frequency
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2wo/n is given by
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AE =
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a result due to M. BELL (Proceedings of the Glasgow Mathematical Association 3, 132, 1957).
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§28
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Anharmonic oscillations
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85
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of the co-ordinates Xi and the velocities Xr. Denoting the coefficients in these
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new sums by dapy and Hapy's we have the Lagrangian in the form
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(28.2)
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a
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a,B,Y
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We shall not pause to write out in their entirety the equations of motion
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derived from this Lagrangian. The important feature of these equations is
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that they are of the form
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(28.3)
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where fa are homogeneous functions, of degree two, of the co-ordinates Q
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and their time derivatives.
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Using the method of successive approximations, we seek a solution of
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these equations in the form
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(28.4)
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where Qa2, and the Qx(1) satisfy the "unperturbed" equations
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i.e. they are ordinary harmonic oscillations:
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(28.5)
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Retaining only the second-order terms on the right-hand side of (28.3) in
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the next approximation, we have for the Qx(2) the equations
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(28.6)
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where (28.5) is to be substituted on the right. This gives a set of inhomo-
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geneous linear differential equations, in which the right-hand sides can be
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represented as sums of simple periodic functions. For example,
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cos(wpt + ag)
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Thus the right-hand sides of equations (28.6) contain terms corresponding
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to oscillations whose frequencies are the sums and differences of the eigen-
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frequencies of the system. The solution of these equations must be sought
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in a form involving similar periodic factors, and so we conclude that, in the
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second approximation, additional oscillations with frequencies
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wa+w
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(28.7)
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including the double frequencies 2wa and the frequency zero (corresponding
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to a constant displacement), are superposed on the normal oscillations of the
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system. These are called combination frequencies. The corresponding ampli-
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tudes are proportional to the products Axap (or the squares aa2) of the cor-
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responding normal amplitudes.
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In higher approximations, when further terms are included in the expan-
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sion of the Lagrangian, combination frequencies occur which are the sums
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and differences of more than two Wa; and a further phenomenon also appears.
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86
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Small Oscillations
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§28
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In the third approximation, the combination frequencies include some which
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coincide with the original frequencies W Wa+wp-wp). When the method
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described above is used, the right-hand sides of the equations of motion there-
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fore include resonance terms, which lead to terms in the solution whose
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amplitude increases with time. It is physically evident, however, that the
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magnitude of the oscillations cannot increase of itself in a closed system
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with no external source of energy.
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In reality, the fundamental frequencies Wa in higher approximations are
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not equal to their "unperturbed" values wa(0) which appear in the quadratic
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expression for the potential energy. The increasing terms in the solution
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arise from an expansion of the type
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which is obviously not legitimate when t is sufficiently large.
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In going to higher approximations, therefore, the method of successive
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approximations must be modified so that the periodic factors in the solution
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shall contain the exact and not approximate values of the frequencies. The
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necessary changes in the frequencies are found by solving the equations and
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requiring that resonance terms should not in fact appear.
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We may illustrate this method by taking the example of anharmonic oscil-
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lations in one dimension, and writing the Lagrangian in the form
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L =
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(28.8)
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The corresponding equation of motion is
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(28.9)
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We shall seek the solution as a series of successive approximations:
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where
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x(1) = a cos wt,
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(28.10)
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with the exact value of w, which in turn we express as w=wotw1)+w(2)+....
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(The initial phase in x(1) can always be made zero by a suitable choice of the
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origin of time.) The form (28.9) of the equation of motion is not the most
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convenient, since, when (28.10) is substituted in (28.9), the left-hand side is
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not exactly zero. We therefore rewrite it as
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(28.11)
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Putting x(1)+x(2), w wotwi and omitting terms of above the
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second order of smallness, we obtain for x(2) the equation
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= aa2 cos2wt+2wowlda cos wt
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= 1xa2-1xa2 cos 2wt + 2wow1)a cos wt.
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The condition for the resonance term to be absent from the right-hand side
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is simply w(1) = 0, in agreement with the second approximation discussed
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