Resonance in non-linear oscillations 87 at the beginning of this section. Solving the inhomogeneous linear equation in the usual way, we have (28.12) Putting in (28.11) X wo+w(2), we obtain the equa- tion for x(3) = - or, substituting on the right-hand side (28.10) and (28.12) and effecting simple transformation, wt. Equating to zero the coefficient of the resonance term cos wt, we find the correction to the fundamental frequency, which is proportional to the squared amplitude of the oscillations: (28.13) The combination oscillation of the third order is (28.14) $29. Resonance in non-linear oscillations When the anharmonic terms in forced oscillations of a system are taken into account, the phenomena of resonance acquire new properties. Adding to the right-hand side of equation (28.9) an external periodic force of frequency y, we have +2x+wo2x=(fm)cos = yt - ax2-Bx3; (29.1) here the frictional force, with damping coefficient A (assumed small) has also been included. Strictly speaking, when non-linear terms are included in the equation of free oscillations, the terms of higher order in the amplitude of the external force (such as occur if it depends on the displacement x) should also be included. We shall omit these terms merely to simplify the formulae; they do not affect the qualitative results. Let y = wote with E small, i.e. y be near the resonance value. To ascertain the resulting type of motion, it is not necessary to consider equation (29.1) if we argue as follows. In the linear approximation, the amplitude b is given 88 Small Oscillations §29 near resonance, as a function of the amplitude f and frequency r of the external force, by formula (26.7), which we write as (29.2) The non-linearity of the oscillations results in the appearance of an ampli- tude dependence of the eigenfrequency, which we write as wo+kb2, (29.3) the constant K being a definite function of the anharmonic coefficients (see (28.13)). Accordingly, we replace wo by wo + kb2 in formula (29.2) (or, more precisely, in the small difference y-wo). With y-wo=e, the resulting equation is = (29.4) or Equation (29.4) is a cubic equation in b2, and its real roots give the ampli- tude of the forced oscillations. Let us consider how this amplitude depends on the frequency of the external force for a given amplitude f of that force. When f is sufficiently small, the amplitude b is also small, so that powers of b above the second may be neglected in (29.4), and we return to the form of b(e) given by (29.2), represented by a symmetrical curve with a maximum at the point E = 0 (Fig. 32a). As f increases, the curve changes its shape, though at first it retains its single maximum, which moves to positive E if K > 0 (Fig. 32b). At this stage only one of the three roots of equation (29.4) is real. When f reaches a certain value f k (to be determined below), however, the nature of the curve changes. For all f > fk there is a range of frequencies in which equation (29.4) has three real roots, corresponding to the portion BCDE in Fig. 32c. The limits of this range are determined by the condition db/de = 8 which holds at the points D and C. Differentiating equation (29.4) with respect to €, we have db/de = Hence the points D and C are determined by the simultaneous solution of the equations 2-4kb2e+3k264+2 0 (29.5) and (29.4). The corresponding values of E are both positive. The greatest amplitude is reached where db/de = 0. This gives E = kb2, and from (29.4) we have bmax = f/2mwod; (29.6) this is the same as the maximum value given by (29.2). §29 Resonance in non-linear oscillations 89 It may be shown (though we shall not pause to do so heret) that, of the three real roots of equation (29.4), the middle one (represented by the dotted part CD of the curve in Fig. 32c) corresponds to unstable oscillations of the system: any action, no matter how slight, on a system in such a state causes it to oscillate in a manner corresponding to the largest or smallest root (BC or DE). Thus only the branches ABC and DEF correspond to actual oscil- lations of the system. A remarkable feature here is the existence of a range of frequencies in which two different amplitudes of oscillation are possible. For example, as the frequency of the external force gradually increases, the ampli- tude of the forced oscillations increases along ABC. At C there is a dis- continuity of the amplitude, which falls abruptly to the value corresponding to E, afterwards decreasing along the curve EF as the frequency increases further. If the frequency is now diminished, the amplitude of the forced oscillations varies along FD, afterwards increasing discontinuously from D to B and then decreasing along BA. b (a) to b (b) f<f b (c) f>tp B C Di A E F € FIG. 32 To calculate the value of fk, we notice that it is the value of f for which the two roots of the quadratic equation in b2 (29.5) coincide; for f = f16, the section CD reduces to a point of inflection. Equating to zero the discriminant t The proof is given by, for example, N.N. BOGOLIUBOV and Y.A. MITROPOLSKY, Asymp- totic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corporation, Delhi 1961. 4 90 Small Oscillations §29 of (29.5), we find E2 = 3X², and the corresponding double root is kb2 = 2e/3. Substitution of these values of b and E in (29.4) gives 32m2wo2x3/31/3k. (29.7) Besides the change in the nature of the phenomena of resonance at fre- quencies y 22 wo, the non-linearity of the oscillations leads also to new resonances in which oscillations of frequency close to wo are excited by an external force of frequency considerably different from wo. Let the frequency of the external force y 22 two, i.e. y = two+e. In the first (linear) approximation, it causes oscillations of the system with the same frequency and with amplitude proportional to that of the force: x(1)= (4f/3mwo2) cos(two+e)t (see (22.4)). When the non-linear terms are included (second approximation), these oscillations give rise to terms of frequency 2y 22 wo on the right-hand side of the equation of motion (29.1). Substituting x(1) in the equation = - using the cosine of the double angle and retaining only the resonance term on the right-hand side, we have = - (8xf2/9m2w04) cos(wo+2e)t. (29.8) This equation differs from (29.1) only in that the amplitude f of the force is replaced by an expression proportional to f2. This means that the resulting resonance is of the same type as that considered above for frequencies y 22 wo, but is less strong. The function b(e) is obtained by replacing f by - 8xf2/9mwo4, and E by 2e, in (29.4): 62[(2e-kb2)2+12] = 16x2f4/81m4w010. (29.9) Next, let the frequency of the external force be 2= 2wote In the first approximation, we have x(1) = - (f/3mwo2) cos(2wo+e)t. On substituting in equation (29.1), we do not obtain terms representing an external force in resonance such as occurred in the previous case. There is, however, a parametric resonance resulting from the third-order term pro- portional to the product x(1)x(2). If only this is retained out of the non-linear terms, the equation for x(2) is = or (29.10) i.e. an equation of the type (27.8) (including friction), which leads, as we have seen, to an instability of the oscillations in a certain range of frequencies. §29 Resonance in non-linear oscillations 91 This equation, however, does not suffice to determine the resulting ampli- tude of the oscillations. The attainment of a finite amplitude involves non- linear effects, and to include these in the equation of motion we must retain also the terms non-linear in x(2): = cos(2wo+e)t. (29.11) The problem can be considerably simplified by virtue of the following fact. Putting on the right-hand side of (29.11) x(2) = b cos[(wo++)+8], where b is the required amplitude of the resonance oscillations and 8 a constant phase difference which is of no importance in what follows, and writing the product of cosines as a sum, we obtain a term (afb/3mwo2) of the ordinary resonance type (with respect to the eigenfrequency wo of the system). The problem thus reduces to that considered at the beginning of this section, namely ordinary resonance in a non-linear system, the only differences being that the amplitude of the external force is here represented by afb/3wo2, and E is replaced by 1/6. Making this change in equation (29.4), we have Solving for b, we find the possible values of the amplitude: b=0, (29.12) (29.13) 1 (29.14) Figure 33 shows the resulting dependence of b on € for K > 0; for K < 0 the curves are the reflections (in the b-axis) of those shown. The points B and C correspond to the values E = To the left of B, only the value b = 0 is possible, i.e. there is no resonance, and oscillations of frequency near wo are not excited. Between B and C there are two roots, b = 0(BC) and (29.13) (BE). Finally, to the right of C there are three roots (29.12)-(29.14). Not all these, however, correspond to stable oscillations. The value b = 0 is unstable on BC, and it can also be shown that the middle root (29.14) always gives instability. The unstable values of b are shown in Fig. 33 by dashed lines. Let us examine, for example, the behaviour of a system initially “at rest” as the frequency of the external force is gradually diminished. Until the point t This segment corresponds to the region of parametric resonance (27.12), and a com- parison of (29.10) and (27.8) gives 1h = 2af/3mwo4. The condition 12af/3mwo3 > 4X for which the phenomenon can exist corresponds to h > hk. + It should be recalled that only resonance phenomena are under consideration. If these phenomena are absent, the system is not literally at rest, but executes small forced oscillations of frequency y. 92 Small Oscillations §29 C is reached, b = 0, but at C the state of the system passes discontinuously to the branch EB. As € decreases further, the amplitude of the oscillations decreases to zero at B. When the frequency increases again, the amplitude increases along BE.- b E E A B C D FIG. 33 The cases of resonance discussed above are the principal ones which may occur in a non-linear oscillating system. In higher approximations, resonances appear at other frequencies also. Strictly speaking, a resonance must occur at every frequency y for which ny + mwo = wo with n and m integers, i.e. for every y = pwo/q with P and q integers. As the degree of approximation increases, however, the strength of the resonances, and the widths of the frequency ranges in which they occur, decrease so rapidly that in practice only the resonances at frequencies y 2 pwo/q with small P and q can be ob- served. PROBLEM Determine the function b(e) for resonance at frequencies y 22 3 wo. SOLUTION. In the first approximation, x(1) = -(f/8mwo2) cos(3wo+t) For the second approximation x(2) we have from (29.1) the equation = -3,8x(1)x(2)2, where only the term which gives the required resonance has been retained on the right-hand side. Putting x(2) = b cos[(wo+)+8] and taking the resonance term out of the product of three cosines, we obtain on the right-hand side the expression (3,3b2f(32mwo2) cos[(wotle)t-28]. Hence it is evident that b(e) is obtained by replacing f by 3,8b2f/32wo², and E by JE, in (29.4): Ab4. The roots of this equation are b=0, Fig. 34 shows a graph of the function b(e) for k>0. Only the value b=0 (the e-axis) and the branch AB corresponds to stability. The point A corresponds to EK = 3(4x2)2-A3)/4kA, t It must be noticed, however, that all the formulae derived here are valid only when the amplitude b (and also E) is sufficiently small. In reality, the curves BE and CF meet, and at their point of intersection the oscillation ceases; thereafter, b = 0.