Mechanics by L.D. Landau & E.M. Lifshitz

Author:L.D. Landau & E.M. Lifshitz
Language: eng
Format: epub
Tags: Physics; Equations of Motion; Collisions; Conservation Laws; Oscillations; Rigid Body; Cannonical Equations

Fig. 32

To calculate the value of/&, we notice that it is the value of/for which the two roots of the quadratic equation in & 2 (29.5) coincide; for/ = /&, the section CD reduces to a point of inflection. Equating to zero the discriminant

t The proof is given by, for example, N. N. BoGOLiuBOvandY. A. Mitropolsky,^}-^-totic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corporation, Delhi 1961.

of (29.5), we find e 2 = 3 A 2 , and the corresponding double root is ko 2 = 2e/3. Substitution of these values of b and e in (29.4) gives

32mWA 3 /3V3M. (29.7)

Besides the change in the nature of the phenomena of resonance at frequencies y x coo, the non-linearity of the oscillations leads also to new resonances in which oscillations of frequency dose to coo are excited by an external force of frequency considerably different from too-

Let the frequency of the external force y x -|coo> i-e. y = -^coo + 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 d) = (4f/3mojo 2 ) cos(|a>o + e)t

(see (22.4)). When the non-linear terms are included (second approximation), these oscillations give rise to terms of frequency 2y x coo on the right-hand side of the equation of motion (29.1). Substituting x {1) in the equation

*< 2 > + 2AxW + coo 2 * (2) + rf> 2 + j8*e>3 = - a*< 1)2 ,

using the cosine of the double angle and retaining only the resonance term on the right-hand side, we have

x®>+ 2Ax< 2 > + co 0 2 * (2) + ox® 2 + fix®* = - (8a/ 2 /9w 2 co 0 4 ) cos(co 0 + 2e)t.

(29.8)

This equation differs from (29.1) only in that the amplitude /of the force is replaced by an expression proportional to / 2 . This means that the resulting resonance is of the same type as that considered above for frequencies y x coo, but is less strong. The function b(e) is obtained by replacing / by - 8a/ 2 /9wco 0 4 , and e by 2e, in (29.4):

& 2 [(2e-K& 2 ) 2 + A 2 ] = 16a 2 /4/81m4co 0 10 . (29.9)

Next, let the frequency of the external force be y = 2u>o + e. In the first approximation, we have x a) = — (//3mco 0 2 ) cos(2co 0 + e)t. On substituting x = x (1) + x (2) 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 proportional to the product x (1) x (2) . If only this is retained out of the non-linear terms, the equation for # (2) is

x® + 2\x®> + co 0 2 *< 2 > = - 2a*<%<2) or

W + 2\x®+a>o 2 \l ^7 cos(2co 0 + e)fl^ 2 ) = 0, (29.

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