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Thus r2 + 2λ + wo2 = 0 , where from r1,2 = −λ ± (λ2 − wo2 ) . We are thus led to the following general solution of the equation of motion x = c1 exp(r1t) + c2 exp(r2t) . Among the roots r we shall look at the following particular cases: (i) λ < wo . One gets complex conjugate solutions. The solution is x = Re Aexp −λt + i (wo2 − λ2 ) , where A is an arbitrary complex constant. The solution can be written of the form x = a exp(−λt) cos(wt + α), where w= (wo2 − λ2) , (28) where a and α are real constants.

Thus, the lowest order nonzero term is proportional to the velocity, and moreover we shall neglect all higher-order terms · fr = −α x , where x is the generalized coordinate and α is a positive coefficient; the minus sign shows the oposite direction to that of the moving system. Adding this force to the equation of motion we get · .. m x= −kx − α x , 60 or · .. x= −kx/m − α x /m . (27) Writing k/m = wo2 and α/m = 2λ; where wo is the frequency of free oscillations of the system and λ is the damping coefficient.

Goldstein, Classical Mechanics, (Addison-Wesley, 1992). • L. D. Landau & E. M. Lifshitz, Mechanics, (Pergammon, 1976). • J. B. T. Thornton, Classical Dynamics of Particles and Systems, (Harcourt Brace, 1995). • W. M. Hollister, The Gyroscope: Theory and application, Science 149, 713 (Aug. 13, 1965). 51 4. SMALL OSCILLATIONS Forward: A familiar type of motion in mechanical and many other systems are the small oscillations (vibrations). They can be met as atomic and molecular vibrations, electric circuits, acoustics, and so on.

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