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By B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)

ISBN-10: 008010021X

ISBN-13: 9780080100210

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Formulate the problem on the electrical vibrations in a conductor, similar to the problem on the longitudinal vibrations of a homogeneous flexible rod, in the following cases: (a) one end of the rod is rigidly fixed, and the other end elastically attached; (b) one end of the rod is free, and the other experiences a resist­ ance proportional to the velocity; (c) one end of the rod is fixed elastically, and the other end moves according to a given law. Estabhsh the necessary and sufficient conditions that the first problem be similar to the second.

Small vibrations are those for which it is possible to neglect squares, products and higher powers of the functions, describing the vibrations, and of their deriva­ tives. 1. Free Vibrations in a Non-resistant Medium; Equations with Constant Coefficients In problems of this group the effect of the force of gravity on the vibrations of particles is assumed to be negligibly small in comparison with the effect of other forces, therefore it is possible to neglect the action of gravityt. 1. The axis Ox is directed along a rod; the characteristic function is assumed to be the displacement u(x, t) along the x-axis of a cross-section, whose ab­ scissa equals χ in the equilibrium state; in other words, at time / the abscissa of this section equals χ = x-\-u(x, t).

134. Solve the preceding problem v^hen the hnear density of the force equals Φ(χ, t) = 0Qsin ωί,Ο < χ < 1,0 < t < + 0 0 , where Φο = const. 135. Find the longitudinal vibrations of a rod 0 < Λ: < /, the end Λ: = 0 of which is rigidly fixed, and the end χ = I, starting at time t = 0, moves according to the law w(/, t) = A sin , 0 < ί < + oo. The medium does not produce a resistance to the vibrations. 136. Find the longitudinal vibrations of a rod 0 < Λ: < / in a non-resistant medium, if the end χ = 0 of the rod is rigidly fixed, and a force is applied to the end χ = I, starting at time t = 0 F(t) = Asmωt, 0 < ί < + oo .

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A Collection of Problems on Mathematical Physics by B. M. Budak, A. A. Samarskii, A. N. Tikhonov, I. N. Sneddon, M. Stark and S. Ulam (Auth.)


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