13

Solutions for finite elastic lumped lines

Analysing the structure of the obtained exact solutions, we have revealed that they contain specific summands and multipliers enabling each solution to satisfy at the same time a few types of differential equations being a part of the modelling system. For finite lines, the solutions have general structure – a ratio of trigonometric functions as sines and cosines whose arguments depend on the line length and parameters, on the frequency of vibration process and on the element location in the line. The completeness of obtained solutions is determined by the following way. On one hand, their assemblage satisfies both homogeneous and heterogeneous systems of the modelling equations; and on the other hand, the solutions cover the entire range of frequencies from zero to infinity.

For free vibrations in a finite line, there is typical the discrete frequency band of permissible vibrations in which the vibration amplitude increases monotonously as the mode number increases, but in lumped lines never reaches the infinite value. As opposed, with forced vibrations in a finite line, as the frequency increases the amplitude multiply reaches the infinite value but never vanishes, and the vibration spectrum is continuous. With it the values of permissible frequencies of free vibrations coincide with the values of resonance frequencies for forced vibrations, and the block structure of solutions is the same.

For finite lumped unfixed-end lines, the last element vibration amplitude is not maximal, as it is conventionally thought. Its value is shifted by the angle . When transiting to distributed lines, this difference disappears. In this connection finite lumped lines cannot be correctly modelled by distributed lines, because with the limiting process a number of features disappears and is non-restored with the reverse transition.

References:

1. Karavashkin, S.B. Exact analytical solution on infinite one-dimensional elastic lumped-parameters line vibration. Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, v.4, 1999, #3, pp.15-23 (Russian)

2. Pain, H.J. The Physics of Vibrations and Waves. Mir, Moscow, 1979; 389 pp. (Russian; from edition: John Wiley and Sons, Ltd. London – New York – Sydney – Toronto, 1976)

3. Krylov, A.N. On some differential equations of mathematical physics. MGITTL, 1950; 368 pp. (Russian)

4. Savin, G.N., Kiltchevsky, N.A. and Putyata, G.V. Theoretical mechanics. Gostechizdat, Moscow, 1963; 610 pp. (Russian)

5. Olkhovsky, I.I. The Course of theoretical mechanics for physicists. Nauka, Moscow, 1970; 447 pp. (Russian)

Ó S.B.Karavashkin, 1999.