Volume 4 (1999), No 4, pp. 5-13

13

Solutions for finite elastic lumped lines

Analysing the structure of the yielded 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 like 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 yielded solutions is determined by the following way. On one hand, their assemblage satisfies both homogeneous and heterogeneous systems of modelling equations; and on the other hand, these solutions cover the whole 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. On the contrary, 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 same.

For finite lumped free-end lines, the last element vibration amplitude is not maximal, as it is conventionally thought. Its value is shifted by the angle . In passing 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 analytic solution for 1D infinite vibrant elastic lumped line. Materials, Technologies, Tools (National Academy of Sciences of Belarus), 4 (1999), 3, 15–23 (Russian)

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

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

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

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

The paper was submitted on 21.06.99