Volume 4 (1999), No 3, pp. 15-23

23

Exact analytic solution for 1D infinite vibrant elastic lumped line

The initial conditions of vibrations are a special matter. As follows from the solutions for forced vibrations, the initial conditions are specified by the external affecting force parameters, which is quite natural. With free vibrations we cannot limit ourselves by the only our wish to establish some or other value of the initial condition, because the vibration pattern is determined by the ‘history’, how these vibrations were caused. Thus, to yield the necessary initial conditions at a specified element of line, it is necessary to form definitely the external excitation of a line that preceded the origin of free vibrations. In their assemblage, the external force parameters and ‘history’ of the present free vibrations form the conditions that we used to name initial.

5. Conclusions

  The carried out study has revealed a number of essential drawbacks in the conventional solution for 1D infinite elastic lumped lines. These drawbacks do not allow considering this solution exact.

The exact analytic solutions that we yielded by our original method show that in general case, in ideal elastic lines, three vibration regimes take place, and they are very different in their peculiarities.

The periodical regime is characterised by along-the-line propagation of progressive non-damping vibrations with the element-to-element phase delay equal to 2 .

The aperiodical regime is characterised by the fast-damping along-the-line process in which the neighbouring elements vibrate in antiphase.

The critical regime is characterised by the antiphase vibration of elements, but depending on the type of line, it can be as damping as non-damping, and in an infinite line this regime never takes place.

All revealed vibration regimes are different forms of the general solution, which remains completely valid the theorem of uniqueness of solution of differential equations.

All three regimes are inherent in both forced and free vibrations in infinite lines.

Even in limits of one regime, patterns of forced and free regimes essentially differ. For example, in the periodical regime, in a semi-finite line, in the first case progressive wave propagates, but in that second the standing waves settle.

The solutions for free vibrations show that the pattern of processes in the line much depends on parameters of the affection preceding the origin of vibrations.

When analysing the structure of yielded exact solutions, we revealed that they contain the specific summands and multipliers allowing each solution to satisfy few different types of differential equations included into the modelling system.

The completeness of yielded solutions is determined in a following way. On one hand, their assemblage satisfies both homogeneous and heterogeneous systems of the modelling equations. On the other hand, the solutions cover the whole range of frequencies from zero to infinity.

References:

1. J.Bazer. Multiple Scattering in ?ne Dimension. J.Soc.Indust.Appl.Math., 12, 3, September, 1964

2. Grigoriev, E.T. and Tultchinskaya, N.B. Joint longitudinal vibrations in inhomogeneous rods with systems of lumped masses elastically connected to them within few sections. – In: Vibrations of complex mechanical systems. Transactions of Academy of Sciences of Ukraine. – Institute of technical mechanics, Kyev, Naukova Dumka, 1990 (Russian)

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

4. Elsgoltz, E. – Differential equations and calculus of variations. Nauka, Moscow, 1969 (Russian).