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

21

Exact analytic solution for 1D infinite vibrant elastic lumped line

For an arbitrary ith equation of the left side of the system

and on the right side

Thus, (37) satisfies all equations of the modelling system (36).

  In the aperiodical regime (38) we can see the typical antiphase vibration of the line elements. But by contrast to the related regime for forced vibrations (7), the amplitude damps only at the interval between the line end and the selected kth element. The pattern of damping is not a fast-damping process but distributing along the whole interval, which is typical for finite lines. In the semi-finite interval i k , the amplitude grows as a complex-power law, because in this interval the power of numerator of fraction in (39) is more than the power of denominator.

The yielded pattern of free vibrations is inconsistent with the conventional understanding. And we should mark that up to now for infinite lumped lines there existed the only solution (2). Nobody expected that there can exist three regimes of vibrations and, the more, basically different patterns of forced and free vibrations. But such behaviour of free vibrations in aperiodical regime, when standing or damping vibrations arose in an infinite line the same as in finite ones – it is the more unexpected in the view of conventional understanding of free vibrations in infinite systems in general. None the less, it is easy to check: the solution yielded in this case is also a solution of homogeneous system (36).

For the left side of the first equation we yield

and for that right 

For left side of the ith equation we yield

and for that right

Thus, mathematically this solution fully satisfies the modelling system.

  For the critical regime, the same as for that aperiodical, we can see considerable difference both in solutions for forced vibrations and in the conception of free vibrations in the line. The expression (39) describes the linearly damping element-to-element vibrations in the interval i k and linearly increasing vibrations in the interval i k . With it the expression (39) also fully satisfies the homogeneous modelling system (36).

An unexpected pattern of vibrations described by solutions (38) and (39) relates to the phenomenon when the infinite energy necessary to excite the infinite line and exciting the overcritical vibrations is brought into the system at an infinite point of a line. Then, by the analogy with finite lines, in those semi-finite the damping antiphase vibrations will propagate from the energy application point to the opposite end of a line. With this conception, the line finiteness does not matter, because the very damping in aperiodical and critical regimes means not scattering but redistribution of the vibration energy along the line. And only in this interpretation the solutions (38) get their physics. The periodical regime does not need such conventionality, though the requirement remains, in order to excite finite free vibrations, the general energy of the line has to be infinite. Out of this stipulation we can state that free vibrations in an infinite line are absent and think the line to be dissipative regarding any finite pulses of the energy.