11

Solutions for finite elastic lumped lines

For the aperiodical regime at >1

Finally, for that critical at =1 the solution has the following form:

The system of equations describing the free vibrations in an investigated line differs from (41) by the force absence in the first equation; it has the form

The given vibrations arise in an elastic system at the condition

Proceeding from (46), the natural frequencies are determined by the expression

and the equation of vibration has the form

In comparison with the line having an unfixed end, in the presented solutions the substantial changes really have taken place, though the block structure in the form of ratio of trigonometric functions has retained.

For the forced vibrations, the changes concern to the resonance frequency, because the conditions of its rise gained the form

The same for the conditions determining the amplitude minimum:

It follows from the conditions (49) and (50) that due to fixing one of the line ends, in the amplitude dependence with respect to the maximums and minimums shift took place, and not by a quarter of a period, as one could expect, but some more, because on the left parts of these expressions we see the value (2n - 1 ), not 2n, as it was in (14) and (15). This difference disappears as the number of line elements growing and as the gradual transition to the distributed line. When reverse transition, this difference is non-restored.

For free vibrations, the amplitude dependence on the mode number is determined by the conditions which we can set up after the following transformation of the amplitude part of solution (48), noting (46):

The obtained expression (51) shows that as the number of mode p increases the denominator of the expression decreases similarly to (33). It means, the amplitude of free vibrations, the same as in a free-end line, increases monotonously with the increasing mode number and reaches the maximum at p_max, but for a lumped line it never reaches an infinite value.

To complete this analysis, make certain that the presented expressions are the solutions. It is simple to do so by their direct substitution to any related equation of the modelling systems, by analogy with the above models. For example, substitute (48) to the last but one equation of (45). We yield: