A Mixed Problem Describing Oscillations of a Nonhomogeneous Rod with Point Massesстатья
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 25 сентября 2013 г.
Аннотация:This paper studies the mixed problem describing
the excitation of longitudinal oscillations in a rod
consisting of several segments each of which has arbitrary
density, Young’s modulus, and length. At the joints of
these segments, concentrated masses may be placed.
We assert that this mixed problem has a unique solution
and give the analytic form of this solution.
NOTATION
Consider a rod consisting of n segments enumer
ated consecutively from 1 to n. Let xi (i = 0, 1, …, n) be
the points such that the ith segment coincides with the
interval [xi – 1, xi]. In this notation, x0 and xn are,
respectively, the left and the right ends of the rod, and
x1, x2, …, xn – 1 are the points of junction. We assume
that the ith segment of the rod has density ρi, Young’s
modulus ki, and wave velocity ai = . We denote the
wave transit time along the ith fragment by ti.
To the segment joints, we attach point masses; at
each point xi (i = 1, 2, …, n – 1), a mass mi ≥ 0 is placed.
We denote the set of junction points at which positive
masses are placed by X and the set of the other junction
points by Y. It is easy to show that the case of a composite rod with masses placed at arbitrary places can be
reduced to the case under consideration: it suffices to
assume that the points at which the masses are
attached are junction points as well (fragments may
have identical characteristics).