We consider one-dimensional longitudinal vibrations
of a rigid rod with a nonuniform cross-section,
fixed at its ends with lumped masses and springs. The
cross-section inertia effects are taken into account on
the basis of the Rayleigh theory. The equation of
motion and the boundary conditions are derived from
Hamilton’s variational principle. The characteristic
equation is constructed and the eigenvalues for the harmonic
vibrations of the rod are calculated. It is shown
that the eigenvalues are bounded from above. Two
types of the orthogonality of the eigenfunctions corresponding
to the eigenvalues are discussed. The Green
function is constructed for the problem of forced vibrations
of the rod governed by a linear fourth-order partial
differential equation, which involves mixed derivatives.
Exact solutions of the rod vibration problems are found
for rods with constant and conical cross-sections.
Rigid isotropic waveguides are often used for generating,
transmitting, and amplifying mechanical vibrations,
for example, in acoustic transducers. Theoretical
investigation of acoustic, mechanical, and electromagnetic
waveguides is usually based on the analysis of
second-order wave equations. This approach is justified
in descriptions of the wave propagation in relatively
thin and long rigid rods. As was shown by Rayleigh [1],
the error due to the neglect of the transverse motion of
the rod is proportional to the square of the ratio of the
characteristic section radius to the length of the rod
(aspect ratio). For a more accurate analysis of the longitudinal
vibrations of a relatively thick and short rod,
the rod deformation in the transverse direction must also be taken into account. The approach to the analysis
of the vibrations of a thick and short rod used in this
study is based on the theory of longitudinal vibrations
of a rod, in which the effects due to the transverse
motion are taken into account (the corresponding mathematical
model is called the Rayleigh rod). The equation
of motion and the boundary conditions for the onedimensional
longitudinal vibrations of the Rayleigh rod
with variable cross section and ends fixed by means of
lumped masses and springs are derived from Hamilton’s
variational principle. As a result, we arrive at a linear
fourth-order partial differential equation with variable
coefficients, which involves mixed derivatives.
Previously, approximate analytical methods, such as
the Galerkin method [2] and the method based on the
expansion of the solution in a power series in the Poisson
coefficient [3], were used for solving this equation.
The frequencies of the natural vibrations of a cylindrical
rod with rigidly fixed ends were determined in [4,
pp. 159, 160]. In this study we use the method of the
separation of variables based on the exact solutions of
the equations of motion of the Rayleigh rod, which
makes it possible to construct the Green function. A
similar approach to an analysis of the longitudinal
vibrations of stepped rigid waveguides described by
second-order wave equations was applied in [5, 6].
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:tut/oai:encore.tut.ac.za:d1001012 |
| Date | 27 February 2007 |
| Creators | Fedotov, IA, Polyanin, AD, Shatalov, MY |
| Publisher | Pleaides Publishing LTD |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | |
| Rights | Pleiades Publishing LTD |
| Relation | Doklady Physics |
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