• Version
• 39 Downloads
• 2.27 MB File Size
• 1 File Count
• November 2, 2016 Create Date
• November 2, 2016 Last Updated

National Advisory Committee for Aeronautics, Report - On Traveling Waves in Beams

The basic equations of Timoshenlco for the motion of vibrating
nonuniform beams, which allow for ejects of transverse shear
deformation and rotary inertia, are presented in several forms,
including one in which the equations are mitten in the directions
of the characteristics. The propagation of discontinuities in
moment and shear, as governed by these equations, is discussed.

Numerical traveling-wave solutions are obtained for some
elementary problems of finite uniform beam for which the prop-
agat'on. velocities of bending and shear discontinuities are
taken to be equal. These solutions are compared with modal
solutions of Timoshenko’s equations and, in some cases, with
exact closed solutions.

The theoretical analysis of transient stresses in aircraft
wings and fuselages subjected to impact loadings has gen-
erally been performed by means of a mode-superposition
method that uses the natural modes of vibration predicted
by the elementary engineering theory of beam bending.
(See, for example, refs. 1 to 3.) For very sharp impact
loadings, however, this approach is known to have certain
shortcomings. For sharp impacts of short duration, many
modes are often required for a satisfactory degree of con—
vergence (see, for example, ref. 4); in addition, the use of
elementary beam theory in the calculation of the higher
modes of vibration is inaccurate because of the neglect of,
among other factors, the effects of transverse shear deforma—
tion and rotary inertia which become increasingly important
for higher and higher modes (ref 5).

A classically recognized alternative to the modal method
of calculating transient stresses in elastic bodies is the travel—
ing-wave approach, which seeks to trace directly the propa»
gation of stresses through the body (ref. 3). Although the
traveling-wave concept has been successfully used to treat
such simple problems as longitudinal and torsional impact
of rods, only recently have serious attempts been made to
study the transient bending response of beams by this
approach.