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National Advisory Committee for Aeronautics, Technical Notes - Transverse Vibrations of Hollow Thin Walled Cylindrical Beams

The variational principle, differential equations, and boundary con—
ditions considered appropriate to the analysis of transverse vibrations
of hOllOW thin-walled cylindrical beams are shown. General solutions
for the modes and frequencies of cantilever and free-free cylindrical
beams of arbitrary cross section but of uniform thickness are given.
The combined influence of the secondary effects of transverse shear
deformation, shear lag, and longitudinal inertia is shown in the form
of curves for cylinders of rectangular cross sectiOn and uniform thick-
ness. The contribution of each of the secondary effects to the total
reduction in the actual frequency is also indicated.

The elementary theory of bending vibration is often inadequate for
the accurate calculation of natural modes and frequencies of hollow, thin—
walled cylindrical beams." Such secondary effects as transverse shear
deformation, shear lag, and longitudinal inertia, which are not considered
in the elementary theory of lateral oscillations, can have appreciable
influence, particularly on the higher modes and frequencies of vibration.
The effects of transverse shear deformation and of rotary (rather than
longitudinal) inertia have been studied by many on the basis of the
original investigations of Rayleigh (reference 1) and Timoshenko (refer—
ence 2). Anderson and Houbolt (reference 3) have presented a procedure
for including the effects of shear lag in the numerical calculation of
modes and frequencies of box beams of rectangular cross Section. How—
ever, there does not appear to exist a general solution for the vibra-
tion of hollow beams that incorporates the influence of all the secondary
effects mentioned.

The purpose of the present paper is_threefold: First, to exhibit
the variational principle, differential equations, and boundary condi—
tions appropriate for the analysis of the uncoupled bending vibration
of hollow thin-walled cylindrical beams; second, to give general solutions
for cantilever and free-free cylinders of arbitrary cross section but of ‘_
uniform thickness; and finally, to show quantitatively the influence of _ u
the secondary effects by means of numerical results for hollow beams of
rectangular cross section; cf various lengths, widths, and depths.