Flexural Vibration of Simply Supported Circular Bars Involving the Second Frequency Spectrum of Timoshenko Beam Theory

摘要

For circular cylinders, the Timoshenko beam theory, which includes both shear deformation and rotary inertia effects, can be applied by introducing the constant 0.9 or 6(1+ν)/(7+6ν) for the shear coefficient κ, where ν is Poisson's ratio. In this paper, we show that for the simply supported circular cylinders, firstly, both the first spectrum and the second spectrum can be determined exactly by the phase velocities of the transverse elastic waves in the Pochhammer-Chree theory. Secondly, from the Timoshenko beam theory, it is shown that the same results can be obtained by the new shear coefficient, which is constructed both by the phase velocities of the transverse elastic waves in the Pochhammer-Chree theory and by the ratio of the radius of the circular section to the wavelength. The new shear coefficient is obtained by a fourth-order equation for the angular frequency.