Calculation of Phase Velocities of Traveling Waves in a Cylindrical Shell Based on the Solution Analysis of Boundary Problem of Forced Oscillations

Abstract

In this paper we consider a moving orthotropic cylindrical shell of rotation. The purpose is to assess the choice of kinematic hypothesis for calculating the phase velocities of cylindrical shells. The comparison was done for the two hypotheses, namely: those of Timoshenko and Kirchhoff-Love. The calculation was performed under the following assumptions: all Poisson's ratios of orthotropic material were taken to be zero; the principal axes of anisotropy coincide with the lines of curvature, the coefficients of mutual influence of forces per unit length and bending moments were taken to be zero, which is valid for sufficiently thin shells. Analysis of the phase velocity of the cylindrical shell has shown that at low frequencies of traveling wave Timoshenko’s hypothesis gives an infinite value of the phase velocity. However, with increasing frequency of the traveling wave phase velocities obtained with different kinematic hypotheses, in the limit approach each other. Additionally, this article presents a numerical calculation of the phase velocity of the traveling waves. Calculation technique developed by V.O. Kaledin is based on the assumption that the traveling (direct and reflected) waves, forming a standing wave, are in superposition at sustained forced vibrations of a shell. Next, the analytical results, obtained for a cylindrical shell with the harmonic disturbing force acting at the front edge, have been compared with the numerical results obtained under the same assumptions. The difference between the numerical and analytical results is less than 1,5%.We note that many of the well-known works mention low accuracy when using the Kirchhoff-Love hypothesis to calculate phase velocities of the second and higher forms in thin cylindrical shells of rotation. This work is soundly refutes this claim and can form the basis of further studies of wave processes in shells of rotation of arbitrary Gaussian curvature using the Kirchhoff-Love hypothesis.

Keywords

• shell
• the phase velocity
• longitudinal waves
• bending waves
• critical frequency
• steady wave