Various polynomial approximations for the wave propagation in hollow cicular cylinder

Introduction

This repository contains Maxima files to perform the symbolic calculation of various polynomial approximations for the (longitudinal) wave propagation in a hollow cicular cylinder.

This comes along with a paper being prepared for publication.

The .mac files are the Maxima inputs. The .m files are the ouput files.

Jacobi modes approximation

See the detailed equations in:

• Brizard, D., E. Jacquelin, et S. Ronel. (2019). ‘Polynomial mode approximation for longitudinal wave dispersion in circular rods’. Journal of Sound and Vibration 439 (january): 388‑97. https://doi.org/10.1016/j.jsv.2018.09.062.

Getting the results for the hollow circular cylinder is only a matter of changing the lower bound for the integral in the set of equations for the plain circular cylinder.

Mirsky approximate theories

The three Maxima files for Mirsky approximate theories correspond to the following articles:

• Herrmann, G., & Mirsky, I. (1956). Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders. Journal of Applied Mechanics, 23(4), 563‑568. https://doi.org/10.1115/1.4011399
• Mirsky, I., & Herrmann, G. (1958). Axially Symmetric Motions of Thick Cylindrical Shells. Journal of Applied Mechanics, 25(1), 97‑102. https://doi.org/10.1115/1.4011695
• Mirsky, I. (1964). Vibrations of Orthotropic, Thick, Cylindrical Shells. The Journal of the Acoustical Society of America, 36(1), 41. https://doi.org/10.1121/1.1918910

These Maxima files are intended to rewrite the given equations with variables W (dimensionless circular frequency) and C (dimensionless velocity).

Naghdi and Cooper

• Naghdi, P. M., and R. M. Cooper. (1956). ‘Propagation of Elastic Waves in Cylindrical Shells, Including the Effects of Transverse Shear and Rotatory Inertia’. The Journal of the Acoustical Society of America 28 (1): 56–63. https://doi.org/10.1121/1.1908222.

The equations provided by Naghdi and Cooper use the variables $\delta=h/\Lambda$ and $c$. The Maxima file ouputs the polynomial with variables $c$ and $\omega$.

Lin and Morgan

• Lin, T. C., & Morgan, G. W. (1956). A Study of Axisymmetric Vibrations of Cylindrical Shells as Affected by Rotatory Inertia and Transverse Shear. Journal of Applied Mechanics, 23(2), 255‑261. https://doi.org/10.1115/1.4011296

The approximate theory from Lin and Morgan already provides the equations with dimensionaless variables $\omega$ and $c$.

It can therefore be directly programmed from the article.