1. Reference problem#

1.1. Geometry#

_images/100002010000019A000000EFE4218732F55AD697.png

Fig. 1. Diagram of pipe geometry

Average radius: \(R=0.050m\)

Pipe thickness: \(h=0.0025m\)

Length: \(L=1m\)

The vectors \(\vec{x}\), \(\vec{\mathrm{\theta }}\), and \(\vec{z}\) represent the axial, circumferential, and radial directions of the pipe.

Displacements in the \(\vec{x}\), \(\vec{\mathrm{\theta }}\), and \(\vec{z}\) directions are noted by \(u\), \(v\), and \(w\).

1.2. Material properties#

The properties of the material from which the pipe is made are:

\(E=2.{10}^{11}\mathrm{Pa}\)

Young’s module

\(\mathrm{\mu }=0.3\)

Poisson’s ratio

\(\mathrm{\rho }=7800\mathit{kg}/{m}^{3}\)

Volume density

1.3. Boundary conditions and loads#

Eigenmodes are calculated under five different boundary conditions:

  • FF: both ends of the pipe are free;

  • SF: one end is free and the other is in simple support;

  • SS: both ends are in simple support;

  • CS: one end with simple support and the other embedded;

  • CC: both ends are recessed.

    where F, S, and C represent simply-supported, clamped, and*free* in English.

    No load.

1.4. Initial conditions#

Not applicable