Reference problem
=====================


Geometry
---------


.. image:: images/10003FF600001F0F00001EAD1B549C2267BB15C4.svg
    :width: 187
    :height: 197


.. _RefImage_10003FF600001F0F00001EAD1B549C2267BB15C4.svg:

It is a cylindrical ring, with an average radius :math:`{R}_{m}=0.369m`, a thickness :math:`t=0.048m` and a length of :math:`L=0.05m`.


Material properties
----------------------

The material is homogeneous, isotropic, and linear elastic. The elastic coefficients are:

:math:`E=185000\mathrm{MPa}` and :math:`\nu =0.3`.

The density is constant and is equal to: :math:`\rho =7800{\mathrm{kg.m}}^{-3}`.


Boundary conditions and loads
-------------------------------------

The structure is free in space.


Order of magnitude of natural frequencies
----------------------------------------

The desired natural modes correspond to Fourier modes of order 2 and 3 of the ring. The frequencies of a ring can be estimated from an analytical model of an Euler curved beam [:ref:`bib1 <bib1>`]. For a Fourier mode of order :math:`n`, the frequency is equal to:

:math:`{f}_{n}=\frac{n({n}^{2}-1)}{2\pi {R}_{m}^{2}}\sqrt{\frac{E{I}_{y}}{m({n}^{2}+1)}}`

where: :math:`{I}_{y}=\frac{L{t}^{3}}{12}` and :math:`m=\rho Lt`

For the ovalization (:math:`n=2`) and trifoliate (:math:`n=3`) modes, the corresponding frequencies are equal to :math:`211.65\mathrm{Hz}` and :math:`598.64\mathrm{Hz}` respectively. The search for natural modes is carried out on band :math:`200-800\mathrm{Hz}` in order to capture these two Fourier modes.


Bibliographical reference
-------------------------


1. Blevins R.D., Formulas for natural frequency and mode shape, N.Y.: Van Nostrand Reynhold Company, 1979, 492 pp.