1. Reference problem#
1.1. Geometry#
It is a cylindrical ring, with an average radius \({R}_{m}=0.369m\), a thickness \(t=0.048m\) and a length of \(L=0.05m\).
1.2. Material properties#
The material is homogeneous, isotropic, and linear elastic. The elastic coefficients are:
\(E=185000\mathrm{MPa}\) and \(\nu =0.3\).
The density is constant and is equal to: \(\rho =7800{\mathrm{kg.m}}^{-3}\).
1.3. Boundary conditions and loads#
The structure is free in space.
1.4. 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 [bib1]. For a Fourier mode of order \(n\), the frequency is equal to:
\({f}_{n}=\frac{n({n}^{2}-1)}{2\pi {R}_{m}^{2}}\sqrt{\frac{E{I}_{y}}{m({n}^{2}+1)}}\)
where: \({I}_{y}=\frac{L{t}^{3}}{12}\) and \(m=\rho Lt\)
For the ovalization (\(n=2\)) and trifoliate (\(n=3\)) modes, the corresponding frequencies are equal to \(211.65\mathrm{Hz}\) and \(598.64\mathrm{Hz}\) respectively. The search for natural modes is carried out on band \(200-800\mathrm{Hz}\) in order to capture these two Fourier modes.
1.5. Bibliographical reference#
Blevins R.D., Formulas for natural frequency and mode shape, N.Y.: Van Nostrand Reynhold Company, 1979, 492 pp.