Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is the one presented in René-Jean's book GIBERT.

By adopting the following notations:


* next beam :math:`x`


* :math:`y` and :math:`z` the flexure movements in planes :math:`\mathrm{xOz}` and :math:`\mathrm{xOy}`


* :math:`S`: section of the beam


* :math:`I`: flexural moment of inertia with respect to the :math:`y` and :math:`z` axes


* :math:`{I}_{x}`: moment of inertia per unit length with respect to the :math:`\mathrm{Ox}` axis


* :math:`\rho`, :math:`E` the characteristics of the material


* :math:`\Omega` girder rotation speed

Singular solutions are governed by the following system of equations:

:math:`\mathrm{EI}\frac{{\partial }^{4}Y}{\partial {x}^{4}}-\rho S{\omega }^{2}Y+i\omega \Omega {I}_{x}\frac{{\partial }^{2}Z}{\partial {z}^{2}}=0`

:math:`\mathrm{EI}\frac{{\partial }^{4}Z}{\partial {x}^{4}}-\rho S{\omega }^{2}Z-i\omega \Omega {I}_{x}\frac{{\partial }^{2}Y}{\partial {z}^{2}}=0`

by complying with the following boundary conditions:

:math:`\{\begin{array}{}Y=Z=0\\ \frac{{\partial }^{2}Y}{\partial {z}^{2}}=\frac{{\partial }^{2}Z}{\partial {z}^{2}}=0\end{array}` in :math:`\{\begin{array}{}x=0\\ x=L\end{array}`

Two families of natural modes are obtained:


1. Retrograde mode:

:math:`{Y}_{1}=-{\mathrm{i.Z}}_{1}=\mathrm{sin}\frac{n\pi x}{L}` with :math:`(\frac{{\omega }_{1}}{{\omega }_{0}})=\sqrt{{\lambda }^{2}+1}-\lambda`


2. Direct mode:

:math:`{Y}_{2}=-{\mathrm{i.Z}}_{2}=\mathrm{sin}\frac{n\pi x}{L}` with :math:`(\frac{{\omega }_{2}}{{\omega }_{0}})=\sqrt{{\lambda }^{2}+1}+\lambda`

by asking:

natural pulsation without rotation: :math:`{\omega }_{0}={(\frac{n\pi }{L})}^{2}\sqrt{\frac{\mathrm{EI}}{\rho S}}`

:math:`\lambda =\frac{1}{2}\mathrm{.}\frac{\Omega {I}_{x}}{\sqrt{\mathrm{EI}\rho S}}` with :math:`{I}_{x}=\frac{\rho S{D}^{2}}{8}` and :math:`I=\frac{\pi {D}^{4}}{64}`


Benchmark results
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4 first natural modes of flexure.


Uncertainty about the solution
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Analytical solution with the Euler beam hypothesis.


Bibliographical references
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René-Jean GIBERT, Vibrations of structures, No. 69 of the EDF R&D collection at EYROLLES, p. 235-237 (1988).