2. Benchmark solution#

2.1. 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 \(x\)

  • \(y\) and \(z\) the flexure movements in planes \(\mathrm{xOz}\) and \(\mathrm{xOy}\)

  • \(S\): section of the beam

  • \(I\): flexural moment of inertia with respect to the \(y\) and \(z\) axes

  • \({I}_{x}\): moment of inertia per unit length with respect to the \(\mathrm{Ox}\) axis

  • \(\rho\), \(E\) the characteristics of the material

  • \(\Omega\) girder rotation speed

Singular solutions are governed by the following system of equations:

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

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

by complying with the following boundary conditions:

\(\{\begin{array}{}Y=Z=0\\ \frac{{\partial }^{2}Y}{\partial {z}^{2}}=\frac{{\partial }^{2}Z}{\partial {z}^{2}}=0\end{array}\) in \(\{\begin{array}{}x=0\\ x=L\end{array}\)

Two families of natural modes are obtained:

  1. Retrograde mode:

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

  1. Direct mode:

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

by asking:

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

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

2.2. Benchmark results#

4 first natural modes of flexure.

2.3. Uncertainty about the solution#

Analytical solution with the Euler beam hypothesis.

2.4. Bibliographical references#

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