2. Reference solution

2.1. Calculation method used for the reference solution

In the local frame of reference of the beam:

\(\frac{{\partial }^{2}{U}_{x}}{\partial {x}^{2}}+\frac{\rho }{E}{\omega }^{2}x=0\) with \(\begin{array}{}{U}_{x}(0)=0\\ \frac{\partial {U}_{x}}{\partial x}(L)={\sigma }_{\mathrm{xx}}(L)=0\end{array}\)

By integrating the previous differential equation, we obtain, in the frame of reference of the beam:

\({U}_{x}(x)=\frac{\rho {\omega }^{2}}{2E}(x{L}^{2}-\frac{{x}^{3}}{3})\) \({U}_{y}={U}_{z}=0\)

The movements of all points of the beam are therefore written in the global coordinate system:

\(\begin{array}{}{U}_{x}(X,Y,Z)=\frac{\rho {\omega }^{2}}{2\sqrt{3}E}(r{L}^{2}-\frac{{r}^{3}}{3})\\ {U}_{y}(X,Y,Z)=\frac{\rho {\omega }^{2}}{2\sqrt{3}E}(r{L}^{2}-\frac{{r}^{3}}{3})\\ {U}_{z}(X,Y,Z)=\frac{\rho {\omega }^{2}}{2\sqrt{3}E}(r{L}^{2}-\frac{{r}^{3}}{3})\end{array}\)

with \(r=\sqrt{{X}^{2}+{Y}^{2}+{Z}^{2}}\)

2.2. Benchmark results

Values of the three movements in the center of the section farthest from the axis of rotation.