2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is the one given in sheet SDLL02 /89 of the guide VPCS which presents the calculation method as follows:
By the dynamic stiffness method, it is shown that the folded beam admits double frequencies, solution of:
\(\begin{array}{ccc}\mathrm{cos}(\lambda )\mathrm{=}0& \mathrm{\Rightarrow }& {\lambda }_{i}\mathrm{=}(\mathrm{2i}\mathrm{-}1)\frac{\pi }{2}\\ {f}_{i}\mathrm{=}\frac{1}{2\pi }\frac{{\lambda }_{i}^{2}}{{L}^{2}}\sqrt{\frac{{\mathit{EI}}_{z}}{\rho A}}& & i\mathrm{=}\mathrm{1,}\mathrm{2,}\mathrm{...}\end{array}\)
For a rectangular cross section, we obtain:
\({f}_{i}\mathrm{=}{(\mathrm{2i}\mathrm{-}1)}^{2}\pi \frac{R}{{\mathrm{8L}}^{2}}\sqrt{\frac{E}{12\rho }}\) \(i=\mathrm{1,}\mathrm{2,}\mathrm{...}\)
This formulation neglects shear force and rotational inertia deformations (Euler-Bernoulli beam).
For clean modes, the shapes are given in guide VPCS. They are normalized to 1 or —1 at the point of greatest amplitude. We only get results for modes 1, 2, 3, 3, 4, 7, and 8. For example, the shapes of the first two eigenmodes are as follows:
Note:
In Code_Aster, when an eigenvalue is multiple, the eigenmodes associated with this eigenvalue, even if they are normalized and orthogonal two by two, are, a prima facie, unpredictable. At the moment, we do not know how to test the form of a multiple mode.
2.2. Benchmark results#
The reference results are the first eight natural frequencies.
2.3. Uncertainty about the solution#
There is no uncertainty about the solution because it is analytical.
2.4. Bibliographical references#
PIRANDA J.: Courses and Tutorials in Structural Vibrations - Mechanical Option - National School of Mechanics and Micromechanics - Laboratory of Applied Mechanics - Besançon (France (1983).)