2. Benchmark solution

2.1. Calculation method used for the reference solution

The reference solution is that given in Timoshenko’s book on the vibration theory of beams and plates ([1]):

Problem 1: Analytical calculation

The flexure equation for non-slender beams gives Timoshenko’s formulation, superimposing the effects of simple bending, shear force deformations, and rotational inertia.

The natural frequencies in traction-compression are given according to this theory by:

\({f}_{i}=\frac{{\lambda }_{i}}{2\pi L}\sqrt{\frac{E}{\rho }}\) with \({\lambda }_{i}=\frac{(\mathrm{2i}-1)}{2}\pi\) \(i=\mathrm{1,2},\mathrm{...}\)

An equivalent formula for the flexure modes is found in reference [1].

Problem 2:

Since the problem has no analytical solution, the solution is established by non-regression results.

The flexure and traction-compression modes are combined.

2.2. Benchmark results

Problem 1:6 first clean modes.

Problem 2: First 5 clean modes.

2.3. Uncertainty about the solution

Problem 1: analytical solution. `

2.4. Bibliographical references

1. S.P. TIMOSHENKO, D.H. YOUNG, W. WEAVER. Vibrations Problems in Engineering. New‑York: Wiley & Sons, 4th edition, p. 415 (1974).