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


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 ([:ref:`1 <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:

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

         An equivalent formula for the flexure modes is found in reference [:ref:`1 <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.


Benchmark results
----------------------

Problem 1:6 first clean modes.

Problem 2: First 5 clean modes.


Uncertainty about the solution
---------------------------

Problem 1: analytical solution. `


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