2. Benchmark solution

2.1. Calculation method used for the reference solution

The reference solution is obtained analytically for a Timoshenko beam, taking into account the shear force deformation and the rotary inertia of the sections. Theoretical aspects are developed in the reference given in 2.4.

Define the following dimensionless quantities:

\({\Omega }_{n}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{n}^{2}\) eigenvalues

\(j=\frac{I}{A{L}^{2}}\) rotary inertia

\(g=\frac{\mathit{EI}}{k\text{'}AG{L}^{2}}\) shear coefficient

The natural frequencies of the first modes of flexure are given by the following expression:

\({\Omega }_{n}=\frac{(g+j){\lambda }_{n}^{2}+1-\sqrt{{(g-j)}^{2}{\lambda }_{n}^{4}+2(g+j){\lambda }_{n}^{2}+1}}{2gj}\)

with

\({\lambda }_{n}=n\pi\), \(n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}\)

The frequencies of the expansion modes are given by:

\({f}_{n}=(\mathrm{2n}-1)\frac{1}{4L}\sqrt{\frac{E}{\rho }}\), \(n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}\)

2.2. Benchmark results

Fashion	Shape	Frequency (\(\mathit{Hz}\))
1	bending	\(115.7\)
2	bending	\(442.2\)
3	extension	\(648.6\)
4	bending	\(931.6\)
5	bending	\(1534.0\)

2.3. Uncertainty about the solution

• Analytical solution.

2.4. Bibliographical references

• ROBERT G., Analytical solutions in structural dynamics, Samtech Report No. 121, Liège, 1996.