2. Benchmark solution

2.1. Calculation method used for the reference solution

• first natural frequency

The data of the problem are chosen in such a way that the stiffness in bending and in extension is large compared to the geometric and centrifugal stiffness. Under these conditions, the value of the first natural frequency is obtained analytically by considering a rigid pendulum.

Taking angle \(\theta\) between the pendulum and the \(X\) axis as the degree of freedom, the equation of motion is written:

\(2L\ddot{\theta }=\mathrm{3g}\mathrm{cos}\theta -{\Omega }^{2}(\mathrm{3a}+\mathrm{2L}\mathrm{cos}\theta )\mathrm{sin}\theta\)

Here we consider the small oscillations \(\Delta \theta\) of the pendulum around a position of static equilibrium \({\theta }_{0}\). By linearizing the equation of motion in the vicinity of this position, we obtain the equation for small disturbances:

\(2L\Delta \ddot{\theta }+\left[\mathrm{3g}\mathrm{sin}{\theta }_{0}+{\Omega }^{2}(\mathrm{3a}\mathrm{cos}{\theta }_{0}+\mathrm{2L}\mathrm{cos}2{\theta }_{0})\right]\Delta \theta =0\)

The pulsation of the first mode is deduced from this:

\(\omega =\sqrt{\frac{\mathrm{3g}}{\mathrm{2L}}\mathrm{sin}{\theta }_{0}+{\Omega }^{2}\left[\frac{\mathrm{3a}}{\mathrm{2L}}\mathrm{cos}{\theta }_{0}+\mathrm{cos}2{\theta }_{0}\right]}\)

This natural pulsation can still be written in the form

\(\omega =\sqrt{\frac{K(\sigma )+K({\Omega }^{2})}{I}}\)

with

\(K(\sigma )=\frac{1}{2}\rho S{L}^{2}g\mathrm{sin}{\theta }_{0}+\rho S{L}^{2}{\Omega }^{2}\left[\frac{a}{2}\mathrm{cos}{\theta }_{0}+\frac{L}{3}{\mathrm{cos}}^{2}{\theta }_{0}\right]\) (geometric stiffness)

\(K({\Omega }^{2})=-\frac{1}{3}\rho S{L}^{3}{\Omega }^{2}{\mathrm{sin}}^{2}{\theta }_{0}\) (centrifugal stiffness)

\(I=\frac{1}{3}\rho S{L}^{3}\) (rotational inertia)

• other natural frequencies

The reference values for frequencies 2 to 6 are obtained numerically using version 7 of software SAMCEF. Two different models were used: 20 shear deformable beam elements and \(20\times 4\) membrane elements with 8 knots. The results obtained in both cases are identical if we limit ourselves to the first 4 significant figures. Since the stiffness corrections are small with respect to the terms of linear stiffness, it can be verified that the frequencies 2 to 6 differ little from the analytical values obtained for a slender beam that is not deformable under the shear force. In fact, the maximum difference between numerical and analytical values does not exceed \(\text{1 \%}\).

2.2. Benchmark results

The first 5 critical loads are ranked in order of increasing modulus.

Mode	Natural frequency ( \(\mathrm{Hz}\) )
1	1.75556
2	100.2
3	324.0
4	674.4
5
6

2.3. Uncertainty about the solution

Analytical solution for the first frequency. Digital solution for others. The estimated tolerance of the numerical results is \(\text{1 \%}\).

2.4. Bibliographical references

Not applicable.