2. Benchmark solution

2.1. Calculation method used for the reference solution

The reference solution is the one given in sheet SDLL04 /89 of the guide VPCS which presents the calculation method as follows:

The equation with the natural pulsations of the complete system is written as:

\(\lambda {r}_{i}L\left[\frac{\mathrm{sin}({r}_{i}a)\mathrm{sin}({r}_{i}b)}{\mathrm{sin}({r}_{i}L)}\mathrm{-}\frac{\mathit{sh}({r}_{i}a)\mathit{sh}({r}_{i}b)}{\mathit{sh}({r}_{i}L)}\right]\mathrm{=}2({\omega }_{i}^{2}\mathrm{-}{\omega }_{c}^{2})\mathrm{/}{\omega }_{c}^{2}\)

with:

\(\lambda \mathrm{=}\frac{{m}_{e}}{\rho AL}\) \({r}_{i}^{4}\mathrm{=}{\omega }_{i}^{2}\frac{\rho A}{EI}\) \({\omega }_{C}\mathrm{=}\frac{{k}_{e}}{{m}_{e}}\) \(a+b\mathrm{=}L\)

In the absence of a secondary system, \({k}_{e},{m}_{e}\mathrm{=}0\), we find the natural frequencies of the slender beam on two supports.

\({f}_{i}\mathrm{=}{i}^{2}\frac{\pi }{2}\frac{1}{{L}^{2}}\sqrt{\frac{\mathit{EI}}{\rho A}}\mathrm{=}{i}^{2}\frac{\pi }{2}\)

When the secondary system is exactly tuned to the first mode of this beam, the new natural frequencies of the system can be obtained by the approximate formulas:

\({f}_{\mathrm{1,2}}^{\mathrm{\ast }}\mathrm{=}(1\mathrm{\pm }0.5\sqrt{\frac{{m}_{e}}{{M}_{1}}}){f}_{1}\mathrm{=}(1\mathrm{\pm }0.5\sqrt{\lambda }){f}_{1}\) \({f}_{3}^{\mathrm{\ast }}\mathrm{\simeq }{f}_{2}\)

with \({M}_{1}\) modal mass of the beam without a secondary system for an eigenmode normalized to 1 at point \(D\).

2.2. Benchmark results

The first two natural frequencies for \(\lambda \mathrm{=}0.\)

The first three natural frequencies for \(\lambda \mathrm{=}0.001\) and \(\lambda \mathrm{=}0.01\).

2.3. Uncertainty about the solution

Less than \(4\lambda \text{\%}\) for the first few modes if the system is tuned to the first mode.

2.4. Bibliographical references

• NOUR - OMID, SACKMAN, KIUREGHIAN. Modal characterization of equipment continuous structure system. Journal of Sound and Vibration, V.88 no. 4, p.459,472 (1983).