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


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:

:math:`\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:

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

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

:math:`{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:

:math:`{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}` :math:`{f}_{3}^{\mathrm{\ast }}\mathrm{\simeq }{f}_{2}`

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


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

The first two natural frequencies for :math:`\lambda \mathrm{=}0.`

The first three natural frequencies for :math:`\lambda \mathrm{=}0.001` and :math:`\lambda \mathrm{=}0.01`.


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

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


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).