Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is the one given in sheet SDLL15 /89 of the guide VPCS, which presents the calculation method as follows:

The problem with non-eccentric mass leads to decoupled modes:


* traction-compression (effect of mass alone),


* twisting (effect of inertia around the neutral fiber),


* flexure in planes :math:`x,y` and :math:`x,z` (effect of mass).

The various natural frequencies are determined with a finite element model of an Euler beam (slender beam).

For the first mode with an eccentric mass, a Rayleigh method gives the approximate formula:

:math:`{f}_{1}\mathrm{=}\frac{1}{2\pi }\sqrt{\frac{3E{I}_{z}}{{L}^{3}({m}_{c}+0.24M)}}`

with :math:`M` = total mass of the beam.

When the mass is eccentric, the flexure :math:`(x,z)` and torsional modes are coupled, as well as the :math:`(x,y)` flexure and traction-compression modes.

For the natural mode, the components at point :math:`B` make it possible to calculate the components at the center of gravity of the mass (point :math:`C`) by:

:math:`\left[\begin{array}{c}{u}_{c}\\ {v}_{c}\\ {w}_{c}\end{array}\right]\mathrm{=}\left[\begin{array}{c}{u}_{B}\\ {v}_{B}\\ {w}_{B}\end{array}\right]+\left[\begin{array}{ccc}0& {z}_{c}& \mathrm{-}{y}_{c}\\ \mathrm{-}{z}_{c}& 0& +{x}_{c}\\ +{y}_{c}& \mathrm{-}{x}_{c}& 0\end{array}\right]\left[\begin{array}{c}{\theta }_{\mathit{xB}}\\ {\theta }_{\mathit{yB}}\\ {\theta }_{\mathit{zB}}\end{array}\right]`

:math:`{u}_{c}\mathrm{=}{u}_{B}\mathrm{=}\mathrm{-}{\theta }_{\mathit{zB}}`

For this test:

:math:`\begin{array}{c}{v}_{c}\mathrm{=}{v}_{B}\\ {w}_{c}\mathrm{=}{w}_{B}+{\theta }_{\mathit{xB}}\end{array}`


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

Case 1:10 first clean modes.

Case 2:8 first clean modes.


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


.. csv-table::

    "Problem 1:", ":math:`\mathit{f1}` analytical solution"
    "", "other frequencies :math:`\mathrm{\pm }1\text{\%}`"
    "Problem 2:", ":math:`\pm 1\text{\%}`"



Bibliographical references
---------------------------


1. Dynamic Analysis Working Group. Commission for the Validation of Structural Analysis Software Packages. French Society of Mechanics. (1988)