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


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

The reference solution is obtained using the finite element calculation software SAMCEF for identical models but with coherent elementary mass matrices.

We recall the analytical solution given in sheet SDLL09 /89 of guide VPCS. The differential equation in bending of the beam in question, in Euler-Bernoulli theory, is written (Euler-Bernoulli theory):

:math:`\frac{{\partial }^{2}(E{I}_{z}\frac{{\partial }^{2}v}{\partial {x}^{2}})}{\partial {x}^{2}}=-\rho A\frac{{\partial }^{2}v}{\partial {t}^{2}}`

where :math:`{I}_{z}` and :math:`A` vary with the abscissa.

The natural frequencies are then of the form:

:math:`{f}_{i}=\frac{1}{2\pi }{\lambda }_{i}(\alpha ,\beta )\frac{{h}_{1}}{{L}^{2}}\sqrt{\frac{E}{12\rho }}`

with :math:`\alpha =\frac{{h}_{0}}{{h}_{1}}=4` and :math:`\beta =\frac{{b}_{0}}{{b}_{1}}=4`.

For this value of :math:`\alpha` and :math:`\beta`, the first values of the series :math:`({\lambda }_{i})` are:


.. csv-table::

    "", ":math:`{\lambda }_{1}` "," :math:`{\lambda }_{2}` "," :math:`{\lambda }_{3}` "," "," :math:`{\lambda }_{4}` "," :math:`{\lambda }_{5}`"
    ":math:`\beta =4` ", "23.289", "73.9", "73.9", "165.23", "299.7", "478.1"



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

The reference results selected are the first 5 natural frequencies of the bending modes.


Uncertainty about the solution
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Analytical solution in Bernoulli beam theory, and numerical solution SAMCEF.


Bibliographical references
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1. H.H. MABIE, C.B. ROGERS, Transverse vibrations of double-tapered cantilever beams- Journal of the Acoustical Society of America, No. 51, p.1771-1774 (1972).