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


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

The reference solution is the one given in sheet SDLL09 /89 of the guide VPCS which presents the calculation method as follows:

Exact calculation by numerical integration of the differential equation of the bending of beams (Euler-Bernouilli theory).

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

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

We get:

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

with:

:math:`\begin{array}{c}\alpha \mathrm{=}\frac{{h}_{0}}{{h}_{1}}\mathrm{=}4\\ \beta \mathrm{=}\frac{{b}_{0}}{{b}_{1}}\mathrm{=}4\mathit{ou}5\end{array}`


.. csv-table::

    "", ":math:`{\lambda }_{1}` "," :math:`{\lambda }_{2}` "," :math:`{\lambda }_{3}` "," "," :math:`{\lambda }_{4}` "," :math:`{\lambda }_{5}`"
    ":math:`\beta \mathrm{=}4` ", "23.289", "73.9", "73.9", "165.23", "299.7", "478.1"
    ":math:`\beta \mathrm{=}5` ", "24.308", "75.56", "75.56", "167.21", "301.9", "480.4"



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

5 first natural modes of flexure.


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

Semi-analytical solution.


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

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).