2. Benchmark solution#
2.1. 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).
\(\frac{{\mathrm{\partial }}^{2}({\mathit{EI}}_{z}\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^{2}})}{\mathrm{\partial }{x}^{2}}\mathrm{=}\mathrm{-}\rho A\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{t}^{2}}\)
where \({I}_{z}\) and \(A\) vary with the abscissa.
We get:
\({f}_{i}\mathrm{=}\frac{1}{2\pi }{\lambda }_{i}(\alpha ,\beta )\frac{{h}_{1}}{{L}^{2}}\sqrt{\frac{E}{12\rho }}\)
with:
\(\begin{array}{c}\alpha \mathrm{=}\frac{{h}_{0}}{{h}_{1}}\mathrm{=}4\\ \beta \mathrm{=}\frac{{b}_{0}}{{b}_{1}}\mathrm{=}4\mathit{ou}5\end{array}\)
\({\lambda }_{1}\) |
|
|
|
|
||
\(\beta \mathrm{=}4\) |
23.289 |
73.9 |
73.9 |
165.23 |
299.7 |
478.1 |
\(\beta \mathrm{=}5\) |
24.308 |
75.56 |
75.56 |
167.21 |
301.9 |
480.4 |
2.2. Benchmark results#
5 first natural modes of flexure.
2.3. Uncertainty about the solution#
Semi-analytical solution.
2.4. 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).