2. Reference solution#

2.1. 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):

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

where \({I}_{z}\) and \(A\) vary with the abscissa.

The natural frequencies are then of the form:

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

with \(\alpha =\frac{{h}_{0}}{{h}_{1}}=4\) and \(\beta =\frac{{b}_{0}}{{b}_{1}}=4\).

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

	\({\lambda }_{1}\)	\({\lambda }_{2}\)	\({\lambda }_{3}\)		\({\lambda }_{4}\)	\({\lambda }_{5}\)
\(\beta =4\)	23.289	73.9	73.9	165.23	299.7	478.1

2.2. Benchmark results#

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

2.3. Uncertainty about the solution#

Analytical solution in Bernoulli beam theory, and numerical solution SAMCEF.

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).