2. Reference solution#

2.1. 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 \(x,y\) and \(x,z\) (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:

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

with \(M\) = total mass of the beam.

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

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

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

\({u}_{c}\mathrm{=}{u}_{B}\mathrm{=}\mathrm{-}{\theta }_{\mathit{zB}}\)

For this test:

\(\begin{array}{c}{v}_{c}\mathrm{=}{v}_{B}\\ {w}_{c}\mathrm{=}{w}_{B}+{\theta }_{\mathit{xB}}\end{array}\)

2.2. Benchmark results#

Case 1:10 first clean modes.

Case 2:8 first clean modes.

2.3. Uncertainty about the solution#

Problem 1:

\(\mathit{f1}\) analytical solution

other frequencies \(\mathrm{\pm }1\text{\%}\)

Problem 2:

\(\pm 1\text{\%}\)

2.4. Bibliographical references#

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