2. Benchmark solution

2.1. Calculation method used for the reference solution

The reference solution is the one obtained using Code_Aster, with a harmonic calculation. The harmonic calculation was itself validated using the results provided in the book by Michel LALANNE and Guy FERRARIS.

The comparison will therefore focus on the results obtained via calculations A and B (transitory calculation, exploitation of steady-state results), and the reference defined by calculation C (harmonic calculation). The load is of the unbalance type, the speed of rotation of the rotor being constant.

The movements of the nodes are written in steady state, as a function of time \(t\), in the form:

\(Y(t)={Y}_{\mathrm{max}}\mathrm{.}\mathrm{cos}(\omega t+{\theta }_{y})\)

\(Z(t)={Z}_{\mathrm{max}}\mathrm{.}\mathrm{cos}(\omega t+{\theta }_{z})\)

With:

\({Y}_{\mathrm{max}}\) and \({Z}_{\mathrm{max}}\) the half amplitudes (in \(m\))

\(\omega\): the speed of rotation of the rotor (in \({\mathrm{rd.s}}^{-1}\))

\({\theta }_{y}\) and \({\theta }_{z}\), the phases of the two signals.

It is quite easy, by visualizing the curves or by editing the result file, to record the half-amplitudes of the movements following \(Y\) and \(Z\). To calculate the phases, it is first necessary to determine the temporal abscissa of an extremum of the sine wave, and only then to deduce the phase. The phase corresponds to the offset with respect to that of the unbalance, which will be defined at time \(t=0\), as being of zero value (force due to the unbalance collinear to the direction \(Y\) at the time \(t=0\)).

We therefore note \({t}_{\mathrm{ymax}}\) (abscissa in time of one extremum of the next movement \(Y\))

So we have: \(Y(t={t}_{\mathrm{ymax}})={Y}_{\mathrm{max}}\mathrm{.}\mathrm{cos}(\omega \mathrm{.}{t}_{\mathrm{ymax}}+{\theta }_{y})={Y}_{\mathrm{max}}\)

\(\mathrm{cos}(\omega \mathrm{.}{t}_{\mathrm{ymax}}+{\theta }_{y})=1\)

\(\omega \mathrm{.}{t}_{\mathrm{ymax}}+{\theta }_{y}=2k\pi\)

from where: \({\theta }_{y}=2k\pi -\omega \mathrm{.}{t}_{\mathrm{ymax}}=\mathrm{Ent}(\omega \mathrm{.}{t}_{\mathrm{ymax}})-\omega \mathrm{.}{t}_{\mathrm{ymax}}\)

The same treatment is carried out for the following displacement \(Z\).

Code_Aster provides directly, in harmonic calculation, the phases and amplitudes of the movements to the nodes.

2.2. Benchmark results

Amplitude of the radial displacements (following \(Y\) and \(Z\)) of the loading node (disk 2) in steady state.

Phase in relation to loading of the radial displacements of the loading node (disk 2) in steady state.

2.3. Uncertainty about the solution

Less than \(\text{1\%}\).

2.4. Bibliographical references

Michel LALANNE and Guy FERRARIS, Rotordynamics, Prediction in Engineering, JOHN WILEY AND SONS (1990).