2. Reference solution#

2.1. Calculation method used for the reference solution#

  • Vibration modes in the plane

For these vibration modes, the equation for bending curved beams by V. Boussinesq (1883), without extending the neutral fiber, leads to:

(2.1)#\[ {f} _ {i} =\ frac {1} {1} {2} {2\ pi}\ sqrt {(\ frac {{i} ^ {2} -1)} ^ {2} -1)} ^ {2}}} {{i}} {{i}} {i} ^ {2} +1}\ times\ frac {{2} +1}\ times\ frac {{i} {\ mathit {EI}} {\ rho {\ mathit {AR}}} {{i}}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}} 4}})}\]

The reference solution is established for thin arcs such as \(\alpha R\ge 100\sqrt{(\frac{{I}_{z}}{A})}\) with \(\alpha\), which is the center angle in radians.

  • Off-plane vibration modes

For transverse vibration modes with rectangular section, the solution was established from the results of two calculation codes, using different formulations.

Natural modes in the plane (polar coordinates \((i,\theta )\))

Off-plane clean modes
Symmetric \({u}_{i}^{\text{'}}=i\mathrm{cos}(i\theta )\) \({v}_{i}^{\text{'}}=\mathrm{sin}(i\theta )\)
_images/10000200000000A1000001E40D47DF5F899015DD.png

\({\theta }_{i}^{\text{'}}=-\frac{1-{i}^{2}}{R}\mathrm{sin}(i\theta )\)

\(i=0\) Set rotation \(i=1\) set translation \(i=2\) \(i=3\)
Antisymmetric \({u}_{i}^{\text{'}}=i\mathrm{sin}(i\theta )\) \({v}_{i}^{\text{'}}=-\mathrm{cos}(i\theta )\) \({\theta }_{i}^{\text{'}}=-\frac{1-{i}^{2}}{R}\mathrm{cos}(i\theta )\)
_images/10000200000000A8000001F385C32AFB0B5097BF.png

\(i=0\) \(i=1\) \(i=2\) \(i=3\) \(i=4\)

 _images/10000200000000AB000002080926F5C1B4FB4007.png

2.2. Reference quantity#

  • \(\mathrm{FREQ}\): frequency

2.3. Reference quantity and result#

Component

Nature of clean mode

Reference \((\mathrm{Hz})\)

\(i\)

order

Modes in the plan

\(\mathrm{FREQ}\)

\(2\)

\(\mathrm{4,5}\)

\(318.36\)

\(3\)

\(\mathrm{6,7}\)

\(900.46\)

\(4\)

\(\mathrm{8,9}\)

\(1726.55\)

\(5\)

\(\mathrm{10,11}\)

\(2792.21\)

Transverse modes

\(\mathrm{FREQ}\)

\(2\)

\(\mathrm{4,5}\)

\(511.\)

\(3\)

\(\mathrm{6,7}\)

\(1590.\)

\(4\)

\(\mathrm{8,9}\)

\(3184.\)

2.4. Bibliographical references#

[1] Structural Calculation Software Validation Guide: SFM, AFNOR technique, ISBN: 2-12-486611-7