OverviewThe purpose of this example is to compare the predicted natural frequencies of a cantilever beam with the standard theoretical result.
Hand CalculationFigure 1 shows a cantilever beam with the following properties:Figure 1  Cantilever beam
A circular cross section of radius, R, 5mm, giving:
(Equation 3b from Table 16.1, Chapter 16, page 765) where:
Note: mass of the beam, M = 1.0 * 7.8537e5 * 2700 = 0.212kg Thus, f_{n} = 2.06037K_{n} Using Roark the values of K_{n} are used to find the the first five modes. These are shown in Table 1.
Finite Element ModelThe finite element model discretises the bar into 10 CBAR elements.The model can be downloaded here. The node numbers are shown overlaid on the elements below: The beam is constrained at the root using all 6 degrees of freedom as shown below: Using LUCID/iron the output from the FE model is shown below. Notes: (i) these results are for a version of LUCID/iron with the CBAR element consistent mass matrix rather than element lumped mass matrix being used. (See Logan for more information); (ii) the model was constrained at one end only, thus there are two bending modes being reported for each frequency; Comparison of ResultsThe theoretical model and FE model results are compared side by side in the table below. This shows that there is very little difference in the numbers being reported by both examples in this case (less than 0.1% difference). In this case the principle source of difference is likely to be roundoff error in the hand calculation.

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