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Example: Natural Frequencies of a Cantilever Beam

Overview

The purpose of this example is to compare the predicted natural frequencies of a cantilever beam with the standard theoretical result.


Try this example using the FREE LUCID/iron application

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Hand Calculation

Figure 1 shows a cantilever beam with the following properties:

Figure 1 - Cantilever beam
  • Length, L = 1 metre (m).

A circular cross section of radius, R, 5mm, giving:
  • a cross sectional area, A, of 7.8537e-5m;
  • area moments of inertia, Izz = Iyy = 4.909e-10m4;
  • beam torsional constant = 9.8175e-10m4;
  • material density, rho = 2700kg/m3; 
  • material elasticity, E = 72 GPa.
Using Roark's Formula for Stress and Strain (7th edition) the modes of vibration of a cantilever beam are as follows:
(Equation 3b from Table 16.1, Chapter 16, page 765)

    

where:
  • Kn is a constant where n refers to the mode of vibration;
  • w = load per unit length including beam weight (Newtons/metre) = A*rho*g = 7.8537e-5 * 2700 * 9.81 = 2.0802 N/m.

Note: mass of the beam, M = 1.0 * 7.8537e-5 * 2700 = 0.212kg

Thus, fn = 2.06037Kn

Using Roark the values of Kn are used to find the the first five modes. These are shown in Table 1.

Mode Kn
 fn (Hz)
1
3.52
7.25
2
22.0
45.3
3
61.7
127.1
4
121
249.3
5
200
412.1
Table 1 - First five modes of the beam

Finite Element Model

The 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:

Node IDs

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 Results

The 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 round-off error in the hand calculation.

Mode Theory
(Hz)
 FE Model
(Hz)
Modeshape
(Note: Displacements not normalised - this is not the peak displacement.)
1 7.25 7.24
2 45.3
 45.4

3 127.1 127.1
4 249.3
249.2
5
 412.1
 412.5


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