First Natural Frequency of a Cantilever Beam under the Stretching Longitudinal Force

Let us consider a cantilever beam. A length of the beam is L. The cross-section of the beam is a rectangle of the width b and the height h.

The beam is clamped on the left end and loaded with the stretching longitudinal  force P on the right end.

Let us use the following initial data: the length L of the beam is 0.5 m, the width b is 0.05 m, and the height h is 0.02 m, the magnitude of the applied force P is 50000 N.
Material characteristics: E = 2.1E+011 Pa, ν = 0.28.
Sought quantity is the first natural frequency of the beam under the load.

The analytical solution appears as:

,
where f1 is the first natural frequency of the cantilever beam, J – the moment of inertia, ρ – the material density, F – the area of the cross section, k1 = 1.875.

Thus, f1* = 85.804 Hz.

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1. Parameters of finite element mesh

Table 2. Result "Frequency"*

 

 

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

 

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