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AutoFEM Analysis Natural Frequencies of a Cantilever Beam |  |
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AutoFEM Analysis Natural Frequencies of a Cantilever Beam | |
Natural Vibration Frequencies of a Cantilever Beam
Given is a cantilevered beam of length L with a rectangular cross-section of width b and height h.
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Sought are the three natural frequencies of the beam.
Assume L = 0.5 m, b = 0.05 m, h = 0.02 m.
The material properties are: Young's modulus E = 2.1E+011 Pа, Poisson's ratio ν = 0.28, the density ρ = 7800 kg / m3.
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The finite element model with restraints |
The analytical solution appears as:
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where fi - natural frequencies, E – the material Young's modulus, J – the moment of inertia, ρ – the material density, F – the area of the cross section, L – the beam length, ki - the factor that depends on the vibration mode ( k1 = 1.875, k2 = 4.694, k3 = 7.855 ).
The results are as follows*:
Table 1. Parameters of finite element mesh
Finite Element Type |
Number of nodes |
Number of finite elements |
quadratic tetrahedron |
395 |
906 |
Table 2. Result "Frequency"*
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Numerical solution |
Analytical solution |
Error δ = 100%*| fi* - fi| / | fi | |
1 |
67.307
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67.0
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0.46 |
2 |
418.736
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420.2
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0.35 |
3 |
1157.75
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1176.7
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1.61 |
Conclusion:
The relative error of the numerical solution compared to the analytical solution is equal to 0,4-1.6% for the first and third forms when using quadratic finite elements and it grows with the increase in the form number.
*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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