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AutoFEM Analysis Bending Vibrations of a Circular Ring |  |
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AutoFEM Analysis Bending Vibrations of a Circular Ring | |
Bending Vibrations of a Circular Ring
Let us consider a circular ring. A radius R of the central line is 1 m (see figure).
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A square cross-section is considerably smaller than the radius R . The length of the side of the square is equal 0.050 m .
The bottom face is restrained along the normal.
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The finite element model with restraints |
The material properties are: the Young's modulus E = 2.0E+011 Pа, Poisson's ratio ν=0.29, the density ρ = 7900 kg / m3.
Analytical solution of this problem is given by:
, 
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Thus, f2 = 31.015 Hz , f3 = 87.723 Hz , f4 = 168.201 Hz, f5 = 272.017 Hz.
After carrying out calculation with the help of AutoFEM, the following results are obtained*:
Table 1. Parameters of finite element mesh
Finite Element Type |
Number of nodes |
Number of finite elements |
quadratic tetrahedron |
1000 |
1767 |
Table 2. Result "Frequency"*
Numerical solution |
Analytical solution |
Error δ = 100%*| fi* - fi| / | fi | |
31.056 |
31.015 |
0.13 |
87.680 |
87.723 |
0.05 |
167.677 |
168.201 |
0.31 |
270.245 |
272.017 |
0.65 |
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Conclusions:
The relative error of the numerical solution compared with the analytical solution not exceed 0.8%.
*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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