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AutoFEM Analysis Bending Vibrations of a Circular Ring (surface) |  |
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AutoFEM Analysis Bending Vibrations of a Circular Ring (surface) | |
Bending Vibrations of a Circular Ring
Let us consider a circular ring. A radius R of the central line is 1 m.
A square cross-section is considerably smaller than the radius R . The length of the side of the square is equal 0.050 m .
Displacement along the normal of the ring is restricted. In-plane rotations are also restricted.
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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:
, 
.
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 |
linear triangle |
1735 |
2958 |
Table 2. Result "Frequency"*
Numerical solution |
Analytical solution |
Error δ = 100%*| fi* - fi| / | fi | |
31.565 |
31.015 |
1.77 |
89.129 |
87.723 |
1.60 |
170.361 |
168.201 |
1.28 |
274.355 |
272.017 |
0.86 |
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*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.
Read more about AutoFEM Frequency Analysis
