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AutoFEM Analysis Determining the First Natural Frequency of a Round Plate |  |
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AutoFEM Analysis Determining the First Natural Frequency of a Round Plate | |
Determining the First Natural Frequency of a Round Plate
Sought is the natural frequency of the first vibration mode of a round plate of radius R and thickness h, clamped along the contour.
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Assume the plate radius equal to R = 0.2 m, the plate thickness h = 0.01 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. Due to the symmetry, we will consider the quarter of the plate and apply the appropriate boundary conditions.
Let us calculate the first natural frequency using, first, tetrahedral finite elements. Obtained results are compared with the analytical solution which is given by:
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where R – plate radius, ρ – density of material, h – thickness of material, D = E h3 / 12 (1-ν2) – flexural stiffness.
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The finite element model with restraints |
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 |
592 |
1622 |
Table 2. Result "Mode 01"*
Numerical solution |
Analytical solution |
Error δ = 100%*| fi* - fi| / | fi | |
635.545 |
633.9 |
0.26 |
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Conclusions:
The relative error of the numerical solution compared to the analytical solution for the first form is equal to 0.3% for tetrahedral finite element (compare with the calculation based on the triangular elements here).
*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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