Natural Vibrations of a Spherical Dome

Let us consider a spherical dome of radius R, clamped along the contour (see figure).

Thickness of wall of the dome h is considerably smaller than its radius R.
Only one quarter of spherical surface will be considered. The bottom edge is fully restrained, the symmetry boundary conditions are applied to the side faces.

Let us use the following data: radius R = 300 mm, thickness h = 3 mm (R / h = 100).
The material properties are: the Young's modulus E = 2.1E+011 Pа, Poisson's ratio ν=0.28, the density ρ = 7800 kg / m3.
Analytical solution of this problem is given by:

fi= ki . ω0 / 2π

,

where E – Young’s modulus, ki – coefficient whose value for the first five natural frequencies is: 0.5457, 0.7377, 0.8563, 0.8598, 0.9034.
Thus, f1 = 1564.7 Hz , f2 = 2115.3 Hz , f3 = 2455.4 Hz, f4 = 2465.4 Hz f5 = 2590.4 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"*

Conclusion:

The relative error of the numerical solution compared with the analytical solution not exceed 1%.

*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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