2. Reference solution#
2.1. Calculation method used for the reference solution#
The reference solution is the one given in sheet SDLS04 /89 of the guide VPCS which presents the analytical solution as follows:
The solution of the frequency determinant established from Bessel functions leads to the formula:
\({f}_{\mathrm{ij}}=\frac{1}{2\pi {R}_{e}^{2}}{\lambda }_{\mathrm{ij}}^{2}\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}\)
With:
\(i\) = number of nodal diameters
\(j\) = number of nodal circles
and \({\lambda }_{\mathrm{ij}}^{2}\) such as:
i j |
0 |
1 |
2 |
3 |
|
0 |
13.0 |
13.3 |
13.3 |
14.7 |
18.5 |
1 |
85.1 |
86.7 |
86.7 |
91.7 |
Flexion mode with 2 nodal diameters and 1 nodal circle: \({f}_{\mathrm{2,1}}=\mathrm{559,09}\mathrm{Hz}\)
2.2. Benchmark results#
8 clean modes.
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
A.W. LEISSA, Vibration of Plates, Document NASA SP160, 1969, pp. 19-30.