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.

Determining the First Natural Frequency of a Round Plate

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 , 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:

,

where R – plate radius, ρ – density of material, h – thickness of material, D = E h3 / 12 (1-ν2) – flexural stiffness.

Determining the First Natural Frequency of a Round Plate, the finite element model with restraints

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
Frequency fi*, Hz

Analytical solution
Frequency fi, Hz

Error δ = 100%*| fi* - fi| / | fi |

635.545

633.9

0.26

Determining the First Natural Frequency of a Round Plate, the result of finite element modelling

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.

 

 

Read more about AutoFEM Frequency Analysis

autofem.com

Return to contents