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> > Validation Case: Modal Analysis of Annular Plate

# Problem Statement

In this case, modal analysis is performed on an annular plate which has its inner diameter fully fixed. Modal analysis is used to determine the natural frequency and mode shapes of the structure. The aim of this FEA analysis is to find the first 6 natural frequencies of the annular plate. It should be noted that since the moment of inertia of the annulus is the same with respect to two coordinate axis, modes involving azimuthal dependencies will be degenerated and will come in pairs: any linear combination of these two modes will also be a valid natural mode.

The comparison of the simulation results against the analytical calculations validates the use of following conditions for modal analysis in Solve:

• Modal Solver
• Restraints

## Geometry

The CAD for the geometry used for this analysis is available as an OnShape Document, where you can download the file by right clicking on the part “Disk”, and choosing the export option. Key dimensions are presented in table 1.

Table 1: Description of model geometry.
SymbolDimensionValue [mm]
2aInner diameter508.0
2ROuter diameter1,016.0
hThickness25.4

## Material Properties

A material model with homogeneous isotropic linear elastic properties was applied to the geometry. Material properties are shown in table 2.

Table 2: Material properties used in model.
SymbolPropertyValue
EYoung modulus210~\text{GPa}
\nuPoisson’s ratio0.3
\rhoDensity7,855~\text{kg} \cdot \text{m}^{-3}

## Boundary Conditions

The inner face of the disk is fully restrained (fixture condition). All other faces are free.

PartFace DescriptionFace index
Part 1Inner faceFace 0

## Meshing

OnScale Solve automatically generates a 3D second-order straight-edge tetrahedral mesh. For this study, the “very fine” mesh quality option was used such that the model had the properties in table 3.

Table 3: Mesh details.
Mesh TierNumber of CellsNumber of VerticesDegrees of Freedom
Very Fine50,04113,262258,087

## Analysis

To perform a modal analysis, the analysis type was selected in the simulator section of the interface and changed the “Mechanical Analysis Type” selection to “modal.” Further configurations could be made to select the number of modes to be calculated and setting the entries definitions to “modes” and number of modes to 6.

# Results

## Analytical Solution

Under plane-stress conditions, the mid-surface displacement of the plate perpendicular to the plate plane has the following general expression for the i-th natural mode with the inner surface clamped [1, Ch. 11]:
Z(r,\theta,t)=\left[b_i \mathcal{J}_i \left(\frac{\lambda_i r}{R}\right) +c_i \mathcal{Y}_i \left(\frac{\lambda_i r}{R}\right) +d_i \mathcal{I}_i \left(\frac{\lambda_i r}{R}\right) +e_i \mathcal{K}_i \left(\frac{\lambda_i r}{R}\right)\right]\cos(n_i \theta)\qquad(1)
where

• \mathcal{J}_i is the Bessel function of first kind and i-th order.
• \mathcal{Y}_i is the Bessel function of second kind and i-th order.
• \mathcal{I}_i is the Modified Bessel function of first kind and i-th order.
• \mathcal{K}_i is the Modified Bessel function of second kind and i-th order.
• b_i, c_i, d_i and e_i are constants determined by the boundary conditions.
• n_i is the number of nodal radii in the i-th mode.

The frequency f_i of each mode is computed as

f=\frac{\lambda_i^2}{2 \pi R^2} \sqrt{\frac{E h^3}{12 \cdot \rho h \cdot (1-\nu^2)}}

where \lambda_i^2 is the solution of a transcendental equation which is out of the scope of this post. Table 4 shows the numerical values for \nu=0.3 and a/R = 1/2. Recall that modes 2 & 3 and 4 & 5 correspond to degenerated pairs.

Table 4: Numerical values of \lambda^2 for different modes [1].
i\lambda_i^2
113.0
213.3
313.3
414.7
514.7
618.5

## Result Comparison

Table 5 lists the modal frequencies calculated from the reference analytical model and FEA using Onscale Solve. Once again, any linear combination of modes 2 & 3 and of modes 4 & 6 is also a natural oscillation mode.

Table 5: Comparison of modal frequencies between plane-stress analytical method and OnScale Solve.
ModeRef. [Hz]Solve [Hz]Difference [%]
13193180.5
23263231.0
33263231.0
43603551.5
53603551.5
64534452.0

The results computed by the solver agree with those obtained from the analytical solution with the error being less than 2% for the worst case.

# References

[1]
R. Blevins, Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold Company, 1979.