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# Validation Case: Modal Analysis of Annular Plate

In this case, modal analysis is performed on an annular plate which has its inner diameter 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.

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

• Modal Solver
• Restraints

## Geometry: Geometry Dimensions

• Part 1: Inner diameter of inner plate = 0.508m
• Part 2: Inner diameter of inner plate = 0.7112m
• Part 3: Inner diameter of outer plate = 0.8636m
• Part 3: Outer diameter of outer plate = 1.016m
• Thickness of all plates = 0.0254m

## Material:

These materials are taken directly from the OnScale library of materials.

Part 1, 2 & 3:

• Young’s Modulus (E) = 200 GPa
• Density (ρ) = 7855 kg/m³
• Poisson’s Ratio (v) = 0.30

Note: All other material properties can be left as their default values.

## Physics:

Mechanical Physics:

• Restraints (Fixture)
• Part 1 – Inner Face – Face 0 ## Meshing:

OnScale Solve automatically generates a 3D second-order tetrahedral mesh. The meshing statistics are:

Mesh Quality: Very Fine

## Analysis:

The analysis type can be selected in the Simulator section.

Mechanical Analysis Type: Modal

Definitions: Modes

Number of Modes: 6

## Analytical Solution:

The plate equation in polar coordinates has the following general solution for the transverse vibration of a circular plate which has elementary, polar symmetric boundary conditions: i = 0, 1, 2, 3…

Where the natural frequency, f, is a function of the dimensionless parameter λ: Where a = Outer radius of outer plate, h = thickness of plate, ɣ = mass per unit area

## Results Comparison:

 Results Analytical Method (Hz) OnScale Solve (Hz) Natural Frequency 1 310.911 312.072 Natural Frequency 2 318.086 317.618 Natural Frequency 3 318.086 317.621 Natural Frequency 4 351.569 349.023 Natural Frequency 5 351.569 349.045 Natural Frequency 6 442.451 437.613
 1st Mode: 2nd Mode: 3rd Mode: 4th Mode: 5th Mode: 6th Mode: ## References:

 R. J. Blevins, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company Inc., 1979, Table 11-2, Case4, pg. 247