VM-LSDYNA-SOLVE-051

VM-LSDYNA-SOLVE-051
Natural Frequencies of a Circular Plate

Overview

Reference:	Blevins, R.J. (1979). Formula for Natural Frequency and Mode Shape. Van Nostrand Reinhold Company Inc., p. 241, table 11-1: Circular Plates, case 3.
Analysis Type(s):	Implicit Vibration Analysis
Element Type(s):	2D Quadrilateral Shell Elements
Input Files:	Link to Input Files Download Page

Test Case

This test case models the vibration of a circular plate with its edge subjected to a clamped condition. The objective is to validate the natural frequencies of the first three modes of vibration (j = 0, 1, 2) for the first harmonic (i = 0). The circular plate has a radius of 17 in and a thickness of 0.5 in. Figure 177 illustrates the domain dimensions and boundary conditions.

This problem is also presented in test case VM181 in the Mechanical APDL Verification Manual.

Figure 177: Test case schematic

The following table lists the material and geometric properties of the test case.

Material Properties	Geometric Properties
Young’s modulus ( ) = 3 ⋅ 10⁷ psi Poisson's ratio () = 0.3 Density () = 7.3 ⋅ 10^-4 lbf-s²/in⁴	Radius () = 17 in Thickness () = 0.5 in

Material Properties

Geometric Properties

Young’s modulus ( ) = 3 ⋅ 10⁷ psi

Poisson's ratio () = 0.3

Density () = 7.3 ⋅ 10^-4 lbf-s²/in⁴

Radius () = 17 in

Thickness () = 0.5 in

Analysis Assumptions and Modeling Notes

For a thin flat circular plate of homogeneous, linear elastic material, the natural frequency of the plate can be calculated as:

(25)

where

is a dimensionless parameter

is the plate radius

is the plate thickness

is the mass per unit area

is the Young's modulus

and

is the Poisson's ratio

For a clamped edge condition, the dimensionless parameters for the first three modes of vibration (j = 0, 1, 2) and the first harmonic (i = 0) are: = 10.22, = 39.77, and = 89.10. Therefore, the natural frequencies for the first three modes are: = 172.64 Hz, = 671.79 Hz, and = 1505.07 Hz.

One part is defined to represent the circular plate, meshed with 2D quadrilateral elements. The plate elements use a fully integrated shell formulation with higher accuracy (ELFORM=-16) and an elastic material card (*MAT_ELASTIC) with the properties listed in the table above. The nodes corresponding to the edge of the plate are grouped using *SET_NODE_LIST, and the keyword *BOUNDARY_SPC_SET is used to define the constraint of this node set (translational and rotational constraint about the three axes). The keywords *CONTROL_IMPLICIT_GENERAL (IMFLAG=1), *CONTROL_IMPLICIT_DYNAMICS (IMASS=0), and *CONTROL_IMPLICIT_EIGENVALUE (NEIG=15) are used to activate the implicit eigenvalue static analysis with fifteen eigenvalues to be extracted.

Figure 178: Model setup in LS-DYNA of the 2D modal analysis of a circular plate

Results Comparison

The visualization of the first mode can be performed by reading the d3eigv file generated for the modal analysis. The first three modes of vibration (j = 0, 1, 2) for the first harmonic (i = 0) are the first, sixth, and fifteenth modes respectively, in the current model. To quantify the error between the theoretical and LS-DYNA results, the natural frequencies , and of the circular plate and their relative errors are calculated and shown in the following table. This comparison verifies the agreement between the natural frequencies.

Results	Target	LS-DYNA Solver	Error (%)
Natural Frequency (Hz)	172.64	173.05	0.24
Natural Frequency (Hz)	671.79	674.38	0.38
Natural Frequency (Hz)	1505.07	1512.94	0.52