Overview

Test Case

This test case models the free vibration of an annular plate, subjected to a free-clamped condition. The objective is to validate the first six natural frequencies of the structure. The thin annular plate has an outer diameter of 40 in, an inner diameter of 20 in, and a thickness of 1 in. The inner surface of the plate is clamped, while the outer surface is free to move. Figure 151 illustrates the domain dimensions and boundary conditions. See the table below for geometric and material properties.

Figure 151: Schematic of the test case, including domain geometry, main dimensions, and boundary conditions

For a thin, flat annular plate of homogeneous, linear elastic material, the mode shapes and natural frequencies are characterized by two integer indices and . The angular mode index   and radial mode index   represent the number of nodal lines in a circular or elliptical structure. The natural frequency of the plate can be calculated as:

(12)

Where is a dimensionless parameter, is the outer radius, is the plate thickness, is the mass per unit area, is the Young's modulus, and is the Poisson's ratio. The parameter is a function of the mode indices   and  , the Poisson's ratio, the ratio between outer and inner radii of the plate, and the boundary conditions of the outer and inner surfaces. The mass per unit area is calculated as the product of the material density and the plate thickness .

The first six modes of the annular plate are and . For a free-clamped condition, = 0.3, and = 0.5, the dimensionless parameters = 13.0, = 13.3, = 14.7, and = 18.5. Therefore, the natural frequencies for the first six modes are 316.67 Hz, 323.97 Hz, 323.97 Hz, 358.08 Hz, 358.08 Hz, and 450.64 Hz.

Analysis Assumptions and Modeling Notes

One part is defined to represent the annular plate, being meshed with 3D hexahedral elements with sizes of 0.125 in (thickness direction), 0.2 in (radial direction), and 0.245 to 0.491 in (circumferential direction). The plate elements use a constant stress solid element formulation (*SECTION_SOLID with ELFORM=1) and an elastic material card (*MAT_ELASTIC) with a density of 7.33 ⋅ 10^-4 lbf-s²/in³, a Young's modulus of 3 ⋅ 10⁷ psi, and a Poisson's ratio of 0.3. The nodes corresponding to the inner surface 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=6) are used to activate the implicit eigenvalue, static analysis with a total of six eigenvalues to be extracted.

Figure 152: Model setup in LS-DYNA of the 3D modal analysis of an annular plate

Results Comparison

The annular plate configuration for the first six modes is shown in Figure 153. The visualization of different modes is performed by reading the d3eigv file, generated for the modal analysis.

Figure 153: Visualization of first six modes of the annular plate

To quantify the error between the theoretical and LS-DYNA results, the first six natural frequencies of the annular plate and their relative errors are calculated and shown in the following table. This comparison verifies the strong agreement between the LS-DYNA results and the reference targets for all frequencies.

Results	Target	LS-DYNA Solver	Error (%)
1st Natural Frequency,  (Hz)	316.67	312.99	-1.16
2nd Natural Frequency, (Hz)	323.97	318.38	-1.73
3rd Natural Frequency, (Hz)	323.97	318.38	-1.73
4th Natural Frequency, (Hz)	358.08	349.68	-2.35
5th Natural Frequency, (Hz)	358.08	349.68	-2.35
6th Natural Frequency, (Hz)	450.64	438.84	-2.62