Overview

Test Case

This simulation models the vibration of a rectangular plate with the two smaller edges subjected to a simply supported condition and one of the other edges subjected to a clamped condition. The objective is to validate the first six natural frequencies of the structure. The rectangular plate has a length of 0.25 m, a width of 0.1 m, and a thickness of 0.005 m. Figure 157 illustrates the domain dimensions and boundary conditions—one fixed long edge with both short edges simply supported.

Figure 157: Schematic of the test case

The table below shows the material and geometric properties.

For a thin flat rectangular plate of homogeneous, linear elastic material, the mode shapes and natural frequencies are characterized by two integer indices, and . The mode indices and represent the number of nodal lines in the length and width direction of a rectangular plate. The natural frequency of the plate can be calculated as:

(14)

where

is a dimensionless parameter.

is the plate length.

is the plate thickness.

is the mass per unit area.

is Young's modulus.

is Poisson's ratio.

The parameter is a function of the mode indices and , Poisson’s ratio, the ratio between the length and width of the plate, and the boundary conditions. is calculated as the product of the material density and the plate thickness . The first six modes of the rectangular plate are , , , , , and . For the current boundary conditions, = 0.3, and l/w = 2.5, the dimensionless parameters = 30.63, = 58.08, = 105.5, = 149.46, = 173.1, and = 182.8. Therefore, the natural frequencies for the first six modes are 595.70 Hz, 1129.55 Hz, 2051.78 Hz, 2906.73 Hz, 3366.48 Hz, and 3555.13 Hz.

Analysis Assumptions and Modeling Notes

The rectangular plate is modeled as a thin plate and meshed with 2D quadrilateral shell elements (size = 0.0025 m). The plate elements use a fully integrated shell formulation with higher accuracy (ELFORM=-16) and an elastic material card (*MAT_ELASTIC) with a density = 7850 kg/m³, Young's modulus = 2 ⋅ 10¹¹ Pa, and Poisson's ratio = 0.3. Nodes at the smaller edges of the plate are grouped using *SET_NODE_LIST. The keyword *BOUNDARY_SPC_SET is used to define the constraint of this node set (translational constraint about the three axes). The nodes corresponding to one of the larger edges are also grouped using *SET_NODE_LIST and constrained using *BOUNDARY_SPC_SET (translational and rotational constraint about the three axes). To perform implicit eigenvalue, static analysis, use the keywords *CONTROL_IMPLICIT_GENERAL (IMFLAG=1), *CONTROL_IMPLICIT_DYNAMICS (IMASS=0), and *CONTROL_IMPLICIT_EIGENVALUE (NEIG=6). These settings designate the extraction of six eigenvalues.

Figure 158: Model setup of the 2D modal analysis of a rectangular plate

Results Comparison

The rectangular plate configuration for the first modes is shown in Figure 159. The visualization of the first six modes is performed by reading the d3eigv file, generated for the modal analysis.

Figure 159: Visualization of the first six modes of the rectangular plate

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

Results	Target	LS-DYNA Solver	Error (%)
1st Natural Frequency, (Hz)	595.70	589.23	-1.09%
2nd Natural Frequency, (Hz)	1129.55	1115.68	-1.23%
3rd Natural Frequency, (Hz)	2051.78	2029.22	-1.10%
4th Natural Frequency, (Hz)	2906.73	2866.90	-1.37%
5th Natural Frequency, (Hz)	3366.48	3324.40	-1.25%
6th Natural Frequency, (Hz)	3555.13	3482.78	-2.04%