Overview

Test Case

This test case models the vibration of a clamped squared plate with a central opening. The objective is to validate the fundamental natural frequency of the structure. The thin square plate has a length of 1.0 m, a circular opening diameter of 0.1 m, and thickness of 0.01 m. The outer surface of the plate is clamped, while the inner surface (opening) is free to move. Figure 160 illustrates the domain geometry, main dimensions and boundary conditions.

Figure 160: Test case schematic

The following table lists the material and geometric properties. The units for the current test case follow the International System of Units.

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

(15)

where is a dimensionless parameter, is the plate length, 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 i and j, the Poisson's ratio, the ratio between the opening diameter and plate length, and the boundary conditions of the outer and inner surfaces. The mass per unit area is calculated as the product of the material density ρ and the plate thickness . For a clamped condition, = 0.3, and = 0.2, the dimensionless parameters = 35.7. Therefore, the natural frequency for the first mode, also known as fundamental frequency, is 43.39 Hz.

Analysis Assumptions and Modeling Notes

One part is defined to represent the square plate with opening, being meshed with 3D hexahedral and tetrahedral elements with sizes of 0.01 m (length directions) and 0.00125 m (thickness 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 7850 kg/m³, a Young's modulus of 2 ⋅ 10¹¹ Pa, and a Poisson's ratio of 0.3. The nodes corresponding to the outer 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=1) are used to activate the implicit eigenvalue, static analysis with one eigenvalue to be extracted.

Figure 161: Model setup in LS-DYNA

Results Comparison

The square plate configuration for the first mode is shown in the figure below. The visualization of the first mode is performed by reading the d3eigv file, generated for the modal analysis.

Figure 162: First mode of the square plate with opening

To quantify the error between the theoretical and LS-DYNA results, the fundamental frequency of the square plate with opening and its relative error is calculated and shown in the following table. This comparison verifies the excellent agreement between the fundamental frequencies.