VM-LSDYNA-SOLVE-043
VM-LSDYNA-SOLVE-043
Fundamental Frequency of an Elliptical Plate
Overview
| Reference: | Blevins, R. D. (1979). Formulas for natural frequency and mode shape. Van Nostrand Reinhold, p.249, table 11.3, case 3. |
| Analysis Type(s): | Implicit Vibration Analysis |
| Element Type(s): | 2D Quadrilateral and Triangular Shell Elements |
| Input Files: | Link to Input Files Download Page |
Test Case
This test case models the vibration of an elliptical plate with its edge subjected to a clamped condition. The objective is to validate the fundamental natural frequency of the structure. The elliptical plate has a major radius of 30 in, a minor radius of 20 in, and a thickness of 1 in. Figure 154 illustrates the domain dimensions and boundary conditions.
Figure 154: Schematic of the test case, including domain geometry, dimensions, and boundary conditions
The model is fully constrained at plate boundary—fixed translational and rotational constraints about the three axes. The table below shows the material and geometric properties. The simulation uses a U.S. Customary unit system, though it can be readily adapted to other consistent unit systems if required.
| Material Properties | Geometric Properties |
|---|---|
|
Density ρ = 7.33 ⋅ 10-4 lbf-s2/in4 Young's modulus E = 3 ⋅ 107 psi. Poisson's ratio ν = 0.30 | Major radius (α) = 30 in Minor radius (b) = 30 in Thickness (h) = 1 in |
For a thin flat elliptical plate of homogeneous, linear elastic material, the mode shapes
and natural frequencies are characterized by two integer indices, i and
j. The angular mode index, i, and radial mode
index, j, represent the number of nodal lines in a circular or elliptical
structure. The natural frequency (
) of the plate can be calculated as:
 | (13) |
Where 
is a dimensionless parameter, 
is the minor 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 i and
j, the Poisson’s ratio, the ratio between the major and minor
radii of the plate, and the boundary conditions of the plate. The mass per unit area,
, is calculated as the product of the material density, 
, and the plate thickness, .
For a clamped condition, = 0.3, and 
= 1.5, the dimensionless parameter 
= 7.567. Therefore, the natural frequency for the first mode, also known as
fundamental frequency, is 184.32 Hz.
Analysis Assumptions and Modeling Notes
One part is defined to represent the elliptical plate, meshed with 2D quadrilateral and triangular elements with size 0.25 in. The plate elements use a fully integrated shell formulation with higher accuracy (ELFORM=-16) and an elastic material card (*MAT_ELASTIC) with a density of 7.33 ⋅ 10-4 lbf-s2/in4, a Young's modulus of 3 ⋅ 107 psi, and a Poisson's ratio of 0.3. 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=1) are used to activate the implicit eigenvalue, static analysis with one eigenvalue to be extracted.
Results Comparison
To quantify the error between the theoretical and LS-DYNA results, the fundamental frequency of the elliptical plate and its relative error is calculated and shown in the following table. This comparison verifies the excellent agreement between the fundamental frequencies.
| Result | Target | LS-DYNASolver | Error (%) |
|---|---|---|---|
Fundamental Frequency,  (Hz) | 184.32 | 184.71 | 0.21 |
The elliptical plate configuration for the first mode is shown in Figure 156. The visualization of the first mode is performed by reading the d3eigv file, generated for the modal analysis.
