Modal Shapes of Completely Free Circular Trigonal Plates

National Science Foundation/Army Research Office

Research Experience for Undergraduates:
Studies in Vibration and Sound

Sally Cook - Colorado State University

Since isotropic symmetry has been the main focus of studies in the past, the goal for this summer was to explore the effects of trigonal symmetry. Isotropic material symmetry has been applied to a range of shapes including circular plates with both simply supported and free edges, as well as cylinders, and square plates. Trigonal is defined as the division of a hexagonal crystal system or the forms belonging to it characterized by a vertical axis of threefold symmetry. One study has been geared towards finite trigonal elastic cylinders but there are many more objects that trigonal properties could be applied to. Throughout this summer I have progressively worked towards successfully applying trigonal symmetry to completely free circular plates.

A program was previously developed to compute the deformed shapes for finite trigonal elastic cylinders. It combines Hamilton's principle and the Ritz method to form a matrix that corresponds with the eigenvalue problem. This program was used to complete much of the research for this particular project. When the properties of the program were changed to reflect the shape of a plate, it was discovered that the trigonal deformed shapes look much like those with isotropic symmetry. To acknowledge the contribution of the trigonal shapes, cosine and sine terms had to be suppressed out based on the number of nodal diameters in each mode for the deformed shape. Suppressing these terms produced the deformed trigonal shapes for each mode and frequency. An 8% decrease in frequency can be seen when the frequencies for the suppressed shapes are compared to those of the original deformed shapes. The deformed shapes that can be seen below represent the trigonal deformed shapes without suppression followed by the suppressed deformed shape, and finally the contour plot of the unsuppressed shape.

Group 1: Mode 213 - Frequency 100824 Hz / 87645 Hz

Group 2: Mode 212 - Frequency 74029 Hz / 75423 Hz

Group 5: Mode 227 - Frequency 164707 Hz / 162798 Hz

By changing parameters in the program designed to compute finite trigonal cylinders, it was possible to produce the deformed shapes of trigonal completely free plates. These results are based on the fact that it was possible to successfully recreate known deformations. Since trigonal symmetry has never been studied with respect to plates there are no patterns or figures to compare these results with. For the purposes of the summer study, we were successful in plotting trigonal deformed shapes.

Funding from the National Science Foundation and the Army Research Office enabled this research. The research team includes Sally Cook (CSU), and Professor Paul Heyliger (CSU).