Plate Theory

We propose a continuous model for rectangular plates that involves a mesh of slender interconnected beams. These beams are disposed orthogonally. The advantage of this model is that simple equations are involved and the behavior of the plate is accurately described. After a brief description of the model, we present an application written in the Python programing language that allows calculating the mode shapes for rectangular plates with various boundary conditions. The mode shapes achieved with the application for a specific plate are found to be similar with those obtained from simulation involving the SolidWorks software. In consequence, we conclude that the proposed model is reliable and the application developed on this base can be used to study the behavior of rectangular plates with different boundary conditions.

We propose in this paper a method to detect cracks in circular plates clamped at the circumference. It consists in analyzing and interpreting the frequency changes that occur due to a circular arc crack. We show that the effect of the crack depends on the strain energy loss, which actually is proportional to the curvature achieved by the affected region in the radial direction. Three types of cracks are considered herein to demonstrate this dependency, differing by their length. Modal analysis is performed involving a professional simulation software for the intact plate, and that with the generated cracks. We found a perfect fit when we compared the curvature of the plate in the radial direction obtained with the analytical relation with the frequency values for the crack displaced along the radius. Therefore, using the curvature of several vibration modes we can obtain damage patterns that can be used to locate circular cracks.

We propose a continuous model for rectangular plates that involves a mesh of slender interconnected beams. These beams are disposed orthogonally. The advantage of this model is that simple equations are involved and the behavior of the plate is accurately described. After a brief description of the model, we present an application written in the Python programing language that allows calculating the mode shapes for rectangular plates with various boundary conditions. The mode shapes achieved with the application for a specific plate are found to be similar with those obtained from simulation involving the SolidWorks software. In consequence, we conclude that the proposed model is reliable and the application developed on this base can be used to study the behavior of rectangular plates with different boundary conditions.

We use the principle of virtual work to derive a higher-order shear and normal deformable theory for a plate comprised of a linear elastic incompressible anisotropic material. The theory does not use a shear correction factor and employs three components of displacement and the hydrostatic pressure as independent variables. For a Kth order plate theory, a set of 4 (K+ 1) coupled equations need to be solved for the (K+ 1) pressures and the 3 (K+ 1) displacements defined on the reference surface of the plate.

The linear behaviour of drum membranes has been extensively studied and is now well understood. Particular attention has been devoted to modal analyses of circular drum heads, and good agreement has been found between experiment and theory. Up to now, however, there has been relatively little investigation into the relevance and nature of nonlinear effects in drum membranes.
Stiff strings and stiff plates, however, have seen a good deal of such investigation: pitch glides and the migration of energy towards higher frequencies are typical phenomena that can be found at high striking amplitudes. Such effects result from nonlinearities that arise due to coupling between transverse and longitudinal wave motion in the material.
It has recently been shown that nonlinear effects, similar to those encountered in stiff strings and plates, can be important for drum membranes, both from a physical and a perceptual point of view. While existing tension-modulation techniques provide a useful starting point for modelling the effects of these nonlinearities, a complete nonlinear model is required for a more accurate, and ultimately more realistic, description.
In this study a nonlinear finite difference time domain model of a tom-tom is used, alongside experimental evidence, to highlight and quantify the relevance of nonlinear phenomena in drum membranes. The model includes geometrical membrane nonlinearities, and full air coupling between the two drum membranes. Experimental evidence is obtained from measurements of internal and external sound pressure fields around the drum.