Analytical Modeling and Vibration Analysis of Partially Cracked Rectangular Plates With Different Boundary Conditions and Loading

Abstract

This study proposes an analytical model for vibrations in a cracked rectangular plate as one of the results from a program of research on vibration based damage detection in aircraft panel structures. This particular work considers an isotropic plate, typically made of aluminum, and containing a crack in the form of a continuous line with its center located at the center of the plate and parallel to one edge of the plate. The plate is subjected to a point load on its surface for three different possible boundary conditions, and one examined in detail. Galerkin’s method is applied to reformulate the governing equation of the cracked plate into time dependent modal coordinates. Nonlinearity is introduced by appropriate formulations introduced by applying Berger’s method. An approximate solution technique—the method of multiple scales—is applied to solve the nonlinear equation of the cracked plate. The results are presented in terms of natural frequency versus crack length and plate thickness, and the nonlinear amplitude response of the plate is calculated for one set of boundary conditions and three different load locations, over a practical range of external excitation frequencies.

Exact free vibration study of rectangular Mindlin plates with all-over part-through open cracks

“…Transverse free vibration of viscoelastic rectangular thin plate with linearly varying thickness and multiple all-over part-through cracks was studied by Wang and Wang [29] using the differential quadrature method. Very recently, a new analytical model was presented by Israr et al [30] for the vibration analysis of cracked rectangular plates subjected to transverse loading at some specified position with different sets of boundary conditions using Galerkin's method.…”

Finite Element Modelling and Analysis of Free Vibration of a Square Plate with Side Crack

“…Huang et al [14] further proposed a new set of admissible functions and investigated the vibration behaviour of rectangular plate with internal crack at arbitrary orientation using Ritz method. Israr et al [15] proposed an analytical model to study the vibration characteristic of isotropic plate containing a crack at the center and parallel to the edge of the plate using line spring model. They applied Galerkin's method and Berger's method to formulate the time dependent nonlinear governing equation of cracked plate and solved the equation by method of multiple scale.…”

An analysis of the effects of the orientation angle of a surface crack on the vibration of an isotropic plate

“…Next Galerkin's method is applied in the usual manner to discretise the partial differential equation in (1) and transform the transverse deflection coordinate w(x, y, t) into time dependent modal coordinates. Subsequently, by employing the Berger formulation, the derived governing equation of motion of the cracked plate is converted into a nonlinear ordinary differential equation model in terms of a chosen modal coordinate, and then by considering the system to be under the influence of weak classical linear viscous damping µ, and the load to be harmonic [2], the final equation of motion for the system takes the form of a specialised Duffing equation, [7] in the form,…”

“…However, no literature contains any substantial references to the study of a plate with cracks of variable orientation undergoing forced vibration. The aim of this research is to extend the forced vibration analysis of the cracked plate discussed by Israr [2] and Israr et al [7] by considering an alternative geometry whereby the crack orientation is variable.…”