Free Vibration Analysis of Composite Plate Using Higher Order Haar Wavelet

In the last few decades, laminated composite plates, due to their high characteristics resistance and hardness, have been used in a variety of designs as structural components. Considering the structural complexities of laminated composite plates, choosing the appropriate method to solve the equation governing the dynamic equilibrium of these plates in order to analysis their vibration behavior has been attracted the attention of many researchers.
In the present research, free vibration of a laminated composite plate, has been investigated using numerical solution. In order to model the behavior of the composite plate, the refined Zigzag theory in which the transverse shear stress distribution is allowed through the thickness of the layers and also satisfies the condition of zero shear stress on the top and bottom surfaces of the composite plate without need to use shear correction factor, have been deployed. In this theory, seven independent displacement variables are considered and according to the continuity condition of displacement between layers, the number of variables will be independent of the number of layers. Principal of virtual work and minimization of the potential energy have been used for deriving the partial differential equations of motion. The higher order Haar wavelet method have been developed to replace displacement variables and numerically solve the partial differential equations and calculating eigenvalues. In most of the similar researches, the use of Haar wavelet numerical solution has mostly been discussed on beams, annular plates and shells, and research in the field of the rectangular composite plate has not been observed. In this paper, the numerical solution is modeled in a simple form and the equations governing the rectangular laminate composite plate have been replaced by linear algebraic series and so, the free vibration of laminate composite plate has been investigated.
Finally, the accuracy and convergence of solution method in analysis of laminated composite plates are evaluated and the results of the proposed method, earned from programming in the Matlab software, have been compared with the results of other solutions methods that mentioned in this paper. The comparison of these results shows good accuracy compared to other solution methods. Meanwhile, by converting multivariable equations of motion into linear algebraic expressions, the results of the proposed method have been obtained without any particular complexity and with a small number of collection points. Achieving the results by the proposed method with the same accuracy as the Haar solution results using smaller matrix dimensions and so reducing the computational cost is another advantage of this proposed method. It can be used instead of the Haar solution method in the numerical solutions of various solid mechanics’ problems. Examining the convergence of this proposed method shows that this solution has a good ability to approach the exact solution by increasing the resolution and collection points. It was also observed that the dimensionless frequency index increases with the increase in the ratio of longitudinal to transverse modulus regardless of the number of layers.