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The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite element solutions.