Antisymmetric laminate vibration

Robert M. Jones, Harold S. Morgan, and James M. Whitney, Buckling and Vibration of Antisymmetrically Laminated Angle-Ply Rectangular Plates, Journal of Applied Mechanics, December 1973, pp. 1143-1144. 

Classical solutions to laminated shell buckling and vibration problems in the manner of Chapter 5 were obtained by Jones and Morgan [6-47]. Their results are presented as normalized buckling loads or fundamental natural frequency versus the Batdorf shell curvature parameter. They showed that, for antisymmetrically laminated cross-ply shells as for plates, the effect of coupling between bending and extension on buckling loads and vibration frequencies dies out rapidly as the number of layers... 

The free vibration frequencies and mode shapes will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different types of lamination will be compared to determine the influence of bend-twist coupling and bending-extension coupling on the vibration behavior. As with the deflection problems in Section 5.3 and the buckling problems in Section 5.4, different simply supported edge boundary conditions will be used in the several problems presented. 

Antisymmetric cross-ply laminates were found in Section 4.3.3 to have extensional stiffnesses A, A. 2, A22 = A.. , and Age, bending-ex-tension coupling stiffnesses and B22 = -Bi., and bending stiffnesses Di2. D22 = l ii- Dgg. The new terms here in comparison to a specially orthotropic laminate are and 622- Because of this bending-extension coupling, the three vibration differential equations are coupled ... 

Figure 5-33 Relative Fundamental Natural Vibration Frequencies of Square Antisymmetric Cross-Ply Laminated Plates (After Jones [5-19])...

Normalized fundamental natural frequencies for the exampie un-symmetrical cross-piy laminated graphite-epoxy piates are shown in Figure 5-40. The vibration resuits are analogous to the buckling resuits in the same manner as explained for antisymmetric iaminates, nameiy, iesser differences for vibration frequencies because of the square-root factor. 

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