Vibration of laminated plates

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. 

Figure 5-30 Vibration of a Simply Supporled Laminated Rectangular Plate...

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

Robert M. Jones, Buckling and Vibration of Rectangular Unsymmetrically Laminated Oross-Ply Plates, AIM Journal, December 1973, pp. 1626-1632. 

Baburaj, V., Matsukai, Y. (1994). A study on the material damping of thin angle-ply laminated plates. Journal of Sound and Vibration, 172(3), 415 19. doi 10.1006/jsvi. 1994.1184... 

Once the orientations of a laminate are specified, the stacking sequence will control the flexural rigidities of a laminated plate. Thus, the stability, vibration, and static bending behavior are controlled by the relative dispersion and thickness distribution of these oriented layers. 

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 equilibrium equations for a beam are derived to illustrate the derivation process and to serve as a review in preparation for addressing plates. Then, the plate equilibrium equations are derived for use in Chapter 5. Next, the plate buckling equations are discussed. Finally, the plate vibration equations are addressed. In each case, the pertinent boundary conditions are displayed. Nowhere in this appendix is reference needed to laminated beams or plates. All that is derived herein is applicable to any kind of beam or plate because only fundamental equilibrium, buckling, or vibration concepts are used. 

In many cases of interest, the vibrating system comprises a number of elements, each of which may have its own loss factor T i and its own contribution Wi to the system vibratory energy. These elements might be, for example, the several layers of a laminated beam or plate. 

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