Reply by Author to P.A.A. Laura

T author appreciates Dr. Laura's kind comments about his paper. His discussion on the "reduction" methods is also very interesting. What follows is basically a comparison between the reduction approach and the approach proposed in Ref. 1 and related works." For almost half a century, numerous investigators, including those mentioned in the comment, have attempted to simplify and understand the mechanics of anisotropic plates, using some form of affine transformations. In the process, parameters wholly or partially equivalent to those proposed in Ref. 1 were defined. Curiously, no one seemed to realize that these parameters are the key to undetstanding the physics of anisotropic problems. While there is no unique approach to defining the parameters, such an exercise should aspire to minimize the efforts of understanding the proper physical trends in anisotropic problems. The reduction method provides an efficient means of obtaining solutions to anisotropic problems if the isotropic solutions exist. The drawbacks include its dependence on isotropic solutions, boundary condition constraints, and, more importantly, the fact that the end results (solutions) are still in terms of the numerous anisotropic elastic constants, whose relative importance and bounds are unknown, resulting in an inevitable obstruction of the physical insight to be acquired from the analysis. The approach proposed in Ref. 1 and other related works," on the other hand, is independent of isotropic solutions and boundary conditions. Furthermore, the reduction in the number of elastic constants and the fact that the resulting new parameters are the generalized forms of their isotropic counterparts, seem to simplify the physics of the problems significantly. A similarity rule approach proposed by Brunelle for some boundary-value problems is basically the equivalent of the reduction method in the affine space. Like the reduction method, it is very useful, but restricted to certain boundaryvalue problems and assumes the existence of a solution (isotropic or an anisotropic). However, unlike the reduction method, the physics of the problems is clearly exposed by this approach. The increasing popularity of anisotropic materials in engineering designs makes maximum physical understanding of anisotropic problems very desirable. In a recent effort toward achieving such a goal, Librescu presents an elaborate treatment of anisotropic and heterogeneous panel flutter problems in the physical space. A view of these problems in the affine space should greatly enhance their physical understanding. With reference to the approximate mode shapes for the clamped or rotation-restrained boundary conditions, proposed by Laura et al., it should be pointed out that based on their accuracy (shown in Ref. 7), they should also be very useful in the affine space because of their relative simplicity. In concluding, it may be worthwhile to point out that the use of affine transformations in complex problems is expanding. For example, Trevino has recently analyzed some nonstationary random processes in an affine space where they are reduced to "stationary" random processes. References 1 Oyibo, G. A., "Flutter of Orthotropic Panels in Supersonic Flow Using Affine Transformations," AIAA Journal, Vol. 21, Feb. 1983, pp. 283-289. Oyibo, G. A., "Unified Panel Flutter Theory with Viscous Damping Effects," AIAA Journal, Vol. 21, May 1983, pp. 767-773. Brunelle, E. J., and Oyibo, G. A., "Generic Buckling Curves for Specially Orthotropic Rectangular Plates," AIAA Journal, Vol. 21, Aug. 1983, pp. 1150-1156. Oyibo, G. A., "Unified Aeroelastic Flutter Theory for Very Low Aspect Ratio Panels," AIAA Journal, Vol. 21, Nov. 1983, pp. 15811587. Brunelle, E. J., "Eigenvalue Similarity Rules for a Class of Rectangular Specially Orthotropic Laminated Plates," presented at the Ninth U.S. National Congress of Applied Mechanics, June 1982. Librescu, L., Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures, Nordhoff International Publishing, Leyden, The Netherlands, 1975. Laura, P.A.A., Luisoni, L. E., and Filipich, C., "A Note on the Determination of the Fundamental Frequency of Vibration of Thin, Rectangular Plates with Edges Possessing Different Rotational Flexibility Coefficient," Journal of Sound and Vibration, Vol. 55, 1967, pp. 327-333. Trevino, G., "On the Bispectrum Concept for Random Processes," Journal of Sound and Vibration, Vol. 90, No. 4, Oct. 1983, pp. 590-594.