Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

The application of Kantorovich's method to a trinomial model of deformation taking into account transverse bending of a plate leads to a connected system of three ordinary differential equations of fourth order with respect to three unknown functions of the longitudinal coordinate and to the coresponding boundary conditions for them at the fixed end and on the free edge. For the approximate calculation of the frequencies and forms of natural vibrations Galerkin's method is used, and as coordinate functions we chose orthogonal Jacobi polynomials with weight function. The dimensionless frequencies depend on the magnitude of the four dimensionless complexes, three of which characterize the anisotropy of the elastic properties of the composite. For the fibrous composites used at present we determined the possible range of change of the dimensionless complexes d₁₆ and d₂₆ attained by oblique placement. The article examines the influence of the angle of reinforcement on some first dimensionless frequencies of a plate made of unidirectional carbon reinforced plastic. It also analyzes the asymptotics of the frequencies when the length of the plate is increased, and it shows that for strongly anisotropic material with the structure []_T the frequencies of the flexural as well as of the torsional vibrations may be substantially lower when flexural-torsional interaction is taken into account.For Communication 1 see [4].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1. pp. 23–33, January–February, 1997.