Dynamic characteristics of skew cantilevered trapezoidal plates

Abstract

Isotropic skew trapezoidal plates exhibit geometrical bending-torsion coupling in their natural modes. This coupling can complicate control design and implementation. In laminated composite plates this coupling can be minimized or eliminated using the right fiber orientation. An optimization technique based on the static deflection of the plates is used to determine the fiber orientation angles which rninimize this coupling and move the torsional modes to higher frequencies. The optimal orientation angles are obtained for different geometries are lay-ups. Introduction K.EW trapezoidal plates can be used to simulate ^swept flexible wings of high speed aircraft. These plates when made of isotropic material exhibit geometric coupling between bending and torsion motions in their natural modes. The existence of the torsional motion in the lower modes decrease the flutter stability region and complicates the vibration control of such wings. Elastic coupling between bending and torsional motions of layered composite plates can be used to eliminate or reduce the geometric coupling in skew trapezoidal plates. Tailoring techniques have been used in the past to optimize the fiber orientation and the lay-up of composite wings for minimum deformation or to increase the first natural frequency of the wing. in this paper we optimize the fiber orientation angles for several lay-up configurations and different geometries to minimize the bending-torsion coupling and move the torsional modes to higher frequencies. Effect of shear deformations on the natural frequencies of composite plates is more pronounced than their effect on isotropic plates due to the low transverse shear modulus of such plates. In most of the studies on the effect of shear deformations on composite plates a generic shear correction factor of 5/6, 2/3 or 7T/-/12 (see for example Lavejoy and Kapania). Shear correction factors, however, are functions of the lamination configuration of the plate. This makes it necessary to estimate these factors for each lay-up configuration and not to assume that a single shear correction factor applies for any plate layup, in this paper we use a method developed by Pai to calculate shear correction factors for different lay-ups. The method utilizes a layer-wise high order shear deformation theory and the * Research Associate, Member AJAA * Assistant Professor, Member AIAA * Graduate Student Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. continuity of the displacement and the shear stresses at the layer interfaces to closely estimate the shear correction factors. The natural frequencies are calculated using the finite element code MSC/NASTRAN. In NASTRAN the shear correction factors are calculated using an energy balance method. The shear correction factors from NASTRAN are compared with the shear correction factors of Pai and the influence on the natural frequencies is evaluated. Figure 1: Geometry of the plate and coordinate svstem NASTRAN Model MSC/NASTRAN finite clement code is used to calculate the natural frequencies and mode shapes of the plates. The geometry of the plate is shown in Figure 1. The plate is clamped at x = 0 and free at all other edges. The finite element mesh used with NASTRAN is shown in Figure 2. CQUAD4 elements PCOMP property cards and MATS material cards are used. In dealing with PCOMP and MATS, NASTRAN converts the material properties to their orthotropic equivalents (MAT2 cards). The Mode shapes are calculated using the normal mode solution (SOL 103) of NASTRAN. Optimization with NASTRAN (SOL 200), however, works only with the static analysis. The optimized fiber orientation angles had to calculated using this static analysis tools. This was achieved by applying two opposite forces in the z-direction at points 46 and 88 as shown in Figure 2. The application of these forces causes static deformation similar to the second mode shape. A rectangular plate deforms in pure bending under the application of these forces. The objective function to be minimized is then defined to be the difference in the vertical displacements of the points on the tip of the wing as follows: