Altan Kayran^ e-mail: akayran@metu.edu.tr Can Serkan ibrahimoglu e-mail: 8i29875@metu,edu.tr Department ol Aerospace Engineering, Middle East Tectinicat University, 06531 Ankara, Turkey Effect of Semi-Geodesic Winding on the Vibration Characteristics of Filament Wound Sheils of Revolution The effect of semigeodesic winding on the free vibration characteristics of filament wound shells of revolution is studied. For this purpose multisegment numerical integration tech- nique is extended to the solution of the free vibration problem of composite shells of revo- lution which are wound along the semigeodesic fiber paths counting on the preset friction used during the winding process. Sample results are obtained for truncated conical and spherical shells of revolution and the effect of preset friction on the vibration characteris- tics of filament wound shells of revolution is particularly analyzed. Results show that when the preset friction is increased natural frequencies of higher circumferential vibra- tion modes also increase irrespective of the initial winding angle, and the circumferential bending stiffness stands out as the dominant parameter governing the natural frequencies of higher circumferential vibration modes. [DOI: 10.1115/1.40039071 Keywords: filament winding, semigeodesic fiber path, shell of revolution, composite materials, vibration characteristics 1 Introduction Laminated composite shells of revolution are widely encoun- tered structures in many applications such as pressure vessels, rocket nozzles, fuselage frames, extemal stores, antenna, etc. The axisymmetric composite shells of revolution are typically manu- factured by the filament winding process by winding organic or inorganic filaments or tapes over a mandrel of required shape. During the filament winding process, filaments are placed in arbi- trary orientation with respect to the geometric axis of the shell of revolution with the fundamental requirement that filaments must be slip-free [1]. Geodesic fiber trajectories are the most frequently used trajectories in the filament winding operation due to the fact that filaments wound along geodesic paths do not require any fric- tion to be stable. For conical shells of revolution, it has been shown that geodesic winding causes the winding angle and the thickness to vary continuously along the meridian of the shell of revolution [2,3]. The starting edge of the winding process, such as small or large radius edge, also has significant effect on the merid- ional variation of the winding angle, thickness and the stiffness coefficients. In filament winding operation, filaments must not necessarily be wound along geodesic paths to be stable. The so-called semi- geodesic fiber paths slightly deviate from the geodesic paths, counting on friction to keep the fiber in its proper position [4]. In case of semigeodesic winding, the tensile force in the fiber also has transverse component which can cause fibers to slip if enough friction is not available. The ratio of the transverse force to the normal force on the fiber is defined as the slippage tendency which must be less than the maximum friction coefficient [1,5]. In case of preset constant slippage tendency, the winding law can be expressed by a differential equation which is more complex than the winding law for geodesic winding [1]. Semigeodesic winding presents alternative fiber paths which can be exploited especially for optimization purposes. Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication In the JOURNAL OF APPLIED MECHANICS. Manuscript received April 23. 2010; final manuscript received March 19, 2011; published online August 24, 2011. Assoc. Editor: Anthony Waas. Most of the previous work on the study of the vibration charac- teristics of composite shells of revolution has been perfortned by taking constant stiffness coefficients or on shells of revolution, with constant radii of curvature, for which the stiffness coefficients and thickness remain constant along the meridian of the shell of revolution. Some examples of these studies include the work of Noor and Peters [6], who used a three-field mixed finite element model for the discretization in the meridional direction, and ana- lyzed cylindrical and toroidal shells of revolution. A semianalytical study of the composite shells of revolution has been performed by Xi et al. [7[, who used conical finite elements and included trans- verse shear deformation in their analysis. A finite element semiana- lytical model for laminated axisymmetric shells is presented by Correia et al. [8]. Ferreria et al. [9] presented a meshless method based on multiquadric radial basis functions for the solution of nat- ural frequencies of cross-ply composite shells. Nguyen-Van et al. [10] investigated the free vibration analysis of laminated shell structures based on FSDT with a novel quadrilateral finite element. A typical study on the effect of the variation of the stiffness coefficients due to geodesic winding on the buckling behavior of filament wound conical shells has been performed by Goldfield and Arbocz [3]. Korjakin et al. [Il] investigated the damped vibrations of laminated conical shells by finite element analysis, and incorporated the effect of the variation of the winding angle and thickness due to geodesic winding on the natural frequencies of laminated conical shells. Park et al. [12] considered the varia- tion of the winding angle in the longitudinal and thickness direc- tion in the dome part of a cylindrical pressure vessel due to semi- geodesic winding and performed finite element analysis of the pressure vessel subject to internal pressure. Recently, Kayran and Yavuzbalkan [13] studied the effect of the variation of stiffness coefficients due to geodesic winding on the free vibration charac- teristics of filament wound shells of revolution by a semianalytical method. For geodesic winding Kayran [14] also studied the effect of the initial winding angle and the starting edge of the winding operation on the undamped free vibration characteristics of com- posite shells of revolution. Blom et al. [15] optimized eigenfre- quencies of conical shells which are assumed to be built using an advanced tow-placement machine, which allows in-plane steering of the fibers, resulting in a variable stiffness structure. Journal of Applied iUlechanics Copyright © 2011 by ASME NOVEMBER 2011, Vol. 78 / 061008-1