1. Figures. 3-4 compare some of the results for the nonconventional boundary conditions 1 with the conventional ones. The ordinate in
2. Figures. 5-6 show the effect of the cone angle (a) on the fundamental frequency ratio (Wc IWi) for cases 1,3,6 and 7 respectively. Both figures are obtained for a circumferential wave number of 3. Since for a=oO, which corresponds to a circular cylindrical shell and where h / is smallest, the effect of transverse shear deformation is the highest. For a greater than 50°, further increase of the cone angle does not have any appreciable effect on the fundamental frequency ratio.
3. Figures. 9-10 show the effect of different lamination arrangements and the relaxing of the large end displacements on the fundamental frequency ratio of a seven layer clamped-clamped shell. As Figure-9 depicts, at low circumferential wave numbers the effect of transverse shear deformation is higher for a shell with axial fiber orientation (0') than for a shell with circumferential fiber orientation (90'). However for high circumferential wave numbers this trend is reversed. Furthermore it was also found that relaxing the large end displacements reduces the effect of transverse shear deformation in both all-axial and all-circumferential fiber orientation cases. Figure-10 shows the ratio of the fundamental frequency (due to improved theory) of a seven layer all-axial fiber shell to a seven layer all-circumferential fiber shell for different circumferential wave numbers and end conditions (cases 3 and 8). In axisymmetric vibration, it was found that the orientation of the layers did not have any effect on the fundamental frequency for either end conditions. This is due to the fact that in axisymmetric motion torsional modes completely uncouple from the bending and extensional modes. For the cases analyzed, the fundamental frequency is seen to be the lowest torsional frequency. For the beam mode vibration (n=l) the fundamental frequency of the axially laminated shell is higher than the circumferentially laminated shell for the clamped-free case. A slight increase was also found for the clamped-clamped case, but since the increase was very small it is not noticable in Figure-10. For higher circumferential wave numbers (n>l) the fundamental frequency of all-circumferential fiber case is significantly higher than all-axial fiber case. Physically this is due to the higher increase of stiffness at higher circumferential wave numbers for a circumferential fiber shell than an axial fiber orientation.
4. Figures. 11-12 show the transverse mode shapes for all-axial fiber (0') and all circumferential fiber (90') cases. It is seen that there is a strong dependence of the mode shape on the circumferential wave number. In each case the position of the maximum displacement shifts toward the larger end of the shell and little transverse motion occurs at the small end with increasing circumferential wave number. Physically, the suppression of transverse displacement near the small end of the cone at large values of n is due to the short distance between consecutive nodal meridians. Thus, this results in high stiffness in this region. This suppression of transverse displacement is more pronounced for a conical shell which is stiffer in the circumferential direction than in axial direction, and this is clearly seen in figures 13 and 14.