Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

832 ELIA EFRAIM AND MOSHE EISENBERGER
1
0.5
0
-0.5
-45 -30 -15 0 15 30 45
Ω1,1
1
0.5
0
u
u
u
u
u
u
-45 -30 -15 0 15 30 45
Ω1,2
1
0.5
0
-0.5
u u
u
u u
-1
-45 -30 -15 0 15 30 45
Ω1,3
1
0.5
0
u u
u
u u
u u
v
v
v
v
v
v
v
v
v
v
v
v
v v
v
v
v v
-0.5
-45 -30 -15 0 15 30 45
Ω1,4
u u
v v
u v
w w
φ
-
w w
w
φ
0.1
0.05
0
-0.05
-0.1
-45 -30 -15 0 15 30 45
ψθ
ψφ
ψ φ
ψ θ
φ φ φ
u wv
-0.2 w
ψθ ψφ
ψ
φ
φ φ -45 -30 -15 0 15 30 45 ψφ
ψθ ψφ
w w
0.1
0
-0.1
w
0.1
ψ
θ
φ φ φ
0
w w
ψθ ψφ
v
-0.1
φ φ φ
-0.2 w w
ψθ ψφ
u
ψ
φ
φ φ -45 -30 -15 0 15 30 45 φ
w
ψθ ψφ
w
φ
ψ θ
φ φ φ
w w
ψθ ψφ
w
ψ
θ
φ φ 0
φ
w w
ψθ ψφ
-0.2
u vφ
φ φ
-0.4 w w
ψθ ψφ ψ
φ
φ φ -45 -30 -15 0 15 30 45 φ
w
ψθ ψφ
Figure 6. Mode shapes of vibrations with one circumferential waves (n = 1) of a completely
free spherical barrel shell with variable thickness (Figure 3, right).
of both transverse shear stresses and rotary inertia are accounted for, in their general forms for isotropic
homogeneous materials. The proposed method shows the following advantages:
(1) Any polynomial variation of the thickness of the shell along the meridian can be considered.
(2) The method is mesh-free, and dividing the surface to many elements doesn’t improve the solution.
No convergence study is necessary to obtain the true results.
(3) The shape functions are derived automatically and they are the exact solutions for the system of the
differential equation of motion with variable coefficients. As a result, the solution for free vibrations
w
ψ