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

STIFFNESS VIBRATION ANALYSIS OF SPHERICAL SHELL SEGMENTS WITH VARIABLE THICKNESS 831
Ω
Ω
Ω
Ω
1
0.5
0
-0.5
Ω0,5
0.1
0.05
-0.05
-45 -30 -15 0 15 30 45
0
Ω0,8
Ω0,8
u u
u u
u u
u u
v v
v v
-45 -30 -15 0 15 30 45
1
0.5
0
-0.5
u u
u u
u u
v v
v v
v v
-45 -30 -15 0 15 30 45
Ω0,10
1
0.5
0
-0.5
u u
u u
v v
v v
v v
v v
-45 -30 -15 0 15 30 45
Ω0,13
u u
v v
v
u w w
φ
w w
φ
-
ψ ψ
ψ φ
w w
φ
φ
ψ ψ
θ θ
w w
φ
φ
0.1
0.05
0
-0.05
0.5
-0.5
-45 -30 -15 0 15 30 45
1
0
ψθ
ψθ
ψφ
ψφ
-45 -30 -15 0 15 30 45
ψθ
ψφ
ψ φ
ψ θ
φ
u w
φ
ψφ
v 0.1
φ φ
0.05 w w
ψθ ψφ
φ
u wφ
w 0 φ
w w-
-0.05
ψθ
ψψφ φ
ψφ
φ φ
w w
-45 -30 -15 0
ψθ
15
ψ
30 45 φ
ψφ
ψ
φ φ
w w
v 0.1
φ φ
0.05 w w
ψθ
ψθ
ψφ
ψφ
φ
ψ
θ
φ
u w w 0
φ φ
- -0.05
ψθ
ψ
φ φ
ψφ
w w
φ
w w
φ
-45 -30 -15 0 15 30 45
Figure 5. The first four mode shapes of pure torsional vibrations (n = 0) of completely
free spherical barrel shell with variable thickness (Figure 3, right).
5, 6, and 7 show the three-dimensional vibrational modes and the exact displacement shape functions
obtained by the present method.
5. Conclusions
The natural frequencies for spherical shells of revolution with different boundary conditions have been
investigated using the Dynamic Stiffness method. This approach is combined with the exact element
method for the vibration analysis of spherical shell segments with curved meridian and variable crosssection.
The analysis uses the equations of the two-dimensional theory of elasticity, in which the effects
ψθ
ψφ
v
ψ
ψ θ
φ