Vibration of Shells

Jawad, Maan H.

doi:10.1115/1.801993.ch15

The governing equations for the vibration of cylindrical shells are based, in part, on the equations derived in Chapter 10. These equations, which are based on symmetric loading, were derived from Fig. 10-2, which is reproduced here as Fig. 15-1a in a more general form. Additionally, non-symmetric loading such as torsional moments and shear strains, Fig. 15-1b, must be considered in the derivation of the equations for shell vibration. Also, deformations u, v, and w as well as inertial forces in the axial, circumferential and radial directions must be taken into consideration. The resulting differential equations in the axial, circumferential and radial directions are

r^{2} \frac{\partial^{2} u}{\partial x^{2}} + \frac{1}{2} (1 - μ) \frac{\partial^{2} u}{\partial θ^{2}} + \frac{r}{2} (1 + μ) \frac{\partial^{2} v}{\partial x \partial θ} - μ r \frac{\partial w}{\partial x} = \frac{ρ}{E} (1 - μ^{2}) r^{2} \frac{\partial^{2} u}{\partial t^{2}}

\frac{r}{2} (1 + μ) \frac{\partial^{2} u}{\partial x \partial θ} + \frac{r^{2}}{2} (1 - μ) \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial θ^{2}} - \frac{\partial w}{\partial θ} = \frac{ρ}{E} (1 - μ^{2}) r^{2} \frac{\partial^{2} v}{\partial t^{2}}

μ r \frac{\partial u}{\partial x} + \frac{\partial v}{\partial θ} - w - k (r^{4} \frac{\partial^{4} w}{\partial x^{4}} + 2 r^{2} \frac{\partial^{4} w}{\partial x^{2} \partial θ^{2}} + \frac{\partial^{4} w}{\partial θ^{4}}) = \frac{ρ}{E} (1 - μ^{2}) r^{2} \frac{\partial^{2} w}{\partial t^{2}}

Where,

k = \frac{t^{2}}{12 r^{2}}

and,

E = modulus of elasticity

r = radius of cylinder

t = thickness of cylinder

u = axial deflection along the x-axis

v = circumferential deformation along the θ-axis

w = radial deformation

μ = Poisson's ratio

ρ = mass density per unit volume (density/acceleration)

15 Vibration of Shells

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