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Chapter 15
Vibration of Shells

Excerpt

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

r22ux2+12(1μ)2uθ2+r2(1+μ)2vxθμrwx=ρE(1μ2)r22ut2
r2(1+μ)2uxθ+r22(1μ)2vx2+2vθ2wθ=ρE(1μ2)r22vt2
μrux+vθwk(r44wx4+2r24wx2θ2+4wθ4)=ρE(1μ2)r22wt2
Where,
k=t212r2
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)

  • 15-1 Cylindrical Shells
  • 15-2 Spherical Shells

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