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Unformatted text preview: Section 4.4 Solid Mechanics Part II Kelly 77 4.4 Rotating Discs 4.4.1 The Rotating Disc Consider a thin disc rotating with constant angular velocity ω , Fig. 4.4.1. Material particles are subjected to a centripetal acceleration 2 ω r a r − = . The subscript r indicates an acceleration in the radial direction and the minus sign indicates that the particles are accelerating towards the centre of the disc. Figure 4.4.1: the rotating disc The accelerations lead to an inertial force (per unit volume) 2 ω ρ r F a − = which in turn leads to stresses in the disc. The inertial force is an axisymmetric “loading” and so this is an axisymmetric problem. The axisymmetric equation of equilibrium is given by 4.3.5. Adding in the acceleration term gives the corresponding equation of motion: ( ) 2 1 ω ρ σ σ σ θθ r r r rr rr − = − + ∂ ∂ , (4.4.1) This equation can be expressed as ( ) 1 = + − + ∂ ∂ r rr rr b r r θθ σ σ σ , (4.4.2) where 2 ω ρ r b r = . Thus the dynamic rotating disc problem has been converted into an equivalent static problem of a disc subjected to a known body force. Note that, in a general dynamic problem, and unlike here, one does not know what the accelerations are – they have to be found as part of the solution procedure. – they have to be found as part of the solution procedure....
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- Fall '11
- DISC, σ θθ, σ rr