Problem_11_35

Problem_11_35 - Problem 11.35 Given: A disk of inner radius...

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Problem 11.35 Given: A disk of inner radius a and outer radius b is subjected to an angular velocity ω . The disk is constrained at r = a , so that the radial displacement u is zero. At r b , the disk is free of applied forces. Derive formulas for the constants of integration C 1 and C 2 as functions of ω , the material properties and radii. Solution: The boundary conditions are: at ra = u0 = at rb = σ rr 0 = Eq. 11.47 with T=0 u 1 ν 2 () 8E ρ r 3 ω 2 C 1 r + C 2 r + = at = 1 ν 2 ρ a 3 ω 2 C 1 a C 2 a + = or C 2 1 ν 2 ρ a 4 ω 2 C 1 a 2 = Eq. 11.49 σ rr E 1 ν 2 3 ν + 1 ν 2 ρ r 2 ω 2 1 ν + C 1 + 1 ν r 2 C 2 = at = σ rr 0 = 1 ν + C 1 1 ν b 2 C 2 3 ν + 1 ν 2 ρ b 2 ω 2 = Eliminate C 2 : 1 ν + C 1 1 ν b 2 1 ν 2 ρ a 4 ω 2 C 1 a 2 3 ν + 1 ν 2 ρ b 2 ω 2 = C 1
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