For torsion the frequency equation ([1],[2],[6]) for a cylindrical beam fixed at x=0 and free at x=L is similar to the axial frequency equation ( ) 2 2 2 2 4 2 1 mL n GJ M K n n n − = = π ω (43) J LK G 2 1 8 π = (44) Knowing E and G, Poisson’s ratio can be determined as 1 2 1 1 − = torsion axial AK JK ν (45) DAMPING After introducing viscous damping into the axial equations of motion [6], the coefficient of elasticity is independent of the rate of loading as determined by the dynamic test. The damping coefficient determines the response due to the rate of loading. ∂ ∂ = + t E a x x x ε ε σ 1 where x u x ∂ ∂ = ε (46) ∂ ∂ ∂ + ∂ ∂ = x t u a x u F EA 2 1 stiffness proportional damping (47) ( , ) 0 2 2 q x t t u a t u m x F + ∂ ∂ − ∂ ∂ = ∂ ∂ mass proportional damping (48) ( , ) 0 2 2 2 3 1 2 2 q x t t u ma t u m x t u EAa x u EA = ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ + ∂ ∂ (49) Let ∑ ∞ = = 1 ( ) ( ) ( , ) n n n x U t u x t φ (50) ( ) ∫ = + + + L n n n n n n n n x q x t dx K U dt dU a M a K dt d U M 0 1 0 2 ( ) ( , ) φ (51) So the modal damping coefficient is n n nC a M a K1 0 + = (52) 0a and 1a should be constant for all modes if the above model is valid. 1a is a property of the material and 0a is a property of the system. 174
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