# L8 - Quick Review Torsion ρφ γ= L ρ radial dist c...

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Unformatted text preview: Quick Review: Torsion ρφ γ= L ρ: radial dist. c: radius L: length γ: shear strain ϕ: angle of twist ρ τ = τ max c Tc Tρ τ max = and τ = J J ρφ γ= L γ max cφ = L τ max Tc = J γ max τ max Tc = = G JG TL φ= JG Analogous to: FL δ= AE Angle of Twist in Elastic Range TL φ= JG => •  Adding all the segments: Ti Li φ =∑ i J iG i φ A / B = φ E / B + φ D / E + φC / D + φ A / C Remember: φ =0 at the wall Gear transmission Torsion: TCD rc = TAB rB Angle of twist: φC rC = φ B rB φC φB Statically Indeterminate Shafts •  From statics, TA + TB = 90 lb ⋅ ft statically indeterminate (cannot determine TA and TB) Need another equation (compatibility eq) compatible deformations- break it into 2 sections: The angle of twist at point B is 0. φ1 + φ2 = 0 φ = φ1 + φ2 TA L1 TB L2 = − =0 J1G J 2G => Now we have 2 equations and 2 unknowns, LJ TA + 1 2 TA = 90 lb ⋅ ft L2 J1 L1 J 2 TB = TA L2 J1 •  Substitute into the original equilibrium equation, Design of Transmission Shafts •  Principal transmission shaft performance specifications are: -  Power -  speed P = Tω = 2πfT P P T= = ω 2πf f = frequency ω=angular frequency=2pf •  allowable shearing stress Tc τ max = J J π3 T (solid shafts) = c= c2 τ max J π T 4 4 = (c 2 − c1 ) = τ (hollow shafts) c 2 2c 2 max Stress Concentrations •  flange couplings, gears and pulleys, and cross-section discontinuities can cause stress concentrations •  Concentration factor: Tc τ max = K J (use small c) 3.41 3.36 3.85 ...
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L8 - Quick Review Torsion ρφ γ= L ρ radial dist c...

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