MECHANICS OF MATERIALS Design of a Transmission Shaft

Transcription

1 Design of a Transmission Shaft If power is transferred to and from the shaft by gears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Normal stresses due to transverse loads may be large and should be included in determination of maximum shearing stress. Shearing stresses due to transverse loads are usually small and contribution to maximum shear stress may be neglected. 8-1 Design of a Transmission Shaft At any section, Mc σ m I Tc τm J where M M y + M z Maximum shearing stress, τ max m + M + T ( τ ) m Mc Tc + I J I J for a circular or annular cross - section, c τmax J σ Shaft section requirement, M + T J max c min τ all 8-1

2 Sample Problem 8. Determine the gear torques and corresponding tangential forces. Find reactions at A and B. Solid shaft rotates at 80 rpm and transmits 0 kw from the motor to gears G and H; 0 kw is taken off at gear G and 10 kw at gear H. Knowing that σ all 50 MPa, determine the smallest permissible diameter for the shaft. Identify critical shaft section from torque and bending moment diagrams. Calculate minimum allowable shaft diameter. 8 - Sample Problem 8. Determine the gear torques and corresponding tangential forces. P 0kW TE πf π ( 8Hz) 10kW TD π ( 8Hz) π ( 8Hz) T 597 N m F E E.7kN re 0.16m 0kW TC 98N m FC 6.6kN 199 N m Find reactions at A and B. Ay 0.9kN By.80kN 597 N m Az 6.kN Bz.90kN FD.9kN 8 -

3 Sample Problem 8. Identify critical shaft section from torque and bending moment diagrams. ( M y + M z + T ) max N m 8-5 Sample Problem 8. Calculate minimum allowable shaft diameter. J c M y + M τ all z + T 157 N m MPa 6 m For a solid circular shaft, J π c m c c m 5.85m d c 51.7 mm 8-6

4 Stresses Under Combined Loadings Wish to determine stresses in slender structural members subjected to arbitrary loadings. Pass section through points of interest. Determine force-couple system at centroid of section required to maintain equilibrium. System of internal forces consist of three force components and three couple vectors. Determine stress distribution by applying the superposition principle. 8-7 Stresses Under Combined Loadings Axial force and in-plane couple vectors contribute to normal stress distribution in the section. Shear force components and twisting couple contribute to shearing stress distribution in the section. 8-8

5 Stresses Under Combined Loadings Normal and shearing stresses are used to determine principal stresses, maximum shearing stress and orientation of principal planes. Analysis is valid only to extent that conditions of applicability of superposition principle and Saint-Venant s principle are met. 8-9 Determine internal forces in Section EFG. Evaluate normal stress at H. Evaluate shearing stress at H. Three forces are applied to a short steel post as shown. Determine the principal stresses, principal planes and maximum shearing stress at point H. Calculate principal stresses and maximum shearing stress. Determine principal planes

6 Determine internal forces in Section EFG. Vx 0 kn M x M y 0 ( 50kN)( 0.10m) ( 75kN)( 0.00m) 8.5kN m M z P 50kN Note: Section properties, A Ix Iz ( 0.00m)( 0.10m) Vz 75kN ( 0kN)( 0.100m) kn m ( 0.00m)( 0.10m) m m ( 0.10m)( 0.00m) m 8-11 ( 8.5kN m)( 0.05m) m Evaluate normal stress at H. P M z a M x b σ y + + A Iz Ix 50kN m m ( kn m)( 0.00m) ( ) MPa 66.0MPa Evaluate shearing stress at H. Q A y [( 0.00m)( 0.05m) ]( 0.075m) m V Q τ z yz Ixt 17.5MPa ( 75kN)( m ) ( m )( 0.00m) 8-1 6

7 Calculate principal stresses and maximum shearing stress. Determine principal planes. τmax R MPa σ max OC + R MPa σ min OC R MPa CY 17.5 tan θ p θ p 7.96 CD.0 θ p 1.98 τmax 7.MPa σ max 70.MPa σ min 7.MPa θ p