MECHANICS OF MATERIALS Design of a Transmission Shaft

Transcription

1 Design of a Transission Shaft If power is transferred to and fro the shaft by ygears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Noral stresses due to transverse loads ay be large and should be included in dt deterination ti of axiu shearing stress. Shearing stresses due to transverse loads are usually sall and contribution to axiu shear stress ay be neglected. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-1

2 Design of a Transission Shaft At any section, Mc σ where M M y + M I Tc τ J Maxiu shearing stress, τ ax σ + τ ( ) Mc I Tc + J for a circular or annular cross - section, I τ ax c J M + T Shaft section requireent, M + T J c τ all 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8 - in ax J

3 Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. Find reactions at A and B. Solid shaft rotates at 480 rp and transits 30 kw fro the otor 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, deterine the sallest perissible diaeter for the shaft. Identify critical shaft section fro torque and bending oent diagras. Calculate iniu allowable shaft diaeter. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-3

4 Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. T F E E P T r 30kW πfπ f π ( 8H ) E E 597 N 0.16 ( 8H) 597 N 3.73kN 0kW T C 398N FC π 10kW T D 199 N F π 8H D ( ) Find reactions at A and B. A B y y 0.93 kn.80 kn A B 6. kn.90 kn 6.63kN.49 kn 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-4

5 Saple Proble 8.3 Identify critical shaft section fro torque and bending oent diagras. ( ) M + M + T y ax N The McGraw-Hill Copanies, Inc. All rights reserved. 8-5

6 Saple Proble 8.3 Calculate iniu allowable shaft diaeter. J c M y + M + T τ all 1357 N MPa 6 3 For a solid circular shaft, J c c π 3 c d c The McGraw-Hill Copanies, Inc. All rights reserved. 8-6

7 Stresses Under Cobined Loadings Wish to deterine stresses in slender structural ebers subjected to arbitrary loadings. Pass section through points of interest. Deterine force-couple syste at centroid of section required to aintain equilibriu. Syste of internal forces consist of three force coponents and three couple vectors. Dt Deterine stress distribution ditibti by applying the superposition principle. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-7

8 Stresses Under Cobined Loadings Axial force and in-plane couple vectors contribute to noral stress distribution in the section. Shear force coponents and twisting couple contribute to shearing stress distribution in the section. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-8

9 Stresses Under Cobined Loadings Noral and shearing stresses are used to deterine principal p stresses, axiu 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 et. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-9

10 Saple Proble 8.5 SOLUTION: Deterine internal forces in Section EFG. Evaluate noral stress at H. Evaluate shearing stress at H. Three forces are applied to a short steel post as shown. Deterine the principle stresses, principal planes and axiu shearing stress at point H. Calculate principal stresses and axiu shearing stress. Deterine principal planes. 006 The McGraw-Hill Copanies, Inc. All rights reserved. 8-10

11 Saple Proble 8.5 SOLUTION: Deterine internal forces in Section EFG. V M x x 30 kn P 50kN 75kN ( 50kN)( ) ( 75kN)( 0.00 ) 8.5kN M y 0 M Note: Section properties, V ( 30kN)( ) 3kN 3 A ( )( ) I x I ( )( ) ( )( ) The McGraw-Hill Copanies, Inc. All rights reserved. 8-11

12 Saple Proble 8.5 Evaluate noral stress at H. σ y P + + A M I a 50kN M I x x b ( 3kN )( 0.00 ) ( 8.5kN )( 0.05) ( ) MPa 66.0MPa Evaluate shearing stress at H. Q τ y A y V I Q t x [( )( 0.045) ]( ) 17.5 MPa 3 ( )( 3) 75kN ( )( ) 006 The McGraw-Hill Copanies, Inc. All rights reserved

13 Saple Proble 8.5 Calculate principal stresses and axiu shearing stress. Deterine principal i planes. τ ax R MPa σ σ ax in tan θ θ p OC OC p R R CY CD τ ax σ σ θ p ax in MPa MPa MPa 70.4MPa 7.4 MPa θ p The McGraw-Hill Copanies, Inc. All rights reserved. 8-13