17. Shaft Design. Introduction. Torsion of circular shafts. Torsion of circular shafts. Standard diameters of shafts

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1 Objetives 17. Shaft Design Compute fores ating on shafts from gears, pulleys, and sprokets. ind bending moments from gears, pulleys, or sprokets that are transmitting loads to or from other devies. Determine torque in shafts from gears, pulleys, sprokets, luthes, and ouplings. Compare ombined stresses to suitable allowable stresses, inluding any required stress redution fators suh as stress onentration fators and fators of safety. Determine suitability of shaft design and/or neessary size of shafting. Introdution Shaft must have adequate torsional strength to transmit torque and not be over stressed. Shafts are mounted in bearings and transmit power through devies suh as gears, pulleys, ams and luthes. Components suh as gears are mounted on shafts using keys. Shaft must sustain a ombination of bending and torsional loads. August 15, August 15, 007 Standard diameters of shafts Torsion of irular shafts Diameter (in.) Upto Diameter inrements (in.) 1/16 to 5 1/8 5 to 8 1/ August 15, 007 August 15, 007 Torsion of irular shafts T L Angle of twist, θ G J θ the angle of twist (radians) T the applied torque (in-lb.) L shaft length (in.) J polar moment on inertia of the shaft ross setion (in ) G shear modulus of elastiity of the shaft material (lb/in ) August 15, August 15,

2 Torsional Shear Stresses Shear Stress in a shaft Torsional shear stress, S S T J J Polar moment of inertia radius of the shaft T Torque d diameter of shaft π d Torque August 15, T Shear stress, S S π D Where T torque D diameter of the shaft Torque August 15, T π S S ores on spur gear teeth t Transmitted fore n Normal fore or separating fore r Resultant fore θ pressure angle n t tan θ r t os θ August 15, ores on spur gear teeth Power, P T n 6,000 or Torque, T t r and r D p / Combining the above we an write t T D p P 6,000 D n August 15, p 6,000 P T n Loads from Bevel gears Loads from Bevel gears An additional axial fore will be ating on the shaft beause of the bevel angle or the pinion it is relatively small, and an be negleted. or the larger gear it will be signifiant and will be larger than the radial separating fore. ore transmitted, n t tan θ os γ θ Pressure angle γ Cone angle Axial ore, a t tan θ sin γ Resultant ore, r t + n or a depending on whihever is larger August 15,

3 Loads from Worm gears Axial Driving Separating Loads from Worm gears Driving fore on the worm gear, t T o Output torque Separating fore, s osφ where λ lead angle ϕ normal pressure angle f oeffiient of frition t sin φ T r o wg osλ - f sin λ August 15, August 15, Loads from Worm gears Loads from Worm gears Axial fore on the worm gear a(gear) where λ lead angle t(gear) ϕ normal pressure angle f oeffiient of frition osφ sin λ + f osλ osφ osλ - f sin λ August 15, August 15, Loads from Belts and Chains or a belt, Total load, t f + b Net driving fore, d f b Driving torque, T d r r effetive radius of pulley or sproket or a hain b 0 Bending of irular shafts Shafts transmit power through gears and pulleys These produe bending load in addition to torsion Use strength of material approah to alulate the reation fores and bending moments August 15, August 15,

4 Bending of irular shafts Bending of irular shafts August 15, August 15, Shaft Design Problems Step 1: Calulate the torque on the shaft from power Step : ind the torsional stress in the shaft Step : Calulate the loads oming from gears, belts or hains Step : Calulate the bending moment due to the ating fores. If neessary ombine the fores. Step 5: Calulate the bending stress in the shaft Step 6: Combine the bending stress and the torsional stress using the theories disussed in hapter August 15, Example Problem 17-1: Design Stresses in Shafts Shaft shown drives a gear set that is transmitting 5 hp at 1750 rpm. Shaft is supported in self-aligning ball bearings and gears are both 10 pith, 0 tooth, 0 spur gears. ind torsional and bending stresses in shaft. August 15, 007 ind the torsional stress in the shaft. ind the torsion in the shaft: irst find Z': (Appendix ) then: Tn hp 6,000 (-6) Z' π D 16 π (.75 in) Z' 16 (17-1) Z'.08 in 6,000 hp T n Ss T Z' (-6) 6,000 (5) T 1750 T 180 in-lb 180 in-lb Ss.08 in Ss 170 lb/in August 15, 007 August 15, 007

5 ind the resultant fore on the shaft: ind the load at the gear pith irle: (1-) Dp NT Pd Dp 0 10 (11-) t r os θ 90 lb r os 0 r 96 lb Dp inhes ind the maximum moment: t T DP (1-) Mm L (Appendix ) (180 in-lb) t in 96 lb (15 in) Mm t 90 lb Mm 60 in-lb August 15, August 15, Combined Stresses in Shafts ind the stress: S M Z As seen in Chap Z π D (Appendix ) π (.75 in) Z Z.01in S M Z 60 in-lb S.01 in S 8780 lb/in August 15, August 15, Combined maximum shear stress τ Maximum ombined shear stress S normal stress S S shear stress This an be rewritten as τ T + D T Torque in the shaft M Maximum moment τ ( ) 1/ M S S 1/ S + Example Problem 17-: Combined Stresses in Shafts rom previous example problem, find the ombined stress using the maximum shear stress theorem: τ Ss + S ½ Substituting stresses from previous example problem: τ (170 lb/in ) lb/in ½ τ 900 lb/in This should be ompared to shear stress allowables. (-5) August 15, August 15,

6 Maximum Normal Stress Theory σ equivalent ombined normal stress S normal stress from bending or axial loads S S shear or torsional stress S σ ± S This an be written as S S + 1/ 1/ [ M + (T M ] σ + D ) Example Problem 17-: Combined Stresses in Shafts rom Example Problem 17-1, find the ombined stress using the maximum normal stress theory: S σ ± Ss 8780 lb / in σ σ 900 lb / in S 1 + Substituting stresses from Example Problem 17-1: + (170 lb / in ) This should be ompared to the normal stress allowable in 1 + August 15, August 15, 007 Solid Cirular shaft Critial speeds of shafts τ ( T M ) D + 1/ σ 1/ [ M + (T M ) ] D + August 15, 007 August 15, 007 Critial speeds of shafts Operating speed should be 0% away from the ritial speed. Vibration frequeny, f is given by 1 k g f π W f frequeny in yles per seond, Hz k fore onstant, fore per inh of defletion g aeleration due to gravity, 86. in./s W weight in pounds, lb August 15, August 15,

7 Change the frequeny to rpm Critial speed, N 60 f Also k is weight divided by defletion W k δ 60 N π W g W δ Shaft with n onentrated loads Rayleigh s s equation is used. N W δ + W δ + W δ W δ W δ + W δ + W δ n n n n W δ N δ August 15, August 15, Example Problem 17-5: Critial Speed ind the estimated ritial speed for the shaft in Example Problem 17-1 (assume the entire shaft diameter is ¾ inh). irst, find defletion: L δ 8 EI I π D 6 π (.75 in) I 6 (Appendix ) (Appendix ) Example Problem 17-5: Critial Speed (ont d.) N N N rpm (17-1) I.016 in δ 96 lb (15 in) 8 (0 x 10 6 lb/in ) (.016 in ) δ.1 inh This is approximate, and additional multiples would exist at 80, 10, and 160 rpm. August 15, August 15, August 15,