Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
10th Edition
ISBN: 9780073398204
Author: Richard G Budynas, Keith J Nisbett
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 15, Problem 1P

An uncrowned straight-bevel pinion has 20 teeth, a diametral pitch of 6 teeth/in, and a transmission accuracy number of 6. Both the pinion and gear are made of through-hardened steel with a Brinell hardness of 300. The driven gear has 60 teeth. The gearset has a life goal of 109 revolutions of the pinion with a reliability of 0.999. The shaft angle is 90°, and the pinion speed is 900 rev/min. The face width is 1.25 in, and the normal pressure angle is 20°. The pinion is mounted outboard of its bearings, and the gear is straddle-mounted. Based on the AGMA bending strength, what is the power rating of the gearset? Use K0 = 1 and SF = SH = 1.

Expert Solution & Answer
Check Mark
To determine

The power rating of gear-set for bending.

Answer to Problem 1P

The power rating for gear-set for bending is 14.06hp.

Explanation of Solution

Write the expression for pitch diameter of gear.

dG=NPd (I)

Here, the pitch diameter of gear is dG, the number of teeth on gear is N and the diametral pitch is Pd.

Write the expression for pitch diameter of pinion.

dP=nPd (II)

Here, the pitch diameter of pinion is dP, the number of teeth on pinion is n and the diametral pitch is Pd.

Write the expression for speed ratio of gear set.

m=Nn (III)

Here, the speed ratio is m.

Write the expression for pitch line velocity.

vt=πdPnP (IV)

Here, the pitch line velocity is vt and the rotation of pinion per unit time is nP.

Write the expression for dynamic factor.

Kv=(A+vtA)B (V)

Here, dynamic factor is Kv and constants are A and B.

Write the expression for constant B.

B=0.25(12Qv)2/3 (VI)

Here, the transmission accuracy number is Qv.

Write the expression for constant A.

A=50+56(1B) (VII)

Write the expression for maximum pitch line velocity.

vt,max=[A+(Qv3)]2 (VIII)

Here, maximum pitch line velocity is vt,max.

Write the expression for size factor for 0.5Pd16teeth/in.

KS=0.4867+0.2132/Pd (IX)

Here, size factor is KS.

Write the expression for load distribution factor.

Km=Kmb+0.0036F2 (X)

Here, load distribution factor is Km, the face width is F and the constant is Kmb.

Write the critical expression for stress cycle factor for pinion.

KLP=1.683(NLP)0.0323 for 3×106NLP1010 (XI)

Here, the stress cycle factor for pinion is KLP and the life goal in rotation for pinion is NLP.

Write the critical expression for stress cycle factor for gear.

KLG=1.683(NLPm)0.0323 for 3×106NLP1010 (XII)

Here, the stress cycle factor for gear is KLG.

Write the expression for reliability factor for bending strength.

KR=0.50.25log(1R) for 0.99R0.999 (XIII)

Here, the reliability factor is KR and the reliability is R.

Write the expression for allowable bending stress number.

Sat=44HB+2100 (XIV)

Here, the allowable bending stress number is Sat and the Brinell hardness is HB.

Write the expression for permissible bending stress for pinion.

SwtP=SatKLPSFKTKR (XV)

Here, the permissible bending stress for pinion is SwtP, the bending safety factor is SF and the temperature factor is KT.

Write the expression for permissible bending stress for gear.

SwtG=SatKLGSFKTKR (XVI)

Here, the permissible bending stress for gear is SwtG.

Write the expression for bending stress for pinion.

stP=WtPFPdKoKvKSKmKxJP (XVII)

Here, the bending stress in pinion is StP, the transmitted load by pinion is WtP, the overload factor is Ko, length wise curvature factor is Kx and geometry factor for bending in pinion is JP.

Write the expression for bending stress for gear.

stG=WtGFPdKoKvKSKmKxJG (XVIII)

Here, the bending stress in gear is StG, the transmitted load on gear is WtG and geometry factor for bending in gear is JG.

Write the expression for transmitted load by pinion.

WtP=33000HPvt (XIX)

Here, the transmitted load by pinion is WtP and the power in hp for pinion is HP.

Write the expression for transmitted load on gear.

WtG=33000HGvt (XX)

Here, the transmitted load on gear is WtG and the power in hp for gear is HG.

Conclusion:

Substitute 60teeth for N and 6teeth/in for Pd in Equation (I).

dG=60teeth6teeth/in=10in

Substitute 20teeth for n and 6teeth/in for Pd in Equation (II).

dP=20teeth6teeth/in=3.333in

Substitute 60teeth for N and 20teeth for n in Equation (III).

m=60teeth20teeth=3

Substitute 3.333in for dP and 900rotation/min for nP in Equation (IV).

vt=π(3.333in)900rotation/min=9423.835in/min(1ft12in)=785.3ft/min

Substitute 6 for Qv in Equation (VI).

B=0.25(126)2/3=0.8255

Substitute 0.8255 for B in Equation (VII)

A=50+56(10.8255)=59.77

Substitute 59.77 for A, 0.8255 for B and 785.3ft/min for vt in Equation (V).

Kv=(59.77+785.3ft/min59.77)0.8255=(1.4688)0.8255=1.374

Substitute 59.77 for A and 6 for Qv in Equation (VIII).

vt,max=[59.77+(63)]2=[59.77+(3)]2=3940ft/min

Substitute 6teeth/in for Pd in Equation (IX).

KS=0.4867+0.2132/6teeth/in=0.5222

Substitute 1.1 for Kmb for one gear straddle-mounted and 1.25in for F in Equation (X).

Km=1.1+0.0036(1.25)2=1.106

Substitute 109rotation for NLP in Equation (XI).

KLP=1.683(109rotation)0.0323=0.862

Substitute 3 for m and 109rotation for NLP in Equation (XII).

KLG=1.683(1093)0.0323=0.893

Substitute 0.999 for R in Equation (XIII).

KR=0.50.25log(10.999)=1.25

Substitute 300 for HB in Equation (XIV).

Sat=44(300)+2100=15300psi

Substitute 15300psi for Sat, 0.862 for KLP, 1 for SF, 1 for KT and 1.25 for KR in Equation (XV).

SwtP=(15300psi)0.8621×1×1.25=10550.88psi

Refer to Figure 15-7 “Bending factor J(YJ) for coniflex straight-bevel gears with a 20° normal pressure angle and 90° shaft angle” to obtain value of JP as 0.249.

Substitute 10550.88psi for StP, 1.25in for F, 6teeth/in for Pd, 1 for Ko, 1.374 for Kv, 0.5222 for KS, 0.249 for JP and 1.106 for Km in Equation (XVII).

10550.88psi=WtP1.25in(6teeth/in)(1)(1.374)(0.5222×1.1061×0.249)WtP=10550.88×1.25×0.2496×1.374×0.5222×1.106lbfWtP=689.71lbf

Substitute 689.71lbf for WtP and 785.3ft/min for vt in Equation (XIX).

689.71lbf=33000HP785.3ft/minHP=689.71×785.333000hpHP=16.41hp

Substitute 15300psi for Sat, 0.893 for KLP, 1 for SF, 1 for KT and 1.25 for KR in Equation (XVI).

SwtG=(15300psi)×0.8931×1×1.25=10930.32psi

Refer to Figure 15-7 “Bending factor J(YJ) for coniflex straight-bevel gears with a 20° normal pressure angle and 90° shaft angle” to obtain value of JG as 0.206.

Substitute 10930.32psi for StG, 1.25in for F, 6teeth/in for Pd, 1 for Ko, 1.374 for Kv, 0.5222 for KS, 0.206 for JG and 1.106 for Km in Equation (XVIII).

10930.32psi=WtG1.25in×6teeth/in×1×1.374×0.5222×1.1061×0.206WtG=10930.32×1.25×0.2066×1.374×0.5222×1.106lbfWtG=591.13lbf

Substitute 591.13lbf for WtG and 785.3ft/min for vt in Equation (XX).

591.13lbf=33000HG785.3ft/minHG=785.3×591.1333000HG=14.06hp

The power rating for a gear-set for bending is defined as minimum of power rating in bending for pinion and gear, so that the bending of teeth on both gear and pinion can be saved.

Thus, the power rating for gear-set for bending is 14.06hp.

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