Mechanics of Materials, 7th Edition
Mechanics of Materials, 7th Edition
7th Edition
ISBN: 9780073398235
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek
Publisher: McGraw-Hill Education
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Chapter 3.5, Problem 75P

Three shafts and four gears are used to form a gear train that will transmit 7.5 kW from the motor at A to a machine tool at F. (Bearings for the shafts are omitted in the sketch.) Knowing that the frequency of the motor is 30 Hz and that the allowable stress for each shaft is 60 MPa, determine the required diameter of each shaft.

Chapter 3.5, Problem 75P, Three shafts and four gears are used to form a gear train that will transmit 7.5 kW from the motor

Expert Solution & Answer
Check Mark
To determine

The required diameter of the shaft AB.

The required diameter of the shaft CD.

The required diameter of the shaft EF.

Answer to Problem 75P

The required diameter of the shaft AB is 15.0mm_.

The required diameter of the shaft CD is 20.4mm_.

The required diameter of the shaft EF is 27.6mm_.

Explanation of Solution

Given information:

The frequency of the motor is 30 Hz.

The power transmitted by the shafts is 7.5 kW.

The allowable shearing stress in each shaft is 60 MPa.

The diameter of the shaft AB is dAB=16mm.

The diameter of the shaft CD is dCD=20mm.

The diameter of the shaft EF is dEF=28mm.

Calculation:

The maximum shear stress in the shaft (τmax) is expressed as shown below:

τmax=TcJ (1)

Here, T is the torque transmitted by the shaft, c is the radius of the shaft, and J is the polar moment of inertia of the shaft.

The power transmitted by the shaft (P) is expressed as follows:

P=(2πf)T (2)

Here, f is the frequency of the shaft.

For shaft AB:

The frequency of the shaft AB is fAB=30Hz.

Substitute 7.5kW for P and 30Hz for f in Equation (2).

7.5kW=2π(30Hz)(T)7.5kW(103W1kW)=2π(30Hz)(T)T=39.789Nm

The polar moment of inertia of shaft AB with radius c is JAB=π2(c)4.

Substitute 60 MPa for τmax, 39.789Nm for T, and π2(c)4 for J in Equation (1).

60MPa=(39.789Nm)(c)π2(c)460MPa×106Pa1MPa=2(39.789Nm)πc3c3=4.2217×107c=7.50×103m

Diameter of the shaft AB is twice the radius of the shaft AB.

dAB=2(7.50×103m)=15.0×103m=15.0×103m×103mm1m=15.0mm

Therefore, the required diameter of the shaft AB is 15.0mm_.

For shaft CD:

The radius at gear B is rB=60mm.

The radius at gear C is rC=150mm.

The frequency of the shaft CD is as follows:

fCD=rBrCfAB=60mm150mm(30Hz)=12Hz

Substitute 7.5kW for P and 12Hz for f in Equation (2).

7.5kW=2π(12Hz)(T)7.5kW(103W1kW)=2π(12Hz)(T)T=99.472Nm

The polar moment of inertia of shaft CD with radius c is JCD=π2(c)4.

Substitute 60 MPa for τmax, 99.472Nm for T, and π2(c)4 for J in Equation (1).

60MPa=(99.472Nm)(c)π2(c)460MPa×106Pa1MPa=2(99.472Nm)πc3c3=1.0554×106c=0.01018m

The diameter of the shaft CD is twice the radius of the shaft CD.

dCD=2(0.01018m)=0.02036m=0.02036m×103mm1m=20.4mm

Therefore, the required diameter of the shaft CD is 20.4mm_.

For shaft EF:

The radius at gear D is rD=60mm.

The radius at gear E is rE=150mm.

The frequency of the shaft EF is as follows:

fEF=rDrEfCD=60mm150mm(12Hz)=4.8Hz

Substitute 7.5kW for P and 4.8Hz for f in Equation (2).

7.5kW=2π(4.8Hz)(T)7.5kW(103W1kW)=2π(4.8Hz)(T)T=248.68Nm

The polar moment of inertia of shaft EF with radius c is JEF=π2(c)4.

Substitute 60 MPa for τmax, 248.68Nm for T, and π2(c)4 for J in Equation (1).

60MPa=(248.68Nm)(c)π2(c)460MPa×106Pa1MPa=2(248.68Nm)πc3c3=2.6386×106c=0.01382m

The diameter of the shaft EF is twice the radius of the shaft EF.

dEF=2(0.01382m)=0.02764m=0.02764m×103mm1m27.6mm

Therefore, the required diameter of the shaft EF is 27.6mm_.

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Students have asked these similar questions
EXERCISE 7 Three shafts and four gears are used to form a gear train that will transmit 7.5 kW from the motor at A to a machine tool at F (Figure 4). (Bearings for the shafts are omitted in the sketch.) Knowing that the frequency of the motor is 30 Hz and that the allowable stress for each shaft is 60 MPa, determine the required diameter of: 150 mm (a) shaft AB. Ans dAB: 15 mm 150 mun (b) shaft CD. Ans dcp: 20.4 mm D. 60 mm 60 (c) shaft EF B. Ans dEF: 27.6 mm
Problem 2: Consider the two gears mounted on the shaft, as shown. Distances are L1 = 100 mm, L2 = 250 mm, and L3 = 175 mm. Diameter of shaft AB is 15 mm, diameter of shaft BC is 20 mm, and diameter of shaft CD is 17.5 mm. Young's modulus for each portion is 200 GPa. Determine the displacement of point D with reference to point A. Also, determine the displacement of point A relative to C. b. Determine the axial strain at any point in shaft segments AB, BC and CD, respectively. If Poisson's ratio for the material of the shafts is v = 0.2, determine the lateral strain in each of the shaft segments. Based on your answers to part (a), do the gears move towards each other, or do they separate from each other? Note that in real gearboxes, movement of gears towards or away from each other is prevented by using thrust bearings. 10 kN 4 kN 5 kN 7 kN 4 kN NY 0I 17.
Solve it correctly please.

Chapter 3 Solutions

Mechanics of Materials, 7th Edition

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