Explanation The free body diagram of Gear F is as follows. Figure-(1) The free body diagram of the shaft is as follows. Figure-(2) Write the expression for tangential force acting on the shaft. F t = T ( c 2 ) (I) Here, the tangential force on the shaft is F t , torque acting on the shaft is T and the diameter of the shaft is c . Write the expression for normal force acting on the shaft. F n = F t tan ϕ (II) Here, F n is the normal force acting on the shaft and ϕ is the pressure angle. Write the expression for torque at gear C . T c = F t ( b 2 ) (III) Here, torque at gear C is T c . Write the expression for load at sprocket B . P = T c ( a 2 ) (IV) Here, the load at sprocket B is P . Write the expression for moment about A in z - direction. ∑ ( M A ) z = 0 ( e + f + g ) R D y − F n ( f + g ) − ( P × e ) = 0 R D y = F n ( f + g ) + ( P × e ) ( e + g + f ) (V) Here, the moment about A in z - direction is ( M A ) z , the reaction at D in y - direction is R D y , the distance between sprocket B and pivot A is e , the distance between sprocket B and gear C is f and the distance between gear C and pivot D is g . Write the expression for moment about A in y - direction. ∑ ( M A ) y = 0 − ( e + f + g ) R D z + F t ( f + g ) = 0 R D z = F t ( f + g ) ( e + g + f ) (VI) Here, the moment about A in y - direction is ( M A ) y and the reaction at D in z direction is R D z . Write the expression for forces in y - direction. ∑ F y = 0 R A y + R D y − F n − P = 0 R A y = F n + P − R D y (VII) Here, the reaction at A in y - direction is R A y . Write the expression for forces in z - direction. ∑ F z = 0 R A z + R D z − F t = 0 R A z = F t − R D z (VIII) Here, the reaction at A in z - direction is R A z . From given figure, the maximum bending moment will be either at B or C . Write the expression for bending moment at B . M B = e R A y 2 + R A z 2 (IX) Here, M B is the bending moment at B . Write the expression for bending moment at C . M C = g R D y 2 + R D z 2 (X) Here, M C is the bending moment at C . Write the expression for bending stress at B . σ B = 32 M B π d 3 (XI) Here, the diameter of the shaft is d and the bending stress at B is σ B . Write the expression for shearing stress at B . τ B = 16 T C π d 3 (XII) Here, the shearing stress at B is τ B . Calculate the maximum principal normal stresses in the shaft. σ max = σ B 2 + ( σ B 2 ) 2 + τ B 2 (XIII) Here, the maximum principal normal stress in the shaft is σ max . Calculate the maximum principal shear stresses in the shaft. τ max = ( σ B 2 ) 2 + τ B 2 (XIV) Here, the maximum principal shear stress in the shaft is τ max . Conclusion: Substitute 100 N ⋅ m for T and 125 mm for c in Equation (I). F t = 100 N ⋅ m ( 125 mm 2 ) = 100 N ⋅ m ( 125 mm ) ( 1 m 1000 mm ) 2 = 1600 N Substitute 1600 N for F t and 20 ° for ϕ in Equation (II). F n = ( 1600 N ) ( tan 20 ° ) = 582.352 N ≈ 582.4 N Substitute 1600 N for F t and 250 mm for b in Equation (III). T c = ( 1600 N ) ( 250 mm ) 2 = ( 1600 N ) ( ( 250 mm ) ( 1 m 1000 mm ) ) 2 = 200 N ⋅ m Substitute 200 N ⋅ m for T c and 150 mm for a in Equation (IV). P = 200 N ⋅ m ( 150 mm 2 ) = 200 N ⋅ m ( 150 mm × 1 m 1000 mm 2 ) ≈ 2667 N Substitute 2667 N for P , 582

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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Design of Shafts and Shaft Components

The shaft is basically defined as the machine element that rotates, is in circular cross-section, it is majorly used to transfer the power from one part of the system to another, or from a power generating machine to a power absorbing machine. The shaft …

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Chapter 3, Problem 77P

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The maximum principal normal stresses in the shaft.

The maximum principal shear stresses in the shaft.

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Chapter 3, Problem 76P

Chapter 3, Problem 78P

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Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)

Ch. 3 - 31 to 34 Sketch a free-body diagram of each...Ch. 3 - 31 to 34 Sketch a free-body diagram of each...Ch. 3 - Sketch a free-body diagram of each element in the...Ch. 3 - 3-1 to 3-4 Sketch a free-body diagram of each...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - 35 to 38 For the beam shown, find the reactions at...Ch. 3 - For the beam shown, find the reactions at the...Ch. 3 - For the beam shown, find the reactions at the...Ch. 3 - Repeat Prob. 36 using singularity functions...

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The maximum principal normal stresses in the shaft. The maximum principal shear stresses in the shaft.

Solution Summary: The author explains the principal normal stresses in the shaft and the maximum principal shear stresses. The free body diagram of Gear F is as follows.

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The maximum principal normal stresses in the shaft. The maximum principal shear stresses in the shaft.

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The maximum principal normal stresses in the shaft.

The maximum principal shear stresses in the shaft.