Explanation Given information: The motor produces torque of 20 N ⋅ m . The inner and outer diameter of the shaft is 30 mm and 50 mm. Calculation: Sketch the free body diagram of the gears as shown in Figure 1. Refer Figure 1. Here, F is force and T ′ is torque. For 30 mm radius of the gear, F ( 0.03 m ) = 20 N ⋅ m F = 666.67 N For 100 mm radius of the gear, T ' = F ( 0.1 m ) (1) Substitute 666.67 N for F in Equation (1). T ' = ( 666.67 N ) ( 0.1 m ) = 66.67 N ⋅ m The shaft is fixed at C , so the torque exists only in the region EF of the shaft. Find the angle of twist of F with respect to E ( ϕ F / E ) using the relation. ϕ F / E = T L J G (2) Here, the torque applied in the shaft is T , the length of the shaft is L , the polar moment of inertia is J , and the shear modulus of elasticity of the material is G . Length of shaft from F to E is 0.6 m. The polar moment of inertia for a hollow shaft of outer radius ( r 0 ) and inner radius ( r i ) is π 2 ( r 0 4 − r i 4 ) . Substitute π 2 ( r 0 4 − r i 4 ) for J in Equation (2). ϕ F / E = T L ( π 2 ( r 0 4 − r i 4 ) ) G = 2 T L ( π ( r 0 4 − r i 4 ) ) G (3) The value of outer radius ( r 0 ) and inner radius ( r i ) of the hollow shaft is 25 mm and 15 mm. Refer the properties of L 2 steel: The value of shear modulus of elasticity of L 2 steel is 75 GPa. Substitute 66.67 N ⋅ m for T , 0.6 m for L , 25 mm for r 0 , 15 mm for r i , and 75 GPa for G in Equation (3). ϕ F / E = 2 T L ( π ( r 0 4 − r i 4 ) ) G = 2 ( 66.67 N ⋅ m ) ( 0.6 m ) ( π ( ( 25 mm ) 4 − ( 15 mm ) 4 ) ) ( 75 GPa ) = 2 ( 66.67 N ⋅ m ) ( 0.6 m ) ( π ( ( 25 mm × 10 − 3 m 1 mm ) 4 − ( 15 mm × 10 − 3 m 1 mm ) 4 ) ) ( 75 GPa × 10 9 N m 2 1 GPa ) = 0

10th Edition

ISBN: 9780134319650

Author: Russell C. Hibbeler

Publisher: PEARSON

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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 5.4, Problem 5.73P

To determine

The angle of twist of F with respect to E

The angle of twist of F with respect to D

The absolute maximum shear stress in the shaft

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Chapter 5.4, Problem 5.72P

Chapter 5.4, Problem 5.74P

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Chapter 5 Solutions

Mechanics of Materials (10th Edition)

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The angle of twist of F with respect to E The angle of twist of F with respect to D The absolute maximum shear stress in the shaft

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The angle of twist of F with respect to E The angle of twist of F with respect to D The absolute maximum shear stress in the shaft

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Chapter 5 Solutions

The angle of twist of F with respect to E

The angle of twist of F with respect to D

The absolute maximum shear stress in the shaft