Concept explainers
The shaft is made of A992 steel and has an allowable shear stress of τallow = 75 MPa. When the shaft is rotating at 300 rpm, the motor supplies 8 kW of power, while gears A and B withdraw 5 kW and 3 kW, respectively. Determine the required minimum diameter of the shaft to the nearest millimeter. Also, find the rotation of gear A relative to C.
The required minimum diameter of the shaft.
The angle of twist of gear A relative to gear C.
Answer to Problem 1RP
The required minimum diameter of the shaft is
The angle of twist of gear A relative to gear C is
Explanation of Solution
Given information:
The allowable shear stress in the shaft is 75 MPa.
The motor supplies power of 8 kW.
Gear A and B withdraws power of 5 kW and 3 kW.
Shaft rotates at 300 rpm.
Calculation:
The expression for the power transmitted
Here, T is the applied torque and
Rearrange Equation (1) to find the torque at A.
Here,
The expression for angular velocity of the shaft
Here, f is the frequency of shaft’s rotation.
Substitute
Substitute 5 kW for
Find the torque at C.
Here,
Substitute 8 kW for
Sketch the internal torque in the segment BC of the shaft as shown in Figure 1.
Sketch the internal torque in the segment AB of the shaft as shown in Figure 2.
Refer Figure 1 and Figure 2.
Segment BC of the shat is subjected to a greater internal torque of
The torsion formula for allowable maximum shear stress in the solid shaft
Here,
The outer radius of the shaft is r.
The polar moment of inertia for a solid shaft of radius
Substitute r for c and
Substitute 75 MPa for
The diameter of the shaft is twice the radius of the shaft. So the value of diameter is 26 mm.
Therefore, the required minimum diameter of the shaft is
Determine the angle of twist
Here, L is the length of the shaft and G is the shear modulus of elasticity of the material.
Rearrange Equation (7) for angle of twist of gear A relative to gear C
Refer the properties of A992 steel.
The value of shear modulus of elasticity of A992 steel is 75 GPa.
The value of radius of the solid shaft is 13 mm.
Substitute
Refer Figure 2.
The torque in the region AB of the shaft is
Refer Figure 1.
The torque in the region BC of the shaft is
Substitute
Therefore, the angle of twist of gear A relative to gear C is
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Chapter 5 Solutions
Mechanics of Materials
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