Concept explainers
A countershaft carrying two V-belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the Light side.
(a) Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed.
(b) Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.
(c) Draw shear-force and bending-moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.
(d) At the point of maximum bending moment, determine the bending stress and the torsional shear stress.
(e) At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.
Problem 3–71*
Dimensions in millimeters.
(a)
The tensions in the belt on pulley
Answer to Problem 71P
The tension in the belt pulley
Explanation of Solution
Write the relationship between tension on the loose side with respect to tension on the tight side.
Here, the tension on the tight side is
Write the equation to balance the tension on the counter shaft.
Substitute
Here, the tension on the tight side of pulley
Conclusion:
Substitute
Substitute
Thus, the tension in the belt pulley
(b)
The magnitude of the bearing reaction forces.
Answer to Problem 71P
The magnitude of bearing reaction force at
Explanation of Solution
Write the expression for magnitude of bearing reaction force at
Here, the tension on tight side of pulley
Write the expression for magnitude of bearing reaction force at
Write the expression for magnitude of bearing reaction force at
Here, the magnitude of bearing force at
Write the expression for magnitude of bearing force at
Here, the magnitude of bearing reaction force at
Calculate the bearing reaction force at
Here, the bearing reaction force at
Calculate the bearing reaction force at
Here, the bearing reaction force at
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the magnitude of bearing reaction force at
(c)
The shear force diagram and bending moment diagram for the shaft.
Answer to Problem 71P
The shear force diagram and bending moment diagram for the shaft in
The shear force diagram and bending moment diagram for the shaft in
Explanation of Solution
The calculations for shear force and bending moment diagram in
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the moment at
The moment at the supports of the simply supported beam is zero.
Calculate the moment at
Here, the moment at
The calculations for shear force and bending moment diagram in
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the moment at
The moment at the supports of the simply supported beam is zero.
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Conclusion:
Substitute
Substitute
Substitute
Substitute
Thus, the shear force diagram and bending moment diagram for the shaft in
Figure-(1)
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the shear force diagram and bending moment diagram for the shaft in
Figure-(2)
(d)
The bending stress at point of maximum bending moment.
The shear stress at point of maximum bending moment.
Answer to Problem 71P
The bending stress at point of maximum bending moment is
The shear stress at point of maximum bending moment is
Explanation of Solution
It is clear from the bending moment diagrams, that the critical location is at
Write the net moment at
Here, the net moment at
Write the torque transmitted by shaft from
Here, the torque transmitted by shaft from
Calculate the bending stress.
Here, the bending stress is
Calculate the shear stress.
Here, the shear stress is
Conclusion:
Substitute
Substitute
Convert the diameter of the shaft from
Substitute
Thus, the bending stress at point of maximum bending moment is
Substitute
Thus, the shear stress at point of maximum bending moment is
(e)
The principal stresses at point of maximum bending moment.
The maximum shear stress at point of maximum bending moment.
Answer to Problem 71P
The principal stresses at point of maximum bending moment are
The maximum shear stress at point of maximum bending moment is
Explanation of Solution
Calculate the maximum principal stress.
Here, the maximum principal stress is
Calculate the minimum principal stress.
Here, the minimum principal stress is
Calculate the maximum shear stress.
Here, maximum shear stress is
Conclusion:
Substitute
Substitute
The principal stresses at point of maximum bending moment are
Substitute
Thus, the maximum shear stress at point of maximum bending moment is
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Chapter 3 Solutions
Loose Leaf for Shigley's Mechanical Engineering Design Format: LooseLeaf
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