Concept explainers
A gear reduction unit uses the countershaft shown in the figure. Gear A receives power from another gear with the transmitted force FA applied at the 20° pressure angle as shown. The power is transmitted through the shaft and delivered through gear B through a transmitted force FB at the pressure angle shown.
(a) Determine the force FB, assuming the shaft is running at a constant speed.
(b) Find 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 3–72*
(a)
The magnitude of the force acting on the wheel B.
Answer to Problem 72P
The magnitude of the force acting on the wheel B is
Explanation of Solution
The figure below shows the free body diagram of pulley A.
Figure-(1)
The figure below shows the free body diagram of pulley B.
Figure-(2)
Calculate the force
Here, the force acting on pulley
Conclusion:
Substitute
Thus, the force
(b)
The bearing reaction forces at the supports.
Answer to Problem 72P
The bearing reaction at
Explanation of Solution
Write the expression for moment about bearing
Here, the reaction force at bearing
Write the equation to balance the forces in
Here, the reaction force at bearing
Write the expression for moment about bearing
Here, the reaction force at bearing
Write the equation to balance the forces in
Here, the reaction force at bearing
Calculate the reaction forces at bearing
Here, the reaction force at bearing
Calculate the reaction forces at bearing
Here, the reaction force at bearing
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the bearing reaction at
Substitute
Thus, the bearing reaction force at
(c)
The shear force and bending moment diagram for the shaft.
Answer to Problem 72P
The shear force and bending moment diagram for the shaft in
The shear force 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 shear force at
Here, the shear force at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
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
Here, the shear force at
Calculate the shear force at
Here, the shear force at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Calculate the moment at
Here, the moment at
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the shear force and bending moment diagram in
Figure-(3)
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the shear force and bending moment diagram in
Figure-(4)
(d)
The bending stress at point of maximum bending moment.
The shear stress at point of maximum bending moment.
Answer to Problem 72P
The bending stress at point of maximum bending moment is
The shear stress at point of maximum bending moment is
Explanation of Solution
Write the net moment at
Here, the net moment 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
Since
Substitute
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 72P
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
Thus, the principal stresses at the 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
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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