Concept explainers
The steel shaft shown in the figure carries a 18-lbf gear on the left and a 32-lbf gear on the right. Estimate the first critical speed due to the loads, the shaft’s critical speed without the loads, and the critical speed of the combination.
Problem 7–32
Dimensions in inches.
The first critical speed of the shaft due to loads.
The critical speed of the shaft without loads.
The critical speed of the combination.
Answer to Problem 32P
The first critical speed of the shaft due to loads is
The critical speed of the shaft without loads is
The critical speed of the combination is
Explanation of Solution
Write the expression for the moment of the inertia.
Here, the diameter of the shaft is
Write the expression for the ratio of the bending moment to inertia.
Here, the bending moment across the shaft is
Write the expression for the ratio of the bending moment to inertia in terms of the spread sheet cell locations.
Substitute
Here, the young’s modulus of the shaft material is
Integrate the Equation (IV) with respect to x.
Substitute
Apply the boundary conditions.
Substitute
Substitute
Substitute
Substitute
Write the expression for the deflection at
Write the expression for the deflection at
Write the expression for the deflection of the shaft due to
Write the expression for the deflection of the shaft due to
Write the expression for the critical velocity due to load.
Here, the acceleration due to gravity is
Write the expression for the weight of the left gear.
Here, the physical constant of the material is
Write the expression for the weight of the right gear.
Write the expression for the critical velocity without load.
Here, the acceleration due to gravity is
Write the expression for the critical speed for the combination using Dunkerley’s equation.
Conclusion:
Draw the diagram for the shaft.
Figure-(1)
The Figure-(1) shows all the dimensions of the shaft.
Calculate the moment of inertia for the part
Substitute
Calculate the moment of inertia for the part
Substitute
Calculate the moment of inertia for the part
Substitute
Since the diameter of the shaft part
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Here, the gravitational constant is
Substitute
Thus, he first critical speed of shaft due to loads is
Refer to table A-5 “Physical constants of the materials.” to obtain the weight density of the steel as
Substitute
Substitute
Draw the free body diagram for the calculated weights.
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the critical speed of shaft without loads is
Substitute
Thus, the critical speed of the combination is
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Chapter 7 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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