Concept explainers
The demonstration of how rapidly Rayleigh’s method converges for the uniform-diameter solid shaft.
Answer to Problem 29P
The Rayleigh method for uniform diameter shaft is converging rapidly by using a static deflection beam equation.
Explanation of Solution
Write the expression for moment of inertia.
Here, the diameter of the shaft is
Write the expression for area of the shaft.
Write the expression for weight of the shaft.
Here, the specific weight is
Write the expression for influence coefficient.
Here, the length of the shaft is
Write the expression for deflection at point 1.
Write the expression for Rayleigh method.
Write the expression for Rayleigh method.
Write the expression for first critical speed.
Draw the diagram for the two elements system.
Figure-(1)
The figure-(1) shows the required dimension.
Write the expression for the deflection at point 1 for two element.
Write the expression for the deflection at point 2 for two element.
Write the expression for Rayleigh method for two element.
Write the expression for Rayleigh method for two element.
Write the expression for first critical speed for two element.
Draw the diagram for the three element system.
Figure-(2)
The Figure-(2) shows all the dimensions for the three elements.
Write the expression for the deflection at point 1 for three elements
Write the expression for the deflection at point 2 for three element.
Write the expression for the deflection at point 2 for three element.
Write the expression for Rayleigh method for three elements.
Write the expression for Rayleigh method for three elements.
Write the expression for first critical speed for three elements.
Conclusion:
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Calculate the square of the deflection at point 1 of element 1.
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Calculate the square of the deflection at point 1 of element 2.
Calculate the square of the deflection at point 2 of element 2.
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Calculate the square of the deflection at point 1 of element 3.
Calculate the square of the deflection at point 2 of element 3.
Calculate the square of the deflection at point 3 of element 3.
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Since the static bending equation is available, and satisfied the moment-free and deflection-free ends, so the convergence is rapid using a static deflection beam equation.
Thus, the Rayleigh method for uniform diameter shaft is converging rapidly by using a static deflection beam equation.
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Chapter 7 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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