Impact Loading on a Shaft A shaft of diameter d and length L has a flywheel (radius of gyration r, weight W, modulus of rigidity G, yield strength in shear Sys) at one end and runs at a speed of n. If the shaft is instantly stopped at the other end, determine a. The maximum shaft angle of twist b.

Question

Impact Loading on a Shaft A shaft

of diameter d and length [latex]L[/latex] has a flywheel (radius of gyration [latex]r,[/latex] weight [latex]W,[/latex] modulus of rigidity [latex]G[/latex], yield strength in shear [latex]S_{ys}[/latex]) at one end and runs at a speed of [latex]n[/latex]. If the shaft is instantly stopped at the other end, determine

a. The maximum shaft angle of twist

b. The maximum shear stress

Given: [latex]d[/latex] = 3 in., [latex]L[/latex] = 2.5 ft, [latex]W[/latex] = 120 lb, [latex]r[/latex] = 10 in., [latex]n[/latex] = 150 rpm

Assumption: The shaft is made of ASTM-A242 steel. So, by Table B.1, [latex]G = 11.5 imes 10^{6}[/latex] psi and [latex]S_{ys}[/latex] = 30 ksi.

Accepted Answer

The area properties of the shaft are

[latex] A=\frac{\pi(3)^2}{4}=7.069 in ^2, \quad J=\frac{\pi(3)^4}{32}=7.952 in ^4 [/latex]

The angular velocity equals

[latex] \omega=n\left(\frac{2 \pi}{60} ight)=150\left(\frac{2 \pi}{60} ight)=5 \pi rad / s [/latex]

a. The kinetic energy of the flywheel must be absorbed by the shaft. So, substituting Equation 4.43 into Equation 4.41, we have

[latex] E_k=\frac{1}{2} I \omega^2 [/latex] (4.41)

[latex] I = mr^{2} [/latex] (4.43)

[latex] \begin{aligned} E_k &=\frac{W \omega^2 r^2}{2 g} \ &=\frac{120(5 \pi)^2(10)^2}{2(386)}=3835 in \cdot lb \end{aligned} [/latex] (a)

From Equation 4.39a,

[latex] \begin{aligned} \phi_{\max } &=\sqrt{\frac{2 E_k L}{G J}}=\left[\frac{2(3835)(2.5 imes 12)}{\left(11.5 imes 10^6 ight)(7.952)} ight]^{1 / 2} \ &=0.05 rad =2.87^{\circ} \end{aligned} [/latex]

b. Through the use of Equation 4.40,

[latex] \begin{aligned} au_{\max } &=2 \sqrt{\frac{E_k G}{A L}}=2\left[\frac{(3835 \left(11.5 imes 10^6 ight)}{(7.069)(2.5 imes 12)} ight]^{1 / 2} \ &=28.84 ksi \end{aligned} [/latex]

Comment: The stress is within the elastic range, 28.84 < 30, and hence assumption 2 of Section 4.7 is satisfied.

Question 4.14: Impact Loading on a Shaft A shaft of diameter d and length L...

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