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22-Mec-B1 Advanced Machine Design · May 2013

Question 3 of 6: Two-gear shaft — fatigue diameter and deflection

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Exams, May 2013 — 07-Mec-B1 Advanced Machine Design. Open book, 3 hours, 100 marks. Part I (Problems 1–2) is compulsory; three of the four Part II problems (3–6) constitute a complete paper. All six problems are solved as a study resource.

Reference texts. R. G. Budynas & J. K. Nisbett, Shigley’s Mechanical Engineering Design, 10th ed. (shafts §7, bolted joints §8, bearings §12, brakes §16); R. L. Norton, Machine Design: An Integrated Approach, 5th ed.; R. C. Hibbeler, Mechanics of Materials, 10th ed. (impact loading, beam deflection); R. C. Juvinall & K. M. Marshek, Fundamentals of Machine Component Design, 5th ed. (journal bearings, friction brakes).

Question 3: Two-gear shaft — fatigue diameter and deflection (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Simply-supported shaft (self-aligning bearings): bearing $A$ at $0$, left gear (250 lb) at $4\text{ in}$, bearing $B$ at $12\text{ in}$, right (overhung) gear (750 lb) at $16\text{ in}$; torque fluctuates $T:-200 \to 400\text{ lb}\cdot\text{in}$; $S_{ut}=108\text{ ksi}$, $S_y=62\text{ ksi}$; design factor $n=2$.

Find. the required shaft diameter $d$ (DE-Goodman fatigue) and the maximum bending deflection.

A B 250 lb 750 lb 4 in 12 in 16 in
Load layout: bearings $A$(0) and $B$(12 in) as simple supports; 250 lb at 4 in, 750 lb overhung at 16 in. The largest bending moment falls at bearing $B$.

Approach. Find the support reactions and the bending-moment envelope; the rotating shaft makes the bending stress fully reversed while the torque fluctuates about a non-zero mean. Apply the DE-Goodman shaft-sizing equation at the critical section (bearing $B$) with the endurance limit adjusted by Marin factors, iterate for $d$, then integrate the elastic curve for the deflection.

  1. Reactions (vertical loads). Taking moments about $A$: $$R_B = \frac{250(4)+750(16)}{12} = 1083\text{ lb}, \qquad R_A = 1000 - 1083 = -83\text{ lb}.$$
  2. Critical bending moment. From the overhang free body (right of $B$), $M_B = -750(16-12)=-3000\text{ lb}\cdot\text{in}$; this exceeds the $-333\text{ lb}\cdot\text{in}$ at the left gear, so bearing $B$ governs. Because the shaft rotates, this bending is fully reversed: $$M_a = 3000\text{ lb}\cdot\text{in}, \qquad M_m = 0.$$
  3. Torque components. $$T_a = \frac{400-(-200)}{2}=300, \qquad T_m = \frac{400+(-200)}{2}=100\text{ lb}\cdot\text{in}.$$
  4. Endurance limit (Marin factors). $S_e' = 0.5S_{ut}=54\text{ ksi}$; machined surface $k_a = 2.70\,S_{ut}^{-0.265}=0.781$; size $k_b\approx0.86$ (converges near $d=1.2\text{ in}$); $k_c=k_d=k_e=1$ (bending, room temp, 50% reliability): $$S_e = k_a k_b\,S_e' \approx 36.4\text{ ksi}.$$
  5. DE-Goodman diameter. With $K_f=K_{fs}=1$ (no fillet/keyway data at $B$): $$d = \left[\frac{16n}{\pi}\!\left(\frac{\sqrt{4(K_fM_a)^2+3(K_{fs}T_a)^2}}{S_e}+\frac{\sqrt{4(K_fM_m)^2+3(K_{fs}T_m)^2}}{S_{ut}}\right)\right]^{1/3}.$$ Iterating (updating $k_b$ each pass) gives $d = 1.19\text{ in}$, so specify $$\boxed{d = 1.25\text{ in }\left(1\tfrac14''\right).}$$ A first-cycle yield check gives $n_y = 3.4 \ge 2$, so yielding does not govern.
  6. Maximum bending deflection. With $d=1.25\text{ in}$, $I=\pi d^4/64 = 0.120\text{ in}^4$ and $E=30\times10^6\text{ psi}$. Integrating $EI\,y''=M(x)$ with $y(0)=y(12)=0$, the largest deflection is at the overhung 750-lb gear: $$\boxed{y_{max} \approx 0.016\text{ in} = 0.40\text{ mm (downward, at the free end).}}$$
Question 3 — results
QuantityValue
Reactions $R_A,\,R_B$$-83\text{ lb},\ 1083\text{ lb}$
Alternating moment $M_a$ (at $B$)$3000\text{ lb}\cdot\text{in}$
Torque $T_a / T_m$$300 / 100\text{ lb}\cdot\text{in}$
Endurance limit $S_e$$36.4\text{ ksi}$
Required diameter$1.19\text{ in} \Rightarrow 1.25\text{ in}$
Max bending deflection$0.016\text{ in} = 0.40\text{ mm}$

Check / assumptions: No fillet, shoulder or keyway geometry is given at the critical section (bearing $B$), so $K_f=K_{fs}=1$ is assumed; a press-fit bearing seat or a profiled keyway would raise $K_f$ to roughly $1.6$–$2.0$ and increase the required diameter accordingly. Reliability is taken at 50% ($k_e=1$); a 99% requirement ($k_e=0.814$) would enlarge $d$ slightly. The DE-Goodman line is used as the (conservative) mean-stress criterion.