Geneva Mechanism & Dial-Indexing Production Line: 20-Bio-B7 Ergonomics, May 2016, Q6

Question 6: Geneva Mechanism & Dial-Indexing Production Line (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.

Check — stated assumptions (per the exam's own Note 1): (1) the dwell portion of the cycle is set equal to the slowest station's required processing time $T_p=4.3\ \text{s}$ — the minimum synchronized cycle at which no station is ever left waiting; (2) “rotational speed of the driven member” follows Groover's usage for this problem type, in which the driven member is the continuously turning Geneva drive (crank) that the motor turns once per cycle, $N=1/T_c$. The star wheel's own average speed ($N/n$) is reported alongside it for completeness. (3) Line efficiency and the proportion of downtime use Groover's definitions, $E=T_c/(T_c+FD)$ and $FD/(T_c+FD)$. The ideal cycle $T_c$ includes the indexing time, because the machine is up (not broken down) while it indexes.

Given.

Quantity	Symbol	Value
Number of stations (slots)	$n$	8
Slowest station processing time	$T_p$	4.3 s
Frequency of line stops	$F$	0.075 stops/cycle
Average downtime per stop	$D$	5 min = 300 s

Find. (a) indexing (handling) time $T_h$; (b) rotational speed of the driven member (Geneva drive); (c) production rate $R_p$; (d) line efficiency $E$; (e) proportion of downtime.

Fig. 6 — continuously-rotating driver pin engages the 8-slot star wheel; smooth entry/exit fixes the driver's indexing rotation to $180^{\circ}-360^{\circ}/n$.

Approach. Use the standard Geneva-wheel geometry to split the driver's rotation into an indexing (motion) angle and a dwell angle, set the dwell time equal to the station bottleneck $T_p$ to get the ideal cycle time, then layer in stoppages using the standard production-line relations $T_c'=T_c+FD$, $R_p=1/T_c'$, $E=T_c/T_c'$.

Geneva indexing/dwell angles. Smooth (impact-free) pin entry and exit requires the crank-to-pin line to be perpendicular to the slot at engagement, which fixes the driver's rotation angle during indexing to $$\theta_{\text{index}}=180^{\circ}-\frac{360^{\circ}}{n}=180-\frac{360}{8}=135^{\circ},$$ leaving a dwell angle of $360-135=225^{\circ}$ (fraction of cycle dwelling $=225/360=0.625$).
Part (a) — ideal cycle and indexing time. The dwell portion must equal the bottleneck station time: $$T_c=\frac{T_p}{0.625}=\frac{4.3}{0.625}=6.88\ \text{s},\qquad T_h=T_c-T_p=6.88-4.3=\boxed{2.58\ \text{s}}.$$
Part (b) — rotational speed. The Geneva drive (the member the motor turns) makes one revolution per cycle, and its dwell arc must last the 4.3 s station time: $$N=\frac{\theta_{\text{dwell}}}{360\,T_p}=\frac{225}{360(4.3)}\ \text{rev/s}=\frac{60}{6.88}\ \text{rpm}=\boxed{8.72\ \text{rpm}}.$$ The star wheel indexes only $1/8$ revolution per cycle, so its own average speed is $N/n=8.72/8=1.09$ rpm (for reference).
Part (c) — average cycle time with downtime. $$T_c'=T_p+T_h+FD=4.3+2.58+0.075(300)=4.3+2.58+22.5=29.38\ \text{s}.$$ Production rate: $$R_p=\frac{3600}{T_c'}=\frac{3600}{29.38}=\boxed{122.5\ \text{pieces/hr}}.$$
Part (d) — line efficiency. Uptime is the whole ideal cycle $T_c$ (processing plus indexing), so $$E=\frac{T_c}{T_c+FD}=\frac{6.88}{29.38}=\boxed{0.2342\ (23.4\%)}.$$
Part (e) — proportion of downtime. $$\frac{FD}{T_c'}=\frac{22.5}{29.38}=\boxed{0.7658\ (76.6\%)}.$$ Check: $E+D=0.2342+0.7658=1.000$, so uptime and downtime together make up the whole average cycle.

Result	Value
(a) Indexing (handling) time $T_h$	2.58 s
(b) Driven member (Geneva drive) speed	8.72 rpm
(b) Star wheel average speed (for reference)	1.09 rpm
(c) Production rate $R_p$	122.5 pieces/hr
(d) Line efficiency $E$	23.4%
(e) Proportion of downtime	76.6%

Concept & Theory

Topic. Geneva-mechanism dial indexing and automated production-line cycle analysis.

Key relations. Geneva indexing angle $\theta_{\text{index}}=180^{\circ}-360^{\circ}/n$; ideal cycle $T_c=T_p+T_h$; average cycle with downtime $T_c'=T_c+FD$; production rate $R_p=1/T_c'$; line efficiency $E=T_c/T_c'$; proportion of downtime $=FD/T_c'=1-E$.

Why this works. The Geneva mechanism converts a uniformly-rotating driver into an intermittent indexing motion purely from its slot geometry, so the dwell fraction of the cycle — during which the stations actually do their work — is fixed by the slot count $n$ alone; layering the standard $T_c'=T_c+FD$ relation on top of that ideal cycle accounts for real-world breakdowns, giving a realistic (much lower) production rate and line efficiency than the ideal, breakdown-free cycle would predict.

Common pitfall. Using the ideal cycle time $T_c=6.88\ \text{s}$ (no downtime) to compute production rate and efficiency — the exam explicitly supplies stoppage data $(F,D)$ that must be folded into $T_c$ first, and skipping this step overstates the production rate by a factor of roughly 4 here. A second trap is computing $E$ from the processing time alone ($T_p/T_c'$ = 14.6%): Groover's line efficiency counts indexing as uptime, and $E$ and the downtime proportion must sum to 1.

Source. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, ch. on Geneva mechanisms and analysis of automated production lines.

Question 6 of 6: Geneva Mechanism & Dial-Indexing Production Line

Question 6: Geneva Mechanism & Dial-Indexing Production Line (20 marks)