Twin Acme power screws raising a sluice gate — torque, speed and power (20 marks): 22-Mec-B1 Advanced Machine Design, December 2018, Q6

Question 6: Twin Acme power screws raising a sluice gate — torque, speed and power (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. Two identical single-start 3-in Acme screws share a sluice gate; the thread table supplies the geometry and the question supplies both friction coefficients, the track-friction allowance and the hoisting speed.

Given data — Problem 6
Quantity	Symbol	Value
Gate weight	$W_g$	50 tons (short tons, 2000 lb each)
Track friction allowance	±$W_f$	+2 tons raising, −2 tons lowering
Number of screws sharing the load	—	2
Nominal (major) screw diameter	$d$	3.000 in
Threads per inch (table, 3-in Acme)	—	2 → pitch $p = 0.500$ in
Mean (pitch) diameter (table)	$d_m$	2.750 in
Thread friction coefficient	$\mu$	0.10
Collar effective diameter / friction	$d_c$ / $\mu_c$	5.0 in / 0.03
Acme thread half-angle	$\alpha$	14.5$^\circ$
Gate travel speed	$v$	2 ft/min = 24 in/min

Find. (a) the raising and lowering torque at each screw, (b) the screw rotational speed, and (c) the motor horsepower per screw for raising.

One of two Acme power screws: the gate load plus track friction is shared by the two screws; each also overcomes collar-bearing friction at $d_c=5$ in.

Approach. Split the gate load (with its track-friction allowance) between the two screws, read the thread geometry off the Acme table, then apply the standard Acme power-screw torque expressions — which differ from square-thread formulas only by the $\sec\alpha$ factor that accounts for the wedging action of the 29° thread — and add the collar-bearing torque separately. Speed follows from the lead, and power from torque times angular velocity.

Thread geometry from the table. The 3.000 in row gives 2 threads per inch, so $p = 1/2 = 0.500$ in, and for a single-threaded screw the lead equals the pitch: $$l = p = 0.500\text{ in}, \qquad d_m = 2.750\text{ in (pitch diameter, table)}$$ The lead angle is $\lambda = \arctan\bigl[l/(\pi d_m)\bigr] = \arctan(0.500/8.639) = 3.31^\circ$, confirming a shallow, self-locking-candidate thread.
Axial load carried by each screw. Track friction adds to the load on the way up and subtracts on the way down; the two screws share it equally. Using the short ton of 2000 lb, $$F_{raise} = \frac{(50+2)\times 2000}{2} = 52\,000\text{ lb}, \qquad F_{lower} = \frac{(50-2)\times 2000}{2} = 48\,000\text{ lb}$$
Acme thread correction factor. The thread flank is inclined at $\alpha = 14.5^\circ$, so the normal force on the flank — and hence the friction — is larger than for a square thread by $\sec\alpha$: $$\sec\alpha = \frac{1}{\cos 14.5^\circ} = 1.0328$$
Raising torque at the thread. Shigley's Acme raising expression is $$T_{R,\text{screw}} = \frac{F d_m}{2}\left(\frac{l + \pi\mu d_m\sec\alpha}{\pi d_m - \mu l \sec\alpha}\right)$$ With $\pi d_m = 8.639$, $\pi\mu d_m\sec\alpha = 0.8923$ and $\mu l\sec\alpha = 0.0516$: $$T_{R,\text{screw}} = \frac{52\,000(2.750)}{2}\left(\frac{0.500+0.8923}{8.639-0.0516}\right) = 71\,500 \times 0.16212 = 11\,593\text{ lb}\!\cdot\!\text{in}$$
Collar torque and total raising torque. The thrust collar is a separate, purely frictional loss acting at the effective collar radius: $$T_c = \frac{F \mu_c d_c}{2} = \frac{52\,000(0.03)(5.0)}{2} = 3900\text{ lb}\!\cdot\!\text{in}$$ $$\boxed{\;T_R = 11\,593 + 3900 = 15\,493\text{ lb}\!\cdot\!\text{in per screw} \;(1291\text{ lb}\!\cdot\!\text{ft})\;}$$ The roller thrust bearing is doing its job: even so, the collar still accounts for a quarter of the total driving torque.
Lowering torque. Lowering reverses the sign of the lead term at the thread, and the load is now the lighter 48 000 lb: $$T_{L,\text{screw}} = \frac{F d_m}{2}\left(\frac{\pi\mu d_m\sec\alpha - l}{\pi d_m + \mu l\sec\alpha}\right) = 66\,000\left(\frac{0.8923-0.500}{8.639+0.0516}\right) = 2980\text{ lb}\!\cdot\!\text{in}$$ $$T_c = \frac{48\,000(0.03)(5.0)}{2} = 3600\text{ lb}\!\cdot\!\text{in}$$ $$\boxed{\;T_L = 2980 + 3600 = 6580\text{ lb}\!\cdot\!\text{in per screw}\;}$$
Self-locking check. The screw thread is self-locking when the friction term exceeds the lead term, $\pi\mu d_m\sec\alpha \gt l$: $$0.8923 \;\gt\; 0.500 \;\checkmark$$ The lowering thread torque is positive, so the gate cannot run away under its own weight even before the collar friction and any brake are counted. That is an essential safety property for a dam gate: a power failure leaves the gate where it is.
Rotational speed. One revolution advances the nut by one lead, so $$N = \frac{v}{l} = \frac{24\text{ in/min}}{0.500\text{ in/rev}}$$ $$\boxed{\;N = 48\text{ rev/min}\;}$$
Motor power per screw for raising. Using the question's own definition, raising torque times angular velocity, with the customary conversion $\text{hp} = T[\text{lb}\!\cdot\!\text{in}]\times N[\text{rpm}]/63\,025$: $$P = \frac{15\,493 \times 48}{63\,025}$$ $$\boxed{\;P = 11.8\text{ hp per screw (8.8 kW)}\;}$$

The overall efficiency of the screw thread alone on raising is $\eta = Fl/(2\pi T_{R,\text{screw}}) = 35.7$ %, which is typical for a self-locking Acme screw — the price paid for the safety of not being back-drivable. Counting the collar loss the figure drops to about 26.7 %. In practice each drive would be specified with a 15 hp motor through a reduction gearbox, giving margin for gate silting, ice loading and a partly seized track.

Final results — Problem 6
Quantity	Symbol	Result
Lead / mean diameter (3-in Acme, 2 tpi, single)	$l$ / $d_m$	0.500 in / 2.750 in
Lead angle	$\lambda$	3.31$^\circ$
Load per screw, raising / lowering	$F$	52 000 lb / 48 000 lb
Thread torque, raising / lowering	$T_{screw}$	11 593 / 2980 lb·in
Collar torque, raising / lowering	$T_c$	3900 / 3600 lb·in
(a) Total raising torque per screw	$T_R$	15 493 lb·in
(a) Total lowering torque per screw	$T_L$	6580 lb·in
Self-locking?	—	Yes ($0.892 \gt 0.500$)
(b) Screw rotational speed	$N$	48 rev/min
(c) Motor power per screw (raising)	$P$	11.8 hp (8.8 kW)
Thread efficiency on raising	$\eta$	35.7 %

Topic. Acme power-screw drive: torque to raise and lower a load, collar friction, self-locking and drive power.

Key relations. $T_R = \dfrac{Fd_m}{2}\dfrac{l+\pi\mu d_m\sec\alpha}{\pi d_m - \mu l\sec\alpha} + \dfrac{F\mu_c d_c}{2}$; $T_L$ flips the sign of $l$ in the numerator and of $\mu l \sec\alpha$ in the denominator; self-locking when $\pi\mu d_m\sec\alpha \gt l$; $N = v/l$; $\text{hp} = TN/63\,025$.

Why this works. Unwrapping one turn of the thread gives an inclined plane of slope $\lambda$; the raising torque is the force needed to push the load up that plane against friction, multiplied by the mean radius. The Acme flank angle tilts the normal force out of the plane of the load, which increases the normal force by $\sec\alpha$ and therefore the friction with it.

Common pitfall. Forgetting that the two screws share the load, which doubles every torque; and applying the track-friction allowance in the wrong sense — it adds when raising but subtracts when lowering, so the raising and lowering calculations use different axial loads. A third trap is treating the pitch as the lead: they are equal only because this screw is single-threaded.

Source. Budynas & Nisbett, Shigley's Mechanical Engineering Design, 11th ed., §8-2 (power screws, Eqs. 8-5, 8-6, 8-7 and Table 8-3 for Acme thread dimensions); Juvinall & Marshek, Fundamentals of Machine Component Design, 6th ed., §10-4.

Question 6 of 6: Twin Acme power screws raising a sluice gate — torque, speed and power (20 marks)

Question 6: Twin Acme power screws raising a sluice gate — torque, speed and power (20 marks)