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)

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 3-inch single-start Acme screws sharing the gate load, each running through a collar thrust bearing.

Find. (a) the raising and lowering torque per screw; (b) the screw rotational speed; (c) the horsepower to raise the gate.

Approach. Split the load between the two screws, add or subtract the track-friction allowance according to the direction of travel, apply the Acme power-screw torque equations (thread term plus collar term), then convert the linear gate speed to screw revolutions through the lead and take the product of torque and angular velocity for power.

1. Load carried by each screw. Raising, the track friction adds to the weight; lowering, it subtracts. With two screws sharing equally and 2000 lb per short ton, $$F_{\text{raise}}=\frac{50+2}{2}\times 2000 = 52\,000\ \text{lb},\qquad F_{\text{lower}}=\frac{50-2}{2}\times 2000 = 48\,000\ \text{lb}$$
2. Thread geometry from the Acme table. The 3.000-in row gives 2 threads per inch, so the pitch is $p = 0.500$ in; the thread is single-start, hence the lead $l = p = 0.500$ in and the pitch diameter is $d_m = 2.750$ in. The Acme thread-angle correction is $$\sec\alpha = \sec 14.5^\circ = 1.0328$$ and the lead angle is $\lambda = \arctan\left[l/(\pi d_m)\right] = \arctan\left(0.5/8.639\right)=3.31^\circ$.
3. Raising torque. The standard Acme screw relation, with the collar term added, $$T_R=\frac{F d_m}{2}\left(\frac{l+\pi\mu d_m\sec\alpha}{\pi d_m-\mu l\sec\alpha}\right)+\frac{F\mu_c d_c}{2}$$ Evaluating the thread group first, $\pi\mu d_m\sec\alpha = \pi\times0.1\times2.75\times1.0328 = 0.8923$ in and $\pi d_m = 8.6394$ in, so $$T_R=\frac{52\,000\times 2.75}{2}\left(\frac{0.5+0.8923}{8.6394-0.0517}\right)+\frac{52\,000\times0.03\times5}{2}$$ $$T_R = 11\,593 + 3900$$ $$\boxed{T_R = 15\,493\ \text{lb}\cdot\text{in} = 1291\ \text{lb}\cdot\text{ft per screw}}$$
4. Lowering torque. Lowering reverses the sense of the thread-friction term: $$T_L=\frac{F d_m}{2}\left(\frac{\pi\mu d_m\sec\alpha-l}{\pi d_m+\mu l\sec\alpha}\right)+\frac{F\mu_c d_c}{2}$$ $$T_L=\frac{48\,000\times2.75}{2}\left(\frac{0.8923-0.5}{8.6394+0.0517}\right)+3600 = 2980+3600$$ $$\boxed{T_L = 6580\ \text{lb}\cdot\text{in} = 548\ \text{lb}\cdot\text{ft per screw}}$$ The lowering torque is positive, which is the arithmetic statement that the screw is self-locking: since $\pi\mu d_m\sec\alpha = 0.892 \gt l = 0.5$, the gate cannot run away under its own weight and no brake is needed on the drive.
5. Rotational speed of the screws. A single-start screw advances one lead per revolution, so $$N = \frac{v}{l}=\frac{24\ \text{in/min}}{0.5\ \text{in/rev}}$$ $$\boxed{N = 48\ \text{rev/min}}$$
6. Power to raise the gate. Using the customary horsepower relation for torque in lb·in and speed in rev/min, $$P = \frac{T_R N}{63\,025}=\frac{15\,493\times 48}{63\,025}$$ $$\boxed{P = 11.8\ \text{hp per screw, i.e. 23.6 hp for the pair}}$$ Cross-checking through angular velocity, $\omega = 48\times2\pi/60 = 5.027$ rad/s and $T_R\omega = 15\,493\times5.027/12 = 6491$ lb·ft/s $= 11.8$ hp, which confirms the constant.
7. Efficiency of the raising duty. Useful work per revolution is $F l$, and the thread torque alone is 11 593 lb·in, so $$e_{\text{screw}}=\frac{Fl}{2\pi T_{R,\text{thread}}}=\frac{52\,000\times0.5}{2\pi\times11\,593}=0.357$$ Including the collar, overall efficiency falls to $52\,000\times0.5/(2\pi\times15\,493)=0.267$. A 3-inch Acme screw at a 3.3° lead angle is inherently inefficient — roughly three quarters of the input goes to friction — which is the price paid for self-locking and for the huge mechanical advantage.