Question 6 of 6: Twin Acme power screws — sluice gate

Question 6: Twin Acme power screws — sluice gate (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 single-thread 3-in Acme screws share a 50-ton gate; thread friction $\mu=0.1$; collar $d_c=5$ in, $\mu_c=0.03$; track friction $\pm2$ tons (adds when raising, subtracts when lowering); lift speed 2 ft/min. Short ton = 2000 lb (stated assumption). From the table, 3-in Acme has 2 tpi, so pitch = lead $l=0.5$ in and pitch (mean) diameter $d_m=2.75$ in; Acme half-angle $14.5^\circ$.

Find. (a) raising and lowering torque per screw; (b) screw rotation speed; (c) motor horsepower per screw to raise.

Approach. Split the 50-ton gate plus track friction between the two screws, giving 26 tons per screw raising and 24 tons lowering. Use the Acme power-screw torque equations (with the $\sec\alpha$ thread-angle correction) plus a collar-friction term; get speed from lead and lift rate; convert raising torque and speed to horsepower.

1. Load per screw. Gate 50 t over two screws = 25 t each; track friction $\pm2$ t total shares as $\pm1$ t but is conventionally applied to the gate weight, giving $W_{up}=26$ t $=52\,000$ lb and $W_{dn}=24$ t $=48\,000$ lb per screw.
2. Raising torque. With the Acme correction $\sec\alpha=\sec14.5^\circ=1.033$, $$T_R=\frac{Wd_m}{2}\!\left(\frac{l+\pi\mu d_m\sec\alpha}{\pi d_m-\mu l\sec\alpha}\right)+\frac{W\mu_c d_c}{2}.$$ Substituting $W=52\,000$ lb gives thread torque 14 593 lb·in plus collar 3900 lb·in, so $$\boxed{T_R\approx15\,493\text{ lb}\cdot\text{in}\ (1291\text{ lb}\cdot\text{ft}).}$$
3. Lowering torque. Flipping the thread-term sign for descent, $$T_L=\frac{Wd_m}{2}\!\left(\frac{\pi\mu d_m\sec\alpha-l}{\pi d_m+\mu l\sec\alpha}\right)+\frac{W\mu_c d_c}{2}\Big|_{W=48\,000}=\boxed{6580\text{ lb}\cdot\text{in}}.$$ Because $\pi\mu d_m\sec\alpha=0.864\gt l=0.5$, the thread term stays positive — the screw is self-locking, so the gate cannot back-drive under its own weight (a safety requirement for a dam gate).
4. Rotation speed. Each turn advances the gate one lead, so at $v=2$ ft/min $=24$ in/min, $$N=\frac{v}{l}=\frac{24}{0.5}=\boxed{48\text{ rpm}}.$$
5. Motor horsepower to raise. $$\text{hp}=\frac{T_R\,N}{63\,025}=\frac{15\,493\times48}{63\,025}=\boxed{11.8\text{ hp per screw}}.$$